Matter Instability in Theories with Modified Induced Gravity

Abstract

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.

2. DGP-Inspired 𝑓(𝑅) Gravity

2.1. The Basic Equations

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 𝑆=𝑑⁴𝑥√𝑓(𝑅)−𝑔2𝜅²+ℒ_𝑚, (2.1) where ℒ_𝑚 is the matter Lagrangian and 𝜅²=8𝜋𝐺. Varying this action with respect to the metric gives 𝐺_𝜇𝜈=𝜅²𝑇^(tot)_𝜇𝜈=𝜅²𝑇^(𝑚)_𝜇𝜈+𝑇^(𝑓)_𝜇𝜈=𝜅²𝑇^(𝑚)_𝜇𝜈+𝑇^(𝑓)_𝜇𝜈𝑓^′, (2.2) where 𝑇^(𝑚)_𝜇𝜈=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^(𝑓)_𝜇𝜈 is the stress-energy tensor attributed to the curvature defined as follows: 𝑇^(𝑓)_𝜇𝜈=12𝑔_𝜇𝜈𝑓(𝑅)−𝑅𝑓^′+𝑓^′;𝛼𝛽𝑔_𝛼𝜇𝑔_𝛽𝜈−𝑔_𝛼𝛽𝑔_𝜇𝜈. (2.3)

By substituting a flat FRW metric into the field equations, one achieves the analogue of the Friedmann equations as follows [30–37]: 3𝑓^′𝐻²=𝜅²𝜌_𝑚+12𝑓(𝑅)−𝑅𝑓^′̇−3𝐻𝑓^′−2𝑓^′𝐻̇=𝜅²𝜌_𝑚+𝑅̇²𝑓^′′′+𝑅̈−𝐻𝑅̇𝑓^′′, (2.4) 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 1𝑆=2𝜅³₅𝑑⁵𝑥√−𝑔ℛ+𝑑⁴𝑥√𝑓(𝑅)−𝑞2𝜅²+ℒ_𝑚, (2.5) where 𝑔_𝐴𝐵 is a five-dimensional bulk metric with Ricci scalar ℛ, while 𝑞_𝑎𝑏 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] 3𝑓^′𝐻²=𝜅²𝜌_𝑚+𝜌^(𝑓)−3𝐻𝑟_𝑐, (2.6) where 𝑟_𝑐=𝐺⁽⁵⁾/𝐺⁽⁴⁾=𝜅²₅/2𝜅² 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: 𝐻̇1+12𝐻𝑟_𝑐𝑓^′𝜅=−²𝜌_𝑚2𝑓^′−𝑅̇²𝑓^′′′+𝑅̈−𝐻𝑅̇𝑓^′′2𝑓^′. (2.7)

To achieve this equation, we have used the continuity equation for 𝜌^(𝑓) as 𝜌̇^(𝑓)+3𝐻𝜌^(𝑓)+𝑝^(𝑓)+𝑅̇𝑓^′′𝑟_𝑐𝑓^′2=𝜅²𝜌_𝑚𝑅̇𝑓^′′𝑓^′2, (2.8) where the energy density and pressure of the curvature fluid are defined as follows: 𝜌^(𝑓)=1𝜅²12𝑓(𝑅)−𝑅𝑓^′̇−3𝐻𝑓^′,𝑝^(𝑓)=1𝜅²̇2𝐻𝑓^′+̈𝑓^′−12𝑓(𝑅)−𝑅𝑓^′. (2.9)

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 𝐺_𝛼𝛽=1𝑀⁶₅𝒮_𝛼𝛽−ℰ_𝛼𝛽, (3.1) where ℰ_𝛼𝛽 (which we neglect it in our forthcoming arguments) is the projection of the bulk Weyl tensor on the brane ℰ_𝛼𝛽=⁽⁵⁾𝐶^𝑀_𝑅𝑁𝑆𝑛_𝑀𝑛^𝑅𝑔^𝑁_𝛼𝑔^𝑆_𝛽, (3.2) where 𝑛^𝑀 is a unit vector normal to the brane, and 𝒮_𝛼𝛽 as the quadratic energy-momentum correction into the Einstein field equations is defined as follows: 𝒮_𝛼𝛽1=−4𝜏_𝛼𝜇𝜏^𝜇_𝛽+112𝜏𝜏_𝛼𝛽+18𝑔_𝛼𝛽𝜏_𝜇𝜈𝜏^𝜇𝜈−1𝑔24^𝛼𝛽𝜏². (3.3)𝜏_𝛼𝛽 as the effective energy-momentum tensor localized on the brane is defined as [25–27] 𝜏_𝛼𝛽=−𝑚²_𝑝𝑓^′(𝑅)𝐺_𝛼𝛽+𝑚²_𝑝2𝑓(𝑅)−𝑅𝑓^′(𝑅)𝑔_𝛼𝛽+𝑇_𝛼𝛽+𝑚²_𝑝∇_𝛼∇_𝛽𝑓^′(𝑅)−𝑔_𝛼𝛽𝑓^′(𝑅). (3.4)

