Matter stability is a necessary condition to have a cosmologically viable model. Modified gravity
in the spirit of f(R) theories suffers from matter instability in some subdomains of the model
parameter space. It has been shown recently that the late-time cosmic speedup can be explained
through an f(R)-modified induced gravity program. In this paper, we study the issue of matter
instability in a braneworld setup with modified induced gravity.

1. Introduction

There are many lines of astronomical evidence supporting the idea that our universe is currently undergoing a speedup expansion [1–4]. Several approaches are proposed in order to explain the origin of this novel phenomenon. These approaches can be classified in two main categories: models based on the notion of dark energy which modify the matter sector of the gravitational field equations and those models that modify the geometric part of the field equations are generally dubbed as dark geometry in the literature [5–13]. From a relatively different viewpoint (but in the spirit of dark geometry proposal), the braneworld model proposed by Dvali, Gabadadze, and Porrati (DGP) [14] explains the late-time cosmic speedup phase in its self-accelerating branch without recourse to dark energy [15, 16]. However, existence of ghost instabilities in this branch of the solutions makes its unfavorable in some senses [17, 18]. Fortunately, it has been revealed recently that the normal, ghost-free DGP branch has the potential to explain late-time cosmic speedup if we incorporate possible modification of the induced gravity on the brane in the spirit of f(R) theories [19–24]. This extension can be considered as a manifestation of the scalar-tensor gravity on the brane. Some features of this extension are studied recently [25–29].

Modified gravity in the spirit of f(R) theories have the capability to provide a unified gravitational alternative to dark energy and inflation [30–37]. A number of viable modified f(R) gravities are proposed in recent years (see [38, 39] and references therein). The cosmological viability of these theories is a necessary condition, and in this respect, there are important criteria for viability such as the fulfillment of the solar system tests. Among these requirements, one of the most important ones is related to the so-called matter instability [40–57] in f(R) gravity. Matter instability is related to the fact that spherical body solution in general relativity may not be the solution in modified theory in general. This instability may appear when the energy density or the curvature is large compared with the average one in the universe, as is the case inside of a star [58]. In a simple term, matter instability means that the curvature inside a matter sphere becomes very large, leading to a very strong gravitational field. It was indicated that such matter instability may be dangerous in the relativistic star formation processes [59–61] due to the appearance of the corresponding singularity. In this respect, for a model to be cosmologically viable, it is necessary to have matter stability in the model. For a detailed study of the issue of matter instability in f(R) theories, see [40–58].

Since the f(R)-modified induced gravity (brane f(R) gravity) has the capability to bring the normal DGP solutions to be self-accelerating, it is desirable to see whether this model is cosmologically viable from matter instability viewpoint. So, this letter is devoted to the issue of matter instability in a brane f(R) gravity.

Modified gravity in the form of f(R) theories is derived by generalization of the Einstein-Hilbert action so that R (the Ricci scalar) is replaced by a generic function f(R) in the action
(2.1)S=∫d4x-g(f(R)2κ2+ℒm),
where ℒm is the matter Lagrangian and κ2=8πG. Varying this action with respect to the metric gives
(2.2)Gμν=κ2Tμν(tot)=κ2(Tμν(m)+Tμν(f))=κ2T~μν(m)+T~μν(f)f',
where T~μν(m)=diag(ρ,-p,-p,-p) is the stress-energy tensor for standard matter, which is assumed to be a perfect fluid and by definition f'≡df/dR. Also, T~μν(f) is the stress-energy tensor attributed to the curvature defined as follows:
(2.3)T~μν(f)=12gμν[f(R)-Rf′]+f′;αβ(gαμgβν-gαβgμν).

By substituting a flat FRW metric into the field equations, one achieves the analogue of the Friedmann equations as follows [30–37]:
(2.4)3f'H2=κ2ρm+[12(f(R)-Rf′)-3Hf′˙]-2f'H˙=κ2ρm+R˙2f′′′+(R¨-HR˙)f′′,
where a dot marks the differentiation with respect to the cosmic time. In the next step, following [28, 29] we suppose that the induced gravity on the DGP brane is modified in the spirit of f(R) gravity. The action of this DGP-inspired f(R) gravity is given by
(2.5)S=12κ53∫d5x-gℛ+∫d4x-q(f(R)2κ2+ℒm),
where gAB is a five-dimensional bulk metric with Ricci scalar ℛ, while qab is an induced metric on the brane with induced Ricci scalar R. The Friedmann equation in the normal branch of this scenario is written as [28, 29]
(2.6)3f'H2=κ2(ρm+ρ(f))-3Hrc,
where rc=G(5)/G(4)=κ52/2κ2 is the DGP crossover scale which has the dimension of [length] and marks the IR (infrared) behavior of the DGP model. The Raychaudhuri equation is written as follows:
(2.7)H˙(1+12Hrcf′)=-κ2ρm2f'-R˙2f′′′+(R¨-HR˙)f′′2f'.

