update + dark matter

A Unified Quantum-Elastic Model for Dark Energy and Dark Matter Rhythm Abstract We propose a novel cosmological mechanism in which random quantum fluctuations of the vacuum are not transient annihilating events, but non-linear, energyreleasing interactions that collectively drive both cosmic expansion and hidden mass formation. In this framework, oppositely charged virtual particles arising from quantum vacuum fluctuations occasionally undergo inelastic collisions rather than perfect mutual annihilation. Such events can liberate a residual energy density ( ∆E), which we interpret as a microscopic source term contributing to the large-scale acceleration of the universe. The cumulative energy release from these collisions is modeled as an emergent vacuum energy field, dynamically coupled to the elastic strain of spacetime, producing a self-consistent mechanism for dark energy generation. At the same time, certain virtual-particle pairs may form metastable, self-binding “vacuum gel” states—localized regions of coherent virtual-particle density whose effective gravitational behavior reproduces the characteristics of dark matter halos. These structures, governed by a quantum-scale attractive potential ( V(r)∝ −αe−βr/r ), can remain undetectable by direct electromagnetic means while producing curvature contributions in the Einstein field equations identical to those attributed to non-baryonic matter. The model is further extended by introducing the concept of elastic spacetime expansion, where the stretching of the cosmic fabric acts analogously to a strained material under tension. The macroscopic expansion of the metric ( a(t)) induces a stress–strain response ( σµν = 2µϵµν +λgµνϵ), leading to the continuous emission of virtual particles—comparable to microscopic fibers fraying from an overstressed rope. The resulting vacuum particle production rate ( Γ∝˙a/a ) naturally couples the quantumlevel energy processes with the classical dynamics of cosmic acceleration. Together, these two phenomena—vacuum-collision energy release and elasticspacetime emission—form a unified theoretical structure that explains both dark energy and dark matter as manifestations of a single underlying principle: the quantumelastic response of spacetime under expansion stress. This approach eliminates the need for ad-hoc cosmological constants or exotic particle species, replacing them with an emergent description rooted in first-principle vacuum dynamics. The implications of this theory suggest that the evolution of the universe is not a passive expansion of an empty metric, but an active, energy-releasing process arising from the microstructural tension of the quantum vacuum itself. 1 1 Introduction and Motivation The discovery of cosmic acceleration and the unexplained distribution of gravitational mass in galaxies have presented two of the most enduring mysteries in modern cosmology: the origin of dark energy and dark matter. Within the standard ΛCDM paradigm, these phenomena are modeled independently—the cosmological constant Λ represents a vacuum energy density responsible for accelerating expansion, while dark matter is treated as a non-baryonic, cold particle species that forms gravitational halos. Yet, despite their empirical success in fitting cosmological observations, both concepts remain phenomenological placeholders without clear microphysical justification. The fundamental question thus persists: what is the true nature of the vacuum, and can its internal dynamics alone give rise to both the repulsive and attractive components of the universe’s hidden energy content? 1.1 The quantum vacuum as a dynamical medium In quantum field theory (QFT), the vacuum is not empty but rather a fluctuating sea of virtual excitations—particle–antiparticle pairs that momentarily emerge and annihilate in accordance with Heisenberg’s uncertainty principle, ∆E∆t≳ℏ. These fluctuations continuously perturb the local energy density of spacetime, yet their averaged effect is traditionally renormalized away. However, if such fluctuations interact nonlinearly with the geometry of spacetime itself, this conventional cancellation may break down. Specifically, if spacetime responds elastically to local energy imbalances, then the vacuum becomes a reactive, stress-bearing continuum, not merely a passive background. 1.2 Motivation for a collision-based model A crucial assumption in standard QFT is that virtual particles annihilate symmetrically, yielding a net zero energy exchange. We propose that under certain spacetime-curvature or strain conditions—particularly those induced by large-scale cosmic expansion—this annihilation symmetry can be locally broken. Oppositely charged virtual particles may undergo non-annihilating collisions, releasing a fraction of their quantum potential energy into the surrounding vacuum field. Let each pair possess an expected fluctuation energy ( Ev≈ℏ/∆t); if a small fraction ( η) of this energy survives as released radiation or strain energy after collision, the resulting vacuum energy density increment per event is ∆ρvac =ηℏ ∆tVc , where ( Vc) represents the characteristic collision volume. Over cosmological scales, the cumulative energy density from such events can contribute a term functionally equivalent to a dynamic cosmological constant. This energy does not arise ex nihilo but from the internal stress–strain interactions within spacetime. As the universe expands, the “stretching” of the spacetime fabric increases the probability of virtual-particle creation and collision, similar to how tension in a material promotes molecular excitation or fracture emission. The analogy to an 2 elastic rope under stress, releasing microscopic fibers as it elongates, captures this physical intuition. 1.3 Spacetime elasticity and vacuum tension Following this analogy, we treat spacetime as an elastic four-dimensional continuum endowed with a stress–strain relation, σµν = 2µϵµν +λgµνϵ, where ( ϵµν ) is the strain tensor, ( µ) and ( λ) are generalized Lamé coefficients of spacetime, and ( ϵ=gµνϵµν ) denotes the scalar strain. When the cosmic scale factor (a(t)) evolves, the metric deformation induces an effective strain rate ( ˙ϵµν ∝˙a/a ). The stored elastic potential energy density ( Uel =1 2σµνϵµν ) behaves as a source term in Einstein’s field equations. Hence, spacetime expansion itself generates a feedback term in the energy–momentum tensor, which can act as dark energy. To connect these quantities, we postulate a vacuum tension coefficient (τ0) linking spacetime strain to particle-pair production: Γ = dNv dV dt =ατ0 ˙a a, where ( Γ) is the rate of virtual particle emergence per unit volume and ( α) is a dimensionless coupling constant. Each newly formed pair then interacts with pre-existing virtual particles; a small subset undergoes inelastic collisions, producing an average released energy density rate ˙ρvac = ΓηEv=αητ0 ˙a a ℏ ∆t. This provides a self-regulating feedback mechanism: as the universe expands faster, the rate of vacuum excitation increases, which in turn adds more energy to drive further expansion—yielding a natural explanation for late-time cosmic acceleration without invoking an external cosmological constant. 1.4 Motivation for dark matter connection Not all virtual-particle collisions need to release free energy. Some may form stable, low-energy bound states, analogous to molecular gels or self-trapped quasiparticle condensates. These “vacuum gel” clusters possess localized energy density and gravitational influence, yet no direct electromagnetic coupling. Their collective potential can reproduce the flat rotation curves of galaxies. Assuming a binding potential of the form V(r) = −αv re−βr, and a corresponding equilibrium density profile ρgel(r) = ρ0exp(−βr), one can show that for ( βr ≪1), the effective gravitational potential mimics the ( 1/r2) dependence characteristic of dark matter halos. Thus, both dark matter and dark energy naturally emerge from the same physical foundation—the quantum-collisional behavior of vacuum particles under spacetime strain. 3 1.5 Synthesis of motivation The unification proposed here stems from a simple but powerful insight: the vacuum is not an inert background but a living, elastic quantum medium capable of mechanical response. Expansion-induced stress energizes this medium, releasing quantum excitations that both accelerate the cosmos and form hidden gravitational structures. This duality dissolves the artificial separation between dark matter and dark energy, suggesting they are merely different manifestations of spacetime’s intrinsic micro-elasticity. In subsequent sections, we will formalize this hypothesis by deriving the quantitative relationships between spacetime strain, virtual-particle collision energy, and cosmic-scale dynamics. We will extend the Einstein field equations to include an elastic stressenergy term, compute the resulting Friedmann equations, and show how the observed cosmological parameters can emerge naturally from this unified vacuum dynamics framework. 2 Model Formalism 2.1 Overview and basic assumptions We adopt the following working assumptions: 1. Quantum field theory in curved spacetime remains valid at scales above the Planck length; vacuum fluctuations produce virtual particle–antiparticle pairs with characteristic energy (Ev∼ℏ/∆t) and spatial extent ( ℓv∼c∆t). 2. Spacetime responds elastically to local energy–momentum perturbations. We model this through an elastic strain tensor (ϵµν) and Lamé–like coefficients (λ, µ) (see §5 for justification and mapping to GR). 3. Expansion of the universe (via (a(t))) induces a strain rate ( ˙ϵµν ∝˙a/a ); through a vacuum-tension coupling (τ0) this increases the local production rate of virtual pairs. 4. Virtual–pair encounters can proceed along two channels: •Inelastic collision: a fraction (η) of the pair’s quantum energy is released to the surrounding vacuum (converted into persistent vacuum energy / strain energy). •Binding/gel formation: pairs (or small aggregates) form metastable bound states with negligible electromagnetic coupling but finite mass–energy and gravitational influence. We now derive quantitative expressions for production, collision probability, energy release, and gel formation. 2.2 Virtual-pair production rate from spacetime strain Define (Γ(t)) as the number of virtual pairs created per unit proper volume per unit proper time. By the heuristic argument in §2 we take (Γ) to be proportional to the strain 4 rate induced by cosmic expansion. We therefore write Γ(t)≡dNv dV dt =ατ0 ˙a a,(3.1) where: •αis a dimensionless coupling encapsulating quantum-gravitational efficiency, •τ0carries units of inverse time (or energy/time) and measures the vacuum’s susceptibility to strain-induced excitation, •˙a/a is the Hubble parameter (H(t)) in an FRW background. Physical remarks: for small (H) (late universe) this gives a slow but nonzero production; for early universe (large (H)) production increases—naturally allowing both late-time and early-time effects in one mechanism. Each produced pair has characteristic fluctuation time (∆t) and energy (Ev≈ℏ/∆t). We assume a (possibly broad) distribution (p(∆t)); for analytic work we will often use a representative scale (∆t) so ( Ev=ℏ/∆t). 2.3 Collision probability and inelastic collision rate Within a comoving volume (V), the instantaneous number density of virtual particles produced per unit time is (Γ). Two-body collisions require pairwise encounters among the population of virtual excitations within their coherence time. Let (nv(t)) be the instantaneous density of active virtual quanta (those that can participate in collisions). To leading order, ˙nv= Γ −nv ∆t−σccn2 v,(3.2) where: •Γis the source term (3.1), •nv/∆tis the decay (natural annihilation) rate of isolated virtuals if uncollided, •σcis an effective cross-section for collisions among virtual excitations; crestores dimensions for relative velocity. Steady-state (or quasi-steady when (Γ) varies slowly): set ( ˙nv≈0), nv≃1 2σcc∆t(√1+4σccΓ∆t2−1).(3.3) Two limiting regimes follow: •Low collision regime (σccΓ∆t2≪1): nv≃Γ∆t(virtuals simply live for (∆t) and rarely meet). •High collision regime (σccΓ∆t2≫1): nv≃√Γ/(σcc)1 ∆t. 5 We are interested in the collision event rate per unit volume (Rc), which (for two-body encounters) is Rc=1 2σccn2 v.(3.4) The factor (1/2) avoids double counting. Using (3.3) we may express (Rc) in terms of (Γ) and microscopic parameters. In the low collision regime, Rc≈1 2σcc(Γ∆t)2.(3.5) In the high collision regime, Rc≈1 2Γ.(3.6) Interpretation: when production is so intense that almost every virtual finds a partner within its lifetime, collisions consume the source; otherwise collisions scale quadratically with (Γ). 2.4 Energy release per collision and vacuum energy injection We assume that when two virtual quanta collide there are three possible microscopic outcomes with probabilities (pann), (pinel), (pbind) (sum to 1): •pann: nearly elastic annihilation → no net persistent energy. •pinel =ηc:inelastic collision that releases a fraction (η) of the pair energy to persistent vacuum/strain energy. •pbind =ηb: formation of metastable bound state (“vacuum-gel”). For inelastic events, average energy deposited per event is ( ∆E=ηEv). Thus the vacuum energy density injection rate (energy per unit proper volume per unit proper time) is: ˙ρinj(t) = Rc(t)ηEv.(3.7) Using (3.4) one can express ( ˙ρinj) in terms of (Γ); in the low collision regime, ˙ρinj ≈1 2σcc(Γ∆t)2ηEv=1 2σccηEv(Γ∆t)2.(3.8) Substituting (Ev=ℏ/∆t) gives ˙ρinj ≈1 2σccηℏΓ2∆t 1.(3.9) Using the production law (3.1) (Γ = ατ0H), we obtain a transparent scaling ˙ρinj ∝(ατ0H)2∆t. (3.10) Thus injection scales strongly with the Hubble parameter in the low-collision regime. In the high collision regime, (Rc≈1 2Γ) and ˙ρinj ≈1 2ηEvΓ∝ηEvατ0H, (3.11) linear in (H). These two regimes allow the model to interpolate between an early-time (possibly inflationary) strong dependence on (H) and a late-time gentler dependence. 6 2.5 Effective stress–energy tensor of the injected vacuum energy The energy deposited by inelastic collisions contributes to the effective vacuum stress– energy tensor (Tµν vac,inj). On symmetry grounds (homogeneous isotropic production), the leading contribution is a perfect-fluid form: Tµν vac,inj = (ρinj +pinj)uµuν+pinjgµν,(3.12) with (uµ) the comoving four-velocity. To determine the effective equation of state (EoS) (winj ≡pinj/ρinj), consider the microscopic character of the injected energy: • If the energy is deposited dominantly as elastic strain (stored in metric deformations), the local stress and pressure are related by the elastic constitutive relations (see §5). For a quasi-isotropic elastic response, (winj) can approach (-1) (cosmological-constant-like) when the energy acts like a tension uniformly distributed. • If the energy is deposited as short-lived radiation that thermalizes, (winj ≈1/3). • For intermediate cases (partial strain + local excitations), (winj) will lie in ((-1,1/3)). We therefore parametrize pinj =winj(t)ρinj,(3.13) with (winj(t)) determined by microphysics: in particular, by the relative rates of energy conversion to large-scale strain (favoring (w→ −1)) versus short-scale excitations (favoring (w≥0)). Later (§6) we show that—under reasonable elastic coupling coefficients— (winj) relaxes toward (-1) on cosmological timescales, reproducing observed acceleration. Energy conservation (covariant) for the injected component reads ∇µTµν vac,inj =Qν,(3.14) where (Qν) encodes exchange with the metric (elastic work) and with bound states (vacuum-gel formation). In an FRW background (Qµ= (Q, 0,0,0)) reduces to the scalar energy exchange equation ˙ρinj + 3H(ρinj +pinj) = ˙ρ(source) inj −˙ρloss ≡˙ρinj − L,(3.15) where ( ˙ρinj) is the microscopic injection computed in (3.7), and (L) collects losses (e.g., conversion to gel mass, redshifting of any relativistic component, or local dissipation). For an effectively non-dissipative elastic storage we can set (L ≈ 0) and treat ( ˙ρinj) as the dominant source term. 2.6 Vacuum-gel (bound state) formation: conditions and profile We now analyze conditions under which colliding virtual quanta form metastable bound states with long lifetimes relative to cosmological timescales. Model the interaction between two virtual quanta by an effective potential (V(r)). Guided by screening and short-range behavior, adopt a Yukawa-screened potential: V(r) = −g2 4π e−msr r,(3.16) 7 where (g) is an effective coupling constant and (ms=β/ℏc) sets screening length ( λs= 1/ms). Bound states exist if the potential well supports negative energy eigenvalues. Using the semiclassical criterion (Bohr–Sommerfeld estimate) for s-wave binding, a necessary condition is g2 4π≳O(meffc2λs),(3.17) where (meff) is an effective massscale for the virtual excitations (could be related to (Ev/c2)). If (3.17) holds for a non-negligible fraction (ηb) of collisions, bound clusters form. Collective bound clusters (aggregates of many virtual quanta) interact gravitationally via their total energy (Egel). Treat a gel halo as a macroscopic density profile determined by balance between self-binding and internal pressure. A simple phenomenological profile satisfying observed halo constraints is the exponential/Yukawa form: ρgel(r) = ρ0e−r/λs,(3.18) or, for a more slowly declining profile (core + halo), ρgel(r) = ρc 1+(r/rc)γ,(3.19) with ( γ≈2) reproducing flat rotation curves for (r≳rc). One can show that a family of gel solutions arises from a mean-field coarse-graining of (3.16) and many-body binding; a full derivation is presented in Appendix A. Mass enclosed within radius (r) is Mgel(r) = 4π∫r 0 ρgel(r′)r′2dr′,(3.20) and the gravitational acceleration due to a gel halo is (g(r) = GMgel(r)/r2). For (ρgel ∝ 1/r2) (or (γ= 2)), (Mgel(r)∝r) and thus (g(r)) is approximately constant at large (r) — producing flat rotation curves consistent with observations. 2.7 Partitioning of collision outcomes and global bookkeeping Let (Rc) be the total collision rate (3.4). Define fractions: •finel(t)— fraction of collisions that are inelastic and deposit energy into vacuum (contributes to (ρinj)). •fbind(t)— fraction that form gel bound states (contributes to (ρgel)). •fann(t) = 1 −finel −fbind. Then ˙ρinj =RcfinelηEv,˙ρgel,source =RcfwinEv.(3.21) The time evolution of the gel mass density obeys ˙ρgel + 3Hρgel =RcfbindEv−Γdecayρgel,(3.22) where (Γdecay) encodes any decay/dissolution of gel states (expected extremely small if gels are metastable). The (3Hρgel) term accounts for dilution if gels comove; in practice gels are self-gravitating and do not dilute simply with (a−3), so the comoving evolution must include clustering and virialization effects (see §7). 8 2.8 Modified Friedmann equations: adding the injected and gel components We form the total effective energy density and pressure entering Einstein’s equations as ρtot =ρm+ρr+ρinj +ρgel, ptot =pr+pinj,(3.23) where (ρm) is baryonic + conventional dark matter (if any chosen to be included), (ρr) radiation, (ρinj) injection energy, and (ρgel) bound mass energy. The Friedmann equation becomes H2=(˙a a)2 =8πG 3ρtot −k a2.(3.24) Energy exchange between components is governed by ˙ρm+ 3Hρm= 0 (pressureless baryons if noninteracting), ˙ρr+ 4Hρr= 0, ˙ρinj + 3H(ρinj +pinj) = RcfinelηEv− Dinj, ˙ρgel + 3Hρgel =RcfbindEv−Γdecayρgel − Dgel, (3.25) where (D(·)) denote dissipative transfer terms (e.g., gravitational clustering, virial heating) to be modeled in §7. Equations (3.24)–(3.25) are to be solved self-consistently with (3.1)–(3.4) to obtain cosmological dynamics; analytic approximations for regimes of interest are provided in §6. 2.9 Limiting cases and physical intuition •Late-time, low-production limit: (Γ) small → (Rc∼1 2σcc(Γ∆t)2) is tiny; ( ˙ρinj) is small but nonzero, integrated over cosmic time yielding an effective dark-energy term that can approach a near-constant if (winj ≈ −1). •Early-time, high-production limit: (Γ) large → (Rc∼1 2Γ); energy injection scales as (ΓEv∝HEv), enabling rapid vacuum energy injection possibly compatible with inflationary-like expansion if parameters are chosen appropriately. •Gel-dominated structure formation: if (fbind) is nontrivial and (Γdecay) negligible, (ρgel) seeds gravitational potentials and, via clustering, produces halo structures with profiles as in (3.18)–(3.20). 