Halo model and halo properties in Galileon gravity cosmologies

1.9 MB·

by  Alexandre Barreira

IPPP/13/103/DCPT/13/208
HalomodelandhalopropertiesinGalileongravitycosmologies
AlexandreBarreira,1,2,∗ BaojiuLi,1 WojciechA.Hellwing,1,3 LucasLombriser,4 CarltonM.Baugh,1 andSilviaPascoli2
1InstituteforComputationalCosmology,DepartmentofPhysics,DurhamUniversity,DurhamDH13LE,U.K.
2InstituteforParticlePhysicsPhenomenology,DepartmentofPhysics,DurhamUniversity,DurhamDH13LE,U.K.
3Interdisciplinary Centre for Mathematical and Computational Modeling (ICM),
University of Warsaw, ul. Pawin´skiego 5a, Warsaw, Poland
4Institute for Astronomy, University of Edinburgh, Edinburgh, EH9 3HJ, U.K.
Weinvestigatetheperformanceofsemi-analyticalmodellingoflarge-scalestructureinGalileongravitycos-
mologies using results from N-body simulations. We focus on the Cubic and Quartic Galileon models that
provideareasonablefittoCMB,SNIaandBAOdata. WedemonstratethattheSheth-Tormenmassfunction
andlinearhalobiascanbecalibratedtoprovideaverygoodfittooursimulationresults. Wealsofindthatthe
4 haloconcentration-massrelationiswellfittedbyapowerlaw.Thenonlinearmatterpowerspectrumcomputed
1 inthehalomodelapproachisfoundtobeinaccurateinthemildlynonlinearregime,butcapturesreasonably
0 well the effects of the Vainshtein screening mechanism on small scales. In the Cubic model, the screening
2 mechanismhidesessentiallyalloftheeffectsofthefifthforceinsidehaloes. InthecaseoftheQuarticmodel,
thescreeningmechanismleavesbehindresidualmodificationstogravity,whichmaketheeffectivegravitational
y
strengthtime-varyingandsmallerthanthestandardvalue.Comparedtonormalgravity,thiscausesadeficiency
a
of massive haloes and leads to a weaker matter clustering on small scales. For both models, we show that
M
there are realistic halo occupation distributions of Luminous Red Galaxies that can match both the observed
large-scaleclusteringamplitudeandthenumberdensityofthesegalaxies.
2

]
O I. INTRODUCTION however, the original action of Ref. [2] leads to higher-order
equationsofmotion,andtherefore,tothepropagationofOs-
C
trogradskighosts[3]. ThisproblemwasfixedinRefs.[4,5]
. Over the past few years, models of modified gravity have
h byintroducingexplicitcouplingsbetweentheGalileonderiva-
attractedmuchattentionasanexplanationtotheobservedac-
p tivetermsandcurvaturetensorsinL andL . This,however,
celerating expansion of the Universe [1]. In these models, 4 5
-
breaks the Galilean shift symmetry. The Galileon model is
o the acceleration is a natural consequence of the breakdown
r ofthe theoryofgeneral relativity(GR)on largescales. This thereforethesectorofthegeneralHorndeskitheory[6]thatis
st contrasts with models like ΛCDM, in which GR is the the- Galileaninvariantinthelimitofflatspacetime.
a
oryofgravityandtheaccelerationiscausedbyamysterious Avitalrequirementofmodifiedgravitymodelsisthatany
[
"darkenergy"component,suchasacosmologicalconstantΛ departurefromGRonsmallscaleshastosatisfythestringent

2 oraslowly-rolling,minimallycoupledscalarfield. Although Solar System tests of gravity [7]. This is usually realized by
v thesetwoclassesofmodelscanbesetuptohaveidenticalex- invokingascreeningmechanismthatdynamicallysuppresses
7
pansionhistories,theywilltypicallydifferinhowmatterand the modifications to gravity in regions where the density is
9
lightreacttothedistributionofgravitationalsources. Hence, high, like the Solar System. Interestingly, the nonlinear na-
4
1 observables associated with the growth rate of structure and tureofthederivativecouplingsinL3−5 thatdrivethemodifi-
. lensinghavethepotentialtotestthelawofgravityoncosmo- cationstogravityisalsowhatallowsthesemodificationstobe
1
logicalscales. suppressedinhighdensityregions. Thisisamechanismthat
0
4 Here, we focus on Galileon gravity models, which were is widely known as the Vainshtein effect [8–10]. The gen-
1 first proposed in Ref. [2]. In particular, Ref. [2] showed that eral picture is as follows. In regions where the overdensity
: thereareonlyfiveLagrangiandensityterms(L ,i = 1,...,5) is small, the nonlinearities are negligible and the perturbed
v i
forasinglescalarfieldϕ(i)whosephysicsisinvariantunder Galileonfieldequationofmotionessentiallybecomesalinear
i
X theso-calledGalileanshifttransformation∂ ϕ → ∂ ϕ+b Poisson-like equation. Therefore, the Galileon field pertur-
µ µ µ
r (wherebµ isaconstantfour-vector);and(ii)whoseequations bationactslikeanextragravitationalpotentialanditsspatial
a
of motion in flat spacetime are up to second order in field gradientgivesrisetoasizeablefifthforce. Ontheotherhand,
derivatives. The Lagrangian terms are characterized by the closertomassivebodies, asthedensitygetshigher, thenon-
power with which ϕ (from hereon in the Galileon field) ap- linear terms become increasingly important, effectively sup-
pears. Thelinear(L )andquadratic(L )termscorrespondto pressingthemagnitudeofthefifthforcesothatnormalgravity
1 2
a linear potential function and to the canonical kinetic term, isfelt.
