On the Magnitude of Dark Energy Voids and Overdensities

0.49 MB·English

by  David F. Mota

Draftversion February1,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/10/03
ON THE MAGNITUDE OF DARK ENERGY VOIDS AND OVERDENSITIES
David F. Mota1, Douglas J. Shaw2 and Joseph Silk3
1 InstituteforTheoretical Physics,UniversityofHeidelberg,69120Heidelberg,Germany
2 DAMTP,CentreforMathematicalSciences, UniversityofCambridge,CambridgeCB30WA,UKand
3 Astrophysics,UniversityofOxford,OxfordOX13RH,UK
Draft versionFebruary 1, 2008
ABSTRACT
We investigate the clustering of dark energy within matter overdensities and voids. In particular,
we derive an analytical expression for the dark energy density perturbations, which is valid both in
8
the linear, quasi-linear and fully non-linear regime of structure formation. We also investigate the
0
possibility of detecting such dark energy clustering through the ISW effect. In the case of uncoupled
0
quintessence models, if the mass of the field is of order the Hubble scale today or smaller, dark
2
energyfluctuations arealwayssmallcomparedto the matter density contrast. Evenwhenthe matter

n perturbations enter the non-linear regime, the dark energyperturbations remain linear. We find that
a virialisedclustersandvoidscorrespondtolocaloverdensitiesindarkenergy,withδ /(1+w) (10−5)
7 J for voids, δφ/(1+w) ∼ O(10−4) for super-voids and δφ/(1+w) ∼ O(10−5) forφa typica∼l vOirialised
cluster. If voids with radii of 100 300Mpc exist within the visible Universe then δ may be as large
as 10−3(1+w). Linear overdensit−ies of matter and super-clusters generally correspoφnd to local voids
]
h in darkenergy; for a typicalsuper-cluster: δ /(1+w) ( 10−5). The approachtakeninthis work
φ
p could be straightforwardly extended to study the clust∼erOing−of more general dark energy models.
-
o Subject headings: Cosmology: Theory, miscellaneous. Relativity. Galaxies: general, Large scale
r structure of the universe.
t
s
a
[ 1. INTRODUCTION a new form of energy (see Dvali, Gabadadze & Porrati
2000; Amarzguioui et al. 2006; Nojiri & Odintsov 2003;
It has been almost a decade since observations of
2 Capozziello, Cardone & Troisi 2005; Koivisto & Mota
Type Ia supernovae (SNe Ia) were first found to sup-
v 2007a,b). It also been suggested that the Universe is
port the notion that our universe is currently undergo-
7 not accelerating at all, but that a large local inhomo-
ing a phase of accelerated expansion (Riess et al. 1998;
2 geneity prevents the SNe Ia data from being correctly
Perlmutter et al. 1999). Since that time, evidence in
2
interpretedintermsofahomogeneousandisotropiccos-
2 favour of this accelerated expansion has strengthened
mologicalmodel(Kolb Matarrese & Riotto2006;Moffat
. significantly as the result of further SNe Ia observations
9 2006). This said, dynamical dark energy (hereafter
(Riess et al.2004,2006a,b;Wood-Vasey et al.2007),im-
0 DDE)isbyfarthemostpopularcandidatetoexplainthe
proved measurements of the cosmic microwave back-
7 currentastronomicaldata. InthesimplestDDE models,
ground (CMB) (Spergel et al. 2003, 2007) and surveys
0 knownasquintessence,darkenergyisassociatedwiththe
oflargescalestructure(LSS)(Adelman-McCarthy et al.
:
v 2006; Tegmark et al. 2006). The precise cause of this energy density of a scalar field with a canonical kinetic
i late-time acceleration, however, remains unknown. structure.
X
Ifdarkenergydoes indeed existthenastronomicalob-
Ifgeneralrelativityisaccurateonastrophysicalscales,
r servations presently provide us with only hints as to its
a then the assumptions of large-scale homogeneity and
nature. We know that today it represents about 70%
isotropy require that the agent responsible for the uni-
of the total energy density of the Universe, and that its
verse’s acceleration, dubbed ‘dark energy’, behave cos-
equation of state (EoS) parameter, w p/ρ, is fairly
mologically as a fluid with negative pressure. The ≡
close to 1: w = 1 0.1 for z < 1, (Riess et al.
standard model of particle physics predicts only one
− − ±
2006a). Furthermore,ifmatter dominatesthe expansion
such fluid: the vacuum energy, or cosmological con-
stant, for which the pressure, p, is always equal to mi- of the Universe for z > 1.8 then w(z > 1) = −0.8+−00..61
nus the energy density, ρ. However, if the vacuum en- (Riess et al. 2006a). Hence, dark energy has negative
ergy density is indeed non-zero then it is generally ex- pressureathigherredshiftswitha98%confidence. These
pected to be of the order of M4 , where M is the bounds on w are entirely consistent with a pure cosmo-
Pl Pl
Planck-mass. This is some 120 orders of magnitude logical constant. Whilst detecting either w = 1 or
6 −
larger than the observed dark energy density. A num- dw/dz = 0 would rule out a cosmological constant, in
6
ber of proposals have therefore been made in the lit- many DDE models significant deviations from w = 1
−
erature for models in which dark energy is dynamical onlyoccuratearlytimesandassuchwouldbedifficultto
andassociatedwithsomenewformofenergy(Wetterich detect. The late time, background cosmology predicted
1988; Peebles & Ratra 1988). In these models, the bymanyDDE modelsisthereforeverysimilartothatof
present small size of the effective cosmological constant a universe with a true cosmologicalconstant.
is by-product of the age of Universe. The accelera- DDE models generally cease to mimic a cosmo-
tion of the Universe might alternatively be explained logical constant in inhomogeneous backgrounds or
bymodifying GeneralRelativity ratherthanpostulating when one considers cosmological perturbation the-
2
ory. In particular, a number of authors have stud- than 10% at the centre of an inhomogeneity with prop-
ied the effect of DDE on the formation of large scale erties similar to that of the local supercluster (hereafter
structure (Corasaniti, Giannantonio & Melchiorri 2005; LSC). If accurate, the results of Dutta & Maor (2007)
Hannestad & Mortsell 2002; Brookfield et al. 2006b; implythatanydeviationsfromw = 1wouldbe signifi-
−
Doran, Robbers & Wetterich 2007). In the vast major- cantlyamplifiedby the presenceofalocaloverdensityof
ity of these works the energy density of dark energy is matter. Furthermore, they suggest that DDE clustering
taken to be homogeneous i.e. it is assumed that DDE mightbe relativelystrongwhenthematterperturbation
does not cluster. The extent to which this assump- goes non-linear.
tion of homogeneity is valid has been the subject of Dutta & Maor (2007) studied DDE clustering when
some interest and much debate in the literature. In the matter perturbation is growing in the linear regime,
an inhomogeneous background, the DDE energy den- which is only accurate when δ 1. They considered
m
≪
sityandEoSparametershouldexhibitsomespatialvari- the evolution δ for a cluster of matter with initial den-
φ
ations. The key issue of how large these variations sity profile (at z = 35) of δ = Aexp( r2/σ2) where
m
should be is however far from settled, particularly when A = 0.1 and σ = 0.01H−1 with H be−ing the initial
i i
the matter perturbation goes non-linear (Maor & Lahav value of the Hubble parameter. However,in the absence
2005;Mota & van de Bruck2004;Linder & White 2005; of any DDE, the linear approximation would generally
Bartelmann, Doran & Wetterich 2005; Nunes & Mota cease to be accurate when z 2.8; additionally the per-
2006; Abramo et al. 2007). In this paper we attempt turbationwouldbeexpectedt≈oturnaroundwhenz 1.0
to settle this issue for uncoupled quintessence models by and virialise when z 0.3. It is therefore far from≈clear
derivingananalyticalexpressionforthedarkenergycon- whether or not the s≈harp late-time growth in δ found
φ
trast, δφ, in by presence of a matter perturbation. Im- by Dutta & Maor (2007) is indeed a physical effect, or
portantly, we do not constrain the matter perturbation just a resultofusing linearizedfield equations outside of
to be small (i.e. linear). realm in which they are valid. Although one might well
Clustering of DDE over scales smaller expect there existsome mapping between the linear and
than about 100Mpc has been the sub- non-linearregimesas it happens in anEinstein-de Sitter
ject of a number of recent articles (Caimmi Universe.
