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ModernPhysicsLetters A
(cid:13)c WorldScientificPublishingCompany
5
1
0
2
Cosmic expansion and structure formation in running vacuum
n cosmologies
a
J
6
1 SpyrosBasilakos∗
Research Centerfor Astronomy and Applied Mathematics, Academy of Athens
]
Soranou Efesiou 4, 11527, Athens, Greece
O
[email protected]
C
.
h
WeinvestigatethedynamicsoftheFLRWflatcosmologicalmodelsinwhichthevacuum
p
energy varies with redshift. A particularly well motivated model of this type is the so-
- calledquantumfieldvacuum,inwhichbothkindoftermsH2andconstantappearinthe
o
effective darkenergy density affecting the evolution of the maincosmological functions
r
t atthe background andperturbationlevels.Specifically, itturns outthat thefunctional
s
formofthequantum vacuumendows thevacuum energyofamilddynamicalevolution
a
[ whichcouldbeobservednowadaysandappearsasdynamicaldarkenergy.Interestingly,
the low-energybehavior isveryclose tothe usual ΛCDMmodel, butitisby nomeans
2 identical. Finally, withinthe frameworkof the quantum fieldvacuum wegeneralize the
v largescalestructureproperties,namelygrowthofmatterperturbations,clusternumber
9 counts andsphericalcollapsemodel.
4
7 Keywords: Cosmology;DarkEnergy;LargeScaleStructure.
3
PACSNos.:98.80.-k,95.35.+d,95.36.+x
0
.
1
0 1. Introduction
5
1 The statistical analysis of various cosmological data (SNIa, Cosmic Microwave
: Background-CMB, Baryonic Acoustic Oscillations-BAOs, Hubble parameter mea-
v
i surementsetc)stronglysuggeststhatweliveinaspatiallyflatuniversethatconsists
X
of 4% baryonic matter, 26% dark matter and 70% some sort of dark energy
r ∼ ∼ ∼
a (hereafter DE) which is necessary to explain the accelerated expansion of the uni-
verse(seeRefs.[1,2]andreferencestherein).Althoughthereisacommonagreement
regarding the ingredients of the universe, there are different views concerning the
possible physical mechanism which is responsible for the cosmic acceleration.
The simplest dark energy candidate corresponds to a cosmological constant. In
the standardconcordance ΛCDM model, the overallcosmic fluid contains baryons,
colddark matter plus a vacuum energy(cosmologicalconstant),that appearsto fit
accuratelythecurrentobservationaldataandthusprovidesanexcellentscenarioto
describetheobserveduniverse.However,theconcordancemodelsuffersfrom,among
other, two fundamental problems: (a) The fine tuning problem i.e., the fact that
∗email:[email protected]
1
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2 Spyros Basilakos
the observed value of the vacuum energy density (ρ = Λc2/8πG 10 47GeV4)
Λ −
≃
is many orders of magnitude below the value found using quantum field theory
(QFT),3 and (b) the coincidence problem,4 i.e., the fact that the matter energy
density and the vacuum energy density are of the same order just prior to the
present epoch, despite the fact that the former is a rapidly decreasing function of
time while the latter is stationary. Such problems have inspired many authors to
propose alternative candidates to the concordance ΛCDM, such as general dark
energy models and modifications of the theory of gravity (for review see Ref.[5]).
Analternativepaththatonecanfollowinordertoalleviatetheaboveproblems
istoconsideratimevaryingvacuum,Λ(t).Withinthisframeworkwedonotneedto
include new fields in nature nor to modify the theory of General Relativity. In this
cosmological scenario the dark energy equation of state parameter w P /ρ ,
DE DE
≡
isstrictlyequalto-1,butthe vacuumenergydensityisnotaconstantbutratherit
isafunctionofthecosmictime.Notethatthereisanextensive(old6,8andnew9–14)
literatureinwhichthetime-evolvingvacuumhasbeenphenomenologicallymodeled
as a function of time in various possible ways, in particular, as a function of the
Hubble parameter.
