Table Of ContentMon.Not.R.Astron.Soc.000,1–8(2012) PrintedJanuary29,2013 (MNLATEXstylefilev2.2)
The contribution of the Warm-Hot Intergalactic Medium
to the CMB anisotropies via the Sunyaev-Zeldovich effect
3
I. F. Suarez-Vel´asquez1⋆, J. P. Mu¨cket1 and F. Atrio-Barandela2
1
0 1Leibniz-Institutfu¨r Astrophysik Potsdam, Ander Sternwarte 16, 14482, Potsdam, Germany
2 2Fisica Teorica, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain
n
a
J Accepted .Received;inoriginalform2012
8
2
ABSTRACT
] Cosmological hydrodynamical simulations predict that a large fraction of all baryons
O residewithin mildly non-linearstructureswith temperaturesin the range105−107K.
C As the gas is highly ionized, it could be detected by the temperature anisotropies
. generatedonthe CosmicMicrowaveBackgroundradiation.We refineourpreviouses-
h
timatesofthethermalSunyaev-Zeldovicheffectbyintroducinganon-polytropicequa-
p
tion of state to model the temperature distribution of the shock heated gas derived
-
o from temperature-density phase diagrams of different hydrodynamical simulations.
r Depending on the specific model, the Comptonization parameter varies in the range
st 10−7 6 yc 6 2×10−6, compatible with the FIRAS upper limit. This amplitude is in
a
agreementwithasimpletoymodelconstructedtoestimatethe averageeffectinduced
[
byfilamentsofionizedgas.Usingthelog-normalprobabilitydensityfunctionwecalcu-
1 late the correlationfunction and the power spectrum of the temperature anisotropies
v generatedbytheWHIMfilaments.Forawiderangeofthe parameterspace,themax-
6 imum amplitude of the radiation power spectrum is (ℓ+1)ℓCℓ/2π = 0.7−70(µK)2
4 atℓ≈200−500.This amplitude scaleswith baryondensity,Hubble constantandthe
5 amplitude ofthe matterpowerspectrumσ8 as[(ℓ+1)ℓCℓ]max/2π∝σ82.6(Ωbh)2.Since
6
the thermalSunyaev-Zeldovicheffecthas a specific frequency dependence, weanalyze
.
1 the possibility of detecting this component with the forthcoming Planck data.
0
3 Key words: Missing baryons, WHIM, Sunyaev-Zeldovich
1
:
v
i
X
1 INTRODUCTION beenidentified(Fukugita& Peebles2004;Shullet al.2011).
r Hydrodynamical simulations predicted that baryons could
a The distribution of baryons in the local Universe is one of
be in the form of shock-heated intergalactic gas in mildly-
the main problems of modern Cosmology. The highly ion-
nonlinear structures with temperatures 105 107K, called
ized intergalactic gas, that evolved from the initial density −
Warm-HotIntergalacticMedium(WHIM).Thebaryonfrac-
perturbations into a complex network of mildly non-linear
tion in this medium could be as large as 40%-50% in
structures in the redshift interval 2 < z < 6 (Rauch 1998;
the local Universe (Cen & Ostriker 1999; Dav´eet al. 1999,
Stockeet al. 2004), could contain most of the baryons in
2001;Danforth & Shull2008;Prochaska & Tumlinson2008;
the universe (Rauchet al. 1997; Schaye 2001; Richteret al.
Smith et al.2011)containingthebulk,ifnotall,theuniden-
2006).Atredshiftsz>2mostbaryonsarefoundintheLyα
tified baryons.
systems detected through absorption lines in the spectra
of distant quasars. With cosmic evolution, the baryon frac- The observational effort has concentrated in searching
tionin thesestructuresdecreases asmorematterisconcen- for the WHIM X-ray signature both in emission and in ab-
trated within compact virialized objects. At low redshifts, sorption. Sol tan (2006) looked for the extended soft X-ray
theLyαsystemsarefilamentswithlowHIcolumndensities emission around field galaxies buthistask was complicated
containing 30% of all baryons (Stockeet al. 2004), while by the need to subtract all systematic effects that could
∼
thematerial in starsandgalaxies containsabout 10% ofall mimic the diffuse signal. More recently, the effort has been
baryons.Anextra5%couldbeintheformofCircumgalactic concentrated insearching for absorption lines duetohighly
Medium around galaxies (Guptaet al. 2012). More impor- ionizedheavyelementsfromthefar-ultraviolettothesoftX-
tantly,atz 0about50%ofallcosmicbaryonshavenotyet ray(seeShullet al.2011,forareview)withpartialsuccess.
