The Layzer-Irvine equation in theories with non-minimal coupling between matter and curvature

The Layzer-Irvine equation in theories with non-minimal coupling between matter and curvature Cl´audio Filipe Vieira Gomes A Thesis presented for the degree of Master in Physics Centro de F´ısica do Porto Departamento de F´ısica e Astronomia Universidade do Porto Portugal July, 2014 Jury: Prof. Miguel Sousa da Costa Prof. Orfeu Bertolami Neto (advisor) Prof. Jorge Tiago Almeida P´aramos Acknowledgements I would like to thank: •my mother for her support for all these years, and specially for this current one; •my friends for keeping reminding me to have some off-work life; •my advisor, Prof. Orfeu Bertolami, for his guidance and helpful support during my work, without which the latter might not have gone that far; •to Prof. Jorge P´aramos for discussion at the initial phases of our study of non-minimally coupled theories between matter and curvature. iii Resumo Neste trabalho, ´e derivada a equa¸c˜ao de Layzer-Irvine no contexto de teorias alternativas da gravita¸c˜ao que envolvem um acoplamento n˜ao m´ınimo entre mat´eria e geometria. Como aplica¸c˜ao, ´e analisado o caso do enxame de gal´axias Abell 586, por ser notoriamente esfericamente sim´etrico e livre de intera¸c˜oes, recorrendo a alguns perfis de densidade. Este trabalho baseia-se no trabalho desenvolvido na Ref. [1]. Palavras-chave: Relatividade Geral, teorias alternativas da gravita¸c˜ao, cosmologia, enxame A586. v Abstract In this work, the Layzer-Irvine equation is derived in the context of alternative gravitational theories with non-minimal coupling between matter and geometry. As an application, the case of the spherically symmetric cluster Abell 586 is analysed, assuming some matter density profiles. This work is based on work developed in Ref. [1]. Keywords: General Relativity, alternative theories of gravity, cosmology, A586 cluster. vii List of Figures 5.1 Function f2(r) for the cluster A586, in terms of the discussed density profiles. .................................. 17 ix Chapter 2 Non-minimal curvature-matter coupling In GR, the action functional is expressed as S=Z[κR+Lm]√−gd4x(2.1) where Ris the scalar curvature, Lmis the matter Lagrangian density, gis the metric determinant and κ=c4/16πG,Gbeing the Newton’s gravitational constant. In the so-called f(R) theories [4–6], the scalar curvature term in the previous action is replaced by an arbitrary function of it: S=Z1 2f(R) + Lm√−gd4x(2.2) More generally, one can think in a non-minimal coupling between curvature and matter [7]: S=Z1 2f1(R) + (1 + f2(R)) Lm√−gd4x, (2.3) where f1(R) and f2(R) are arbitrary functions of the scalar curvature, R. One notes that, by setting f1(R)=2κRand f2(R) = 0, General Relativity is recovered. Varying the action with respect to the metric yields the field equations [7]: FRµ ν−1 2δµ νf1−(gµσ∇σ∇ν−δµ ν)F= (1 + f2)Tµ ν,(2.4) 3 Chapter 2. Non-minimal curvature-matter coupling 4 with Fi≡dfi/dR(i= 1,2), F≡F1+ 2F2Lm, and Tµν is the energy-momentum tensor of matter defined as Tµν =−2 √−g δ(√−gLm) δ(gµν)(2.5) The Bianchi identities for the Einstein tensor, ∇µGµ ν= 0, and the identity (2∇ν−∇ν2)Fi=Rµν∇µFi(2.6) imply for the expression of Eq.(2.4): ∇µTµ ν= (Lmδµ ν−Tµ ν)∇µln (1 + f2).(2.7) This is one of the fundamental features of the model (2.3) - the non-conservation of the energy-momentum tensor. This property induces an extra force acting on a test particle, which is orthogonal to the fluid four-velocity and can be expressed for a perfect fluid as: fµ=1 ρ+pF2 1 + f2 (Lm+p)∇νR+∇νphµν,(2.8) where hµν =gµν +uµuνis the projection operator. Chapter 3 Perturbed Friedmann-LemaˆıtreRobertson-Walker model We now aim to achieve the prime objective of this work, which is to derive the Layzer-Irvine equation for the non-minimal coupling model described by the action Eq. (2.3). To do so, we follow closely the derivation performed in Refs. [17,19,23,26]. First, we shall consider that the Universe is well described by a perfect fluid, whose energy-momentum tensor reads Tµν = (ρ+p)uµuν+pgµν , where uµ= (1, ui) is the four-velocity under the condition uµuµ=−1 . We also admit an homogeneous and isotropic spacetime described by the Robertson-Walker metric,γij, whose perturbations are given by the line element ds2=−(1 + 2Φ) dt2+a2(t) (1 −2Ψ) γijdxidxj.