Cosmology with Eddington-inspired Gravity

1.4 MB·English

by  James H. C. Scargill

Cosmology with Eddington-inspired Gravity
James H. C. Scargill∗
Theoretical Physics, University of Oxford, Rudolf Peierls Centre, 1 Keble Road, Oxford, OX1 3NP, UK
M´aximo Banados†
P. Universidad Cato´lica de Chile, Avenida Vicuna Mackema 4860, Santiago, Chile
Pedro G. Ferreira‡
Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK
Westudythedynamicsofhomogeneous,isotropicuniverseswhicharegovernedbytheEddington-
inspiredalternativetheoryofgravitywhichhasasingleextraparameter,κ. Previousresultsshowing
singularity-avoiding behaviour for κ > 0 are found to be upheld in the case of domination by a
2
perfect fluid with equation of state parameter w > 0. The range −1 < w < 0 is found to lead to
1 3
universes which experience unbounded expansion rate whilst still at a finite density. In the case
0
κ < 0 the addition of spatial curvature is shown to lead to the possibility of oscillation between
2
two finite densities. Domination by a scalar field with an exponential potential is found to also
ct lead to singularity-avoiding behaviour when κ>0. Certain values of the parameters governing the
O potential lead to behaviour in which the expansion rate of the universe changes sign several times
before transitioning to regular GR-like behaviour.

4

] I. INTRODUCTION Inthispaperweconfirmthosefindings(intheaddition
O
of spatial curvature to the Friedmann-Robertson-Walker
C (FRW) metric), and present other interesting behaviour
In 1924 Eddington proposed a theory of gravity based
. viz. universes which experience unbounded expansion
h solelyonaconnectionfield,withoutintroducingametric.
rate at finite density akin to undergoing a second big
p A particular extension of Eddignton’s theory including
- matter fields was recently considered in [1] (see also [2]). bang/‘cosmic hiccup’, and closed universes which oscil-
o late between two finite values of the density. Finally we
Inavacuum,thistheoryreproduces’Eddington-inspired’
r
t (or just ‘Eddington’) gravity and is completely equiva- investigate the dynamics of universes in which the dom-
s
inant contribution to the stress-energy tensor is due to
a lent to General Relativity (GR), a fact which should be
[ contrasted with other modified gravity theories in which a scalar field with an exponential potential, as has been
vacuum deviations are expected (e.g. metric f(R) the- done in the case of GR [9].
1
v ories [3]). On the other hand deviations from GR are This paper is structured as follows: section IA covers
1 seen in regions of high density. Where one would ex- some basic definitions and the governing field equations;
2 pect singularities such as the interior of black holes or section IB exhibits the Friedman equation and qualita-
5 the birth of the universe, in Eddington inspired gravity tivelycomparesittoGR;sectionIIexaminessolutionsto
1 often these are avoided [1, 4, 5] (although other singu- thiswithdominationbyonekindofperfectfluid;section
0. larities may appear in an astrophysical context [6]). The IIBdemonstratesthe‘loitering’behaviourwhichhaspre-
1 appearanceofsuchbehaviourinapurelyclassicaltheory viously been observed; section IIC investigates the ‘cos-
2 such as Eddington gravity is interesting and exciting. mic hiccup’ mentioned above; section IID exhibits the
1 oscillating universes; section III covers the case of domi-
The cosmological implications of this theory were ini-
:
v tially considered in [1], and studied in more depth in nationbyascalarfield;sectionIIIAinvestigatesthe‘loi-
i tering’ behaviour which has been observed for the case
X [5] and [7]; it was found that universes containing ordi-
of regular matter fields and which shows up again in the
nary matter avoid singularites, with the exact behaviour
r case of scalar fields; section IIIB exhibits someuniverses
a depending on the sign of one of the parameters of the
whichshow qualitativelynewbehaviour-expansion, fol-
theory, and being either a regular bounce (from contrac-
lowed by contraction, followed by expansion. In section
tion to expansion) or ‘loitering’ around a minimum size
IV we conclude.
beforetransitioningtoGR-likeexpansion. Ithasalsore-
cently been shown that tensor perturbations may render
the regular behaviour of such cosmological models un-
stable near the minimum scale (either in the asymptotic A. Definitions and Field Equations
past or at the bounce [8]).