To achieve this equation, we have used the continuity equation for ρ(f) as
(2.8)ρ˙(f)+3H(ρ(f)+p(f)+R˙f′′rc(f′)2)=κ2ρmR˙f′′(f′)2,
where the energy density and pressure of the curvature fluid are defined as follows:
(2.9)ρ(f)=1κ2(12[f(R)-Rf′]-3Hf′˙),p(f)=1κ2(2Hf′˙+f′¨-12[f(R)-Rf′]).

After presentation of the necessary field equations, in what follows we study the issue of matter instability and cosmological viability in this setup.

3. The Issue of Matter Instability

In 4 dimensions, the conditions f'(R)>0 and f′′(R)>0 are necessary conditions for f(R) theories to be free from ghosts and other instabilities [40–44, 62]. In our brane f(R) scenario, in addition to f(R), there is another piece of information in the action (2.5) (the DGP character of the model) which should be taken into account when discussing the issue of instabilities. To study possible instabilities in this setup, we proceed as follows: variation of the action (2.5) with respect to the metric yields the induced modified Einstein equations on the brane
(3.1)Gαβ=1M56𝒮αβ-ℰαβ,
where ℰαβ (which we neglect it in our forthcoming arguments) is the projection of the bulk Weyl tensor on the brane
(3.2)ℰαβ=(5)CRNSMnMnRgαNgβS,
where nM is a unit vector normal to the brane, and 𝒮αβ as the quadratic energy-momentum correction into the Einstein field equations is defined as follows:
(3.3)𝒮αβ=-14ταμτβμ+112τταβ+18gαβτμντμν-124gαβτ2.ταβ as the effective energy-momentum tensor localized on the brane is defined as [25–27]
(3.4)ταβ=-mp2f′(R)Gαβ+mp22[f(R)-Rf′(R)]gαβ+Tαβ+mp2[∇α∇βf′(R)-gαβf′(R)].

Following [40–44], the trace of (3.1), which can be interpreted as the equation of motion for f'(R), is obtained as
(3.5)R=56rc2([2f-Rf′]2+9(□f′)2+6Rf′□f′-12f□f′)+53rc2mp2(Rf′-2f+3□f′)T+524M56T2.

We parameterize the deviation from Einstein gravity as
(3.6)f(R)=R+ϵφ(R),
where ϵ is a small parameter with the dimension of an inverse-squared length, and φ is arranged to be dimensionless. Typically, ϵ≈H02≃10-66(eV)2 (see, for instance, [42]). By evaluating □f' as
(3.7)□f'=ϵ□φ′(R)=ϵ(φ′′′∇αR∇αR+φ′′□R),
(3.5) can be rewritten as follows:
(3.8)R=56rc2([φ′′′∇αR∇aR+φ′′□R][φ′′′∇αR∇αR+φ′′□R]2[R+ϵ(2φ-Rφ′)]2+9ϵ2[φ′′′∇αR∇aR+φ′′□R]2+6ϵ(ϵ(Rφ′-2φ)-R)[φ′′′∇αR∇aR+φ′′□R][φ′′′∇αR∇αR+φ′′□R]2)+524M56T2+53rc2mp2(R(ϵφ′-1)-2ϵφ+3ϵ[φ′′′∇αR∇aR+φ′′□R])T.

This equation to first order of ϵ gives
(3.9)R=524M56T2+56rc2([R2+2ϵR(2φ-Rφ′)]-6ϵR[φ′′′∇αR∇aR+φ′′□R])+53rc2mp2(R(ϵφ′-1)-2ϵφ+3ϵ[φ′′′∇αR∇aR+φ′′□R])T.