(3.1) From now on, we consider the choice of the Lagrangian density as Lm=−ρ, which is the most suitable for describing bound systems as discussed in Ref. [29]. The other possible choice, Lm=p, is not very useful, since we assume a pressureless Universe.1Defining the potential velocity in terms of the components of the 41In the context of theories with non-minimal coupling between matter and curvature, one breaks the degeneracy on the Lagrangian density choice that existed in General Relativity [29]. We can 5 Chapter 3. Perturbed Friedmann-Lemaˆıtre-Robertson-Walker model 6 velocity as ui=−∂ivand computing the first order perturbation in the components δTi 0of the stress tensor for a matter dominated epoch, ρ≈ρm, we get [30]: ˙v+˙ Φcv= Φ + δΦc,(3.3) where Φc= ln (1 + f2). This expression can be rewritten in terms of the four-velocity as ˙ui=−∇r(Φ + δΦc−v˙ Φc).(3.4) We shall make the assumption that the flow velocity associated to the expansion rate of the Universe is much smaller than the typical peculiar velocities of cosmic structures. Then ui≈a˙xi≡vm i . Under this condition, Eq. (3.4) can be expressed in a more convenient form ∂ ∂t (avm) = −a∇rΦ + δΦc−˙ Φcv.(3.5) The evolution of matter density perturbations is given in the Fourier space by [30] ˙ δρm+ 3Hδρm= 3 ˙ Ψρm−k2 a2 v aρm,(3.6) where H= ˙a/a is the expansion rate. In the configuration space, using the notation σm≡δρm, only considering peculiar velocities and in the subhorizon approximation (k/a > H) we can write ˙σm+ 3Hσm=−1 a∇x·(ρm−→ vm).(3.7) easily see that the non-conservation of the energy-momentum tensor, Eq. (2.7), strongly depends on the Lagrangian density. For instance, for a pressureless Universe, p≃0, such that ∇νp≃0, the extra force, Eq. (2.8], vanishes for Lm=p, whilst for the other possible, Lm=−ρ, gives fµ=−∇ν(1 + f2(R)) hµν ,(3.2) which is, in general, different from zero. Chapter 3. Perturbed Friedmann-Lemaˆıtre-Robertson-Walker model 7 Finally, from the time component of the non-conservation of the energy-momentum tensor, for a pressureless (w= 0) Universe with Lagrangian density L=−ρm, then2 ˙ρm+ 3Hρm= 0.(3.9) 2The generalisation of the previous result is as follows [31] ˙ρm+ 3H(1 + w)ρm=F2 1 + f2 (α−1) ρm˙ R,(3.8) where α=     1,L=−ρm −w, L=p so that the Lagrangian density has the form L=−αρm(see Ref. [29] for a thorough discussion) and w=p/ρmis the equation of state parameter. Chapter 4 The Layzer-Irvine equation We are now able to derive the Layzer-Irvine equation. We start by contracting Eq. (3.5) with a−→ vmρmd3r, for r=ax, and then integrating over the volume, we get: Zρma−→ vm ∂ ∂t (a−→ vm)d3r=−Za2−→ vmρm∇rΦ + δΦc−˙ Φcvd3r. (4.1) Using Eq. (3.9), the left hand side of Eq. (4.1) can be expressed as ∂ ∂t (a2K), where K≡1/2Rρmv2 md3ris the kinetic energy associated with the peculiar velocity. In its turn, the right hand side can be evaluated performing an integration by parts: −Za2−→ vmρm∇rΦ + δΦc−˙ Φcvd3r= =−Z∇ra2−→ vmρmΦ + δΦc−˙ Φcvd3r+ZΦ + δΦc−˙ Φcv∇r·a2−→ vmρmd3r =−ZΦ + δΦc−˙ Φcva2( ˙σm+ 3Hσm)d3r. (4.2) In the last equality, the first integral corresponds to a total derivative, which therefore vanishes. Moreover, we have resorted to Eq. (3.7). Collecting the results, we get ∂K ∂t + 2HK =−Z(Φ + δΦc−˙ Φcv)∂ ∂t σmd3r.(4.3) We will require that each potential satisfies Poisson’s equation. We shall define the autocorrelation function f(−→ r) of the matter density perturbation field, σm, as 8 Chapter 4. The Layzer-Irvine equation 9 in Ref. [17] as Dσm(−→ r , t)σm(−→ r0, t)E=σ2 mf|−→ r−−→ r0|.(4.4) From which we can define some astrophysical and cosmological scales. We should also note that hσm(−→ r , t)i= 0. Additionally, we use that ∂ ∂t 1 |r−r0|=−H |r−r0|.(4.5) Since we require that the potentials satisfy the Poisson’s equation, then any of them can be expressed in terms of the matter density perturbation as ϕ=−GZσm(r0, t) |r−r0|d3r0.