# Gauss–Bonnet models with cosmological constant and non zero spatial curvature in \(D=4\)

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## Abstract

In the present paper the possibility of eternal universes in Gauss-Bonnet theories of gravity in four dimensions is analysed. It is shown that, for zero spatial curvature and zero cosmological constant, if the coupling is such that \(0<f'(\phi )\le c \exp (\frac{\sqrt{8}}{\sqrt{10}}\phi )\), then there are solutions that are eternal. Similar conclusions are found when a cosmological constant turned on. These conclusions are not generalized for the case when the spatial curvature is present, but we are able to find some general results about the possible nature of the singularities. The presented results correct some dubious arguments in Santillan (JCAP 7:008, 2017), although the same conclusions are reached. On the other hand, these past results are considerably generalized to a wide class of situations which were not considered in Santillan (JCAP 7:008, 2017).

## 1 Introduction

One of the main interests in higher derivative gravity theories is that they can describe inflation by the addition of a higher order curvature to the Einstein–Hilbert action [1, 2]. This is achieved without the addition of dark energy or scalar fields. An important role in this context is played by the Gauss-Bonnet invariant, since it appears in QFT renormalization in curved space times [3]. In addition, the Gauss-Bonnet term arises in low-energy effective actions of some string theories. For instance, the tree-level string effective action has been calculated up to several orders in the \(\alpha '\) expansion in [4, 5, 6, 7, 8, 9, 10]. The result is that there is no moduli dependence of the tree-level couplings. However, one loop corrections to the gravitational couplings have been considered in the context of orbifold compactifications of the heterotic superstring [11, 12]. It has been shown in that reference that there are no moduli dependent corrections to the Einstein term while there are non trivial curvature contributions. They appear as the Gauss-Bonnet combination multiplied by a function of the modulus field.

The results described above partially motivated the study of cosmological consequences of the Gauss-Bonnet term. In four dimensions, this term does not have any dynamical effect. However, when this term is non-minimally coupled with any other field such as a scalar field \(\phi \), the resulting dynamics is non trivial. Several cosmological consequences has been exploited in recent literature, and we refer the reader to [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and references therein. But the aim of the present letter is not focused in inflationary aspects of the theory, instead in the characterization of singular and eternal solutions of the theory. It is important to mention that there exist preliminary works on this subject, examples are given in [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. In particular, the results of [36, 37, 38] suggest the existence of singular solutions as well as regular solutions. The singular solutions are confined to an small portion of the phase space, while the non singular fill the rest. This situation is different than in GR, where the Gauss–Bonnet term is absent, and the powerful Hawking–Penrose theorems apply [59, 60]. Related work can be found in [61, 62].

In the present work we are going to provide evidence for these claims, when a cosmological constant is turned on or when the spatial curvature is vanishing. In addition, some partial results about the case with \(k=\pm 1\) will be also presented.

This work is organized as follows. In Sect. 2, generalities of Gauss-Bonnet models are briefly reviewed. Section 3 reviews some general arguments given in [63] for zero spatial curvature and vanishing cosmological constant. These arguments are valid for any model of these characteristics. Section 4 contains an analysis of eternal universes which avoid some dubious arguments in [63]. The calculations presented in this section are particularly explicit, since they are an important part of this paper. In Sect. 5 the results of Sect. 4 are generalized to the case where the cosmological constant is turned on. In Sect. 6, the results of Sect. 3 are generalized for the case in which the scalar curvature *k* is turned on. The obtained results are not as universal as the ones in Sect. 3, but some partial conclusions concerning the possible type of singularities can be obtained. In Sect. 7 the results of Sect. 4 are partially generalized to the case where the spatial curvature is turned on. Section 8 contains a discussion of the obtained results and open perspectives.

