Generalized Swiss-cheese cosmologies: Mass scales

0.38 MB·English

by  Cédric Grenon

Generalized Swiss-cheese cosmologies: Mass scales
C´edric Grenon∗ and Kayll Lake†
Department of Physics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6
(Dated: January 11, 2010)
We generalize the Swiss-cheese cosmologies so as to include nonzero linear momenta of the asso-
ciated boundary surfaces. The evolution of mass scales in these generalized cosmologies is studied
for a variety of models for the background without having to specify any details within the local
inhomogeneities. We find that the final effective gravitational mass and size of the evolving inho-
mogeneities dependson theirlinear momenta buttheseproperties are essentially unaffected bythe
details of thebackground model.
0
1
0 I. INTRODUCTION A. Junction Conditions
2

n To establish notation, let us write the metrics of the
a The Swiss-cheese models give us (noninteracting) in-
spacetimes in the form [7]
J homogeneities in a cosmological setting that are exact V±

1 solutions to Einstein’s equations. As a result, the mod- ds2 =ds2 (x1,x2)+R2(x1,x2)dΩ2, (1)
1 elshavebecome astandardconstruction[1]andarevery ± Γ± ± ± ± ±

widely studied [2]. The classical Einstein-Straus vacuole where the signature of the two-surfaces Γ is zero and
] ±
O model, which requires a comoving boundary surface, is dΩ2 is the metric ofaunittwo-sphere,whichwewrite in
unstable. Many subsequent Swiss-cheese models have the usual form dθ2+sin2(θ)dφ2. We need consider only
C
also assumed that the associated boundary surfaces re- one boundary surface. The metric on Σ can be written
.
h main comoving, but it is well known that this need not in the form
p be the case. Some general studies of this issue go back
- many years [3]. Some specific examples of noncomoving ds2 =R2(τ)dΩ2 dτ2, (2)
o Σ −
r boundary surfaces include the study of Vaidya-type in-
st homogeneities [4] and the evolution of “density waves” where τ is the proper time on Σ. Since Σ is, by assump-
a in the Lemaˆıtre-Tolman model [5]. Here we examine tion, geodesic, there is only one nonvanishing indepen-
[ the role of the linear momentum of a boundary surface dent component of the extrinsic curvature Kαβ. This
3 in a Robertson-Walker background. Within the context can be written in the form (e.g. [3])
v of these generalized models, we study the evolution of
1 the effective gravitational mass and size of the inhomo- K2 =R2 R˙2+1 2M± , (3)
4 geneities for a variety of well-knownbackgroundmodels. θθ± − R
2 (cid:18) (cid:19)
Wefindthatwhereasthemomentumplaysacentralrole,
0 where . d/dτ and are the effective gravitational
the details of the backgroundmodel are relatively unim- ±
. ≡ M
0 portant. masses of ±. The invariant properties of were first
V M
1 explored by Hernandez and Misner [8] who wrote the
9 function in the form
0
: R3
v = θφ, (4)
i M 2 Rθφ
X
r II. MODEL CONSTRUCTION where is the Riemann tensor of . See also [9–12]
a for furtRher discussion [13]. From (3)V, and the continu-
ity conditions, we arrive at the central condition of the
Thetheoryofhypersurfacesinspacetimeiswellestab- model:
lished(see,for example,[6])andwedo notreproduceall
= , (5)
the necessary machinery here. Rather, we go directly to − +
M M
theessentialingredientsofthemodel. Themodelconsists
a statement which is independent of the coordinates
ofrandomlydistributednonintersectingsphericalbound-
(x1,x2).
ary surfaces Σ in a Robertson-Walker background with ± ±
eachparticleonΣexecutingradialtimelikegeodesicmo-
tion.