Now we consider a small region of spacetime in the weak-field regime in which curvature and the metric can locally be approximated by
(3.10)R=-κ2T+R1,gab=ηab+hab,
respectively, where R1 is curvature perturbation, and ηab is the Minkowski metric. In this case, the metric can be approximately taken as a flat one, so □=∂t2-∇2 and (∇αR)(∇αR)=R˙2-(∇R)2. Now, (3.9) up to first order of R1 gives (we set for simplicity κ2=1)
(3.11)R1-T=524M56T2+56rc2[T2-2TR1+4ϵφR1-4ϵφT-2ϵφ′T2+4ϵTφ′R1]+5ϵrc2T[φ′′′(R)(T˙2-2T˙R˙1-(∇T)2+2∇R1∇T)+φ′′(R)(R¨1-T¨-∇2R1+∇2T)]-5ϵrc2R1×[φ′′′(R)(T˙2-(∇T)2)+φ′′(R)(∇2T-T¨)]+5rc23mp2×(3ϵ[φ′′′(R)(T˙2-2T˙R˙1-(∇T)2+2∇R1∇T)+(R1-T)(ϵφ′-1)-2ϵφ+φ′′(R)(R¨1-T¨-∇2R1+∇2T)])T.

This relation can be recast in the following suitable form:
(3.12)R¨1-∇2R1+φ′′′φ′′(T˙2-(∇T)2+2∇R1∇T-2T˙R˙1)-[110ϵrc2Tφ′′+1ϵφ′′-13Tφ′′(φ+92Tφ′)]R1-12Tφ′′[φ′′′(T˙2-(∇T)2)+φ′′(∇2T-T¨)]R1=T¨-∇2T-(R¨1-∇2R1+(∇2T-T¨))+13φ′′(Tφ′+2φ)-1ϵφ′′(110rc2+5T3).

The coefficient of R1 in the fourth term on the left-hand side is the square of an effective mass defined as
(3.13)meff2=-110ϵrc2Tφ′′-1ϵφ′′+13Tφ′′(φ+92Tφ′)-12Tφ′′[φ′′′(T˙2-(∇T)2)+φ′′(∇2T-T¨)].

This quantity is dominated by the term 1/ϵφ′′ due to the extremely small value of ϵ needed for these theories. It is therefore obvious that the theory will be stable (i.e., meff2>0) if φ′′=f′′<0, while an instability arises if this effective mass squared is negative, that is, if φ′′=f′′>0. Based on this fact and as an example, the f(R)=R+γR-n function (with γ as a positive quantity) in the spirit of normal DGP braneworld scenario suffers from a matter instability for n>0 and n<-1. For this kind of f(R) function, γ plays the same role as ϵ in (3.6) and is supposed to be positive (note that ϵ is a small parameter with dimension of an inverse length squared), so φ(R)=R-n. The condition for matter stability φ′′=f′′<0 leads to -1<n<0.

As another important point, we focus on the stability of the de Sitter accelerating solution under small homogeneous perturbations in the normal branch (see [28, 62, 63] for a similar argument). It is useful to rewrite the Friedmann equation corresponding to a de Sitter brane with Hubble rate H0 in a form that exhibits the effect of an extra dimension on a 4D f(R) model as follows [28, 62, 63]:
(3.14)H02=H(4)2+1-1+(2/3)f0′f02f0′2.

Subscript 0 stands for quantities evaluated in the de Sitter space time. We also note that the de Sitter brane is described by the scalar factor a(t)=a0exp(ξt) which leads to R0=12ξ2. H(4)2 is defined as
(3.15)H(4)2=f0′6f0.

Therefore, the presence of the extra dimension implies a shift on the Hubble rate. One can perturb the Hubble parameter up to the first order as δH=H(t)-H0. Also by a perturbed Friedmann equation based on (2.6), one can achieve an evolution equation for δH as [64]
(3.16)δH¨+3H0δH˙+(Meff)2δH=0.

The stability condition for the de Sitter solution in the DGP normal branch is positivity of the effective mass squared, (Meff)2>0. Now (Meff)2 can be written as the sum of three terms (Meff)2=m(4)2+mback2+mpert2. In the 4D version of the f(R) gravity, this summation reduces to [64]
(3.17)meff2=m(4)2=f0′2-2f0f′′3f0′f′′.

In the braneworld version, we except the crossover distance to affect the effective mass. In this respect, mback2 is a purely background effect due to the shift on the Hubble parameter with respect to the standard 4D case, while mpert2 is a purely perturbative extradimensional effect [28, 63]. In our setup, these quantities are defined as follows:
(3.18)mback2=-2rc2f′02[1-1+23rc2f0′2f0],mpert2=-f′03f′′[1-1+23rc2f0′2f0]-1.

The de Sitter brane is close to the standard 4D regime as long as H02~H(4)2, which leads to
(3.19)|3(1-1+(2/3)f0′f0)f0f0′|≪1.

