Emergent Universe Scenario, Bouncing and Cyclic Universes in Degenerate Massive Gravity
Shou-Long Li, H. Lü, Hao Wei, Puxun Wu and Hongwei Yu
Department of Physics and Synergetic Innovation Center for Quantum Effect and Applications, Hunan Normal University, Changsha 410081, China
School of Physics, Beijing Institute of Technology, Beijing 100081, China
Center for Joint Quantum Studies, School of Science, Tianjin University, Tianjin 300350, China
We consider alternative inflationary cosmologies in massive gravity with degenerate reference metrics and study the feasibilities of the emergent universe scenario, bouncing and cyclic universes. We focus on the construction of the Einstein static universe, classes of exact solutions of bouncing and cyclic universes in degenerate massive gravity. We further study the stabilities of the Einstein static universe against both homogeneous and inhomogeneous scalar perturbations and give the parameters region for a stable Einstein static universe.
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General relativity (GR), as a classical theory describing the non-linear gravitational interaction of massless spin-2 fields, is widely accepted at low energy limit. Nevertheless, there are still several motivations to modify GR, based on both theoretical considerations (e.g. [1, 2]) and observations (e.g.[3, 4].) One proposal, initiated by Fierz and Pauli , is to assume that the mass of graviton is nonzero. Unfortunately, the interactions for massive spin-2 fields in Fierz-Pauli massive gravity have long been thought to give rise to ghost instabilities . Recently, the problem has been resolved by de Rham, Gabadadze and Tolley (dRGT) , and dRGT massive gravity has attracted great attention and is studied in various areas such as cosmology [7, 8, 9, 10] and black holes [11, 12]. We refer to e.g. [13, 14, 15] and reference therein for a comprehensive introduction of massive gravity.
There are several extensions of dRGT massive gravity for different physical motivations, such as bi-gravity , multi-gravity , minimal massive gravity , mass-varying massive gravity , degenerate massive gravity  and so on . Thereinto, the degenerate massive gravity was initially proposed by Vegh  to study holographically a class of strongly interacting quantum field theories with broken translational symmetry. Later this theory has been studied widely in the holographic framework [22, 23, 24, 25] and black hole physics [26, 27, 28, 29, 30]. However, the cosmological application of this theory is few. Recently, together with suitable cubic Einstein-Riemann gravities and some other matter fields, degenerate massive gravity was used to construct exact cosmological time crystals  with two jumping points, which provides a new mechanism of spontaneous time translational symmetry breaking to realize the bouncing and cyclic universes that avoid the initial spacetime singularity. It is worth noting that the higher derivative gravities are indispensable for the realization of cosmological time crystals. However, if we consider only the infrared modification of GR, we can study instead the feasibilities of bouncing and cyclic models in degenerate massive gravity without the higher-order curvature invariants.
Actually, it is valuable to investigate alternative inflationary cosmological models within the standard big bang framework, because traditional inflationary cosmology [32, 33, 34, 35] suffers from both initial singularity problem  and trans-Planckian problem . By introducing a mechanism for a bounce in cosmological evolution, both the trans-Planckian problem and initial singularity can be avoided. The bouncing scenario can be constructed via many approaches such as matter bounce scenario , pre-big-bang model , ekpyrotic model , string gas cosmology , cosmological time crystals  and so on [42, 43, 44]. The cyclic universe, e.g. , can be viewed as the extension of the bouncing universe since it brings some new insight into the original observable universe . Another direct solution to the initial singularity proposed by Ellis et al. [47, 48], i.e., emergent universe scenario, is assuming that the universe inflates from a static beginning, i.e., the Einstein static universe, and reheats in the usual way. In this scenario, the initial universe has a finite size and some past-eternal inflation, and then evolves to an inflationary era in the standard way. Both horizon problem and the initial singularity are absent due to the initial static state. Actually, these alternative inflationary cosmologies have been studied in different class of massive gravities. The bouncing and cyclic universes have been studied in mass-varying massive gravity . The emergent scenario has been also studied in dRGT massive gravity [50, 51] and bi-gravity [52, 53]. To our knowledge, these alternative inflationary models have not been studied in degenerate massive gravity. For our purpose, we would like to study the feasibilities of emergent universe, bouncing and cyclic universes in massive gravity with degenerate reference metrics.
The remaining part of this paper is organized as follows. In Sec. 2, we give a brief review of the massive gravity and its equations of motion. In Sec. 3, we study the emergent universe in degenerate massive gravity with perfect fluid. First we obtain the exact Einstein static universe solutions in sevaral cases. Then we give the linearized equations of motion and discuss the stabilities against both homogeneous and inhomogeneous scalar perturbations. We give the parameters regions of stable Einstein static universes. In Sec. 4, we construct exact solutions of the bouncing and cyclic universes in degenerate massive gravity with a cosmological constant and axions. We conclude our paper in Sec. 5.
2 Massive gravity
In this section, following e.g. , we briefly review massive gravity. The four dimensional action of massive gravity is given by
where is Plank mass and we assume in the rest discussion, is the action of matters, is the Ricci scalar, represents the determinant of , represents the mass of graviton, are free parameters and are interaction potentials which can be expressed as follows,
where the regular brackets denote traces, such as . is given by
where is a fixed symmetric tensor and called reference metric, which is given by
where is the Minkowski background and are the Stückelberg fields introduced to restore diffeomorphism invariance . In the limit of , massive gravity reduces to GR. The equations of motion are given by
Generally, all the Stückelberg fields are nonzero in massive gravity and the rank of the matrix (2.5) is full, i.e. . In Ref. , there are two spatial nonzero Stückelberg fields which break the general covariance in massive gravity. The matrix has rank 2 thus being degenerate. The massive gravity with degenerate reference metrics is called degenerate massive gravity. For our purpose, we set only the temporal Stückelberg field to equal to zero. It follows that massive gravity we consider in this paper has degenerate reference metrics of rank 3. And the unitary gauge of the corresponding Stückelberg fields is defined simply by . So are given by 
in the basis , where is a positive constant.