In this article we reconsider the so called quantum field vacuum (see next sec-
tion). We would like to stress that this vacuum model was previously investigated
byourteaminmanyaspects. Thelayoutofthearticleisthefollowing.Insection2,
we present the main cosmological ingredients of the quantum field vacuum model.
Insection3weprovidethepowerspectrumandthelineargrowthofmatterpertur-
bations.In sections 4 and5 we discuss the cluster number counts and the spherical
collapse model. Finally, we draw our conclusions in section 6.
2. Background expansion - Observational constraints
Thenatureofthecurrentvacuummodel9–12 isessentiallyconnectedwiththerenor-
malizationgroup(RG)inquantumfieldtheory(QFT).Inthiscontext,theevolution
of the vacuum is written as
Λ(H)=Λ +3ν(H2 H2). (1)
0 − 0
where Λ Λ(H ) = 3Ω H2 and ν is interpreted in the RG framework as a
0 ≡ 0 Λ 0
“β-function”ofQFTincurvedspacetime,whichdeterminestherunningofthecos-
mologicalconstant. Regarding the dynamical role of ν it has been found that since
ν 1(for arecentdiscussionsee Ref.[13]andreferencestherein)atlowredshifts
| |≪
themodelbecomesalmostindistinguishablefromthecosmicconcordancemodel.In
other words the ν-parameter endows the vacuum energy of a mild dynamical evo-
lution which could be observed nowadays and appears as dynamical dark energy.
The low-energy behavior is thus very close to the concordance model, but it is by
no means identical.
Naturally, the next step here is to derive the Friedmann equations assuming of
course a running vacuum. This procedure is perfectly allowed by the Cosmological
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Dynamics inΛ(H) cosmologies 3
Principle embedded in the FLRW metric. In this context the Friedmann equations
are written
a˙
8πGρ 8πGρ +Λ=3 =3H2 , (2)
tot m
≡ a
(cid:18) (cid:19)
8πGp 8πGp Λ= 2H˙ 3H2 (3)
tot m
≡ − − −
where the overdotdenotes derivativewith respectto cosmic time t. Notice that the
Bianchiidentitiesthatinsurethe covarianceofthetheory,poseanenergyexchange
between vacuum and matter for G=const.
ρ˙ +3(1+ω )Hρ = ρ˙ . (4)
m m m Λ
−
Combining equations (2), (3) and (4), we provide the basic differential equation
that governs the dynamics of the Universe
3 (1+ω )
H˙ + (1+ω )H2 = m Λ(H). (5)
m
2 2
Since we are in the matter era we set ω 0.
m
≡
Inserting eq.(1) into eq.(5) we can integrate the latter in order to obtain the
Hubble parameter as a function of time:
Ω ν 3
Λ
H(t)=H − coth H (Ω ν)(1 ν)t . (6)
0 0 Λ
1 ν 2 − −
r − (cid:20) (cid:21)
p
where Ω = 1 Ω and H is the Hubble constanta. Using H = a˙/a the cosmic
Λ m 0
−
time, t(a), is given by
2 Ω˜
t(a)= sinh 1 Λ a3(1 ν)/2 (7)
3Ω˜1Λ/2(1−ν)H0 − sΩ˜m −
where we have introduced
Ω Ω ν
Ω˜ = m , Ω˜ = Λ− . (8)
m Λ
1 ν 1 ν
− −
Inverting eq.(7) we easily determine the scale factor a = a(t). Substituting the
cosmic time into (6), one can provide the normalized Hubble parameter
H2(a)
E2(a)= =Ω˜ +Ω˜ a 3(1 ν) (9)
H2 Λ m − −
0
where the cosmological parameters obey the standard cosmic sum rule, namely
Ω˜ +Ω˜ =1=Ω +Ω .
m Λ m Λ
aFor the comoving distance and for the dark matter halo mass (see section 4) we use the tradi-
tionalparametrizationH0=100hkm/s/Mpc.Ofcourse,whenwetreatthepowerspectrumshape
parameterΓweutilizeh≡hPlanck=0.673[2].
