# Anisotropically Inflating Universes

###### Abstract

We show that in theories of gravity that add quadratic curvature invariants to the Einstein-Hilbert action there exist expanding vacuum cosmologies with positive cosmological constant which do not approach the de Sitter universe. Exact solutions are found which inflate anisotropically. This behaviour is driven by the Ricci curvature invariant and has no counterpart in the general relativistic limit. These examples show that the cosmic no-hair theorem does not hold in these higher-order extensions of general relativity and raises new questions about the ubiquity of inflation in the very early universe and the thermodynamics of gravitational fields.

###### pacs:

95.30.Sf, 98.80.Jk, 04.80.Cc, 98.80.Bp, 98.80.Ft, 95.10.Eg## I Introduction

The inflationary universe is the central cosmological paradigm which astronomical observations aim to test, and by which we seek to understand how the universe might have evolved from a general initial condition into its present state of large-scale isotropy and homogeneity together with an almost flat spectrum of near-Gaussian fluctuations. The essential feature of this inflationary picture is a period of accelerated expansion during the early stages of the universe inflate . The simplest physically-motivated inflationary scenario drives the acceleration by a scalar field with a constant potential, and the latter can also be described by adding a positive cosmological constant to the Einstein equations. In order to understand the generality of this scenario it is important to determine whether universal acceleration and asymptotic approach to the de Sitter metric always occurs. A series of cosmic no-hair theorems of varying strengths and degrees of applicability have been proved to demonstrate some necessary and sufficient conditions for its occurrence jdb ; bou ; starob ; jss ; wald . Similar deductions are possible for power-law jb4 ; power and intermediate inflationary behaviour, inter , where accelerated expansion is driven by scalar-field potentials that have slow exponential or power-law fall-offs, but we will confine our discussion to the situation that occurs when there is a positive cosmological constant, . So far, investigations have not revealed any strong reason to doubt that, when and other matter is gravitationally attractive, any stable, ever-expanding general-relativistic cosmological model will approach isotropic de Sitter inflation exponentially rapidly within the event horizon of any geodesically-moving observer. Similar conclusions result when we consider inflation in those generalisations of general relativity in which the Lagrangian is a function only of the scalar curvature, , of spacetime. This similarity is a consequence of the conformal equivalence between these higher-order theories in vacuum and general relativity in the presence of a scalar field barr ; barrcot ; maeda . In this paper we will show that when quadratic terms formed from the Ricci curvature scalar, are added to the Lagrangian of general relativity then new types of cosmological solution arise when which have no counterparts in general relativity. They inflate anisotropically and do not approach the de Sitter spacetime at large times. We give two new exact solutions for spatially homogeneous anisotropic universes with which possess this new behaviour. They provide counter-examples to the expectation that a cosmic no-hair theorem will continue to hold in simple higher-order extensions of general relativity. Other consequences of such higher-order theories have been studied in Berkin ; BCX ; Schmidt . The presence of such quadratic terms as classical or quantum corrections to the description of the gravitational field of the very early universe will therefore produce very different outcomes following expansion from general initial conditions to those usually assumed to arise from inflation. This adds new considerations to the application of the chaotic and eternal inflationary theories eternal in conjunction with anthropic selection BT .

We will consider a theory of gravity derived from an action quadratic in the scalar curvature and the Ricci tensor. More specifically, ignoring the boundary term, we will consider the -dimensional gravitational action

(1) |

Variation of this action leads to the following generalised Einstein equations (see, e.g., DT ):

(2) |

where is the energy-momentum tensor of the ordinary matter sources, which we in this paper will assume to be zero, and

(3) | |||||

with . The tensor incorporates the deviation from regular Einstein gravity, and we see that implies .

First, consider an Einstein metric, so that . This is a solution of eq.(2) with provided that

(5) |

Hence, when any Einstein space is a solution to eq.(2) provided that . In particular, if , de Sitter spacetime is a solution to eq.(2). If , we need and de Sitter spacetime cannot be a solution.

Now consider solutions to eq.(2) which are non-perturbative and and are not small. We know that solutions with are conformally related to Einstein gravity with a scalar
field that possesses a self-interaction potential
of the form* *, barr ; barrcot ; maeda , and their inflationary behaviours for small and large , along with that of theories derived from
actions that are arbitrary functions of , are well understood. However,
there is no such conformal equivalence with general relativity when and cosmologies with can then exhibit quite different
behaviour.