respectively. The remaining cubic (L ), quartic (L ) and
3 4 Many recent papers have studied the cosmological prop-
quintic(L )termscontainnonlinearderivativeself-couplings
5 erties of the covariant Galileon model at the linear level
of the Galileon field, which are behind the modifications to
in perturbation theory [11–22]. In particular, the work of
gravity. InaFriedmann-Robertson-Walker(FRW)spacetime,
Refs. [11, 12] has shown that the Galileon model can fit
the Cosmic Microwave Background (CMB) data better than
ΛCDM, because it allows for less power on the largest an-
gularscales, whichisslightlyfavouredbythemeasurements
∗Electronicaddress:[email protected] from the WMAP [23] and Planck [24] satellites. However,
2
Refs. [11, 12] have also discovered that the predicted ampli- configurationsoftheGalileonmodelsweconsider.InSec.III,
tudeofthelinearmatterpowerspectumoftheGalileonmod- we present the equations of the halo model of the nonlinear
elsthatbestfittheCMBdataissubstantiallylargerthanthat matter power spectrum. We also describe the Sheth-Tormen
inΛCDM.ThisraisedthepossibilitythattheGalileonpredic- formulaeforthehalomassfunctionandlinearhalobias[29–
tions may be in tension with the observed large-scale galaxy 31],andtheNavarro-Frenk-White(NFW)[33]concentration
distribution[25],whichweinvestigatehere. parameter of dark matter haloes. In Sec. IV, we show our
The above papers considered only linear perturbation the- main results, where we compare and calibrate the analytical
orywhich,bydefinition,ignorestheeffectsoftheintrinsically predictions with the results of the N-body simulations. We
nonlinear Vainshtein screening. Moreover, a realistic com- alsousethecalibratedformulaeforthehalomassfunctionand
parison between theory and observations is also subject to a linearbiastoconductahalooccupationdistributionanalysis
proper understanding of how the distribution of galaxies and of Luminous Red Galaxies (LRGs). Finally, we summarize
theirhosthaloesisbiasedrelativetotheunderlyingdarkmat- anddrawourconclusionsinSec.V.
terdensityfield. Toaddresstheseuncertainties,oneneedsto Unless otherwise specified, we assume the metric conven-
studythenonlinearevolutionofthematterandGalileonfield tion (+,−,−,−) and work in units in which the speed of
perturbations. The most accurate way to do this is through light c = 1. Greek indices run over 0,1,2,3 and we use
N-body simulation. References [26] and [27] performed the 8πG = κ = M−2 interchangeably, where G is Newton’s
Pl
first N-body simulations of the covariant Galileon model of constantandMPlisthereducedPlanckmass.
Ref. [4]. These two works focused on the so-called Cubic
(L )andQuartic(L )Galileonmodels(seeSec.II),re-
2−3 2−4
spectively. They found that the nonlinearities of the Vain- II. THEGALILEONMODEL
shtein screening, although very noticeable on small scales,
do not have an impact on the clustering of matter on scales
A. Action
k (cid:46) 0.1h/Mpc. N-body simulations of the most general
Quinticmodel(L )aremorechallengingtoperform. Nev-
2−5 TheactionofthecovariantGalileonmodel[4]isgivenby
ertheless, by studying the behavior of the fifth force assum-
ingsphericalsymmetryinthequasi-staticlimit,Ref.[28]has
stohopwronvtihdaetpthheyseiqcaulatsioolnustioofnsmwothioennothfethdeenQsuitiyntpicermtuorbdaetliofanisl S =(cid:90) d4x√−g(cid:34) R − 1(cid:88)5 c L −L (cid:35), (1)
become of order unity. Consequently, the study of nonlinear 16πG 2 i i m
i=1
structureformationintheQuinticmodelisnolongerofinter-
est. By making use of the spherical collapse model and the whereRistheRiccicurvaturescalarandgisthedeterminant
excursion set theory formalism, Ref. [28] also estimated the ofthemetricg . L describesthemattercontentoftheuni-
µν m
predictedhalomassfunctionandlinearhalobiasintheQuar- verse,whichinthismodelisminimallycoupledtothemetric
tic model. The indications were that the halo bias could be andGalileonfields. Theparametersc arerealdimension-
1−5
smallerthaninstandardΛCDMmodels.However,becauseof lessconstantsandthefiveLagrangiandensityterms,fixedby
the simplified treatment, no decisive quantitative statements the Galilean invariance in flat spacetime, ∂ ϕ → ∂ ϕ+b ,
µ µ µ
couldbemade. aregivenby
Here,ourgoalistoputtheexcursionsettheorypredictions
ofRef.[28]ontoamorequantitativelevel. Inparticular, by L1 =M3ϕ,
comparing to the results of N-body simulations, we analyse L =∇ ϕ∇λϕ,
2 λ
the performance of the fitting formulae based on the ellip- 2
L = (cid:3)ϕ∇ ϕ∇λϕ,
soidal collapse of structures to describe the halo mass func- 3 M3 λ
tion and linear bias [29–31]. We anticipate that these fitting 1 (cid:104)
formulaecanmatchthesimulationresultsverywell,provided L4 = M6∇λϕ∇λϕ 2((cid:3)ϕ)2−2(∇µ∇νϕ)(∇µ∇νϕ)
their adjustableparameters arecalibrated against thesimula- (cid:105)
−R∇ ϕ∇µϕ/2 ,
tions. Wealsomeasureandfitthehaloconcentrationparame- µ
ter. Ultimately,weusethesehalopropertiestodeterminethe 1 (cid:104)
nonlinearmatterpowerspectrumcomputedinthehalomodel L5 = M9∇λϕ∇λϕ ((cid:3)ϕ)3−3((cid:3)ϕ)(∇µ∇νϕ)(∇µ∇νϕ)
approach [32]. The study of these simplified analytical for- +2(∇ ∇νϕ)(∇ ∇ρϕ)(∇ ∇µϕ)
µ ν ρ
mulae helps us to understand better the physical picture of
(cid:105)
thehalodistribution,whichisoftenhiddeninthebrute-force −6(∇ ϕ)(∇µ∇νϕ)(∇ρϕ)G , (2)
µ νρ
calculationsofaN-bodysimulation. Theseformulaealsoal-
lowforanefficientexplorationofthemodelsparameterspace, inwhichM3 ≡ M H2, whereH istheHubbleexpansion
Pl 0 0
whichwouldnotbepossiblewiththetime-consumingN-body rate today. In this model, the nonlinear coupling of the co-
simulations. variant derivatives of ϕ induces interactions between partial
This paper is organized as follows. In Sec. II, we display derivatives of g and ϕ (a process known as kinetic grav-
µν
theaction,thecosmologicalparametersandtheequationsthat ity braiding [34–36]). This is why this model is a modified
govern the gravitational interaction in spherically symmetric gravitymodel. Inadditiontotheseinteractions,therearealso
3
direct couplings to the Ricci scalar, R, and the Einstein ten-
TABLEI.ParametersoftheGalileonmodelsstudiedinthispaper.