2007; Balaguera-Antol´ınez,Mota & Nowakowski In this paper we investigate a similar problem to
2006; Mainini 2005; Percival 2005; Wang that considered by Dutta & Maor (2007), i.e. the clus-
2006; Nunes, da Silva & Aghanim 2005; tering of uncoupled quintessence on sub-horizon scales.
Balaguera-Antol´ınez,Mota & Nowakowski 2007). There are, however, two important differences between
Most attention has been focussed on models our approach to this problem and the one taken by
in which the DDE couples to baryonic and/or Dutta & Maor (2007):
dark matter since DDE clustering is expected to
be strongest in such theories (Amendola 2000; 1. Firstly, we do not use numerical simulations but
Brookfield et al. 2006a; Manera & Mota 2006; instead use the method of matched asymptotic ex-
Pettorino, Baccigalupi & Mangano 2005). In other pansions (MAEs) to develop a analytical approxi-
works, a more phenomenological approach has been mation to δ .
φ
taken and the observables associated with DDE cluster-
ing have been parametrized. In particular, it has been 2. Secondly,we donot requirethe density contrastof
shown that inhomogeneities in dark energy could pro- the matter perturbation, δ , to be small. Indeed,
m
duce detectable signaturesonthe CMB(Weller & Lewis ouranalysisandresultsremainvalidevenafterthe
2003; Koivisto & Mota 2006). virialisation of the matter overdensity.
Recently,Dutta & Maor(2007) consideredthe growth
ofspatialDDE perturbationsinthe absence ofanymat- Ourapproximationforδ isaccurateprovidedthatδ is
φ m
ter coupling. They considered only those circumstances onlynon-linear(& (1))onsub-horizonscales. Thispro-
where both the density contrast of matter, δm, and that vision is consistentOwith observations. For simplicity we
oftheDDE,δφ,weresmallenoughtobetreatedasalin- take the matter perturbation to be sphericalsymmetric.
earizedperturbationsaboutahomogeneousandisotropic We also require that gravity is suitably weak whenever
cosmological background. They studied the simplest the inhomogeneity is non-linear i.e. GM/R 1. We re-
classofquintessencemodelswheredarkenergyisassoci- statetheserequirementsinarigorousfashion≪later,how-
ated with the slow-roll of a scalar field φ down a poten- evertheyareessentiallyequivalenttothestatementthat
tialV(φ);φisminimallycoupledtogravity. Theauthors gravity is approximately Newtonian over scales smaller
linearized the full field equations for φ, the matter and thanH−1. SinceDDEclusteringinthelinearregimehas
the metric and solved the resulting system numerically. beendealtwith ingreatdetailby Dutta & Maor(2007),
Intriguingly they found that, at late times, a local over- our main focus in this paper is on whatoccurs when the
density of matter corresponded to a local under-density, matter perturbation goes non-linear.
or void, of dark energy. Conversely, a void in the mat- This paper is organizedas follows: in Section 2 we de-
ter was seen to produce a local DDE overdensity. Al- scribe our model for an inhomogeneous spacetime and
though the DDE density contrast, δφ, is initially very the DDE. We state the equations that must be satisfied
smallcomparedto the matter density contrast,δm, they by the metric quantities and the dark energy scalar field
found that, when δm (1), δφ (10−2). δφ was φ. In Section 3 we introduce the method of matched
∼ O | | ∼ O
alsoobservedtobegrowingmorequicklythanδm atlate asymptotic expansions (MAEs). This method relies the
times. TheirresultssuggestthatDDEclusteringmayin- existence of locally small parameters,and we state what
duceapositivecorrectionto thevalueof(1+w)ofmore these are for our model and interpret them physically.
3
We alsonote what constraintsthe smallness ofthese pa- WetakethemattercontentoftheUniversetobeamix
rameters places upon our analysis. In Section 4 we ap- of irrotational dust and dynamical dark energy, which
ply the method of MAEs to the evolution of dynamical is described by a scalar field φ. For simplicity we re-
dark energy perturbations. We derive a simple equa- strict ourselves to considering only spherically symmet-
tion for the DDE density contrast, δ , in terms of the ricspacetimesforwhichthe mostgenerallineelementin
φ
peculiar velocity of matter particles, δv, and the mat- comoving coordinates is:
ter density contrast, δm. Importantly this equation is ds2 = dt2 U(t,r)dr2 R(t,r)2 dθ2+sin2θ dϕ2 .
equally as valid for δ & (1) (‘the non-linear regime’) − −
as it is when δm m(1)O(‘the linear regime’). In Sec- We make the definitions U(t,r) ≡(cid:0)R,2r(t,r)/Q(t,r) a(cid:1)nd
≪ O k(t,r) 1 Q(r,t). This coordinate choice is unique
tion 5 we use our results to study the evolution of δ in
φ up to t≡ t−+t , and r r′(r). With these definitions
the linear (δm ≪ 1), quasi-linear (δm ∼ O(1)−O(10)) the2-sp→heres t0,r =con→sthavesurfacearea4πR2(t,r),
and fully non-linear (δ 1) regimes. We compare our
m { }
≫ and in this sense R(t,r) represents the ‘physical radial
analytical results with those found numerically in the
coordinate’. The energy-momentum tensor of pressure-
linear regime by Dutta & Maor (2007). In Section 6 we
less dust is given by
consider the spatial profile of the DDE density contrast
for realistic astrophysicalinhomogeneities such as voids, T(m) =diag ε(m),0,0,0 ,
ab
supervoids, clusters and superclusters. We conclude in
and the energy-momentum(cid:16)tensor of th(cid:17)e scalar field, φ,
Section 7 with a discussion of our results and their ob-
is
servational implications. We also note how our analysis
1
might be extended to include even more general DDE T(φ) =∂ φ∂ φ g ∂ φ∂cφ V(φ) .