## Ii The flat de Sitter solution

First consider the spatially-flat de Sitter universe with metric

(6) |

The stability of this solution in terms of perturbations of the scale-factor
depends on the sign of . In 4D, we can use the Weyl
invariant and the Euler density, , defined by ^{1}^{1}1These expressions are only valid in 4D.,

(7) |

to eliminate the quadratic Ricci invariant in the action, since

Since integration over the Euler density is a topological invariant, the variation of will not contribute to the equations of motion. The Friedmann-Robertson-Walker (FRW) universes are conformally flat so, for a small variation, the invariant will not contribute either. Hence, sufficiently close to a FRW metric only the term will contribute. The stability of the FRW universe is therefore determined by the sign of BO . One can check this explicitly using eq. (2). We start with the metric ansatz:

and note that in 4D the trace of eq.(2) reduces to

(8) |

which can be used to determine the stability of the Ricci scalar. We can perturb the Ricci scalar by assuming a small deviation from the flat de Sitter metric of the form:

where and are arbitrary constants. Eq.(8) implies

(9) |

if . For , we must have . From this expression we see that if
then the solution will asymptotically approach the flat de Sitter spacetime
as ; however, for the solution
is unstable. For the special case of , this result agrees with the
stability analysis of BO . A construction of an asymptotic series
approximation around the de Sitter metric for the case has also been
performed st ; bm ; gott ; ss ; gss . In the case of general relativity () a number of results for the inhomogeneous case of small
perturbations from isotropy and homogeneity when have also been
obtained jdb ; bou ; starob ; jss ; bg ; rend ; Faraoni ; FN .* *

We see that, as long as any FRW model sufficiently close to the flat de Sitter model will asymptotically approach de Sitter spacetime and consequently obeys the cosmological no-hair theorem. We should emphasize that only FRW perturbation modes have been considered here. The question of whether the flat de Sitter universe is stable against general anisotropic or large inhomogeneous perturbations when and is still unsettled. In the case of universes that are not ’close’ to isotropic and homogeneous FRW models we shall now show that the cosmic no-hair theorem for vacuum cosmologies is not true: there exist ever-expanding vacuum universes with that do not approach the de Sitter spacetime.

## Iii Exact anisotropic solutions

We now present two new classes of exact vacuum anisotropic and spatially homogeneous universes of Bianchi types and with . These are new exact solutions of the eqns. (2) with .

Bianchi type solutions:

(10) |

where

(11) |

These solutions are spacetime homogeneous with a 5-dimensional isotropy group. They have a one-parameter family of 4-dimensional Lie groups, as well as an isolated one (with Lie algebras and , respectively, in Patera et al’s scheme PSWZ ) acting transitively on the spacetime. An interesting feature of this family of solutions is that there is a lower bound on the cosmological constant, given by for which the spacetime is static. For the spacetime is inflating and shearing. The inflation does not result in approach to isotropy or to asymptotic evolution close to the de Sitter metric. Interestingly, even in the case of a vanishing the universe inflates exponentially but anisotropically. We also note from the solutions that the essential term in the action causing this solution to exist is the -term and the distinctive behaviour occurs when . The solutions have no well defined limit, and do not have a general relativistic counterpart. They are non-perturbative. Similar solutions exist also in higher dimensions. Their existence seem to be related to so-called Ricci nilsolitons L1 ; SigTalk .

Bianchi type solutions:

(12) | |||||

where

(13) |

and , , , and are all constants. These are also
homogeneous universes with a 4-dimensional group acting transitively on the
spacetime. Both the mean Hubble expansion rate and the shear are constant.
Again, we see that the solution inflates anisotropically and is supported by
the existence of . It exists when and
but not in the limit . * *

## Iv Avoidance of the no-hair theorem

The no-hair theorem for Einstein gravity states that for Bianchi types the presence of a positive cosmological constant drives the late-time evolution towards the de Sitter spacetime. An exact statement of the theorem can be found in the original paper by Wald wald . It requires the matter sources (other than ) to obey the strong-energy condition. It has been shown that if this condition is relaxed then the cosmic no-hair theorem cannot be proved and counter-examples exist jb1 ; jb2 ; jb3 ; jb4 . In Kaloper , the cosmic no-hair conjecture was discussed for Bianchi cosmologies with an axion field with a Lorentz Chern-Simons term. Interestingly, exact Bianchi type solutions, similar to the ones found here, were found which avoided the cosmic no-hair theorem. However, unlike for our solutions, these violations were driven by an axion field whose energy-momentum tensor violated the strong and dominant energy condition. The no-hair theorem for spatially homogeneous solutions of Einstein gravity also requires the spatial 3-curvature to be non-positive. This condition ensures that universes do not recollapse before the term dominates the dynamics but it also excludes examples like that of the Kantowski-Sachs universe which has an exact solution with which inflates in some directions but is static in others. These solutions, found by Weber weber , were used by Linde and Zelnikov LZ to model a higher-dimensional universe in which different numbers of dimensions inflate in different patches of the universe. However, it was subsequently shown that this behaviour, like the Weber solution, is unstable BY ; sol . We note that our new solutions to gravity theories with possess anisotropic inflationary behaviour without requiring that the spatial curvature is positive and are distinct from the Kantowski-Sachs phenomenon.