sor, G , in L and L , respectively. These latter two cou-
µν 4 5 Ω ,Ω ,Ω ,h,n ,andτ are,respectively,thepresent-dayfrac-
plingsareneededsothattheequationsofmotionarekeptup r0 b0 c0 s
tionalenergydensityofradiation(r),baryons(b)andcolddarkmat-
tosecond-orderinfieldderivativesincurvedspacetimes,such
ter (c), the dimensionless present-day Hubble expansion rate, the
as the one described by the FRW metric [4]. However, this primordial scalar spectral index and the optical depth to reioniza-
processcomesatthecostofbreakingtheGalileanshiftsym- tion.ThescalaramplitudeatrecombinationA referstoapivotscale
s
metry. k=0.02Mpc−1.Theuniverseisspatiallyflatinthesemodels.The
We will set the potential term L1 to zero (i.e. c1 = 0) parametersc2,c3,c4,c5arethedimensionlessconstantsthatappear
as we are only interested in cases where the acceleration is intheactionEq.(1)andρϕ,i/ρm,i istheratiooftheGalileonand
drivensolelybykineticenergyterms. Notethatinthiscase, totalmatter(m)energydensitiesatzi. WealsoshowtheGalileon
thevalueofϕbecomesirrelevantforthephysicsofthemodel field time derivative ϕ¯˙ic13/3 at zi , the age of the Universe and the
since the action contains only ∂ϕ terms. One can then con- present-dayvalueofσ8.Onlyinthistable,thesubscript"i"refersto
quantitiesevaluatedatz = z = 106. ThepredictedCMBangular
sider three classes of Galileon models. The Cubic Galileon i
powerspectrumandthelinearmatterpowerspectrumofthesemodel
model is the simplest and encompasses L and L only; the
2 3 parameterscanbefoundinFig.1ofRef.[28].
QuarticGalileonmodelconsidersL ;andfinally,theQuin-
2−4
ticGalileonisthemostgeneralwhichincludesalloftheLa-
Parameter CubicGalileon QuarticGalileon
grangiantermsL . ItisworthnotingthatnoneoftheCu-
2−5
bic,QuarticorQuinticmodelspossessaΛCDMlimitforthe
Ω h2 4.28×10−5 4.28×10−5
expansion history of the Universe. Recently, Ref. [28] has r0
Ω h2 0.02196 0.02182
shown that, if the density perturbations become higher than b0
Ω h2 0.127 0.126
O(1), then the equation of motion of the scalar field in the c0
h 0.731 0.733
QuinticmodelsthatprovideagoodfittotheCMBdatafails
n 0.953 0.945
s
toadmitrealphysicalsolutionsforthespatialgradientofϕin τ 0.0763 0.0791
spherically symmetric quasi-static configurations. Naturally, log(cid:2)1010A (cid:3) 3.154 3.152
s
thispreventsonefromstudyingnonlinearstructureformation log[ρ /ρ ] −4.22 −37.39
ϕ,i m,i
intheQuinticmodel. Asaresult,inwhatfollows,wealways c /c2/3 −5.38 −4.55
2 3
assumethatc = 0andfocusonlyontheCubicandQuartic c 10 20
5 3
Galileonmodels. c /c4/3 0(fixed) −0.096
4 3
ByvaryingtheactionofEq.(1)withrespecttogµν andϕ, c5/c53/3 0(fixed) 0(fixed)
oneobtainsthemodifiedEinsteinequationsandtheGalileon
field equation of motion, respectively. These equations are ϕ¯˙ c1/3 1.10×10−9 1.54×10−20
i 3
lengthy and therefore we do not show them in this paper Age(Gyr) 13.748 13.770
(the interested reader can find them in Eqs. (A1-A7) of σ (z=0) 0.997 0.998
8
Ref. [11]). The background and the linearly perturbed equa-
tionshavealsobeenderivedandpresentedinpreviouspapers
(e.g.Refs.[11,12]),andwealsoabstainfromshowingthem
here. Inthispaper,wesimplylayoutthenonlinearequations
thatdeterminethemodificationstogravityinsphericalmatter TABLEII.Valuesofthecriticalinitialoverdensityforthespherical
overdensities. collapse to occur at a = 0.6, a = 0.8, a = 1.0, extrapolated to
a = 1.0 with the ΛCDM linear growth factor. This extrapolation
is done purely to enable the resulting values of δ in the Galileon
c
model to be more easily compared to results from other models in
B. Fifthforcesolutions
theliterature.Thevaluesofδ areshownforΛCDMandthemodels
c
ofTableIII,andwereobtainedbyfollowingthestrategypresented
Tobuildsomeintuitionabouttheresultsthatarepresented inRef.[28].