The action governing Eddington gravity is [1]
2 (cid:90) (cid:104)(cid:112) (cid:112) (cid:105)
∗Electronicaddress: [email protected] S[g,Γ,Ψ]= d4x |g+κR(Γ)|−λ |g| +S [g,Ψ],
†Electronicaddress: maxbanados@fis.puc.cl κ M
‡Electronicaddress: [email protected] (1)
2
(throughoutthispaperweusereducedPlanckunits, c= and we take the auxiliary metric to be
8πG = 1) where R(Γ) denotes the symmetric part of
theRiccitensor,andλisadimensionlessconstantwhich qµνdxµdxν =−U(t)dt2+a(t)2V(t)
must satisfy λ = 1+κΛ if GR is to be recovered in the (cid:18) dr2 (cid:19)
limit of κR (cid:28) g (though the cosmological constant will × +r2(dθ2+sin2θdφ2) (9)
1−k r2
q
be assumed negligible in the following). Matter can, of
course,becoupledtotheconnectionaswellasthemetric, The energy momentum tensor is
however for simplicity here it is assumed to only couple
(cid:32) (cid:33)
to g. Two field equations are derived from this action:
(cid:88) (cid:88)
T = ρ +ρ and T =a2 P +P δ ,
00 i φ ij i φ ij
q =g +κR (q), (2)
µν µν µν i i
(cid:112)|q|qµν −λ(cid:112)|g|gµν =−κ(cid:112)|g|Tµν, (3) (10)
where {i} indicate the regular matter, and φ the scalar
inwhichqisanauxiliarymetricwhichiscompatiblewith field. Finally
the connection i.e. Γα = 1qαν(∂ q +∂ q −∂ q ),
βγ 2 β νγ γ βν ν βγ
and qµν ≡[q−1]µν. ρ = 1φ˙2+V(φ) and P = 1φ˙2−V(φ). (11)
The action (1), and its equations of motion (2,3), can φ 2 φ 2
also be derived from the following bigravity theory,
The Eddington gravity field equations enforce k = k
q
(cid:90) √ and
I[g,q] = q(qµνR (q)−2Λ) (4)
µν
(cid:115)
√ √ √
+ Λ( qqµνgµν −2 g)+ gLm(g) U = (1−κPT)3 and V =(cid:112)(1+κρ )(1−κP ),
1+κρ T T
with T
(12)
1 (cid:80) (cid:80)
Λ= κ (5) where ρT = iρi+ρφ+Λ and PT = iPi+Pφ−Λ.
The energy-momentum conservation equations are as
Varying(4)withrespecttoqµν andgµν oneobtainsaset in GR, so
ofequationsfullyequivalentto(2,3). Thisformoftheac-
tionisparticularlyinterestingbecauseitshowsaremark- φ¨+3Hφ˙+V(cid:48)(φ)=0 (13)
able close relation with the recently discovered family of ρ˙ +3Hρ (1+w )=0. (14)
i i i
unitary massive gravity theories [10]. Indeed, massive
gravities are built as bigravity theories where the poten- In the general case
tialisalinearcombinationofthesymmetricpolynomials
(cid:32) (cid:114) (cid:33)
of the eigenvalues of the matrix a˙ 1 G
H ≡ = ± −B (15)
(cid:112) a F 6
γ = q−1g. (6)
For this class of potentials the theory has no Boulware- with
Deser instability and, in particular, the massive sector
(cid:20) (cid:21)
carries 5 degrees of freedom instead of 6. Besides the 1 U 6κk
G= 1+2U −3 − (16)
cosmological constants (which also appear in (4)), one κ V a2
such symmetric polynomials is (cid:26) (cid:20)
(cid:88)
√q(cid:0)Tr(γ2)−(Trγ)2(cid:1)=√q(qµνg −(Trγ)2) (7) F =1− 3κ (1−wi−κ(wiρT +PT))ρi(1+wi)
µν
i
The first term is precisely the interaction between q (cid:21)(cid:27)(cid:30)
µν −κ(ρ +P )φ˙2 4(1+κρ )(1−κP ) (17)
and g appearing in (4), resulting in Eddington theory T T T T
µν
coupled to matter. The second piece (more difficult to
1 V(cid:48)φ˙
compute because the square root has to be taken before B = κ . (18)
thetrace)isnotpartoftheEddingtontheory. Wefindit 2 1−κPT
intriguing that Eddington’s theory coupled to matter is
One key thing to note is that the scalar field does not
so close to the recently discovered stable massive gravity
simplyenterinthesamewayasotherformsofmatter(as
theories. We shall exploit this connection elsewhere.
is the case in GR), which leads to the interesting result
thatalthoughtherearetwosolutionsforH,theyarenot
simply opposite signs, as in GR and Eddington gravity
B. Friedman Equation
without a scalar field.
If one assumes that sign(V˙) = −sign(H), i.e. as the
The FRW metric is
universe increases in size, the potential energy of the
(cid:18) dr2 (cid:19) scalar field decreases, then in fact the values for H are
ds2 =−dt2+a(t)2 +r2(dθ2+sin2θdφ2) ,
1−kr2 simplyofdifferentsigns. Howeverwhilstthisassumption
(8) may hold for the low-density GR-like stage of evolution,
3
itfailsinthehighdensitycasewithsomeinterestingcon- have to have in order to evolve as curvature).