ab a b − ab 2 c −
models. (cid:18) (cid:19)
Throughout the paper we use units where c=1. The Einstein equations for this metric read:
1
2. THEMODEL Gab =Rab− 2Rgab =κ Ta(bm)+Ta(bφ) , (1)
2.1. Geometrical Set-Up where κ=8π. The tt-componen(cid:16)t of Eq. (1) g(cid:17)ives:
Our aim in this paper is to derive an expression for R2R+kR =R R2κε(m)+R R2κ 1φ˙2 (2)
the DDE density contrast, δ = δε(φ)/ε(φ), inside an ,t ,r ,r ,r 2
φ c (cid:18)
over-density,or under-density, of matter. Henceforth we (cid:0) (cid:1) Q Q˙
use ε(i) to represent the energy density of a component +2R2 φ2,r+V(φ) +RR,rR,tQ,
i. We are particularly interested in those cases where ,r (cid:19)
the matter perturbation is non-linear i.e. δε(m)/ε(m) & and the tr-component of the Einstein equations is:
Otre(1a)t.mAenltthoofugthhewdeeanismitytopreermtuaribnastuioitna,bwlyegdenoemraalkicne othuer RR,r QQ˙ =−κφ˙φ,r. (3)
following simplifying assumptions: The rr-component of Eq. (1) reads:
R2R+kR +Q R= (4)
We assume sphericalsymmetry. We briefly discuss ,t ,t ,t
•
the extent to which the relaxation of this assump-
tionwouldaffectourresultsinSection7below,and (cid:0) (cid:1) κ 1φ˙2+ Q φ2 V(φ) R R2.
− 2 2R2 ,r − ,t
concludethatthequalitativenatureofourfindings (cid:18) ,r (cid:19)
would be unaffected. ¿From T(m)a =0, it follows that:
b;a
F(r) Q(r,t)
We define a ‘physical radial coordinate’, R, by the κε(m) = , (5)
• requirement that a spherical surface with physical Rp,rR2
radius R has surface area 4πR2. We assume that where F(r) is anarbitraryconstantof integration. If we
allcurvatureinvariantsareregularatR=0i.e. we define r sothat R(r,t )=r for some t=t , then F(r)=
i i
do notconsiderthosecasesinwhichthereisacen- 2M (r)/ Q(r,t )whereM (r)isthemassinsideashell
i,r i i
tralblackhole. We arguebelow,however,thatour of radius r at t=t .
i
results are still accurate even when there is a cen- ¿From Tp(φ)a =0 one obtains:
b;a
tral black-hole, provided they are only applied at
(cid:3)φ=V (φ). (6)
radii that are large comparedto the Schwarzschild − ,φ
Eq. (6) is subject to the boundary conditions: φ = 0
radius of the black hole. ,r
at R=0, and φc(t)=limr→∞φ(r,t), where φc(t) is the
Finally weassumethatforradiismallerthansome solution of Eq. (6) in the cosmological background. We
• R H−1 gravityis suitably weak inthe inhomo- make the following definitions:
0
≪ r
geneous region; H is the Hubble parameter of the
2M(r,t)= F(r) Q(r,t)dr,
background spacetime. We define what we mean
by weak rigorouslyin Section 3. For radiiR>R0, Z1r0(t) p
we require that the matter density contrastis .1. ε(φ)= φ˙2+V(φ ),
This assumption holds for most realistic models of c 2 c c
collapsing overdensities provided that the radius δε(φ)=1φ˙2+ Q φ2 +V(φ) ε (t),
the overdense region is less than about 0.1/H. 2 2R2 ,r − c
,r
φ
2.2. Einstein’s Equations δε˜(φ)=δε(φ) R φ˙ ,r,
,t
− R
,r
4
wherer (t)isdefinedbyR(r (t),t)=0. Wealsodefine:
0 0
1
P(φ)(t)= φ˙ (t) V(φ ),
c 2 c − c
1 Q
δP(φ)(r,t)= φ˙2+ φ2 V(φ) P(φ)(t).
2 2R2 ,r− − c
,r
Integrating Eqs. (3) and (5) and using Eqs. (3) and (6)
we find that
2M(r,t) 1
R2 = k(r,t)+ + R2κε(φ)(t)
,t − R 3 c
κ r
+ R (z,t)R2(z,t)δε˜(φ)(z,t) dz, (7)
,r
R
Zr0(t)
and
M(r,t) 1 Fig. 1.— Sketch showing the interior, exterior and intermedi-
R,tt =− R2 − 6R κε(cφ)+3κPc(φ) − (8) ate regions. The interior is defined to be the region where the
inner approximation is valid, and the exterior is the region where
(cid:16) (cid:17)
κ r 1 the outer approximation is applicable. In order for the matching
R (z,t)R2(z,t)δε˜(φ)(z,t) dz+ RκδP(φ) procedure towork, wemustrequirethat the exterior and interior
"2R2 Zr0(t) ,r 2 # regionsoverlapinsomeintermediateregion. Wemustalsorequire
the exterior and interior regions are contiguous, so that one may
As R we must recover the FRW background cos- movesmoothlyfromtheexterior,throughtheintermediateregion,
→ ∞ totheinterior.
mology which implies:
1
lim M(r,t) κε(m)(t )a3(t )r3,
r→∞ ∼2 c 0 0
weuseunitswherec=1,hencethefirstexpressionabove
lim k(r,t) k0r2, is in fact the dimensionless quantity δv/c 1. Our
r→∞ ∼ ≪
assumptionrequiresthatwheneverHR& (1),wehave
where t0 is an arbitrary time, a(t) is the scale factor δ1/HR 1. This condition is equivalenOt to requiring
of the FRW background, and k0 is the curvature of the t|hat the| ≪mean matter density contrast, δ¯(R,t), inside
background. In line with current observations, and be-
the sphere with radius R (1/H) or greater, is small
cause it greatly simplifies the calculations, we take the ∼ O
enough that it may be treated as a linear perturbation
cosmological background to be flat and set k = 0. We
0 about the cosmological background. The mean matter
do not attempt to solve the Einstein or DDE equations perturbation is however allowed to be non-linear, δ¯ &
exactly, but instead we develop asymptotic approxima-
(1), when HR 1 provided R 1. We also define
,t
tions to the true solutions which are accurate so long as O ≪ | |≪
a number of parameters, which we define and interpret δ = 1R2κ(ε(m) ε(m)) 1R2ε(m)δ .
below, remain small. 2 3 − c ≡ 3 c m
We require δ 1. This again require that δ & 1
3. SMALLPARAMETERSANDTHEMATCHED 2 ≪ m
only when HR 1 i.e. the matter perturbation is only
ASYMPTOTICEXPANSIONS.
≪
non-linear on sub-horizon scales. We note that δ
One can think of the small parameters approach and ((R2 +k) H2R2), andsogenerally δ 1provi2de∼d
the matched asymptotic expansions as an expansion in O ,t − | 1|≪
δ 1.
the Newtonian potential. Although it is perhaps fairer 2 ≪
In what follows we assume that δ and δ are small
to say that is closest to the context of General Relativ- 1 2
everywhere. These assumptionscanbe checkedonce one
ity. The differences are not obvious at leading order but
specifiesinitialconditionsforthematteroverdensity,but
at next to leading order one will see that what appears
they generally hold very well whenever the scale of the
in the equation for the acceleration is not necessarily
inhomogeneous region is <0.1/H.
what one would expect to appear in the Non-relativistic
Wenowdefinetwoover-lappingregionswhichweshall
regime.
refer to as the interior and the exterior.
As statedaboveinSection2.1,we requirethatgravity
is ‘suitably weak’ inside the density perturbation. By 3.1. The Interior Region
suitably weak we mean that
The interior region is defined by
δ 1,
1
| |≪ δ HR 1.