inthesubsequentsections,itisinstructivetolookatthemod-
ifications to gravity in the limit of spherical symmetry, for
Model a=0.6 a=0.8 a=1.0
whichanalyticalsolutionscanbederived.Inthiscase,accord- δ δ δ
c c c
ing to Ref. [28], the total gravitational force in the Galileon
modelisobtainedbysolvingthefollowingequations ΛCDM 2.346 1.907 1.677
QCDM 2.224 1.780 1.560
Cubic
Φ, Ω δa−3+A (ϕ, /χ)+A (ϕ /χ)2 CubicGalileon 2.219 1.767 1.537
χ = m0 1 χ 2 χ , (3) LinearizedCubicGalileon 2.213 1.749 1.500
χ A
4
Ψ, B (Φ, /χ)+B (ϕ, /χ)+B (ϕ, /χ)2 QCDM 2.226 1.784 1.565
χ = 0 χ 1 χ 2 χ , (4) Quartic
χ B QuarticGalileon 2.236 1.805 1.593
3
LinearizedQuarticGalileon 2.196 1.728 1.492
(cid:20)ϕ, (cid:21) (cid:20)ϕ, (cid:21)2
0=η δ+(η δ+η ) χ +η χ
01 11 10 χ 20 χ
4
FIG.1.TimeanddensitydependenceoftheeffectivegravitationalstrengthG ofEq.(7)fortheCubic(leftpanel)andQuartic(rightpanel)
eff
Galileonmodels.ThecolourscalebarsattherightofeachpanelshowthevalueofG /G.Thecolorscaleisthesameforbothpanels.
eff
(cid:20)ϕ, (cid:21)3 paper, whenever we refer to the Cubic and Quartic Galileon
+η χ , (5)
30 χ modelswemeantheseparticularparametersets.
The modifications to gravity can be quantified in terms of
where χ ≡ aH02r (with r being the comoving radial coordi- aneffectivegravitationalstrengthGeff definedas
nate and a the cosmic scale factor) and δ ≡ ρ /ρ¯ −1 is
m m
the matter density contrastof atop-hat sphericaloverdensity
G Ψ, /χ Ψ, /χ
w.r.t. themeanbackgrounddensityρ¯m. ThequantitiesAi,Bi eff(a,δ)= χ = χ . (7)
G Ψ,GR/χ Ω δ/(2a3)
and ηab depend only on time and can be found in Ref. [28]. χ m0
The two gravitational potentials Φ and Ψ are defined by the
In Ref. [28], this quantity is shown for the Quartic model as
perturbed spatially-flat FRW line element in the Newtonian
a function of time and density. In Fig. 1, for completeness,
gauge
werepeatthesamefigurebutincludealsothecorresponding
result for the Cubic Galileon. In both models, the deviation
ds2 =(1+2Ψ)dt2−a(t)2(1−2Φ)γ dxidxj, (6) of Geff/G from unity arises only at late times: a (cid:38) 0.6 and
ij a(cid:38)0.5fortheCubicandQuarticmodels,respectively.Inthe
where γ = diag[1,1,1]. Step by step, for fixed a and δ, linearregime(|δ|(cid:28)1),G increaseswithtimeinbothmod-
ij eff
onecansolveEq.(5)analyticallyforϕ, /χ,andplugtheso- els, but it does so more pronouncedly in the Cubic Galileon:
χ
lutionintoEqs.(3)and(4)todeterminethetotalforcegiven at a = 1, G is roughly 90% larger than G in the Cubic,
eff
by Ψ, /χ. Note that we have written Eqs. (3), (4) and (5) butonly40%largerintheQuartic. Inhigher-densityregions
χ
assuming c = 0. If c (cid:54)= 0, Eq. (5) becomes a sixth-order (δ (cid:38) 1), however, the qualitative pictures of these two mod-
5 5
algebraic equation for ϕ, /χ, which does not admit an ana- elsbecomedistinct. Focusingforinstanceonthepresentday
χ
lytical solution and has to be solved numerically. Moreover, (a = 1),intheCubicmodel,G /Gapproachesunityasthe
eff
inthecaseoftheCubicGalileon,theequationssimplifyeven densitycontrastincreases.Thisshowstheeffectofthescreen-
furthersinceη ,η ,A ,B ,B vanishandB =B =−2. ing mechanism in high-density regions. On the other hand,
11 30 2 1 2 0 3
Inparticular,intheCubicmodelΦ=Ψ. in the Quartic model, at a = 1, G does not approach the
eff
Table I lists the model parameters we consider. These standardvalue,butinsteadbecomesroughly40%smallerand
wereobtainedbyfollowingthestepsinRef.[12],whichused time-varying(thetimevariationcanalsobeseeninFig.4of
datafromtheWMAP9-yrresultsforthetemperaturefluctu- Ref.[27]). Thisresultputsthemodelintoseveretensionwith
ations power spectrum of the CMB [23], type Ia supernovae SolarSystemtestsofgravitythatconstrainthetimevariation
(SNIa)fromtheSNLS3-yrsample[37]andbaryonicacous- of G /G to be very small [7]. The weaker gravity in the
eff
ticoscillations(BAO)measurementsfromthe6dF[38],SDSS Quartic Galileon model follows from modifications induced
DR7 [41] and BOSS [39] galaxy surveys. These correspond by the time-varying A , B and B terms in Eqs. (3) and
4 0 3
alsototheparametersusedtoruntheN-bodysimulationsin (4). These terms do not depend on the spatial gradients of
Refs.[26]and[27]. ThepredictedCMBangularpowerspec- theGalileonfield,andtherefore,cannotbesuppressedbythe
trumandthelinearmatterpowerspectrumofthesemodelpa- Vainshteinmechanism. Theoriginofthesethreecoefficients
rameterscanbefoundinFig.1ofRef.[28]. Intherestofthe canbetracedbacktothedirectcouplingoftheGalileonfield
5
withtheRicciscalarRinL4,whichisnecessarytoavoidthe B. Halomassfunction
presence of ghosts. In the case of the Cubic Galileon, these
threetermsareconstantandtheproblemdoesnotarise. Thehalomassfunctioncanbeexpressedas
The physical picture depicted in Fig. 1 indicates that the
modelling of halo properties can be very different in these
dn(M) ρ¯
two models, in particular because of the different behavior dlnM = m0f(S)dS, (11)
in high-density regions. In the remainder of the paper, we dlnM M
focuson thesedifferences, ignoringfornow thefact thatthe
whereSdenotesthevarianceofthelineardensityfieldfiltered
time-varyingG inhigh-densityregionsisputtingtheQuar-
eff onsomecomovingdistancescaleR,
ticmodelintohugeobservationaltension.
