DRSN XIX: Drifted Spectral Applications I. Cosmology, Cosmic Superstrings and String Phenomenology (De Rerum Spectrale Natura, Report XIX, Version 1.0)

DSRN XIX: DRIFTED SPECTRAL APPLICATIONS I Cosmology, Cosmic Superstrings and String Phenomenology De Rerum Spectrale Natura series REPORT XIX (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro October 2025 •Drifted FRW cosmology from the universal spectral potential V(s) = αs2+βs4. •Cosmic (super)strings with drifted tension and modified network dynamics. •Gravitational-wave signatures sensitive to drift evolution. •Brane–antibrane inflation interpreted through the drift dictionary. •Internal drift and string-inspired phenomenology in the spectral framework. The Drifted Spectral Universe: Applications to FRW Cosmology, Cosmic Superstrings and Brane Inflation J. Pinho-da-Cruz 1 1 Department of Mechanical Engineering, University of Aveiro (Dated: October 2025) We develop the first set of physical applications of the Drifted Spectral Universe established in DSRN I–XVIII. The drift deformation of Dirac-type operators induces a universal quartic spectral potential that drives FRW cosmology, cosmic superstring dynamics, drifted tension evolution, gravitational-wave backgrounds, brane–antibrane inflation and internal string phenomenology. Connections to pulsar timing array signals, drifted network evolution, brane– antibrane inflation dictionaries, and drifted internal Yukawa structures are established within a unified operator-theoretic framework. Keywords: Spectral Action; rDrift Geometry; FRW Cosmology; Cosmic Superstrings; Time-varying Tension; Gravitational Waves; Brane–Antibrane Inflation; String Phenomenology; Noncommutative Geometry. CONTENTS I. Introduction 3 II. Drifted FRW Cosmology 4 III. Cosmic Strings and Cosmic Superstrings under Drift 5 IV. Brane–Antibrane Inflation and Drift Geometry 6 V. String Phenomenology in the Drifted Spectral Universe 6 VI. Unified Operator-Theoretic Interpretation 7 VII. Conclusions 7 A. Drifted FRW Cosmology (Technical Details) 7 B. Drifted Cosmic String Networks and Time-Varying Tension 8 3 C. Brane–Antibrane Inflation and the Spectral Drift Dictionary 9 D. Internal Dirac Operator and Yukawa Drift 9 References 10 I. INTRODUCTION The Drifted Spectral Universe developed in DSRN I–III establishes a unified operator-theoretic framework in which geometry, fields and dynamics arise from a controlled similarity deformation of Dirac-type operators, Ds=esϕ D e−sϕ, preserving principal symbol and ellipticity while modifying lower-order terms. A universal quartic spectral potential, V(s)=αs2+βs4, β > 0, emerges from the heat-kernel expansion of D2 s and admits a non-trivial minimum s∗ , yielding a positive vacuum energy and a dynamically stable condensate [1–3]. The foundational operator geometry and spectral action were introduced in DSRN I [ 1 ], extended to worldsheet and unified constructions in DSRN II [ 2 ], generalised to supersymmetric drift geometry in DSRN III [ 3 ], and synthesised at the level of a complete operator picture in DSRN XVIII [ 4 ]. These works provide the analytic backbone for applications to cosmology, strings and phenomenology developed here. a. Canonical dependency. All operator-theoretic structures used in this report are established in DSRN I–III and fixed canonically in DSRN XVII. No new foundational definitions are introduced here. In this report we present the first set of physical applications of the drifted framework: (i) FRW cosmology and the emergence of late-time acceleration from the spectral condensate, following standard cosmological notation [ 5 ]; (ii) cosmic strings and cosmic superstrings with drifted, timevarying tension and their implications for gravitational-wave backgrounds, including recent pulsartiming-array interpretations [ 8 , 9 ]; (iii) brane–antibrane inflation via a clean dictionary between the inflaton modulus and the drift parameter, within the standard string-theoretic setting [ 6 , 7 ]; (iv) internal string-inspired phenomenology, where drift acts on the finite Dirac operator to generate stable Yukawa hierarchies [10,11]. 