# Cosmological perturbations and stability of nonsingular cosmologies with limiting curvature

###### Abstract

We revisit nonsingular cosmologies in which the limiting curvature hypothesis is realized. We study the cosmological perturbations of the theory and determine the general criteria for stability. For the simplest model, we find generic Ostrogradski instabilities unless the action contains the Weyl tensor squared with the appropriate coefficient. When considering two specific nonsingular cosmological scenarios (one inflationary and one genesis model), we find ghost and gradient instabilities throughout most of the cosmic evolution. Furthermore, we show that the theory is equivalent to a theory of gravity where the action is a general function of the Ricci and Gauss-Bonnet scalars, and this type of theory is known to suffer from instabilities in anisotropic backgrounds. This leads us to construct a new type of curvature-invariant scalar function. We show that it does not have Ostrogradski instabilities, and it avoids ghost and gradient instabilities for most of the interesting background inflationary and genesis trajectories. We further show that it does not possess additional new degrees of freedom in an anisotropic spacetime. This opens the door for studying stable alternative nonsingular very early Universe cosmologies.

###### pacs:

98.80.Cq, 04.50.Kd, 98.80.-k, 04.20.Cv, 04.20.Jb^{†}

^{†}thanks: Vanier Canada Graduate Scholar

## I Introduction

One of the biggest problems with the classical theory of general relativity is the occurrence of singularities, which are inevitable under realistic assumptions [1, 2, 3] and which signify the breakdown and the incompleteness of the theory. The big bang singularity in cosmology and the singularity at the center of a black hole are two well-known instances of singularities in general relativity that one would like to resolve. These singularities often find themselves in regions of high density, high energy, and high curvature, where one may expect the breakdown of the classical theory and the emergence of quantum behavior. For this reason, there is hope that a quantum theory of gravity would provide the resolution to the otherwise pathological classical singularities.

Without a proper theory of quantum gravity, one may approach the problem with an effective theory that could mimic the low-energy behavior of the full quantum theory. The effective theory could be constructed with one or more new degrees of freedom, e.g. a new scalar field. This allows one to study nonsingular theories of the very early Universe within effective field theory (EFT) [4, 5, 6] as is done in, for instance, bouncing cosmologies [7, 8, 9, 10] or genesis scenarios [11, 12, 13, 14, 15, 16] with a generalized scalar field such as in Horndeski [17] and generalized Galileon [18] theories, whose equivalence was first proved in [19]. Alternatively, one may attempt to modify the Einstein-Hilbert gravity action to include higher-order curvature terms (e.g., [20, 21, 22]). Interestingly, this can serve as the basis for implementing the limiting curvature hypothesis, which seeks to incorporate the idea of a fundamental limiting length (that would be realized in the full theory of quantum gravity) into the effective theory for gravity.

In line with special relativity where the speed of light is bounded from above and quantum mechanics where the Heisenberg uncertainty relation holds, the idea of the limiting curvature hypothesis comes from the fact that one may expect quantum gravity to possess a fundamental length scale below which no measurement can be made and on which all physical observables must be smeared out. Presumably, this fundamental scale is at least of the order of the Planck length , although it could be larger. Taking , it is straightforward to see that if all curvature invariants are bounded throughout the spacetime manifold (, , , , etc., where is the Ricci tensor, is the Ricci scalar, is the Weyl tensor, and is the covariant derivative), then the spacetime is nonsingular. Indeed, in the well-known cases of the big bang and black hole singularities, some of the physically measurable curvature invariants such as , , and blow up; hence finding theories in which all invariants are bounded is certainly a necessary condition for constructing nonsingular cosmologies.

