Modified gravity: walk through accelerating cosmology ††thanks: Work supported in part by Global COE Program of Nagoya University (G07) provided by the Ministry of Education, Culture, Sports, Science & Technology and by the JSPS Grant-in-Aid for Scientific Research (S) # 22224003 and (C) # 23540296 (S.N.); and MINECO (Spain), FIS2010-15640 and AGAUR (Generalitat de Catalunya), contract 2009SGR-345 (S.D.O.). This paper is based on the lecture given by S.D. Odintsov at the 7th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, Belgrade, 9-19.09.2012.
We review the accelerating (mainly, dark energy) cosmologies in modified gravity. Special attention is paid to cosmologies leading to finite-time future singularities in , and modified gravities. The removal of the finite-time future singularities via addition of -term which simultaneously unifies the early-time inflation with late-time acceleration is also briefly mentioned. Accelerating cosmology including the scenario unifying inflation with dark energy is considered in gravity with Lagrange multipliers. In addition, we examine domain wall solutions in gravity. Furthermore, covariant higher derivative gravity with scalar projectors is explored.
PACS numbers: 04.50.Kd, 95.36.+x, 98.80.-k
It is observationally implied that the current expansion of the universe is accelerating. Provided that the universe is homogeneous, two representative approaches to account for the current cosmic acceleration exist. The first is to assume the existence of the so-called dark energy whose pressure is negative (for a recent review, see, e.g., ). The second is to consider that a gravitational theory would be modified at the large distance scale. The simplest theory is gravity (for reviews, see, for example, ).
In this paper, we examine the accelerating (dark energy) solutions of modified gravity which may produce future singularities. We concentrate on reviewing the results in Refs. [3, 4, 5, 6] on theoretical aspects of modified gravity theories with presenting dark energy components. In particular, we study the finite-time future singularities in , and gravity theories [3, 7, 8], where is the Ricci scalar, with and the Ricci tensor and the Riemann tensor, respectively, is the Gauss-Bonnet invariant, and is an arbitrary function of and . This is a generalized gravity theory including both and gravity theories. We also discuss the removal of the finite-time future singularities in gravity via addition of -term which simultaneously leads to the unification of early-time inflation with late-time acceleration . In the frameworks of or theory, the corresponding term may be different, of course . We note that as related studies, the finite-time future singularities [10, 11, 12] and the realization of the phantom phase including the crossing of the phantom divide  have also been examined. Furthermore, the features of the finite-time future singularities in non-local gravity , modified teleparallel gravity  and its extended analysis in loop quantum cosmology (LQC)  has recently been investigated. In addition, dark energy in the context of gravity with Lagrange multipliers  is considered. We also present domain wall solutions in gravity . Moreover, covariant higher derivative gravity with scalar projectors  is explained. We use units of and denote the gravitational constant by with the Planck mass of GeV.
The paper is organized as follows. In Section 2, we explore accelerating cosmologies leading to the finite-time future singularities in , and gravity theories. In Section 3, we study dark energy in the framework of gravity with Lagrange multipliers. In Section 4, we examine domain wall solutions in gravity. In Section 5, we investigate covariant higher derivative gravity with scalar projectors. Finally, conclusions are presented in Section 6.
2 Finite-time future singularities in , and gravity theories
The action of gravity is , where is the determinant of the metric tensor and is the matter Lagrangian. This is a generic theory including both and gravities. We take the flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric . The Hubble parameter is given by , where the dot denotes the time derivative of . In the FLRW background, with the gravitational field equations we find that the effective (i.e., total) energy density and pressure of the universe read and , respectively. For the action in Eq. (1), we obtain
where and , and and are the energy density and pressure of matter (which has been assumed to be a perfect fluid).
2.2 Finite-time future singularities
Provided that the Hubble parameter is written as
where , , , and are constants, is the time when a finite-time future singularity occurs, and . In what follows, we consider the case of . We note that even if and is a non-integer value, in the limit some derivative of diverges and hence the scalar curvature becomes infinity . Moreover, since the case of leads to a de Sitter space, we suppose .
The finite-time future singularities are classified into four types . Type I (“Big Rip” ): In the limit , , and . The case that and are finite values at  is included. This happens for and . In this paper, we regard the singularities for as “Big Rip” and those for as “Type I”. (ii) Type II (“sudden” ): In the limit , , and . This occurs for . (iii) Type III: In the limit , , and . This appears for . (iv) Type IV: In the limit , , , , and higher derivatives of diverge. The case that and/or become finite values at is also included. This is realized if but is not any integer number. Here, and are constants.