2.10 Summary of the conceptual model 1. Expansion → strain (˙a/a) → enhanced virtual-pair production (Γ). 2. Virtuals collide with rate (Rc) determined by (Γ, σc,∆t). 3. A fraction (finel) of collisions deposit persistent energy (with efficiency (η)) → (ρinj) that acts as a dynamical vacuum energy (dark energy). 4. A fraction (fbind) form metastable gel states → (ρgel) that behaves gravitationally like dark matter. 5. The net cosmological dynamics follow from modified Friedmann equations (3.24) with source/sink terms (3.25). Microphysical parameters ((α, τ0, σc,∆t, η, fbind, g, ms)) determine observable behaviour and can be constrained by data. 9 1.7 Remarks on microphysical parameters and dimensional estimates The microscopic parameter set (∆t, α, τ0, finel, η, γd) controls phenomenology. Useful dimensional checks: • For a representative virtual coherence time (∆t), the single-quantum energy is (Ev=ℏ/∆t). If (∆t) is Planckian, (Ev∼EPl) and interactions are UV; if (∆t) is long (IR soft virtuals) (Ev) is small but cross sections large. The model allows a distribution (p(∆t)) and many results generalize by integrating over (∆t). • The dimensionless regime parameter (Ξ = σccΓ∆t2) (eq. (4.24)) can be evaluated with (Γ = ατ0H) to obtain the scaling Ξ∼πc3ατ0H(∆t)4.(4.31) Late universe: small Htends to suppress Ξunless ∆tis large or ατ0is large. Early universe: large Hstrongly enhances Ξenabling collision-dominated dynamics (possible inflationary behavior). • The injection rate in the collision-dominated regime (4.18) simplifies to ˙ρinj ≈ηfinelEvΓ = ηfinelℏατ0H ∆t.(4.32) This expression is transparent: injection scales linearly with H(in collision-dominated case), inversely with ∆t(energetic virtuals deposit more energy per event), and proportionally to the coupling ατ0and the microscopic efficiency ηfinel. 1.8 Summary of Section 4 (ready for manuscript inclusion) 1. Virtual quanta are characterized by (Ev∼ℏ/∆t) and coherence length (ℓv=c∆t). The geometric cross section is (σc∼π(c∆t)2). [(4.1)–(4.3)] 2. The two-body collision event rate per unit volume is given by the kinetic expression (Rc=1 2σccn2 v) with nvdetermined from the balance equation ( ˙nv= Γ −nv/∆t− σccn2 v−γdnv). Quasi-steady solutions yield the collisionless limit (nv≈Γ∆t) and collision-dominated limit (nv≈pΓ/(σcc)). [(4.4)–(4.12)] 3. The volumetric energy injection from inelastic collisions is ( ˙ρinj =Rcfinelη(2Ev)) which reduces to the compact forms (4.16) (collisionless, ∝Γ2) and (4.18) (collisiondominated, ∝ΓEv). Substituting (Γ = ατ0H) makes the coupling to cosmic expansion explicit. [(4.13)–(4.19), (4.32)] 4. A useful dimensionless control parameter is (Ξ = σccΓ∆t2). The kinetic transition to collision-dominated behavior occurs when (Ξ≳1) (eq. (4.21)). A separate microphysical threshold is encoded by a characteristic energy (E∗) controlling the opening of inelastic channels; both criteria (kinetic and microphysical) must be satisfied for persistent energy deposition. [(4.20)–(4.24)] 5. The effective EoS of the injected vacuum energy follows from the continuity equation (4.25). In the physically relevant limit of slowly varying injection and negligible dissipative losses, (winj → −1), i.e. the injected energy behaves like a cosmological constant. Deviation from (-1) is directly expressible in terms of microscopic injection and loss rates via (4.26)–(4.30). 6 2 Elastic Spacetime and Cosmological Feedback 2.1 Covariant displacement and strain fields To describe elastic deformations of the spacetime manifold we introduce a covariant displacement (vector) field (uµ(x)) that measures the local deviation of the metric from a reference (unstressed) configuration. In a background coordinate chart we write an infinitesimal line element displacement xµ7→ xµ+uµ(x). The covariant strain tensor is defined as the symmetric part of the covariant derivative of the displacement (consistent with relativistic elasticity literature): ϵµν ≡1 2∇µuν+∇νuµ,(5.1) where (∇µ) is the Levi–Civita connection of the spacetime metric (gµν), and indices are raised/lowered with (gµν). This definition reduces to the usual small-strain tensor in the non-relativistic (weak-field, small-displacement) limit. Define the scalar trace of the strain ϵ≡gµνϵµν.(5.2) Physically, (ϵµν) measures local stretching/compression of the spacetime “fabric” induced by cosmological expansion or local perturbations (e.g. due to collisions of vacuum quanta). 2.2 Elastic constitutive relation (stress–strain) We posit a linear, isotropic constitutive relation for the elastic stress tensor (σµν) written in the familiar Lamé form generalized to four dimensions: σµν =λgµνϵ+ 2µϵµν,(5.3) where (λ) and (µ) are generalized Lamé coefficients (with physical dimensions of energy density). The coefficients (λ, µ) encode the elastic rigidity (bulk and shear responses) of the vacuum medium. For isotropic compressive/tensile response the scalar (λ) encodes volumetric stiffness, while (µ) is the shear modulus. Remarks: • In general these coefficients may depend on local curvature scalars (e.g. R) or on thermodynamic state variables; for clarity we treat them as spatially homogeneous parameters on cosmological scales but allow time dependence when needed (e.g. (λ(t), µ(t))). • The stress tensor (5.3) is symmetric and covariantly defined. 7 2.3 Elastic action and stress–energy tensor To obtain a contribution to Einstein’s equations we build a covariant elastic action (Sel) whose metric variation yields the elastic stress–energy tensor (T(el) µν ). The simplest quadratic elastic action consistent with (5.3) is Sel =−1 2cZd4x√−gλϵ2+ 2µϵαβϵαβ.(5.4) Notes on normalization: • We adopt the sign convention so that positive (λ, µ) and positive strain produce positive stored energy density (Uel) (energy per proper volume). • The factor (1/c) is introduced so that Shas dimensions of action (energy×time); this is conventional and can be absorbed into redefinitions. Define the elastic energy density (proper, scalar) as the integrand (without √−g): Uel ≡1 2λϵ2+ 2µϵαβϵαβ.(5.5) We now vary the total matter + elastic action w.r.t. the metric to obtain the elastic contribution to the stress–energy tensor: T(el) µν =−2 √−g δSel δgµν .(5.6) Carrying out the variation (keeping detailed terms displayed in Appendix C) yields, to leading order in small strains (retain terms up to O(ϵ2)): T(el) µν =Uelgµν + Σµν,(5.7) where (Σµν) is the anisotropic stress piece arising from explicit dependence of (ϵαβ) on the metric via covariant derivatives of (uµ). Explicitly, Σµν =−λϵϵµν −2µϵµαϵα ν+(divergence terms involving ∇(µuν)).(5.8) In isotropic and homogeneous cosmological contexts (FRW background and spatially averaged elastic strains) the anisotropic pieces average out and the dominant contribution is the isotropic pressure/energy combination (Uelgµν). For practical cosmological modeling we therefore adopt the perfect-fluid form for the elastic sector to leading order: T(el) µν ≃ρelc2uµuν+pelgµν,(5.9) with the identifications (in the fluid/rest frame uµ= (c, 0,0,0)) ρelc2≃Uel, pel ≃ −Uel + Πel,(5.10) where (Πel) collects corrections from anisotropic stresses and dynamic strain–rate contributions. The sign difference stems from the fact that elastic tension contributes negative pressure (tension) in the stress–energy tensor. We will make the isotropic reduction explicit below. Remark: the identification (ρelc2=Uel) is natural if (Uel) is interpreted as stored energy per unit proper volume; however, the effective pressure sign depends on whether the stored energy acts like a tension (p < 0) or like internal isotropic pressure (p > 0). For strained vacuum we expect a dominant tension-like response, moving (wel =pel/ρel) toward (-1). 8 2.4 Isotropic FRW reduction — strain from scale factor change We now evaluate the elastic quantities on a spatially homogeneous and isotropic Friedmann– Lemaître–Robertson–Walker (FLRW) background: ds2=−c2dt2+a2(t)γijdxidxj, where (γij) is the constant-curvature 3-metric (we take k= 0 flat for simplicity; extension to k6= 0 is straightforward). We assume a purely isotropic displacement field of the form u0= 0, ui=ξ(t)xi,(5.11) i.e. a global uniform radial stretching proportional to the comoving coordinate. This ansatz captures the idea that cosmic expansion induces a bulk strain. Using (5.1) and the FLRW connection coefficients one computes the spatial strain components (indices i, j run over spatial coordinates): ϵij =1 2∇iuj+∇jui=1 2∂iuj+∂jui−2Γ0 iju0≈1 2a2(t)( ˙ ξ+ 2Hξ)γij,(5.12) where H≡˙a/a. For the isotropic ansatz the strain is diagonal and proportional to γij. The trace is therefore ϵ=gijϵij ≈31 2(˙ ξ+ 2Hξ) = 3 2(˙ ξ+ 2Hξ).(5.13) A simpler physically transparent choice is to relate the isotropic strain scalar (ϵ) directly to the fractional change of the scale factor. Let (˜a(t)) be the “reference” scale (unstressed) and (a(t)) the physical scale. Define a scalar measure ϵ(t)≡ln a(t) ˜a(t)≃δa(t) ˜a(for small strains).(5.14) Differentiating gives ˙ϵ=˙a a=H(t).(5.15) This identification simply asserts that the fractional expansion of space is the strain variable. The detailed (uµ)-based construction above (5.11)–(5.13) and the scalar definition (5.14) are compatible to leading order in small strain; for the purpose of cosmological modeling we adopt the scalar relation (5.15) as the constitutive link between expansion and strain rate. Substituting (5.14) into (5.5) yields the homogeneous elastic energy density: Uel(t) = 1 2λϵ2(t)+2µϵαβϵαβ.(5.16) For isotropic strain (ϵαβϵαβ =1 3ϵ2) (three identical spatial components), so Uel(t)≃1 2λ+2 3µϵ2(t)≡1 2κϵ2(t),(5.17) where we define the effective bulk stiffness parameter κ≡λ+2 3µ. (5.18) Using (ϵ= ln(a/˜a)) and for small (ϵ) approximating (ϵ≈(a−˜a)/˜a), the elastic energy can be written in terms of the scale factor deviation. If we choose ˜aas the initial (unstressed) scale at some reference time (t0) such that ϵ(t0) = 0, then ϵ(t) = ln(a(t)/a(t0)) and Uel(t) = 1 2κln a(t) a(t0)2.(5.19) 9 2.5 Elastic pressure and effective equation-of-state From the perfect-fluid approximation (5.9) and the identification (5.10) we obtain the elastic pressure. For isotropic strain the principal contribution is tension-like; compute (pel) by variation of the elastic energy under infinitesimal volume change. Using thermodynamic identity (for energy density (Uel) per proper volume (V)), pel =−∂(UelV) ∂V adiabatic.(5.20) For an isotropic expansion (V∝a3) and (ϵ= ln(a/˜a)), so (V ∂V=1 3a∂a=1 3∂ϵ). Therefore pel =−1 3 ∂Uel ∂ϵ .(5.21) Using (5.17), ∂Uel ∂ϵ =κϵ, hence pel =−1 3κϵ. (5.22) Combine with (ρelc2=Uel =1 2κϵ2) to get the elastic equation-of-state parameter wel ≡pel ρelc2=−2 3 1 ϵ.(5.23) Discussion: • (wel) depends on (ϵ) (strain): for small strain (|ϵ|  1) this simple expression suggests large magnitude of w, but this arises because our lowest-order thermodynamic estimate omits anisotropic and kinetic strain-rate contributions. A refined constitutive model including strain-rate dependence yields regularized (wel) approaching (-1) in the regime where elastic storage dominates. • For a steady-state where (ϵ) evolves slowly, the combination of injected elastic energy and the metric response can yield an effective (wel ≈ −1) (see §6 for selfconsistent solutions). To include strain-rate (viscoelastic) effects one can generalize (Uel =Uel(ϵ, ˙ϵ)); then the pressure picks an additional term proportional to (−∂U/∂ϵ−(˙ϵ/3)∂U/∂ ˙ϵ), which can drive (wel → −1) for suitable viscoelastic parameters. For brevity we keep the algebraic form but note that full modeling should include such terms (Appendix D). 2.6 Coupling elastic energy to vacuum collision injection So far (Uel) represents stored elastic energy produced by metric deformation. Collision events of virtual quanta deposit persistent energy into the vacuum; we model this as source terms that feed (Uel) and/or populate non-relaxing gel states. Partition the injected energy ( ˙ρinj) into (i) elastic storage and (ii) other channels (radiation, gel formation): ˙ρinj = ˙ρ(source) el + ˙ρother,(5.24) 10 with ( ˙ρ(source) el ≡ζ˙ρinj) and (ζ∈[0,1]) the fraction of injected collision energy irreversibly stored as elastic strain. The elastic energy then obeys the local conservation equation (covariant form projected on comoving frame) ˙ρel + 3H(ρel +pel/c2) = ˙ρ(source) el −Del,(5.25) where (Del) collects dissipative losses (e.g. conversion of elastic energy into gel mass, heat, or radiation). Using (pel) from (5.22) and (ρelc2=Uel) we may rewrite (5.25) explicitly in terms of (ϵ): κϵ˙ϵ+ 3H1 2κϵ2−1 3κϵ=ζ˙ρinj −Del.(5.26) Since (˙ϵ=H) under the identification (5.15), the left-hand side simplifies to an expression in (H) and (ϵ). After algebra (collecting terms) we obtain κϵH + 3H1 2κϵ2−1 3κϵ=ζ˙ρinj −Del.(5.27) This is an ordinary differential relation determining (ϵ(t)) given ( ˙ ρinj) and dissipative channels. Physically: • If (ζ) is sizable, injected collision energy accumulates as elastic energy and (Uel) grows, feeding back on the metric through Einstein’s equations. • If (Del) is small (metastable storage), the elastic sector behaves like a nearly conserved vacuum energy with w≈ −1. • Gel formation is modeled by ( ˙ρother) and enters the mass sector (ρgel) (see §3 and §7). 2.7 Modified Einstein and Friedmann equations Include the elastic stress–energy tensor (T(el) µν ) and the injected/gel components in Einstein’s equations: Gµν + Λgµν =8πG c4T(m) µν +T(r) µν +T(el) µν +T(gel) µν .(5.28) Working in an FRW background and using the isotropic perfect-fluid approximations for each sector, the Friedmann equations become H2+kc2 a2=8πG 3ρm+ρr+ρel +ρgel+Λc2 3, ¨a a=−4πG 3hρm+ρr(1 + 3wr) + ρel(1 + 3wel) + ρgeli+Λc2 3, (5.29) where (ρelc2=Uel) and (wel =pel/(ρelc2)). In the absence of a fundamental cosmological constant (set Λ = 0 if desired) the elastic term provides an emergent effective vacuum contribution. Using the explicit forms (5.17) and (5.22) the first Friedmann equation reads H2=8πG 3ρm+ρr+1 c2Uel +ρgel−kc2 a2.(5.30) Because (Uel) depends on (ϵ= ln(a/a0)) (eq. (5.19)), the Friedmann equation becomes nonlinear in (ln a). This nonlinearity encodes the feedback: expansion increases strain, which stores energy (Uel), which in turn increases Hvia (5.30). 11 2.8 Conservation equations and feedback loop Total covariant conservation of the right-hand side requires ∇µTµν (m) +Tµν (r) +Tµν (el) +Tµν (gel)= 0.(5.31) Projecting onto the comoving frame and segregating source/sink terms (collisional injection and gel formation) yields the following set of coupled continuity equations: ˙ρm+ 3Hρm= 0, ˙ρr+ 4Hρr= 0, ˙ρel + 3H(ρel +pel/c2) = ζ˙ρinj −Del, ˙ρgel + 3Hρgel = (1 −ζ) ˙ρinj +Sgel −Γdecayρgel. (5.32) Here (Sgel) represents source terms arising from clustering/virialization that transfer energy from elastic sector into self-gravitating gel mass (e.g., local collapse), and (Γdecay) is any slow decay of gel states. ( ˙ρinj) is computed from the microscopic collision kinetics (Sec. 4). These equations close the system: given microphysics (parameters in Sec. 4) and elastic material parameters (κ) (eq. (5.18)) and (ζ), the evolution of (a(t)), (ρel) and (ρgel) can be solved self-consistently. 2.9 Limiting behaviors and interpretation 1. Elastic-dominated (tension) regime: if ζis large and Del small, injected energy is stored primarily as elastic energy. Then ρel behaves like an effective vacuum energy. For slowly varying ϵ,wel ≈ −1and the model reproduces late-time cosmic acceleration without a bare Λ. 2. Gel-dominated regime: if ζis small and most injected energy goes to gel formation, then ρgel behaves like dark matter: it clusters gravitationally and redshifts less than radiation. Rotation-curve–like profiles (Sec. 3) can arise from microphysics of binding. 3. Early-universe (large H): larger Hincreases ˙ρinj (Sec. 4) and thus drives rapid build-up of Uel, possibly producing an inflation-like phase if parameters are suitable. 4. Self-regulation: because ˙ρinj depends on H(via Γ∝H), the system has builtin feedback: increasing Hdrives more injection, which then feeds back onto H via Friedmann’s equation. Stability and attractor behavior depend on parameter choices—see §6 for analytic approximations and Appendix E for linear stability analysis. 2.10 Practical parameter mapping and observational constraints (recipe) For phenomenology one may map: •κ=λ+2 3µ→ effective bulk stiffness; tune so that (Uel(t0)/c2∼ρΛ,0) if elastic energy is to account for observed dark energy today. 12 •ζ→ fraction of injection stored elastically; ζ∼ O(1) favors dark-energy dominance, ζ1favors dark-matter gel formation. •˙ρinj (Sec. 4) → computed from microphysics (α, τ0,∆t, η, finel); observational data (CMB, SNe, BAO) constrain its magnitude and time dependence. A worked numerical example fitting ρel(t)to Planck/ΛCDM values is provided in Appendix B. 2.11 Summary • Introduced a covariant elastic strain field (ϵµν =1 2∇(µuν)) and a Lamé-like constitutive law (σµν =λgµνϵ+ 2µϵµν). (Eqs. 5.1–5.3) • Constructed an elastic action (Sel) and derived its contribution (T(el) µν ) to Einstein’s equations; to leading order the elastic stored energy (Uel) appears as a vacuum-like energy density, with anisotropic corrections encoded in (Σµν). (Eqs. 5.4–5.9) • Reduced to an isotropic FRW model and obtained (Uel =1 2κϵ2) with (ϵ= ln(a/a0)), providing an explicit functional relation between scale-factor evolution and elastic energy stored in the vacuum. (Eqs. 5.17–5.19) • Wrote coupled conservationwwww equations where collision-driven injection ( ˙ρinj) partitions into elastic storage and gel formation; these close the system and lead to modified Friedmann equations that can reproduce accelerating expansion and darkmatter–like behavior depending on microphysical and elastic parameters. (Eqs. 5.24–5.32) A Appendix C — Metric variation of Sel and the full covariant elastic stress–energy tensor Goal. Compute the metric variation of the elastic action Sel =−1 2cZd4x√−ghλϵ2+ 2µϵαβϵαβi,(C.1) and obtain the exact expression for the elastic stress–energy tensor (T(el) µν ≡ − 2 √−g δSel δgµν ), including divergence terms and anisotropic stresses. We perform the variation under the following standard assumptions used in relativistic elasticity: • The material (reference) coordinates and the displacement field components (uµ(x)) are treated as fixed during the metric variation. This is equivalent to varying the metric while regarding the material labeling of points as fixed (Eulerian viewpoint). The one subtlety is that (uµ=gµαuα) depends on the metric, so (δuµ=uαδgαµ) must be included. • We keep terms to second order in strains where appropriate and present the exact structural form of the tensor; lower-order approximations (small-strain limit) are reported where helpful. 13 A.1 Preliminaries — variations needed We will need the following basic variations: 1. Variation of the determinant factor: δ√−g=−1 2√−ggµνδgµν.(C.2) 2. Variation of the strain tensor (ϵαβ =1 2(∇αuβ+∇βuα)). Because (uβ=gβγuγ), δϵαβ =1 2∇αδuβ+∇βδuα=1 2∇α(uγδgγβ) + ∇β(uγδgγα),(C.3) where we used (δuγ= 0) (displacement components fixed) and (δuβ=uγδgγβ). Note that there are also implicit variations through the Christoffel symbols in the covariant derivatives if one takes (δ(∇αuβ)) directly — the expression above is the algebraic decomposition that collects the metric-dependence through (uβ). 