(cid:90)
S(R)≡σ2(R)=4π k2P W˜2(k,R)dk. (12)
III. HALOMODELOFTHENONLINEARMATTER k,lin
POWERSPECTRUM
Here, W˜ (k,R) = 3(sin(kR)−kRcos(kR))/(kR)3 is the
Inthissection,wedescribethehalomodelofthenonlinear Fourier transform of the filter function, which we take as a
matterpowerspectrum,aswellasthehalopropertiesthatare top-hatinrealspace. Themassenclosedbythefilterisgiven
neededasinput. Inparticular,wedefineandpresentthehalo by
massfunction,linearhalobiasandhalodensityprofiles.
M =4πρ¯ R3/3. (13)
m0
A. Halomodel
In Eq. (11), f(S)dS is associated with the fraction of
the total mass that resides in haloes whose variances fall
In the halo model approach, one of the main premises is
within[S,S+dS](orequivalently,whosemassesfallwithin
that all matter in the Universe is in bound structures. Thus, [M −dM,M])1.
the two-point correlation function of the matter density field
Motivatedbytheellipsoidalcollapseofoverdenseregions,
can be decomposed into the contributions from the correla-
Refs.[29–31]proposedthefollowingexpressionforf(S)
tions between mass elements that belong to the same halo
(the1-haloterm)andtodifferenthaloes(the2-haloterm). In
termsofthematterpowerspectrum,thiscanbewrittenas(see (cid:114) q δ (cid:34) (cid:18)qδ2(cid:19)−p(cid:35) (cid:20) δ2(cid:21)
f(S)=A c 1+ c exp −q c ,
Ref.[32]foracomprehensivereview)
2πS3/2 S 2S
(14)
P =P1h+P2h, (8)
k k k
whereδ ≡δ (z)isthecriticalinitialoverdensityforaspher-
c c
where
ical top-hat to collapse at redshift z, extrapolated to z = 0
with the ΛCDM linear growth factor 2. The values of δ (z)
c
(cid:90) M dn(M) for the Cubic and Quartic Galileon models are shown in Ta-
P1h = dM |u(k,M)|2,
k ρ¯2m0 dlnM ble II. Note that for consistency, Pk,lin in Eq. (12) is also
theinitialpowerspectrumofthespecificmodel(saytheCu-
P2h =I(k)2P , (9)
k k,lin bic or the Quartic Galileon models), evolved to z = 0 with
are,respectively,the1-haloand2-haloterms,with the ΛCDM linear growth factor. The choice of parameters
(q,p) = (1,0) leads to the Press-Schechter mass function
(cid:90) 1 dn(M) [42],whoseshapeandamplitudearemotivatedbythespher-
I(k)= dM b (M)|u(k,M)|. (10) ical(ratherthanellipsoidal)collapseofmatteroverdensities.
ρ¯ dlnM lin
m0 However, Refs. [29–31] found that the choice of parameters
In the above expressions, k is the comoving wavenum- (q,p)=(0.75,0.30)providesamuchmoreaccuratedescrip-
ber;ρ¯ isthepresent-daybackgroundmatterdensity;P tionofthemassfunctionmeasuredfromN-bodysimulations
m0 k,lin
is the matter power spectrum obtained using linear the- ofΛCDMmodels. ForGalileongravitymodels,itisnotnec-
ory;dn(M)/dlnM denotesthecomovingnumberdensityof essarily true that this choice of (q,p) parameters also results
haloes per differential logarithmic interval of mass (we shall
refertothisquantityasthemassfunction);b (M)isthelin-
lin
earhalobias;u(k,M)istheFouriertransformofthedensity
profileofthehaloestruncatedatthesizeofthehaloandnor- 1Notethatforfixedcosmologicalparameters,thequantitiesS,RandM
malized such that u(k → 0,M) → 1. In order to compute canberelatedtooneanotherviaEqs.(12)and(13). Intheremainderof
thepaper, weusethesequantitiesinterchangeablywhenreferringtothe
thematterpowerspectrumofEq.(8),onehastomodelthese
scaleofthehaloes.
quantitiesfirst. Thisisdoneintheremainderofthissection. 2Thisextrapolationisdonepurelytoensurethattheresultingvaluesofδc
We follow the notation of Ref. [28], to which we refer the intheGalileonmodelcanbemorereadilycomparedtoresultsfromother
readerforfurtherdetailsofthederivation. modelsintheliterature.
6
inagoodfittoN-bodyresults. Inthenextsection,werecal- ThemassoftheNFWdensityprofile,M ,canbeobtained
∆
ibrate these two parameters to our simulations of the Cubic byintegratingEq.(17)uptosomeradiusR (themeaningof
∆
and Quartic Galileon models. The normalization constant A thesubscript∆willbecomeclearlater)
(cid:82)
isfixedbyrequiringthat f(S)dS = 1. Themassfunction
computed using Eqs. (11) and (14) is known as the Sheth-
Tormenmassfunction. (cid:90) R∆
M = dr4πr2ρ (r) (18)
∆ NFW
0
C. Linearhalobias =4πρ R∆3 (cid:20)ln(1+c )− c∆ (cid:21),
s c3 ∆ 1+c
∆ ∆
Thelinearhalobiasparameterb(M)[43]quantifiesthedif-
ference between the clustering amplitude of haloes of mass wherewehaveusedtheconcentrationparameter
M and that ofthe underlying total dark matter field on large
scales(k (cid:28)1h/Mpc),
R
c = ∆ (19)
δhalo(M)=b(M)δmatter, (15) ∆ rs
where δhalo and δmatter represent, respectively, the density (nottobeconfusedwiththeciparametersintheactionofthe
contrast of the distribution of haloes of mass M and of the GalileonmodelEq.(1)).
total dark matter field. Equation (23) is only valid when
Inoursimulations,thehalomassisdefinedas
|δ | (cid:28) 1,i.e.,onlargecosmologicalscales. Onsmaller
matter
scales, where the matter overdensity is larger, higher order
termsareneeded[44].