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4 Spyros Basilakos
Concerningtheevolutionofmatterdensitythesituationisasfollows.Combining
eq.(1), (2) and (4) we arriveat ρ˙ +3Hρ =3νHρ . Integratingthe latter (using
m m m
ρ˙ =aHdρ /da) we find
m m
ρ (a)=ρ a 3(1 ν), (10)
m m0 − −
where ρ is the matter density at the present time (a = 1), and therefore Ω =
m0 m
ρ /ρ , where ρ = 3H2/8πG is the current critical density. In fact, defining
m0 c0 c0 0
Ω (a) ρ (a)/ρ (a) it is easy to see, with the help of (10) and the definition of
m m c
≡
E(a), that
Ω a 3(1 ν)
m − −
Ω (a)= . (11)
m E2(a)
Lastly, upon inserting (10) in (4) and integrating once more in the scale factor
variable,wearriveatthe explicitexpressionforthe evolutionofthe vacuumenergy
density:
νρ
Λ(a)=Λ +8πG m0 a 3(1 ν) 1 . (12)
0 − −
1 ν −
− h i
Obviously,forν =0the currenttime varyingvacuummodelreducesto the concor-
dance Λ cosmology as it should.
Using the above normalized Hubble parameter [see eq.(9)], a joint statistical
analysis,involvingthelatestcosmologicaldata[SNIa,15 A(z)ofBAOs16andPlanck
CMB shift parameter2,18] is implemented. Since the explored cosmological models
(Λ and ΛCDM) contain different number of free parameters, as a further sta-
RG
tistical test we use the (corrected) Akaike Information Criterion (AIC) relevant
to our case (N /k > 40),19 which is given, in the case of Gaussian errors, as
tot
follows: AIC = χ2 +2k, where χ2 = χ2 +χ2 +χ2 is the overall χ2-
t,min t SNIa BAO CMB
function and k is the number of free parameters. From the statistical viewpoint,
a smaller value of AIC points to a better model-data fit. Within this framework,
we need to make clear that small differences in AIC are not necessarily signif-
icant and therefore, it is important to derive the model pair difference, namely
∆AIC= AIC AIC . The larger the value of ∆AIC, the higher the evidence
y x
− | |
againstthemodelwithlargervalueofAIC,withadifference ∆AIC 2indicating
| |≥
a positive such evidence and ∆AIC 6 indicating a strong such evidence, while
| | ≥
a value 2 indicates consistency among the two comparison models. For the con-
≤
cordance ΛCDM cosmology, if we impose ν = 0 and minimizing with respect to
Ω we find Ω =0.291 0.011 with χ2 (Ω )/dof 567.5/586(AIC 569.5).
m m ± t,min m ≃ Λ≃
For comparison, we provide the results of Planck: Ω h2 = 0.1426 0.0025, with
m
±
h =0.673 0.012 which implies Ω =0.315 0.016. Notice that throughout
Planck m
± ±
the paper we set Ω Ω = 0.291. In the case of the Λ model we find that
m,Λ m RG
≡
theoveralllikelihoodfunctionpeaksatΩ =0.282 0.012andν =0.0048 0.0032
m
± ±
with χ2 (Ω ,ν) 563.8 (AIC 567.8) for 585 degrees of freedom.13 It turns
t,min m ≃ RG≃
out that the current vacuum model appears to fit slightly better than the ΛCDM
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Dynamics inΛ(H) cosmologies 5
the observational data. Still, the ∆AIC=AIC AIC values (ie., 2) indi-
Λ RG
| | | − | ≤
catethatthecosmologicaldataaresimultaneouslyconsistentwiththecosmological
models. Note that for the rest of the paper we will restrict our present analysis to
the choice of (Ω ,ν)=(0.282,0.0048).
m
Table1. StatisticalresultsfromfittingSNIa+BAO+CMBshiftdata.The1st
column indicates the model. The 2nd and 3nd columns provide the best fit
parameters (Ωm,ν). The 4nd and 5nd columns listthe statistical significance
of the fit. Finally, the 6th and 7th columns shows the estimated values of
(σ8,δc).Noticethatinsection4weprovidearelevantdiscussionregardingδc.