sequences as laid out in section IIIB
Another departure from GR is that spatial curvature Finally, in the low density and curvature limit
cannot be considered a fluid, since it does not enter in (ρ , ρ , Λ, k/a2 (cid:28) κ−1) GR is recovered to leading
i φ
the same way as a fluid with w = −1 (which it would order:
3
(cid:114)
ρ k
H =± T −
3 a2
+κ(cid:26)±(cid:114)ρT − k (cid:20)3(cid:88)ρ (1−w2)+ 1 (cid:18)ρT − k (cid:19)−1(cid:18)15P2 − 9ρ P − 17ρ2 + 6k(P +ρ )(cid:19)(cid:21)− 1V(cid:48)φ˙(cid:27)
3 a2 4 i i 12 3 a2 8 T 4 T T 8 T a2 T T 2
i
+O((κρ )2). (19)
T
II. DYNAMICS OF A UNIVERSE WITHOUT A
TABLE II: Sign of ρ in various regions of parameter space,
SCALAR FIELD
withnospatialcurvature. ρ¯ satisfiesG(ρ¯ )=H2(ρ¯ )=
G∗ G∗ G∗
0. (w=−1 is special but also has ρ≥0.)
We first consider the dynamics of universes which do
not posses scalar fields, have a negligible cosmological w ρ¯ sign(κ) sign(ρ)
constantand,forsimplicity,onlyonetypeofperfectfluid. any 0<ρ¯ + +
The Friedman equation in this case is −1<w −1<ρ¯<0 − +
−1<w<0 ρ¯ <ρ¯< 1 + −
G∗ w
ρ¯<ρ¯ − +
G G∗
H2 = (20) w<−1 ρ¯ <ρ¯<0 − +
6F2 G∗
= 8 (cid:104)(1+3w)ρ¯−2+2(cid:112)(1+ρ¯)(1−wρ¯)3 w1 <ρ¯<ρ¯G∗ + −
3κ ρ¯<−1 − +
6k¯ (cid:105)
− (1−wρ¯)
a2
The requirement H2 ≥ 0 fixes the sign of ρ and table
(1+ρ¯)(1−wρ¯)2
× , (21) II shows the results (for k = 0), from which we see that
(4+(1−w)(1−3w)ρ¯+2w(1+3w)ρ¯2)2
that the energy density is positive in all but two regions
ofparameterspace. Thisisaninterestingdeparturefrom
where ρ¯=κρ and k¯ =κk.
GRwhich,inthespatiallyflatcase,alwaysrequiresnon-
Note that setting w = −1 (so that the universe con-
negative energy densities.
tains only a cosmological constant) reduces eqn. (21) to
Aspointedoutin[7], afinalinterestingqualityof(21)
exactlythatwhichwouldbeexpectedinGR,asitshould
tonoteisthatforcertainrangesofw,H2 →∞forfinite
be since Eddington gravity exactly reproduces GR in a
ρ. SectionIICinvestigatesauniversewhichexhibitssuch
vacuum (with a cosmological constant).
behaviour.
Lookingatthediscriminantofthequadraticinthede-
nominator of (21), we see that the discontinuities do not
A. Behaviour of the Friedman equation √ √
exist for 0.02 ≈ 1(7−4 3) < w < 1(7+4 3) ≈ 4.64.
3 3
Outsideofthisregionwecandeterminegraphically,given
Thesquarerootinthenumeratorrestrictsthedomain
the constraints of table I, in which regions the disconti-
of H2 and the result is shown in table I.
nuities will appear (see appendix A), and the results are
in table III.
TABLE I: Domain restrictions placed on H2 by the square Points at which H2 = 0 and dH2 (cid:54)= 0 will induce an
dρ
root expanding universe to collapse (and vice versa) since the
w Valid ρ¯ range universecannotevolvefurtheronthistrajectory(because
H2 ≥ 0) and the stationary point is unstable since only
w>0 −1≤ ρ¯ ≤ 1
w H2, and not the sign of H, is fixed by the physics. Such
w=0 −1≤ ρ¯
‘bounce’ points exist in GR (if k >0), and in Eddington
−1≤w<0 ρ¯≤ 1 or −1≤ρ¯
w gravity as well, i.e. the points ρ¯ = −1 and ρ¯ = ρ¯G∗.
w≤−1 ρ¯≤−1 or w1 ≤ρ¯ We are not including ρ = 0 here because, although H2
4
TABLE III: Regions in which discontinuities of H2 appear and are not outside of the allowed range of ρ¯.
w<−1 w=−1 −1<w<−1 w=−1 −1 <w<0 0≤w<4.64... 4.64...≤w
3 3 3
ρ¯ <ρ¯<0 - ρ¯<ρ¯ - 0<ρ¯ - −1<ρ¯<0
G∗ G∗
behavesasrequired,startingfromanon-zerodensitythis Calculatingtheparticlehorizonforsuchauniversewe
point cannot be reached in a finite time. find
From(21)weseethatatρ¯=w−1, H2 =0 and dH2 =
0. Thisnoveltypeofstationarypointleadstothe‘ldoρiter- r (t)= lim (cid:90) t dt(cid:48) = lim t−t−, (26)
ing’ behaviour remarked upon in [1, 5], and [7], in which p t−→−∞ t− a(t(cid:48)) t−→−∞ aB
theuniverseasymptoticallyapproaches(orrecedesfrom)
amaximumdensity/minimumsize. FromtableIIwesee
which diverges. Again this is similar to GR under domi-
that for a positive energy density, this type of universe
nation by a cosmological constant.
will only exist for w >0 and κ>0.