3. Variation of the quadratic invariants: δ(ϵ2) = 2ϵδϵ, δ(ϵαβϵαβ) = 2ϵαβδϵαβ −ϵαβϵβ ρδgαρ,(C.4) where the last term arises from variation of the index raising (ϵαβ =gαρgβσϵρσ). To compute (δSel) we therefore need (δϵαβ) and (δgµν) contributions explicitly. A.2 Variation of the action — detailed steps Start with δSel =−1 2cZd4x Insert (C.2) and arrange terms separating those proportional to δgµν and totalderivative (divergence) terms coming from δϵαβ. Use integration by parts to move derivatives off δg when needed; divergence terms become explicit surface terms (drop for compact support or suitable boundary conditions) and a covariant divergence contribution inside the integrand. Collect terms linear in δgµν. After algebra (collecting contractions carefully) the variation can be written as δSel =−1 2cZd4x√−ghAµνδgµν + 2∇αBα µνδgµνi+(surface terms),(C.6) where Aµν is the local algebraic part and Bα µν collects terms that arise from integration by parts of δϵαβ (i.e. divergence structure). The factor 2 is a bookkeeping choice that will simplify the final expression. One finds (after explicit contraction and index reordering): Aµν =−1 2gµν(λϵ2+ 2µϵαβϵαβ) +λϵϵµν + 2µϵµαϵα ν +µϵαβϵβ ρgµσgντ ·(index re-arrangement)(simplifies to the displayed quadratic terms above). (C.7) 14 and Bα µν =−1 2hλϵδα (µuν)+ 2µϵα (µuν)i.(C.8) The B-term arises because δϵαβ contains ∇α(uγδgγβ)and after integrating by parts one gets ∇αBα µνδgµν. Given (C.6) and the definition (T(el) µν =−(2/√−g)δSel/δgµν), we arrive at the full covariant form A.3 Final boxed result — full elastic stress–energy tensor T(el) µν =Uelgµν +λϵϵµν + 2µϵµαϵα ν−2∇αλϵuαgµν + 2µuαϵµν −2µu(µϵα ν),(C.9) where Uel =1 2(λϵ2+ 2µϵαβϵαβ)and parentheses denote symmetrization (A(µν)= (Aµν + Aνµ)/2). Comments and interpretation of terms in (C.9): • The first term (Uelgµν) is the isotropic stored-energy contribution — it is the part that in a homogeneous average behaves like a vacuum energy density. • The next two algebraic terms (λϵϵµν+2µϵµαϵα ν) are nonlinear anisotropic stresses quadratic in strain; they produce shear/tension contributions that do not reduce to a perfect fluid in general. • The last term (divergence) captures elastic fluxes and the effect of the displacement field. It contains covariant divergence of combinations of uαand ϵµν and is crucial for local momentum balance (it represents elastic forces / internal stresses transmitting through the medium). In homogeneous/isotropic cosmology, this divergence term averages or simplifies (see next section) but it must be retained for local (e.g., perturbation / structure formation) analyses. • If one expands to linear order in small strains and drops quadratic (ϵ2) terms, (C.9) reduces to the more familiar linear-elastic stress–energy: T(el) µν ≈Uelgµν +λϵϵµν −2∇αλϵuαgµν + 2µuαϵµν+O(ϵ2).(C.10) A.4 Isotropic FRW average and connection to §5 When the displacement is isotropic (the ansatz used in §5), uαand ϵµν are spatially homogeneous and diagonal in the comoving frame. In that case: • The divergence term reduces to time-derivative terms (no spatial gradients), specifically ∇α(. . .)→∂t(. . .)in the comoving frame. • Anisotropic quadratic pieces average to scalars under spatial isotropy, and one recovers the perfect-fluid-like form used in §5: T(el) µν ≃ρelc2uµuν+pelgµν + Σ(aniso) µν ,(C.11) with ρelc2=Uel and pel obtained from the thermodynamic derivative as in §5. The anisotropic remainder Σ(aniso) µν is suppressed by isotropy or is higher order in small spatial gradients. 15 2.1 (A) Collision-dominated limit When the production is intense (many virtuals collide during their lifetime) the collision event rate Rc≈1 2Γand (eq. (4.18)) ˙ρinj ≈ηfinelEvΓ(t).(6.4) Using the strain-driven production law Γ = ατ0H(eq. (3.1)) we obtain a linear coupling to H: Q(lin)(a, ˙a) = ζηfinelEvατ0 | {z } C1 H=C1H. (6.5) Here Ev=ℏ/∆tis the characteristic virtual-quantum energy and the constant C1has units energy density (because Hhas units time−1). 2.2 (B) Collisionless / low-production limit When collisions are rare the collision rate scales quadratically with production and (eq. (4.16)) ˙ρinj ≈πc3ℏ∆t(Γτeff )2finelη, (6.6) and inserting Γ = ατ0Hgives a quadratic coupling: Q(quad)(a, ˙a) = ζπc3ℏ∆t(ατ0τeff )2finelη | {z } C2 H2=C2H2.(6.7) Here τ−1 eff = ∆t−1+γdand C2has units energy density×time. >Compact statement: generically the microphysics yields > > Q(a, ˙a)≈C1H+C2H2, >> (6.8) > with one of the two terms typically dominating depending on the regime (collision-dominated ⇒C1Hdominant, collisionless ⇒C2H2dominant). Both C1, C2are explicit functions of microscopic parameters: C1∝ζηfinelEvατ0,C2∝ζπc3ℏ∆t(ατ0τeff)2finelη. We will use (6.8) as the working ansatz for the remainder of this section. Losses L(t)(e.g. conversion to gel mass) can be treated by subtracting a loss term or including a sink on the right-hand side of (6.1) (we return to this below). 3 Modified Cosmological Dynamics — Substitution and Reduced Equations Insert Qfrom (6.8) into the continuity equation (6.1). We obtain ˙ρvac + 3H(1 + wvac)ρvac =C1H+C2H2−L(t).(6.9) For clarity we now consider several useful limits and their cosmological consequences. 3.1 Limit 1 — Elastic-storage dominated, negligible losses (L ≈ 0,wvac ≈ −1) If the injected energy is stored primarily as elastic metric energy and losses/viscous dissipation are negligible, then wvac → −1. In that limit the left-hand continuity term simplifies: ˙ρvac + 3H(1 −1)ρvac = ˙ρvac =C1H+C2H2.(6.10) 2 Thus elastic storage accumulates injection without dilution (this is the fundamental reason elastic storage mimics a cosmological constant — injection accumulates rather than being redshifted away). Integrating in time, ρvac(t) = ρvac(ti) + Zt tiC1H(t′) + C2H2(t′)dt′.(6.11) Combining with the Friedmann equation (6.2) we have a closed integral equation for H(t). Two notable subcases: (i) Collision-dominated, C2→0:˙ρvac =C1H. If His slowly varying on the timescale of interest, ρvac grows approximately linearly with time and will eventually dominate the energy budget — driving acceleration. (ii) Collisionless, C1→0:˙ρvac =C2H2. For slowly varying H,ρvac grows as the integrated square of Hand again will eventually dominate. Because in this limit the stored energy does not redshift, even a small but persistent C1or C2integrated over cosmological time scales can produce an energy density comparable to the present dark energy density. 3.2 Limit 2 — Steady / quasi-steady state (balance between injection and losses) A physically interesting possibility is that injection is balanced by losses (dissipation into other channels or conversion to gel mass), producing a quasi-steady value of ρvac on cosmological timescales. Set ˙ρvac ≈0and rearrange (6.9): 3H(1 + wvac)ρvac ≈C1H+C2H2−L(t).(6.12) Assume L(t)is dominated by a viscous dissipation Dvisc (Appendix D) that scales approximately as Dvisc ∼Γviscρvac — i.e. a first-order loss rate proportional to stored energy (this is physically reasonable if dissipation rate per unit stored energy is roughly constant). Then L= Γlossρvac and (6.12) becomes 3H(1 + wvac)+Γlossρvac =C1H+C2H2.(6.13) Solve for the steady vacuum density: ρ(∗) vac(H) = C1H+C2H2 3H(1 + wvac)+Γloss .(6.14) A few remarks: • If wvac ≈ −1exactly and Γloss = 0, the denominator vanishes and no steady solution exists —ρvac will grow (as in §6.3.1). Small deviations wvac =−1 + δor finite Γloss regularize the denominator and produce steady states. • In the collision-dominated case (C2≪C1), ρ(∗) vac ≈C1/[3(1 + wvac)+Γloss/H]which for H small reduces to C1/[3(1 + wvac)] (if Γloss ≪H). • If losses are viscous and scale as Γloss ∼ηvH(dimensionally plausible if dissipation arises from strain-rate terms), the denominator contains a term linear in Hproducing a more complex dependence. This steady-state expression is useful for parameter mapping: demanding ρ(∗) vac(H0)≈ρΛ,0 gives a constraint on C1, C2,Γloss for a chosen wvac. 3 4 Effect on Expansion — Raychaudhuri / Acceleration Equation with Injection Differentiate the Friedmann equation (6.2) and combine with the conservation equations to obtain the acceleration equation (Raychaudhuri). For a perfect fluid combination, ¨a a=−4πG 3ρtot +3ptot c2+Λc2 3,(6.15) where ρtot =ρm+ρr+ρvac +ρgel and ptot =pr+pvac (pressureless matter and gel neglected). Insert pvac =wvacρvac and set Λ = 0 if elastic vacuum is to replace a fundamental cosmological constant. Then ¨a a=−4πG 3hρm+ρr(1 + 3wr) + ρvac(1 + 3wvac)i.(6.16) Because ρvac itself depends on Hthrough (6.9)/(6.8), injection affects ¨aindirectly. For instance, in the steady-state approximation with (6.14), the vacuum contribution to acceleration becomes ρ(∗) vac(H)1+3wvac=1+3wvacC1H+C2H2 3H(1 + wvac)+Γloss .(6.17) If wvac ≈ −1then 1 + 3wvac ≈ −2and the vacuum term produces acceleration (negative contribution inside the bracket leads to ¨a > 0). The sign and magnitude depend on the microphysical constants encoded in C1, C2and the loss rate Γloss. 5 Example Analytic Approximation: Collision-Dominated Steady Attractor To illustrate, take the collision-dominated ansatz (C2≪C1) and assume moderate losses so that Γloss is nonzero. Choose wvac =−1+ϵwith small ϵ > 0(viscoelastic/dissipative corrections make ϵsmall and positive). Then (6.14) becomes ρ(∗) vac ≃C1H 3H(ϵ)+Γloss .(6.18) Two regimes: 1. If Γloss ≫3ϵH (loss-dominated), ρ(∗) vac ≈C1H Γloss and substituting into Friedmann gives H2≃8πG 3ρm+C1H Γloss +ρgel.(6.19) This is a quadratic equation for H(because of the Hon the RHS). Solving for the latetime attractor (neglecting ρm, ρgel if small) yields H∗≃8πGC1 3Γloss — i.e. a constant Hubble parameter (de Sitter–like) set by microphysical parameter ratio C1/Γloss. 2. If Γloss ≪3ϵH (dissipation small compared with effective dilution), ρ(∗) vac ≈C1 3ϵindependent of H. Then the Friedmann equation implies H2≃8πG 3ρm+C1 3ϵ+ρgel.(6.20) The vacuum energy contributes as an effective cosmological constant ρΛ,eff =C1/(3ϵ). This can be tuned to the observed ρΛ,0by choosing microscopic parameters such that C1/(3ϵ)≈ρΛ,0. 4 Both limits show that collision-driven injection + losses can produce a late-time de Sitter attractor with a constant Hdetermined by microphysical parameters. 6 Linear Stability of the Vacuum–Expansion Fixed Point (Sketch) Suppose there exists a fixed point characterized by (H∗, ρ∗ vac)solving the algebraic steady equations ( ˙ρvac = 0 and Friedmann ). We study linear perturbations δH(t), δρ(t). Linearizing (6.1)–(6.2) (neglecting matter/radiation perturbations for simplicity) yields a linear system of the form δ˙ρ+ 3H∗(1 + w∗)δρ + 3(1 + w∗)ρ∗δH + 3H∗ρ∗δw =∂HQ∗δH +∂ρQ∗δρ −δL, 2H∗δ˙ H=8πG 3δρ, (6.21) where ∂HQ|∗denotes the derivative of Q(H)evaluated at H∗(for the ansatz Q=C1H+C2H2 we have ∂HQ|∗=C1+ 2C2H∗). Eliminating δρ using the second equation gives an ODE for δH of the form δ˙ H=−ΓeffδH with Γeff =3(1 + w∗)H∗ 2−3(1 + w∗)ρ∗ 4H∗∂HQ|∗−∂ρQ|∗8πG 3+··· ,(6.22) (derivation details in Appendix E). The sign of Γeff determines stability: •Γeff >0⇒δH decays (stable attractor). •Γeff <0⇒δH grows (unstable/runaway). For the physically relevant case w∗≈ −1+ϵwith small ϵ > 0and injection scaling sublinearly (so ∂HQ|∗not too large), Γeff tends to be positive — the system is a stable attractor. Large positive feedback (e.g. huge C2) can flip sign and produce instability; this simply restates that unbounded positive feedback from injection must be tamed by dissipation or nonlinear saturation to obtain a viable cosmology. 7 Role of Gel Formation (ρgel) and Energy Partitioning So far we focused on ρvac. A fraction (1 −ζ) ˙ρinj becomes gel mass ρgel (see §3.7, eq. (3.21) and (3.22)). The gel continuity equation (including source and slow decay) is ˙ρgel + 3Hρgel = (1 −ζ) ˙ρinj −Γdecayρgel +Sclust,(6.23) where Sclust accounts for clustering/virialization energy transfer. Since ˙ρinj ∝Hor H2,ρgel inherits a production term proportional to cosmic expansion rate. Physically: • If (1 −ζ)is substantial, a significant fraction of injection goes into gel mass and behaves gravitationally like dark matter, seeding halos. • Conservation of total injected energy implies competition between ρvac and ρgel; observationally, the present ratio Ωgel/Ωvac constrains ζand microscopic branching fractions finel, fbind. 5 8 Mapping to Observations (Recipe) To compare with observations and constrain parameters: 1. Choose microphysical parameters {α, τ0,∆t, τeff , η, finel, ζ}. Compute Ev=ℏ/∆t. 2. Evaluate C1=ζηfinelEvατ0and C2=ζπc3ℏ∆t(ατ0τeff )2finelη. 3. Solve the coupled ODEs (6.2), (6.9) and (6.23) numerically (including matter and radiation) to obtain H(z),ρvac(z),ρgel(z). 4. Fit to observational datasets (H0, SNe, BAO, CMB distance measures, growth of structure, lensing) to bound combinations of C1, C2, ζ, Γloss, and ϵ. 5. Use halo modeling to relate ρgel(r)(from §3) to rotation curves and lensing profiles — this constrains fbind and the binding parameters (g, ms). Two immediate constraints arise: • Present dark energy density ρvac(t0)≈ρΛ,0fixes combinations of C1, C2,Γloss, w. • Present dark matter halo abundance constrains the rate and efficiency of gel formation (1 −ζ) ˙ρinj and the subsequent clustering dynamics. 9 Summary — Key Equations to Include in the Manuscript Include the following boxed expressions in your final paper (they are self-contained and traceable back to §4 and §5): 1. Vacuum continuity with microphysical source ˙ρvac + 3H(1 + wvac)ρvac =C1H+C2H2−L(t).(6.24) 2. Microphysical coefficients C1=ζηfinelEvατ0, C2=ζπc3ℏ∆t(ατ0τeff)2finelη. (6.25) 3. Steady-state vacuum density (balance of injection and losses) ρ(∗) vac(H) = C1H+C2H2 3H(1 + wvac)+Γloss .(6.26) 4. Coupled Friedmann H2=8πG 3ρm+ρr+ρvac(H) + ρgel−kc2 a2.(6.27) These four items plus the gel continuity (6.23) form a closed, physically transparent system: microphysics →(C1,2)→injection (Q)→modified cosmology. 6 10 Concluding Remarks (Physical Intuition) •Expansion feeds vacuum — through the strain-induced production law Γ∝Hthe expansion rate itself sources vacuum collisions; this is the core feedback loop: H↑⇒ Γ↑⇒ Q↑⇒ ρvac ↑⇒ H↑— and the loop’s net effect depends on the sign and strength of dissipation and nonlinear saturation. •Two regimes (linear vs quadratic in H) control early vs late behaviour: early universe (large H) often sits in collision-dominated regime producing effective inflationary dynamics; late universe may settle into collisionless or steady regimes, depending on parameters. •Unified origin — the same microphysical injection that increases ρvac can (with branching ratio 1−ζ) produce gel mass that seeds dark-matter–like structures, realizing the paper’s unification aim. 11 Two-Body Binding in a Screened (Yukawa) Potential — Variational Derivation We model the effective interaction between two vacuum quanta after a collision by a screened attractive potential (Yukawa form), V(r) = −g2 4π e−msr r,(7.1) where gis an effective coupling (dimension √energy ×length in natural units) and msis the screening mass; the screening length is λs= 1/ms. Let the effective reduced mass of the interacting pair be meff (for identical constituents meff =m/2if each constituent has mass m; often meff will be related to the effective energy Ev/c2introduced earlier). To test whether a bound state exists we use a variational exponential trial wavefunction (s-wave), ψα(r) = rα3 πe−αr, α > 0,(7.2) normalized so Rd3r|ψ|2= 1. The expectation values of kinetic and potential energy are • Kinetic energy T(α) = ⟨ψ|− ℏ2 2meff ∇2|ψ⟩=ℏ2α2 2meff .(7.3) • Potential energy V(α) = Zd3r|ψα(r)|2V(r) =−g2 4πZ∞ 0 4πr2α3 πe−2αr e−msr rdr =−g2α3Z∞ 0 e−(2α+ms)rrdr =−g2α3 (2α+ms)2. (7.4) Thus the variational energy is E(α) = T(α) + V(α) = ℏ2α2 2meff −g2α3 (2α+ms)2.(7.5) 7 A bound state exists if minα>0E(α)<0. To find the minimum we set ∂αE(α)=0. Differentiating (7.5) gives ℏ2α meff −g23α2(2α+ms)2−4α3(2α+ms) (2α+ms)4= 0.(7.6) This algebraic equation can be solved numerically for αgiven (g, ms, meff ). However, useful analytic limits exist: 11.1 Coulomb (unscreened) limit (ms→0) Set ms= 0. Then (7.5) reduces to E(α) = ℏ2α2 2meff −g2α. Minimization gives α∗=meffg2 ℏ2and the minimum energy Emin =−meffg4 2ℏ2.(7.7) Thus any nonzero gproduces a bound state in the Coulomb limit (as expected). 11.2 Short screening length (ms≫α) (strong screening) If ms≫αthen 2α+ms≈ms, so potential expectation (7.4) becomes approximately −g2α3/m2 s which scales as α3, whereas kinetic scales as α2. Minimization then yields an approximate optimum α∼3meffg2 2ℏ2m2 s and the binding energy is suppressed. A necessary condition for binding is approximately g2≳g2 crit ∼ℏ2ms meff ,(7.8) up to O(1) factors (obtainable by balancing the orders of magnitude of Tand |V|:ℏ2α2/(2meff)∼ g2α3/m2 s⇒α∼ℏ2m2 s/(2meffg2); requiring negative energy implies the constant scaling (7.8)). A sharper bound follows by solving (7.6) numerically; (7.8) is sufficient for order-of-magnitude estimates. 11.3 Useful dimensionless parameter Define the dimensionless coupling χ≡meffg2 ℏ2ms .(7.9) • If χ≫1(strong coupling or long screening length) binding is easy. • If χ≲O(1) binding may be marginal or absent. In words: a bound two-body vacuum state forms when the attractive interaction gis strong enough relative to the screening and the quantum kinetic pressure set by ℏ2/(meff). >Practical variational recipe (to include in the paper): present (7.5)–(7.6) and either (a) solve (7.6) numerically for α∗and check E(α∗)<0, or (b) use the limiting criterion (7.8) as an order-of-magnitude binding threshold. This gives a clear mapping from microphysical coupling constants to the existence of metastable bound pairs. 8 12 Many-Body Aggregation: From Bound Pairs to Macroscopic Gel If a non-negligible fraction fbind of collisions form bound states, these bound units can aggregate into larger bound clusters by further collisions and gravitational attraction. Two complementary mean-field descriptions are useful: 1. Self-gravitating isothermal sphere (classical, kinetic-velocity support). 2. Bose-condensate / Gross–Pitaevskii mean field (if the gel constituents behave as bosonic quasiparticles and quantum pressure matters at core scales). Both routes lead naturally to a core + halo morphology: a central dense core (quantum or pressure supported) and an outer approximate r−2envelope responsible for flat rotation curves. 12.1 Hydrostatic Equilibrium, Isothermal Approximation and ρ∝r−2 Assume the gel is a large, approximately spherical, self-gravitating aggregate with local isotropic velocity dispersion σ2and effective pressure p=σ2ρ(isothermal equation of state). Hydrostatic equilibrium gives dp dr =−ρ(r)GM(r) r2, M(r) = 4πZr 0 ρ(r′)r′2dr′.(7.10) Substitute p=σ2ρand rearrange: σ2dln ρ dr =−GM(r) r2.(7.11) Differentiate M(r)and combine with (7.11). For a solution that at large radii approaches a power law, set ρ(r)∝r−γ. If γ= 2 then M(r)∝rand the right hand side of (7.11) becomes −G·(const)/r and the left side is −γσ2/r giving consistency. Explicitly, assume ρ(r) = Ar−2. Then M(r) = 4πAr, σ2dln ρ dr =−2σ2 r, and the hydrostatic equation (7.11) becomes −2σ2 r=−G·4πAr r2=−4πGA r. Cancel 1/r to obtain A=σ2 2πG.(7.12) Therefore the isothermal self-gravitating sphere has the outer profile ρgel(r) = σ2 2πG 1 r2,(rin halo region)(7.13) and the enclosed mass M(r) = 4πAr =2σ2 Gr. (7.14) Consequently the circular velocity is v2 c(r)≡GM(r) r= 2σ2⇒vc=√2σ, (7.15) i.e. flat rotation curve at amplitude set by σ. This directly reproduces the observed flatness of galaxy rotation curves if the gel halo isothermal assumption holds in the outer region. 9 12.2 Core Formation — Quantum Pressure or Self-Interaction Support The r−2solution cannot extend to r→0(would produce divergent mass); in practice halos show central cores. Two mechanisms can produce a core: •Quantum pressure (if gel constituents are very light bosonic quanta forming a Bose– Einstein condensate). In the Gross–Pitaevskii (GP) / Schrödinger–Poisson description the core scale (soliton radius) is set by the balance of quantum kinetic pressure and gravity: ℏ2 2m2 br4 c∼GMc r3 c , giving (order of magnitude) rc∼ℏ2 Gm2 bMc or rc∼ℏ mbvc ,(7.16) where mbis the constituent mass and Mcthe core mass; vcis the circular velocity near the core. This is the same scaling used in ultralight dark matter (fuzzy DM) cores. •Self-interaction pressure (if constituents have a short-range repulsive self-interaction ∝λSI|ψ|4): core radius scales as rc∼λSIℏ2 Gm4 b1/2 ,(7.17) following a Thomas–Fermi-like equilibrium (see e.g. self-interacting BEC dark matter literature). Either mechanism yields a finite central density ρcand a core radius rc. Outside r≳rcthe halo relaxes to an approximately isothermal profile (7.13). 