3
≡ ≪
where
We define the inner limit of a quantity F(r,t) to be
δ =δv R HR.
1 ≡ ,t− Lint(F)=limδ3→0F(r,t). The inner limit can be imag-
PhysicallyR isthevelocityofaparticleofdustin t,R ined as the limit in which the cosmological background
,t
{ }
coordinates, where R is the physical radial coordinate, density of matter is taken to zero, and the cosmologi-
andHRisthevelocitythatsuchaparticlewouldhavein cal horizon is taken to infinity. In the interior we con-
the cosmologicalbackground. The smallparameterδ = struct asymptotic approximations to quantities in the
1
δv is therefore the peculiar velocity of a matter particle limit HR 0. We also expand in δ and δ . We re-
1 2
→
withrespecttothecosmologicalbackground. Recallthat fer to this as the inner approximation.
5
3.2. The Exterior Region two regions. We then assume that there is an interme-
diate regionwhere both of these conditions hold. Which
In the exterior region, spacetime is required to be
implies that in the intermediate region: δ 1 and
homogeneous and isotropic at leading order. In other | 1| ≪
δ 1. So the assumption that δ ,δ 1 everywhere
words,the matterperturbationmustbe linearinthe ex- 2 1 2
| |≪ ≪
ensures that we have an intermediate region. We could
terior:
infactrelaxthisbutthecalculationswouldbecomemore
δ δ /δ 1,δ = δ 1.
4 ≡| 1 3|≪ 5 | m|≪ difficult: often still doable, but in almost all physically
From the form of R , it is clear that δ 0 implies interesting cases totally unnecessary. For instance the
,t 5
→
δto4 →be0L.eWxt(eFd)efi=nleimthδe5→ex0tFer(iro,rt)l.imAitsoinfathqeuainntteirtyioFr l(irm,tit) rwealardxaatniodnatlolow|δs1|on<e1toagnoda|lδl2t|h≪e w1a,yisupvetroyastbrlaaicgkhthfoolre-
we also expand in δ and δ . We construct asymptotic horizon (Shaw & Barrow 2006a,b,c).
1 2
approximationstoquantitiesintheexteriorregioninthis
4. EVOLUTIONOFDYNAMICALDARKENERGY
exterior limit and refer to them as the outer approxima-
PERTURBATIONS
tions.
In this section we find an asymptotic approximation
3.3. Matching and the Intermediate Region tothe DDE densitycontrast,δφ. The discussionis fairly
technical,andreadersmoreinterestedintheresultsthan
We are primarily concerned with the behaviour of the
themachineryusedtoderivethemmayprefertofocuson
DDE in the interior region. In Section 4, we find the
the statement, discussion and application of our results
inner approximation to φ by solving Eq. (6) order by
in Section 4.3 and following.
order in the interior limit. We cannot, however, ap-
The DDE is described by the field φ which satisfies:
ply both of the boundary conditions on φ directly to
the inner approximation. This is because the condition (cid:3)φ=V,φ(φ).
−
limR→∞φ = φc(t) must be applied at R = , a point We solve this equation by constructing an asymptotic
∞
that is very clearly in the exterior and not in the in- approximationtoφinthesmallparameterδ . Wewrite:
1
terior region. As a result, the inner approximation will
φ φ (t)+δφ(t,r)(1+ (δ )),
containambiguousconstantsofintegration. Fortunately, c 1
∼ O
this ambiguity can be lifted by matching the inner and where δφ (δ ). Before solving for φ, we make the
1
outerapproximationstoφifthereexistssomeintermedi- dependenc∼e oOf the metric on δ explicit by transforming
1
ateregion wherebothapproximationsaresimultaneously to a new radial coordinate, ρ = R(t,r)/a(t), where a(t)
valid. This matching of the inner approximation to the is the scale factor of the Friedmann-Robertson-Walker
outeroneisreferredtoasthemethodofmatchedasymp- (FRW) cosmological background. In t,ρ coordinates
totic expansions (MAEs). It relies on the fact that, in the metric is: { }
anygivenregion,theasymptoticexpansionofaquantity
1 k(r,t) δ2(r,t) dt2 δ (r,t)a(t)dtdρ
is unique (for a proof see Hinch 1991). Thus if the inner ds2= − − 1 +2 1
and outer approximations of φ are both valid in the in- (cid:0) 1−k(r,t) (cid:1) 1−k(r,t) −
termediateregion,theymustbeequalinthatregion. For a2(t)dρ2
the methodofMAE tobe applicable wemust, ofcourse, a2ρ2 dθ2+sin2θdϕ . (9)
1 k(r,t) −
require that an intermediate region exists. A necessary −
(cid:8) (cid:9)
condition for an intermediate region to exist is that for Consider (cid:3)φc(t) in this metric:
−
some range of R: 1Q˙ φ˙ (t) ∂
(cid:3)φ (t)=V (φ )+ φ˙ (t) c ρ2δ .
HR 1, 3δ /κε(m)(t)R2 = 3δ (t)/κ = 1. − c ,φ c 2Q c − a(t)ρ2∂ρ 1
≪ | 2 c | | m | ≪
This becomes a necessary and sufficient condition if the whereQ˙ isevaluatedatconstantrasopposedto(cid:0)cons(cid:1)tant
only boundary of the interior region is in the exterior ρ or R and is thus given by Eq. (3). It follows that δφ
one, and vice versa (see Figure 1 for an illustration). If satisfies:
the scale of the inhomogeneity, R , is taken to be the
0 (cid:3)δφ(1+ (δ ))=(V (φ) V (φ )) (10)
largest value of R for which 3δ /κε(m)(t)R2 >0.3, and − O 1 ,φ − ,φ c
| 2 c | 1 φ˙ (t)
then then an intermediate region generally exists pro- + κφ˙2(t)ρδφ + c ρ2δ .
vided that R .0.1H. 2 c ,ρ a(t)ρ2 1 ,ρ
0
In resume, in this section we have considered spher- (cid:0) (cid:1)
We now solve for δφ in both the interior and exterior
ically symmetric backgrounds and far from black hole
regions.
horizons. We have basically performed an expansion
in the Newtonian potential at leading order, at least 4.1. Inner Approximation
as far as the evolution of scalar field perturbations are
We note that 1κφ˙2R2 . (δ2), and that k
concerned. Notice, however, that this method can be 2 c O 3 | | ∼
straightforwardly extended to allow for deviations from O(δ3,δ12,δ2), so that
spherical symmetry (Shaw & Barrow 2006a,b,c). ρ2
To be clear, we should point out that the expansion ∂ a3(t)δφ + ρ2δφ (V (φ) V (φ ))a2ρ2
− a t ,t ,ρ ,ρ ∼ ,φ − ,φ c
in δ and δ , which is a lot like expanding in the New-
1 2
tonian potential, is not really the key to our method. (cid:0) (cid:1)+φ˙c(cid:0)(t)a ρ(cid:1)2δ1 ,ρ+O(δ1δφ,δ2δφ,δ32δφ).