Model Ωm ν χ2t,min AIC σ8 δc
ΛCDM 0.291±0.011 0 567.5 569.5 0.829 1.675
ΛRG 0.282±0.012 0.0048±0.0032 563.8 567.8 0.758 1.644
3. CDM Power Spectrum - growth factor
The CDM power spectrum is given by P(k)=P knT2(k), where T(k) is the CDM
0
transfer function and n 0.9603 following the recent analysis of the Planck data.2
≃
Concerning the form of T(k), we use that of Eisenstein & Hu20 T(k) = L0 ,
L0+C0q2
where q k, L = ln(2e + 1.8q), e = 2.718 and C = 14.2 + 731 . Here
≡ Γ 0 0 1+62.5q
Γ is the shape parameter17 which is written as: Γ = Ω h exp( Ω
m Planck b
− −
√2h Ω /Ω ). The value of Γ, which is kept constant throughout the model
Planck b m
fitting procedure, is estimated using the Planck results Ωb = 0.0222h−Pl2anck with
h =0.672.
Planck
Another important quantity here is the rms fluctuations of the linear density
field on mass scale M :
h
σ2(M,z)=σ2(z) 0∞kns+2T2(Ωm,k)W2(kR)dk , σ (z)=σ D(z) . (13)
8 R0∞kns+2T2(Ωm,k)W2(kR8)dk 8 8D(0)
where σ σ(R ,0) isRthe rms mass fluctuation on R =8h 1 Mpc scales. At this
8 8 8 −
≡
pointwewouldliketoremindthereaderthatD(z)isthegrowthfactor(seebelow),
W(kR)=3(sinkR kRcoskR)/(kR)3 andR=(3M /4πρ )1/3 withρ =Ω ρ
h m0 m0 m c0
−
denotes the mean matter density of the universe at the presenttime (ρ =2.78
m0
×
1011Ω h2M Mpc 3). In this work we use the Planck prior, namely σ = 0.829.
m − 8,Λ
⊙
Of course, in order to use σ properly along the current vacuum model we need to
8
rescale the value of σ by
8
1/2
σ =σ D(0) P0 0∞kns+2T2(Ωm,k)W2(kR8)dk , (14)
8 8,Λ
DΛ(0)"P0,ΛR0∞kns+2T2(Ωm,Λ,k)W2(kR8)dk#
where P /P = (Ω /Ω )2. BRased on the aforementioned observational con-
0 0,Λ m,Λ m
straintsand eq.(14)we find σ =0.758for the Λ model.In Table 1,one maysee
8 RG
amorecompactpresentationofthe cosmologicalparametersincludingtheδ which
c
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6 Spyros Basilakos
is the linearly extrapolated density threshold above which structures collapse (see
section 4).
Now we focus on the basic equation which describes the evolution of matter
fluctuations within the context of the previously described vacuum model. At sub-
horizon scales the corresponding time evolution equation (in the linear regime) for
the matter density contrast D δρ /ρ , in a pressureless fluid, is given by (see
m m
≡
Refs. [8, 21]):
D¨ +(2H +Q)D˙ 4πGρ 2HQ Q˙ D =0, (15)
m
− − −
where in our case Q(t) = Λ˙/ρ . I(cid:16)t becomes clear, tha(cid:17)t the interacting vacuum
m
−
energyaffectsthegrowthfactorviathefunctionQ(t).Inthecaseofnoninteracting
DE models,[Q(t)=0],the aboveequation(15)reducesto the usualtime evolution
equation for the mass density contrast.22,23 For the concordance Λ cosmology it is
easy to prove that the solution of eq.(15) is22
5Ω E(a) a dx
m
D (a)= . (16)
Λ 2 x3E3(x)
Z0
Fig. 1. Right Panel: Comparison of the observed and theoretical evolution of the growth rate
f(z)σ8(z).ThedashedlinecorrespondstotheΛCDM(σ8,Λ=0.829)andthesolidoneisforΛRG
(σ8=0.758). Left Panel:Theevolutionofthegrowthindexofmatter perturbations.