Unfortunately, bounding an infinite number of curvature invariants is rather nontrivial. Indeed, there are well-known instances where low-order curvature invariants are bounded while higher-order invariants are still unbounded (e.g., while ). This is where the limiting curvature hypothesis comes in handy. The hypothesis states that [23, 24, 25, 26, 27] if one finds a theory that allows a finite number of curvature invariants to be bounded by an explicit construction (e.g., and ), and when these invariants take on their limiting values, then any solution of the field equations reduces to a definite nonsingular solution (e.g., de Sitter space), for which all curvature invariants are automatically bounded. We note, though, that the assumptions of the limiting curvature hypothesis generally do not ensure that solutions avoid singularities when curvature scalars are not on their limiting values.

The limiting curvature hypothesis has been used and tested in the context of black hole physics [27, 26, 28, 29, 30, 31, 32, 33, 34, 35]. In this context, the geometry outside the black hole horizon is described by the usual Schwarzschild metric, but inside the event horizon, the black hole singularity is replaced with a nonsingular de Sitter spacetime, which, in turn, could be the source of a new “baby” Friedmann universe. Similarly, in a cosmological context [36, 37, 38, 30, 31, 39], a nonsingular universe can be constructed, in vacuum, such that it is asymptotically de Sitter in the past and Minkowski in the future (or vice versa). This is in line with Penrose’s vanishing Weyl tensor conjecture [40] (see also the discussion in Ref. [41]), which states that the Weyl tensor should vanish at the beginning of the universe, since de Sitter space has . With the addition of matter sources, one can obtain asymptotically de Sitter and Friedmann cosmologies, remaining nonsingular throughout cosmic time. Recent works also show that the ideas of limiting curvature could allow one to construct nonsingular bouncing cosmologies [42, 43, 44].

In this paper, we want to revisit nonsingular cosmological models that make use of an effective theory for gravity in which the limit curvature hypothesis is realized. It was shown in Refs. [36, 37] that interesting background cosmologies can be found within this framework by constructing a theory in which the curvature invariants and are bounded. However, these studies did not explore the cosmological perturbations [45] for the action containing the above curvature invariants. Recent developments in nonsingular cosmology within EFT [4, 5, 6] have shown that it is often rather difficult to avoid instabilities in the cosmological perturbations (e.g., see Refs. [46, 47, 48] for no-go theorems within Galileon and Horndeski theories; see, also, Refs. [49, 50, 51]). For this reason, one could tend to believe that nonsingular models constructed as in Refs. [36, 37] are going to be very unstable at the perturbation level, thus rendering the models unviable.

In this work, we will show that the naive models of Refs. [36, 37] are indeed generically unstable. We will see that minimal extensions in which one also includes the Weyl tensor squared, , in the curvature invariants are more robust, i.e. there are fairly large regions of parameter space that are stable. Yet, there do not seem to exist nonsingular cosmological solutions that remain stable throughout cosmic history, and moreover, the theory will be shown to be equivalent to a theory of gravity (where is the Gauss-Bonnet term), which has unavoidable ghosts [52]. We will then construct a completely new curvature-invariant function and show that it allows for stable nonsingular cosmological solutions throughout time. There will remain some difficulties though in constructing a physically relevant model in certain cases.

The outline of this paper is as follows. In Sec. II, we will review the construction of a nonsingular cosmology in which the limiting curvature hypothesis is realized, set up the action for the theory, and find the background equations of motion. In particular, we will discuss two specific scenarios: an inflationary scenario and a genesis scenario. We will then study the cosmological perturbations, determine the general stability conditions and check them for specific models. The equivalence with gravity will also be demonstrated. In Sec. III, we will construct a new model with a new curvature scalar, derive the resulting cosmological perturbations and the stability conditions, and comment on the case of an anisotropic background. We will then discuss the recovery of vacuum Einstein gravity and Friedmann cosmology with the addition of matter sources. We will end with a summary of the results and a discussion in Sec. IV.

