Table Of ContentMon.Not.R.Astron.Soc.000,??–??(2010) Printed11January2011 (MNLATEXstylefilev2.2)
Evolution of Spherical Overdensity in Thawing Dark
Energy Models
1 N. Chandrachani Devi1⋆ and Anjan A Sen,1†
1
1CenterFor Theoretical Physics, Jamia MilliaIslamia, New Delhi 110025, India
0
2
n
11January2011
a
J
9
ABSTRACT
We study the general evolution of spherical over-densities for thawing class of dark
]
O energy models. We model dark energy with scalar fields having canonical as well as
non-canonical kinetic energy. For non-canonical case, we consider models where the
C
kinetic energy is of the Born-Infeld Form. We study various potentials like linear,
.
h inverse-square, exponential as well as PNGB-type. We also consider the case when
p darkenergyishomogeneousaswellasthecasewhenitisinhomogeneousandvirializes
- togetherwithmatter.Ourstudyshowsthatmodelswithlinearpotentialinparticular
o
with Born-Infeld type kinetic term can have significant deviation from the ΛCDM
r
t modelintermsofdensitycontrastatthe timeofvirialization.Althoughourapproach
s
a is a simplified one to study the nonlinear evolution of matter overdensities inside the
[ cluster and is not applicable to actual physical situation, it gives some interesting
insights into the nonlinear clustering of matter in the presence of thawing class of
2
dark energy models.
v
4 Key words: Cosmology:Dark Energy, Thawing Model, Scalar fields, Spherical Col-
9
lapse.
0
4
.
3
0
0 1 INTRODUCTION Liddle& Scherrer 1999; Steinhardt 1999). These scalar
1 field models can be broadly classified into two categories
One of the most significant discoveries in cosmology in re-
: depending upon the form of their potentials: fast roll and
v cent years is the fact that our universe is currently going
slow roll models termed as freezing and thawing models in
i through an accelerated expansion phase(Riess 2004; Knop
X the literature(Caldwell & Linder 2005). In case of fast roll
2003). This can have far reaching implications for funda-
r models, the potential is steep resulting the scalar field to
mental theories of physics. This late time acceleration of
a track thebackgroundfluid and is subdominant for most of
the universe can be due to the presence of an exotic fluid
the evolution history. Only at late times, the field becomes
with large negative pressure known as dark energy or due
dominant and drives the acceleration of the universe. Such
to the modification of gravity itself. The simplest candi-
solutions are also known as trackers.
date of dark energy is provided by cosmological constant
Λ with equation of state parameter w = 1. However, Then there are the slow-roll models for which the field
−
the ΛCDM model is plagued with fine tuning and cosmic kinetic energy is much smaller than its potential energy.
coincidence problems (See Copeland (2006); Sami (2009); Usuallyithassufficientlyflatpotentialsimilartoaninflaton.
Sahni& Starobinsky (2000); Padmanabhan (2003); Linder Atearlytimes,thefieldisnearlyfrozenatw= 1duetothe
−
(2008);Frieman(2008);Caldwell(2009);Silvestri(2009)for large Hubble friction. Its energy density is nearly constant
a nice review). with anegligible contributiontothetotalenergy densityof
Scalar field models mimicking a variable Λ can theuniverse.Butasradiation/matterrapidlydilutesdueto
alleviate the fine tuning and coincidence problems the expansion of the universe and the background energy
and provide an interesting alternative to cosmolog- density becomes comparable to scalar field energy density,
ical constant(Ratra & Peebles 1988; Caldwell 1998; thefieldbreaksawayfromitsfrozenstateandevolvesslowly
totheregionwithw> 1.However,inthiscase,themodel
−
needs some degree of fine tuningof the initial conditions in
order toachieve a viable late timeevolution.
⋆ email:[email protected]
† email:[email protected] Recent observations suggest that the equation of state
(cid:13)c 2010RAS
2
parameter for dark energy does not significantly deviate simulations to accurately measure the effect of dark energy
from w = 1 around the present epoch (Wood-Vasey on theclustering properties of matterand thishas been in-
−
2007; Davis 2007). This type of equation of state can be vestigated in anumberof papers(Maccio 2004;Baldi2008;
easily obtained in dynamical models represented by thaw- Courtin 2010). The simplest analytical approach to study
ing scalar fields. Motivated by this fact, Scherrer and Sen the nonlinear clustering of matter is the spherical collapse
(Scherrer& Sen2008a)examinedquintessencemodelswith formalism firstdevelopedbyGunnandGott(Gunn & Gott
nearly flat potentials satisfying the slow-roll conditions. It 1972). It describes how a small spherical over density
wasshownthatundertheslow-rollconditions,ascalarfield decouplesfrom thebackgroundevolution,slows down,then
with a variety of potentials V(φ) evolves in a similar fash- eventually turns around and collapses. It is generally as-
ion and one can derive a generic expression for the equa- sumed that thecollapse is not complete so the system does
tion of state for all such scalar fields. Similar results were not reach the singularity, instead it eventually virializes
later established for the case of phantom (Scherrer & Sen and stabilizes to a finite size. Combining this with the
2008b)andtachyonscalarfields(Ali2009;Sen2010).Itwas Press-Schechter formalism (Press & Schechter 1974), one
demonstrated that under slow-roll conditions, all of them canhaveamodelforformation ofstructureswhichpredicts
haveidenticalequationofstateandhencecannotbedistin- the abundances of virialized objects as a function of mass.
guished, atleast, at the level of background cosmology. The Thismodelcangiveinsightintothephysicsofstructurefor-
crucialassumptionforarrivingatthisimportantconclusion mation whichcan beultimately used foradetailed N-Body
was the fulfillment of the slow-roll conditions for the field simulation. There are numerous studies of the spherical
potentials. collapse formalism to include both homogeneous and
InarecentpaperbySenetal.(Sen2010),assumptionof inhomogeneous dark energy (Lahav 1991; Shaw & Mota
slowrollhasbeenrelaxed,butitwasassumedthatthescalar 2008; Manera & Mota 2006; Lokas & Hoffman 2001;
field is of thawing type i.e it is initially frozen at w = 1 Basilakos & Voglis 2007; Schaefer & Koyama 2004;
duetolargeHubbledamping.Withthesechoicestheevo−lu- Bartelmann 2006; Basilakos 2003; Horellou & Berge
tionofavarietyofscalarfieldmodelshavingbothcanonical 2005; Maor & Lahav 2005; Mota & Van deBruck 2004;
and non-canonical kinetic terms have been studied. Obser- Nunes& Mota 2006;Wang & Steinhardt1998;Wang2006;
vational quantities like Hubble parameter, luminosity dis- Weinberg& Kamionkowski 2003;Zeng 2005).