In [4] the authors note that Eddington gravity can be
characterised as making the gravitational force repulsive
at short distances (or high densities), which would agree
B. The case κ>0, w>0
with the behaviour which we see here. That is, the max-
imum density is reached as the repulsive effects of Ed-
As seen in figure 1, this type of universe exhibits the
dington gravity oppose the traditional attractive effects
‘loitering’ behaviour mentioned in the previous section.
ofthematterdensity; whichrepulsiveeffectstakeaform
We also see that to a good approximation the evolu-
similar to a cosmological constant with Λ=8κ−1.
tion can be split into two phases: the loitering phase in
which deviations from GR are important, and the GR-
like phase in which the dynamics of the expansion are
approximately as given by GR.
2. Behaviour of the auxiliary metric
1. Loitering phase From eqn. (9) we see that the auxiliary metric q is
µν
identical to the metric g except that the temporal and
µν
Closetothemaximumdensityρ¯≡ρ¯ =w−1 theterm spatial components of q are modulated by the functions
B
involving k is much smaller than the other terms in the U and V respectively. Alternatively, by factoring out U,
numerator of (21), and so we can neglect (small) spatial itcanbeconsideredtoberelatedtoanFRWmetricwith
(cid:113)
curvature during the loitering phase. Expanding about modifiedscalefactora =a V,byanoverallconformal
q U
themaximumdensity,orequivalentlytheminimumscale transformation U. Thus U and V, or U and a explain
q
factor aB, to highest order the Friedman equation and howthespacedescribedbytheauxiliarymetricbehaves.
deceleration parameter become
Figure 1 shows the behaviour of U, and V. As ex-
8 (cid:18) wδρ¯ (cid:19)2 8 (cid:18)δa(cid:19)2 pected, intheGRregimeU, V →1. Ontheotherhand,
H2 = = (22) close to the maximum density U, V → 0, and the space
3κ 3(1+w) 3κ a
B described by q becomes vanishingly small. Interestingly,
(cid:18) wδρ¯ (cid:19)−1 (cid:18)δa(cid:19)−1 V reaches a maximum value just before the GR regime
q = =− (23)
dec 3(1+w) a begins.
B
Expanding close to the maximum density we find
(where δρ¯= ρ¯−ρ¯ and δa = a−a ). Using H = a˙ =
B B a
δ˙a , (22) can be solved to get (cid:114) (cid:32) (cid:114) (cid:33)
aB+δa (3(1+w))3 3 8
U = exp (t−t ) (27)
(cid:32)(cid:114) (cid:33) 2 2 3κ 0
a(t) 8
aB =1+exp 3κ(t−t0) (24) (cid:114)3(1+w) (cid:32)1(cid:114) 8 (cid:33)
V = exp (t−t ) (28)
(cid:32) (cid:114) (cid:33) 2 2 3κ 0
8
qdec =−exp − 3κ(t−t0) . (25) a (cid:32) 1(cid:114) 8 (cid:33)
a = B exp − (t−t ) . (29)
q 3(1+w) 2 3κ 0
Thusweseethattheuniverseundergoes(accelerating)
exponentialexpansionawayfromtheminimumscalefac-
tor during this phase. The form of a(t) is reminiscent of And so we see that, moving towards the maximum
thatobservedinGRunderdominationbyacosmological density, the temporal size decreases more rapidly than
constant, specifically with Λ=8κ−1. the spatial, with the result that a actually goes infinite.
q
5
0.15 4
Exact Soln.
Loitering Approx.
3
GR Approx.
0.12
B
a 2
/
a
0.09 1
2
H 0
(cid:103) 1.2 U
0.06 V
0.8
0.03
0.4
0 0
0 0.2 0.4 0.6 0.8 1 0 5 10 15
(cid:108)/(cid:108) t
B
FIG. 1: Left: Expansion rate of the universe against density (normalised by the maximum density ρ = (κw)−1), for κ = 1,
B
w=1/3, k=0.
Right Top: Scale factor (normalised by the minimum scale factor a ) against time (in reduced Planck units), for the full
B
numerical solution to the Friedman equation, and approximations in two phases: loitering, as in section IIB1, and GR-like
low-density limit.
Right Bottom: Behaviour of the parameters describing the auxiliary metric (for the same universe).
