Advances in High Energy Physics

Volume 2012 (2012), Article ID 679156, 12 pages

doi:10.1155/2012/679156

Research Article

Matter Instability in Theories with Modified Induced Gravity

Kourosh Nozari¹ and Faeze Kiani²

¹Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

²Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran

Received 22 September 2012; Accepted 6 November 2012

Academic Editor: George Siopsis

Copyright © 2012 Kourosh Nozari and Faeze Kiani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Matter stability is a necessary condition to have a cosmologically viable model. Modified gravity in the spirit of 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 -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 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 theories have the capability to provide a unified gravitational alternative to dark energy and inflation [30–37]. A number of viable modified 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 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 theories, see [40–58].

Since the -modified induced gravity (brane 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 gravity.

2. DGP-Inspired Gravity

2.1. The Basic Equations

Modified gravity in the form of theories is derived by generalization of the Einstein-Hilbert action so that (the Ricci scalar) is replaced by a generic function in the action

where is the matter Lagrangian and . Varying this action with respect to the metric gives

where is the stress-energy tensor for standard matter, which is assumed to be a perfect fluid and by definition . Also, is the stress-energy tensor attributed to the curvature defined as follows:

By substituting a flat FRW metric into the field equations, one achieves the analogue of the Friedmann equations as follows [30–37]:

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 gravity. The action of this DGP-inspired gravity is given by

where is a five-dimensional bulk metric with Ricci scalar , while is an induced metric on the brane with induced Ricci scalar . The Friedmann equation in the normal branch of this scenario is written as [28, 29]

where is the DGP crossover scale which has the dimension of and marks the IR (infrared) behavior of the DGP model. The Raychaudhuri equation is written as follows:

To achieve this equation, we have used the continuity equation for as

where the energy density and pressure of the curvature fluid are defined as follows:

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 and are necessary conditions for theories to be free from ghosts and other instabilities [40–44, 62]. In our brane scenario, in addition to , 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

where (which we neglect it in our forthcoming arguments) is the projection of the bulk Weyl tensor on the brane

where is a unit vector normal to the brane, and as the quadratic energy-momentum correction into the Einstein field equations is defined as follows:

as the effective energy-momentum tensor localized on the brane is defined as [25–27]

Following [40–44], the trace of (3.1), which can be interpreted as the equation of motion for , is obtained as

We parameterize the deviation from Einstein gravity as

where is a small parameter with the dimension of an inverse-squared length, and is arranged to be dimensionless. Typically, (see, for instance, [42]). By evaluating as

(3.5) can be rewritten as follows:

This equation to first order of gives

Now we consider a small region of spacetime in the weak-field regime in which curvature and the metric can locally be approximated by

respectively, where is curvature perturbation, and is the Minkowski metric. In this case, the metric can be approximately taken as a flat one, so and . Now, (3.9) up to first order of gives (we set for simplicity )

This relation can be recast in the following suitable form:

The coefficient of in the fourth term on the left-hand side is the square of an effective mass defined as

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

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 in a form that exhibits the effect of an extra dimension on a 4D model as follows [28, 62, 63]:

Subscript stands for quantities evaluated in the de Sitter space time. We also note that the de Sitter brane is described by the scalar factor which leads to . is defined as

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 . Also by a perturbed Friedmann equation based on (2.6), one can achieve an evolution equation for as [64]

The stability condition for the de Sitter solution in the DGP normal branch is positivity of the effective mass squared, . Now can be written as the sum of three terms . In the 4D version of the gravity, this summation reduces to [64]

In the braneworld version, we except the crossover distance to affect the effective mass. In this respect, is a purely background effect due to the shift on the Hubble parameter with respect to the standard 4D case, while is a purely perturbative extradimensional effect [28, 63]. In our setup, these quantities are defined as follows:

The de Sitter brane is close to the standard 4D regime as long as , which leads to

The assumption that we are slightly perturbing the Hilbert-Einstein action of the brane, that is, , and also the positivity of the effective gravitational constant on the brane at late time, that is, , implies that 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 and ) is guaranteed if which leads to the following condition (for more details, see [28, 63, 64]):

Based on this condition, function in the spirit of the normal DGP braneworld exhibits a de Sitter stability if the following condition is satisfied:

Figure 1 shows the behavior of versus and . The de Sitter phase is stable in this setup if . 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 that the corresponding de Sitter phase is not stable.