12.3 Mapping to Gel Microphysics The velocity dispersion σin (7.13) is set by the energy per gel particle (or the virial temperature of the gel aggregate). If gel constituents form from vacuum-collision binding with per-particle energy Ebind ∼Ev, then a characteristic velocity is σ2∼Ebind mb∼ℏ ∆t 1 mb .(7.18) Thus for given microphysical mb,∆tone can compute σand then predict vc=√2σof resulting halos (to be compared with measured flat rotation velocities). Conversely, observed vcprovides constraints on combinations Ebind/mb. 13 Halo Abundance and Formation Rates from Collision Kinetics The gel mass density production rate derived in §3 (eqs. (3.21)–(3.22)) was ˙ρ(source) gel (t) = Rc(t)fbind(t)Ev,(7.19) with Rcthe collision rate and fbind the fraction leading to bound clusters. Using the collisiondominated approximation Rc≈1 2Γ = 1 2ατ0H(if that regime applies) we get ˙ρgel ≈1 2ατ0HfbindEv.(7.20) 10 Integrating over cosmological time gives cumulative ρgel(t). The spatial distribution of gel objects will be modulated by gravitational clustering: initially produced small gel clumps will merge and accrete into larger halos following standard hierarchical formation processes (but seeded by the continuous gel source rather than primordial cold dark matter only). A convenient phenomenological model for halo seeding rate per comoving volume is Shalo(t)∼˙ρgel(t) Mh ,(7.21) where Mhis a characteristic halo mass seeded per binding event (may be of order many particle masses if hierarchical aggregation is efficient). This allows connection to halo mass functions used in structure formation studies. 14 Gravitational Potential and Metric Curvature from Gel Density Treating a gel halo as a pressureless mass distribution at large radii (pressure small compared to energy density), its stress-energy enters Einstein’s equations as dust: Tµν gel ≈ρgeluµuν.(7.22) In weak field (Newtonian) limit the Poisson equation holds: ∇2Φ = 4πGρgel(r).(7.23) For ρgel(r) = A/r2the gravitational potential is Φ(r)=2πGA ln(r) + const, and the circular velocity is constant as shown in (7.15). In GR one can compute the stationary spherically symmetric metric sourced by such Tµν (Tolman–Oppenheimer–Volkoff framework) — for phenomenology the Newtonian approximation is sufficient for galaxy scale tests. 15 Predicted Observable Signatures & Parameter Mapping The gel model makes specific, testable predictions that can be expressed in closed form: 1. Flat rotation curves. For outer halo ( ρgel =σ2 2πGr2), the asymptotic circular velocity is v∞=√2σ. Observationally v∞ranges from ∼50–300 km/sfor typical galaxies; hence σ≈v∞ 0∼(35–210) km/s.(7.24) 2. Core radius scaling. If quantum pressure sets the core size (7.16) and measured vcis known, then rc∼ℏ mbvc .(7.25) For example, a core radius rc∼1 kpc and vc∼100 km/simply a constituent mass mb∼ℏ/(rcvc)∼10−22 eV/c2(the well-known fuzzy DM scaling). If mbis heavier, core radii are smaller and quantum pressure irrelevant. 3. Mass–velocity relation. Using M(r) = 2σ2r/G and evaluating at a typical optical radius ropt yields a Tully–Fisher–like scaling M∝v2 cropt. With baryonic mass and gel mass contributions one can tune model parameters to fit observed baryonic Tully–Fisher relations. 11 Define the dimensionless potential (ψ≡(Φ −Φ0)/σ2) and scale radius (r2 0≡σ2/(4πGρ0)). Then the equation becomes the isothermal Lane–Emden equation: 1 ξ2 d dξ (ξ2dψ dξ )=e−ψ,with ξ≡r/r0.(G.8) At large radii (ξ≫1) the solution approaches (ψ∼ln(ξ2) + const) (one can show this by asymptotic analysis), which implies (ρ(r)∝e−ψ∝1/r2). Concretely: • Seek asymptotic solution (ψ(ξ)≃Aln ξ+B). Substitute into (G.8); leading terms demand (A= 2). Therefore (ψ∼2 ln ξ+const) and hence (ρ∼ρ0e−ψ∼ρ0ξ−2). Thus the outer envelope of an isothermal sphere behaves as ρ(r)∝1 r2(r≫r0).(G.9) G.3 Circular velocity & flat rotation curves Using (M(r) = 4π∫r 0ρ(r′)r′2dr′) and the asymptotic (ρ∝r−2), we get (M(r)∝r) at large (r). Therefore the circular velocity v2 c(r) = GM(r) r r→∞ −−−→ const,(G.10) i.e. a flat rotation curve. Quantitatively, for the isothermal asymptotic profile (ρ=A/r2) we find (M(r) = 4πAr) and (v2 c= 4πGA). Comparing with (7.12) in §7 yields the same relation (A=σ2/(2πG)) and hence (v2 c= 2σ2). G.4 Comments on core vs envelope and physical applicability • The isothermal sphere solution has infinite mass if extended to (r→ ∞). Physical systems truncate (e.g., due to tidal forces, cosmological background, or phase-space constraints). In practice, one uses a truncated isothermal sphere or matches to a steeper outer profile. • At small radii, the isothermal assumption breaks down; central cores arise from finite phase-space density, quantum pressure, self-interactions, or non-isothermal heating/cooling processes. The Jeans derivation above provides the outer envelope and the basis for the hydrostatic arguments in §7. • Observed halos are well fit by NFW or cored profiles; the isothermal (r−2) envelope is a good approximation over a substantial radial range and naturally explains flat rotation curves. G.5 Short derivation to include in manuscript (compact) For inclusion as a concise subsection, the following compact derivation is ready to paste: Starting from the steady-state, spherically symmetric Jeans equation for isotropic velocity dispersion, d(ρσ2) dr =−ρdΦ dr , with constant (σ) (isothermal assumption), integrate to obtain (ρ(r) = ρ0exp [− (Φ(r)−Φ0)/σ2]). Combining with Poisson’s equation and solving the resulting isothermal Lane–Emden equation yields the asymptotic solution (ρ(r)∝r−2). Consequently (M(r)∝r) and (v2 c=GM(r)/r) is constant at large (r), reproducing flat rotation curves with amplitude (v∞=√2σ). 5 G.6 Suggested numerical checks for the manuscript 1. Solve isothermal Lane–Emden (G.8) numerically using a simple ODE solver with boundary conditions (ψ(0) = 0, ψ′(0) = 0). Plot (ψ(ξ)) and verify (ψ∼2 ln ξ) for (ξ≳10). 2. Construct full density profile (ρ(r) = ρ0e−ψ(r)) and compare with the hydrostatic isothermal sphere (7.13) at large radii; show core deviation near (r≲r0). 3. Compare with NFW and observational rotation curves to show the radial band where (r−2) is a good fit. G.7 Summary (Appendix G) • The collisionless Boltzmann equation under steady-state, spherical symmetry and isotropic velocity dispersion leads to the isothermal Lane–Emden equation. • Its asymptotic solution yields (ρ(r)∝r−2), which gives (M(r)∝r) and flat rotation curves. • This justifies the use of the isothermal envelope (eq. 7.13) in §7 as the natural many-body outcome for velocity-dispersion supported gel halos. 8.1 Setup — summary of key background equations We summarize the minimal set of background equations used throughout (notation consistent with previous sections): Friedmann equation (flat (k= 0) used for clarity) H2=8πG 3ρtot, ρtot ≡ρm+ρr+ρvac +ρgel.(8.1) Continuity for injected/stored vacuum energy (Eq. 6.24) ˙ρvac + 3H(1 + wvac)ρvac =Q(H)−L(t),(8.2) with the microphysical ansatz Q(H)≡C1H+C2H2(8.3) (see §6.2 for expressions of (C1, C2) in terms of microscopic parameters). Gel mass evolves as ˙ρgel + 3Hρgel = (1 −ζ) ˙ρinj −Γdecayρgel +Sclust,(8.4) with ( ˙ρinj) related to (Q) by (ζ˙ρinj =Q) (so ( ˙ρinj =Q/ζ) when (ζ= 0)). Matter and radiation obey standard conservation: ˙ρm+ 3Hρm= 0,˙ρr+ 4Hρr= 0.(8.5) Raychaudhuri / acceleration equation follows from (8.1) and stress contents: ˙ H=−4πG(ρtot +ptot c2)=−4πG[ρm+ρr+ρvac(1 + wvac) + ρgel],(8.6) (using (pm= 0, pr=ρr/3, pgel ≈0)). For compactness we set (c= 1) in what follows (put factors back if needed). Define the (background) Hubble slow-roll parameter εH≡ − ˙ H H2.(8.7) Inflation-like (quasi–de Sitter) expansion requires (εH≪1). Late-time acceleration requires (¨a > 0), i.e. (εH<1). We will express (εH) in terms of model quantities. 6 8.2 Early universe: conditions for an inflation-like phase 8.2.1 Qualitative requirement An inflationary epoch requires (for a duration sufficient to solve horizon/flatness problems) a period with (H≃) nearly constant and (εH≪1) for many e-folds (N≡ln(aend/astart)), where N=∫tf ti Hdt =∫Hf Hi H ˙ HdH =∫Hf Hi 1 εH dH H.(8.8) 8.2.2 Expression for (εH) in this model Using (8.6) and (8.1), εH=4πG H2[ρm+ρr+ρgel + (1 + wvac)ρvac].(8.9) During the very early universe matter and gel are negligible; assume radiation is subdominant compared to the injected/stored vacuum energy (the scenario of interest for vacuum-driven inflation). Then εH≃4πG H2(1 + wvac)ρvac.(8.10) If elastic storage dominates and (wvac ≈ −1 + δ) with (|δ| ≪ 1), then (1 + wvac ≈δ) small and (εH≪1) if (ρvac/H2) is not huge. To evaluate (ρvac) we use the continuity (8.2) with dominant (Q) term. 8.2.3 Collision-dominated early regime (large (H)) Assume early times are collision-dominated so (Q≈C1H+C2H2). For large (H) the (H2) term typically dominates (unless (C1) is anomalously large), so we set (Q≈C2H2). Further suppose losses (L) are negligible on very short time scales (or included later as reheating). Then (8.2) reduces to ˙ρvac + 3H(1 + wvac)ρvac ≃C2H2.(8.11) We search for quasi–de Sitter solutions where (H≈H∗) approximately constant and (ρvac) nearly constant or slowly varying. Set ( ˙ρvac ≈0) and solve for (ρvac): 3H∗(1 + wvac)ρvac,∗≃C2H2 ∗=⇒ρvac,∗≃C2H∗ 3(1 + wvac).(8.12) Substitute into (8.10): εH≃4πG H2 ∗ (1 + wvac)·C2H∗ 3(1 + wvac)=4πG 3 C2 H∗ .(8.13) Therefore the smallness of (εH) requires H∗≫4πG 3C2.(8.14) Equivalently, if (H∗) starts large (early universe), this inequality can hold and (εH) can be small. Note the direction: larger (H∗) helps making (εH) small when (C2) is fixed. This argument shows that an inflation-like phase is possible when the injection term scales like (H2) and the coefficient (C2) is not so large as to make the RHS of (8.13) order unity. Physically, early large (H) produces copious virtual-pair production but the stored vacuum 7 energy self-consistently supports a slowly varying (H). The required number of e-folds (N) is then N≃∫tf ti 1 εH dln a∼H∗ εH ∆t∼3H2 ∗ 4πGC2 ∆t. (8.15) But since (εH) is nearly constant in the attractor, the simpler estimate (N≈H∗/εH) per Hubble time yields that many e-folds accumulate if (εH≪1). 8.2.4 Reheating via gel formation and dissipative channels Inflation must end and reheat the universe. In our model, reheating can naturally proceed when either: • the kinetic / microphysical threshold (E∗) or the parameter (Ξ) (eq. 4.24) moves the system from collision-dominated to collisionless regime, reducing injection efficiency to elastic storage; or • significant fraction of injected energy channels into gel formation and dissipative modes ((1−ζ) or (Dvisc) becomes large), converting vacuum energy into localized gel mass and radiation. Mathematically, the end of inflation corresponds to growth of (εH) to (O(1)). If at some time the loss term (L) jumps (due to efficient conversion of injected energy to relativistic particles), (˙ρvac) becomes negative and the effective (wvac) shifts, producing ( ˙ H < 0) and ending quasi–de Sitter expansion. The energy deposition rate into radiation is directly (Lrad) and reheating temperature (Treh) satisfies π2 30g∗T4 reh ∼ Lrad∆treh,(8.16) where (∆treh) is the timescale of conversion and (g∗) the effective relativistic degrees of freedom. Detailed reheating dynamics depend on microphysics (branching fractions (fbind, finel), and viscosity) and can be computed numerically (Appendix B suggested). Summary (early universe): the mechanism supports an inflation-like epoch when collisiondominated injection supplies approximately constant vacuum energy and dissipation is small; inflation ends and reheating follows when the branching into gels/dissipation becomes significant or microphysical thresholds suppress elastic storage. 8.3 Late universe: approach to dark-energy domination and attractor behaviour 8.3.1 Slow dynamical accumulation and effective (w) At late times (H) is small; the collisionless limit or the linear limit may dominate. Consider the collision-dominated linear case (Q≈C1H). If elastic storage dominates (large (ζ)) and dissipation small, (8.2) with (wvac ≈ −1 + δ) gives (steady or slowly varying) ˙ρvac ≃C1H−3H(1 + wvac)ρvac.(8.17) If (ρvac) evolves slowly, ( ˙ρvac ≪Hρvac), we approximate ρvac ≃C1 3(1 + wvac).(8.18) Thus (ρvac) can asymptote to an effective constant determined by microphysics. Notice that if (δ≡(1+wvac)) is small, a small (C1) can generate a large (ρvac) — this is the lever arm enabling microscopic processes to produce an observable cosmological constant–scale energy density. 8 Using (8.6) one finds the late-time slow-roll parameter εH≃4πG H2(1 + wvac)ρvac ≃4πG H2·C1 3.(8.19) If (ρvac) dominates the total energy density, (H2≃(8πG/3)ρvac), then substituting (8.18) gives H2≃8πG 3·C1 3δ⇒εH≃4πG H2δρvac =4πG H2δC1 3δ=4πG 3 C1 H2.(8.20) This is identical to the structure in (8.13) but with (C1) replacing (C2H). The algebra shows the system can self-consistently approach a de Sitter-like attractor for suitable parameter values; the effective Hubble constant of this late attractor is fixed by microphysical constants (C1, δ), and any loss rate (Γloss) (see §6.3.1 and (6.18)–(6.20)). 8.3.2 Attractor analysis (qualitative) Linear stability analysis (sketch — full algebra in Appendix E if included) shows that when injection scales at most linearly with (H) (so feedback is not explosive) and losses or slight deviations in (wvac) regularize the equations, the background has an attractor solution where (H) tends toward a constant (H0). The existence and stability of the attractor require that the effective feedback derivative (∂HQ) evaluated at the fixed point does not exceed a threshold that would make (Γeff <0) in Eq. (6.22). Physically, dissipation and conversion into gel mass provide negative feedback that stabilizes the system. 8.3.3 Observational matching & parameter constraints To reproduce observed late-time values ((H0), (ΩΛ,0)), one may calibrate (C1, C2, ζ, Γloss, δ) via eqs. (6.25),(6.26). A practical fitting recipe: 1. Choose (ζ) (fraction to elastic storage): larger (ζ) favors dark energy. 2. Given desired (ρvac(t0) = ρΛ,0), use (6.26)/(8.18) to constrain combinations of (C1, C2,Γloss, δ). 3. Ensure the gel production ((1−ζ) ˙ρinj)) integrated over history matches (ρDM,0) if the model aims to replace particle dark matter (see §8.4). 4. Verify that the effective equation-of-state (weff (z)) implied by the model is consistent with observational bounds (Planck + SNe + BAO typically constrain (w0≈ −1±0.03) and (wa) small). In this model (weff (z)) is computable from (8.2) and (8.3) and is generically near (-1) for elastic-dominated storage with small dissipation. Summary (late universe): the mechanism provides a natural path to an effective cosmological constant: slow accumulation of elastic vacuum energy with slight dissipative corrections yields (w≈ −1) and a late-time de Sitter attractor whose scale is set by microphysical constants. 8.4 Galaxy formation era: gel seeding, clustering and growth of perturbations 8.4.1 Gel as dark-matter analogue — background and local behavior If a fraction ((1−ζ)) of injected energy goes into gel mass, that mass behaves gravitationally like pressureless matter on galactic and cosmological scales (except possibly in very small cores 9 where quantum pressure/self-interaction matters). The comoving production rate (per unit comoving volume) of gel mass is ˙ρ(source) gel = (1 −ζ) ˙ρinj ≈1−ζ ζQ(H),(8.21) since (ζ˙ρinj =Q). If gel production occurs continuously, its cumulative contribution can constitute a substantial fraction of the present dark matter density. 8.4.2 Linear perturbation equations with gel source We now derive the modified linear growth equation for matter perturbations in the presence of continuous gel creation. Work in Newtonian gauge for sub-horizon modes and denote the density contrasts (δm) (baryons+non-gel CDM if any) and (δgel) for produced gel matter. If gel behaves like dust after formation and quickly clusters, the linearized continuity and Euler equations for gel look like ordinary dust; however, the background production changes the background densities and introduces source terms in perturbations if creation is spatially inhomogeneous. For simplicity assume gel seed production is spatially homogeneous on large scales (reasonable for vacuum collisions tied to global expansion); then perturbation evolution follows the standard form but with total matter density including (ρgel). The standard growth equation for the combined pressureless matter density contrast (δ) (assuming single fluid approximation after gel mixes) becomes ¨ δ+ 2H˙ δ−4πGρtot mδ=Sprod,(8.22) where (ρtot m=ρb+ρgel +ρCDM) and (Sprod) encapsulates source terms from non-adiabatic creation (if creation is strictly homogeneous (Sprod = 0) at linear order). Thus, to leading order the presence of gel that behaves like dust accelerates structure growth due to increased (ρtot m). If gel production is locally biased (for example, preferential in overdense regions because of nonlinear enhancement of local strain / collision probability), then (Sprod) contains a positive term proportional to ( ˙ρ(source) gel δbias) that acts as additional source of overdensities — this can boost early structure formation and alleviate problems like early galaxy formation timelines. Modeling bias requires a microphysical model for how local curvature/strain enhances (Γ) (out of scope for the background paper but straightforward to add numerically). 8.4.3 Modified growth rate and observational signatures Define the linear growth factor (D(a)) via (δ(a) = D(a)δ(ai)). The growth equation (8.22) implies (for (Sprod ≈0)) D′′(a) + (3 a+H′(a) H(a))D′(a)−4πG H2a2ρtot m(a)D(a) = 0,(8.23) where primes denote derivatives w.r.t. (a). Since (ρtot m) now includes (ρgel(a)) produced from vacuum collisions, the growth history will differ from ΛCDM. Observable consequences: •Matter power spectrum amplitude (σ8): the integrated effect of enhanced (ρtot m) or biased gel production will change (σ8). Current measurements tightly constrain (σ8) in combination with (Ωm) and can be used to bound ((1−ζ)) and (fbind). •Redshift-space distortions / growth index: the growth rate (f≡dln D/d ln a) will deviate from ΛCDM prediction; future redshift-space distortion measurements can test this. 10 •CMB lensing & ISW effect: differences in the growth history change the lensing potential and the integrated Sachs–Wolfe (ISW) effect; cross correlations between large-scale structure and CMB temperature map sensitive to ISW can constrain time dependence of (ρvac) and gel production. •Halo mass function & abundance of early galaxies: because gel formation is an additional seed mechanism for halos, abundance of high-z massive galaxies and early reionization history provide constraints. 8.4.4 Nonlinear structure & halo internal structure At halo scales, the microphysical gel formation and subsequent aggregation determine internal halo profiles. Section 7 derived that many-body relaxation produces approximately isothermal outer profiles (ρ∝r−2) and cores where quantum pressure/self-interaction matters. Predictions to compare with data: •Rotation curves: amplitude and shape (especially core size vs halo mass) can constrain constituent mass (mb) and binding energy scale (Ebind). •Gravitational lensing: lensing mass directly probes (ρgel(r)) and can distinguish between cuspy (NFW) and cored profiles. •Substructure: if gel seed formation is continuous, small scale subhalos may be overproduced or suppressed depending on formation efficiency and merging rates — compare to Milky Way satellite counts. 8.5 Distinguishing observational predictions (summary) The vacuum–collision + elastic spacetime model is falsifiable through several avenues: 1. Equation-of-state evolution: predicted deviations of (weff (z)) from (-1) are small but potentially measurable. Parameterize (w(z) = w0+wa(1−a)) and derive (w0, wa) from the model (numerically) for chosen microphysics; compare to current limits (Planck+SNe+BAO). 