For the purpose of our study it is important to solve eq.(15) in order to investi-
gatethe matterfluctuationfieldofthe vacuummodel(1)inthe linearregime.Here
we present the main steps (for more details see Ref. [11]). First of all we change
variables from t to a new one according to the transformation
3
y =coth H (Ω ν)(1 ν)t . (17)
0 Λ
2 − −
(cid:20) (cid:21)
p
Also from equations (6), (7) and (9) we get the following useful relations:
1 ν Ω
y = − E(a), y2 1= m a−3(1−ν). (18)
Ω ν − Ω ν
r Λ− Λ−
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Dynamics inΛ(H) cosmologies 7
Utilizing (17) and (18) we find, after some non-trivial algebra, that equation (15)
becomes
3β2(y2 1)2D′′ +k(β)y(y2 1)D′ 2[g(β)y2 ψ(β)]D =0, (19)
− − − −
a solution of which is
D(y)= (y2 1)4−6β9βyF 1 + 1,3, 1 + 3, 1 (20)
C − 3β 2 2 3β 2 −y2 1
(cid:18) − (cid:19)
where β 1 ν, primes denote derivatives with respect to y, the quantity F
≡ −
is the hypergeometric function, k(β) = 2β(6β 5), g(β) = (2 β)(3β 2) and
− − −
ψ(β) = β(4 3β). Now inserting eq.(18) into eq.(20) and using eq.(8), we finally
−
obtain the growth factor D(a) as a function of the scale factor:
D(a)=C1a9β2−4E(a)F 31β + 21,32,31β + 23,−ΩΩ˜˜mΛ a3β! (21)
where C is an integration constant to be adjusted by an initial condition.b
1
∝C
Furthermore,a crucialrole in structure formationstudies plays the growthrate
of clustering which is defined as f(a) = dlnD Ωγ (a) where γ is the growth
dlna ≃ m
index. It has been shown that for those dark energy models which have a slow
varying equation of state parameter, the growth index γ is well approximated by
γ 3(w−1) (see Refs. [24, 25, 26]), or γ 6/11 in the case where w = 1 (ΛCDM
≃ 6w 5 ≃ −
model).−Now since we know the analyticalform of the growthfactor D(a) one may
directly compute the evolution of f(a).
In order to test the performance of the Λ vacuum model at the perturbation
RG
level, we utilize the recent growthrate data for which their combination parameter
of the growth rate of structure, f(z), and the redshift-dependent rms fluctuations
of the linear density field, σ (z), is available as a function of redshift, f(z)σ (z).
8 8
ThetotalsamplecontainsN =16entries(ascollectedby Ref.[27]-seetheirTable
1 and references therein). In the right panel of figure 1 we present the growth data
(solid points) together with the predicted f(z)σ (z) for the running vacuum model
8
(solid line) and ΛCDM (dashed line). We find that the concordance Λ cosmology
(with Ω =0.291) reproduce the growth data with χ2 /dof 20.02/15implying
m min ≃
that the ΛCDM modelcannot simultaneouslyaccommodatethe Planck priorsand
the growthdata (see also Ref. [28]). On the other handfor the Λ model we have
RG
χ2 /dof 8.42/15. Performing the AIC information criterion for small sample
min ≃
size, namely AIC = χ2min +2k+ 2Nk(kk−11) we find ∆AIC= AICΛ −AICRG ≃ 11.6
which means that the growth data fa−vo−r the current running vacuum scenario.
Let us finish this section with a relevant discussion regarding the growth index
γ.Aswehavealreadydescribed,wecanexpressthelineargrowthrateofclustering
in terms of the growth index. Specifically, from dlnD Ωγ (a) we easily obtain:
dlna ≃ m
γ(z)≃ ln[−l(n1Ω+mz)(dz)ldnzD], where z =a−1(z)−1. Based on the aforesaid observational
bWenormalizethegrowthfactoratlargeredshifts(z≫1),namelyD(z)≃1/(1+z).