The assumption that we are slightly perturbing the Hilbert-Einstein action of the brane, that is, f0~R0>0, and also the positivity of the effective gravitational constant on the brane at late time, that is, f0′>0, implies that f0f0′≫1 for the last inequality. Now the stability of the de Sitter accelerating solution under small homogeneous perturbations in the normal DGP branch of the model (since mback2>0 and mpert2<0) is guaranteed if meff2>m(4)2 which leads to the following condition (for more details, see [28, 63, 64]):
(3.20)f0′2<4f0f0′′.

Based on this condition, f(R)=R+γR-n function in the spirit of the normal DGP braneworld exhibits a de Sitter stability if the following condition is satisfied:
(3.21)R0n+1-n(3n+4)γ2R0-(n+1)<nγ(6+4n).

Figure 1 shows the behavior of M≡meff2-m(4)2 versus n and γ. The de Sitter phase is stable in this setup if n≲-4.12. This is shown with more resolution in Figure 2. Note that the matter stability in this induced gravity braneworld scenario occurs in those values of n that the corresponding de Sitter phase is not stable.

The behavior of M≡meff2-m(4)2 versus n and γ.

Stability of the de Sitter phase. The de Sitter phase is stable for n≲-4.12.

4. Summary and Conclusion

Matter stability is a necessary condition for cosmological viability of a gravitational theory. Recently, it has been shown that the normal, non-self-accelerating branch of the DGP cosmological solutions self-accelerates if the induced gravity on the brane is modified in the spirit of f(R) gravity. In this letter, we have studied the issue of matter stability in an induced gravity, brane-f(R) scenario. We obtained the condition for matter stability in this setup via a perturbative scheme, and we applied our condition for an specific model of the type f(R)=R+γR-n. For this type of modified induced gravity, the matter is stabilized on the brane for -1<n<0. We have also studied the stability of the de Sitter phase for this type of modified induced gravity. For this type of the modified induced gravity, the de Sitter phase is stable for n≲-4.12. Albeit for these values of n, matter is not stable. So, these types of modified induced gravity are not suitable candidates for late-time cosmological evolution. We note however that other types of modified induced gravity such as f(R)=Rnexp(η/R) with η a constant have simultaneous matter stability and stable de Sitter phase in some subspaces of the model parameter space (see [65]).

Acknowledgment

This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under Research Project no. 1/2782-37.