## Ii Nonsingular cosmology with limiting curvature

### ii.1 Setup of the theory and background evolution

The approach taken in Refs. [36, 37] to implement the limiting curvature hypothesis consists of introducing a finite number of nondynamical scalar fields , or Lagrange multipliers, such that the action takes the form

(1) |

where the ’s are functions of curvature invariants that can depend on , , , and combinations and derivatives thereof. Accordingly, given an appropriately chosen potential , one can rewrite this action into a general effective theory of gravity. By virtue of the Lagrange multipliers, a given potential imposes constraints on the ’s, hence the idea that the right choice of can naturally bound the curvature invariants and satisfy the limiting curvature hypothesis asymptotically. We will give examples where this is realized below.

As is done in Refs. [36, 37], we are going to consider two nondynamical scalar fields and start with a general action of the form

(2) |

where is the action for possible matter sources. At this point, we do not make any assumption on the functional form of and , but we want them to scale as , so let us require that we recover a certain limit at the background level:

(3) |

The superscript refers to the metric of a flat^{1}^{1}1It is straightforward to generalize this to include curvature
(see Ref. [37]). Friedmann-Lemaître-Robertson-Walker (FLRW) universe,

(4) |

where is the lapse function and is the scale factor.
Accordingly, is the Hubble parameter, and a dot is a derivative with respect to physical time, .
At the background level, we can further ask that and .
The original action then becomes^{2}^{2}2We omit the superscript for and when it is clear that they
represent background quantities.

(5) |

Varying with respect to and yields the equations of motion (EOMs)

(6) |

and

(7) |

respectively, where we set the lapse function to . Letting , where is the stress-energy tensor associated with the matter action and where is the energy density and is the pressure, one can then vary the background action with respect to to find

(8) |

again setting . Finally, varying with respect to gives (setting once more)

(9) |

Let us denote and for shorthand notation from here on. We can then summarize the set of EOMs as

(10) | ||||

(11) | ||||

(12) | ||||

(13) |

In the limit where , we note that we recover the usual Friedmann equations, as expected. Also, in the limit where , one obtains the EOMs in vacuum. Demanding that for all values so that is real and looking at an expanding universe (so ; this could be generalized to a contracting universe with , in which case a minus sign would appear in certain equations), we can write the EOMs as

(14) | ||||

(15) | ||||

(16) |

Furthermore, the equations can be written in the following form:

(17) | ||||

(18) | ||||

(19) |

#### ii.1.1 Example of inflationary scenario

To solve the background EOMs, one needs to specify a form for the potentials and . As a first example, one can consider

(20) | ||||

(21) |

which is inspired from Ref. [37], and as we will see, this gives rise to a nonsingular inflationary scenario. In vacuum (), the EOMs are given by

(22) | ||||

(23) |

We plot these functions in Fig. 1. As we can see in the left-hand plot, the Hubble parameter is finite everywhere: as , the spacetime is asymptotically Minkowski (; recall that we are in vacuum), whereas when , the spacetime is asymptotically de Sitter since . Similarly, looking at the right-hand plot, is finite everywhere, and it is asymptotically vanishing as . Accordingly, this verifies the limiting curvature hypothesis as in Ref. [37].

We note at this point that since we regard our theory as a low-energy effective theory of a possible quantum theory of gravity, there should be a cutoff scale beyond which the EFT is no longer valid. The model here includes only two dimensionful parameters: and . Therefore, the cutoff scale should naively be given by these parameters as for a given integer , and in particular, it should be at least of the order of . Determining the exact value for involves a rather nontrivial computation for the given theory, but since the energy scale of our cosmological solutions is always less than by construction, the validity of EFT is naturally ensured.

The phase-space diagram for the model is plotted in Fig. 2,
where the arrows show the vectors with components computed from Eqs. (15)
and (16) (in vacuum with ) with the potentials (20) and (21).
We highlight a specific trajectory in green for illustration. In this case, the universe starts
asymptotically at and and ends asymptotically at
and , so as we saw from Fig. 1,
the universe starts in a de Sitter spacetime and ends in a Minkowski spacetime.^{3}^{3}3With the addition of matter sources,
it would end in a FLRW spacetime as shown in Ref. [37].