2.3 gravity with finite-time future singularities
By taking , the action in Eq. (1) becomes that of gravity. With the method to reconstruct modified gravity [10, 21, 22], for the Hubble parameter to be represented in Eq. (3) we explore gravity models in which finite-time future singularities can appear. By introducing two proper functions and of a scalar field , which we regard as the cosmic time , we rewrite the term in the action in Eq. (1). In this case, by varying the action with respect to we acquire . In principle, by solving this equation we have the relation . If we substitute it into the above form of , we find and hence the original action is found again. We express the scale factor as with a constant and a proper function. Here, we neglect the contribution from matter because when the finite-time future singularities appears, the energy density of dark energy components are completely dominant over that of matter. In this case, the gravitational field equations yield
Accordingly, if we find the solutions and of these equations, by plugging those into with we obtain the concrete form of . We acquire the followings consequences.
(a) [Big Rip]: For or , with , whereas if , .
(b) [Type I]: .
(c) [Type III]: .
(d) [Type II () and Type IV ( but is not an integer)]: . Here, “” means the asymptotic behavior in the limit .
It is remarkable that adding -term to such a theory, one removes future singularity. (Note that gravity was proposed as inflationary model in Ref.  (celebrated Starobinsky inflation) and was used for the first unified inflation-dark energy modified gravity proposed in Ref. ). Hence, we not only remove singularities by adding term but also unify the dark energy era with inflation in such a way (for realistic models of such unification, see ). Several viable gravity models which unify inflation with dark energy and do not contain the finite-time future singularities are listed below :
2.4 gravity with finite-time future singularities
With the same method as in gravity in Section 2.3, it is possible to execute the reconstruction of gravity models in which the finite-time future singularities occur. The action of gravity  is described by Eq. (1) with . In this case, the gravitational field equations yield
The results are as follows .
(a) [Big Rip]: For , with and constants. If , with a positive constant.
(b) [Type I]: .
(c) [Type III], [Type II], [Type II] and (but is not integer) [Type IV]: .
(d) [Type II] (this is a special value in this case): . We remark that the finite-time future singularities appearing in the limit can be removed by the additional term , where is a constant, and and . Furthermore, the finite-time future singularities emerging in the limit can be cured by adding the term , where is an integer .
2.5 gravity with finite-time future singularities
Using the similar procedure in Section 2.3, we reconstruct the form of leading to the finite-time future singularities. With proper functions , and of a scalar field , which we indentify with , we represent the term in the action in Eq. (1) as . Varying this action with respect to , we find . By solving this equation, we obtain . Combining this and the above representation , we acquire . It follows from the gravitational field equations, the conservation law, , and that
For , can be described as with and generic functions of and . We show the results.
(a) [Big Rip]: For or , we find with . Here, , and are constants. There also exists the following model: , where . Here, and are constants.
(b) [Type I]: with , where is a constant.
(c) [Type II (), Type III (), and Type IV ( but is not an integer)]: . In Ref. , it has been examine that the finite-time future singularities can be removed by the term , where and are positive integers.
3 Dark energy from gravity with the Lagrange multipliers
In this section, we study gravity with the Lagrange multiplier field. With and arbitrary functions of , the action is expressed as
where is the Lagrange multiplier field and yields a constraint equation . The variation of the action in Eq. (10) with respect to leads to the gravitational field equation as
where is the covariant derivative and is the covariant d’Alembertian. For the de Sitter space-time, which realize the current cosmic accelerated expansion, i.e., the dark energy dominated stage, the scalar curvature is a positive constant value and hence the Ricci tensor becomes . In this case, from the above constraint equation and Eq. (11), we have . Moreover, in the flat FLRW background, the above constraint equation reads . For , this equation can be solved in terms of as . Provided that the form of is given by the analysis of the observational data, is able to be reconstructed so that the evolution of can be reproduced. It follows from that presents the evolution of , and by solving this equation inversely, we can find . Thus, we acquire with . We note that is an arbitrary function of . As an example, we consider with leading to , where and are constants. In this case, the accelerated expansion of the universe or power-law inflation happens. We have , from which we also acquire . Using these relations, we obtain . As another example, we examine the case that is described by with and positive constants. In the limit , , and therefore the universe asymptotically approaches the de Sitter space-time. In this case, we can regard that in the limit , inflation in the early universe occurs, whereas that in the limit , the late-time cosmic acceleration happens. We also have . As a consequence, for the above , this gravity model with the constraint originating from the Lagrange multiplier can be a unified scenario between inflation and dark energy era, although it should carefully be studied whether the reheating stage after inflation can be realized. Furthermore, does not affect cosmological evolution of the universe and influences only the correction of the Newton law. Thus, cosmology is determined only by the form of .