tance and quantities related to the Baryon Acoustic Oscil- Inthiswork,wesystematically studythespherical col-
lation (BAO)measurement havebeenstudiedfor varietyof lapsemodelforthawingtypescalarfielddarkenergymodels
potentials. It was shown that given the current error bars for various potentials. We consider the canonical as well as
fordifferentobservationaldatarelatedwithbackgroundcos- noncanonicalkineticenergyforthescalarfields.Weassume
mology,itisdifficulttodistinguishdifferentthawingmodels the dark energy to be homogeneous as well as inhomoge-
fromΛCDM.ItwasalsoshownthatlowerthevalueofΩ neouswhenitalsovirializestogetherwithmatterinsidethe
m0
parameter, higher the chance to distinguish and there is a sphericaloverdensity.Wecalculatethematterdensitycon-
redshiftrangebetweenz 0.4 1wherethedeviationfrom trast at turnaround as well as at virialization and compare
ΛCDM is maximum irres∼pectiv−e of themodel considered. theresult with thecorresponding ΛCDM model.
The plan of the paper is as follows: in section 2 we
Dark energy not only affects background expansion
describe the background evolution of thawing dark energy
rate and thereby modifying the observables like distance-
models for ordinary scalar field as well as for tachyons. In
redshift relation, it also affects the growth of struc-
section3westudytheevolutionoflinearandnonlinearmat-
ture. On large scales, dark energy also clusters which in
terdensityperturbationassumingthedarkenergytobeho-
turn affects the matter clustering. But at small scales,
mogeneous. In section 4 we describe the spherical collapse
although dark energy is more or less smoothly dis-
formalism for a general scalar field dark energy model and
tributed, it provides an extra Hubble drag thereby slow-
study the virialization process in this scenario and discuss
ing down the growth rate of matter perturbations. The
the results obtained. Finally, we draw our conclusions in
behavior of linear perturbation in a scalar field and
section 5.
its effect on large scale structure of the universe have
been studied by a number of authors (Bartolo (2004);
Hu (2005); Gordon & Wands (2005); Avelino (2008);
Unnikrishnan (2008a); Gordon & Hu (2004); Zimdahl
(2005); Unnikrishnan (2008b); Jassal (2009, 2010)). The
2 BACKGROUND EVOLUTION FOR
impact has been studied for cosmic microwave back-
THAWING DARK ENERGY MODELS
ground (Macorra 2003; Baccigalupi & Acquaviva 2006;
Acquaviva& Baccigalupi 2006; Giovi 2005), galaxy red- Asdiscussedintheintroduction,thawingdarkenergymod-
shift surveys (Eisenstein (1998); Seo& Eisenstein (2003); elsarecharacterizedinsuchawaythatintheearlyuniverse
Haiman (2000); Jain & Taylor (2003); Bernstein & Jain thescalar field is frozen by very large Hubble damping due
(2004); Allen (2004)), cross-correlation of the inte- totheexpansionoftheuniverse.Astheuniverseexpandsthe
grated Sachs-Wolf effect (Pogosian 2005; Corasaniti 2005; HubbleparameterdecreasessotheHubbledampingandthe
Giannantonio 2008)as well as in neutral hydrogen surveys scalarfieldstartsevolvingslowly downitspotential.There-
(Bharadwaj 2009). fore, the equation of state initially starts with w = 1 and
−
In the non-linear regime one has to perform N-Body slowlydepartsfromthisvalueinthelatertime.Inthispaper
(cid:13)c 2010RAS,MNRAS000,??–??
3
weconsiderbothordinaryscalarfieldwithcanonicalkinetic We assume Ω = 0.75 at the present epoch for all chosen
φ0
term aswell astachyontypescalar fieldhavingBorn-Infeld valuesof Γ.
typekineticterm. We also consider the Pseudo-Nambu Goldstone Boson
(PNGB)model(Frieman1995).(Forarecentdiscussion,see
Ref.(Abrahamse 2007) and references therein). This model
is characterized bythepotential
2.1 Thawing Scalar field
V(φ)=m4[cos(φ/f)+1]. (8)
Inwhatfollows, weshallassumethatthedarkenergyisde-
scribedbyaminimally-coupledscalarfield,φ,withequation Alam et al., (Alam 2003) have previously considered such
of motion type of potential to see whether dark energy is decaying or
not.
φ¨+3Hφ+dV/dφ=0 (1)
Without any lose of generality, we choose f =1. Here,
where theHubbleparameter H is given by theconstant m is related to the mass of thefield.
a˙
H = = 8πGρ/3. (2)
a
(cid:16) (cid:17) p 2.2 Thawing Tachyon Model
Hereρisthetotalenergydensityintheuniverse.Wemodel
a flat universe containing only matter and a scalar field, so In recent years, tachyon-like scalar field having ki-
that Ω +Ω =1. netic energy of Born-Infeld form has generated lot
φ M
Equation (1) indicates that the field rolls downhill in of interests in cosmology. There have been several
thepotentialV(φ)butitsmotionisdampedbyatermpro- investigations using this type of field as dark en-
portional to H. The equation of state parameter w is given ergy (Copeland 2005; Tsujikawa & Sami 2004; Sami
by w =p /ρ where the pressure and density of the scalar 2004; Felder 2002; Abramo& Finelli 2003; Bagla 2003;
φ φ
field havethe forms Aguirregabiria & Lazkoz 2004). In what follows, we form
the similar set of equations for tachyon field as described
φ˙2
p = V(φ), (3) abovefor ordinary scalar field.
