Anisotropically Inflating Universes

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by  John D. Barrow

Anisotropically Inflating Universes
John D. Barrow∗ and Sigbjørn Hervik†
∗DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilbeforce Road, Cambridge CB3 0WA, UK.
†Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5 Canada.
E-mail: [email protected], [email protected]
(Dated: February 7, 2008)
We show that in theories of gravity that add quadratic curvature invariants to the Einstein-
Hilbertactionthereexistexpandingvacuumcosmologies withpositivecosmological constantwhich
donotapproachthedeSitteruniverse. Exactsolutionsarefoundwhichinflateanisotropically. This
behaviourisdrivenbytheRiccicurvatureinvariantandhasnocounterpartinthegeneralrelativistic
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 theubiquityof inflation in thevery
early universeand the thermodynamicsof gravitational fields.
6
PACSnumbers: 95.30.Sf,98.80.Jk,04.80.Cc,98.80.Bp,98.80.Ft,95.10.Eg
0
0
2 I. INTRODUCTION fromtheRiccicurvaturescalar,R Rµν areaddedtothe
µν

n Lagrangianofgeneralrelativitythennewtypesofcosmo-
a logicalsolutionarisewhenΛ>0 whichhavenocounter-
J The inflationary universe is the central cosmological
parts in general relativity. They inflate anisotropically
0 paradigm which astronomical observations aim to test, and do not approach the de Sitter spacetime at large
3 and by which we seek to understand how the universe
times. We give two new exact solutions for spatially ho-
might have evolved from a general initial condition into
mogeneous anisotropic universes with Λ > 0 which pos-
3 its present state of large-scale isotropy and homogeneity
sess this new behaviour. They provide counter-examples
v
together with an almost flat spectrum of near-Gaussian
7 to the expectation that a cosmic no-hair theorem will
fluctuations. The essential feature of this inflationary
2 continuetoholdinsimplehigher-orderextensionsofgen-
picture is a period of accelerated expansion during the
1 eral relativity. Other consequences of such higher-order
1 early stages of the universe [1]. The simplest physically- theories have been studied in [13, 14, 15]. The presence
1 motivated inflationary scenario drives the acceleration
of such quadratic terms as classical or quantum correc-
5 by a scalar field with a constant potential, and the lat-
tions to the description of the gravitational field of the
0 ter can also be described by adding a positive cosmo-
/ very early universe will therefore produce very different
c logical constant to the Einstein equations. In order to
outcomes following expansion from general initial condi-
q understand the generality of this scenario it is impor-
tions to those usually assumed to arise from inflation.
- tant to determine whether universal acceleration and
r This adds new considerations to the application of the
g asymptotic approach to the de Sitter metric always oc-
chaotic and eternal inflationary theories [16] in conjunc-
: curs. A series of cosmic no-hair theorems of varying
v tion with anthropic selection [17].
i strengths and degrees of applicability have been proved
X
to demonstrate some necessary and sufficient conditions
We will consider a theory of gravity derived from an
r for its occurrence [2, 3, 4, 5, 6]. Similar deductions
a action quadratic in the scalar curvature and the Ricci
are possible for power-law [7, 8] and intermediate in-
tensor. More specifically, ignoring the boundary term,
flationary behaviour, [9], where accelerated expansion is
we will consider the D-dimensional gravitationalaction
driven by scalar-fieldpotentials that have slow exponen-
tial or power-law fall-offs, but we will confine our dis-
cussion to the situation that occurs when there is a pos-
itive cosmological constant, Λ > 0. So far, investiga- 1
tions have not revealedany strong reasonto doubt that, SG = dDx |g| R+αR2+βRµνRµν −2Λ .