B. Background Geodesics
The Robertson-Walker geometry (excluding the Ein-
∗[email protected] stein static subcase) has a Killing algebra of dimension
†[email protected] 6 (3 translations and 3 rotations). Since Σ is in radial
2
motion,weareinterestedintheconstantsofmotiongen- and
eratedbytranslationalinvariance. Writethebackground
in the form Gαβnαnβ +Λ=0. (15)
dr2
ds2 =a2(t) +r2dΩ2 dt2, (6) From (6) and (14) we find r˙ = 0 and from (6) and (15)
+ ǫ2(r) − we find 8πp = 0 where p is the comoving isotropic pres-
(cid:18) (cid:19)
sure. As a result, in the Einstein-Straus model, Σ must
whereǫ(r) √1 kr2 withk = 1,0. Thetranslational
≡ − ± beexactlycomovingandthebackgroundmustbeexactly
Killing vectors are then given by
dust. If either ofthese conditions do notholdthen (5)is
sin(θ) necessarilyviolated. Without resortingto surfacelayers,
ξα =ǫ cos(θ)∂ ∂ , (7)
1 r − r θ we can take the view that the culprit is the assumption
(cid:18) (cid:19) that is exactly vacuum. That is, (13) is too strong
−
V
a condition to impose on . In other Swiss-cheese cos-
−
1 cos(φ) V
ξα =ǫ sin(θ)sin(φ)∂ + cos(θ)sin(φ)∂ + ∂ , mologies, (13) is not imposed on −. A widely used ex-
2 r r θ sin(θ) φ ample is to assume that is duVst. However, typically
(cid:20) (cid:18) (cid:19)(cid:21) V−
(8) Σ is takenascomovinga priori andso we arrivebackat
and (14)and(15)which, wewouldlike toemphasize,arenot
1 sin(φ) necessary conditions for a boundary surface.
ξα =ǫ sin(θ)cos(φ)∂ + cos(θ)cos(φ)∂ ∂ .
3 r r θ − sin(θ) φ
(cid:20) (cid:18) (cid:19)(cid:21)
(9)
D. Generalized Model
Consider a radial geodesic with tangent uα = r˙∂ +t˙∂
r t
anddefine the constants ξαu and 2 2. It
Cn ≡ n α D ≡ nCn In this paper the only condition imposed on is (5)
follows directly from (7)–(9) that −
V
P andweconcentrateontheevolutionofΣin which,for
+
2ǫ2 the sake of clarity, we assume is spatially flVat. It follows
r˙2 = D , (10)
a4 from (10) and (11) that [18]
so that from the timelike condition uαu = 1 we have
α
− ∞ 2(1+x)2 dx
2 r (z)= D , (16)
t˙2 =1+ Da2. (11) Σ Zz s1+D2(1+x)2H(x)
Equations (10) and (11) should be well known [14]. Our wherewehavesettheinitialconditionsbyr (t=0)=0;
Σ
purpose here is to explain the physical meaning of . It the Hubble function H is given, as usual, by 1da; and,
D a dt
is the total linear momentum of Σ [15]. withoutlossingenerality,wehaveseta (today)=1. We
0
have chosen as an independent variable z =1/a 1 and
−
so for universes with a big bang that do not recollapse
C. Swiss-cheese 1<z < .
− ∞
The standard Swiss-cheese inhomogeneous cosmology
takes = 0. Of these models, the Einstein-Straus case, III. BACKGROUND MODELS
D
which sets as vacuum, is the most well known. The
−
V
modelhasseenverywide application[1], butitisknown Inthisworkweexploremassscalesassociatedwithvar-
to be unstable: to aspherical perturbations [16] and to ious models for the background specified by H(z). The
perturbations in the condition (5) which could lead to inhomogeneitiesconsideredherearegovernedbyonefree
the development of surface layers [17]. A clear way to parameter,the total linear momentum . The first task
D
see the instability in this model at a primitive level is to at hand then is to establish a reasonable range of values
look at the necessary conditions for a boundary surface: for this parameter.