(4.6) Bearing this in mind, the right hand side of Eq. (4.3) can be expressed as −Zϕ∂ ∂t(σmd3r) = GZ∂ ∂t(σmd3r)Zσ0 m |r−r0|d3r0 =GZ∂ ∂t(σ0 md3r0)Zσm |r−r0|d3r, (4.7) where σm≡σm(−→ r , t) and σ0 m≡σm(−→ r0, t). Now, recalling the result (4.5), the expression (4.7) can be written as GZ∂ ∂t(σ0 md3r0)Zσm |r−r0|d3r=−(˙ Uϕ+HUϕ),(4.8) where Uϕ≡ −G 2Z Z σmσ0 m |r−r0|d3rd3r0=1 2Zϕ σmd3r. (4.9) Note that the non-minimal coupling effects on the gravitational coupling in the case of clusters are negligible, so that the effective gravitational constant, as defined in Ref. [30], obeys Geff ≈G. Now we can write the Layzer-Irvine equation in the form ∂ ∂t(K+UΦ+UδΦc−˙ Φcv) + H(2K+UΦ+UδΦc−˙ Φcv) = 0,(4.10) which can be rearranged into a more convenient form: Chapter 4. The Layzer-Irvine equation 10 ∂ ∂t(K+U+UNMC ) + H(2K+U+UNMC)=0,(4.11) with U≡UΦand UNMC ≡UδΦc−˙ Φcv=1 2ZδΦc−˙ Φcvσmd3r. (4.12) We see that the non-minimal coupling between matter and geometry induces an extra term in the standard generalised cosmic virial theorem, which can account for the ”dark components” effects on several systems. For a relaxed astrophysical system which no longer evolves in time, we get a generalised version of the virial theorem for these gravitational theories: 2K+U+UNMC = 0.(4.13) From this equation we can proceed to analyse clusters of galaxies and impose some constraints on the non-minimal model. Clearly, any deviation from the usual virial ratio K/U =−1/2 can be expressed in terms of the quotient: UNMC U=−2K U−1.(4.14) Chapter 5 The Abell 586 cluster We consider now the well known relaxed cluster Abell 586, following up the procedure developed in Refs. [19, 22]. We assume the obvious cases of the top-hat and isothermal spheres density profiles. In order to test the sensitivity of the results, we adopt tentatively the Navarro-Frenk-White (NFW) density profile [32], even though this is known to be a profile obtained from N-body simulations for galaxies within the Cosmological Standard Model, ΛCDM (which assumes that dark energy is parametrised by a cosmological constant, Λ, dark matter is taken to be non-relativistic). The NFW model is, therefore, somewhat inacurate for clusters. As we shall see, results for UNMC are dependent on the density profile choice, even though not strongly so. It is relevant to bear in mind that the considered density is exclusively baryonic. 5.1 Top-hat density profile In this case, one assumes that the kinetic and potential energy densities are well described by [19] ρK≃9 8π M R3σ2 v,(5.1) ρW≃ − 3 8π G M2 hRiR3,(5.2) 11 5.2. Navarro-Frenk-White density profile 12 where Me R are the total baryonic mass and radius of Abell 586, which include galaxies and intra-cluster gas, σvis the velocity dispersion and hRiis the mean intergalactic radius. Since the case we are studying has spherical symmetry, the total volume is simply V= 4πR3/3, and the ratio between total peculiar kinetic and potential energies is the same as the ratio of the energy densities, thus: K U≡ρK ρW =−3σ2 vhRi G M .(5.3) 5.2 Navarro-Frenk-White density profile The Navarro-Frenk-White model is very useful in realistic N-body simulations within the ΛCDM paradigm. It is characterised by the energy density [32]: ρ(r) = ρ0 r r01 + r r02,(5.4) where ris the distance from the centre, ρ0and r0are the density and shape parameters, respectively. As described in Ref. [22], the total mass and mean radius can be calculated by integrating Eq. (5.4) over the volume: M= 4πZR 0 ρ(r)r2dr = 4πr3 0ρ0ln 1 + R r0−R R+r0,(5.5) hRi=r0hR r0−2 ln 1 + R r0+R R+r0i hln 1 + R r0−R R+r0i.(5.6) We point out that r0can be numerically calculated from the mean radius, hRi. Thus, the density parameter, ρ0, is immediatly solved numerically. From these quantities we can now estimate the kinetic and potential energy densities assuming a constant average velocity, σv[22] : ρK=9 8π M R3σ2 v,(5.7) Chapter 6. Conclusions 19 different density profiles (top-hat, Navarro-Frenk-White, and isothermal spheres), the ratio between the non-minimal coupling potential energy and the baryonic energy potential, UNMC/U, is of the order of ∼7. One can also conclude from the values in Table 5.2 that the velocity dispersion value from X-ray luminosity is not very reliable, as discussed before in Ref. [22]. 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