2κ
when Λ > 0 and other matter is gravitationally attrac- ZM
p (cid:0) (cid:1)(1)
tive, any stable, ever-expanding general-relativistic cos-
Variationofthisactionleadstothe followinggeneralised
mological model will approach isotropic de Sitter infla-
Einstein equations (see, e.g., [18]):
tionexponentiallyrapidlywithintheeventhorizonofany
geodesically-moving observer. Similar conclusions result
whenweconsiderinflationinthosegeneralisationsofgen-
eralrelativity inwhichthe Lagrangianis a function only G +Φ +Λg =κT , (2)
µν µν µν µν
of the scalar curvature, R, of spacetime. This similarity
is a consequence of the conformal equivalence between
these higher-order theories in vacuum and general rela-
tivity inthe presenceofa scalarfield[10, 11, 12]. Inthis where T is the energy-momentum tensor of the ordi-
µν
paper we will show that when quadratic terms formed nary matter sources, which we in this paper will assume
2
to be zero, and Since integration over the Euler density is a topological
invariant, the variation of E will not contribute to the
1
G ≡R − Rg , (3) equations of motion. The Friedmann-Robertson-Walker
µν µν µν
2
(FRW)universesareconformallyflatso,forasmallvari-
Φµν ≡ ation, the invariant C Cµνρσ will not contribute ei-
µνρσ
1 ther. Hence, sufficiently close to a FRW metric only the
2αR R − Rg +(2α+β)(g (cid:3)−∇ ∇ )R
µν 4 µν µν µ ν R2 term will contribute. The stability of the FRW uni-
(cid:18) (cid:19)
verseisthereforedeterminedbythesignof(3α+β)[19].
1 1
+β(cid:3) Rµν − Rgµν +2β Rµσνρ− gµνRσρ Rσρ, Onecancheckthisexplicitlyusingeq. (2). Westartwith
2 4
(cid:18) (cid:19) (cid:18) (cid:19) the metric ansatz:
(4)
Λ
with (cid:3) ≡ ∇µ∇µ. The tensor Φµν incorporates the de- ds2 =−dt2+e2b(t) dx2+dy2+dz2 , H =r3,
viation from regular Einstein gravity, and we see that and note that in 4D th(cid:0)e trace of eq.(2)(cid:1)reduces to
α=β =0 implies Φ =0.
µν
First,consideranEinsteinmetric,sothatR =λg . −R+2(3α+β)(cid:3)R+4Λ=0, (8)
µν µν
This is a solution of eq.(2) with T =0 provided that
µν whichcanbe usedto determine the stability of the Ricci
λ scalar. We can perturb the Ricci scalar by assuming a
Λ= [(D−4)(Dα+β)λ+(D−2)]. (5) smalldeviationfromtheflatdeSittermetricoftheform:
2
b(t)=Ht+b eλ1t+b eλ2t+O(e2λit),
Hence, when D = 4 any Einstein space is a solution to 1 2
eq.(2) provided that Λ = (D −2)λ/2. In particular, if where b and b are arbitrary constants. Eq.(8) implies
1 2
Λ > 0, de Sitter spacetime is a solution to eq.(2). If
Λ=0, we need λ=0 and de Sitter spacetime cannot be 3H 2
λ =− 1± 1− , (9)
a solution. 1,2 2 s 9H2(3α+β)!
Now consider solutions to eq.(2) which are non-
perturbative and α and β are not small. We know that if (3α + β) 6= 0. For (3α + β) = 0, we must have
solutions with β = 0,α 6= 0 are conformally related to b1 = b2 = 0. From this expression we see that if
Einstein gravity with a scalar field φ = ln(1 + 2αR) (3α+β) > 0 then the solution will asymptotically ap-
that possesses a self-interaction potential of the form proach the flat de Sitter spacetime as t → ∞; however,
V(φ) = (eφ −1)2/4α, [10, 11, 12], and their inflation- for (3α+β) < 0 the solution is unstable. For the spe-
ary behaviours for small and large |φ|, along with that cial case of β = 0, this result agrees with the stability
of theories derived from actions that are arbitraryfunc- analysis of [19]. A construction of an asymptotic series
tions of R, are well understood. However, there is no approximation around the de Sitter metric for the case
such conformal equivalence with general relativity when β =0hasalsobeenperformed[20,21,22,23,24]. Inthe
β 6=0andcosmologieswithΛ>0canthenexhibitquite caseofgeneralrelativity(α=β =0)anumberofresults
different behaviour. for the inhomogeneous case of small perturbations from
isotropy and homogeneity when Λ > 0 have also been
obtained [2, 3, 4, 5, 25, 26, 27, 28].