## 2 Gauss–Bonnet equation

*G*being the Gauss-Bonnet invariant

## 3 Models without potential with flat spatial metric

### 3.1 General analysis

We have made the analogous analysis when a cosmological constant \(\Lambda \ne 0\) is turned on. But we have obtained no universal conclusions in this case. In fact, when the curvature is turned on, the resulting inequalities are analogous to the ones above but with the replacement \(C_0\rightarrow C_0+\Lambda t\). In particular, cyclic cosmologies are allowed when the cosmological constant is turned on.

### 3.2 The behavior of the scalar field

*H*is finite, then \(\dot{\phi }\rightarrow 0\). Also, if \(f'(\phi )\ne 0\) then \(\dot{\phi }\rightarrow 0\) when \(H\rightarrow \pm \infty \). Instead, if \(f'(\phi )=0\) then it may happen that \(\dot{\phi }\rightarrow \pm \infty \) when \(H\rightarrow \pm \infty \), as follows from (3.19). For this reason, a coupling \(f'(\phi )\) which never reaches a zero will be chosen. An example of this may be a coupling \(f'(\phi )>0\) with a global minimum \(f'_m>0\). In addition, this condition implies that \(\dot{\phi }\rightarrow 0\) when \(H\rightarrow 0\) and \(f'(\phi )\) is finite, this is directly seen by use of (6.93).

Note however, that there may be an indetermination when \(H\rightarrow 0\) and \(f'(\phi )\rightarrow \pm \infty \). Consider first the case \(H^2 f'(\phi )\rightarrow c\), with *c* a constant. Then \(H^3f'(\phi )\rightarrow 0\) and from (3.20) it is clear that \(\dot{\phi }\rightarrow 0\). If instead \(H^2 f'(\phi )\rightarrow 0\) then \(H^3f'(\phi )\rightarrow 0\) and again it is seen from (6.93) and that \(\sqrt{1+x}\sim \sqrt{x}\) for \(x>>1\) that \(\dot{\phi }\rightarrow 0\) in this case. Finally, when \(H^2 f'(\phi )\rightarrow \pm \infty \) it is seen from the same formula and that \(\sqrt{1+x}\sim 1+x/2\) for \(x<<1\) that \(\dot{\phi }\rightarrow 0\) again. In other words, there is no way in which \(\dot{\phi }\rightarrow \pm \infty \) if the coupling \(f'(\phi )\) is never zero.

*t*.

## 4 The possibility of eternal universes

Consider again the Eqs. (3.9)–(3.11) with the branch in which \(\dot{\phi }\) is bounded, that is, the branch described by (3.20). In the following, it will be assumed that \(f'(\phi )\) and \(f''(\phi )\) are not divergent for any finite value of \(\phi \). Suppose that the universe falls into a singularity at a given finite time, which can be chosen when \(t\rightarrow \pm 0\), by a shift of time. The choice \(t\rightarrow 0\) is for simplicity, the singularity may be at any value \(t_0\) by a choice of a convenient parametrization. Our aim is to find situations in which this assumption gives a contradiction. In these situations, the universe will be eternal.

The analysis given in the present section avoid some dubious arguments presented in [63], although similar conclusions are obtained. The dubious argument is the one below the formula (3.21) of that paper. We suspect that this formula may be a trivial one due to a numerical computer error. Thus, the present analysis will avoid such types of arguments. Since the results of this section are crucial, the calculations will be as explicit as possible and will not rely in any computer algorithm. All the tools to be used in the following sections are all analytical and its validity can be seen directly.

## 5 The case of unbounded scalar field acceleration \(\ddot{\phi }\) and \(H\rightarrow \pm \infty \)

*H*is finite.