That is just a simplification. The key is that we take:
In many cases δφ is small e(cid:0)noug(cid:1)h so that
δ = HR 1 in the interior region and δ 1 in
3 m
the exterior≪region. Thatis, everythingis linear i≪n those (V (φ) V (φ ))a2ρ2 V (φ )R2δφ,
,φ ,φ c ,φφ c
− ∼
6
andV (H2)sothatφ (t)evolvesovercosmological where the neglected terms are (δ2,δ2,δ δ ,δ ) times
time-s,φcaφle∼s.OIn these cases ict is clear that: f(η,ρ′). After some algebra andOinte1gra3tio1n3we2then ar-
(V (φ) V (φ ))a2ρ2 (δ2δφ), rive at:
,φ − ,φ c ∼O 3 δφ(t,R)(1+ (δ ,δ ,δ )) φ˙ (t) Rδ (t,R′) dR′
and so this term can be dropped at leading order. Al- O 1 2 3 ∼ c ∞ 1
though it will often be the case that V,φ(φ)−V,φ(φc) ≈ +F(η−ρ)R+G(η+ρ). R (14)
V (φ )δφ we do not have to require such a strong as-
,φφ c wherethepreviousassumptionrelatingtotheTaylorex-
sumption in order to justify ignoring the effect of the
pansionoff canbeseentobeequivalenttothestatement
potential to leading order in the interior approximation. that δ¨ (t,R)R2 δ . This assumption will generally
Instead, we make the far weaker assumption that: 1 ≪ 1
break down if there is some initial instant, t = t say,
i
A p|Vhy,φs(iφcacl+ly-δvφia)b−leVD,φD(φEc)t|Rhe2o/rδyφf∼orOw(hδi1c,hδ2t,hδi3s).requ(i1r1e)- wlehntentoδ1δ12=, δ032., δA2t≪t ≫1, wtihhicohweisvecre,rtδ¨a1iRn2ly≪theδ1caisseeqinuitvhae-
ment does not hold would be difficult to construct. For interior region.
instance,ifthisconditiondidnotholdthenevenif,atone Generally, the functions and are related to the
F G
instant,V (φ ) (H2)orsmaller,asmustberequired initial conditions on δφ. Because we have required
for φc to,φevoclve∼oOver cosmological time scales, shortly R−2 ≪H−1 our analysis will break down at early times
afterwards small changes in φc would cause V,φ(φc) to since generally HR−2 → ∞ as t → 0. As a result, we
grow to be many orders of magnitude greater than H2. cannot generally determine and by simply speci-
F G
fying some initial conditions onδφ, at t = t say, and
Thiswouldresultinφ evolvingovertime-scalesthatare i
c
applying them to the interior approximation since the
muchshorterthanthe Hubble time. The only DDE the-
interior approximationby not be valid when t=t . The
ories that we can imagine that could accommodate this i
requirement that δφ(R = 0) can however be applied to
and still be compatible with observations would involve
the interior approximation to give (η) = (η). To
φ being held almost completely fixed at a minimum of
F −G
the order at which we work, (η) itself should be fixed
V(φ ), in which case the DDE would be almost entirely
c F
by matching to the outer approximation.
indistinguishable from a cosmologicalconstant.
To leading order in the interior, we would actually be 4.2. Exterior Solution and Matching
justified in setting a(t) = const as a˙ρ = HR = δ
1
1. However, since it might not be completely clear th≪at In order for φc = limR→∞φ(t,R) we must have
δφ(t,R) 0 as R . In the exterior region, space-
thesetermsmaybeignored,andbecausewecansolvethe
→ → ∞
time is FRW to leading order in δ and δ , and the
δφ equation without having to ignore them, we continue 1 2
equation describing for the leading order behaviour of
to include them.
u=a(t)δφ reads:
We define dη = dt/a(t) and u = aδφ, and note that
ρ2a,ηη/a=H˙R2+H2R2 (δ3), so that: u + 1 ρ2u = V (φ ) H˙ H2 a2u+
∼O − ,ηη ρ2 ,ρ ,φφ c − −
1
−u,ηη+ ρ2 ρ2u,ρ ,ρ (1+O(δ1,δ2,δ3))∼ 1κφ(cid:0)˙2(t)a2(cid:1)ρu(cid:16)+ φ˙c(t)a2 ρ2δ .(cid:17) (15)
(cid:18) 1 (cid:0) (cid:1) (cid:19) 2 c ,ρ ρ2 1 ,ρ
φ˙c(t)ρ2 a(t)ρ2δ1 ,ρ. (12) where dη = dt/a(t). Solving this eq(cid:0)uatio(cid:1)n is far from
straightforward. Fortunately, however, our main inter-
which has as a(cid:0)solution(cid:1)
estinnotinhowδφbehavesintheexteriorregionbutin
1 ∞ 1 f(η(t) X,ρ′) interiorregionwheretheperturbationinthematterden-
δφ(t,ρ) ρ′2 dρ′ ds − +
∼−2 a(t)X(ρ,ρ′,s) sity canbe non-linear,andso it is notnecessaryto solve
Z0 Z−1 Eq. (15) in the exterior region so long as we know how
(η ρ) (η+ρ)
F − + G + (δ δφ,δ δφ,δ δφ), (13) its solutions behave in some intermediate region where
1 2 3
a(t)ρ a(t)ρ O δ = HR 1. We require that δφ 0 as the mat-
3
≪ →
where f(η,ρ) = φ˙c(t)a(t) ρ2δ (t,ρ) and X(ρ,ρ′,s) = ter perturbation is removed i.e. δ1 → 0. We have also
ρ2 1 ,ρ required that δ1 drop off faster than 1/R for R > R−2
fielρd2+anρd′2w−e2dρisρc′su;ssFtahnedm(cid:0)Gfurerpthreesre(cid:1)nmtowmaevnetsairniltyh.eTscoalgaor wthhiserientRe−rm2eidsiaintethreegiinotne,ritohreresgoilountioi.nes. oHfRE−q.2 ≪(151). fIonr
fpurther, we make the following reasonable assumptions which δφ 0 as R have the following form:
→ →∞
about the behaviour of δ1. We assume that there exists R K(η)
some R−2, in the interior region, such that for R>R−2 δφ(t,R)∼φ˙c(t) δ1(t,R′)dR′+ R +
in the interior region, δ1 decreases faster than 1/R2 as ZC(η)
R ;f thendecreasesfasterthan1/R3 forR>R−2. (δ1δφ,δ3δφ,δ2δφ), (16)
→∞ O
This assumptionensuresthatdominantcontributionsto whereC(η) (1/H)orgreater. Wehaveassumedthat
the integral in Eq. (13) comes from values of ρ′ that t is larger co∼mOpared to any initial instant when one one
are well inside the interior region. Note that generally or more of δ = 0, δφ = 0 or δφ˙ = 0 hold. With these
f(η,ρ) varies over conformal times-scales of order 1/aH 1
assumptions, matching to the interior region gives:
and δ /aδ˙ . Provided that δ is not momentarily zero,
1 1 1 K(η)= (η)+ (η)=0,
these scales are much larger than X . √2R−2/a so we F G
can Taylor expand f(η X,ρ′) as and ∞
f(η−X,ρ′)=f(η,ρ−′)+f,η(η,ρ′)X +O(f,ηηX2), 2Fa,(ηη()η) =−φ˙c(t)ZC δ1(t,R′)dR′.