φ 2 −
The tachyon field is specified by the Dirac-Born-Infeld
φ˙2 (DBI)typeofaction(Sen2002a,b;Garousi2000;Bergshoeff
ρ = +V(φ) (4)
φ 2 2000; Kluson 2000; Kutasov & Niarchos 2003):
Defining λ = 1 dV and Γ V d2V/ dV 2, one can
−V dφ ≡ dφ2 dφ = V(φ) 1 ∂µφ∂ φ√ gd4x. (9)
form an autonomous system of equations in(cid:0)volv(cid:1)ing the two S Z − − µ −
observablesΩ andγ =(1+w)togetherwiththeparameter p
φ
λ. This is derived by (Scherrer& Sen 2008a)as InFRWbackground,thepressureandenergydensityofthe
tachyon field φ are given by
′
γ = 3γ(2 γ)+λ(2 γ) 3γΩ , (5)
φ
Ω′φ = −3(1−γ−)Ωφ(1−Ωφ−), p (6) pφ =−V(φ)p1−φ˙2 (10)
λ′ = −√3λ2(Γ−1)pγΩφ. (7) ρφ = V1(φ)φ˙2 (11)
Given the initial conditions for γ, Ω and λ, one can −
φ p
solve this system of equations numerically for different po- The equation of motion which follows from (9) is
tentials. Weare interested in thawingmodels i.e models for V′
which the equation of state is initially frozen at w = 1. φ¨+3Hφ˙(1 φ˙2)+ (1 φ˙2)=0 (12)
− − V −
Hence,wetakeγ =0initially forourpurpose.Alsoonehas
tochoosetheinitialvalueλ .Itwasearliershownthatifλ where H is the Hubble parameter and here prime denotes
i i
dV
issmall(whichisequivalenttoassumingtheslow-rollcondi- the derivative with respect to φ . Now assuming λ= dφ
tionssimilartoinflation)thescalarfieldevolutionissimilar −V23
d2V
to that of cosmological constant Λ (Scherrer& Sen 2008a). and Γ = V dφ2 , we can also form an autonomous system
Here we assume that the slow-roll conditions are strongly (ddVφ)2
of equations involving Ω , γ = (1+w) and λ for tachyon
broken i.e λ = 1 in order to have a behavior which is in φ
i
(Ali2009):
principledifferentfromΛ.Ingeneralthecontributionofthe
scalar field to thetotal energy density of theuniverseis in-
significant at early times, nevertheless one has to fine-tune γ′= 6γ(1 γ)+2 3γΩ λ(1 γ)45 (13)
φ
theinitial value of Ωφ in order to haveits correct contribu- − −Ω′ =3Ωp(1 γ)(1−Ω ) (14)
tion at present. This is the fine tuning one needs to have φ φ − − φ
in athawing model. With theseinitial conditions we evolve λ′= 3γΩ λ2(1 γ)41(Γ 3) (15)
the above system of equations from redshift z = 1000 (or −p φ − − 2
a = 10−3) till the present day z = 0 (a = 1). We consider We choose the similar set of initial conditions to solve
varioustypesofpotentialse.gV =φ, V =φ2, V =eφ and this system of equations for tachyon as we describe earlier
V =φ−2, characterized by Γ=0, 1, 1 and 3 respectively. for ordinary scalar field. We also assume the same set of
2 2
(cid:13)c 2010RAS,MNRAS000,??–??
4
-0.60 2.0
-0.65
-0.70 1.5
Lz -0.75 HLa
HW -0.80 ∆nl 1.0
-0.85 HLa,
-0.90 ∆l
0.5
-0.95
0.0 0.5 1.0 1.5 2.0
0.0
z 0.0 0.2 0.4 0.6 0.8 1.0
a
Figure 1. Plot of equation of state w vs. redshift for different Figure 2. Variation of linear and nonlinear density contrast δ
scalar field and tachyon models. Solid curves represent different as a function of scale factor for scalar field and tachyon models.
TachyonmodelswithV(φ)=φ,φ2,eφ,φ−2respectivelyfromtop DashedlineisforΛCDMmodel,solidlineisforscalarfieldwith
tobottom,Dashedcurvesfromtoptobottomrepresentdifferent linear potential and dash-dotted line is tachyon field with linear
scalar field models with same potentials as in tachyon. Dotted potential.DottedlineisforscalarfieldwithPNGBtypepotential.
curverepresentsPNGBmodel.Ωm0=0.25. Theuppersetisthesolutionforfullnonlinearequationgivenby
eqn (16), where as the lower set is for linearized equation (17).
potentials V = φ, V = φ2, V = eφ and V = φ−2, charac- Ωm0=0.25.
terized by Γ =0, 1, 1 and 3 respectively and Ω = 0.75.
2 2 φ0
Onecansolvethesetwoequationsnumerically.Tofixtheini-
tialconditions,weassumethatinearlytimes,theuniverseis
matter dominated with negligible dark energy contribution
2.3 Background Result which is typical for thawing model. For the matter domi-
nated regime, δ a and dδ 1 fixesour initial conditions.
Letusnowseethebehaviorofthedifferentdarkenergymod- ∼ da ∼
In figure 2, we plot the evolution of both linear as well as
els that we have considered above. In Figure 1, we plot the
nonlinear density contrast as a function of scale factor. We
behaviorsof theequation of stateparameter w for different
show the behaviors for scalar as well as tachyon field with
thawingmodels.Itshowsthattheequationofstateofdiffer-
linear and PNGB potential. To compare the results with
ent fields with different potentials behavedifferently as one
ΛCDM,wealsoplotthecorrespondingbehaviorforΛCDM
approachesthepresentdayalthoughinthepasttheirbehav-
case. As one can see from this plot, that although the lin-
iors are almost identical. This is not surprising as we have
earmatterdensitycontrastsfordifferentthawingmodelsdo
assumedtheviolation ofslow-roll condition,i.eλ 1.
initial ∼ not deviate much from that of the ΛCDM model but the
Withslow-rollconditionsatisfied,i.e,λ <<1,itwasshown
i nonlinear density contrast has substantial deviation from
earlier that models with different potentials have the iden-
ΛCDM model for both scalar as well as tachyon field. We
ticalw(a)bothforscalarandtachyonfields(Scherrer & Sen
shouldmentionthatalthoughweshowtheresultsforlinear
2008a,b;Ali2009).Althoughweshowequationsofstatefor
and PNGB potentials, for other potentials, the behaviors
some models which are more than w = 0.8 at present,
− are similar but with lesser deviations from ΛCDM model.
these models are practically ruled out by current observa-
tional data.