[Gαn uβ]=[Gαn nβ]=0, (12)
β α β α
where Gα is the Einstein tensor, [Ψ] (Ψ Ψ ) , A. The Range in D
and uα aβnd nα are the tangent and nor≡mal v+ec−tors−to|ΣΣ
respectively. By assumption, satisfies To establish a reasonable range in we evaluate
V− r (0) from (16) for three standard moDdels: ΛCDM,
Gαβ +Λδβα =0, (13) ΛΣCDM+noninteractingradiation(seetheAppendix)and
noninteracting matter and radiation without Λ. The re-
where Λ is the cosmological constant. As a result, the
sults are shown in Fig.1. The top curve is included so as
followingtwoconditionsmustbesatisfiedin ifΣcon-
+
V to show the effect of ignoring the background radiation,
stitutes a boundary surface:
and the two bottom curves are included so as to show
Gαn uβ =0, (14) the effect of ignoring Λ and the effect of changing the
β α
3
integrationupper limit to a redshift where radiationcan B. Flat standard cosmological model (ΛCDM)
be neglected, here z = 1500. The middle curve shows
the size of the hole for the ΛCDM+noninteracting ra- Inthe flatΛCDMmodel,darkenergyisthe cosmolog-
diation at earlier time rΣ(100). The wide difference in ical constant, and as usual we have
size r betweenthe top andbottom curves showsthe ef-
Σ
2
fect of adding radiation on the evolution of the surfaces H(z)
=Ω (1+z)3+Ω +Ω (1+z)4, (17)
for a given value of . From the results of Fig. 1, we H m Λ R
D (cid:18) 0 (cid:19)
havechosentoincluderadiation(seetheAppendix)toall
wheretheΩrefertocurrentvalues. Notethatwehaveex-
backgroundmodelsH(z). Throughoutthepaper,wede-
plicitlyincludedradiationdensityΩ (seetheAppendix)
R
and it is included implicitly in all the following back-
ground models. The influence of Ω on r is shown in
m Σ
Fig. 3. The WMAP5 [19] value of Ω = 0.279 0.015
m0 ±
100 has limited effect onthe final size comparedto the value
of the linear momentum in Fig. 2. To compare the dif-
ferent backgrounds, we can plot the mass ratio between
the background model and the ΛCDM background
80 ¯ . From (4) it followMs tΣhat
MΣ
r 3
MΣ = Σ . (18)
¯ r¯
60 MΣ (cid:18) Σ(cid:19)
Throughout, for (18), we use the same value for the
linear momentum ( = ¯). For the ΛCDM model,
D D
the mass ratio (18) is shown in Fig. 4 where is
40
M
determined by Ω = 0.33 and different values of
m0
are shown. In Fig. 5 we plot the mass ratio (18)
D
fortheΛCDMmodelforthesamevaluesofΩ asFig.3.
m0
20
0
0 1 2 3 4 5 200
FIG. 1: The abscissa is the dimensionless fraction D/c ×
10−5 and the ordinate gives r (0) in units of Mpc. The
Σ 150
top curve is the ΛCDM model. The middle curve is
ΛCDM+noninteracting radiation (with the curve just below
giving r (100) for comparison). Thesecond tobottom curve
Σ
is the noninteracting matter and radiation without Λ, and
the bottom curve is the ΛCDM with integration upper limit 100
going to z =1500. We have adopted the values Ωm0 =0.27,
h≡H /100=0.72 and T =2.73.
0 0
fine quantities associated with a flat “standard” ΛCDM 50
model as “ ¯ ” when using the following values: for the
matterenergydensity,Ω =0.27,radiationenergyden-
m0
sity(includingrelativisticneutrinos),Ω 8 10−5and
R0 ∼ ×
cosmological constant Ω , regardless of the value of our
Λ 0
free parameter . Using the ΛCDM background, Fig. 2 0 200 400 600 800 1000
D
shows the evolution of r¯ with total linear momentum
Σ
in the order of 10−5. These inhomogeneities begin
D ∼
with r¯ ( ) = 0 and grow mostly during the radiation FIG. 2: Variation in the evolution of r¯ with redshift from
Σ ∞ Σ
epoch. We are interested in ranges of the parameter variations in D for a ΛCDM background. We have used
D
where surfaces r reach up to 100 Mpc. From Figs. 1 h = 0.72 and T = 2.73, and like all “¯” quantities we have
and 2, this correΣspond to a ran∼ge 10−5 . .5 10−5. used Ωm0 = 0.207 and ΩR0 = 8×10−5. These surfaces start
Insubsequentfiguresweuse =5 10−5uDnlesss×pecified with r¯Σ(z = ∞) = 0. The value of the linear momentum D
otherwise. D × determines thefinal size of theseinhomogeneities.
4
D. Flat dark energy with variable equation of state
(VES)
105
Allowing the equation of state to vary with time
(VES), and using the parameterization[21]
z
100
w(z)=w +w , (21)
0 a
1+z
we have
95 H(z) 2 =Ω (1+z)3+Ω (1+z)3(1+w0+wa)
m de
H (22)
(cid:18) 0 (cid:19)
e−3wa(z/(1+z)).
×
90 The situation is examined in Fig.8. Whereas the mass
ratio decreases with increasing w , there is no evidence
a
that w +w >0 [22].