7
Wehaverequireddropoffofδ meansthat,toleadingor- This expression for δ is our main result and it is valid
1 φ
der, we can set C = . This implies that the 2 /a(η) for both linear and non-linear sub-horizon matter per-
,η
∞ F
issub-leadingorderinthe interiorapproximationandso turbations. In order to evaluate the above expression,
may be neglected. one must simply specify the peculiar velocity, δv, of the
Thus to leading order in the interior region we find: matter particles andthe matter density contrastδ¯. This
exterior is valid in the interior region at late times com-
R
δφ(t,R) φ˙ (t) δv(R′,t)dR′(1+ (δ ,δ ,δ )). paredto anyinitialinstance whenδφ=0 and/orδφ˙ =0
∼ c Z∞ O 1 2 3 i.e. |δv¨|R2 ≪|δv˙|R≪|δv|.
NotethatH andwwilldependonthemassofthefield
Where δv = R HR is the peculiar radial velocity
,t − m,sointhatsense,equation(18)isnotcompletelymodel
of the matter particles relative to the expansion of the
independent. However, all the model dependent terms
backgroundUniverse. Thisapproximationisvalidinthe
come from the homogeneous and isotropic background
interior region at late times compared to any initial in-
cosmology. Sotheexpressionforδ ismodelindependent
stance when δφ = 0 and/or δφ˙ = 0 which is equivalent φ
provided mR 1. Which is true if m O(H) and
to δv¨R2 δv˙ R δv . Note that, to this order, the c ≪ ∼
sur|fac|es of≪co|nst|ant≪φ|are|surfaces of constant: HRc ≪1 as assumed.
The key point is that large deviations in δ from the
φ
∞ predictionusingaLCDMbackgroundcouldonlyoccurif
tφ =t− δvdR′. δv and δm were predicted to change by anorder of mag-
ZR nitude or more. As far as we know, however, structure
Weusethisasymptoticapproximationforδφtoevaluate formation is compatible with LCDM over the scales of
the DDE density contrast on surfaces of constant t in clusters and so it seems unlikely that such large devia-
the matter rest frame. In the rest frame of the matter tions occur.
particles: As we shall see below the DDE perturbation is gen-
1 Q erally small compared to the matter perturbation, so to
ε(φ) = φ˙2+ φ2 +V(φ),
2 2 ,R leading order one may neglect any DDE perturbations
when evaluating δ¯and δv.
and the local inhomogeneity in ε(φ) is therefore: It is interesting to note that the sign of δ depends
φ
on the relative magnitudes of δ¯ and δv: From equation
R
δε(φ) φ˙2(t) 3H δv(R′,t)dR′+(δv(R,t))2 (18)onededucesthattheformationoflocaloverdensities
∼ c − Z∞ or local voids of dark energy arise from two competing
effects:
R
+ R (H˙ +H2)R′ dR′ . (17)
Z∞ (cid:16) ,tt− (cid:17) ! • Tδvh/eHDRrag=eff0e.ct:Ifasosnoeciahtaesdawnithovdeerdvieantsieonrsegfrioomn,
Thecorrectionstermsare (δ ,δ ,δ )timessmallerthan then it expands more slowly. In agreement with
1 2 3
the leading order term. SiOnce φ˙2 =(1+w)ε(φ) we have Dutta & Maor (2007), we find this drag effect to
c c produce a local underdensity of dark energy.
R
δ (1+w) 3H δv(R′,t)dR′+(δv(R,t))2 The Pull effect: associated with deviations from
φ ∼ − Z∞ • δm = 0. An overdensity of matter pulls the dark
energy(and everythingelse) towardsit. So matter
R
+ R (H˙ +H2)R′ dR′ , and fields tend to clump around it. As a result,
,tt
Z∞ (cid:16) − (cid:17) ! tchreeapteusllaenffeocvtermdeeannsistythoaftDaEn.overdensityofmatter
R is the acceleration of the matter particles in (t,R)
,tt
coordinates. Eq. (9) gives: So there is one effect that pulls the dark energy in and
another that pushes it out. As we will see, in the linear
δM
R R H˙R2 H2R2 + (δ3δ , δ2δ2, δ δ2). regime the drag effect grows more quickly than the pull
,tt − − ∼− R O 3 1 1 3 2 3 effect, so one gets a dark energy void. The onset of non-
where δM(R,t) = M(R,t)− 12ΩmH2R3 = 21ΩmH2R3δ¯ lrienaecahrastmruacxtuimreumforamnadtitohne,nhotowetveenrd, ctoau1seast|δlavt/eHtRim|etso,
is the mass contrast at time t inside the sphere with
whereasδ justkeepsongrowing. Sothepulleffectdom-
physical radius R. m
inates at late times, so one ends up with an overdensity
4.3. Discussion of dark energy.
We have assumed above that the mass of the DDE
InthissectionwefoundthattheDDEdensitycontrast scalar field is small compared to the inverse length scale
canbe expressed,to leadingorder,interms ofthe radial of the cluster. Provided this is δ in uncoupled DDE
peculiar velocity, δv , the mean density contrast, δ¯ and theoriesisindependent ofthedetailφsoftheorydescribing
the unperturbed DDE equation of state parameter w: the dark energy.
∞ In the next section we use Eq. (18) to evaluate δφ
δ (1+w) 3H δv(R′,t)dR′+(δv)2 in both the linear and non-linear regimes. We find that
φ
∼ +1Ω H(cid:18)2 ∞ZRδ¯(R′,t)R′dR′ . (18) |sδcφa(leRo=ft0h)e|c∼luOste(Ωr,maHnd2Rδ¯cc2luδ¯sctluisst)t,hwemheeraenRmcaitsttehredernadsiitayl
2 m ZR (cid:19) contrast in R≤Rc.
8
5. CLUSTERINGOFDYNAMICALDARKENERGY Evolution of central density contrasts in the linear regime
5.1. Dynamical Dark Energy Clustering in the Linear 100 | δφ/(1+w) |
Regime δ
m
Althoughoneofourmainaiminthisworkwastostudy
DDE clustering in the non-linear regime, our analysis ast10−2
is also perfectly valid in the linear regime, i.e. when ntr
o
hδmori≪zon1,aptrtohveidiendsttahnattwthheenmwatetewrisinhhtoomeovgaelunaeitteyδis.suWbe- nsity c10−4
φ e
d
found above that δφ is given by Eq. (18). In the linear DE
regime δv HR, and so the term proportional δv2 D
in Eq. |(18|)≪is small compared to the other two terms 10−6
and should be neglected. δ (r,t) is the matter density HR(z=30)=0.03
m δ (Rc=0, z=30) = 0.02
contrast,andwedefine δ (t)to be its Fouriertransform. m
k
In the linear regime (in comoving-coordinates): 10−1800 101
Redshift: z
δ˙ = δv,
m −∇· x 10−5 Evolution of DDE density contrast in the linear regime
1.5
and so:
HR(z=30)=0.03
δvk =−iHf(ka)aδk, 1 δm(Rc=0, z=30) = 0.02
w
wfohrmereoffδ(av). =Wedrlencδokg/ndizlentahaatnd3ΩδvmkHis2δ¯tRhe/2Fo=urδiMer/tRra2n=s- δast: /1+φ 0.5
∂δΦ/∂R where δΦ is the leading order perturbation to ntr
the Newtonian potent2iaδlΦ. δ=Φ4iπsGgδive.n by: ensity co 0
∇ m E d −0.5
The Fourier transform of δ¯(R,t)R is therefore: iaδ /k, DD
k
and the Fourier transform of the DDE density contrast, −1
δ , in the linear regime is given by:
φ
−1.5
3 a2Ω H2δ 100 101
δ(lin) (1+w)g(a) m k. (19) Redshift: z
φk ∼−2 k2
where
Fig. 2.— Evolution of the DDE and matter density contrasts
2f(a)
in the linear regime for an overdensity that begins to collapse at
g(a)= 1. (20)
Ωm − z = zi = 30. We have taken Ωm(z = 0) = 0.27. Initially the
matter density profile is taken to be Gaussian with δm(zi,r =
Motivated by observations, we assume (1 + w) to 0)=0.02andHirc=0.03. Thiscorrespondsδm(r=0)=0.3and
be small at late times, and (1 Ωm) to be small at Rc = arc ≈ 31h−1Mpc today. |δφ| is always small compared to
early times. To a first approxim−ation then we approx- δm. Atearlytimes,theDDEdensitycontrastispositivehoweverit
mustbenotedthatourapproximationisonlyvalidinthiscasefor
imate f(a) by its ΛCDM value. At late times a very
z.29. Atlatetimesitisnegative. Wecanseethat |δφ/(1+w)|
goodapproximationto f(a) in the ΛCDM model is Ω0.6 is growing at late times in the linear regime; indeed it is in fact
m
(Peebles 1980; Lahav et al. 1991). Note that corrections growingfasterthanδm forz<0.23.