3 EVOLUTION OF DENSITY 4 THE SPHERICAL COLLAPSE MODEL
PERTURBATIONS
The spherical collapse model, which has a long history in
The matter density contrast is defined by δ = (ρmc cosmology, is a simple and a fundamental tool for under-
−
ρm)/ρm, where ρmc is theperturbed matter energy density standing how a small spherical patch of over density forms
andρm isthebackgroundmatterenergydensity.Assuming a bound system via gravitational instability (Gunn & Gott
that the dark energy does not cluster, the evolution of the 1972). As we consider flat, homogeneous and isotropic cos-
matter density contrast is governed by theequation mologies, drivenbynon relativistic matteranddarkenergy
a˙ 4 δ˙2 with the equation of state, pφ = w(a)ρφ with pφ < 0, the
δ¨+2aδ˙−4πGρm(1+δ)δ− 3(1+δ) =0. (16) equations that describe our background universe are given
by
Inthelinearregime,onecanignorehigherorderterminthe
a˙ 2 8πG
above equation and approximate it as: = (ρ +ρ ) (18)
a 3 m φ
(cid:16) (cid:17)
a˙
δ¨+2aδ˙−4πGρmδ=0. (17) and
(cid:13)c 2010RAS,MNRAS000,??–??
5
z z z z
9 coll 9 coll 9 coll 9 coll
8 V( φ) = φ 8 V( φ) = φ 2 8 V( φ) = φ 8 V( φ) = φ 2
ζ 7 7 ζ ζ 7 ζ 7
6 6 6 6
590 2 950 2 590 1 zcoll 2 3 950 1 zcoll 2 3
8 V( φ) = Exp (φ) 8 V( φ)= φ−2 8 V( φ) = Exp (φ) 8 V( φ)= φ−2
ζ 7 7 ζ ζ 7 ζ 7
6 6 6 6
950 2 50 z 2 50 1 2 3 50 1 2 3
coll
8 PNGB [t]
ζ 7
6 Figure4.Overdensityatturnaroundvs.collapseredshift(zcoll)
for different tachyon field models with the same potentials as in
5
[t] 0 zcoll 2 fig.3forΩm0=0.25.Thelinescorrespondtothesameasinfig.3.
Figure3.Overdensityatturnaroundvs.collapseredshift(zcoll)
for different scalar field models with different potentials V(φ)= represents the equation of state of the dark energy inside
φ,φ2,eφ,φ−2 for Ωm0 = 0.25. In each figure, solid curve repre- thecluster.Withsomeinitialoverdensity,thesphericalover
sentsinhomogeneous case,dashedcurverepresentshomogeneous dense region will expand until it reaches the maximum ra-
caseanddotdashedrepresentsΛCDMmodel.ScalarfieldPNGB dius(turnaroundpoint,R˙ =0)andthenbeginstocollapse.
model is also compared with ΛCDM model in the bottom left After that the sphere virializes forming a bound system.
figure.
With differentinitial density,it happenslocally at different
regionsoftheuniverse.Now,afterperforming thefollowing
a¨ 4πG transformations:
= [(3w(a)+1)ρ +ρ ] (19)
a − 3 φ m a R
x= andy= (25)
where a(t) is the scale factor, ρm = ρm0a−3 is the back- at Rt
ground matterdensityandρφ =ρφ0f(a)is thedark energy theequationofbackgroundevolutionandthatofthespher-
density,with ical perturbation become:
1 1+w(u) x˙2=H 2Ω [Ω (x)x]−1 (26)
f(a)=Exp[3 du]. (20) t m,t m
Z (cid:18) u (cid:19)
a and
The Friedmann equation can be written more simply as
H2Ω ζ
H2 ≡ (a˙/a)2 = H02E(a)2 and H0 is the Hubble constant y¨=− t2m,t (cid:20)y2 +νyI(x,y)(cid:21) (27)
with
E(a)= Ω a−3+Ω f(a) 1/2 , (21) where
m0 φ0
(cid:2) (cid:3) [1+3w(R(y))]f(R(y)) Clustered DE
while Ω = (8πGρ )/3H2 is the matter density param- I(x,y)= f(at) (28)
m0 m0 0 (cid:26) [1+3w(x)]f(x) Homogeneous DE
eter and Ω = (8πGρ )/3H2 is the corresponding dark
φ0 φ0 0
energyparameteratthepresentepochwithΩm0+Ωφ0 =1. with
The Ωm(a) and Ωφ(a) evolvewith thescale factor as ν = ρφ,t = 1−Ωm,t . (29)
Ω (a)= Ωm0a−3 and Ω (a)= Ωφ0f(a). (22) ρm,t Ωm,t
m E2(a) φ E2(a) Subscript “t” denotes the turn around time. In order
to solve the above set of equations, we have used the fact
Once we know the solution for Ω (a)and γ(a) by solving
φ that the mass of the forming cluster is conserved: ρ R3 =
either (5)-(7) or (13)-(15), we can easily find the behavior mc
ρ R3 and definethe relation:
of the normalized Hubbleparameter in terms of Ω as mc,t t
φ
E2(a)= H2(a) = 1−Ωφ0a−3, (23) ρmc=ρmc,t RR −3= ζρym3,t (30)
H2 1 Ω (cid:16) t(cid:17)
0 − φ
Here ζ is the matter density contrast at turnaround which
and the equation of state of dark energy w(a)=γ(a) 1.
− is defined as
The evolution of a spherical over dense patch of radius
R(t) in the presence of dark energy is given by the Ray- ζ ρmc,t = Rt −3. (31)
chaudhuriequation: ≡ ρm,t (cid:16)at(cid:17)
R¨ 1 1 The function R(y) is given by R(y)= R y = ζ−1/3a y and
= 4πG w(R)+ ρ + ρ (24) t t
R − h(cid:16) 3(cid:17) φc 3 mci Ωm(x) by
whereρφc andρmc arethedarkenergydensityandthemat- 1
Ω (x)= . (32)
ter density inside the spherical cluster respectively. w(R) m 1+νx3f(x)
(cid:13)c 2010RAS,MNRAS000,??–??
6
z z z z
600 coll 600 coll 600 coll 600 coll
500 V( φ ) = φ (a) 500 V( φ) = φ2 (b) 500 V( φ ) = φ (a) 500 V( φ) = φ2 (b)
∆vir340000 ∆vir340000 ∆vir340000 ∆vir340000
200 200 200 200
1600000 2 1600000 2 1600000 zcoll 2 1600000 zcoll 2
500 V( φ) = Exp ( φ) (c) 500 V(φ) = φ−2 (d) 500 V( φ) = Exp ( φ) (c) 500 V(φ) = φ−2 (d)
∆vir340000 ∆vir340000 ∆vir340000 ∆vir340000
200 200 200 200
100 100 100 100
0 2 0 2 0 2 0 2
600 z
coll
500 PNGB (e) [t]
∆vir340000 Figure 6. The plot of the non-linear density contrast at viri-
200 alization as the function of collapse redshift(zcoll) for different
1000 2 tachyonfieldmodelsforΩm0=0.25withthesamepotentials as
[t] zcoll infig.5.Thelinescorrespondtothesameasinfig.5.