4π
By following the same steps as in Ref. [28], one can M = ∆ρ¯ R3, (20)
∆ 3 c0 ∆
straightforwardly show that the Sheth-Tormen halo bias is
givenby
i.e., M is the mass enclosed by the comoving radius R ,
∆ ∆
withinwhichthemeandensityis∆timesthecriticaldensity
b(M)=1+g(z)(cid:18)qδc2/S−1 + 2p/δc (cid:19),(16) oftheUniversetoday,ρ¯c0.Inthispaperweconsider∆=200,
δ 1+(qδ2/S)p butfornowletuskeepthediscussionasgeneralaspossible.
c c
By combining Eqs. (18) and (20), one finds ρ as a function
s
with g(z) = DΛCDM(z = 0)/DModel(z), where D(z) is ofc :
∆
thelineargrowthfactorofaspecificmodel. Thelatterisde-
fined as δ (z) = D(z)δ (z )/D(z ). Provided the
matter matter i i
(q,p) parameters are calibrated to fit the mass function, the
1 (cid:20) c (cid:21)−1
linear halo bias b(M) should, according to the excursion set ρ = ∆ρ¯ c3 ln(1+c )− ∆ . (21)
theory logic, give automatically a reasonably good fit to the s 3 c0 ∆ ∆ 1+c∆
simulation results. This is one of the well-known lessons of
Refs.[29–31]fortheCDMfamilyofmodels. Theresultsin All that is needed to fully specify the NFW profile is to de-
thenextsectionshowthatthisremainstruefortheCubicand terminethevalueofr ,whichisdonebydirectfittingtothe
s
QuarticGalileonmodels. halodensityprofilesmeasuredfromthesimulations.Inthelit-
erature,however,ithasbecomemorecommontospecifythe
concentration-mass relation c (M ), instead of the equiva-
∆ ∆
D. Halodensityprofiles lent values of r . Previous studies [45–48] have found that
s
theconcentration-massrelationiswelldescribedbyapower
Weassumethattheradialprofileofthedarkmatterhaloes law function. The parameters of the power law, however,
isoftheNFWtype[33]3 seemtohaveasizeablecosmologydependence,evenfordif-
ferentchoicesofcosmologicalparametersinΛCDMmodels
ρ (see e.g. Ref. [47]). In the next section, we will see that the
ρNFW(r)= r/r [1+sr/r ]2, (17) c∆(M∆) relation in Galileon models can also be well fitted
s s by a power law, but with fitting parameters that differ con-
whereρ andr areoftencalledthecharacteristicdensityand siderably from those obtained for ΛCDM. Having found the
s s
thescaleradiusofthehalo. c∆(M∆)relationfromthesimulations,thentheNFWdensity
profilebecomescompletelyspecifiedbythemassM ofthe
∆
halo.
Finally,becausewhatentersEqs.(9)and(10)istheFourier
3Nottobeconfusedwiththetop-hatprofileassumptionusedinthespherical transformoftheprofiles,u(k,M),andnottheprofilesthem-
collapsetoobtainthevaluesofthecriticaldensityδc. selves,wesimplymentionthatitispossibletoshowthat
7
inwhichtheexpansionhistoryisthesameasintheothertwo
TABLEIII.SummaryofthethreevariantsoftheCubicandQuartic
variants. This variant is used as a base model to identify the
Galileon models studied in this paper. All variants have the same
modificationstostructureformationthatarisefromthemod-
backgroundexpansionhistory,butdifferintheforcelaw.
ified force law, excluding those that arise from the modified
expansion rate compared to ΛCDM. The properties of these
Model Expansionhistory Forcelaw
threevariantsaresummarizedinTableIII.
The simulations were performed in a box of size L =
Fullmodel Galileon GR+Screenedfifthforce
200Mpc/h, with N = 5123 dark matter particles and grid
Linearmodel Galileon GR+linearfifthforce p
refinementcriteriaN = 8. Foreachofthemodelvariants,
QCDM Galileon GR th
wehavesimulatedfivedifferentrealizationsoftheinitialden-
sityfield,bychoosingdifferentrandomseeds.Thisallowsfor
statistical averaging, which we use to construct errorbars for
the simulation results by measuring the variance within the
(cid:90) R∆ sinkrρ (r) differentrealizations. Althoughinthispaperweshowresults
u (k,M)= dr4πr2 NFW
NFW kr M onlyforoneboxsize,wenotethatinRefs.[26,27]thesame
0 ∆
(cid:26) modelsweresimulatedusingdifferentboxsizesandparticle
sin(kr )
=4πρ r3 s [Si([1+c ]kr )−Si(kr )]numbers,andgaveconvergedresults.
s s M ∆ s s
cos(kr )
+ s [Ci([1+c ]kr )−Ci(kr )]
M ∆ s s B. Massfunction
(cid:27)
sin(c kr )
− ∆ s , (22)
M(1+c∆)krs In Figs. 2 and 3, we show our results for the cumulative
massfunctionoftheCubicandQuarticGalileonmodels, re-
(cid:82)x (cid:82)∞
whereSi(x) = dtsin(t)/tandCi(x) = − dtcos(t)/t. spectively. Thesewereobtainedwiththephase-spacefriends-
0 x
Notethat,indeed,u(k →0,M)→1,asrequired. of-friendshalofindercodeRockstar[52]. Throughout,we
use M and M interchangeably to denote halo mass. The
200
symbolswitherrorbarsindicatethesimulationresultsandthe
IV. RESULTS dashed lines show the mass function predicted by Eqs. (11)
and(14)usingthevaluesofδ fromTableIIandthestandard
c
Inthissection,wetestthepredictionsoftheformulaepre- Sheth-Tormenparameters(q,p) = (0.75,0.30). Onecansee
sentedinthelastsectionwiththeresultsfromN-bodysimu- that the mass function computed in this way fails to provide
lationsoftheCubic[26]andQuartic[27]Galileonmodels. a reasonable description of the simulation results. In terms
of the relative difference w.r.t. QCDM, the standard Sheth-
Tormen prediction gets the correct qualitative trend, but sig-
A. Summaryofthesimulations nificantly underestimates the effects of the modifications to
gravityseeninthesimulations.