3 Emergent universe scenario
In this section, we consider the realization of emergent universe scenario in the context of degenerate massive gravity. We consider only spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric because the Stückelberg fields in degenerate massive gravity are chosen in a spatially flat basis. On the other hand, based on the latest astronomical observations [55, 56], the universe is at good consistency with the standard spatially-flat case. In the following discussion, we assume that the matter field is composed of perfect fluids. Firstly we construct the Einstein static universe in several cases. Then we study the stability against both homogeneous and inhomogeneous scalar perturbations.
3.1 Einstein static universe
The spatially flat FLRW metric is given by
The energy-momentum tensor corresponding to perfect fluids is given by
where and represent the energy density and pressure respectively, is the constant equation-of-state (EOS) parameter, and velocity 4-vector is given by
where dot denotes the derivative with respect to time. For the sake of obtaining the Einstein static universe, we let scale factor and . We request  to avoid the ghost excitation from massive gravity. The energy density can be solved from the Friedmann equation (3.4),
The Einstein static universe solution is given by . Because there are several parameters in the Eq. (3.8), we will discuss them in different cases.
3.1.1 Case 1: , ,
In this case, Eq. (3.8) reduces to a simple linear equation. The Einstein static solution is given by
Case (1.1): For the solution is given by
Case (1.2): For , the solution is given by
3.1.2 Case 2: ,
In this case, Eq. (3.8) reduces to a quadratic equation. The Einstein static solutions are given by
Case (2.1): For , and
the solution is given by
Case (2.2): For , and
the solution is given by
The existence of requires the following two cases:
Case (2.3): For , and
the solution is given by
Case (2.4): For , and
the solution is given by
3.1.3 Case 3:
In this case, Eq. (3.8) can be rewritten as
For and , there are three real solutions which are given by
For and , there is one real solution which is given by
For , there is one real solution which is given by
For , there is one real solution which is given by
There are three free parameters and in the solutions. It is hard to analyze the parameters region of existence of all six solutions analytically. Instead we analyze the existence regions numerically and plot the parameters regions of the existence of all solutions in Fig. 1.
We find that the solutions and cannot exist.
In the previous subsection, we study the existence of the Einstein static universe in massive gravity with degenerate reference metrics. However, the emergent scenario does not thoroughly solve the issue of big bang singularity when perturbations are considered. For example, although the Einstein static universe is stable against small inhomogeneous perturbations in some cases [57, 58, 59, 60], the instability exists in previous parameters range against homogeneous perturbations . So it is valuable to explore the viable Einstein static universe by considering both homogeneous and inhomogeneous scalar perturbations. Actually, the stabilities of the Einstein static universe has been studied in various modified gravities, for examples, loop quantum cosmology , theory [63, 65, 64], theory [66, 67], modified Gauss-Bonnet gravity [68, 69], Brans-Dicke theory [70, 71, 72, 73], Horava-Lifshitz theory [74, 75, 76], brane world scenario [77, 78, 79], Einstein-Cartan theory , gravity , Eddingtong-inspired Born-Infeld theory , Horndeski theory [83, 84], hybrid metric-Palatini gravity  and so on [86, 87, 88, 89, 90, 91, 92]. We refer to e.g.  and reference therein for more details of stability of the Einstein static universe. In the following discussions, we would consider the stabilities of the Einstein static universe against both homogeneous and inhomogeneous scalar perturbations in degenerate massive gravity.
3.2.1 Linearized Massive Gravity
Now we study the linear massive gravity with degenerate reference metrics. We use the symbols bar and tilde representing the background and the perturbation components of the metric respectively. First, we obtain the linearized equations of motion The perturbed metric can be written as
where is the background metric which is given by Eq. (3.1) with and is a small perturbation. For our purpose, we consider scalar perturbations in the Newtonian gauge. is given by
where and are functions of . For scalar perturbations, it is useful to perform a harmonic decomposition , Now the indexes are lowered and raised by the background metric unless otherwise stated. By using the relation , the inverse metric is perturbed by
So the perturbed can also be written as
According to Eq. (2.4), we have , i.e.,
So we have
where “ 0 ” and “ ” denote time and space components respectively and the same index does not mean the Einstein rule. For perfect fluids, the perturbations of energy density and pressure are and respectively. The perturbations of velocity are given by
where and are also functions of . The perturbed energy momentum tensor is given by
It is useful to perform a harmonic decomposition of ,
In these expressions, summation over co-moving wavenumber are implied. The harmonic function satisfies 
where is Laplacian operator and is separation constant. For spatially flat universe, we have where the modes are discrete [60, 69]. Substituting Eqs. (3.30) and (3.42) into (3.37), after some algebra, we find
where satisfies a second order ordinary differential equation
To analyse the stabilities of the Einstein static universe in massive gravity with a degenerate reference metric, we require the condition of existence of the oscillating solution of Eq. (3.45) which is given by
In the following discussions, we study the parameters region satisfying reality conditions (3.6) and (3.7), and stability condition (3.47) for the Einstein static flat universes against both homogeneous and inhomogeneous perturbations in different cases.
3.2.2 Case 1: , ,
The stabilities of the Einstein static universe (3.11) require