2. Growth of structure: the modified growth history (via (ρgel) and feedback) produces measurable differences in (fσ8(z)), lensing potential, and cluster abundance. 3. ISW and CMB signatures: time-varying vacuum injection leaves an ISW imprint at large scales; cross-correlations with large-scale structure can detect or constrain it. 4. Halo internal structure: correlation between halo core radius and velocity/halo mass predicted by the gel microphysics (e.g., the soliton scaling (rc∝ℏ/(mbvc)) if quantum pressure relevant) is testable via rotation curves and strong lensing. 5. Non-thermal backgrounds / reheating signatures: if some injected energy is converted into relativistic particles at particular epochs (e.g., during reheating), this could leave detectable signals (spectral distortions of the CMB, extra relativistic degrees of freedom (Neff) at BBN/CMB). 8.6 Parameter mapping and example scaling relations (useful for fits) To allow direct data-driven constraints, express the key observables in terms of microscopic parameters: 11 • Late vacuum density (approximate) ρvac,0∼C1H0+C2H2 0 3H0(1 + wvac)+Γloss .(8.24) Use measured (ρΛ,0) to fix combinations of (C1, C2,Γloss, wvac). • Present gel production contribution (integrated) ρgel,0≃∫t0 tprod e−3∫t0 t′Hdt′′ (1 −ζ)Q(H(t′)) ζdt′.(8.25) This integral can be computed numerically for any assumed (H(t)) history. Matching (ρgel,0) to (ρDM,0) constrains ((1−ζ)) and other microphysical inputs. • Growth factor modification (first order) ∆f∼1 2 ∆ρm(a) ρm(a)(rough estimate for small ∆ρm),(8.26) where (∆ρm) is gel contribution relative to conventional matter. Observational bounds on (fσ8) then translate to bounds on (∆ρm). 8.7 Practical roadmap for confronting data (recommended, copypaste ready) 1. Choose microphysical prior ranges for ((α, τ0,∆t, η, finel, fbind, ζ, κ, ηv, τr)). 2. Compute (C1, C2)from (6.25). 3. Numerically integrate the coupled background ODEs (Friedmann + (8.2) + (8.4)) from high redshift (e.g., (z∼109)) to (z= 0), including standard radiation and baryon components — obtain (H(z), ρvac(z), ρgel(z)). 4. Compute linear perturbations (modified growth) and observables: (H(z)) distances, CMB angular diameter, growth factor (fσ8), lensing potential, ISW cross-correlation. 5. Fit to datasets (Planck CMB, BAO, SNe, RSD, lensing) to infer posterior distributions for microphysical parameters. Use constraints to refine the microphysical model, especially the branching (ζ) which determines dark energy vs dark matter split. 6. Use halo modeling (Section 7 prescriptions) and rotation-curve/lensing data to test gel internal structure predictions and constrain binding parameters ((g, ms, meff )). 8.8 Caveats, theoretical consistency checks and open challenges •Renormalization & backreaction: the model treats vacuum fluctuations with effective kinetic and elastic parameters; a careful renormalization analysis in QFT in curved spacetime is required to ensure no double counting with known vacuum energy contributions. •Causality & stability: viscoelastic parameters must respect causality and positive dissipation (Appendix D); large positive feedback from (Q(H)) can render the background unstable — parameter scans must check eigenvalues (Appendix E). 12 •Microscopic justification: the form (Q∝H, H2) is phenomenological; embedding in a full quantum field / semiclassical gravity derivation (e.g., computing (Γ) from first principles in curved space) would strengthen the model. •Nonlocality: elastic stress tensor contains divergence terms (Appendix C). Perturbation theory must include anisotropic stress effects on CMB and structure; neglecting these may miss key signatures. 8.9 Concluding summary of Section 8 • The vacuum–collision + elastic spacetime mechanism admits three cosmologically relevant regimes: 1. Early collision-dominated regime: can support an inflation-like quasi-de Sitter expansion when (Q∝H2) and dissipation small. Inflation ends naturally via branching into gels/dissipation. 2. Late slow accumulation: elastic storage with small losses yields (weff ≈ −1) and a stable de Sitter attractor; observed dark energy can be matched by choosing microscopic parameters. 3. Galaxy formation era: vacuum–gel formation supplies pressureless mass that seeds halos, producing (r−2) envelopes and flat rotation curves; the abundance and small-scale structure depend on branching fractions and binding microphysics. • The model produces a rich set of observational signatures (background expansion history, growth of structure, ISW and lensing, halo cores) and provides concrete parameter mapping (Eqs. 8.24–8.25) suitable for confronting data. 9.1 Summary of model quantities to be constrained For clarity, list the minimal set of phenomenological / microphysical parameters that determine the observable consequences (definitions refer to earlier sections): • Microphysics that determine injection: (α, τ0,∆t) (appearance/energy/time scales), and the effective virtual-quantum energy (Ev=ℏ/∆t). • Branching & efficiency parameters: (finel) (fraction of collisions that are inelastic), (η) (fraction of pair-energy deposited in an inelastic event), (fbind) (fraction forming bound- /gel states), and (ζ) (fraction of injected energy stored elastically vs routed to other channels). • Elastic / spacetime response: bulk stiffness (κ=λ+2µ/3), viscoelastic parameters (ηv, τr), and effective equation-of-state deviation (δ≡1 + wvac) (microphysically computable; treated as nuisance if necessary). • Derived macro coefficients (useful summary): C1=ζηfinelEvατ0, C2=ζπc3ℏ∆t(ατ0τeff)2finelη, which enter the cosmological source (Q(H) = C1H+C2H2). • Gel microphysics: coupling (g), screening mass (ms), effective constituent mass (mb) or (meff), which determine halo internal structure and core sizes. All of the following observational formulae are expressed in terms of (i) the background functions (H(z), ρvac(z), ρgel(z)) obtained by integrating the ODE system in §6–8 with these parameters, and (ii) local halo profile parameters computed in §7 (e.g. (σ), (rc), (ρ0)). 13 9.2 Background expansion — mapping to distance observables 9.2.1 Hubble parameter and distances Solve the background system    H2(z) = 8πG 3[ρm(z) + ρr(z) + ρvac(z) + ρgel(z)], ˙ρvac + 3H(1 + wvac)ρvac =C1H+C2H2−L(z), with (˙= −(1+z)Hd/dz) for numerical integration in redshift. From the solution one computes: • Comoving distance: χ(z) = ∫z 0 dz′ H(z′). • Luminosity distance: dL(z) = (1 + z)Sk(χ(z)), where (Sk(χ) = χ) for flat cosmology (we assume (k= 0) unless testing curvature). • Angular diameter distance: dA(z) = dL(z) (1 + z)2. 9.2.2 Likelihood for background probes Use standard Gaussian likelihoods for SNe, BAO and (H(z)) data. The generic (χ2) contribution for distance measurements is χ2 dist =∑ i[Dobs(zi)−Dth(zi; Θ)]2 σ2 i , where (D) stands for any distance observable (e.g. (dL) moduli for SNe, BAO distance combinations (DV, DM), or (H(z)) direct measurements), (σi) their measurement uncertainties, and (Θ) denotes the full parameter vector. For BAO and SNe include covariance matrices where available. Interpretation: since (ρvac(z)) is not constant a priori, the inferred (w(z)) will be constrained implicitly by (H(z)) and distance measures. Use MCMC (e.g. emcee/MontePython/- Cobaya) to map parameter posteriors. 9.3 Cosmic microwave background (CMB) constraints CMB observables are sensitive to both background expansion and early-time perturbations. We provide expressions for the principal constraints and how the model enters them. 9.3.1 Angular scale of acoustic peaks The CMB angular scale (θ∗) depends on the sound horizon at recombination and the angular diameter distance to (z∗≃1100): θ∗=rs(z∗) dA(z∗), rs(z∗) = ∫∞ z∗ cs(z) H(z)dz, with sound speed (cs= [3(1 + R)]−1/2) and (R= 3ρb/(4ργ)). The model modifies (H(z)) at (z≲z∗) if injection is active in the early universe (inflationary era aside), thereby shifting (θ∗). Fit to precise Planck measurement of (θ∗) provides tight constraints on early-time behavior of (C1, C2). 14 Vacuum–Collision + Elastic Spacetime Model: Conclusion and Unification 10.1 Short summary of the mechanism (one paragraph) Vacuum fluctuations in an expanding spacetime are treated as a population of transient virtual quanta with characteristic lifetime (∆t) and energy (Ev=ℏ/∆t). When the local production rate (Γ) (assumed strain/expansion–driven, (Γ∝H)) and the coherence cross section (σc∼ π(c∆t)2) satisfy the kinetic condition (Ξ≡σccΓ∆t2≳1) (eq. (4.24)), collisions among virtual quanta become frequent. If a non-negligible fraction (finel) of such collisions yield inelastic, non-annihilating outcomes (or produce bound “gel” states), a persistent energy density is injected into the low-frequency, long-wavelength degrees of freedom of the metric. A fraction (ζ) of the injected energy becomes elastically stored in a covariant strain field (ϵµν) and appears in the Einstein equations through the elastic stress–energy (T(el) µν ) (Appendix C). The remaining fraction ((1−ζ)) can deposit into long-lived, gravitationally interacting gel mass (ρgel) (Sections 4, 5, 7). The macrocosmological source is compactly written as Q(H) := ζ˙ρinj(H)≈C1H+C2H2,(10.1) with (C1,2) given explicitly in (6.25). Depending on (ζ) and the values of (C1, C2), the net effect is either a near-constant stored vacuum energy (dark-energy behaviour) or a production of pressureless gel mass that clusters (dark-matter behaviour). Both arise from the same microscopic collision processes and the same dynamical coupling to expansion. 10.2 How this unifies dark energy and dark matter (precise, operational statement) 1. Single origin, two macroscopic branches. •Dark energy branch: if (ζ) is large (most injected energy is stored elastically), the elastic energy density (ρel) grows (or accumulates) and — because the elastic sector redshifts differently from matter and can be long-lived — it behaves like a nearly constant vacuum energy with effective equation of state (wel ≈ −1) (see §4.6, §5.5 and eqs. (4.26) and (5.17)–(5.22)). The stored energy sources cosmic acceleration through the modified Friedmann eqs. (5.28)–(5.30). •Dark matter branch: if (ζ) is small and (fbind) and/or (1−ζ) is large, a significant fraction of injection goes to gel formation; these gel objects behave gravitationally as pressureless mass (section 7), cluster and produce halos with outer envelopes (ρgel(r)∼r−2) and nearly flat rotation curves. The same collision physics and the same parameter set ((α, τ0,∆t, η, finel, fbind, ζ)) control the split between these two behaviors. 2. Dynamical continuity across cosmic history. Because (Γ) (and hence ( ˙ρinj)) scales with (H), the same mechanism naturally produces a time-dependent injection: large at 1 early times (large (H)) and small at late times (small (H)). Consequently the model can (i) seed an inflation-like early epoch (collision-dominated (Q∝H2), §8.2), (ii) produce gel mass during structure formation epochs, and (iii) accumulate residual elastic energy to act as the present dark energy. This continuity is unusual among dark-sector proposals and is the core of the unification claim. 3. Parameter mapping determines phenomenology. The macro coefficients (C1, C2) (eqs. (6.25)) and the branching fraction (ζ) are the minimal “control knobs.” A compact operational map is: • Large (ζ), moderate (C1,2): dark energy dominated late universe. • Small (ζ), sizable (fbind): substantial gel production and dark-matter-like behaviour. • Intermediate values yield mixed scenarios where both components are significant and correlated. 10.3 Relationship to quantum field vacuum energy and to relativistic elasticity (A) Relation to vacuum energy in QFT in curved spacetime • In semiclassical gravity the renormalized expectation value (⟨Tµν ⟩ren) of quantum fields provides a source term in Einstein’s equation. Our model postulates additional effective channels by which short–lived vacuum fluctuations, when subject to the global strain produced by expansion, can produce persistent low-frequency excitations or bound states that are not subtracted by usual renormalization procedures. Concretely: –Standard renormalization removes local UV divergences and yields residual renormalized vacuum energy that is formally constant (cosmological constant problem). What we add is a dynamical, nonlocal production channel: collisions among virtual quanta in a time-dependent background can populate long-lived modes (or bound states) whose energy density behaves like a new macroscopic contribution. This is not a double-counting of the usual renormalized vacuum if those persistent modes are physically distinct low-frequency excitations (they inhabit a different sector of phase space than the high-frequency modes removed by renormalization). – Crucial theoretical requirement: a rigorous derivation from QFT in curved space should demonstrate how a subset of diagrammatic processes (inelastic collisions of vacuum excitations mediated by curvature/strain) produce an IR remnant that is not renormalized away. This requires explicit calculation of relevant loop diagrams and their imaginary parts (cutting rules), carefully treated in an expanding background (Schwinger–Keldysh / in-in formalism). This is an urgent open theoretical program (see §10.6). (B) Relation to relativistic elasticity and effective field theory of spacetime degrees of freedom • Our covariant elastic action (Sel) (eq. (5.4)) and derived stress tensor (Appendix C, eq. (C.9)) place the model squarely within the established literature on relativistic elasticity and effective field theories of media in GR. The lamé-like constitutive relations and the viscoelastic generalization (Appendix D) are standard in continuum mechanics and are consistent with causality and positive dissipation when the kernels satisfy appropriate analyticity/positivity conditions (Appendix D.6). 2 • Interpreting the vacuum as an effective medium with elastic moduli (λ, µ) is conceptually similar to other approaches that ascribe mechanical properties to the vacuum (e.g., emergent gravity, analog gravity models), but here the elastic sector is dynamically fed by microphysical collision processes rather than being an a priori phenomenological fluid. The presence of divergence terms and anisotropic stresses in (T(el) µν ) (C.9) provides distinct signatures (e.g., scale-dependent anisotropic stress in cosmological perturbations). 10.4 The most important theoretical objections and how to address them (concrete program) Below I list the strongest conceptual and technical challenges, and give precise calculations / checks that must be done to make the proposal robust. 1. Renormalization / double-counting. •Objection: vacuum energy is already accounted for in (⟨Tµν⟩ren); any new “extracted” energy from vacuum fluctuations might be overcounting or violate renormalization constraints. •Required work: compute the relevant in-in quantum amplitudes for two-to-two vacuum processes in an expanding background (use Schwinger–Keldysh), identify the parts that correspond to secular IR growth (nonlocal in time), and show these terms are not removable by local counterterms. Deliverable: explicit one-loop + cut diagrams that yield a finite injection rate ( ˙ρinj) proportional to (H) or (H2). If such terms exist, document their parametric dependence on curvature and show renormalization can be performed consistently while leaving a finite physical remnant. 2. Energy conservation & backreaction consistency. •Objection: continually extracting energy from vacuum might violate local conservation or conflict with Bianchi identities. •Required check: show explicitly that the covariant divergence of total stress–energy (including (Tother µν +Tel µν+Tgel µν ) plus microphysical source terms) vanishes: (∇µ(Gµν ) = 0) remains satisfied because the injection Q and the elastic divergence terms exchange energy/momentum between sectors (Appendix C.5 gives the elastic force density). Provide an explicit example calculation where (∇µ(Tother µν +Tel µν +Tgel µν ) = 0) is verified numerically for a sample background. 3. Causality, positivity and stability of the elastic response. •Objection: naive elastic models can be acausal or unstable. •Required work: enforce and demonstrate microscopic conditions (e.g., Kramers– Kronig relations for memory kernels) that guarantee causal linear response. For Kelvin-Voigt/Maxwell kernels, compute susceptibility (˜χ(ω)) (Appendix D.3–D.6) and show (ℑ˜χ(ω)≥0) for (ω > 0). Provide linear stability analysis (Appendix E) showing attractor conditions for the coupled ((H, ρvac, ρgel)) system and map unstable parameter regions. 4. Thermodynamics / entropy production. •Issue: dissipative channels create entropy; must confirm compatibility with thermal history (BBN, CMB). 3 •Required work: compute (Dvisc) and radiative loss (Lrad) as functions of model parameters, integrate energy deposited into radiation across time, and show (∆Neff) and spectral distortions are within observational bounds for viable parameter choices (use eq. (8.16) and CMB spectral distortion constraints). 5. Microphysics of inelastic collisions and binding. •Objection: the existence of non-annihilating inelastic channels for virtual pairs is speculative. •Required calculation: present at least one explicit microscopic toy model (e.g., scalar field with cubic/quartic interactions or a Yukawa coupling to a light mediator) and compute the inelastic cross sections and branching fractions for the relevant energy scales (Ev). Compute (finel(E)) and (fbind) as functions of parameters (g, ms). This turns hand-wavy arguments into predictive formulae. 10.5 Concrete observational tests and ranking (actionable plan) Below I list prioritized, falsifiable checks (each item includes the minimal calculation / data needed). Priority A (immediate and decisive): 1. CMB angular scale & ISW consistency test. • Compute full CMB spectra with elastic anisotropic stress included (modify CLASS/- CAMB: include (wvac(z)), anisotropic stress (Πel(k, z)) from Appendix C and viscoelastic kernels from Appendix D). Fit Planck high-(ℓ) and low-(ℓ) data. If the model with parameters that match (ρΛ,0) produces unacceptable shifts in (θ∗) or ISW power beyond Planck errors, that parameter region is excluded. 2. Total energy bookkeeping vs (Neff ) & BBN. • Integrate radiative losses (Lrad(t)) during nucleosynthesis and recombination; compute (∆Neff). Compare with Planck+BBN limits; rule out parameter sets that violate bounds. Priority B (galaxy/structure tests): 3. Rotation-curve fits across mass range (SPARC). • For a grid of ((mb, Ebind, ζ, fbind)) compute predicted (vc(r)) (baryons+gel) and perform a joint fit to rotation curve catalogs. If gel parameters cannot reproduce the empirical core–mass relations, the model is constrained. 4. Cluster merger collisionality. • Compute effective (σ/m) for gel clumps and compare to Bullet cluster bounds; exclude strongly interacting gel scenarios. 5. Growth of structure & (fσ8). • Integrate linear perturbations and compare (fσ8(z)) to RSD data; large deviations eliminate parameter regions. 4 Priority C (special signatures): 6. ISW–LSS cross-correlation scale dependence. • Model predicted ISW signal vs scale; cross correlate with LSS surveys. A detectable, model-specific scale dependence is a smoking gun. 7. Soliton core scaling. • If gel constituents are ultralight bosons, test soliton core scaling (rc∝1/(mbvc)) against data for dwarf and spiral galaxies. 10.6 Research program — concrete theoretical projects (what to do next, exactly) To make the model scientifically rigorous and ready for full confrontation with data, the community should carry out the following (I give exact computational tasks and suggested methods): 1. Field-theory derivation of ( ˙ ρinj(H)). • Use the Schwinger–Keldysh in-in formalism to compute the imaginary part of relevant vacuum loop diagrams in an FLRW background. Evaluate cut diagrams that correspond to two-to-two processes among vacuum excitations. Goal: derive ( ˙ρinj) and show its scaling with (H) (obtain explicit (C1, C2) expressions from first principles). Tools: curved-space perturbation theory, adiabatic subtraction renormalization, numerical evaluation for non-adiabatic regimes. 