## 6 The case of unbounded scalar field acceleration \(\ddot{\phi }\), \(H^2<\infty \)*and*\({\dot{H}}\rightarrow \pm \infty \)

## 7 The case with bounded scalar field acceleration \(\ddot{\phi }\)

*H*is finite and \({\dot{H}}\rightarrow \pm \infty \). But this clearly does not satisfy (4.34). In addition, if

*H*and \({\dot{H}}\) are finite, there is no singularity and the universe is eternal. Thus, the only possibility for having a singular curvature is that \(H\rightarrow \pm \infty \) and \({\dot{H}}\rightarrow \pm \infty \). By taking into account this, the last equation gives the following necessary condition

*H*near the singularity as

*h*(

*t*) being a function of time that goes to zero faster than linearly. The condition (4.39) is then

*h*(

*t*) goes to zero faster than linearly, the last equation can be satisfied only if \(c=1\) and

*g*(

*t*) a function that goes to zero at \(t\rightarrow 0\), not necessarily analytical. Therefore

*m*(

*t*) goes to zero faster than \(t^5\), and is not necessarily analytic. The conclusion is that, near the singularity

*q*(

*t*) going to zero at \(t\rightarrow 0\). The condition \(H^2({\dot{H}}+H^2)\rightarrow cte\) at \(t\rightarrow 0\) becomes

*s*(

*t*) containing the terms that go to zero even faster than \(t^{2+\epsilon }\). One may consider the possibility that \(\ddot{f}\) goes to zero not like any power law, for instance as \(\ddot{f}(\phi )=-1+\alpha t^{2}u(t)\) with \(u(t)\rightarrow 0\) as \(t\rightarrow 0\) and non analytical. But we will argue below that this is not the case. The last condition may be integrated to give

*r*(

*t*) goes to zero faster than \(t^{3+\epsilon }\). Finally, the fact that

*w*(

*t*) goes to zero faster than linearly. All the obtained expressions are valid in an small interval near \(t=0\).

*r*(

*t*) goes faster than \(t^5\) and is not necessarily analytic.

*W*(

*t*) is the primitive of

*w*(

*t*), and grows faster than quadratically. Both (4.49) and (4.50) combined give that

*w*(

*t*) grows faster than linearly, and that its primitive

*W*(

*t*) goes faster than quadratically. Now, the third term in (4.51) is cubic, and since the right hand does not have any cubic term, it should be cancelled somehow. It can not be cancelled by the fourth or the fifth term, since the behavior of

*w*(

*t*) or

*W*(

*t*) described above makes these terms of higher order than three. The sixth term also is of higher order. But it can be cancelled by second term by assuming that

*V*(

*t*) contains at least cubic terms. Thus, \(w_2(t)\) goes to zero faster than \(t^3\) and therefore \(W_2(t)\) goes to zero faster than \(t^4\). From here it is seen that the fourth and the fifth term of (4.51) go like

## 8 Models with flat spatial metric and cosmological constant \(\Lambda >0\) turned on

## 9 The case of unbounded scalar field acceleration \(\ddot{\phi }\) and \(H\rightarrow \pm \infty \)

*H*is finite.

## 10 The case of unbounded scalar field acceleration \(\ddot{\phi }\), \(H^2<\infty \) and \({\dot{H}}\rightarrow \pm \infty \)

However, there is a further possibility, that is, that \(H\rightarrow 0\) and \({\dot{H}}\rightarrow \pm \infty \). But it is easy to see from Eqs. (5.53)–(5.55) that the case \(H=0\) is not allowed when the cosmological constant \(\Lambda \) is turned on. Thus, for divergent acceleration \(\ddot{\phi }\rightarrow \pm \infty \) there is no singularity in this branch.

## 11 The case with bounded scalar field acceleration \(\ddot{\phi }\)

^{1}shows that

*s*(

*t*) containing the terms that go to zero faster than quadratically. Thus

*r*(

*t*) goes to zero faster than \(t^{3}\). The formula (4.48) is also unchanged. By taking into account

*r*(

*t*) goes faster than \(t^5\) and is not necessarily analytic. However, arguments analogous to the ones giving (4.51) show that

## 12 Spatial curvature \(k=\pm 1\) turned on

*t*.