To explore the Newton law, we take as the Einstein-Hilbert term and add matter. In this case, for , from Eq. (11) we find the Einstein equation with the energy-momentum tensor of matter Its trace equation reads , where is the trace of . Moreover, the constraint equation is given by . Since this is not always met, we should modify the constraint equation as . Thus, this implies that the action with the constraint coming from the Lagrange multiplier field and matter should be described by
For the case of the vacuum such that , the constraint equation is . If , e.g., the first example of shown above, this is equivalent to the constraint equation derived from the action in Eq. (10). In this case, there exist two types of the solutions in the constraint equation . One is and the other is presented by . On the small scales of, e.g., the solar system and galaxies, the solution would be the first solution of so that the Newton law can be recovered. On the other hand, in the bulk of the universe, the solution should be in order that the cosmic evolution can be realized. It is not so clear whether the first solution on the small scales of the solar system and galaxies and the second one in the bulk universe can be connected in the intermediate scales.
4 Domain wall solutions in gravity
In this section, we investigate a static domain wall solution and reconstruct an gravity model with realizing it .
4.1 Static domain wall solution in a scalar field theory
To begin with, we study a static domain wall solution in a scalar field theory. We suppose that the following dimensional warped metric , and that the scalar field only depends on . In this background, the metric in the -dimensional Einstein manifold is , defined by . In addition, for , the space is the de Sitter one, for , it is the anti-de Sitter, and for , it is the flat. Following the procedure proposed in Ref. , it has been demonstrated that a static domain wall solution can exist in a scalar field theory  (a developed study on a static domain wall solution in a scalar field theory has also been executed in Ref. ). We investigate the action , where is a function of the kinetic term of a scalar field and is a potential of . In the above dimensional warped metric, with the and components of the Einstein equation, we obtain the expressions of and . Using these expressions, the energy density is described as . As an example, we consider , where and are constants. In this case, the distribution of reads . Accordingly, the energy density of is localized at and thus a domain wall is made. We note that a condition for to be localized is in the limit .
4.2 Reconstruction of the form of
In the dimensional warped metric, the component and the trace of components of the gravitational field equation read
where the prime denotes the derivative with respect to of , and and . We examine an explicit form of with leading to a domain wall solution for the case that matter is absent. For the model , with the relation , can be described as a function of , , and eventually we find . By plugging this equation into Eqs. (13) and (14) and eliminating , Eqs. (13) and (14) can be expressed as differential equations in terms of . Here, it is enough to analyze Eq. (13) because Eq. (14) is not independent of Eq. (13). As a result, Eq. (13) can be rewritten to
To solve the above relation of in terms of , by defining and expanding exponential terms in the limit , we take only the first leading terms in terms of . We find with and , where and are constants. Finally, for , Eq. (15) can be described by with and , where and constants described by the model parameters , , and . We acquire a general solution of this equation as , where , and are arbitrary constants. Here, the subscriptions of correspond to the sign “” on the right-hand side of this equation. In the model , at the distribution of the energy density is localized and therefore a domain wall is realized as shown above. Consequently, for an exponential model of gravity, a domain wall can appear at .
4.3 Effective (gravitational) domain wall
Next, with the reconstruction method [21, 22], we explore an effective (gravitational) domain wall in gravity. With the same procedure as in Section 2.3, we study the action of gravity given by . Using two proper functions and of a scalar field , we represent the term . The variation over yields . Solving this equation with respect to leads to , by substituting which into the action in Eq. (1) with , we acquire the action of gravity as . In the dimensional warped metric shown in Section 4, for the case that depends only on , it follows from the gravitational field equation with the choice of and (i.e., the flat space), we have
where the prime denotes the derivative with respect to of . For a model and with , and constants, we acquire
In the range where is large, we take and impose the boundary condition that in the limit , the universe asymptotically approaches flat as . As a result, we obtain . From this expression, we see that performs a non-trivial behavior at . Hence, it can be considered that an effective (gravitational) domain wall could appear at . Moreover, by using the representation , we acquire an integration expression of as