PerlmutterS.AlderingG.GoldhaberG.KnopR. A.NugentP.CastroP. G.DeustuaS.FabbroS.GoobarA.GroomD. E.HookI. M.KimA. G.KimM. Y.LeeJ. C.NunesN. J.PainR.PennypackerC. R.QuimbyR.LidmanC.EllisR. S.IrwinM.McMahonR. G.Ruiz-LapuenteP.WaltonN.SchaeferB.BoyleB. J.FilippenkoA. V.MathesonT.FruchterA. S.PanagiaN.NewbergH. J. M.CouchW. J.Measurements of Ω and Λ from 42 high-redshift SupernovaeRiessA. G.FilippenkoA. V.ChallisP.Observational evidence from supernovae for an accelerating universe and a cosmological constantSpergelD. N.First Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parametersHinshawG.WMAP CollaborationFive-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: data processing, sky maps, and basic resultsCopelandE. J.SamiM.TsujikawaS.Dynamics of dark energySahniV.StarobinskyA.Reconstructing dark energyPadmanabhanT.Dark energy and gravityKleinertH.SchmidtH.-J.Cosmology with curvature-saturated gravitational Lagrangian R/1+l4R2NojiriS.OdintsovS. D.Introduction to modified gravity and gravitational alternative for dark energySotiriouT. P.FaraoniV.f(R) theories of gravityCapozzielloS.FrancavigliaM.Extended theories of gravity and their cosmological and astrophysical applicationsDurrerR.MaartensR.Dark energy and dark gravity: theory overviewCapozzielloS.SalzanoV.Cosmography and large scale structure by f(R) gravity: new resultshttp://arxiv.org/abs/0902.0088DvaliG.GabadadzeG.PorratiM.4D gravity on a brane in 5D Minkowski spaceDeffayetC.Cosmology on a brane in Minkowski bulkLueA.The phenomenology of Dvali-Gabadadze-Porrati cosmologiesKoyamaK.Ghosts in the self-accelerating universede RhamC.TolleyA. J.Mimicking Λ with a spin-two ghost condensateSahniV.ShtanovY.Braneworld models of dark energyLueA.StarkmanG. D.How a brane cosmological constant can trick us into thinking that w < -1ChimentoL. P.LazkozR.MaartensR.QuirosI.Crossing the phantom divide without phantom matterLazkozR.MaartensR.MajerottoE.Observational constraints on phantomlike braneworld cosmologiesMaartensR.MajerottoE.Observational constraints on self-accelerating cosmologyBouhmadi-LopezM.Phantom-like behaviour in dilatonic brane-world scenario with induced gravityNozariK.PourghassemiM.Crossing the phantom divide line in a Dvali-Gabadadze-Porrati-inspired F(R,phi) gravitySaavedraJ.VasquezY.Effective gravitational equations on brane world with induced gravity described by f(R) termBorzouA.SepanjiH. R.ShahidiS.YousefiR.Brane f(R) gravityBouhmadi-LopezM.Self-accelerating the normal DGP branchNozariK.KianiF.Dynamical-screening and the phantom-like effects in a DGP-inspired f(R, phi) modelNojiriS.OdintsovS. D.Modified f(R) gravity consistent with realistic cosmology: from a matter dominated epoch to a dark energy universeAmendolaL.PolarskiD.TsujikawaS.Are f(R) dark energy models cosmologically viable?NojiriS.OdintsovS. D.Modified gravity and its reconstruction from the universe expansion historyCapozzielloS.NojiriS.OdintsovS. D.TroisiA.Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phaseAmendolaL.GannoujiR.PolarskiD.TsujikawaS.Conditions for the cosmological viability of f(R) dark energy modelsAmendolaL.TsujikawaS.Phantom crossing, equation-of-state singularities, and local gravity constraints in f(R) modelsFayaS.NesserisS.PerivolaropoulosL.Can f(R) modified gravity theories mimic a LambdaCDM cosmology?LiB.BarrowJ. D.Cosmology of f(R) gravity in the metric variational approachNojiriS.OdintsovS. D.Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant modelsDe FeliceA.TsujikawaS.f(R) theoriesDolgovA. D.KawasakiM.Can modified gravity explain accelerated cosmic expansion?NojiriS.OdintsovS. D.Modified gravity with negative and positive powers of curvature: unification of inflation and cosmic accelerationFaraoniV.Matter instability in modified gravityNojiriS.OdintsovS. D.Modified gravity with lnR terms and cosmic accelerationNojiriS.OdintsovS. D.Newton law corrections and instabilities in f(R) gravity with the effective cosmological constant epochChibaT.1/R gravity and scalar-tensor gravityNojiriS.OdintsovS. D.Modified Gauss-Bonnet theory as gravitational alternative for dark energyHuW.SawickiI.Models of f(R) cosmic acceleration that evade solar-system testsStarobinskyA. A.Disappearing cosmological constant in f(R) gravityApplebyS. A.BattyeR. A.Do consistent F(R) models mimic general relativity plus Λ?TsujikawaS.Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraintsDeruelleN.SasakiM.SendoudaY.“Detuned” f(R) gravity and dark energyCognolaG.ElizaldeE.NojiriS.OdintsovS. D.SebastianiL.ZerbiniS.Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansionNojiriS.OdintsovS. D.Modified f(R) gravity unifying R^{m} inflation with the ΛCDM epochLinderE. V.Exponential gravityDe FeliceA.TsujikawaS.Construction of cosmologically viable f(G) gravity modelsDe FeliceA.TsujikawaS.Solar system constraints on f(G) gravity modelsDe FeliceA.MotaD. F.TsujikawaS.Matter instabilities in general Gauss-Bonnet gravityBambaK.NojiriS.OdintsovS. D.Time-dependent matter instability and star singularity in f(R) gravityKobayashiT.MaedaK.-I.Relativistic stars in f(R) gravity, and absence thereofKobayashiT.MaedaK. I.Can higher curvature corrections cure the singularity problem in f(R) gravity?DevA.JainD.JhinganS.NojiriS.SamiM.ThongkoolI.Delicate f(R) gravity models with a disappearing cosmological constant and observational constraints on the model parametersTsujikawaS.Modified gravity models of dark energyBouhamdi-LopezM.f(R) brane cosmologyhttp://arxiv.org/abs/1001.3028FaraoniV.NadeauS.Stability of modified gravity modelsNozariK.KianiF.Cosmological dynamics with modified induced gravity on the normal DGP branchhttp://arxiv.org/abs/1008.4240