Following [40–44], the trace of (3.1), which can be interpreted as the equation of motion for 𝑓^′(𝑅), is obtained as 5𝑅=6𝑟²_𝑐2𝑓−𝑅𝑓^′2+9□𝑓^′2+6𝑅𝑓^′□𝑓^′−12𝑓□𝑓^′+53𝑟²_𝑐𝑚²_𝑝𝑅𝑓^′−2𝑓+3□𝑓^′5𝑇+24𝑀⁶₅𝑇². (3.5)

We parameterize the deviation from Einstein gravity as 𝑓(𝑅)=𝑅+𝜖𝜑(𝑅), (3.6) where 𝜖 is a small parameter with the dimension of an inverse-squared length, and 𝜑 is arranged to be dimensionless. Typically, 𝜖≈𝐻²₀≃10⁻⁶⁶(eV)² (see, for instance, [42]). By evaluating □𝑓^′ as □𝑓^′=𝜖□𝜑^′(𝑅)=𝜖𝜑^′′′∇^𝛼𝑅∇_𝛼𝑅+𝜑^′′□𝑅, (3.7) (3.5) can be rewritten as follows: 5𝑅=6𝑟²_𝑐𝑅+𝜖2𝜑−𝑅𝜑^′2+9𝜖²𝜑^′′′∇^𝛼𝑅∇_𝑎𝑅+𝜑^′′□𝑅²+6𝜖𝜖𝑅𝜑^′−2𝜑−𝑅𝜑^′′′∇^𝛼𝑅∇_𝑎𝑅+𝜑^′′+5□𝑅24𝑀⁶₅𝑇²+53𝑟²_𝑐𝑚²_𝑝𝑅𝜖𝜑^′−1−2𝜖𝜑+3𝜖𝜑^′′′∇^𝛼𝑅∇_𝑎𝑅+𝜑^′′□𝑅𝑇. (3.8)

This equation to first order of 𝜖 gives 5𝑅=24𝑀⁶₅𝑇²+56𝑟²_𝑐𝑅²+2𝜖𝑅2𝜑−𝑅𝜑^′−6𝜖𝑅𝜑^′′′∇^𝛼𝑅∇_𝑎𝑅+𝜑^′′+5□𝑅3𝑟²_𝑐𝑚²_𝑝𝑅𝜖𝜑^′−1−2𝜖𝜑+3𝜖𝜑^′′′∇^𝛼𝑅∇_𝑎𝑅+𝜑^′′□𝑅𝑇. (3.9)

Now we consider a small region of spacetime in the weak-field regime in which curvature and the metric can locally be approximated by 𝑅=−𝜅²𝑇+𝑅₁,𝑔_𝑎𝑏=𝜂_𝑎𝑏+ℎ_𝑎𝑏, (3.10) respectively, where 𝑅₁ is curvature perturbation, and 𝜂_𝑎𝑏 is the Minkowski metric. In this case, the metric can be approximately taken as a flat one, so □=𝜕²_𝑡−∇² and (∇^𝛼𝑅)(∇_𝛼𝑅)=𝑅̇²−(∇𝑅)². Now, (3.9) up to first order of 𝑅₁ gives (we set for simplicity 𝜅²=1) 𝑅₁5−𝑇=24𝑀⁶₅𝑇²+56𝑟²_𝑐𝑇²−2𝑇𝑅₁+4𝜖𝜑𝑅₁−4𝜖𝜑𝑇−2𝜖𝜑^′𝑇²+4𝜖𝑇𝜑^′𝑅₁+5𝜖𝑟²_𝑐𝑇𝜑^′′′(𝑅)𝑇̇²−2𝑇̇𝑅̇₁−(∇𝑇)²+2∇𝑅₁∇𝑇+𝜑^′′(𝑅)𝑅̈₁−𝑇̈−∇²𝑅₁+∇²𝑇−5𝜖𝑟²_𝑐𝑅₁×𝜑^′′′(𝑅)𝑇̇²−(∇𝑇)²+𝜑^′′(𝑅)∇²𝑇−𝑇̈+5𝑟²_𝑐3𝑚²_𝑝×3𝜖𝜑^′′′(𝑅)𝑇̇²−2𝑇̇𝑅̇₁−(∇𝑇)²+2∇𝑅₁∇𝑇+𝑅₁−𝑇𝜖𝜑^′−1−2𝜖𝜑+𝜑^′′(𝑅)𝑅̈₁−𝑇̈−∇²𝑅₁+∇²𝑇𝑇. (3.11)