0 a
85
0 5 10 15 20 E. Flat DGP models
TheflatDvali-Gabadadze-PorratiDGPmodel[23](see
FIG. 3: Evolution of r with redshift for some values of the also [24]) is a one parameter model from brane theory
matterdensityΩm. WeΣhaveusedaΛCDMbackgroundwith where Ωr =1/(4r2H02) is the dimensional parameter de-
atotallinearmomentumofD=5×10−5 andwehaveextrap- termined by the scale length r which governs the transi-
olatedthesizesoftheseinhomogeneitiestotheinfinitefuture tionfrom4Dto5Dbehavior. Forthismodel,theHubble
(z =−1). The dotted line shows the size at the present red- parameter is given by
shift (z = 0). We have used values of Ωm consistent with
the 99.9% confidence interval of [20] to show the effect of H(z) 2 2
Ωm0. For comparison, the latest WMAP5 [19] results give H = Ωm(1+z)3+Ωr+ Ωr , (23)
Ωm0 =0.279±0.015. (cid:18) 0 (cid:19) (cid:16)p p (cid:17)
Define
∆ / ¯ 1. (19)
M≡MΣ MΣ −
For r r¯ then ∆ 3∆r, where ∆r = (r /r¯ ) 1
Σ ∼ Σ M ∼ Σ Σ −
is the size difference for a fixed . This can be gleaned
D
from Figs. 2–5.
C. Flat dark energy with constant equation of state
This model (which we designate by CES) uses a con-
stant arbitraryvalue for the equationof state parameter
w p/ρ so that the Hubble function becomes
≡
2
H(z)
=Ω (1+z)3+Ω (1+z)3(1+w). (20)
m de
H
(cid:18) 0 (cid:19)
In Fig.6 we show the ratio / ¯ for w = 0.7 and
MΣ MΣ −
w = 1.3correspondingtoa 99.9%confidence level[20].
−
The mass difference ∆ between the two models is less
M
than2%atmaximum. Thedifferencewouldbenegligible
usingWMAP5[19]wherewisconstrainedto 0.97 0.06 FIG. 4: Evolution of ratio M /M¯ with redshift for some
as can be seen in Fig.7 where we have plotte−d the±mass valuesofD=D¯ whereMisgiΣvenbΣytheΛCDMwith Ωm =
ratio for different values of w with Ω =0.27. 0.33 and M¯ is our standard ΛCDM model.
m
5
FIG. 5: Evolution of ratio M /M¯ for ΛCDM with D = FIG.7: RatioM /M¯ forD=5×10−5 fordifferentvalues
Σ Σ Σ Σ
5×10−5 for some values of Ωm. ofwwithΩm =0.27fortheconstantequationofstatemodel.
FIG. 6: The same as Fig. 4 where M now has the constant
equation of state model (CES) for thebackground.
FIG.8: ThesameasFig.7withthevariableequationofstate
(VES) model as background for M. Only the region wa <1
is of physical interest.
where Ω = 1 2√Ω . The 99.9% confidence level of
m r
−
[20] was used for the values of Ω in order to show the
r
mass ratio (18) for the DGP model as given in Fig. 9.
Subsequent models showing the same type of behavior
as Fig. 9 over the redshift range 1 < z < 20 will be summarize in Table I.
−
6
TABLE I: Model properties
Model Ωm Ωde 1st parameter 2nd parameter MΣ/M¯Σ(z)
z=−1 z =20
AFF 0.27 1−Ω α=0.02333 0.696 0.692
0.27 1−Ω α=−0.01667 1.245 1.248
CHA A=0.7 γ =0.25 0.778 0.783
A=0.75 γ =−0.1 1.171 1.166
IDE 0.73 ξ=2.3 w=−1 1.306 1.297
0.73 ξ=3.3 w=−1 0.9192 0.9211
NADE n=2.5 0.859 [25] 0.864
n=3.1 1.1125 [25] 1.1118
FIG.9: SameasFig.7;MisgivenfortheDvali-Gabadadze- FIG.10: RatioM /M¯ evaluatedatz=−1asafunctionof
Σ Σ
Porrati (DGP) model. ξforw=−1,D=5×10−5 andΩ =0.73fortheinteracting
de
dark energy (IDE) model.