Figure 1: The behavior of versus and .

Figure 2: Stability of the de Sitter phase. The de Sitter phase is stable for .

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 gravity. In this letter, we have studied the issue of matter stability in an induced gravity, brane- 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 . For this type of modified induced gravity, the matter is stabilized on the brane for . 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 . Albeit for these values of , 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 with 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.

References

S. Perlmutter, G. Aldering, G. Goldhaber et al., “Measurements of and from 42 high-redshift Supernovae,” Astrophysical Journal Letters, vol. 517, no. 2, pp. 565–586, 1999.

A. G. Riess, A. V. Filippenko, P. Challis, et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” Astrophysical Journal, vol. 116, pp. 1009–1038, 2003.

D. N. Spergel, “First Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters,” Astrophysical Journal, vol. 148, pp. 175–194, 2003.

G. Hinshaw and WMAP Collaboration, “Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: data processing, sky maps, and basic results,” Astrophysical Journal, vol. 180, pp. 225–245, 2009.

E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,” International Journal of Modern Physics D, vol. 15, no. 11, pp. 1753–1935, 2006.

V. Sahni and A. Starobinsky, “Reconstructing dark energy,” International Journal of Modern Physics D, vol. 15, no. 12, pp. 2105–2132, 2006.

T. Padmanabhan, “Dark energy and gravity,” General Relativity and Gravitation, vol. 40, no. 2-3, pp. 529–564, 2008.

H. Kleinert and H.-J. Schmidt, “Cosmology with curvature-saturated gravitational Lagrangian ,” General Relativity and Gravitation, vol. 34, no. 8, pp. 1295–1318, 2002.

S. Nojiri and S. D. Odintsov, “Introduction to modified gravity and gravitational alternative for dark energy,” International Journal of Geometric Methods in Modern Physics, vol. 4, no. 1, pp. 115–145, 2007.

T. P. Sotiriou and V. Faraoni, “ theories of gravity,” Reviews of Modern Physics, vol. 82, no. 1, pp. 451–497, 2010.

S. Capozziello and M. Francaviglia, “Extended theories of gravity and their cosmological and astrophysical applications,” General Relativity and Gravitation, vol. 40, no. 2-3, pp. 357–420, 2008.

R. Durrer and R. Maartens, “Dark energy and dark gravity: theory overview,” General Relativity and Gravitation, vol. 40, no. 2-3, pp. 301–328, 2008.

S. Capozziello and V. Salzano, “Cosmography and large scale structure by f(R) gravity: new results,” http://arxiv.org/abs/0902.0088

G. Dvali, G. Gabadadze, and M. Porrati, “4D gravity on a brane in 5D Minkowski space,” Physics Letters B, vol. 485, no. 1–3, pp. 208–214, 2000.

C. Deffayet, “Cosmology on a brane in Minkowski bulk,” Physics Letters B, vol. 502, pp. 199–208, 2001.

A. Lue, “The phenomenology of Dvali-Gabadadze-Porrati cosmologies,” Physics Reports, vol. 423, no. 1, pp. 1–48, 2006.

K. Koyama, “Ghosts in the self-accelerating universe,” Classical and Quantum Gravity, vol. 24, no. 24, pp. R231–R253, 2007.

C. de Rham and A. J. Tolley, “Mimicking Λ with a spin-two ghost condensate,” Journal of Cosmology and Astroparticle Physics, vol. 2006, article 004, 2006.

V. Sahni and Y. Shtanov, “Braneworld models of dark energy,” Journal of Cosmology and Astroparticle Physics, vol. 2003, article 014, 2003.

A. Lue and G. D. Starkman, “How a brane cosmological constant can trick us into thinking that w < -1,” Physical Review D, vol. 70, no. 10, Article ID 101501, 2004.