The simulations presented in this paper were performed It is not completely surprising that the use of the standard
with the ECOSMOG code [49], which is a modified version Sheth-Tormen parameters (q,p) = (0.75,0.30) fails in the
of the RAMSES code [50] that includes extra solvers for the Galileonmodel, sincethesewerechosentofitΛCDMsimu-
scalardegreesoffreedomthatappearinmodifiedgravitythe- lations [29–31]. The ellipsoidal collapse motivates a depar-
ories. The code solves the equation of motion of the scalar turefrom(q,p) = (1,0)(whichcorrespondstothespherical
field by performing Gauss-Seidel iterative relaxations on an collapse case), but the magnitude of this departure is deter-
adaptivelyrefinedgrid.Thegridisrefinedwheneverthenum- mined by fitting to numerical results. Very crudely, one can
berofparticleswithinagridcellexceedssomeuser-specified saythatthefitted(q,p)parametersabsorbsomeoftheuncer-
threshold,N .Thisensuresthathigh-densityregionsaresuf- taindetailsofthenonlinearstructureformation,whichcannot
th
ficientlywellresolved,whilesavingcomputationalresources beaccuratelydescribedbytheellipsoidalcollapse. Inmodels
inregionswherethedensityislower. Formoredetailsabout thatdiffersignificantlyfromΛCDM,liketheCubicorQuartic
the code implementation, in particular in Galileon cosmolo- Galileonmodels, itistobeexpectedthatthespecificsofthe
gies,wereferthereadertoRefs.[26,27,49,51]. ellipsoidalcollapseshouldalsobedifferent. Inpractice, this
The results that follow correspond to three variants of the translatesintodifferentvaluesforthe(q,p)parameters. The
CubicandQuarticGalileonmodels. Wecallthelinearmodel solidlinesinFigs.2and3showthemassfunctionpredicted
the one in which the screening mechanism is artificially set usingthevaluesofδ fromTableIIandthebest-fitting(q,p)
c
tozero,bylinearizationoftheEinsteinandtheGalileonfield parameters to the simulation results. The latter were deter-
equations. Thefullmodelis,asthenameindicates,thecom- minedforeachvariantoftheCubicandQuarticGalileonmod-
pletemodelwithoutanymodifications. Comparingthesetwo elsata = 0.6,a = 0.8anda = 1. Theirvaluesareshownin
variantsallowsonetoseetheeffectsofthescreeningmecha- TableIV.Byallowing(q,p)todifferfromthestandardvalues,
nism.TheothervariantiscalledQCDM,andcorrespondstoa one sees that the analytical predictions of Eqs. (11) and (14)
modelwherethereisnocontributionfromthefifthforce,but can actually provide an extremely good fit to the simulation
8
FIG.2. TheupperpanelsshowthecumulativemassfunctionofthethreevariantsoftheCubicmodelata = 0.60,a = 0.80anda = 1.00.
The triangles with errorbars show the simulation results considering only haloes (and not subhaloes) with mass M > 100M , where
200 p
M = Ω ρ¯ L3/N istheparticlemass. ThesolidlinescorrespondtothecumulativemassfunctionpredictedusingEqs.(11)and(14),
p m0 c0 p
withthebest-fitting(q,p)parameterstothesimulationresultsgiveninTableIV.Thedashedlinesarecomputedinthesamewayasthesolid
lines,butwiththestandardSheth-Tormenparametervalues(q,p)=(0.75,0.30).Forreference,theSheth-Tormencumulativemassfunction
foraΛCDMmodelwithWMAP9parameters[23]isshownbytheblackdashedcurveintheupperpanels. Thecolorschemeindicatedin
thefigureappliestothelinesandsymbols. Inthelowerpanels,therelativedifferenceofthesimulationresultsw.r.t. theQCDMsimulations
resultsisshown,andtherelativedifferenceoftheanalyticalpredictionsisplottedw.r.t.totheanalyticalpredictionsoftheQCDMmodel.Also
inthelowerpanels,thesolidredanddashedredlinesarebothzero,bydefinition.
TABLEIV.Best-fittingSheth-Tormen(q,p)parameterstothesimulationresultsforthevariantsoftheCubicandQuarticGalileonmodels
ata = 0.60,a = 0.80anda = 1.00. Theuncertaintyinthevaluesofq andpis∆ = 3.5×10−3 and∆ = 1.5×10−3,respectively.
q p
Theseparametersweredeterminedbyminimizingthequantity(cid:80) |nsims(>M )/nST(>M ,q,p)−1|,wherensimsisthecumulativemass
i i i
functionmeasuredinthesimulations,nST istheanalyticalresultgivenbytheSheth-Tormenmassfunctionandtheindex’i’runsoverthe
numberofbinsinthecumulativemassfunction.