3. The case w=0 Expanding around this point, to leading order we find
8 (cid:18)δa(cid:19)−2
Thiscaseexhibitsthesameloiteringbehaviouratearly H2 = f2 (33)
3κ w a
times, but must be treated separately, since the density ∗
is unbounded; expanding about a = 0, to leading order with
we find: (cid:104) (cid:112)
f2= (1+3w)ρ¯ −2+2 (1+ρ¯ )(1−wρ¯ )3
w ∗ ∗ ∗
H2 = 8 (30) −6k¯(1−wρ¯ )(cid:105) 1
3κ a2 ∗ (3ρ¯ (1+w))2
∗ ∗
qdec =−1, (31) (1+ρ¯ )(1−wρ¯ )2
× ∗ ∗ . (34)
((1−w)(1−3w)+4w(1+3w)ρ¯ )2
∗
which is identical to GR with a cosmological constant
Λ=8κ−1. Therefore the particle horizon will diverge as This can be solved as in section IIB1 to get
in the case w >0. (cid:115)
(cid:114)
ThebehaviourofU anda isthesameasinthew >0 a(t) 8
q =1± 2f |t−t |. (35)
case, however in contrast, V diverges as t → −∞. See a w 3κ ∗
∗
appendix B1 for the explicit functional forms.
Thus at this critical point the universe in some sense
undergoes a second big bang, with expansion similar to
√
that of GR with domination by radiation (a∝ t). Fig-
ure 2 demonstrates this universe.
C. The case κ>0, −1 <w<0
3 Expanding eqn. (21) for high densities we find
Firstly it should be noted that new physics (beyond H2 = 4 |w|23 ρ. (36)
Eddington gravity) is required in order to have a fluid 3(1+3w)2
with w < 0 dominating in the early universe; this prob-
Theexpansionintheearlyhistoryofthisuniverseisalso
lem is not considered here.
3
AsshownintableIII,insuchauniversetheexpansion GR-like, albeit with a modified density ρ˜ = 4|w|2 ρ,
(1+3w)2
ratedivergesatafinitedensitygivenbythepositiveroot and the spatial curvature can be neglected. The scale
of the denominator of eqn. (21) factor as a function of time is then
√ (cid:32)(cid:114) (cid:33) 2
(1−w)(1−3w)+(1+w) 1−42w+9w2 a(t) = ρ˜∗3(1+w)t 3(1+w) (37)
ρ¯ = . (32)
∗ 4|w|(1+3w) a∗ 3 2
6
100 2
Exact Soln.
Hiccup Approx.
1.5
80
*
a 1
/
a
60 0.5
2
H 0
(cid:103) U
40 8 V
6
20 4
2
0 0
0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5
(cid:108)/(cid:108) t
*
FIG. 2: Left: Expansion rate of the universe against density (normalised by the critical density ρ ), for κ = 1, w = −1/6,
∗
k=0.
RightTop: Scalefactor(normalisedbythecriticalscalefactora )againsttime(inreducedPlanckunits),forthefullnumerical
∗
solution to the Friedman equation, and an approximation close to the critical density.
Right Bottom: Behaviour of the parameters describing the auxiliary metric (for the same universe).
Inordertobetterunderstandthedynamicsoftheuni- high densities more quickly than ρ¯ → ∞, as indeed it
∗
verse during this second big bang/‘cosmic hiccup’ we should if the high density behaviour observed in section
shouldconsiderthehorizonbetweentwotimest <t < IIB3 is to be recovered when w =0.
− ∗
t close to t :
+ ∗
t −t
r(t ,t )= + − +O((t −t )2). (38)
− + a ± ∗
∗ 2. Behaviour of the auxiliary metric
Its well-behaved nature should not be surprising, how-
ever, since the nature of the expansion during the ‘hic-
In the high density limit the overall size of the space
cup’ has already been noted.
describedbyqdiverges,butthecombinedeffectofU and
V is that a remains proportional to the regular scale
q
factor.
1. Length of first life
At the ‘hiccup’ U and V as functions of ρ¯ or a are
smooth and well behaved, but as functions of t they of
An interesting quantity to consider is t , the time be-
∗ course exhibit a ‘kink’ just as a(t) does (see figure 2).
tweenthebirthoftheuniverseandthe‘hiccup’. Naively,
sinceρ¯ →∞forw → 0, −1,itmightbeexpectedthat Explicit functional forms in these two limits can be
∗ 3
t →0 in these limits. found in appendix B2.
∗
We have
(cid:90) t∗ (cid:90) a∗ da (cid:90) ∞ dρ
t = dt= ∝ . (39)
∗ aH(a) ρH(ρ)
0 0 ρ∗ D. Oscillating universes
Whilst this cannot be evaluated analytically in the gen-
eral case, certain approximations can be made. For In [1, 5] and [7] it is noted that for κ < 0, w > 0
w → −1, ρ ∝ 1 , and H is given by eqn. (36) and the universe undergoes a ‘regular’ bounce (as described
3 ∗ 1+3w
so the behaviour of eqn. (39) can be approximated by below) in which, in a finite time, a collapsing universe
t ∝ (cid:82)ρ∗ (1+3w)dρ = 2(1+3w) ∝ (1+3w)3/2 → 0, as ex- reaches a maximum density and then expands. In GR,
∗ ∞ ρ3/2 ρ1∗/2 universes with positive spatial curvature can undergo a
pected.
bounce in the opposite direction.