Figure5.Plotofthenon-lineardensitycontrastatvirialization
as the function of collapse redshift (Zcoll) for different Scalar at turnaround for different models tend toward the fiducial
field models with V(φ)=φ,φ2,eφ,φ−2 for Ωm0 =0.25. In each value of 5.6 for Einstein-de Sitter universe. For structures
figure, solid curve represents inhomogeneous case, dashed curve
which are collapsing around present time i.e z 0, the
represents homogeneous case and dotdashed represents ΛCDM coll ∼
overdensitiesatturnaroundfordifferentscalarandtachyon
model. Scalar field PNGB model is also compared with ΛCDM
modelsarehigherthanthecorrespondingΛCDMvalue.Also
modelinthebottom lastfigure.
thevaluesforthehomogeneouscaseareslightlyhigherthan
the inhomogeneous case and the difference is more promi-
Typically thescalar fieldswe consider here, haveextremely nentfortachyonmodelwithlinearpotential.Forothercases
smallmasses(oftheorderofpresentdayHubblescaleinnat- thedifferences are extremely small.
ural units). This is necessary to have nearly flat potentials Justtomention,weshowalltheplotsassumingΩ =
m0
for thescalar fieldsat present day.Due to this, thescale of 0.25.ForothervaluesofΩ ,thebehaviorsaresimilarwith
m0
fluctuationsforthesescalarfieldsareextremelylarge,mak- lesserdeviation from ΛCDMfor higherΩ andviceversa.
m0
ing it a smoothly distributed field within the horizon scale.
So it is safe to assume thedark energy to be homogeneous.
4.1 VIRIALIZATION IN THE SPHERICAL
Butitisstillinterestingtoconsiderthecasewherethedark
MODEL
energyclustersalongwiththedarkmatterandavoidtheen-
ergynon-conservationproblemexaminedin(Maor & Lahav In general the spherical collapse formalism leads to a point
2005).Henceinoursubsequentcalculations,weassumeboth singularity as the final state of the system. But physically
thecases mentioned in equation (28). the objects go through a virialization process and stabilize
When dark energy is clustered together with the mat- to a finite size. Such process of virialization is not built in
ter, one can solve analytically the system of equations (26) the spherical collpase formalism but we have to put it by
and (27)as the function I(x,y) in eq.(28) depends only on hand in order to ensure virialization.
y. So we easily calculate the density contrast ζ at the turn Therearenumberofthingsthatenterinthisprocessof
around epoch by integrating the equations (26) and (27) virialization as we describe below.
and applying the boundary conditions (dy/dx)x=1 =0 and Firstofall,thetotalenergyhastobeconservedalways,
yx=0 = 0. The corresponding integral equation which gov- hencethe total energy at thetime of turnaround should be
ernsthebehaviorofζ,foraflatcosmological modelisgiven equaltothatatthetimeofvirialization. Attheturnaround
below: the potential energy only contributes to the total energy
1 y 1/2 of the system whereas the point of virialization is defined
dy= as where the virial theorem holds i.e the kinetic energy is
Z (cid:20)ζ+νyP(y) (ζ+P(1)ν)y(cid:21)
0 − related to the potential energy T = 1(r∂U) , where T
1 vir 2 ∂r vir
[xΩ (x)]1/2dx (33) and U are the kinetic and potential energy respectively. In
Z m oursubsequentcalculationsweinvestigatetwospecificcases
0
mentioned below:
In the case of homogeneous dark energy, one can not
follow thisanalyticalprocedureandhastosolvethesystem In the first case, we assume the dark energy is inho-
•
of equations (26) and (27) numerically with the boundary mogeneous and it virializes together with the matter. This
conditions (dy/dx) =0 and y =1. assumption affects the way one calculates the turnaround
x=1 x=1
In figure 3 and 4, we show the matter density con- point, as mentioned in equation (28) ( the case “Clustered
trast at turnaround, ζ as a function of collapsed redshift D.E”). On the other hand, the assumption that the dark
for scalar and tachyon models with different potentials. To energy takes part in the virialization process demands that
compareourresultwithΛCDMmodel,wealsoshowthecor- the virial theorem should hold for the total kinetic and po-
respondingζforΛCDMmodel.Athigherz ,overdensities tential energy of the system (i.e matter + dark energy).
coll
(cid:13)c 2010RAS,MNRAS000,??–??
7
Model Potential Case a b n
φ inhom 0.88±0.02 0.34±0.03 1.34±0.07
hom 0.88±0.02 0.42±0.02 1.25±0.05
φ2 inhom 0.855±0.004 0.383±0.006 1.10±0.02
Tachyon hom 0.853±0.004 0.432±0.005 1.06±0.01
φ−2 inhom 0.8472±0.0003 0.3867±0.0005 1.001±0.001
hom 0.8462±0.0001 0.4254±0.0002 0.9824±0.0005
eφ inhom 0.850±0.001 0.387±0.002 1.032±0.005
hom 0.848±0.001 0.429±0.001 1.007±0.003
φ inhom 0.862±0.008 0.37±0.01 1.16±0.03
hom 0.859±0.007 0.43±0.01 1.11±0.02
φ2 inhom 0.852±0.003 0.333±0.004 1.08±0.01
hom 0.850±0.002 0.431±0.003 1.024±0.005
Scalar φ−2 inhom 0.8460±0.0002 0.3865±0.0003 0.9867±0.0007
hom 0.8453±0.0006 0.4241±0.0008 0.972±0.002
eφ inhom 0.8480±0.0007 0.387±0.001 1.011±0.002
hom 0.8469±0.0002 0.4275±0.0003 0.9905±0.0006
PNGB inhom 0.858±0.006 0.381±0.008 1.12±0.02
hom 0.855±0.005 0.434±0.007 1.08±0.01
Table1. Valuesofthefittingparametersa,bandcbyfittingequation(44)withtheexactresultsgivenbyequation(40)forscalarand
tachyon models.Wehavealsoshowntheerrorsfordifferentparametersat95%confidence level
Thus, applying the energy conservation together with this, Assumingthatatthecollapsepoint,thesystemhasviri-
we get(Basilakos & Voglis 2007) alisedfully,thedensitycontrastatvirializationasafunction
of z and Ω is given by
1 coll m0
T = U +U (34)
c −2 Gc φc ρ ζ a 3
∆ = mc,f = f , (40)
Tc,f +UGc,f +Uφc,f =UGc,t+Uφc,t (35) vir ρm,f λ3 (cid:16)at(cid:17)
where Tc is the kinetic energy and UGc = 3GM2/5R wheretherelationbetweenaf andat canbeestimatedfrom
−
is the potential energy for matter, M being the matter theequation,
mass contained in the spherical over density. Uφc = (1+ dt 1
3w)ρ 4πGMR2 is the potential energy associated wi−th the = , (41)
φc 10 da H(a)a
dark energy inside thespherical over density.Herethesub-
scripts“f”and“t”indicatethevirializationandturnaround togetherwiththeconditionthatthetimeneededtocollapse
time respectively and “c” for inside the cluster. Using the is twice the turn-aroundtime, tf =2tt, i.e.