At this point, one may wonder how the given scenario evades the singularity theorems of [53, 54, 55] regarding the past incompleteness of inflationary cosmology. First, it is important to recall that the singularity theorem for inflation [53, 54] is proved under the assumption that gravity is given by the Einstein-Hilbert action and that inflation is driven by matter obeying the null energy condition. Our higher derivative gravity terms, when taken to the matter side of the equations of motion, act as matter violating the null energy condition. Hence the theorem does not apply in our setup. We also note that some of the past directed geodesics would have finite affine length (in agreement with the situation in Ref. [55]), but this is simply due to the fact that the flat FLRW chart does not cover the entire de Sitter space. One can extend the spacetime so that all geodesics are complete as in the case of de Sitter space. Thus, our inflationary universe is free from initial singularities.

#### ii.1.2 Example of genesis scenario

As another example, let us consider

(24) | ||||

(25) |

In vacuum, the EOMs become

(26) | ||||

(27) |

We plot these functions in Fig. 3. As we can see in the left- and right-hand plots, the Hubble parameter and its time derivative are again everywhere finite: as or , the spacetime is asymptotically Minkowski with , and as . We note that is now reached when . Thus, this is another type of scenario that verifies the limiting curvature hypothesis, namely a genesis scenario, which starts in Minkowski space rather than de Sitter space.

The phase-space diagram for this model is plotted in Fig. 4,
where again, the arrows show the vectors with components computed from Eqs. (15)
and (16) (in vacuum with ) with the genesis potentials (24) and (25).
We highlight different trajectories in green, red, black, and purple for illustration.
All of these curves either start or end at , which corresponds to Minkowski spacetime.
However, the green and red curves are pathological trajectories since they either end or start at ,
at which point it can be shown that . Accordingly, from Eq. (15),
one finds at that point. More interestingly, the black and purple curves start and end
at , and they turn around at some minimal value, so they never reach the “singularity” at .
Also, these trajectories always have .
When looking at the left-hand plot of Fig. 3, these trajectories suggest that the universe starts
in the far right at (Minkowski spacetime), rolls up to the left but does not reach ,
and rolls back down to Minkowski spacetime again. In light of a structure formation scenario for the very early universe,
one would like to have some form of reheating^{4}^{4}4For instance, reheating could occur via
gravitational particle production [56, 57] (see Refs. [16, 58] for examples
of gravitational particle production in nonsingular cosmologies).
near the maximal value that the Hubble parameter reaches. Thus, the universe would start as Minkowski spacetime, but it would end
as a radiation- and then matter-dominated FLRW spacetime.

### ii.2 Cosmological perturbations and stability analysis

We now turn to the study of the cosmological perturbations for the action given by Eq. (II.1). At this point, one needs to specify the form of the curvature-invariant functions and . Motivated by Refs. [36, 37], let us take

(28) |

where at this point is just some real constant. In a flat FLRW background, these curvature invariants reduce to

(29) |

under the assumption^{5}^{5}5We note that the requirement restricts our attention to the regions of phase space in which
for the examples given in Secs. II.1.1 and II.1.2. that ,
which was the hypothesis of Eq. (3)
that allowed us to find the general background EOMs
in Sec. II.1.
We note that the background expressions for the curvature invariants
do not depend on the constant since flat FLRW spacetime is conformally flat,
so the term proportional to the Weyl tensor squared does not affect the dynamics
of background spacetime.