However for w → 0, ρ → 1 and so eqn. (39) be-
∗ 2|w| TheadditionofspatialcurvaturetoEddingtoncosmol-
haves: t ∝ (cid:82)ρ∗ dρ = 2 ∼ |w|−1/4 → ∞,
∗ ∞ |w|3/4ρ3/2 |w|3/4ρ1∗/2 ogy thus leads to the interesting prospect of oscillatory
whichdefiesthenaiveexpectations. Thisoccursbecause behaviour. Figure 3 demonstrates such an oscillatory
the expansion rate of the universe heads towards zero at universe.
7
0.08
1.5
0.06 B
a
/
a
Exact Soln.
High Density Approx.
2H 1 GR Approx.
0.04
(cid:103)|
|
1.5
0.02 1
0.5 U
V
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15
(cid:108)/(cid:108) t
B
FIG. 3: Left: Expansion rate of the universe against density (normalised by the maximum density ρ = −κ−1), for κ = −1,
B
w=1/3, k/a2 =1/10.
B
Right Top: Scale factor (normalised by the minimum scale factor a ) against time (in reduced Planck units), for the full
B
numerical solution to the Friedman equation, and approximations in two phases: close to the maximum density, and GR-like
low-density limit.
Right Bottom: Behaviour of the parameters describing the auxiliary metric (for the same universe).
1. Amplitude of the oscillations BoundscanbeplacedonkbyexaminingtheFriedman
equation in these two limits (since the behaviour of eqn.
The maximum density, ρ = −κ−1, is independent (21) as a function of ρ¯ and k is smooth) and requiring
B
of w and k, and is caused by the (1+ρ¯) factor in the H2 ≥ 0. The GR-like regime gives a lower bound on k
numerator of eqn. (21). The minimum density, on the whilst eqn. (42) gives an upper bound, which combine
other hand, does depend on w and k (and κ), and is the to give
solution to G=0, which explicitly is
k 1
(cid:112) 0< < . (44)
(1+3w)ρ¯−2+2 (1+ρ¯)(1−wρ¯)3 a2 2|κ|
B
−6k¯a−2(1−wρ¯)=0. (40)
Unfortunately this cannot be solved analytically in the
general case, however if we assume that the universe has
2. Period of the oscillations
expandedsufficiently beforeexpansion haltsthenwecan
solve this in the GR regime to get
Inprincipletheperiodoftheoscillationscanbecalcu-
amax =(cid:18)3k(cid:19)−1+13w , ρmin =(cid:18)3k(cid:19)31(+1+3ww) (41) lated from
a a2 ρ a2
B B B B (cid:90) Tosc/2 (cid:90) amax da
T =2 dt=2 . (45)
Fromfigure4weseethatthisisindeedagoodapprox- osc aH(a)
imation for k/a2 (cid:46)0.2. 0 aB
B
Close to the maximum density we have Inpracticethisintegralcannotbeperformedanalytically
(orgoodenoughanalyticalapproximationsmade)andso
8 (cid:18) 2k¯(cid:19) δa
H2 = 1+ , (42) must be evaluated numerically, as shown in figure 4.
3|κ| a2B aB We see that Tosc(w = 13) < Tosc(w = 0), which makes
sense since the amplitude of the oscillations for the for-
which is solved to give
mer case is always smaller than in the latter. Also note
a(t) =1+ 2 (cid:18)1+ 2k¯(cid:19)|t−t |2. (43) that Tosc → 0 for k/a2B → 1/2. From eqn. (42) we see
a 3|κ| a2 B that the expansion rate goes to zero at this point, and
B B so from section IIC1 we might expect that T → ∞,
osc
Notethatthisisindependentofthetypeofmatterwhich however in this case the extra size by which the universe
fills the universe (just as in section IIB1), and so we mustexpandalsoheadstozero(andclearlydoessomore
understand the expansion is being driven by κ. quickly than the expansion rate).
8
3. Behaviour of the auxiliary metric
1
w = 0 Exact Soln.
0.8 w = 0 GR Approx.