above formulation, one can obtain a cubic equation which af 1 at 1
relates theratio between thefinal (virial) R and the turn- da=2 da. (42)
f Z H(a)a Z H(a)a
around radius R , defined as thecollapse factor λ= Rf: 0 0
t Rt In case of a Λ cosmology we get an analytical solution:
2n λ3 (2+n )λ+1=0 (36)
1 − 2 sinh−1(a 3/2√ν )=2sinh−1(a 3/2√ν ), (43)
f 0 t 0
where
where ν = (1 Ω )/Ω . The ratio between the scale
n = (3w(a )+1)Ωφ0f(af) (37) factorsc0onverge−stomth0eEinms0teindeSittervalue(af)=22/3
1 − f ζΩ a−3 at
m0 t at high redshifts.
and Infigure5and6,weshowthebehaviorofmatterdensity
n2 =−(3w(at)+1)ζΩΩφ0f(aa−t3) (38) caofnutnracstitoantovfirciaollilzaaptsieodnr∆edvisrhfioftrzdciofflel.reTnhtitshpawarianmgemteordepllsayass
m0 t acrucial roleasan observationaltool whichcan bedirectly
with ρmc,t being the matter density inside the sphere at appliedtoPress-Schechtertheoryforcomovingnumberden-
the turn around time while ρ is the background matter
m,t sityofvirializedobjects.Thebehaviorisconsistentwiththe
density at thesame epoch.
fact that with z , ∆ decreases and settles to the value
coll vir
In the second case, we assume that the dark energy is 18π2 for Einstein de-Sitteruniverse.
•
homogeneous and it does not virialize inside the cluster. In
Oncevirialized,thedensitycontrastinsidetheclusteris
thiscase,togettheturnaroundpoint,weuseeqn(28)(with
slightlyhigherinthehomogeneouscasecomparedtothein-
the choice “Homogeneous DE”). On the other hand, as the
homogeneous case. This is consistent with the result at the
darkenergydoesnotvirialze,itsonlyeffectistocontribute
turnaround where also similar things happen. This means
tothepotentialenergyofthesystem.Soapplyingtheenergy
thatin thehomogeneous case, whereonly mattervirializes,
conservation, we get(Maor & Lahav 2005)
the density inside the clustered objects are higher. This is
R∂U inagreementwiththeresultobtainedbyMaor(Maor2006)
U +U + Gc =[U +U ] (39)
φc Gc 2 ∂R Gc φc t for dark energy with constant equation of state. There are
h if
(cid:13)c 2010RAS,MNRAS000,??–??
8
600 600
Homogeneous Inhomogeneous
500 Ω = 0.25 500 Ω = 0.25
m m
400 400
r r
vi vi
∆ ∆
300 300
200 200
100 100
0 1 2 3 0 1 2 3
z z
coll coll
Figure 7.Plot of the fitting function for the density contrast at virialization∆vir vs the collpase redshift(zcoll) fordifferent models.
Fromtoptobottomrepresentstachyonwithlinearpotential,scalarfieldwithlinearpotentialandscalarfieldwithPNGBpotential.For
each case, thesmooth lineisforthe fitting equation givenbyeqn (44) withtheparameters mentioned intable1, whereas the dots are
generatedbytheexactnumericalresultfor∆vir calculatedforeachmodel.
alsofewinterestingresults.Thedifferencein∆ forapar- Next,wegiveafittingformulafor∆ asafunctionof
vir vir
ticularthawingmodelwiththecorrespondingΛCDMmodel collapsed redshift z which is given by
coll
islargerforscalarandtachyonfieldwithlinearpotentialas
∆ (z )=18π2(a+b Θ(z )n) (44)
well as for scalar field with PNGB potential. Also tachyon vir coll ∗ coll
field with linear potential has the largest deviation from where Θ(z ) = 1 1. In Table 1, we show
ΛCDM.Thispredictsthatthesemodelsareeasiertodistin- coll Ωm(zcoll) −
the values of the fitting parameters a,b and n for different
guishfromΛCDMmodelbyobservingabundancesofbound
models assuming Ω =0.25. We also quote the the corre-
m0
objects.Forscalarfieldmodel,thediferencebetweenhomo-
spondingerrors for theseparameters at the95% confidence
geneousandinhomogeneous modelishighestforthepoten-
level.
tial V(φ) = 1 . It is interesting to note that this potential
φ2 In figure 7, we show accuracy of our fitting function
also can act as a tracker or freezing model. For tachyon
for thawing models with linear as well as PNGB potential
model, these two cases are hardly distinguishable for all of
potential. For other potentials it also works equally well.
thepotentials.