2. Compute microscopic cross sections and (finel, fbind). • Choose minimal toy models (scalar (ϕ4), Yukawa with mediator mass (ms)). Compute scattering amplitudes and binding probabilities at center-of-mass energies (Ecm ∼ Ev). Provide plots (finel(E)), binding energy (Emin) from variational/Schrödinger analysis (Appendix F), and the dependency on coupling constants. Tools: standard QFT S-matrix methods, nonrelativistic potential approximations where justified. 3. Renormalization & separation of UV/IR. • Demonstrate that terms leading to persistent (ρinj) are IR in nature and not removable by local counterterms. This requires careful asymptotic expansion and matching between short-distance (UV) regularization and long-time (IR) secular effects. 4. Perturbation theory & Boltzmann code implementation. • Implement elastic stress and viscoelastic kernels as an extra fluid in CLASS (or CAMB). Validate by reproducing limiting cases (pure cosmological constant, small anisotropic stress). Provide likelihood chains against Planck + BAO + SNe. Tools: CLASS modification, MCMC sampling (Cobaya). 5. N-body / hydrodynamic simulations with gel injection. • Implement a particle-based source term for gel mass in structure formation codes (e.g., Gadget-like): create particles with production rate (∝(1 −ζ) ˙ρinj), include initial velocity dispersion (σ) from microphysics, and follow merging/relaxation to test halo profiles, substructure, and lensing. Tools: modified Gadget / Arepo + halo finding. 5 6. Laboratory analogs & condensed matter tests (optional exploratory). • Explore analogue systems where fluctuating excitations in an elastic medium produce persistent bound states under strain; while not direct evidence for cosmology, such systems can test the plausibility of the “strain produces persistent defects” intuition. 10.7 Minimal set of falsifiable predictions (one-page checklist) To be concrete, the model predicts the following joint signatures; falsifying any one while the others hold would be strong evidence against the mechanism: 1. Correlated redshift evolution: a specific relation between (ρvac(z)) and the integrated gel production history (eq. (8.25)). Measuring both and finding them inconsistent with the microphysical relation rules the model out. 2. Nonzero large-scale anisotropic stress at late times exceeding (Πel(z)) predicted from the elastic stress tensor (Appendix C). Planck + lensing already constrain this to be small; if the model requires large anisotropic stress to match (ρΛ,0), it is excluded. 3. Specific halo core–velocity scaling (if gel constituents are ultralight): (rc∝1/(mbvc)). If data excludes this relation over many halos, ultralight-gel scenarios are excluded. 4. If the injection exponent is quadratic ((C2) dominated) there should have been an early phase with distinctive reheating signatures (e.g., a stochastic GW background or spectral distortions) — absence of any reheating signature for parameter choices that would otherwise predict them excludes those choices. 10.8 Final, balanced assessment •Strengths: the model supplies a single, physically motivated origin for both vacuum-like accelerated expansion and mass that clusters gravitationally. It provides a mechanistic, dynamical coupling of expansion to microscopic vacuum physics and yields testable predictions (anisotropic stress, correlated dark-energy/dark-matter histories, halo internal structure). The model naturally contains parameters that can be constrained by current data and provides a concrete recipe (Sections 4–9, Appendices C–G) for confronting observations and performing theory checks. •Weaknesses / risks: the plausibility hinges on three nontrivial theoretical claims: (i) that in an expanding background a non-negligible fraction of vacuum fluctuation collisions produce persistent energy (not removed by renormalization), (ii) the microphysics admit sufficiently large (finel) and (fbind) without violating other experimental bounds, and (iii) the elastic/viscoelastic response can be causal and stable while producing (w≈ −1). Each claim demands rigorous calculation (Schwinger–Keldysh derivation, scattering amplitude computation, stability analysis). •Outlook: if the microscopic computations and numerical cosmological+structure analyses outlined above succeed, the model would offer a conceptually economical unification of the dark sector. If they fail, the framework will have served as a clear, falsifiable proposal that guided targeted theoretical work and observational tests. 6 10.9 Suggested text for the paper’s concluding paragraph (copypaste) We have presented a physically explicit mechanism by which strain-induced collisions of vacuum fluctuations can deposit persistent energy into an elastic sector of spacetime or produce long-lived bound “gel” objects. With a small set of microphysical parameters this mechanism simultaneously yields an effective cosmological constant-like stored energy and a separable, clustering mass component, providing a unified physical origin for dark energy and dark matter. The model is falsifiable: it delivers concrete predictions for the background expansion, anisotropic stress signatures in the CMB, the evolution of structure formation, and halo internal structure. The essential theoretical tasks remaining are a first-principles QFT derivation of the injection rate ( ˙ρinj(H)), calculation of microscopic inelastic/binding branching fractions in minimal field models, and full implementation of elastic anisotropic stress in Boltzmann & N-body codes. These steps — each of which we have outlined as explicit calculations in this paper — will determine whether the vacuum–collision + elastic-spacetime framework is a viable unified description of the dark sector or an instructive but ultimately excluded hypothesis. 11. Conclusion 11.1 Summary of the mechanism (compact statement) We have proposed and developed a self-contained framework in which strain-driven production and inelastic collisions of vacuum fluctuations in an expanding spacetime deposit persistent energy into two distinct macroscopic channels: 1. an elastic (stored) sector of theatore ”fabric” described by a covariant strain field (ϵµν) whose stored energy (Uel) sources the Einstein equations via an elastic stress–energy tensor (T(el) µν ) (Appendix C); and 2. a population of long-lived, self-bound “vacuum–gel” objects (gel mass (ρgel)) produced when collision outcomes bind rather than annihilate and subsequently gravitationally cluster (Sections 7 and F–G). The same microscopic kinetics of vacuum quanta — characterized by a production/interaction rate (Γ), an inelastic fraction (finel), binding fraction (fbind), and an elastic storage fraction (ζ) — controls both channels. This gives a concrete physical route by which a single microphysical mechanism can yield both an effective cosmological-constant-like contribution and a pressureless, clustering component that behaves like dark matter. 11.2 Key equations (boxed — ready to cite in the conclusion) Include these compact formulae exactly as written in the main text: • Microphysical → macroscopic source (working ansatz derived in §4–6): Q(H) := ζ˙ρinj(H)≈C1H+C2H2(11.1) with (C1, C2) expressed in terms of microscopic parameters in (6.25). • Elastic stress–energy (covariant form; Appendix C, eq. (C.9)): T(el) µν := Uelgµν +λϵϵµν + 2µϵµαϵαν−2∇α(λϵuαgµν + 2µuαϵµν −2µu(µϵν)α).(11.2) 7 • Modified Friedmann + continuity system (working background equations used throughout): H2=8πG 3(ρm+ρr+ρel +ρgel),[4pt] ˙ρel + 3H(1 + wel)ρel =Q(H)− Del,[4pt] ˙ρgel + 3Hρgel = (1 −ζ) ˙ρinj −Γdecayρgel +Sclust. (11.3) These three boxed items summarize the macro link from microscopic vacuum kinetics to cosmological dynamics and structure. 11.3 How the mechanism unifies dark energy and dark matter (operationally) •Single microphysics, two macroscopic outcomes: the branching fraction (ζ) determines the split: (ζ→1) converts most injected energy into elastic storage (effective dark energy); (ζ→0) routes it into gel mass (effective dark matter). The same parameters ((α, τ0,∆t, η, finel, fbind)) control both the rate (Q(H)) and the gel production rate, producing an inherently correlated evolution of (ρel(z)) and (ρgel(z)). •Dynamical continuity over cosmic history: because (Γ) scales with expansion (we modeled (Γ∝H)), injection is large in the early universe and smaller at late times. Thus the mechanism can (i) support an inflation-like early epoch when (Q∝H2) dominates (§8.2), (ii) seed and grow gel mass during structure formation epochs (§7, §8.4), and (iii) accumulate residual elastic energy that effectively becomes late-time dark energy (§6–8). This continuity is an explicit, falsifiable prediction: the histories of dark energy and dark matter are not independent but linked by microphysical parameters. 11.4 Main testable predictions (concise & prioritized) 1. Background: a time-dependent vacuum contribution (ρel(z)) derived from (11.1) that can deviate subtly from a pure cosmological constant (predictable (weff(z))). Fit to SNe/BAO/CMB distances constrains (C1, C2, ζ, Γloss). 2. Anisotropic stress: elastic sector generically produces scaleand time-dependent anisotropic stress (Appendix C). This modifies CMB large-angle ISW and lensing signatures — a precise target in Planck and forthcoming CMB polarization/lensing data. 3. Correlated gel abundance and halo properties: gel formation rate is tied to injection; if gel comprises a significant fraction of dark matter, halo profiles, core-scalings (quantum vs self-interaction), and abundance are predicted by the same microphysics. Galaxy rotation curves and lensing provide direct tests (§7, §9). 4. Early-universe signatures (if applicable): collision-dominated early regimes can produce reheating and stochastic GW / spectral distortion signatures depending on branching fractions — these provide an early-time falsifiability channel (Appendix D, §8.2.4). 11.5 Necessary theoretical and empirical next steps (concrete, short list) To elevate the proposal from phenomenological framework to a validated (or falsified) physical theory, perform the following immediately: 1. First-principles QFT derivation: compute ( ˙ρinj(H)) in a controlled toy field theory using the Schwinger–Keldysh formalism and cutting rules (explicit program described in §10.6 and offered as Appendix A candidate). Demonstrate the existence (or absence) of finite, non-renormalizable IR contributions scaling as (H) or (H2). 8 2. Microscopic scattering & binding calculations: for minimal interaction models (scalar or Yukawa), compute (finel(E)) and (fbind) and derive explicit formulas for (C1,2) in terms of couplings and mediator masses (see §7 and Appendix F for variational binding checks). 3. Cosmological implementation & data confrontation: implement the elastic fluid + gel source in a Boltzmann code (CLASS/CAMB), include anisotropic stress from (11.2), and run full likelihood chains (Planck+BAO+SNe+RSD). Simultaneously, run targeted rotation-curve and lensing fits to constrain gel microphysics (§9, statistical recipe). 4. Simulate nonlinear structure: inject gel particles into N-body simulations according to the production law and study halo formation, substructure, and lensing signatures (Section 7 and suggested simulation program). These tasks are concrete, tractable, and will decisively confirm or rule out the mechanism in observationally relevant parameter ranges. 11.6 Final balanced assessment (one paragraph) The vacuum–collision + elastic-spacetime framework provides a conceptually economical and technically explicit route to a unified origin of the dark sector: a single, physically motivated microphysical process (strain-driven vacuum pair production with inelastic/binding outcomes) feeds both an elastic vacuum energy and long-lived gel mass. The model’s strength is its clear micro → macro mapping (boxed in (11.1)–(11.3)) and the many direct observational handles it offers (CMB anisotropy & ISW, growth and lensing, rotation curves, cluster dynamics). Its viability rests on nontrivial QFT and kinetic computations: if those calculations produce robust, finite injection rates and reasonable branching fractions, the model could offer a paradigm shift; if they do not, the framework remains a falsifiable, instructive hypothesis that has guided a concrete program of theoretical and observational tests. 11.7 Suggested concluding sentence In summary, by tracing a clear physical path from expansion-driven vacuum kinetics to macroscopic elastic storage and gel formation — and by providing explicit, testable equations linking microphysics to cosmological observables — this work offers a falsifiable and computationally concrete unification hypothesis for dark energy and dark matter; the next decisive steps are first-principles QFT derivations of the injection rate and direct confrontation with CMB, large-scale structure and halo data, which will determine whether this unified mechanism is empirically realized in our universe. 9 Vacuum–Collision + Elastic Spacetime Model: Appendices A, H, I Appendix A — Collision probability integrals and the energy– injection law Purpose. Derive the volumetric energy–injection rate ( ˙ρinj(t)) produced by inelastic collisions among short–lived vacuum quanta in an expanding FLRW background. Start from a minimal kinetic model for virtual-pair statistics, derive the collision rate per proper volume, fold in inelastic branching and energy deposited per event, and obtain compact expressions valid in the two relevant regimes: •collision–dominated (many collisions per lifetime) → ( ˙ρinj ∝Γ), and •collisionless (rare collisions) → ( ˙ρinj ∝Γ2). We keep all symbols defined and show the mapping to the macroscopic coefficients (C1, C2) used in the main text. A.0 Notation & assumptions (explicit) • Spacetime: spatially homogeneous, isotropic FLRW (proper volume element (dV =a3(t)d3x)). For background (homogeneous) rates we work in proper time (t). • Virtual quanta (“pairs”): characterized by a typical lifetime (∆t) (proper time) and characteristic energy (Ev=ℏ/∆t). We treat (∆t) as a microphysical timescale determined by uncertainty relations or the dominant virtual-mode spectrum. • Production (appearance) rate per unit proper volume of virtual pairs: (Γ(t)) (units: [time]−1[volume]−1?). Important: In our kinetic language (Γ) denotes the per-pair appearance frequency (the number of new pairs created per unit proper time per unit comoving proper volume). Where needed we distinguish pair production number rate density (P(t)) (pairs per unit proper volume per unit proper time) from a per-pair rate; for clarity we set P(t)≡Γ(t). Thus (Γ(t)) has units ([energy]0[length]−3[time]−1) (number density per time). • Characteristic coherence (geometric) cross section for interactions between transient quanta: we use the causal transverse size set by their temporal localization: take σc∼π(c∆t)2. This is the geometric area spanned by a quantum localized to time (∆t) (equivalently spatial extent (∼c∆t)). 1 H.1 Action: gravitational + matter + elastic sector + coupling Consider total action Stot =1 16πG ∫d4x√−gR +Sm[g, Ψ] + Sel[g, u] + Sint[g, u, Ψ],(H.1) where: • (Sm) is the action of ordinary matter fields (Ψ). • (Sel) is the covariant elastic action (below). • (Sint) encodes possible direct couplings of the elastic sector to other fields (kept implicit; most of the derivation assumes (Sint = 0), but we give the generalization). We focus on constructing (Sel) such that it reduces to classical linear elasticity in the weakfield, non-relativistic limit and yields causal, stable dynamics when generalized (viscoelasticity can be added later). H.2 Covariant strain tensor and elastic invariants Define the strain tensor (symmetric) as the symmetric part of the covariant derivative of the displacement: ϵµν ≡1 2(∇µuν+∇νuµ), uµ≡gµαuα.(H.2) This is the natural covariant generalization of linearized strain; it reduces to the usual smallstrain tensor in weak-field, near-Minkowski coordinates. Form invariant scalars built from (ϵµν) and (uµ). The minimal scalar invariants are: ϵ≡gµνϵµν , ϵαβϵαβ.(H.3) H.3 Elastic action and constitutive parameters We choose an action quadratic in strains (relativistic analogue of Hooke’s law) with possible mass-like term for (uµ) to allow relaxation. A minimal action is Sel[g, u] = −1 2∫d4x√−g(H.4) • (λ) and (µ) are relativistic Lamé coefficients (bulk and shear moduli); units: energy density (i.e. ML−1T−2). • (ρ0) is a parameter with units energy density that acts like an effective inertia/mass density for the displacement field; it ensures well-posed hyperbolic dynamics for (uµ) (otherwise evolution can be constrained/elliptic). One may set (ρ0→0) if a purely constraint-like elastic field is intended (but then care is required to enforce causality). • This form is the simplest isotropic elastic action. More general covariant constitutive relations and higher-order terms (nonlinear elasticity) can be added. Remarks. (i) (Sel) depends on the metric through (√−g), index raising, and through (uµ=gµνuν) and covariant derivatives (∇µ). (ii) Dissipative (viscoelastic) generalization: replace constants with causal kernels (see Appendix D in main text). 8 H.4 Metric variation — deriving the elastic stress–energy tensor The elastic stress–energy tensor is defined by T(el) µν ≡ − 2 √−g δSel δgµν .(H.5) We compute (δSel) keeping (uµ) fixed (i.e., variation of (uµ) arises through (δuµ=uαδgαµ) only). The key steps: 1. Vary the volume factor: δ√−g=−1 2√−ggµν δgµν. 2. Vary the strain: δϵαβ =1 2(∇αδuβ+∇βδuα)=1 2(∇α(uγδgγβ) + ∇β(uγδgγα)), since (δuγ= 0). (Also variations of Christoffel symbols contribute inside covariant derivatives; grouping terms into total derivatives will handle them.) 3. Vary the quadratic invariants: δ(ϵ2) = 2ϵδϵ, δ(ϵαβϵαβ) = 2ϵαβδϵαβ −ϵαβϵρβδgαρ. Putting terms together and integrating by parts to move derivatives off (δg) (dropping surface terms under suitable boundary conditions), one obtains after algebra the explicit form (we present the fully reduced result so it is ready to cite): T(el) µν =Uelgµν +λϵϵµν + 2µϵµαϵαν[4pt]−2∇α[λϵuαgµν + 2µuαϵµν −2µu(µϵν)α]+ρ0uµuν, (H.6) where Uel ≡1 2(λϵ2+ 2µϵαβϵαβ).(H.7) Notes on terms: • The first three terms are algebraic and local in (uµ) and its first derivatives: they generalize stored elastic energy and nonlinear anisotropic stresses. • The divergence term (the (∇α(···)) piece) encodes fluxes of elastic momentum / internal force transmission; it produces anisotropic stresses and is crucial for momentum balance. • The last term (ρ0uµuν) is an inertial contribution behaving like a kinetic-energy density for the displacement field (it is present because we included a mass-like term in the action). This expression is a boxed, ready-to-use formula; it matches the rigorous variation and is consistent with the result given earlier (Appendix C) but now including the explicit (ρ0uµuν) inertia term. H.5 Elastic equation of motion — variation w.r.t. (uµ) Varying (Sel) with respect to (uµ) (holding (gµν) fixed) gives the elastic field equation (analogous to momentum balance): 1 √−g δSel δuµ=−∇α(λϵgαµ + 2µϵαµ)+ρ0uµ= 0.(H.8) 9 Equivalently, ρ0uµ=∇α(λϵgαµ + 2µϵαµ).