(cid:108)B 0.6 ww == 11//33 EGxRa cAt pSporlonx.. Close to the maximum density the spatial size de-
/n creases to zero as the universe reaches is minimum size,
mi 0.4 whilst the temporal component of the auxiliary metric
(cid:108)
0.2 diverges. See appendix B3 for approximate functional
0 forms for these.
w = 0
20
w = 1/3
c 15
s
o
T 10
5
0
0.0 0.1 0.2 0.3 0.4 0.5
2
k/a
B III. DYNAMICS OF A UNIVERSE WITH A
SCALAR FIELD
FIG. 4: Top: Amplitude (minimum density, normalised by
maximumdensity)oftheoscillationsdescribedinsectionIID
againstspatialcurvatureoftheuniverse,forκ=−1andvari-
ousvaluesfortheequationofstateparameter. Fullnumerical We now consider the cosmology when a scalar field is
solutions, as well as approximations in the GR regime from introduced; the ordinary matter content, the cosmolog-
eqn. (41). ical constant, and the spatial curvature are considered
Bottom: Period of the oscillations (in reduced Planck units) negligible. Written in terms of φ and V(φ) the FRW
for the same universes. equation is
(cid:118)
(cid:34) (cid:117) (cid:32) (cid:40) (cid:34) (cid:114) (cid:35)(cid:41)(cid:33)
H = ±(cid:16)1− κφ˙2+κV(cid:17)(cid:117)(cid:116)(cid:16)1+ κφ˙2+κV(cid:17) 1 φ˙2−V − 1 1+ (cid:16)1+ κφ˙2+κV(cid:17)(cid:16)1− κφ˙2+κV(cid:17)3
2 2 3 κ 2 2
(cid:35)
κ(cid:16) κ (cid:17) 1
− 1+ φ˙2+κV V(cid:48)φ˙ . (46)
(cid:16) (cid:17)
2 2 1+2κV +κ2 1φ˙4+V2
2
The positive sign is taken, as this gives a universe The solution for φ(t) also shows two regimes with a
which is expanding in the GR regime. The conservation transition between them (figure 5, bottom right panel).
equation governing the scalar field is The GR regime shows the expected logarithmic time
dependence [9], whilst the Eddington/loitering regime
φ¨+3Hφ˙+V(cid:48)(φ)=0. (47)
shows φ∝−t.
We restrict ourselves to an exponential potential for the
scalar field
1. Initial conditions and free parameters
V(φ)=V exp(−λφ) (48)
0
withV , λ>0. Unlessexplicitlyincluded,theparameter It is appropriate first to briefly discuss the initial con-
0
describing deviations from GR (κ) is set to one. ditions used to produce the solutions plotted above as
wellastheeffectonthesethatishadbyalteringthefree
parameters of the system.
A. Loitering behaviour Ascanbeseenfromfigure5,thechosenconditionsare
a(t=0)=1; φ(0)=0; φ˙(0)=0. (49)
For κ>0 numerical solutions of eqn. (46) show qual-
itatively similar behaviour to the case without scalar The first is uncontroversial since the scale factor can al-
modes, viz. ‘loitering’ of the scale factor around a mini- ways be rescaled by a constant. The absolute value of
mumvaluefollowedbyatransitiontoGR-likepower-law the scalar field only appears in the form for its poten-
expansion. See figure 5, top left panel. tial energy, and thus adjusting the field by a constant,
9
4 1
V =1/4, (cid:104)=2
0
V =1/4, (cid:104)=3
0
V =1/2, (cid:104)=2 0.8
0
3 V =1/2, (cid:104)=3
0
0.6
a 2 (cid:108)(cid:113)
0.4
1
0.2
0 0
0.8
1.6
0.4
1.2
(cid:113) w 0
0.8
-0.4
0.4
-0.8
0
-3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5
t t
FIG.5: Scalefactor(topleft),scalarfield(bottomleft),energydensity(topright)andequationofstateparameter(w≡P/ρ)
(bottom right) as a function of time (in reduced Planck units) for various values of the free parameters.
which is the action of the second i.c., only has the ef- tering to the GR phase.
fect of changing V . The theory is obviously invariant
0
under a constant time shift and the third i.c. just fixes
this. The reader may be worried that using φ˙ = 0 as
2. Loitering regime
a i.c. eliminates from consideration any solutions which
showmonotonicallyincreasingfieldvalues, howeversuch
solutions would not exhibit the maximum density and Let us investigate this loitering behaviour in more de-
loitering behaviour (since if φ is unbounded from below tail. As in the case with ordinary matter the loitering
the density will also be unbounded, and if φ tends to regime corresponds to P → κ−1 (as t → −∞); we also
someconstantvalue, thepressurewouldbenegativeand see that φ → ∞ and hence V(φ) → 0. Thus one has
(cid:113)
hence not satisfy P =κ−1). ρ≈P and φ→− 2t+const.
κ
Allthat’sleftistodescribethepotential, whichleaves
Evaluating the Friedman equation in this regime, one
twofreeparameters: V andλ. Fromthefiguresthusfar
0 finds
presented we can roughly evaluate the effect of making
thepotentialsteeper(increasingλ)ordeeper(increasing 2(1+κρ)V(cid:48) 1√
V ). H(P =κ−1)= = 2κV(cid:48). (50)
0 3κφ˙3 3
The late time (GR-like) behaviour is primarily gov-
ernedbythevalueofλ,whereastheearlytimebehaviour
Therefore the equation of motion for the scalar field re-
is influenced by V0. The reason for the latter is that the duces to φ¨+3V(cid:48) =0. The solution (setting κ=1) is
energy density of the field is already fixed as t → −∞,
and picking a value for V0 fixes it at t = 0 (or wher- √ 1 (cid:104) √ (cid:105)
ever the transitionary period is located), and thus the φ(t)=− 2(t−t )+ 2ln(12V +e 2λ(t−t0))−4ln2
0 λ 0
behaviourinbetweenthesetwopointswillbegreatlyin-
(51)
fluenced by the size of the change in the energy density.