Knowing that models with linear potential (for both
Given thefact that themass of the dark energy has to tachyon and ordinary scalar fields) as well as models with
beextremelysmall,itismostlikelythatthefieldresponsible PNGB type potential deviate more from ΛCDM model in
fordarkenergyhastobehomogeneousandshouldnotclus- comparisontootherpotentials,westudythesizeofthecol-
terinsidethevirialized objects.Ifthisisthescenariowhich lapsedobjectsforthesepotentialsforhomongeneousaswell
isphysicallyrelevant,ourresultsshowthatscalaraswellas as for inhomogeneous cases. In figure 8, we show the evolu-
tachyonfielddarkenergy modelscan bedistinguished from tionoftheperturbationcollapsing radiiatthepresenttime
theΛCDMmodelbyobservingtheabundancesofboundob- (z =0)forthesepotentials.Theradiushasbeennormal-
coll
jects since the virialized density contrast for homogeneous ized to the turnaround radius R . It has been plotted with
t
cases deviate sufficiently from the ΛCDM model. respect to the scale factor normalized to the turnaround
We should mention, that the final results for ∆ , de- scale factor for ΛCDM model. The collapse to singularity
vir
pend on both the background evolution as well as the fact isavoided assuming that afterthevirialization, thesize be-
that dark energy may cluster. The fact that the ∆ is comes constant. From the figure 8, for homogeneous case,
vir
slightly smaller for inhomogeneous case than the homoge- virialization occurs earlier than ΛCDM model whereas for
neous one for all potentials, shows that the inhomogeneous inhomogeneouscase,ithappenslater.Thismeans,homoge-
dark energy acts against the matter clustering. But as the neous models are more suitable for producing older bound
model is quite nonlinear and involves rigorous numerical objcets than ΛCDM. This gives a slight edge to homoge-
computations,itisverydifficulttopredictthepercentageof neousdark energy models overthe ΛCDM model.
effectscomingfromthebackgroundevolutionaswellasfrom Although, in figure 8, the virialized radii for different
dark energy clustering which may be an useful information models are practically indistinguishable but still there are
for actual N-bodysimulations. slight differences. First of all, the virialized radii for differ-
(cid:13)c 2010RAS,MNRAS000,??–??
9
Homogeneous Inhomogeneous
1 Ωm = 0.25 1 Ωm = 0.25
0.8 0.8
Rt Rt
0.6 0.6
/ /
R R
0.4 0.4
0.2 0.2
0 0
0 0.4 0.8 1.2 1.6 0 0.4 0.8 1.2 1.6
a/a(w=-1) a/a(w=-1)
t t
Figure 8.VariationoftheradiusRnormalizedtotheturnaround radiusRt w.r.t.thescalefactor normalizedtothe turnaroundscale
factorfortheΛCDMmodel.SolidlineisforΛCDMwhereasthedashed,dash-dottedanddottedareforscalarfieldwothlinearpotential,
tachyon withlinearpotential andscalarfieldwithPNGBpotential.
ent scalar and tachyon models are smaller than the ΛCDM linearpotentialbothforstandardscalar fieldsandtachyon,
model, whichin turnconfirmsthatobjects aremoredenser together with scalar field with PNGB type potential have
in thesemodels thanΛCDM model. This isconsistent with significant deviation from ΛCDM model. We have shown
the results shown in figure 5 and 6. Secondly between in- thatthesizeofthevirialised object,thedensitycontrast at
dividual models, virialised radius for tachyon model with turnaround as well as at virialization differ significantly for
linear potential is smaller than the other two models, e.g, thesemodelsfromΛCDMmodel.Thisisconsistentwiththe
scalarfieldwithlinearpotentialandscalarfieldwithPNGB resultsobtainedbySenetal.(Sen2010)forthawingmodels
potential. This is also consistent with behaviour of virial- considering the background evolution. Also the deviations
ized density contrast shown in figure 7. Also the virialized are enhanced in the homogeneous dark energy case where
radiiininhomogeousmodelsareslightlylargerthantheho- matter only virializes inside the cluster, thereby making it
mogneouscounterpart,makingthehomogeneousonedenser more probable to be distinguished from ΛCDM. Given the
than inhomogeneous one. This is also consistent with what fact that the mass of the dark energy field has to be ex-
we show in figure5 and 6. tremely small, it is safe to assume that the dark energy re-
mains homogeneous for most of the relevant scales. Hence
the evolution of spherical overdensities in large scale struc-
ture formation can be a useful tool to distinguigh model
5 CONCLUSION
from ΛCDM.
In this work, we study the evolution of spherical overdensi- We have also derived the fitting formula for ∆vir as
ties for a general class of thawing dark energy models. We functionofcollapsed redshift fordifferentmodelsandtabu-
considerbothordinary scalarfieldsaswellastachyonfields lated thevaluesof the fittingparameters for different mod-
having noncanonical kinetic energy. We consider variety of els assuming the dark energy to be homogeneous as well
potentialsforthefields.Instudyingtheevolutionofspheri- as inhomogeneous. Although we have not shown explicitly,
caloverdensities,weconsiderthedarkenergytobehomoge- if one varies Ωm0, deviations are higher for values smaller
neousaswellasinhomogeneous.Forthehomogeneouscase, than Ωm0 = 0.25 and vice versa. This is because smaller
we assume only matter virializes inside the cluster whereas thevalueof Ωm0, higher thecontribution from dark energy
for the inhomogeneous case, we assume both matter and at late times and hence one needs more matter inside the
dark energy virializes inside the cluster. cluster for objects collapsing at late times.
Our main motivation is to see whether one can dis- Animportant conclusion onecan draw from thisstudy
tinguish thawing dark energy models from ΛCDM model isthatthawingmodelswithlinearpotentialcanhavesignif-
studyingtheevolutionofsphericaloverdensity.Ourresults icant deviation from ΛCDM model. Now, scalar fields with
showthatalthoughallthemodelsbecomeindistinguishable linearpotentialhavereceivedlittleattentionasaviabledark
for objects collapsing earlier, for objects collapsing around energy candidate (Kratochvil 2004). But it certainly de-
present time, some of the thawing models deviate signifi- serves more attention, especially for the homogeneous case.
cantly from the ΛCDM model. To be specific, models with Finallly,wewanttostressthatwestudyatoymodelfor
(cid:13)c 2010RAS,MNRAS000,??–??