4 The emphasis throughout is on a unified operator description: cosmological dynamics, string networks, inflationary phases and internal flavour structures are controlled by the same spectral mechanism, without introducing ad hoc scalar sectors or explicit mass scales. Technical derivations supporting the main text are collected in the appendices. This is Report XIX of the De Rerum Spectrale Natura collection. II. DRIFTED FRW COSMOLOGY In Fig. 1we summarise the operator–cosmology architecture linking the drifted Dirac operator to FRW evolution through the universal spectral potential. Standard cosmological conventions and background equations follow Mukhanov [5]. Drifted Dirac Ds Drifted Lichnerowicz D2 s FRW geometry a(t), H(t) Universal potential V(s) Cosmic eras stiff→rad→matter→DE FIG. 1. Drifted spectral cosmology architecture: drift induces a universal potential whose minimum yields late-time acceleration. We consider the spatially flat FRW metric ds2=−dt2+a(t)2dx2, H = ˙a/a, and assume a homogeneous drift field s=s(t). The effective potential V(s)=αs2+βs4, β > 0, α < 0, drives the dynamics via ¨s+ 3H˙s+V′(s)=0, V ′(s)=2αs + 4βs3. 5 a. Remark. The drift parameter labels an exact similarity flow of operators (Kato type-(A)) and should not be interpreted as a coupling-space renormalisation group flow. For generic initial conditions with ˙s2≫V ( s ), the drift sector behaves as a stiff fluid with equation of state ws≃ 1and energy density ρs∝a−6 . As expansion proceeds, Hubble friction damps ˙sand the system relaxes toward the spectral condensate s∗=r−α 2β, V ′′(s∗)=−2α > 0, yielding ws→ − 1and a positive late-time vacuum energy. Detailed derivations and stability analysis are given in Appendix A. III. COSMIC STRINGS AND COSMIC SUPERSTRINGS UNDER DRIFT In Fig. 2we outline the drifted spectral structure of cosmic (super)string networks. Standard network dynamics is described by the velocity-dependent one-scale (VOS) model; here we emphasise how drift modifies the effective tension and hence the evolution. Recent interpretations of pulsartiming-array data motivate time-varying tension scenarios [8,9]. Cosmic (Super)String Network Network model (L(t), v(t)) Drifted tension µ(t)↔s(t) GW output Ωgw(f) PTA connection (nanohertz band) FIG. 2. Drifted spectral interpretation of cosmic (super)string networks. Drift controls the effective tension and modifies the network evolution and GW spectrum. In the drifted framework the effective string tension becomes a function of the drift field, µ(t)=µ0f(s(t)) , 6 so that network parameters acquire controlled time dependence. This provides a spectral underpinning for models connecting cosmic (super)strings to gravitational-wave backgrounds in the nanohertz band. Technical network equations and the drifted extension of the VOS dynamics are collected in Appendix B. IV. BRANE–ANTIBRANE INFLATION AND DRIFT GEOMETRY In Fig. 3we present the correspondence between standard brane–antibrane inflation and spectral drift geometry. In conventional string models the inflaton field Φis identified with the brane separation modulus, and the effective four-dimensional action takes the form SΦ=Zd4x√−g1 2(∂Φ)2−Vbrane(Φ), with Vbrane determined by warping, fluxes and Coulombic interactions [6,7]. D3–D3 system Separation modulus Φ Drift parameter s(t) Inflation potential Veff FIG. 3. Brane–antibrane inflation viewed through spectral drift: the inflaton modulus Φis identified with the drift parameter s. In the drifted spectral framework inflation is driven by the universal potential V ( s ) = αs2 + βs4 , yielding a geometric inflationary mechanism without the introduction of additional scalar sectors. A precise dictionary between brane separation and spectral drift is given in Appendix C. V. STRING PHENOMENOLOGY IN THE DRIFTED SPECTRAL UNIVERSE Internal degrees of freedom are encoded in the finite Dirac operator DF of the spectral triple. Drift acts by a bounded commutator deformation, (DF)s=DF+s[DF, ϕF], which induces linear deformations of Yukawa matrices, Ys=Y+s[Y, ϕF]. 7 Such deformations naturally generate hierarchical textures while preserving stability, as guaranteed by positive quartic spectral contributions. This mechanism provides a string-inspired explanation of flavour hierarchies consistent with modern constructions of the Standard Model from string theory [10,11]. A self-contained treatment of the internal sector is presented in Appendix D. VI. UNIFIED OPERATOR-THEORETIC INTERPRETATION The applications developed in this report admit a unified interpretation. The same drift mechanism: (i) generates a universal cosmological potential driving FRW evolution, (ii) controls cosmic (super)string tension and network dynamics, (iii) provides a geometric realisation of inflation, (iv) induces