F. Flat interacting dark energy
G. Cardassian models
It is natural to consider the coupling between dark
energy and matter and there are many explicit coupling
The modified polytropic Cardassianmodels (hereafter
procedures considered in the literature. Here we use the
CAR)arethreeparametermodelsthatmodifytheFried-
parameterization of [26] to write
mann equation in a flat, matter-dominated universe in
ordertoallowacceleration. TheHubblefunctionisgiven
H(z) 2 1 −3(w/ξ)
=(1+z)3 1 Ω 1 . by (see [20, 27])
H − de − (1+z)ξ
(cid:18) 0 (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
(24)
H(z) 2 1/q
=Ω (1+z)3 1+(Ω−q 1)(1+z)3q(n−1) .
Figure 10 shows the mass ratio (18) for the interacting H m m −
dark energy (IDE) model with w = 1, Ω = 0.73 and (cid:18) 0 (cid:19) (cid:16) ((cid:17)25)
de
=5 10−5 atz = 1overarange−ofξ consistentwith Theevolutionofthe massratio(18)withredshiftforthe
D × −
[26]. The mass ratio (18) does not change significantly Cardassian models is shown in Fig. 11 for Ω = 0.27
m
over the redshift range 1 < z < 20 as can be seen in and =5 10−5 and values of the parameters n and q
− D ×
Table I. surrounding the 99.9% confidence level of [20, 27].
7
FIG.11: SameasFig.7butnowMisgivenforthecardassian FIG. 12: Same as Fig. 10 but with A = 0.67 and 0.78 as
(CAR) model. a function of the parameter γ for the Chaplygin (CHA) gas
model.
H. Flat Chaplygin gas
A dark fluid that combines dark matter and dark en- 1.6
ergy,wheretheequationofstateisp= A/ργ [28],gives
−
the generalized Chaplygin gas equation (CHA)
H(z) 2 1/(1+γ) 1.4
= A+(1 A)(1+z)3(1+γ) . (26)
H −
(cid:18) 0 (cid:19) (cid:16) (cid:17)
Since the Chaplygin gas is a two parameter model we
1.2
choose two extreme values for A for Fig. 12 in which we
plot the mass-ratio (18) at z = 1 as a function of γ.
−
The values chosenhere follow the 99.9%confidence level
of [20]. As with the IDE model, the mass ratio for the 1
Chaplygin gas model is nearly constant over the range
1<z <20. See Table I.
−
0.8
I. Flat affine equation of state
0.6
The assumptionthatdarkenergyanddarkmatterare
a single dark component that can be modeled by the -0.01 0 0.01 0.02 0.03
affine equation of state p = p +αρ (AFF) gives rise to
0
the Hubble function [29]
FIG.13: SameasFig.10butfortheaffine(AFF)parameter
2 model as a function of α.
H(z)
=Ω˜ (1+z)3(1+α)+Ω , (27)
m Λ
H
(cid:18) 0 (cid:19)
where Ω˜ (ρ ρ )/ρ . The ratio (18) for the AFF J. New agegraphic dark energy (NADE)
m 0 Λ c
≡ −
model is shown in Fig. 13 at z = 1 as a function of α
−
usingthe99%confidencelevelof[29]forα. Complemen- A single-parametermodel where the energy density of
tary information is available in Table I. quantum fluctuations of Minkowski spacetime ρ is in-
q
8
cluded in the Hubble equation gives [30] where from (16)
H(z) 2 Ω (1+z)3 2(1+z)2 1
= m0 , (28) r′ (z)= D (32)
(cid:18) H0 (cid:19) s 1−Ωq(z) Σ −s1+D2(1+z)2H(z)
where Ωm0 = 1− Ωq(0) and the evolution of Ωq(z) is and equivalently for the ΛCDM model (with a bar).
given by Therefore the mass ratio is extremal for
dΩ 2 r r′ H¯
q = Ω (1 Ω ) 3(1+z)−1 Ω . (29) Σ = Σ = = MΣ . (33)
dz − q − q − n q r¯ r¯′ H ¯
(cid:20) (cid:21) Σ Σ (cid:18)MΣ(cid:19)
p
Themassratio(18)forthismodelispresentedinFig.14 An extremum at z = 0 would imply that r = r¯ and
Σ Σ
with values of n consistent with the likelihood values in that = ¯ . Equation(31)canalso be expressedas
MΣ MΣ
[30].