10
nonlinear evolution of over density where we use simplified Corasaniti P .S., Giannantonio T. & Melchiorri A., 2005,
assumption of spherical over-densities. In order to compare Phys.Rev.D,71, 123521
the theoretical results with observational data, one has to Courtin J., et al., 2010, (arXiv:1001.3425)
giveawaythissimplifiedassumptionandhastoperformfull Davis T. M., et al., 2007, ApJ, 666, 716
N-bodysimulations to calculate accurately thedark energy Eisenstein D.J., et al., 1998, ApJ., 494, L1
signatures in nonlinear structure formation. But even with Felder G. N., Kofman L. & Starobinsky A., 2002, JHEP,
thissimplifiedassumption,wefindsomeinterestingfeatures 0209, 026
in the nonlinear evolution of matter over-densities in the Frieman A., et al., 1995, Phys. Rev.Lett., 75, 2077
thawing class of dark energy models. This may be helpful Frieman J., TurnerM. & HutererD.,2008, Ann.Rev.As-
for more detailed N-bodysimulations. tron.Astrophys., 46, 385
Garousi M. R., 2000, Nucl.Phys. B, 584, 284
Giannantonio T., et al., 2008, Phys. Rev.D,77, 123520
6 ACKNOWLEDGMENT Giovi F., et al., 2005, Phys.Rev. D,71, 103009
Gordon C. & Hu. W., 2004, Phys. Rev.D, 70, 083003
The authors acknowledge the financial support provided
Gordon C. & Wands D.,2005, Phys. Rev.D,71, 123505
by the University Grants Commission, Govt. Of India,
Gunn J. E. & Gott J. R., 1972, ApJ,176, 1
through the major research project grant (Grant No: 33-
Haiman Z., et al., 2000, Astrophys. J., 553, 545
28/2007(SR)).
Horellou C. & Berge J., 2005, MNRAS,360, 1393
Hu W., 2005, Phys.Rev.D, 7 1, 047301
Jain B. & Taylor A., 2003, Phys.Rev.Lett., 91, 141302
REFERENCES Jassal H.K., 2009, Phys.Rev. D,79, 127301
Jassal H.K., 2010, Phys.Rev. D,81, 083513
Abrahamse A., et al., 2007, Phys. Rev.D,77, 103503
Kluson J., 2000, Phys. Rev.D,62, 126003
Abramo L. R. W. & Finelli F., 2003, Phys. Lett. B, 575,
Knop R. A.,et al., 2003, ApJ,598, 102
165
Kratochvil J., et al., 2004, JCAP, 0407, 001
Acquaviva V. & Baccigalupi C., 2006, Phys. Rev. D, 74,
Kutasov D. & Niarchos V., 2003, Nucl. Phys.B, 666, 56
103510
Lahav O., Lilje P. B., Primack J. R. & Rees M. J., 1991,
AguirregabiriaJ.M. &LazkozR.,2004,Phys.Rev.D,69,
MNRAS,251, 128
123502
Liddle A. R. & Scherrer R. J, 1999 Phys. Rev. D, 59,
Alam U., Sahni V. & Staronbinsky A.A., 2003, JCAP,
023509
0304, 002
Linder E. V., 2008, Gen. Rel. Grav., 40, 329
Ali A., Sami M. & Sen A.A., 2009, Phys. Rev. D, 79,
Lokas E. L. & Hoffman Y.,2001, (astro-ph/018283)
123501
Maccio A.V., et al., 2004, Phys. Rev.D, 69, 123516
Allen S. W., et.al., 2004, MNRAS.,353, 457
Macorra A. dela., 2003, Phys.Rev.D, 67, 103511
AvelinoP.P.,Beca L.M.G. &MartinsC.J.A.P.,2008,
Manera M. & Mota D.F., 2006, MNRAS,371, 1373
Phys.Rev.D, 77, 101302
Baccigalupi C. &AcquavivaV.,2006, (astro-ph/0606069) Maor I, 2006, (astro-ph/0602441).
Bagla J.S.,Jassal H.K.&Padmanabhan.T.,2003, Phys. Maor I. & Lahav O. , 2005, JCAP, 7, 3
Rev.D,67, 063504 Mota D.F. & van deBruck C., 2004 A & A,421, 71
Baldi M., et al., 2008, (arXiv:0812.3901) NunesN. J. & Mota D. F., 2006, MNRAS,368, 751
Bartelmann M., Doran M. & Wetterich C., 2006, A& A, Padmanabhan T., 2003, Phys. Rep.,380, 235
454, 27 Pogosian L., et al., 2005, Phys. Rev.D,72, 103519
BartoloN.,CorasanitiP.S.,LiddleA.R. &MalquartiM., Press W. H.& SchechterP., 1974, ApJ,187, 425
2004, Phys. Rev.D, 70, 043532 Ratra B. & Peebles P. J. E., 1988, Phys. Rev.D, 37, 3406
Basilakos S., 2003, ApJ,590, 636 Riess A. G., et al., 2004, ApJ, 607, 665
Basilakos S. & Voglis N., 2007, MNRAS,374, 269, SahniV.&StarobinskyA.A.,2000,Int.J.Mod.Phys.D,
Bergshoeff E. A.,et al., 2000, JHEP, 0005, 009 9, 373
Bernstein G. M. & Jain B., 2004, ApJ., 600, 17 SamiM.,SavchenkoN.&ToporenskyA.,2004,Phys.Rev.
BharadwajS., SethiS.K.&SainiT.D.,2009,Phys.Rev. D,70, 123528
D,79, 083538 Sami M., 2009, (arXiv:0904.3445)
Caldwell R. R., Dave R. & Steinhardt P. J., 1998, Phys. Schaefer B. M. & Koyama K., 2008, MNRAS,385, 411
Rev.Lett., 80, 1582 ScherrerR.J.&SenA.A.,2008a,Phys.Rev.D,77,083515
Caldwell R. R. & Kamionkowski M., 2009 ScherrerR.J.&SenA.A.,2008b,Phys.Rev.D,78,067303
(arXiv:0903.0866) Sen A., 2002a, JHEP , 0204, 048
CaldwellR.R. &LinderE.V.,2005,Phys.Rev.Lett.,95, Sen A., 2002b, Mod. Phys.Lett. A, 17, 1797
141301 Sen S., Sen A. A.& Sami M., 2010, Phys.Lett. B, 686, 1
Copeland E. J., Garousi M. R., Sami M., Tsujikawa S., Seo H.J. & Eisenstein D. J., 2003, ApJ, 598, 720
2005, Phys. Rev.D, 71, 043003 Shaw D.J. & Mota D.F., ApJS.,2008, 174, 277
CopelandE.J.,SamiM.&Tsujikawa S.,2006,Int.J.Mod. Silvestri A. & Trodden M., 2009, Rep. Prog. Phys., 72,
Phys.D, 15, 1753 096901
(cid:13)c 2010RAS,MNRAS000,??–??