L. P. Chimento, R. Lazkoz, R. Maartens, and I. Quiros, “Crossing the phantom divide without phantom matter,” Journal of Cosmology and Astroparticle Physics, vol. 2006, article 004, 2006.

R. Lazkoz, R. Maartens, and E. Majerotto, “Observational constraints on phantomlike braneworld cosmologies,” Physical Review D, vol. 74, no. 8, Article ID 083510, 8 pages, 2006.

R. Maartens and E. Majerotto, “Observational constraints on self-accelerating cosmology,” Physical Review D, vol. 74, Article ID 023004, 6 pages, 2006.

M. Bouhmadi-Lopez, “Phantom-like behaviour in dilatonic brane-world scenario with induced gravity,” Nuclear Physics B, vol. 797, pp. 78–92, 2008.

K. Nozari and M. Pourghassemi, “Crossing the phantom divide line in a Dvali-Gabadadze-Porrati-inspired F(R,phi) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2008, article 004, 2008.

J. Saavedra and Y. Vasquez, “Effective gravitational equations on brane world with induced gravity described by f(R) term,” Journal of Cosmology and Astroparticle Physics, vol. 4, article 013, 2009.

A. Borzou, H. R. Sepanji, S. Shahidi, and R. Yousefi, “Brane f(R) gravity,” Europhysics Letters, vol. 8, no. 2, Article ID 29001, 2009.

M. Bouhmadi-Lopez, “Self-accelerating the normal DGP branch,” Journal of Cosmology and Astroparticle Physics, vol. 11, article 011, 2009.

K. Nozari and F. Kiani, “Dynamical-screening and the phantom-like effects in a DGP-inspired f(R, phi) model,” Journal of Cosmology and Astroparticle Physics, vol. 7, article 010, 2009.

S. Nojiri and S. D. Odintsov, “Modified f(R) gravity consistent with realistic cosmology: from a matter dominated epoch to a dark energy universe,” Physical Review D, vol. 74, Article ID 086005, 13 pages, 2006.

L. Amendola, D. Polarski, and S. Tsujikawa, “Are dark energy models cosmologically viable?” Physical Review Letters, vol. 98, no. 13, Article ID 131302, 4 pages, 2007.

S. Nojiri and S. D. Odintsov, “Modified gravity and its reconstruction from the universe expansion history,” Journal of Physics, vol. 66, Article ID 012005, 2007.

S. Capozziello, S. Nojiri, S. D. Odintsov, and A. Troisi, “Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase,” Physics Letters Section B, vol. 639, no. 3-4, pp. 135–143, 2006.

L. Amendola, R. Gannouji, D. Polarski, and S. Tsujikawa, “Conditions for the cosmological viability of f(R) dark energy models,” Physical Review D, vol. 75, no. 8, Article ID 083504, 2007.

L. Amendola and S. Tsujikawa, “Phantom crossing, equation-of-state singularities, and local gravity constraints in f(R) models,” Physics Letters B, vol. 660, no. 3, pp. 125–132, 2008.

S. Faya, S. Nesseris, and L. Perivolaropoulos, “Can f(R) modified gravity theories mimic a LambdaCDM cosmology?” Physical Review D, vol. 76, Article ID 063504, 2007.

B. Li and J. D. Barrow, “Cosmology of gravity in the metric variational approach,” Physical Review D, vol. 75, no. 8, Article ID 084010, 13 pages, 2007.

S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from theory to Lorentz non-invariant models,” Physics Reports, vol. 505, no. 2–4, pp. 59–144, 2011.

A. De Felice and S. Tsujikawa, “f(R) theories,” Living Reviews in Relativity, vol. 13, p. 3, 2010.

A. D. Dolgov and M. Kawasaki, “Can modified gravity explain accelerated cosmic expansion?” Physics Letters B, vol. 573, pp. 1–4, 2003.

S. Nojiri and S. D. Odintsov, “Modified gravity with negative and positive powers of curvature: unification of inflation and cosmic acceleration,” Physical Review D, vol. 68, Article ID 123512, 10 pages, 2003.