tothisfittingformulaeoccurforquintessence-likemodels
(Wang & Steinhardt1998). However,thesearegenerally
toosmalltogreatlyaffectourresults. Hencewetakethe regime) if there is a local void of matter. If an initial
ΛCDMexpressionforsimplicity. Evenwhenitisnotac- mean density perturbation δ (z ) begins to collapse at
k i
ceptable to approximate f(a) by its ΛCDM, we do not z =z when a=a =1, Ω =Ω and H we have:
i i m mi i
expectthisapproximationtogreatlyalterthequalitative
1
nature of our results or the order of magnitude of δ .
φ δ (22)
Transforming back into real space we have: k≈3Ω0.4+2
mi
(1+w)g(a)Ω a2H2 ∞ Ω 0.2 Ω 0.5
δ(lin) m r′δ¯(r′,t)dr′, (21) 3Ω0.4 m a+2 mi a−3/2 δ (z ),
φ ∼− 2 mi Ω Ω k i
Zr (cid:18) mi(cid:19) (cid:18) m (cid:19) !
Our expression for δ(lin) is independent of the mass of and so
φk
φ. This is because we have assumed that m (H)
φ ∼ O 9(1+w)Ω H2 Ω 0.2
in the linear regime and that the matter perturbation δ(lin) mi i Ω0.4(2Ω−0.4 1) m
is small compared with H−1. The mass of the scalar φk ∼ 6Ωm0.4i +4 mi m − (cid:18)Ωmi(cid:19)
field therefore has only a negligible effect on the scales
0.5 5/2
over DDE clustering occurs. It is clear that the sign of 8 Ωmi 1+z δk(zi)
.
δφk is the same as the sign of −g(a)δk. At late times −3(cid:18)Ωm (cid:19) (cid:18)1+zi(cid:19) ! k2
g(a) > 0, and so a linear local overdensity of matter
(0 < δm 1) corresponds to a DDE void. Similarly, Our expressionfor δφ was derived under the assumption
there is a≪DDE overdensity at late times (in the linear that δv¨R2 δv˙ R δv . As result it will break
| | ≪ | | ≪ | |
9
down as we approach z = z . More precisely, we find then
i
that our approximation is valid at R= 0 for a &a (1+
H2HiiiRst−h2e(zini)it)iai.lev.alzue.ofzaHppan≡dzRi(−12−(zi2)HisiRth−e2i(nziit)i)alwvihaeluree δi(r)= 34δr0r3c3√πerf(rrc)−2rrce−(r/rc)2)
of R−2. ¿From this expression we can see that a matter and:
overdensity corresponds initially to a DDE overdensity
(δφ(lin) > 0), however this DDE overdensity becomes a δφ(lin)(r =0,t)≈−9(1+w2)0δ0Hi2rc2(2Ωm−0.2−Ω0m.2),
DDE void at some z = z , and δ(lin) < 0 for z <z .
crit φ crit
at late times.
We see that z is given by:
crit
We plot the evolution of δ (r = 0) and δ (r = 0)
m φ
1+z 6 3Ω0.4(z ) 2/5 for a Gaussian initial matter density perturbation in
1+czrit ≈Ω−m0.6(zcrit) − m8 crit , the linear regime in Figure 2. We have assumed that
i (cid:18) (cid:19) δm(r = 0,zi) = 0.02 at zi = 30, and Hirc = 0.03,
where we have taken Ωmi 1. If Ωm(zcrit) = 1, which which corresponds to and δm(r = 0,z = 0) = 0.3 and
should be a good approxim≈ation for 1.8 . z . 3000, we Harc =0.010todayi.e. Rc =arc 31h−1Mpc. Forthis
≈
have z =(3/8)2/5(1+z ) 1. For this analysis to be choiceofinhomogeneity,ourapproximationforδφ isonly
crit i
hvaolliddswHeimRu−s2t(zhia)v.ez0cr.i2t4<. zFaop−rpiwnhhiocmhowgietnheΩitimes(ztchriat)t≈ar1e g(2o0o0d7f)osra1w+izn.th0e.i9r4n(1u+mzeir)ic≈al2s9im. JuulasttioanssD,uwtetafi&ndMthaoart
largerthanthis atz =zi we do not expect zcrit to be an close to z = zi, δφ/(1+w) > 0. It then becomes nega-
accurate approximation to the redshift when the DDE tive,andatlatetimesitcontinuestogrowmorenegative,
density contrast changes sign. and it appears as if it might overtake δm at some point
Note that z does not depend on the size of the in- inthefuture. However,asweshallshowbelow,thisdoes
crit
homogeneity, although, of course, we must have z . not occur and δφ remains small at all times. Note that
crit | |
zapp for our analysis to be valid, which implies that at the apparent increase in δφ at early times is due in part
z =zcrit, HR−2 .0.2. to fact that our approximation breaks down as one ap-
proachesz =z . Ifonewishestostudytheevolutionofδ
If Ω = 1 exactly then δ(lin)/(1+w) const at late i φ
m φ → allthewaybacktoz =zi,thenumericalapproachtaken
times. If Ωm <1, then δφ/(1+w) grows likes: by Dutta & Maor (2007) provides accurate results. Our
| |
realfocusinthisworkis,however,notonwhatoccursat
δ
φ 2Ω−0.2 Ω0.2, very early times in the linear regime, but how the DDE
1+w ∝ m − m
(cid:12) (cid:12) density contrastevolveswhen the matter inhomogeneity
(cid:12) (cid:12) goes non-linear. We consider this below.
at late times in(cid:12) the l(cid:12)inear regime. This implies that
(cid:12) (cid:12)
δ /(1 + w) tends to a constant in the linear regime
φ
i|f Ω = 1,|and is growing for Ω < 1. At late times 5.2. Dynamical Dark Energy Clustering in the
m m
in the linear regime then: δ /δ Ω−0.4a−1. If we Quasi-Linear Regime
| φ m| ∝ m
take Ωm = 0.27 today then |δφ(lin)/(1+w)| is predicted maWtteerbepgeinrtuourbraitnivoensteigxaittsiotnhoeflhinoewarδφreegviomlveesbywhceonnstihde-
to grow faster than δ when z = 0.23. Dutta & Maor
m
ering a matter perturbation that is in the weakly non-
(2007) also found that at late times in the linear regime
linear or quasi-linear regime i.e. 1 < δ¯ . 10. Eqs.(3)
δ /(1+w) growsfasterthanδ . Thisisbehaviourwere
| φ | m and(5) giveM(r,t)/R M(r)/R−+ (δ3δ , δ2δ2, δ δ2).
tocontinueintothenon-linearregimeitwould,ofcourse, ∼ O 3 1 1 3 2 3
We suppose that at some initial time, t , R =HR and
implythat δ /(1+w) wouldultimatelygrowtobevery i ,t
φ
| | R=r. M(r)isthenthemassinsidetheshellwithradius
large and indeed dominate over δ . As we show in this
m
r at t=t . We write:
paper, however, before this can happen the matter per- i
turbation goes non-linear and this slows the growth of Ω (t )
|δ/(1+w)|. M(r)= m2 i Hi2r3(1+δi(r)).