January 20, 2015 2:19 WSPC/INSTRUCTION FILE basilvac
8 Spyros Basilakos
constraintswefindthat the growthindex atthe presenttime becomes γ 0.58
ΛRG ≃
whichissomewhathigherwithrespecttotheΛCDMvalue,namelyγ 6/11.Inthe
≃
left panel of figure 1 we show the growth index evolution for the Λ and ΛCDM
RG
models,respectively.Fromthe comparisonwe observethat the growthindex of the
Λ vacuum model deviates from the concordance model. In particular, there is a
RG
visibledeviationfromaboveinalltheredshiftrange.Thisdeviationbecomesatthe
level of 10% for z 1.
≤
Fig.2. LeftPanel:FractionaldifferenceδN/NΛ(solidline)intheclusternumbercountsbetween
the ΛRG model and the reference ΛCDM model. The dotted curves correspond to 2σ Poisson
uncertainties.NoticethatweusethepriorsofTable1.RightPanel:Intheinnerpanelweprovide
therelativedeviation(1−λ/λΛ)%ofthecollapsefactorforvariousvacuum modelswithrespect
to the Λ solution. In the outer panel we observe the evolution of the virial density. The lines
corresponds to the following vacuum models: (i) ΛRGH (solid), (ii) ΛRGC1 (dashed line, νs =
0.002) and (iii) ΛRGC2 (dotted line, νs = −0.002). Notice that the solid points represent the
concordance ΛCDMcosmology.
4. Comparison with Cluster Halo Abundances
Another choice in order to discriminate the models, that has been extensively used
so far in the literature, is to estimate the theoretical predictions of the models for
the cluster-sizehalo redshiftdistributions andto confrontthem with the data.The
halo abundances predicted by a large variety of time varying vacuum models have
been compared with those corresponding to the ΛCDM model.11–13,29 We utilize
the Press and Schechter30 formalism, which determines the fraction of matter that
has formed bounded structures as a function of redshift. Mathematical details of
suchatreatmentcanbefoundalsoinRef.[13];hereweonlypresentthebasicideas.
The number density of halos,n(M,z), with masseswithin the range (M, M+δM)
is given by:
ρ¯ dlnσ 1
−
n(M,z)dM = f(σ)dM , (22)
M dM
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Dynamics inΛ(H) cosmologies 9
whereρ¯=ρ is the meanbackgroundmass density (seesection3).Inthe original
m0
Press-Schechter (PSc) formalism, f(σ) = f (σ) = 2/π(δ /σ)exp( δ2/2σ2),
PSc c − c
δ is the linearly extrapolated density threshold above which structures collapse,31
c p
whileσ2(M,z)isthemassvarianceofthesmoothedlineardensityfield,extrapolated
to redshift z at which the halos are identified [see eq.(13)]. The mass variance
depends on the power-spectrum of density perturbations in Fourier space, P(k),
for which we use here the CDM form according to Ref. [20], and the values of
the baryon density parameter, the spectral slope and Hubble constant according
to the recent Planck results.2 Although the originalPress-Schechtermass-function,
f ,wasshowntoprovideagoodfirstapproximationtothatprovidedbynumerical
PSc
simulations,itwaslaterfoundtoover-predict/under-predictthenumberoflow/high
mass halos at the present epoch.32,33 More recently, a large number of works have
providedbetterfittingfunctionsoff(σ),someofthembasedonaphenomenological
approach. In the present study, we adopt the one proposed by Reed et al. [34].
In order to compare the mass function predictions of the different cosmological
models, it is important to use for each model the corresponding value of σ and δ
8 c
(see Table 1). It is well known that for the usual Λ cosmology we have δ 1.675
c
≃
(see Ref.[35]). For the Λ model it has been found that δ 1.644 (see Ref. [13]).
RG c
≃
Giventhehalomassfunctionfromeq.(22)wecannowderiveanobservablequantity
which is the redshift distribution of clusters, (z), within some determined mass
N
range, say M M/h 1M M = 1016. This can be estimated by integrating
1 − 2
≤ ⊙ ≤
over mass the expected differential halo mass function, n(M,z):
dV M2 dV dr z du
(z)= n(M,z)dM, =Ω r2(z) , r(z)=c , (23)
s
N dz dz dz H(u)
ZM1 Z0
where dV/dz is the comoving volume element and Ω is the solid angle.
s
In left panel of figure 2, we present the fractional difference between the Λ
RG
modelandΛCDMnamely,δ / ,withmassesintherange1013.4h 1 .M/M .