V. Faraoni, “Matter instability in modified gravity,” Physical Review D, vol. 74, no. 10, Article ID 104017, 4 pages, 2006.

S. Nojiri and S. D. Odintsov, “Modified gravity with terms and cosmic acceleration,” General Relativity and Gravitation, vol. 36, p. 1765, 2004.

S. Nojiri and S. D. Odintsov, “Newton law corrections and instabilities in f(R) gravity with the effective cosmological constant epoch,” Physics Letters B, vol. 652, pp. 343–348, 2007.

T. Chiba, “1/R gravity and scalar-tensor gravity,” Physics Letters B, vol. 575, pp. 1–3, 2003.

S. Nojiri and S. D. Odintsov, “Modified Gauss-Bonnet theory as gravitational alternative for dark energy,” Physics Letters B, vol. 631, no. 1-2, pp. 1–6, 2005.

W. Hu and I. Sawicki, “Models of f(R) cosmic acceleration that evade solar-system tests,” Physical Review D, vol. 76, Article ID 064004, 2007.

A. A. Starobinsky, “Disappearing cosmological constant in f(R) gravity,” JETP Letters, vol. 86, no. 3, pp. 157–163, 2007.

S. A. Appleby and R. A. Battye, “Do consistent models mimic general relativity plus ?” Physics Letters B, vol. 654, no. 1-2, pp. 7–12, 2007.

S. Tsujikawa, “Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints,” Physical Review D, vol. 77, Article ID 023507, 13 pages, 2008.

N. Deruelle, M. Sasaki, and Y. Sendouda, ““Detuned” f(R) gravity and dark energy,” Physical Review D, vol. 77, Article ID 124024, 5 pages, 2008.

G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini, “Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion,” Physical Review D, vol. 77, Article ID 046009, 11 pages, 2008.

S. Nojiri and S. D. Odintsov, “Modified f(R) gravity unifying Rm inflation with the ΛCDM epoch,” Physical Review D, vol. 77, Article ID 026007, 7 pages, 2008.

E. V. Linder, “Exponential gravity,” Physical Review D, vol. 80, no. 12, Article ID 123528, 6 pages, 2009.

A. De Felice and S. Tsujikawa, “Construction of cosmologically viable f(G) gravity models,” Physics Letters B, vol. 675, pp. 1–8, 2009.

A. De Felice and S. Tsujikawa, “Solar system constraints on f(G) gravity models,” Physical Review D, vol. 80, Article ID 063516, 15 pages, 2009.

A. De Felice, D. F. Mota, and S. Tsujikawa, “Matter instabilities in general Gauss-Bonnet gravity,” Physical Review D, vol. 81, Article ID 023532, 9 pages, 2010.

K. Bamba, S. Nojiri, and S. D. Odintsov, “Time-dependent matter instability and star singularity in f(R) gravity,” Physics Letters Section B, vol. 698, no. 5, pp. 451–456, 2011.

T. Kobayashi and K.-I. Maeda, “Relativistic stars in gravity, and absence thereof,” Physical Review D, vol. 78, no. 6, Article ID 064019, 9 pages, 2008.

T. Kobayashi and K. I. Maeda, “Can higher curvature corrections cure the singularity problem in f(R) gravity?” Physical Review D, vol. 79, Article ID 024009, 9 pages, 2009.

A. Dev, D. Jain, S. Jhingan, S. Nojiri, M. Sami, and I. Thongkool, “Delicate f(R) gravity models with a disappearing cosmological constant and observational constraints on the model parameters,” Physical Review D, vol. 78, no. 8, Article ID 083515, 2008.

S. Tsujikawa, “Modified gravity models of dark energy,” Lecture Notes in Physics, vol. 800, pp. 99–145, 2010.

M. Bouhamdi-Lopez, “f(R) brane cosmology,” http://arxiv.org/abs/1001.3028

V. Faraoni and S. Nadeau, “Stability of modified gravity models,” Physical Review D, vol. 72, Article ID 124005, 10 pages, 2005.

K. Nozari and F. Kiani, “Cosmological dynamics with modified induced gravity on the normal DGP branch,” http://arxiv.org/abs/1008.4240