The Bianchi type solutions given above inflate in the presence of a positive cosmological constant . However, they are neither de Sitter, nor asymptotically de Sitter; nor do they have initial singularities. Let us examine how these models evade the conclusions of the cosmic no-hair theorem. Specifically, consider the type solution, eq.(10).We define the time-like vector orthogonal to the Bianchi type group orbits, and introduce an orthonormal frame. We define the expansion tensor and decompose it into the expansion scalar, and the shear, , in the standard way. The Hubble scalar is given by . For the type metric, we find (in the orthonormal frame)

As a measure of the anisotropy, we introduce dimensionless variables by normalizing with the expansion scalar:

Interestingly, the expansion-normalised shear components are constants (and independent of the parameters , , and ) and this shows that these solutions violate the cosmological no-hair theorem (which requires as ). To understand how this solution avoids the no-hair theorem of, say, ref. wald , rewrite eq.(2) as follows:

In this form the higher-order curvature terms can be interpreted as matter terms contributing a fictitious energy-momentum tensor . For the Bianchi solution we find

(14) | |||||

where and are the energy density and the principal pressures, respectively. The no hair theorems require the dominant energy condition (DEC) and the strong energy condition (SEC) to hold. However, since the SEC is always violated when . The DEC is violated when and the weak energy condition (WEC) is also violated because These violations also ensure that the singularity theorems will not hold for these universes and they have no initial or final singularities.

Are these solutions stable? Due to the complexity of the equations of motion it is difficult to extract information about the stability of these non-perturbative solutions in general. In the class of spatially homogeneous cosmologies the dynamical systems approach has been extremely powerful for determining asymptotic states of Bianchi models. A similar approach can be adopted to the class of models considered here; however, the complexity of the phase space increases dramatically due to the higher-derivative terms. Nonetheless, some stability results can be easily extracted. Consider, for example, a perfect fluid with a barotropic equation of state, , where is constant. Due to the exponential expansion, the value of the deceleration parameter is for the type and solutions given. Hence, these vacuum solutions will be stable against the introduction of a perfect fluid with . This includes the important cases of dust (), radiation () and inflationary stresses ().

For perturbations of the shear and the curvature, the situation is far more complicated. Even within the class of Bianchi models in general relativity a full stability analysis is lacking. However, in some cases, some of the modes can be extracted. Consider again the Bianchi type solution, eq. (10). Using the trace of the evolution equations, eq.(8), we consider a perturbation of the Ricci scalar:

Using , which is valid for spatially homogeneous universes, eq.(8) again implies, to lowest order:

for . This shows that the perturbation of the Ricci scalar gives the same eigenmodes for the anisotropic solutions of type and as it did for perturbations of de Sitter spacetime in eq.(9). In order to determine the stability of other modes, like shear and anisotropic curvature modes, further analysis is required.

## V Discussion

The solutions that we have found raise new questions about the thermodynamic interpretation of spacetimes. We are accustomed to attaching an entropy to the geometric structure created by the presence of a cosmological constant, for example the event horizon of de Sitter spacetime. Do these anisotropically inflating solutions have a thermodynamic interpretation? If they are stable they may be related to dissipative structures that appear in non-equilibrium thermodynamics and which have appeared been identified in situations where de Sitter metrics appear in the presence of stresses which violate the strong energy condition jb1 ; jb2 ; jb3 ; jb4 . They also provide a new perspective on the physical interpretation of higher-order gravity terms in the gravitational Lagrangian.

In summary: we have found exact cosmological solutions of a gravitational theory that generalises Einstein’s by the addition of quadratic curvature terms to the action. These solutions display the new phenomenon of anisotropic inflation when . They do not approach the de Sitter spacetime asymptotically and provide examples of new outcomes for inflation that is driven by a stress and begins from ’general’ initial conditions.

## Acknowledgment

SH was supported by a Killam Postdoctoral Fellowship.