Weconcludeouranalysisofthelinearregimebynoting
that Eq. (21) implies that at late times: δ (r) is interpreted as the initial mean matter density
i
contrast. In the full non-linear regime, which we con-
(lin)
δ (r =0,t) (23) siderinthe nextsubsection,weareonlyabletoevaluate
φ ≈
R(r,t) and hence, via Eq. (18), δ , analytically in a
3C (1+w) φ
− δ 2 (2Ω0m.6−Ωm)H2(ar−2)2δ¯(r−2,t), matter-dominatedbackground(Ωm =1). In many cases
of interest, however, δ (1), e.g. superclusters and
m
∼ O
where r−2 is defined the smallest value of r for which voids. Although the linear approximation fails for such
∂lnδ¯/∂lnr 2,andC (1)dependsontheprecise objects, a good leading approximation to the true mean
δ
form of δ¯. ≥− ∼O density contrast of matter, δ¯, in the range, 1<δ¯.10
−
Wehaveshownthatinthelinearregime δ atthecen- is given by:
φ
tbryeaoffathcteoirnhofomabooguetne(i1ty+iswa)(lwHaayrs−s2m)2a;llHerat|rh−a2|ni|sδ¯(rro−u2g,htl)y| 1+δ¯(r,t)≈ 1−2δ¯lin(r,t)/3 −3/2,
equalto the physicalsize ofthe inhomogeneityasafrac-
where δ¯ is the linea(cid:0)r mean density(cid:1)contrast. This
tionofthehorizonsize. Iftheinitialdensityperturbation lin
approximation improves as Ω 0 and is exact for
is Gaussian i.e. m →
Ω = 0. When this approximation holds we say that
m
δ (z )=δ exp( r2/r2) we are in the quasi-linear regime.
m i 0 − c
10
Evolution of central density contrasts for a typical supercluster Evolution of central density contrasts for a typical void
102
| δφ/(1+w) | 100
δ
100 m
10−2
10−2
ast ast
Density contr1100−−64 Density contr1100−−64 δ| φδ/m(1 |+w)
10−8 R(z=0)=15 h−1 Mpc 10−8 R(z=0)=15 h−1 Mpc
δc(R=0, z=0) = 2 δc(R=0, z=0) = −0.93
m m
10−10 10−10
100 101 100 101
Redshift: z Redshift: z
x 10−5 Evolution of DDE density contrast for a typical supercluster x 10−5 Evolution of DDE density contrast for a typical void
0 1
R(z=0)=15 h−1 Mpc R(z=0)=15 h−1 Mpc
cδ (R=0, z=0) = 2 0.9 δ (cR=0, z=0) = −0.93
m m
−0.5 0.8
w) w)
+ +
δast: /(1φ −1 δast: /(1φ 00..67
ontr ontr 0.5
nsity c −1.5 nsity c 0.4
e e
E d E d 0.3
D D
D D
−2 0.2
0.1
−2.5 0
100 101 100 101
Redshift: z Redshift: z
Fig. 3.—EvolutionoftheDDEandmatterdensitycontrastsfor Fig. 4.—EvolutionoftheDDEandmatterdensitycontrastsfor
asuperclusterwhichtodayhasacorematteroverdensityof2and a typical void which today has a core matter density contrast of
acoreradiusof 15h−1Mpc. We have take Ωm =0.27today. |δφ| −0.93andacoreradiusof15h−1Mpc. WehavetakenΩm=0.27
isalways≪δm,andδφ/(1+w)<0whichcorrespondstoaDDE today. |δφ|isalways≪|δm|,andδφ/(1+w)>0whichcorresponds
void. As in the linear regime, |δφ/(1+w)| grows monotonically toaDDEoverdensity. Asinthelinearregime,|δφ/(1+w)|grows
withtime, andat latetimes |δφ/(1+w)|is growingmorequickly monotonicallywithtimeandatlatetimes|δφ/(1+w)|isgrowing
thanδm. morequicklythan|δm|.
Inthissectionwefindδ (δ¯)forinhomogeneitiesinthe
φ
quasi-linearregime. We areagainassumingthatanyde- UsingEq. (18),wefindthatthedarkenergydensitycon-
viations from the ΛCDM model for structure formation trast,δφ, inthe quasi-linearregimeis wellapproximated
due the background DDE evolution are sub-leading or- by δ(ql) where:
φ
der i.e. (1 Ω )(1+w) is small. If this is not the case
m
tvhaelind,obuurtet−xhpereesvsoioluntifoonr oδφf δ¯inantdermδvswoofuδ¯ldanchdaδnvgei;sesvteilnl δφ(q(l1)+≡w)Ω H2 ∞ 3f(a)((1+δ¯)2/3 1) (25)
still the order of magnitude of δφ would not be greatly m dR′R′ − δ¯
effected. − 2 ZR (cid:18) Ωm − (cid:19)
δ¯(rT,ht)etpheucsu:liar velocity, δv = R,t − HR, is related to +(1+w)ΩmH2R2 f(a)2 ((1+δ¯)2/3 1)2.
4 Ω −
(cid:18) m (cid:19)
δ¯ R 1
δv = ,t HRf(a)((1+δ¯)2/3 1), For 0.27 Ωm 1, the first term in Eq. (25) is neg-
−3(1+δ¯) ≈−2 − ative defini≤te for ≤all positive values of δ¯ for which the
where, as above,f(a) Ω0.6 at late times. In the quasi- quasi-linear approximation holds (0 < δ¯. 10) and pos-
linear regime the phys≈icalmradius, R(r,t), of a shell with itive definite for 1 < δ¯ < 0. The second term in Eq.
−
mass M(r) is given by: (25) is clearly positive definite but vanishes at R = 0.
The relative magnitude of the two terms depends on the
1/2 1/3
2δ (r,t) 2M(r) choice of initial density profile. However, at R = 0 it is
lin
R=a 1− 3 a3Ω H2 clear that an overdensity of matter, δ¯> 0, corresponds
(cid:18) (cid:19) (cid:18) i mi i (cid:19) to a dark energy void, δ (R = 0,t) < 0. Similarly, a
1/2 φ
=ar 1 2δlin(r,t) (1+δ (r))1/3. (24) void of matter corresponds to a local DDE overdensity
− 3 i at R = 0. In Figures 3 and 4 respectively show the
(cid:18) (cid:19)