Model a=0.60 a=0.80 a=1.00
(q,p) (q,p) (q,p)
QCDM (0.699,0.336) (0.727,0.349) (0.791,0.354)
Cubic
CubicGalileon (0.699,0.334) (0.720,0.346) (0.770,0.349)
LinearizedCubicGalileon (0.685,0.326) (0.692,0.308) (0.734,0.301)
QCDM (0.671,0.339) (0.692,0.349) (0.713,0.354)
Quartic
QuarticGalileon (0.713,0.359) (0.840,0.389) (1.024,0.407)
LinearizedQuarticGalileon (0.649,0.316) (0.671,0.316) (0.692,0.321)
9
FIG.3.SameasFig.2,butfortheQuarticGalileonmodel.
results in the entire mass range probed. Similarly, Ref. [53] whosedefinitiondiffersindifferentmodels. Inthispaper,we
has found that, by appropriately adjusting the values of pa- arecomparingthemassM ofEq.(13)withthevaluesofM
200
rametersintheTinkermassfunction[54],thenthelattercan measuredfromthesimulations.Onedoesnotexpectthesetwo
matchthesimulationresultsofamodelthatalsoemploysthe mass definitions to be exactly the same, but nor would one
VainshteineffectbutwithaΛCDMexpansionhistory. expect them to differ significantly. These ambiguities in the
In the case of the Cubic model, one sees that the screen- massdefinitioncan, anyway, beabsorbedinthefittedvalues
ingmechanismworkswellatsuppressingtheenhancementin oftheSheth-Tormen(q,p)parameters.Weexpectthesefitting
the number density of haloes. For instance, the linear vari- parameterstoslightlychangewithdifferentmassdefinitions.
antpredictsanenhancementinthenumberdensityofhaloes However,notethatthisisalsothecasefortheΛCDMmodel,
withM ∼1014M /hata=1ofabout45%,whereasinthe andisnotpeculiartoGalileongravity.
(cid:12)
case of the full model, in which the screening is at play, the
enhancement is smaller than 10%. On the other hand, in the
caseoftheQuarticmodel,theoverallweakeningofgravityin C. Linearhalobias
thefullvariant(cf.Fig.(1))leadstoasignificantsuppresionin
thenumberdensityofcollapsedobjects. Inparticular,haloes Inoursimulations,wemeasurethehalobiasbyevaluating
withM ∼1014M(cid:12)/hare∼50%lessabundantcomparedto theratio
QCDM.Inthecaseofthelinearizedvariant,thesamemassive
haloesare∼30%moreabundantw.r.t. QCDM4.
P (k,M)
Before proceeding, a comment should be made about the b(k,M)= hm , (23)
definitionofhalomassinthesimulationsandintheanalytical P(k)
formulaepresentedinSec.III.Assumingmassconservation,
where P(k) is the total matter power spectrum and
Eq. (13) can be associated with the virial mass of the halo,
P (k,M) is the halo-matter cross spectrum for haloes of
hm
mass M. We used a Delaunay Tessellation field estimator
code[57,58]toobtainthehaloandmatterdensityfieldsfrom
4See also Refs. [55, 56] for N-body simulations of a phenomenological whichwecomputedthesepowerspectra. Inthenumeratorof
modelwithaYukawa-likefifthforce,forwhichsimilarresultsarefound. Eq. (23), we consider the cross power spectrum, rather than
10
FIG.4.LinearhalobiasofthethreevariantsoftheCubic(upperpanels)andQuarticGalileon(lowerpanels)modelsata=0.60,a=0.80and
a = 1.00. Thetriangleswitherrorbarsshowthesimulationresultsconsideringonlyhaloes(andnotsubhaloes)withmassM > 100M ,
200 p
where M = Ω ρ¯ L3/N is the particle mass. The solid and dashed lines correspond to the linear halo bias parameter of Eq. (16)
p m0 c0 p
computedwiththe(q,p)parametersfromTableIVand(q,p) = (0.75,0.30), respectively. ThelinearhalobiasforaΛCDMmodelwith
WMAP9parameters[23]isshownbytheblackdashedlines.Thecolorschemeindicatedinthefigureappliestothelinesandsymbols.
thehalo-halocountspowerspectrum,toreducetheimpactof toreproducetheresultsfromN-bodysimulations[29–31].
shot noise on our measurements. Our estimate for the linear
halobiasisgivenbytheasymptoticvalueofb(k,M)onlarge
scales (small k). The result is shown in Fig. 4 for the Cubic
D. Halooccupationdistributionanalysis
(upper panels) and Quartic (lower panels) Galileon models.
In Fig. 4, one sees that Eq. (16) provides a good description
ofthelinearhalobiasseeninthesimulationsifoneusesthe As indicated by the values of σ8 ∼ 1 in Table I, the am-
(q,p)parametersthatbest-fitthemassfunctionofthesimula- plitudeofthelinearmatterpowerspectrumintheCubicand
tions(solidlines).Thisshowsthattheexcursionsettheoryap- Quartic Galileon models is higher than in standard ΛCDM
proachandthestepsinvolvedinthederivationofEq.(16)are models, for which σ8 ∼ 0.82 [23]. Consequently, it is in-
stillvalidintheCubicandQuarticGalileonmodels.However, teresting to investigate if the enhanced clustering power in
theuseofthebest-fitting(q,p)parametersdoesnotleadtoa theGalileonmodelsisstillconsistentwiththeobservedlarge
significant improvement over the use of the standard Sheth- scale clustering of the host haloes of Luminous Red Galax-
Tormen values, (q,p) = (0.75,0.30) in matching the simu- ies (LRGs) of the SDSS DR7 [25]. The screening mecha-
lationresults. Thelinearhalobiasseemstobelesssensitive nismcouldpotentiallysuppresspartoftheenhancement,but
thanthehalomassfunctiontotheexactchoiceof(q,p). This Refs.[26,27]haveshownthattheimpactoftheVainshteinef-
canbeunderstoodasthelinearhalobiasiscomputedasthera- fectisnegligibleonsufficientlylargescales(k (cid:46)0.1h/Mpc).
tiooftwomassfunctions[28,43],andconsequently,someof On the other hand, the result of Fig. 4 shows that massive
thedependenceonthevaluesof(q,p)cancelstosomeextent. haloes in Galileon cosmologies can be less biased than in
Notethatdespitetheweakersensitivitytotheexactchoiceof ΛCDM, which effectively suppresses the halo power spec-
theSheth-Tormenparameters,thesemuststilldifferfromthe trum. As a result, a robust comparisson between theory and
Press-Schechter limit (q,p) = (1,0), which is known to fail observations requires an exploration of this degeneracy be-
tween the enhanced linear growth of structure and the lower