#### ii.2.1 Tensor modes

Let us begin by studying the tensor fluctuations. We start by perturbing the metric linearly as

(30) |

where the perturbation tensor is transverse and traceless, i.e. (adding the last term on the right-hand side does not change the linear equations but simplifies the derivation). We define the Fourier components of by

(31) |

where represents the polarization basis. Given the curvature-invariant functions of Eq. (II.2), we can now perturb Eq. (II.1) to second order with the above metric to find

(32) |

where

(33) |

and the coefficients and will be specified shortly. We pause here to note that, in general, since the second-order action in the tensor sector has nondegenerate higher-derivative terms (), there appear to be ghost degrees of freedom according to Ostrogradski’s theorem. The only way to avoid those Ostrogradski ghosts would be if were to vanish identically (). We see that this is not possible for a generic real constant , but if one sets , then the model is safe with regards to Ostrogradski instabilities. The original models of Refs. [36, 37] did not include in their curvature invariants at all, so they had . Accordingly, the above implies that these models are inherently unstable. Yet, the addition of the Weyl tensor squared in the invariants with a specific prefactor () avoids this conclusion while having no effect on the background evolution. Still, it does not mean that the theory is necessarily free of all types of instabilities as we will see shortly.

The other coefficients of Eq. (II.2.1) are given by

(34) | ||||

(35) |

The criteria to avoid ghost and gradient instabilities are and , respectively. By using the background EOMs (see Eq. (14), which can be rewritten as , Eq. (15), and Eq. (16) in the case for vacuum with ), the conditions can be written solely in terms of the fields , and their potentials , as

(36) | ||||

(37) |

These conditions will be tested for the different models of Secs. II.1.1 and II.1.2 shortly.

#### ii.2.2 Vector modes

We shall consider vector fluctuations in the following gauge, where

(38) |

Here, the vector perturbations satisfy . The Fourier components of are then defined by

(39) |

where are orthogonal spatial vectors perpendicular to . The second-order action for vector perturbations becomes

(40) |

where and . Accordingly, when , which sets to avoid Ostrogradski ghosts, it turns out that as well, and as a result, there are no dynamical vector modes.

#### ii.2.3 Scalar modes

We shall then focus on the scalar fluctuations in the spatially flat gauge, where , , and

(41) |

The second-order action for scalar modes is then given by

(42) |

where and^{6}^{6}6We
omit the subscript from the perturbation variables when it is clear that they represent the Fourier components.
.
The matrix is given by^{7}^{7}7As before, we omit the superscript for and
when it is understood that they represent background quantities.

(43) |

where stands for symmetric components. Since no time derivatives of , , and appear in the second-order action with , these modes are nondynamical. Then, these variables can be eliminated by their equations of motion. After removing the nondynamical modes, the resulting action can be written solely in term of as follows:

(44) |

where

(45) | ||||

(46) |

with

(47) |

We do not write down the form of the other coefficients because they are not so relevant in the following stability analysis.

In the small scale limit (), one finds

(48) | ||||

(49) |

where and are simply two real positive constants. Thus, the ghost instability in the scalar sector is avoidable when it is absent in the tensor sector, i.e. when is satisfied. In addition, the gradient instability is absent when , so when

(50) |

By using the vacuum background EOMs, this condition can be rewritten as

(51) |

Regarding gradient instabilities which can occur on sub-Hubble scales (), the general procedure is to check for the stability of modes within the validity range of the EFT, i.e. for (see for instance Ref. [10]), where for our models . The condition given by Eq. (51) can be viewed as the one which ensures that the shortest wavelength modes () do not suffer from gradient instabilities. However, since the perturbed action exhibits a modified dispersion relation, i.e., since Eq. (46) has terms of order , , , and , longer wavelength modes (longer than , but still smaller than ) could still suffer from gradient instabilities. As long as the duration for such gradient instabilities is not too long though, their amplification remains controllable in comparison to the smaller wavelength modes which easily blow up (within a time scale of the order of ). This is why we only focus on the stability of the shortest wavelength modes.

#### ii.2.4 Stability analysis

In summary, with , we saw that there is no Ostrogradski instability. Then, we derived three conditions given by Eq. (36), which determines when the model is free of ghost instabilities in the tensor and scalar sectors, and Eqs. (