′ 2 1
MΣ =3δ3 D δ 1 , (34)
¯ 0 1+ 2r (0)H { 0− }
(cid:18)MΣ(cid:19)z=0 r D Σ 0
1.2 where δ3 = (0)/ ¯ (0) andr (0) is obtained from
0 MΣ MΣ Σ
(16) for the model investigated.
(cid:0) (cid:1)
InEq.(16), the influence of inthe firstfactorofthe
integralconvergesto1for 2(1D+z)2 1or1+z 1/
D ≫ ≫ D
1.1 and converges to (1+z) when 1+z 1/ . In effect
thensince 10D−5, isunimportant≪atearDlytimesfor
D ∼ D
r and r¯ .
Σ Σ
1 V. CONCLUSION
We have introduced a new generalized Swiss-cheese
model which does not assume a priori that the asso-
ciated boundary surfaces are comoving. In order to
0.9
quantify evolving inhomogeneities, we have considered
geodesicboundariescharacterizedbytheirlinearmomen-
tum . For the size of the inhomogeneities we are inter-
estedDin, the physical values of /c are 10−5. For a
D ∼
0.8 givenlinearmomentum,wehavefoundthattheinhomo-
2.6 2.8 3
geneities grow almost independently of the background
model(withtheinclusionoftheradiationdensityparam-
FIG. 14: Same as Fig. 7; M is given for the new agegraphic eter Ω ). As shown in Fig.2 these inhomogeneities are
R0
dark energy (NADE)model [25]. almostattheirfullsizebythedecoupling(z 1100). For
∼
a redshift ofz .2, correspondingto highredshift super-
novae, the inhomogeneities considered here are growing
very slowly as is shown in see Fig.4.
IV. DISCUSSION
A property of the mass ratio (18) seen in all relevant ACKNOWLEDGMENTS
figures is simply
CGis supportedby the OntarioGraduate Scholarship
H¯(z)≷H(z) ≷ ¯ . (30)
⇔MΣ MΣ in Science and Technology and the Fonds qu´eb´ecois de
recherche sur la nature et les technologies. KL is sup-
As can be seen in Fig. 6, 7 and 11, the mass ratio (18) ported by a grant from the Natural Sciences and Engi-
appears to have an extremum at z = 0. (This feature neering Research Council of Canada. Portions of this
is present in all plots of the mass ratio as a function of work were made possible by use of GRTensorII[32].
z but is not visible in all figures because of the chosen
scale.) In fact, the extremum can not happen at z = 0.
From (16) and (18), and defining ′ =d/dz, we find APPENDIX: RADIATION
MΣ ′ =3rΣ2 r′ r¯ r¯′ r , (31) TheintegrationinEq.(16)requiresahighredshiftcon-
¯ r¯4 Σ Σ − Σ Σ tributionfromradiation. See,forexample,[19,31]. This
(cid:18)MΣ(cid:19) Σ
(cid:8) (cid:9)
9
was added to all the Hubble background models H(z) first right hand side term is the contribution from the
as noninteracting species a−4. The energy density for CMB photons and the second term is the contribution
∝
radiation ρ is from relativistic neutrinos. The energy density of radia-
R
tion is therefore:
4
ρR =aBTC4MB"1+ 87(cid:18)TCTMνB(cid:19) Nν# , (35) ΩR0 = ρρRc00 ≃8×10−5 (36)
where a = 4σ/c in which σ is the Stefan-Boltzmann’s where ρ is the critical energy density; the subscript 0 is
B c
constant, T =2.725K is the temperature of the cos- used to specify present values (z = 0). Note that before
CMB
micmicrowavebackground(CMB),T =(4/11)1/3T the electron-positron pairs annihilated (z 1010) the
ν CMB
∼
is the temperature parameter for the relativistic neutri- temperature of the neutrinos and CMB radiation was in
nos after the annihilation of electron-positron pairs and equilibrium T = T . This contribution in Eq. (35)
ν CMB
N =3isthe standardnumberofneutrinofamilies. The wasnegligibleinourintegrationof(16)andwasignored.
ν
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18, 148 (1946). A typical modern version of this model Theor. Phys. 117, 229 (2007), [arXiv:gr-qc/0605120];
replaces the vacuum with dust. See, for example, [2] V. Marra, E. Kolb, S. Matarrese, and A. Riotto,
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