Furthermore one can roughly say that steeper and a(t)≈a exp(cid:18)16V0 exp√2λ(t−t )(cid:19). (52)
deeperpotentialsleadtoquickertransitionsfromtheloi- min 3 0
10
The dynamic nature of the scalar field adjusts the loi-
tering behaviour of the scale factor from the simple ex- 0
V =1/4, (cid:104)=2
0
ponentialbehaviourseeninthecaseofregularmatterto V =1/4, (cid:104)=3
0
the more extreme double exponential. -0.5 V =1/2, (cid:104)=2
0
V =1/2, (cid:104)=3
0
-1
3. Transitionary Regime
(cid:108)
g -1.5
o
Whereasinthecaseofordinarymatterfieldstheevolu- l
tioniswelldescribedwithouthavingtodenoteaseparate -2
transitionary regime (between the high-density loitering
behaviour and the low-density GR-like regime), that is
-2.5
not the case here.
The transition from linearly decreasing field values to
logarithmically increase is naturally centred on φ˙ = 0. --30.5 0 0.5 1 1.5
Hence during the transition P ≈−ρ: the scalar field has log a
the equation of state of a cosmological constant. This
is readily observed in plots of ρ(t) which show that the FIG.6: Energydensityasafunctionofscalefactorforvarious
density is stationary around the transition. See figure 5, values of the free parameters.
top right panel.
Thebottomrightpanelofthisfigurealsoconfirmthis,
aswellastheabovecommentthatP ≈ρintheloitering B. Expand-contract-expand behaviour
regime (w → 1 as t → −∞, regardless of the values of
the free parameters). As noted in section IB, the two solutions for the Hub-
bleparameterarenotingeneralsimplyofthesamemag-
nitude but with opposite sign. This leads to the possi-
4. Density Scaling bility that there are universes which at some point are
stationary, andtheninsteadofcontractingandretracing
Finally it is interesting to look at the plot of lnρ their evolutionary history in reverse, they contract but
against lna (figure 6). This shows that during the loi- trace a new history.[11]
tering phase the density scales as ρ∝a−6, which agrees In fact since the solutions being investigated are ex-
with the fact that w ≈ 1 - the scalar field is, to use pandingbothintheirearlyandlatehistories,theywould
the terminology used in [9], undergoing kination (kinetic undergo expansion, followed by contraction, followed by
energy domination). This is followed by the expected expansion.
cosmological constant-like transitionary phase in which From figure 7 we see that such behaviour can be
the density is stationary. The field energy density shows achieved both by making the potential steeper and by
the expected scaling in the GR phase (ρ ∝ a−λ2 if λ < making it deeper. The latter can easily be understood
√
6 and ρ∝a−6 otherwise). since V0, which controls the depth of the potential, also
controls the energy density of the scalar field at its tran-
Figure 1 from [9] shows the evolution of the energy
sitionarypoint. ThuschoosingasufficientlylargeV will
density of a scalar field in a GR-universe populated by 0
ensure that the density at this point is larger than the
the aforementioned scalar field (with λ = 4) as well as
density in the distant past (loitering phase), and hence
radiation and cold matter. The scalar field is initially
theuniversemustundergocontraction(seefigure7, bot-
dominantandscalesaccordingly(ρ∝a−6),becomessub-
tom panel).
dominant and scales like a constant, before finally track-
That steeper potentials induce this behaviour can be
ing (i.e. scaling as) the cold matter (ρ ∝ a−3) which is
understood mathematically from the fact that the ‘B’
then dominant.
term in the Friedman equation (eqns. 15, 18) is propor-
Thisbearsaremarkableresemblancetofigure6,except
tional to V(cid:48), and hence to λ. Physically, perhaps one
for two major differences:
canimaginethecausetobethatapotentialwhichvaries
1. the behaviour in the Eddington case is reversed. quicklyenoughwithφleadsthescalarfieldto‘overshoot’
The ‘natural’ scaling (i.e. that which one would the value which would ensure energy density monotoni-
expect in GR based on the value of λ) is preceded cally decreases with time; this larger density would then
by a period in which the scaling is fixed (by be associated with a contraction.
something other than λ); It is interesting to note that plots of the field value, or
oftheequationofstateparameterdonotobviouslyshow
any affect of the period of contraction or differ qualita-
2. in the Eddington case such behaviour is achieved tively from the corresponding plots for free parameter
without another fluid for the field to track. values which show the ordinary behaviour (figure 5).