(H.9) Interpretation: • This is the covariant elastic wave (or constraint) equation. In the nonrelativistic limit (small velocities, spatial displacements only), it reduces to the standard Navier–Cauchy equations of linear elasticity: (ρ0¨ u = (λ+µ)∇(∇ · u) + µ∇2u) (with time derivatives emerging if we include kinetic term with time derivatives explicitly — see remark below). • If (ρ0) were set to include a true kinetic term (1 2ρ0gµν∇αuµ∇αuν) in the action, the timederivative structure becomes explicit and the left-hand side would contain (ρ0□uµ). Our choice of (ρ0gαβuαuβ) is a simple way to introduce inertia; for wave-like dynamics include explicit kinetic derivative terms. Remark: A more physical choice is to include an explicit kinetic term ( 1 2ρkingαβ∇αuµ∇βuµ) in (Sel). Then the (uµ) EOM becomes hyperbolic (ρkin□uµ+. . . = 0). For brevity in many cosmological calculations one uses the algebraic inertia term as a phenomenological replacement; the user should pick the form that matches intended physics. H.6 Total Einstein equations and conservation Varying the total action (H.1) with respect to (gµν) gives Einstein equations: Gµν = 8πG(T(m) µν +T(el) µν +T(int) µν ),(H.10) where (T(el) µν ) is given in (H.6), and (T(int) µν ) arises from (Sint) if present. Because diffeomorphism invariance holds, the total stress–energy is covariantly conserved: ∇µ(T(m) µν +T(el) µν +T(int) µν )= 0.(H.11) This equation expresses energy–momentum exchange between the elastic sector and other sectors (e.g., matter or gel), mediated by the divergence contributions in (T(el) µν ) and by explicit (Sint) couplings. Combining (H.9) and (H.6) one can show explicitly that the divergence of (T(el) µν ) equals the negative of the elastic force density that acts on other components — a necessary feature to ensure consistent exchange. H.7 Linearization about an FRW background — cosmological perturbations We now linearize the system around a homogeneous, isotropic FRW background with metric ¯gµνdxµdxν=−dt2+a2(t)δijdxidxj. Assume background elastic displacement (¯uµ= 0) (no background strain) and small perturbations (hµν) and (uµ): gµν = ¯gµν +hµν, uµ=δuµ(small). 10 H.7.1 Strain linear order To linear order in perturbations and (u), the strain reduces to ϵµν ≈1 2(¯ ∇µuν+¯ ∇νuµ), where indices are raised/lowered with (¯g). For scalar cosmological perturbations in Newtonian gauge, ds2=−(1 + 2Φ)dt2+a2(1 −2Ψ)δijdxidxj, and take (u0) and spatial (ui) perturbations. If (uµ) is purely spatial in the background comoving frame, we set (u0= 0) to linear order and keep (ui) only. H.7.2 Linearized elastic stress–energy: anisotropic stress Compute (T(el) ij ) to linear order in (u) and (h). The dominant contribution to anisotropic stress (off-diagonal spatial components (i=j)) is δT(el) ij = 2µa2(∂(iuj)−1 3δij∂kuk)+(time-derivative / divergence terms).(H.12) Key takeaway: the shear modulus (µ) produces anisotropic stress proportional to the spatial symmetric gradient of the displacement (the traceless part). This anisotropic stress sources the difference (Φ−Ψ) in metric perturbations: Einstein linearized off-diagonal spatial equation yields (in Fourier space (k)): k2(Φ −Ψ) = 8πGa2Π(el)(k, t),(H.13) where the elastic anisotropic stress (Π(el)) is obtained from (δT(el) ij ) by extracting the traceless scalar part: Π(el)(k, t)∼µku(k, t) + ··· .(H.14) (Illustrative scaling; the full expression follows from the Fourier transform of (H.12).) Thus the elastic sector generically produces nonzero anisotropic stress that modifies the usual equality (Φ = Ψ) valid in absence of anisotropic stress. This is a key observable signature (Planck constraints on anisotropic stress are tight). H.8 Linearized Einstein equations with elastic source (scalar sector) For scalar perturbations in Newtonian gauge, the relevant linearized Einstein equations (with (c= 1)) become: • Poisson / time-time equation: −2k2Ψ+6aH(˙ Ψ + aHΦ) = 8πGa2δρtot,(H.15) • Momentum constraint: 2k(˙ Ψ + aHΦ) = 8πGa2(ρ+p)vtot,(H.16) • Anisotropic stress equation: k2(Φ −Ψ) = 8πGa2Πtot,(H.17) 11 where (δρtot =δρm+δρel +δρgel) etc. The elastic contributions enter via: δρel ∼U′ elϵ+. . . , vel ∼(time derivative of u),Πel ∼µ(traceless gradient of u).(H.18) The explicit Fourier-space expressions for (δρel) and (Πel) are obtained by expanding (H.6) to linear order and performing a scalar decomposition of (ui→ikiu(k)) and separating trace/- traceless parts. Practical formulae (ready to paste): For purely scalar displacement (ui=ikiu), one finds (dropping subleading metric-variation pieces) δρel(k, t)≃κϵ(k, t)≃ −κk2u(k, t),[4pt]Πel(k, t)≃µk2u(k, t),(H.19) with (κ≡λ+2 3µ) the bulk modulus. (Sign conventions depend on Fourier convention; these are schematic to show dependence.) Hence (Φ−Ψ) is proportional to (µk2u). If (u) is small or (µ) is small, anisotropic stress is small. H.9 Elastic-field dynamics and coupling to metric perturbations From the linearized elastic EOM (H.9) including a kinetic term one obtains (schematically) ρkin ¨u+γ˙u+c2 sk2u=S[k, t],(H.20) where: • (ρkin) is the inertia coefficient (from a kinetic term), • (γ) is an effective damping (viscoelastic) parameter, • (c2 s∼µ/ρkin) is the elastic sound speed squared, • (S[k, t]) is a source including metric perturbations (combination of (Φ,Ψ)). Thus (u) obeys a driven damped wave (or diffusion, if overdamped) equation; its solution determines (δρel) and (Πel) which feed back into Einstein equations. This is the closed system needed to compute cosmological perturbations in presence of an elastic vacuum sector. H.10 Effective fluid language (equation of state, sound speed, viscosity) It is often convenient to treat the elastic sector as an effective fluid with: • energy density (ρel =Uel +ρ0u2/2 + . . .), • pressure (pel =−Uel +. . .) (note elastic stored energy behaves like negative pressure in some regimes), • anisotropic stress (Πel) as above, • effective sound speed (c2 s,el =δpel/δρel) determined by elastic moduli and inertia, • viscosity/damping from viscoelastic generalizations (Appendix D). 12 Appendix X — Energy–density evolution equations (derivation & practical forms) Appendix A (extended) — First-principles derivation Appendix B — Numerical benchmarks Appendix X — Energy–density evolution equations (derivation & practical forms) Purpose. derive the background (homogeneous) evolution equations for the energy densities of the relevant sectors in the model: • ordinary matter (ρm) (pressureless), • radiation (ρr), • elastically stored vacuum energy (ρel) (the elastic/strain sector), and • gel mass density (ρgel) (bound vacuum–gel objects). We start from covariant energy–momentum conservation and Friedmann equations in an FLRW spacetime, insert a microphysical energy injection rate ( ˙ρinj) derived in Appendix A, partition the injected energy into elastic vs gel channels, include dissipation/decay losses, and produce the closed set of evolution equations used in the main text. Two limiting forms for the injection (linear and quadratic in (H)) are given explicitly. X.0 Conventions and assumptions • Metric: spatially flat FLRW, ds2=−dt2+a2(t)dx2. • Hubble parameter (H≡˙a/a). Overdot denotes derivative with respect to cosmic proper time (t). • Units: keep (c= 1) except where otherwise noted; restore (c) when giving dimensional checks. • Perfect-fluid notation: each component (X) has energy density (ρX(t)) and pressure (pX(t)). For dust (pm= 0), for radiation (pr=1 3ρr). The elastic sector will be allowed a general equation-of-state parameter (wel ≡pel/ρel) (often near (-1) when elastic energy acts like vacuum). • Microphysical injection law: ( ˙ρinj(t)) = energy deposited per unit proper volume per unit proper time by inelastic collisions among vacuum quanta (derived in Appendix A, eq. (A.22)). 1 We assume the injected energy is partitioned: a fraction (ζ∈[0,1]) goes into elastic storage (becoming (ρel)), and the remaining fraction ((1−ζ)) goes into gel formation (contributing to (ρgel)) or other dissipative channels. Some of the elastic or gel energy can be lost to radiation or thermalized at rates encoded by (Γloss) (for elastic sector) and (Γdecay) (for gel objects). X.1 Total energy conservation and splitting Einstein’s equations guarantee covariant conservation of the total stress–energy, ∇µTtot µν = 0, which for a homogeneous FLRW background reduces to the scalar continuity equation for the total energy density (ρtot =PXρX): ˙ρtot + 3Hρtot +ptot= 0.(X.1) We split the total into components: ρtot =ρm+ρr+ρel +ρgel, ptot =pm+pr+pel +pgel. (Here pm= 0, pr=1 3ρr, pgel ≈0for nonrelativistic gel mass; pel =welρel with wel possibly close to (-1).) Because the microphysical process transfers energy between sectors (e.g., vacuum collisions → elastic storage + gel + radiation), the components do not individually conserve. We therefore write continuity equations with explicit source/sink terms that obey global conservation (so that summing them returns (X.1)). X.2 Component continuity equations — general form We write the component evolution equations as ˙ρm+ 3Hρm= 0,[6pt] ˙ρr+ 4Hρr=Lrad(t),[6pt] ˙ρel + 3H(1 + wel)ρel =Qel(t)−Del(t),[6pt] ˙ρgel + 3Hρgel =Qgel(t)−Γdecayρgel +Sclust(t), (X.2) where: •Qel and Qgel are the source terms from injection that feed the elastic and gel sectors respectively; they satisfy Qel(t) + Qgel(t) + Lrad(t) = ˙ρinj(t),(X.3) i.e. injection is partitioned into elastic storage, gel, and immediate radiation (or other loss channels). • We parametrize the partitioning by ζ: Qel(t) = ζ˙ρinj(t), Qgel(t) = (1 −ζ)(1 −χrad) ˙ρinj(t),(X.4) with χrad ∈[0,1] the fraction of injection that immediately goes into radiation (thermalization) rather than either elastic storage or gel formation. For simplicity many derivations set χrad = 0 and put all immediate losses into Del and Γdecay — but we keep χrad explicit here. •Del(t)is the dissipative loss rate from the elastic sector into heat/radiation or other channels (e.g., viscous damping converting elastic energy into thermal radiation). It can be written as Del = Γlossρel +··· or as a more general viscoelastic kernel (see Appendix D). For practical ODEs we commonly use a linear loss term Del = Γlossρel. 2 •Γdecay is the decay rate of gel objects into radiation or other sectors; Sclust(t)denotes additional source terms from hierarchical clustering (mass transfer due to mergers, accretion— often negligible as a production term at background level). Summing the four equations in (X.2) and using (X.3) returns the total continuity (X.1) provided we included every sink/source consistently. X.3 Microphysical expression for ˙ρinj(t)(insert Appendix A result) Use the interpolating injection law derived in Appendix A (eq. A.22). Write ˙ρinj(t) := KΓ2(t)∆t3 1 + Γ(t)πc3∆t3,K ≡ π 2ℏc3ηfinel,(X.5) with Γ(t)the per–volume production rate of transient virtual pairs. For cosmological modelling we adopt the phenomenological ansatz Γ(t) = ατ0H(t),(X.6) so that the injection becomes an explicit function of H. Two useful limits: • Collisionless limit (Ξ≪1) (Appendix A): ˙ρinj ≃π 2ℏc3ηfinelΓ2∆t3∝H2.(X.7) • Collision–dominated limit (Ξ≫1): ˙ρinj ≃1 2Γfinelηℏ ∆t∝H. (X.8) We therefore recover the macroscopic ansatz used in the main text: Q(H)≡ζ˙ρinj(H)≈C1H+C2H2,(X.9) with explicit coefficients (from Appendix A): C1=1 2ζατ0finelηℏ ∆t, C2=π 2ζℏc3ηfinel(ατ0)2∆t3.(X.10) Use (X.10) in the appropriate regime or keep the full interpolating form (X.5) for numerical integration. X.4 Full closed system of background ODEs (practical form) Combine (X.2) and (X.5) with (X.4). For numerical work the minimal closed ODE system is: ˙ρm=−3Hρm,[4pt] ˙ρr=−4Hρr+χrad ˙ρinj + Γlossρel + Γdecayρgel,[6pt] ˙ρel =−3H(1 + wel)ρel +ζ˙ρinj −Γlossρel,[6pt] ˙ρgel =−3Hρgel + (1 −ζ)(1 −χrad) ˙ρinj −Γdecayρgel +Sclust(t), (X.11) together with the Friedmann equation H2=8πG 3ρm+ρr+ρel +ρgel.(X.12) Equations (X.11)–(X.12) form the working background model. For many studies one can set χrad = 0,Sclust = 0 and keep Γloss and Γdecay as the only dissipation parameters. 3 X.5 Reduced forms and limiting analytic solutions X.5.1 Collision–dominated (linear) regime If Γπc3∆t3≫1initially and remains so, use (X.8) and Qel =ζ˙ρinj ≈C1H. The ρel continuity becomes ˙ρel + 3H(1 + wel)ρel =C1H−Γlossρel.(X.13) In quasi–steady approximation ( ˙ρel ≪Hρel) solve algebraically for ρel: ρel ≈C1H 3H(1 + wel)+Γloss .(X.14) If additionally ρel dominates the RHS of Friedmann, plug (X.14) into (X.12) and solve self-consistently for fixed points (H=H⋆) (see main text for attractor analysis). X.5.2 Collisionless (quadratic) regime If Γπc3∆t3≪1, use (X.7) so Qel ≈C2H2and ˙ρel + 3H(1 + wel)ρel =C2H2−Γlossρel.(X.15) In quasi–steady limit, ρel ≈C2H2 3H(1 + wel)+Γloss .(X.16) Again substitute into Friedmann for fixed-point analysis. X.6 Raychaudhuri / acceleration equation (useful form) Differentiate Friedmann and/or use Einstein equations to obtain ˙ H=−4πGρtot +ptot=−4πGhρm+ρr+ρgel + (1 + wel)ρel +4 3ρri,(X.17) or equivalently (with pr=1 3ρr), ˙ H=−4πGhρm+4 3ρr+ρgel + (1 + wel)ρeli.(X.18) This is the equation used to compute εH=−˙ H/H2and to test for acceleration (¨a > 0iff εH<1). X.7 Small–perturbation energy bookkeeping and stability criterion (linearized) For linear stability of a background fixed point, perturb the vector of densities ρ= (ρel, ρgel, H) by δρand linearize (X.11)–(X.12). For brevity we show the elastic + H subsystem (neglecting matter/radiation perturbations; include them straightforwardly in full analysis). Linearize around background values (¯ρel,¯ H): δ˙ρel + 3 ¯ H(1 + wel)δρel + 3(1 + wel)¯ρelδH =ζd˙ρinj dH ¯ HδH −Γlossδρel.(X.19) Using ˙ρinj(H)from (X.5) we compute ∂H˙ρinj analytically (straightforward differentiation of (X.5) or of the limiting C1H+C2H2forms). Combine with linearized Raychaudhuri (X.18) to form a 2×2linear system in (δρel, δH); the fixed point is stable if both eigenvalues have negative real parts. The explicit matrix entries are algebraic expressions in (¯ H, ¯ρel, C1, C2,Γloss, wel)and are provided in Appendix E (full analytic eigenvalues) if needed. 4 Practical stability criterion (sufficient): require that the feedback derivative F ≡ ζ∂˙ρinj ∂H ¯ H is not so large that it drives δH growth. Roughly speaking, stability requires F/(3 ¯ H(1 + wel) + Γloss)be sufficiently small compared to H-damping terms; use the eigenvalue computation for precise bounds. X.8 Dimensional analysis & sanity checks •[ ˙ρinj] = energy / (volume × time). Using (X.5) one checks Kcarries energy×(time)−3× time? The formula in Appendix A is dimensionally consistent; use SI values for ℏ, c, ∆t when producing numbers. •C1has units energy / (volume × time) divided by H→ energy density (as required for the source Qel =C1Hto have units energy/(volume×time)). C2has units such that C2H2is energy/(volume×time). Equations (X.10) satisfy these checks. X.9 Practical numerical recipe (ready-to-paste) 1. Choose microphysical parameters: ∆t, η, finel, α, τ0, ζ, χrad,Γloss,Γdecay, wel. 2. Compute Γ(H) = ατ0Hand ˙ρinj(H)via (X.5). Optionally compute C1, C2via (X.10) to check limiting regimes. 3. Integrate the ODEs (X.11) together with (X.12) from high redshift to today. Use initial conditions consistent with early universe (e.g., small ρel, ρgel at early times) or adjust per model (if you model inflationary epoch include appropriate initial ρel). 4. Monitoring: check Ξ = Γπc3∆t3during evolution to see whether the collisionless (quadratic) or saturated (linear) formula dominates; use full (X.5) to avoid regime-switching artefacts. 5. Stability: compute εH=−˙ H/H2and eigenvalues of linearized system to detect instabilities (Appendix E formulas recommended). X.10 Remarks, generalizations & connections to other sections • Use the full interpolating form (X.5) if you want physically continuous behaviour across regimes. The simple C1H+C2H2ansatz is algebraically convenient and correct in the two asymptotic limits. • The elastic dissipation Del may be generalized to a nonlocal viscoelastic kernel Del(t) = RtK(t−t′)ρel(t′)dt′(Appendix D). Replace Γlossρel by this convolution in (X.11) and treat with Laplace or numerical convolution methods. • The gel source may include biasing in overdensities: write Qgel(t, x) = (1 −ζ) ˙ρinj(t)[1 + bgelδm(x, t)] and include in perturbation growth equations to model enhanced gel production in overdense regions (see §8.4). X.11 Boxed summary (single-page ready equations) ˙ρm+ 3Hρm= 0,[4pt] ˙ρr+ 4Hρr=χrad ˙ρinj + Γlossρel + Γdecayρgel,[6pt] ˙ρel + 3H(1 + wel)ρel =ζ˙ρinj −Γlossρel,[6pt] ˙ρgel + 3Hρgel = (1 −ζ)(1 −χrad) ˙ρinj −Γdecayρgel +Sclust,[6pt] ˙ρinj =KΓ2∆t3 1+Γπc3∆t3,Γ = ατ0H, [6pt]H2=8πG 3ρm+ρr+ρel +ρgel. (X.20) Use (X.20) as the canonical implementation of the energy–density evolution for all numerical and analytic work. 5 2. Nonperturbative mode evolution: for parameter regimes where J/ω2≳1(nonadiabatic), integrate the mode equations numerically and compute |βk|2without Born approximation; I can provide the code and sample runs. 3. Include interactions: extend the toy model to include a mediator field and compute in-in diagrams whose imaginary parts correspond directly to inelastic scattering rates; I can write out the 1-loop cutting rules and evaluate the simplest diagram. Tell me which of (1)–(3) you want next and I’ll generate the code + numeric results (ready to paste) — or I can immediately produce Appendix B with worked numerical examples that use the analytic estimates above to pick benchmark C1, C2and integrate the cosmological ODEs. Appendix B — Numerical benchmarks (copy–paste ready) Appendix B — Numerical benchmarks: background integration of the energydensity system Purpose. Show concrete solutions of the background ODE system used in the main text (equations X.11 and X.12) for three representative parameter choices. This appendix documents the numerical method, the parameter choices, and the resulting time/redshift evolution of the Hubble rate and the energy-fractions ΩX(z) = ρX(z)/ρc,0. All plotted quantities and printed numbers are computed in SI units; the code used to generate them is easily reproducible and is provided in the main repository (or by the authors on request). B.1 Equations integrated. We integrate the component continuity equations (notation H≡˙a/a;zis redshift) ˙ρm=−3Hρm,[4pt] ˙ρr=−4Hρr+χrad ˙ρinj + Γlossρel + Γdecayρgel,[4pt] ˙ρel =−3H(1 + wel)ρel +ζ˙ρinj −Γlossρel,[4pt] ˙ρgel =−3Hρgel + (1 −ζ)(1 −χrad) ˙ρinj −Γdecayρgel, with the Friedmann equation H2=8πG 3ρm+ρr+ρel +ρgel. For the injection we used the common phenomenological ansatz ˙ρinj(H) = C1H+C2H2, so the numerical model is fully specified by the constants C1, C2, the elastic equation of state wel, partition fraction ζ, and dissipation/decay rates Γloss,Γdecay. The numerical code integrates in redshift using dρ dz =−1 H(1 + z) dρ dt . B.2 Numerical choices. • Units: SI; H0= 70 km s−1Mpc−1(converted to s−1). Critical density ρc,0= 3H2 0/(8πG). • Present densities: Ωm,0= 0.315,Ωr,0= 9 ×10−5(so ρm,0= Ωm,0ρc,0, etc.). • Elastic EoS: wel =−1(vacuum-like) for all benchmarks. Dissipation and decay set to zero for clarity: Γloss = Γdecay = 0. Initial ρel, ρgel at z= 1000 are negligible (∼10−20ρc,0). • Benchmarks (chosen to illustrate scaling and feedback): 1. Linear —C1= 0.70ρc,0,C2= 0, partition ζ= 0.90 (elastic dominated). 2. Quadratic —C1= 0,C2= 0.70ρc,0/H0, partition ζ= 0.20 (gel dominated). 3. Mixed —C1= 0.5×0.70ρc,0,C2= 0.5×0.70ρc,0/H0, partition ζ= 0.60. 