This relation can be recast in the following suitable form: 𝑅̈₁−∇²𝑅₁+𝜑^′′′𝜑^′′𝑇̇²−(∇𝑇)²+2∇𝑅₁∇𝑇−2𝑇̇𝑅̇₁−110𝜖𝑟²_𝑐𝑇𝜑^′′+1𝜖𝜑^′′−13𝑇𝜑^′′9𝜑+2𝑇𝜑^′𝑅₁−12𝑇𝜑^′′𝜑^′′′𝑇̇²−(∇𝑇)²+𝜑^′′∇²𝑇−𝑇̈𝑅₁=𝑇̈−∇²𝑇−𝑅̈₁−∇²𝑅₁+∇²𝑇̈+1𝑇−3𝜑^′′𝑇𝜑^′−1+2𝜑𝜖𝜑^′′110𝑟²_𝑐+5𝑇3. (3.12)

The coefficient of 𝑅₁ in the fourth term on the left-hand side is the square of an effective mass defined as 𝑚²_eff1=−10𝜖𝑟²_𝑐𝑇𝜑^′′−1𝜖𝜑^′′+13𝑇𝜑^′′9𝜑+2𝑇𝜑^′−12𝑇𝜑^′′𝜑^′′′𝑇̇²−(∇𝑇)²+𝜑^′′∇²𝑇−𝑇̈. (3.13)

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., 𝑚²_eff>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]: 𝐻²₀=𝐻²₍₄₎+1−1+(2/3)𝑓^′₀𝑓₀2𝑓^′₀². (3.14)

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. 𝐻²₍₄₎ is defined as 𝐻²₍₄₎=𝑓^′₀6𝑓₀. (3.15)

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] 𝛿𝐻̈+3𝐻₀𝛿𝐻̇+𝑀_eff²𝛿𝐻=0. (3.16)

The stability condition for the de Sitter solution in the DGP normal branch is positivity of the effective mass squared, (𝑀_eff)²>0. Now (𝑀_eff)² can be written as the sum of three terms (𝑀_eff)²=𝑚²₍₄₎+𝑚²_back+𝑚²_pert. In the 4D version of the 𝑓(𝑅) gravity, this summation reduces to [64] 𝑚²_eff=𝑚²₍₄₎=𝑓^′₀²−2𝑓₀𝑓^′′3𝑓^′₀𝑓^′′. (3.17)

In the braneworld version, we except the crossover distance to affect the effective mass. In this respect, 𝑚²_back is a purely background effect due to the shift on the Hubble parameter with respect to the standard 4D case, while 𝑚²_pert is a purely perturbative extradimensional effect [28, 63]. In our setup, these quantities are defined as follows: 𝑚²_back2=−𝑟²_𝑐𝑓^′2₀1−21+3𝑟²_𝑐𝑓^′₀²𝑓₀,𝑚²_pert𝑓=−^′03𝑓^′′1−21+3𝑟²_𝑐𝑓^′₀²𝑓₀⁻¹. (3.18)

The de Sitter brane is close to the standard 4D regime as long as 𝐻²₀∼𝐻²₍₄₎, which leads to 31−1+(2/3)𝑓^′₀𝑓₀𝑓₀𝑓^′₀≪1. (3.19)

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, 𝑓^′₀>0, implies that 𝑓₀𝑓^′₀≫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 𝑚²_back>0 and 𝑚²_pert<0) is guaranteed if 𝑚²_eff>𝑚²₍₄₎ which leads to the following condition (for more details, see [28, 63, 64]): 𝑓^′₀²<4𝑓₀𝑓₀^′′. (3.20)

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: 𝑅₀^𝑛+1−𝑛(3𝑛+4)𝛾²𝑅₀^−(𝑛+1)<𝑛𝛾(6+4𝑛). (3.21)

Figure 1 shows the behavior of 𝑀≡𝑚²_eff−𝑚²₍₄₎ 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.

Figure 1 

The behavior of 𝑀≡𝑚2eff−𝑚2(4) versus 𝑛 and 𝛾.

Figure 2 

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)=𝑅^𝑛exp(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.