*t*the Hubble constant

*H*is positive \(H>0\) and that it is approaching a singularity at \(t_0>t>0\). Suppose that the singularity comes from a behavior of the lapse function of the form \(a(t)\sim c(t_0-t)^\alpha \). Then \(H\sim \alpha /(t-t_0)\), and clearly \(H>0\) only if \(\alpha <0\), since \(t<t_0\). It is convenient to express (6.80) as follows

*H*gives

*I*is the value of the integral, which is finite since \(1/a^2(t)\) does not have any singularity. But the important point to remark is the following. The first term on the left hand is positive and has the singular behavior \(1/(t_0-t)^2\), which is more explosive than the term \(1/(t_0-t)\) of the right hand. The only way that the inequality (6.86) can be fulfilled is that the second and the third term cancel this behavior. But clearly, none of them can do the job. Note that, during all the reasoning above, the value of \(C_0\), the value of

*k*or the behavior of the coupling \(f(\phi )\) was immaterial.

*H*and remembering that \(H<0\) it is found that

*a*(

*t*) do not have zeros. Denote its value as

*I*. Then the last bound is

## Proposition

The Gauss–Bonnet cosmology without potential and with spatial curvature \(k=\pm 1\) or \(k=0\) does not admit solutions for which there is a regime falling into a singularity of the form \(a(t)\sim c/(t_0-t)^\beta \), with \(\beta >0\), neither in the past or future, no matter the explicit form of the coupling \(f(\phi )\).

We have also considered the other two complementary cases, namely \(H<0\) falling into a power law in the past and \(H<0\) falling into a singularity in the future. But the bounds that we found depend on the behavior of \(\dot{\phi }\) and we can find no conclusions in this case.

### 12.1 The negative branch of the scalar field

*c*a constant, then \(\dot{\phi }\rightarrow 0\). Also, if \(f'(\phi )\ne 0\) then \(\dot{\phi }\rightarrow 0\) when \(H\rightarrow \pm \infty \) and

*a*is finite and non zero. The same holds when \(H\rightarrow \pm \infty \) and

*a*goes to zero or infinite. In addition, if \(k=-1\) and \((H^2+\frac{k}{a^2})\rightarrow 0\), with

*H*and

*a*finite, then \(\dot{\phi }\rightarrow 0\) as well. In addition when \(H\rightarrow 0\) it may be possible to have

*c*a constant. In this case, \(\dot{\phi }\rightarrow \pm \infty \). But this limit implies that \(H\rightarrow 0\) and \(H^2/a^2\rightarrow c'\), with \(c'\) another constant. Thus \({\dot{a}}^2/a^4\) tends to a constant value. Therefore \(a\sim t^{-1}\) near this limit, and this contradicts that \(H\rightarrow 0\). Another possible dangerous limit is \(H\rightarrow 0\), \(a\rightarrow 0\) and \(f'(\phi )\rightarrow \infty \) in such a way that \(H^2 f'(\phi )^2/a^2\rightarrow 0\), since for this limit \(\dot{\phi }\rightarrow \infty \). We suggest however that this limit do not take place. In fact, the limit \(H\rightarrow c\) implies near this region that \(a(t)\sim \exp (ct)\), and \(a(t)\ne 0\) even when \(c\rightarrow 0\).

*t*.

## 13 The possibility of eternal universes for \(k\ne 0\)

In the present section the possibility of having eternal solutions is considered, when the spatial curvature \(k=\pm 1\) is turned on. However, the results obtained below are less general than the ones of the previous sections. In fact, the analysis when the spatial curvature *k* is turned on is more difficult than the case \(k=0\).

As before, it is assumed that \(f'(\phi )\) is never zero and is never divergent for any finite value of \(\phi \). In other words, there are no vertical asymptotes at finite \(\phi \) values. Furthermore, we will be working in the branch for which \(\dot{\phi }\) is bounded for any finite time.