Λ −
1016h 1. Recall that we deNnotNe the deviations of the number counts of a gi⊙ven
−
vacuummodelwithrespecttotheΛCDMasδ = .Thedottedlinesshown
Λ
N N−N
in the left panel of figure 2 correspond to 2σ Poisson uncertainties, which however
do not include cosmic varianceand possible observationalsystematic uncertainties,
that would further increase the relevant variance. Specifically, the prediction for
δ / is negativeforallpoints,evenfor thoseinthe 1σ band.Thus,the running
N N ±
vacuum model tends to predict a smaller number of counts as compared to the
ΛCDM.Thefractionaldecreasecanbeassignificantas30 60%.We alsofindthat
−
the redshift variation of the differences between the Λ cosmology and ΛCDM
RG
modelismainlyaffectedbyvariationsinthevaluesofσ andδ (formorediscusion
8 c
see Ref. [13]). Therefore, we have verified that there are observational signatures
thatcanbeusedtodifferentiatethe Λ modelfromtheΛCDMandpossiblyfrom
RG
a large class of DE models.
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10 Spyros Basilakos
5. The spherical collapse model
In this section we generalize the spherical collapse model within the variable Λ(H)
cosmological model, in order to understand non-linear structure formation in such
scenariosandinvestigatethe differenceswiththe correspondingexpectationsofthe
concordance ΛCDM cosmology. Practically, one may start from the Raychaudhuri
equation which is valid either for the entire universe or for homogeneous spherical
perturbations [by replacing the scale factor with radius R(t)]
R¨ 4πG
= (ρ 2ρ ) , (24)
ms Λs
R − 3 −
where ρ and ρ refer to the corresponding values of the matter and vacuum
ms Λs
energy densities in the spherical patch susceptible of ulterior collapse.
Now let us first define some basic quantities of the problem. In particular, we
call α the scale factor of the universe where the overdensity reaches at its max-
t
imum expansion (i.e. when R˙ = 0) and α the scale factor in which the sphere
c
virializes,implying thata cosmicstructure hasformed.Also, R andR denote the
t c
corresponding radii of the spherical overdensity, the former being the turnaround
(or “top hat”) value at the point of maximum size, and the latter refers to the
eventual situation when the sphere has already collapsed and virialized. We would
like to remind the reader that due to the coupling between the running vacuum
and matter one would expect that the matter density in the spherical overdensity
should obey the same power law as the background matter ρ (a) a 3(1 ν) (see
m − −
∝
eq.10). Thus, ρ R 3(1 ν) denotes the matter density in the spherical patch.
ms − −
∝
Regarding,the vacuum energy density in the spherical region, ρ we focus on two
Λs
situations:(i)thevacuumenergyremainshomogeneousandonlythecorresponding
matter virializes (this scenario holds for the Λ cosmology); and (ii) the case with
clustered vacuum energy, assuming that the whole system virializes (both matter
and vacuum components).
In the regime where the vacuum is allowed to cluster we have ρ (R) =
Λs
Λ (R)/8πG.Therefore,insuchasituationitcouldbepossible,onnon-linearscales,
s
tohaveaninteractionbetweendarkmatteranddarkenergywithadifferentν than
the background value. We also assume that the general functional form that de-
scribesthe behaviorofthe vacuumenergydensity insidethe sphericalperturbation
obeys a similar equation as that of eq.(12):
νsρms,t R −3(1−νs)
Λ (R)=Λ +8πG 1 . (25)
s 0
1−νs "(cid:18)Rt(cid:19) − #
where ν is not necessarily equal to the backgroundν.
s
Using the basic differential equations [see eq.(2) and (24)] and performing the
following transformations x=a/a and y =R/R we arrive at
t t
x˙2 =Ht2Ωm,t x−1+3ν +rx2I(x) (26)
(cid:2) (cid:3)