II. THE FLAT DE SITTER SOLUTION We see that, as long as (3α+β)>0, any FRW model
sufficiently close to the flat de Sitter model will asymp-
Firstconsiderthe spatially-flatde Sitter universewith totically approach de Sitter spacetime and consequently
metric obeys the cosmological no-hair theorem. We should em-
phasize that only FRW perturbation modes have been
Λ consideredhere. The questionofwhether the flatde Sit-
ds2 =−dt2+e2Ht dx2+dy2+dz2 , H = .
dS 3 ter universe is stable againstgeneralanisotropic or large
r
(cid:0) (cid:1) (6) inhomogeneous perturbations when α 6= 0 and β 6= 0 is
Thestabilityofthissolutionintermsofperturbationsof stillunsettled. Inthecaseofuniversesthatarenot’close’
the scale-factor depends on the sign of (3α+β). In 4D, toisotropicandhomogeneousFRWmodelsweshallnow
we can use the Weyl invariant and the Euler density, E, show that the cosmic no-hair theorem for Λ>0 vacuum
defined by [40], cosmologies is not true: there exist ever-expanding vac-
uum universes with Λ > 0 that do not approach the de
1
C Cµνρσ = R Rµνρσ −2R Rµν + R2, Sitter spacetime.
µνρσ µνρσ µν
3
E = R Rµνρσ −4R Rµν +R2, (7)
µνρσ µν
III. EXACT ANISOTROPIC SOLUTIONS
to eliminate the quadratic Ricci invariant in the action,
since
We now present two new classes of exact vac-
αR2+βR Rµν = 1(3α+β)R2+ β (C Cµνρσ −E). uum anisotropic and spatially homogeneous universes of
µν 3 2 µνρσ
3
Bianchi types II and VI with Λ > 0. These are new cosmological constant drives the late-time evolution to-
h
exact solutions of the eqns. (2) with (α,β)6=(0,0). wards the de Sitter spacetime. An exact statement of
Bianchi type II solutions: the theorem can be found in the original paper by Wald
[6]. It requires the matter sources (other than Λ) to
obey the strong-energy condition. It has been shown
a 2
ds2 =−dt2+e2bt dx+ (zdy−ydz) +ebt(dy2+dz2), that if this condition is relaxed then the cosmic no-hair
II 2
h i (10) theorem cannot be proved and counter-examples exist
where [7, 32, 33, 34]. In [35], the cosmic no-hair conjecture
wasdiscussedforBianchicosmologieswithanaxionfield
11+8Λ(11α+3β) 8Λ(α+3β)+1
a2 = , b2 = . (11) with a Lorentz Chern-Simons term. Interestingly, exact
30β 30β BianchitypeII solutions,similartotheonesfoundhere,
These solutions are spacetime homogeneous with a 5- were found which avoided the cosmic no-hair theorem.
dimensional isotropy group. They have a one-parameter However, unlike for our solutions, these violations were
family of 4-dimensional Lie groups, as well as an iso- drivenby anaxionfield whose energy-momentumtensor
latedone(with Lie algebrasAq andA1 ,respectively, violatedthe strong and dominant energy condition. The
4,11 4,9
no-hair theorem for spatially homogeneous solutions of
in Patera et al’s scheme [29]) acting transitively on the
Einstein gravity also requires the spatial 3-curvature to
spacetime. An interesting feature of this family of solu-
be non-positive. This condition ensures that universes
tions is that there is a lower bound on the cosmological
constant, given by Λ = −1/[8(α+3β)] = −a2/8 for do not recollapse before the Λ term dominates the dy-
min
namics but it also excludes examples like that of the
which the spacetime is static. For Λ > Λ the space-
min Kantowski-Sachs S2 × S1 universe which has an exact
time is inflating and shearing. The inflation does not
solution with Λ > 0 which inflates in some directions
resultinapproachtoisotropyortoasymptoticevolution
but is static in others. These solutions, found by Weber
close to the de Sitter metric. Interestingly, even in the
[36], were used by Linde and Zelnikov [37] to model a
case of a vanishing Λ the universe inflates exponentially
higher-dimensional universe in which different numbers
butanisotropically. Wealsonotefromthe solutionsthat
of dimensions inflate in different patches of the universe.