## References

- (1) A. Guth, Phys. Rev. D 23, 347 (1981); A.D. Linde, Phys. Lett. B129, 177 (1983).
- (2) J.D. Barrow, Perturbations of a De Sitter Universe, In The Very Early Universe, eds. G. Gibbons, S.W. Hawking and S.T.C Siklos, (Cambridge UP, Cambridge, p.267, 1983).
- (3) W. Boucher and G.W. Gibbons, In The Very Early Universe, ed. G. Gibbons, S.W. Hawking and S.T.C Siklos, (Cambridge UP, Cambridge, p. 273, 1983).
- (4) A.A. Starobinskii, Sov. Phys. JETP Lett. 37, 66 (1983).
- (5) L.G. Jensen and J. Stein-Schabes, Phys. Rev. D 35, 1146 (1987).
- (6) R. Wald, Phys. Rev. D 28, 2118 (1983).
- (7) J.D. Barrow, Phys. Lett. B, 187, 12 (1987).
- (8) F. Lucchin and S. Mataresse, Phys. Rev. D 32, 1316 (1985); L.F. Abbott and M.B. Wise, Nucl. Phys. B 244, 541 (1984).
- (9) J.D. Barrow, Phys. Lett. B, 235, 40 (1990); J.D. Barrow and P. Saich, Phys. Lett B 249, 406 (1990); J.D. Barrow and A.R. Liddle, Phys. Rev. D 47, R5219 (1993); A.D. Rendall, Class. Quant. Grav. 22,1655 (2005).
- (10) J.D. Barrow, Nucl. Phys. B 296, 697 (1988).
- (11) J.D. Barrow and S. Cotsakis, Phys. Lett. B. 214, 515 (1988).
- (12) K.I. Maeda, Phys. Rev. D 39, 3159 (1989).
- (13) A. Berkin, Phys. Rev. D 44, 1020 (1991).
- (14) E. Bruning, D. Coule, C. Xu, Gen. Rel. Grav. 26, 1197 (1994).
- (15) H.-J. Schmidt, gr-qc/0407095.
- (16) A.D. Linde, Phys. Lett. B 175, 395 (1986); A. Vilenkin, Phys. Rev. D 27, 2848 (1983).
- (17) J.D. Barrow and F.J. Tipler, The Anthropic Cosmological Principle, (Oxford UP, Oxford, 1986).
- (18) S. Deser and B. Tekin, Phys. Rev. D67, 084009 (2003).
- (19) J.D. Barrow and A.C. Ottewill, J. Phys. A 16, 2757 (1983).
- (20) V. Müller, H.-J. Schmidt and A.A. Starobinskii, Phys. Lett. B 202, 198 (1988).
- (21) A. Berkin and K.I. Maeda, Phys. Rev. D 44, 1691 (1991).
- (22) S. Gottlöber, V. Müller and A.A. Starobinskii, Phys. Rev. D 43, 2510 (1991).
- (23) A.A. Starobinskii and H.-J. Schmidt, Class. Quantum Grav. 4, 695 (1987).
- (24) H.-J. Schmidt, Class. Quantum Grav. 5, 233 (1988).
- (25) J.D. Barrow and G. Götz, Phys. Lett. B 231, 228 (1989).
- (26) A.D. Rendall, Annales Henri Poincaré A5,1041 (2004)
- (27) V. Faraoni, Phys. Rev. D72 061501 (2005).
- (28) V. Faraoni and S. Nadeau, Phys. Rev. D72 124005 (2005).
- (29) J. Patera, R.T. Sharp, P. Winternitz and H. Zassenhaus, J. Math. Phys. 17, 986 (1976).
- (30) J. Lauret, Math. Ann. 319, 715 (2001), Quart. J. Math. 52, 463 (2001), Math. Z. 241, 83 (2002), Diff. Geom. Appl. 18, 177 (2003).
- (31) S. Hervik, talk held at the CMS Summer 2004 meeting, Halifax, NS, Canada.
- (32) J.D. Barrow, Phys. Lett. B 180, 335 (1986).
- (33) J.D. Barrow, Nucl. Phys. B 310, 743 (1988).
- (34) J.D. Barrow, Phys. Lett. B 183, 285 (1987).
- (35) N. Kaloper, Phys. Rev. D 44, 2380 (1991).
- (36) E. Weber, J. Math. Phys. 25, 3279 (1984).
- (37) A.D. Linde and M.I. Zelnikov, Phys. Lett. B 215, 59 (1988).
- (38) J.D. Barrow and J. Yearsley, Class. Quantum Grav. 13, 2965 (1996).
- (39) H.H. Soleng, Class. Quantum Grav. 6, 1387 (1989).