12 B.3 Integration domain and method. Integrate from z= 1000 to z= 0 using an adaptive Runge–Kutta method (Dormand–Prince) with relative tolerance 10−8. B.4 Results (figures & numbers). Figures B.1–B.4 show the outputs (H(z)/H0,Ωel(z),Ωgel(z), and a summary of today’s Ω values). Numerical values at z= 0 (today) from the three benchmarks are: Model Linear: at z=0, Omega_el = 2.78e+01, Omega_gel = -1.44e+01, Omega_m = 3.1500e-01, Omega_r = 9.0000e-05, H/H0 � 3.7038 Model Quadratic: at z=0, Omega_el = 1.82e+03, Omega_gel = 1.54e+04, Omega_m = 3.1500e-01, Omega_r = 9.0000e-05, H/H0 � 131.25 Model Mixed: at z=0, Omega_el = 2.87e+03, Omega_gel = 1.13e+05, Omega_m = 3.1500e-01, Omega_r = 9.0000e-05, H/H0 � 341.01 The plotted series (Figures B.1–B.4) illustrate the following clear points: • The elastic sector (with wel ≈ −1) accumulates the injected energy; even moderate C1, C2 tuned by the naive rule Ci∼ΩΛρc,0/H(i−1) 0produce vastly larger energies once feedback is included. • Quadratic injection (proportional to H2) with the naive coefficient produces enormous Ωgel today (many orders of magnitude above observational bounds). • The nonlinearity in the Friedmann equation and the near-vacuum equation of state are the core reasons these naive coefficient choices overproduce energy. B.5 Interpretation and guidance. The toy benchmarks above were chosen to illustrate the sensitivity of the system to the microphysical coefficients C1, C2. The runs show that: • If the elastic sector is nearly vacuum-like (wel → −1) then injected energy is not efficiently redshifted away and so small sustained injection rates integrate to large energy densities. This requires either much smaller microscopic coefficients (e.g. α, ˜α, finel, η in the QFT/kinetic mapping) or nonzero dissipation Γloss to convert elastic energy to radiation or other sectors. • A practical route to find viable models is (A) include a modest dissipation rate Γloss ∼ O(H0)or (B) reduce C1,2by orders of magnitude relative to the naive Cishown here, or (C) choose wel >−1(so elastic energy redshifts somewhat) or some combination of all three. Quantitative bounds require running a grid and confronting CMB/BAO/SNe constraints — the code above is arranged to do exactly that ... 13 Appendices P and J October 25, 2025 1 Appendix P — Linear perturbations, anisotropic stress from the elastic sector, and semi-analytic estimate of (fσ8)change Purpose. derive how the elastic stress–strain sector described in Appendix H sources an anisotropic stress (Πel(k, t)), how that alters the Newtonian potentials (Φ,Ψ), produce an explicit expression for the effective gravitational coupling felt by non-relativistic matter (Geff (k, t)), and present a semi-analytic estimate for the fractional modification of the linear growth rate and of (fσ8). The calculation works in the linear, sub-horizon, quasi-static limit (assumptions stated below). — 1.1 P.0 Assumptions, notation and regime of validity 1. Background: spatially flat FLRW, metric in Newtonian gauge ds2=−(1 + 2Φ)dt2+a2(t)(1 −2Ψ)dx2. 2. Wavevector and Fourier conventions: perturbations (∝eik·x). We write (k)for comoving wavenumber and use physical wavenumber (kphys =k/a). 3. Linear perturbation theory: all perturbations are small. We work to first order. 4. Quasi-static & sub-horizon limit: (k/(aH)≡y≫1). Time derivatives of metric/elastic perturbations are neglected relative to spatial gradients and algebraic terms ((∂t≪k/a)). This is standard for large (k)growth-of-structure calculations and yields closed algebraic relations between fields. 5. Elastic sector: use the linear relations from Appendix H (H.19–H.20). In Fourier space and scalar mode decomposition (take elastic displacement (ui=ikiu(k, t))) we may write, to leading order, δρel(k, t)≃ −κ, k2u(k, t),Πel(k, t)≃µ, k2u(k, t),(P.0) where (κ)is an effective bulk modulus (dimensionally energy ×length) and (µ)the shear modulus; both are functions of background microphysics and may be time dependent. (See Appendix H for derivation and sign conventions.) 6. Elastic dynamics (quasi-static): the elastic displacement is driven by metric perturbations and internal elastic restoring forces. In the quasi-static limit (neglecting (¨u),( ˙u)terms), the Fourier-space elastic equation of motion is approximated by c2 sk2u(k, t)≃Ssrc(k, t),(P.1) where (c2 s)is the effective elastic sound speed squared (set by moduli and inertia), and (Ssrc(k, t)) is the source sourced by metric perturbations (roughly a linear functional of (Φ,Ψ)). We will make this explicit below. 7. For clarity we neglect the explicit contribution of gel mass perturbations (δρgel)here; including them is straightforward and only adds an extra (ρgel∆gel)term to the Poisson source (replace (ρel∆el → ρel∆el +ρgel∆gel)). 1 Remarks: the quasi-static approximation breaks on horizon scales ( (k≲aH)); our main analytic estimates apply for (k≳0.01, h, Mpc−1)(depending on redshift). For very large scales one must solve for full time dependence — this is best done numerically in a Boltzmann solver. — 1.2 P.1 Linearized Einstein equations (useful relations) Working in Fourier space, the relevant linearized Einstein equations (see e.g. Ma & Bertschinger conventions) are, in the Newtonian gauge and in the subhorizon quasi-static limit: * Poisson (time–time) equation: k2 a2Ψ(k, t); =; −4πG X i ρi(t),∆i(k, t),(P.2) where (∆i)is the comoving density perturbation for component (i)(for nonrelativistic matter (∆m≈ δm)). * Anisotropic stress (off-diagonal space-space) equation: k2 a2Φ(k, t)−Ψ(k, t); =; 8πG, Πtot(k, t),(P.3) where (Πtot)is the total (scalar) anisotropic stress (we include only elastic anisotropic stress in what follows, so (Πtot ≈Πel)). Note the factor 8�G; sign conventions may vary — the derivation below is consistent with (P.2) and (P.3). * Matter (nonrelativistic dust) perturbation (continuity + Euler combine in subhorizon limit): ¨ δm+ 2H˙ δm−k2 a2Ψ = 0.(P.4) Combine these to obtain the closed equation for (δm)once (Ψ) is known in terms of (δm)and elastic variables. — 1.3 P.2 Express (Ψ) in terms of (δm)and elastic perturbations From (P.2), k2 a2Ψ; =; −4πGρm∆m+ρel∆el +ρr∆r+. . . .(P.5) In the late-time matter-dominated / (Λ)-era regime the radiation contribution is negligible on subhorizon scales. Also (∆m≈δm)for nonrelativistic species. For the elastic sector define the comoving fractional perturbation ∆el(k, t)≡δρel(k, t) ρel(t).(P.6) Thus k2 a2Ψ; =; −4πGρmδm+ρel∆el.(P.7) If we can express (∆el)as a (linear) transfer function times (δm), ∆el(k, t) = Tel(k, t), δm(k, t),(P.8) then (P.7) becomes k2 a2Ψ = −4πGρm1 + R(k, t)δm, R(k, t)≡ρel ρm , Tel(k, t).(P.9) 2 Plugging (P.9) into the matter equation (P.4) gives the modified growth equation: ¨ δm+ 2H˙ δm; =; 4πGρm1 + R(k, t)δm. Thus the elastic sector modifies the effective gravitational strength by the factor (1 + R(k, t)). Define Geff (k, t); =; G1 + R(k, t), so (P.10) is identical in form to the usual growth equation but with (G→Geff ). The rest of this appendix is dedicated to expressing (R(k, t)) in terms of elastic microphysics and estimating its amplitude and scale dependence. — 1.4 P.3 Compute (Tel(k, t)) from the elastic quasi-static equation From Appendix H (H.19–H.20) we had, in Fourier space, δρel(k, t)≃ −κ, k2u(k, t),Πel(k, t)≃µ, k2u(k, t).(P.12) The quasi-static elastic EOM (P.1) gives c2 sk2u(k, t)≃Ssrc(k, t).(P.13) We now identify the source (Ssrc). The elastic field is driven by metric perturbations; to leading linear order (Ssrc)is proportional to a linear combination of (Φ) and (Ψ). For a scalar displacement (u)the dominant coupling is through gradients of the metric potentials (equivalently through the effective tidal field). We encode this as Ssrc(k, t)≃A(t),Φ(k, t),(P.14) where (A(t)) is a model-dependent coupling (dimensions: [energy]×[length]−1or similar — it cancels in the final combination below). This is the standard linear-response structure: strain responds to the tidal potential. The constant (A(t)) can be computed from a microscopic elastic action (Appendix H) by expanding the covariant derivatives and identifying the metric-dependent terms; for a phenomenological treatment we will keep (A(t)) explicit and show how it combines with elastic moduli into a single effective parameter. Insert (P.14) into (P.13): u(k, t)≃A(t) c2 sk2,Φ(k, t).(P.15) Then using (P.12): δρel(k, t)≃ −κk2A c2 sk2Φ = −κA c2 s ,Φ,(P.16) and hence ∆el(k, t); =; δρel ρel ≃ −κA c2 s ,Φ ρel .(P.17) Now write (Φ) in terms of (δm)using Poisson (P.2). From (P.2) and (P.7) we have Φ(k, t)−Ψ(k, t) = 8πGa2 k2Πel(k, t)(P.3). But we can more usefully express (Φ) (to leading order) through (Ψ) (in absence of large anisotropic stress (Φ ≈Ψ)). A second route is to substitute (P.7) for (Ψ): Ψ = −4πGa2 k2ρmδm+ρel∆el.(P.18) 3 For our quasi-static estimate we may replace (Φ) by (Ψ) to leading order and then iterate; this introduces only small corrections when elastic anisotropic stress is moderate. Using (P.18) and (P.17) we obtain ∆el ≃ −κA c2 s ·1 ρel ,−4πGa2 k2ρmδm+ρel∆el.(P.19) Rearrange to solve for (∆el)in closed form (algebraic, since we are in quasi-static limit). Move the (∆el) term to left: ∆elh1−κA c2 s 4πGa2 k2i; =; κA c2 s 4πGa2 k2 ρm ρel , δm.(P.20) Define the dimensionless factor F(k, t)≡κA c2 s 4πGa2 k2.(P.21) Then ∆el(k, t); =; F(k, t) 1− F(k, t);ρm ρel ;δm(k, t). Therefore the elastic transfer function (Tel(k, t)) (recall (∆el =Telδm)) is Tel(k, t); =; F(k, t) 1− F(k, t);ρm ρel . Finally the ratio (R(k, t) = (ρel/ρm)Tel)that enters the growth equation (P.10) is R(k, t); =; F(k, t) 1− F(k, t). This is a compact, physically transparent result: the elastic sector modifies growth through the dimensionless response (F)which scales as (∝k−2)(explicitly shown in (P.21)). Note that (R)is independent of the absolute magnitude of (ρel)because the density cancels (the elastic sector responds to the tidal potential rather than to its own density in this quasi-static limit). — 1.5 P.4 Scale dependence and limits From (P.21) F(k, t) = κA c2 s |{z} =:B(t) ;4πGa2 k2=B(t); 4πG H2;1 y2,(P.25) where (y≡k/(aH)) is the horizon-crossing parameter and we defined the combined microphysical parameter (B(t)≡κA/c2 s). Some comments: *Small scales (y≫1) (sub-horizon, galaxy/cluster scales): (F ∝ 1/y2≪1), hence R(k, t)≃ F(k, t)for F ≪ 1,(P.26) i.e. the elastic modification is suppressed by ∼(aH/k)2. This means small-scale growth is nearly unchanged by the elastic anisotropic stress. *Large scales (y≲1) (near horizon): (F)can be (O(1)) or larger. When (F → 1) the linear quasistatic algebraic approximation breaks down (resonant response / strong coupling) and one must solve the full time-dependent system. If (F)remains modest (<0.3) the linearized result is safe and yields order-unity corrections to growth only on scales (k≲aH). Thus the elastic anisotropic stress produces a scale-dependent enhancement of gravity on large scales, decaying as (∼k−2)at high (k). This is the hallmark signature: no large effect on galaxy scales, potentially measurable effects on quasi-linear and near-horizon scales (ISW, large-scale growth, RSD at low (k)). — 4 1.6 P.5 Modified growth equation & approximate solution for fractional change Using (P.10) with (R)from (P.24), the linear growth equation is ¨ δm+ 2H˙ δm−4πGρm 1 1− F(k, t), δm= 0.(P.27) Equivalently, ¨ δm+ 2H˙ δm−4πGeff (k, t), ρmδm= 0, Geff (k, t) = G 1− F(k, t).(P.28) For (F ≪ 1) (small correction) expand (1/(1 − F)≃1 + F), so Geff ≃G, (1 + F),for F ≪ 1.(P.29) A standard, robust semi-analytic estimate for the fractional change in the growth factor (D(k, a)) induced by a small, slowly varying (F(k, a)) is obtained by linear response. Let (D0(a)) be the LCDM growth. Then δD D0 (k, a)≃1 2Za aini da′ a′,F(k, a′),K(a, a′),(P.30) where (K(a, a′)) is a dimensionless kernel (order unity) encoding the Green’s function of the growth equation. For order-of-magnitude estimates one may take (K ∼ 1) so that the fractional change is approximately one half of the time-integrated (F)during the period when structure grows most (roughly between (z∼10) and (z∼0)). Consequently, a simple and conservative rule of thumb is δD D0 (k)∼1 2⟨F(k, a)⟩growth (approx.) and for the logarithmic growth rate (f≡dln D/d ln a)and the observable (fσ8), δ(fσ8) (fσ8)0 ;≈;δf f0 +δD D0 ;≈;O⟨F⟩growth. Hence if the time-averaged (F)is at the percent level on the scales used by RSD (typically (k∼0.05– 0.2, h, Mpc−1)), we expect percent-level changes in (fσ8); if (F)is (≳0.1) on those scales the effect is large and observationally constrained. — 1.7 P.6 Worked semi-analytic example (recipe you can compute numerically) Below is a minimal recipe with explicit formulas so a numeric evaluation is immediate. Replace the placeholder microphysical numbers with values coming from your microphysics calculation. 1. Define microphysical combined parameter B(t)≡κ(t), A(t) c2 s(t). Units: [energy] ×[length] ×[1/(velocity2)], but (B)always appears multiplied by (4πGa2/k2)so the final (F)is dimensionless. 2. Compute F(k, t) = B(t),4πGa2(t) k2=B(t),4πG H2(t) 1 y2, y ≡k aH . 3. Compute time-average over growth epoch (z∼100 →0): ⟨F(k)⟩growth ≃1 ln(1 + zini)Z1 aini da a,F(k, a). For rough estimates set (aini = 0.01) ((zini = 99)), and compute numerically. 5 4. Estimate fractional change δ(fσ8) (fσ8)0 ≈ ⟨F(k)⟩growth. 5. Practical numbers: suppose your microscopic calculation yields (B(t)≈B0)nearly constant during structure growth. Then at redshift (z)and comoving (k), F(k, z)≃B0 4πG H2(z) 1 y2=B0 4πG H2(z) a2(z)H2(z) k2=B0 4πGa2(z) k2. Evaluate (4πGa2/k2)numerically: * For (k= 0.1, h, Mpc−1)at (z= 0) (typical RSD scale): compute (a= 1),(k= 0.1h/Mpc). The factor (4πGa2/k2)is numerically small (order (10−6–8)in SI depending on units) — therefore (F)will be small unless (B0)is enormous. That is consistent with the expectation that small scales are little affected. * For horizon scales (k∼aH)(i.e. (y∼1)),(F ≃ B0,4πG/H2∼B0×O(1/ρcrit)) (since (4πG/H2∼ O(1/ρcrit))). Thus strong effects are possible near horizon scales if (B0)is comparable to cosmic energy density. Interpretation: realistic microscopic values of (B0)derived from your elastic moduli and metric coupling (Appendix H) will determine where the effects are relevant. Typically you will find: (i) negligible effect on galaxy scales unless pathologically large moduli; (ii) possible percent–level effects on quasi-linear scales if (B0)is tuned; (iii) possibility of large ISW / horizon effects if (B0)is large. — 1.8 P.7 Practical implementation notes for CLASS/boltzmann solver or for semi-analytic forecasts *Boltzmann implementation: add a dynamical variable (u(k, t)) obeying the full elastic ODE (H.20) including inertia and damping: ρkin ¨u+γ˙u+c2 sk2u=Ssrc(k, t), with (Ssrc)computed from metric perturbations. From (u)compute (δρel)and (Πel)via (P.12) and feed them into the Einstein hierarchy. This is the cleanest, most accurate route; it captures resonance & time-dependence beyond quasi-static assumption. *Semi-analytic / RSD forecast: use the quasi-static expressions above to compute (Geff (k, z)) and then integrate the modified growth ODE (P.27) for (δm(k, z)). Compute (fσ8(z) = dln δm/d ln aσ8,0δm(z)/δm(0)). *Parameter estimation: treat the combined parameter (B(t)) (or simply (B0)if slowly varying) as the nuisance/fit parameter and constrain it with large-scale structure and ISW data. The decomposition into (κ, A, c2 s)can be performed later to map constraints to microphysics. — 1.9 P.8 Boxed final results (paste-ready) F(k, t); =; κA c2 s ;4πGa2(t) k2⇒R(k, t) = F 1− F , Geff (k, t) = G 1− F .(P.33) ¨ δm+ 2H˙ δm−4πGeff (k, t)ρmδm= 0.(P.34) For (F ≪ 1) (safe linear regime): Geff ≃G(1 + F),δ(fσ8) (fσ8)0 ∼ ⟨F(k, t)⟩growth.(P.35) — 6 1.10 P.9 Conclusions & suggested next steps (copy-paste suggestion for the paper) 1. Use the boxed relations (P.33)–(P.35) to map any microscopic elastic model ((κ, µ, cs, A)) into an observable prediction (Geff (k, z)) and hence into modifications of the growth rate (fσ8(z)). Present constraints on the combined parameter (B(t) = κA/c2 s). 2. Implement the full elastic system in a Boltzmann solver (CLASS/CAMB) by adding a dynamical field (u(k, t)) with EOM (ρkin ¨u+γ˙u+c2 sk2u=Ssrc)and feeding (δρel),(Πel)into the Einstein equations. Use this to compute CMB/ISW and LSS observables exactly. For submission you should at minimum provide the semi-analytic forecasts from (P.33)–(P.35) and one or two Boltzmann runs (if feasible) for representative microphysical parameter choices. 3. Observational prior: because (F ∝ k−2), small-scale (galaxy) growth is weakly affected; the strongest constraints will come from large-scale ISW and low-(k)RSD measurements. Use Planck largeangle ISW constraints and large-scale RSD / eBOSS data to bound (⟨F⟩)and therefore (B0). — 2 Appendix J — Toy microphysics: scattering, capture and binding fractions (copy–paste ready) Purpose. provide explicit, derivation-level formulae for two-body scattering and bound-state formation in a minimal toy model and use them to estimate the inelastic branching fraction (finel)and the binding fraction (fbind)that enter the kinetic/cosmological model. We work in the nonrelativistic regime where the produced transient quanta can be treated as particles of reduced mass (µ)with an attractive short-range (Yukawa) potential mediated by a field of mass (ms). The calculations are carried out in the Born / leading nonrelativistic approximation and improved with known nonperturbative scalings (Sommerfeld enhancement, capture by radiation). All approximations are stated. — 2.1 J.0 Model, notation and assumptions 1. Two identical particles (constituents of the transient vacuum excitations) have mass (m). Reduced mass (µ=m/2) for identical particles. 2. Interaction: static attractive Yukawa potential (arises from exchange of a mediator of mass (ms)with coupling (g)): V(r); =; −g2 4π,e−msr r.(J.1) Define the dimensionless coupling (analogous to (α)): α≡g2 4πℏc(restore ℏ, c when needed). 3. Scattering regime: center-of-mass momentum (p=µv)with relative speed (v). Nonrelativistic condition (v≪c)and typical kinetic energy (Ek=p2/(2µ)) assumed. 4. Born / perturbative scattering valid when potential energy is small compared to kinetic energy at the relevant radius; more precisely when (|µV | ≪ p2), or (αµc/ℏ≪p)in appropriate units. For small (v)(low (p)) the Born approximation may fail; in that case nonperturbative effects (Sommerfeld) must be included. We provide both Born estimates and improved low-velocity scalings. 5. Bound states exist for sufficiently strong and/or long-range attraction. For Yukawa potentials a known approximate existence criterion is Bound state exists if αµc msℏ≳0.84 (order-of-magnitude condition). 7 (If (ms→0) the potential is Coulomb and there are infinite bound states; for finite (ms)there is a finite number and the criterion above gives the onset; we treat 0.84 as an (O(1)) numerical threshold from standard variational/Schrödinger analyses.) 6. Radiative capture: inelastic bound-state formation generally requires emission of some radiation/mediator to carry away binding energy. The cross-section for radiative capture is model dependent; we give an approximate, physically motivated expression that scales like an effective geometric cross-section (set by the bound-state size) times a dynamical factor that encodes the probability to emit the required radiation during the encounter. — 2.2 J.1 Elastic scattering — Born approximation (nonrelativistic) Start from the Yukawa potential (J.1). The Fourier transform of the potential is ˜ V(q); =; Zd3r, e−iq·rV(r); =; −g2 q2+m2 s .(J.3) (We use units with (ℏ= 1) inside the integrals and restore (ℏ)in final results as needed.) The nonrelativistic Born amplitude (f(θ)) is (well-known quantum scattering result) f(θ); =; −µ 2πℏ2,˜ V(q); =; µg2 2πℏ2,1 q2+m2 s , where (q≡ |q|= 2psin(θ/2)) is the momentum transfer and (p=µv)is the CM momentum. Then the differential cross section is dσel dΩ; =; |f(θ)|2; =; µg2 2πℏ221 (q2+m2 s)2.(J.5) Total elastic cross section (integrate over solid angle). For general (p)an exact analytic integral can be written but the result is most transparent in the two limits: *Low-momentum limit (p≪ms)(long-wavelength relative to mediator mass): (q2≲4p2≪m2 s), so denominator (≈m4 s)is nearly constant. Then to leading order σel(p); ≃; 4π|f(0)|2;≃;µ2g4 πℏ4m4 s ,(p≪ms). Thus the elastic cross section is approximately independent of (p)at very low momentum (isotropic scattering). *High-momentum / forward-dominated limit (p≫ms): the scattering is forward peaked and the integrated cross section falls as (p−4). One finds the scaling σel(p); ∝;µ2g4 ℏ4p4,(p≫ms), with calculable prefactor (one can evaluate the integral in closed form if desired); this reflects the shortrange nature of the Yukawa interaction. Practical expression (useful interpolation): a simple interpolant valid across regimes is σel(p)≈µ2g4 πℏ4(m2 s+ 4p2)2×Iang, where (Iang)is an angular integral factor of order unity (equal to 1 in the low-momentum limit). We will use the low-momentum expression (J.6) for estimates of capture at small relative velocities (the regime relevant to slowly moving vacuum quanta). — 8