the essential term in the action causing this solution to
exist is the βR Rµν-term andthe distinctive behaviour However,itwassubsequentlyshownthatthisbehaviour,
µν
liketheWebersolution,isunstable[38,39]. Wenotethat
occurs when α = 0. The solutions have no well defined
our new solutions to gravity theories with β 6=0 possess
β →0 limit, and do not have a generalrelativistic coun-
anisotropicinflationarybehaviourwithoutrequiringthat
terpart. They are non-perturbative. Similar solutions
thespatialcurvatureispositiveandaredistinctfromthe
exist also in higher dimensions. Their existence seem to
Kantowski-Sachsphenomenon.
be related to so-called Ricci nilsolitons [30, 31].
The Bianchi type solutions given above inflate in the
Bianchi type VI solutions:
h
presenceofapositivecosmologicalconstantΛ. However,
they are neither de Sitter, nor asymptotically de Sitter;
ds2 =−dt2+dx2 nor do they have initial singularities. Let us examine
VIh
+e2(rt+ax) e−2(st+ah˜x)dy2+e+2(st+ah˜x)dz2 ,(12) howthesemodelsevadetheconclusionsofthecosmicno-
hair theorem. Specifically, consider the type II solution,
h i eq.(10).We define the time-like vector n = ∂/∂t orthog-
where
onal to the Bianchi type II group orbits, and introduce
8βs2+(3+h˜2)(1+8Λα)+8Λβ(1+h˜2) an orthonormal frame. We define the expansion tensor
r2 = ,
8βh˜2 θµν = nµ;ν and decompose it into the expansion scalar,
θ ≡ θµ and the shear, σ ≡ θ −(1/3)(g +n n ),
8βs2+8Λ(3α+β)+3 µ µν µν µν µ ν
a2 = . (13) in the standard way. The Hubble scalar is given by
8βh˜2 H = θ/3. For the type II metric, we find (in the or-
and r, s, a, and h˜ are all constants. These are also ho- thonormal frame)
mogeneous universes with a 4-dimensional group acting 1
θ =2b, σ = diag(0,2b,−b,−b).
transitively on the spacetime. Both the mean Hubble µν
6
expansion rate and the shear are constant. Again, we
As a measure ofthe anisotropy,we introduce dimension-
see that the solution inflates anisotropically and is sup-
less variables by normalizing with the expansion scalar:
ported by the existence of β 6= 0. It exists when α = 0
and Λ=0 but not in the limit β →0.
3σ 1 1 1
µν
Σ = =diag 0, ,− ,− .
µν
θ 2 4 4
(cid:18) (cid:19)
IV. AVOIDANCE OF THE NO-HAIR THEOREM Interestingly, the expansion-normalised shear compo-
nents are constants (and independent of the parameters
The no-hair theorem for Einstein gravity states that α, β, and Λ) and this shows that these solutions vio-
for Bianchi types I − VIII the presence of a positive late the cosmological no-hair theorem (which requires
4
σ /θ → 0 as t → ∞). To understand how this solu- equations,eq.(8),weconsideraperturbationoftheRicci
µν
tion avoids the no-hair theorem of, say, ref. [6], rewrite scalar:
eq.(2) as follows:
R≈4Λ+r eλ1t+r eλ2t.
1 2
G =T , T ≡−Λg −Φ +κT .