*a*taking any finite value \(a_0\). No singularity will appear in this situation.

Consider now the possibility that \(H^2<\infty \), \(\ddot{\phi }\rightarrow \infty \) and \(a\rightarrow 0\). If \(H \rightarrow H_0\) then, near the singularity, \(a\sim \exp (H_0 t)\) which contradicts our hypothesis that \(a\rightarrow 0\). Thus, the only possibility is \(H_0=0\). Thus, as \(H\rightarrow 0\) and \(a\rightarrow 0\), a simple inspection shows that the equation (6.74) is never satisfied. If instead, one consider the same situation but with \(a\rightarrow \pm \infty \), this case reduce to the one with \(k=0\) for which, as shown in previous sections, there are no singularities.

*c*a constant. But the term in parenthesis is strictly positive since \(1+{\dot{f}}H>0\). But our assumption is that \({\dot{H}}\rightarrow \pm \infty \), so the requirement is impossible to satisfy.

Finally, one has to check the case \(H\rightarrow H_0\), \({\dot{H}}\rightarrow \pm \infty \) and \(a\rightarrow 0\). As we saw above, this means that \(H_0=0\). The Eq. (6.74) may be satisfied when \(1+{\dot{f}}H\rightarrow 0\). But as \(H\rightarrow 0\), one has that \(1+{\dot{f}}H \rightarrow 1\), which is a contradiction.

The analysis given above is valid for \(\ddot{\phi }\rightarrow \pm \infty \). It is impossible to have a singularity when \(k=1\) in this case. However, the situation for finite \(\ddot{\phi }\) is more difficult to analyse than for the case \(k=0\). The reason is that the analysis made in (4.39)–(4.44) get much more complicated when the term \(k/a^2\) is turned on. Thus, we have obtained no conclusions in this case. In addition, for \(k=-1\), the term \(H^2+k/a^2\) can be zero if a potential singularity takes place at \({\dot{a}}=1\) and \(a\rightarrow 0\), since \(H^2+k/a^2\rightarrow 0\). This zero appears multiplying the factor (6.74) and complicates the analysis of the singularity. We hope to overcome these technical difficulties in a future.

## 14 Discussion

The results of the present work are the following. For a Gauss Bonnet model without cosmological constant and zero spatial curvature, if \(0<f'(\phi )\le c \exp (\frac{\sqrt{8}}{\sqrt{10}}\phi )\), and the scalar field is in some specific branch described in the text, then exists a large class of solutions that are eternal. These conclusions were also obtained when the cosmological constant is turned on. It is important however to emphasize that if the scalar field is in other branch, then the presented conclusions do not apply and in fact singular solutions may appear. The appearance of this solutions do not contradict the well known Hawking singularity theorems, since it is not necessarily true that these theories may be considered as GR coupled with matter satisfying the strong energy conditions.

The analysis when the spatial curvature *k* is turned on is more complicated. The problem is that for \(k=0\) the resulting differential system only involves the Hubble constant *H* and the scalar field, while for \(k=\pm 1\) the system involves also the scale factor, and this complicates the analysis considerably. However, some partial results about the singularities were found, independently of the form of the coupling \(f(\phi )\). These results are collected in the proposition of Sect. 6 in the text, and exclude under certain circumstances some singularities in the scale factor as \(a(t)\sim c/(t-t_0)^\beta \) with \(\beta >0\). This result is independent on the form of the coupling \(f'(\phi )\). We hope to overcome some technical problems and to obtain results related to the case \(k=\pm 1\) in a near future.

## Footnotes

- 1.
Which are also valid in this case, as the equation of motion for \(\phi \) are unchanged by the presence of a cosmological constant. Note that in presence of a potential \(V(\phi )\) there appears a term proportional to \(V'(\phi )\), but for a constant potential (a cosmological constant) this term do not contribute.

## Notes

### Acknowledgements

J.O.M and O.P.S are supported by the CONICET.

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