stable hierarchies in the internal fermionic sector. Cosmology, strings and phenomenology are thus governed by a single spectral deformation of the Dirac operator. VII. CONCLUSIONS We have presented the first set of physical applications of the Drifted Spectral Universe: FRW cosmology, cosmic (super)strings and gravitational-wave backgrounds, brane–antibrane inflation, and string-inspired internal phenomenology. All sectors are controlled by a universal spectral mechanism arising from drift geometry. This completes Applications I and prepares the ground for DSRN XX, devoted to amplitudes, combinatorial superstrings and computational complexity. Appendix A: Drifted FRW Cosmology (Technical Details) We collect the technical derivations supporting Sec. II. We follow standard cosmological notation and conventions as in Ref. [5]. Consider the spatially flat FRW metric ds2=−dt2+a(t)2dx2, H = ˙a/a, and assume a homogeneous drift field s = s ( t ). The effective Lagrangian density for the drift sector is Ls=−1 2˙s2−V(s), V (s)=αs2+βs4, β > 0, α < 0. The corresponding energy density and pressure are ρs=1 2˙s2+V(s), ps=1 2˙s2−V(s). 8 These enter the Friedmann equations together with radiation and matter, H2=8πG 3(ρs+ρr+ρm),˙ H=−4πG (ρs+ps+ρr+pr+ρm+pm). Variation with respect to s(t)yields the drifted Klein–Gordon equation, ¨s+ 3H˙s+V′(s)=0, V ′(s)=2αs + 4βs3. At early times, for generic initial conditions with ˙s2≫V ( s ), the drift sector behaves as a stiff fluid with equation of state ws≃ 1and scaling ρs∝a−6 . As expansion proceeds, Hubble friction damps ˙sand the system relaxes toward the spectral condensate s∗=r−α 2β, V ′′(s∗)=−2α > 0. Linearising around s∗ shows that fluctuations are exponentially damped, so that ws→ − 1and ρs→V(s∗)>0, yielding a dynamically stable late-time acceleration. Appendix B: Drifted Cosmic String Networks and Time-Varying Tension This appendix summarises the drifted extension of standard cosmic string network dynamics. For reviews of cosmic string phenomenology and network evolution see, e.g., Refs. [8,9]. In the velocity-dependent one-scale (VOS) model the network is characterised by a correlation length L(t)and an RMS velocity v(t), obeying dL dt =HL(1+v2) + cv 2,dv dt = (1 −v2)k L−2Hv, where cand kencode loop production and curvature effects. In the drifted spectral framework the effective string tension becomes a function of the drift field, µ(t)=µ0f(s(t)) , so that the effective parameters of the network acquire a controlled time dependence, c→c ( s ( t )) and k→k ( s ( t )). Consequently, loop production, small-scale structure and the gravitational-wave output are modulated by drift evolution. The stochastic gravitational-wave background produced by the network therefore becomes Ωgw(f)−→ Ωgw(f;s(t)) , providing a natural spectral origin for deviations from scale invariance in the nanohertz band probed by pulsar timing arrays. 9 Appendix C: Brane–Antibrane Inflation and the Spectral Drift Dictionary In standard string-theoretic models of brane–antibrane inflation the inflaton Φis identified with the brane separation modulus, and the effective four-dimensional action takes the form SΦ=Zd4x√−g1 2(∂Φ)2−Vbrane(Φ), where Vbrane encodes Coulombic interactions, warping and flux corrections [6,7]. In the drifted spectral framework inflation is instead driven by the universal drift field s ( t )with action Ss=Zd4x√−g1 2(∂s)2−V(s), V (s) = αs2+βs4. This yields the clean identification Φ←→ s . The slow-roll parameters associated with the drift potential are ϵ=1 2V′(s) V(s)2 , η =V′′(s) V(s). For suitable drift profiles and initial conditions one finds ϵ≪ 1and |η| ≪ 1, realising a sustained inflationary phase. Inflation ends dynamically as s ( t )exits the slow-roll regime and relaxes toward the spectral condensate s∗ , providing a geometric reinterpretation of reheating through spectral relaxation rather than brane annihilation. This dictionary clarifies how standard brane–antibrane inflation scenarios arise as a particular realisation of a more universal operator-theoretic mechanism. Appendix D: Internal Dirac Operator and Yukawa Drift In spectral and noncommutative models of particle physics the finite Dirac operator DF encodes fermion masses, Yukawa couplings and flavour structure. At the effective four-dimensional level the fermionic action reads SF=Zd4x√−g⟨ψ, (DM⊗1+γ5⊗DF)ψ⟩. Spectral drift acts on the internal sector by a bounded commutator deformation, (DF)s=DF+s[DF, ϕF],