µν µν µν µν µν µν
Using(cid:3)R=−(R¨+θR˙),whichisvalidforspatiallyhomo-
In this form the higher-order curvature terms can be geneous universes, eq.(8) again implies, to lowest order:
e e
interpreted as matter terms contributing a fictitious
energy-momentum tensor Tµν. For the Bianchi II so- 3H 2
λ =− 1± 1− ,
lution we find 1,2 2 s 9H2(3α+β)!
e
for(3α+β)6=0. Thisshowsthattheperturbationofthe
T = 1diag 5b2−a2,−3b2+3a2,−7b2−a2,−7b2−a2
µν 4 Ricciscalargivesthesameeigenmodesfortheanisotropic
= diag(ρ,p ,p ,p ). (14)
(cid:0) 1 2 3 (cid:1) solutions of type II and VIh as it did for perturbations
e
of de Sitter spacetime in eq.(9). In order to determine
where ρ and p are the energy density and the principal
eie e e the stability of other modes, like shear and anisotropic
pressures,respectively. Thenohairtheoremsrequirethe
curvature modes, further analysis is required.
dominantenergycondition(DEC)andthestrongenergy
e e
condition (SEC) to hold. However, since ρ˜+p˜ +p˜ +
1 2
p˜ = −3b2 < 0 the SEC is always violated when b 6= 0.
3 V. DISCUSSION
The DEC is violated when ρ˜ < 0 and the weak energy
condition (WEC) is also violated because ρ˜+p˜ = ρ˜+
2
p˜ = −(a2+b2)/2 <0.These violations also ensure that The solutions that we have found raise new questions
3
about the thermodynamic interpretation of spacetimes.
the singularitytheoremswillnotholdforthese universes
We are accustomed to attaching an entropy to the geo-
and they have no initial or final singularities.
metric structure created by the presence of a cosmolog-
Are these solutions stable? Due to the complexity of
ical constant, for example the event horizon of de Sitter
the equations of motion it is difficult to extractinforma-
spacetime. Do these anisotropically inflating solutions
tion about the stability of these non-perturbative solu-
have a thermodynamic interpretation? If they are sta-
tions in general. In the class of spatially homogeneous
ble they may be related to dissipative structures that
cosmologies the dynamical systems approach has been
appear in non-equilibrium thermodynamics and which
extremely powerful for determining asymptotic states of
haveappearedbeenidentified insituationswhere de Sit-
Bianchi models. A similar approach can be adopted to
ter metrics appear in the presence of stresses which vi-
the class of models considered here; however, the com-
olate the strong energy condition [7, 32, 33, 34]. They
plexity of the phase space increases dramatically due to
also provide a new perspective on the physical interpre-
the higher-derivative terms. Nonetheless, some stabil-
tation of higher-order gravity terms in the gravitational
ity results can be easily extracted. Consider, for exam-
Lagrangian.
ple, a perfect fluid with a barotropic equation of state,
In summary: we have found exact cosmological solu-
p = wρ, where w is constant. Due to the exponen-
tionsofa gravitationaltheorythat generalisesEinstein’s
tial expansion, the value of the deceleration parameter
is q ≡ −(1 + H˙/H2) = −1 for the type II and VI by the addition of quadratic curvature terms to the ac-
h
tion. These solutions display the new phenomenon of
solutions given. Hence, these vacuum solutions will be
anisotropic inflation when Λ>0. They do not approach
stable against the introduction of a perfect fluid with
the de Sitter spacetime asymptotically and provide ex-
w > −1. This includes the important cases of dust
amples of new outcomes for inflation that is driven by a
(w = 0), radiation (w = 1/3) and inflationary stresses
p=−ρstressandbeginsfrom’general’initialconditions.
(−1<w <−1/3).
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 Acknowledgment
analysis is lacking. However, in some cases, some of the
modescanbeextracted. ConsideragaintheBianchitype SH was supported by a Killam Postdoctoral Fellow-
II solution, eq. (10). Using the trace of the evolution ship.
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