∼
For example, Dietrich et al. (2012) have recently reported
the detection of a large-scale filament between two clusters
⋆ E-mail:[email protected];[email protected]; [email protected] of galaxies.
2 I. F. Suarez-Vel´asquez, J. P. Mu¨cket, and F. Atrio-Barandela
While absorption lines will identify individ- d
ual systems, complementary techniques are needed
to study the overall properties of the WHIM. To
LOS
this purpose, Atrio-Barandela & Mu¨cket (2006) and
Atrio-Barandela et al. (2008) suggested that the WHIM
distribution would generate temperature anisotropies on
D(z)
the Cosmic Microwave Background (CMB) due to the
thermal (TSZ) and kinematic (KSZ) Sunyaev-Zeldovich
α
effect (Sunyaev& Zeldovich 1972, 1980), opening a new
observational window to study the gas distribution at low
redshifts(G´enova-Santoset al.2009).Hallman et al.(2007)
O
found that about a third of theSZ fluxwould begenerated
by unbound gas, but their estimated amplitude was lower Figure1.Geometricalmodelofasimplefilamentwithconstant
than the prediction of our model. Our computation as- density and temperature. The filament has a size L, width d, is
sumed that the weakly non-linear matter distribution was located at redshift z at a comoving distance D(z), subtends an
angle α from the observer and is randomly oriented in space,
described by a log-normal probability distribution function
forminganangleθ withrespecttothelineofsight.
PDF (Coles & Jones 1991; Choudhuryet al. 2001) and
the gas followed a polytropic equation of state. While the
IGM at redshifts z > 2 behaves like a polytrope, at low
redshiftsthepolytropicequationof stateisnotlongervalid where Te and ne are the electron temperature and electron
since the gas is heated mostly by shocks to temperatures number density, dl is the proper distance along the los and
about 105K < T < 107K at density contrasts δ < 100. y0 =σTkB/mec2, with σT, kB, me Thomson cross section,
In the regions where the gas is shock-heated, small-scale Boltzmann constant and electron mass, respectively. While
density perturbations can be neglected due to the high the KSZ effect has the same frequency dependence as the
pressure gradients (Klar & Mu¨cket 2010), simplifying the intrinsic CMB anisotropies, the change with frequency of
treatment. In this paper we shall improve our previous TSZ effect, G(x), is different from that of any other known
calculations by using equations of state derived from foreground.ItisnegativeintheRayleigh-Jeansandpositive
hydrodynamical simulations to provide a more physical intheWienregions,beingnullclosetoν =217GHzandcan
description of the WHIM. Briefly, in section 2 we estimate bemoreeasilyseparatedfromothercontributionsthanKSZ
the order of magnitude contribution of a population of anisotropies. Since they are more easily detectable, here we
WHIM filaments to the Comptonization parameter based shall discuss only theTSZ contribution of theWHIM.
on a crude geometrical model; in section 3 we describe our An order of magnitude estimate of the WHIM TSZ ef-
log-normal model and we detail the differences with our fect can be obtained by considering the effect on the CMB
previous treatments; in section 4 we compute the CMB radiation ofasinglefilamentwith constantelectron density
ComptonizationparameterduetotheWHIMandinsection neandconstanttemperatureTe.Thegeometricalconfigura-
5 the correlation function and angular power spectrum; in tion of a filament of size L and thicknessd is shown in Fig-
section 6we present ourresults and,finally,in section 7 we ure1.Forafilamentrandomlyoriented ω (θ,φ)inspace,
summarize ourconclusions. let θ be the angle between the filament a≡nd the los. The
Comptonization parameter induced in the radiation cross-
ing thisfilament is
n T d
2 ESTIMATE OF THE MEAN y=y0 ceoseθ , (3)
COMPTONIZATION PARAMETER DUE TO
where d/cosθ is the projected thickness of the filament.
THE WHIM
The Comptonization parameter averaged over all possible
The temperature anisotropies due to the inverse Compton azimuths φ is
scattering of CMB photons by the free electrons have only
n T d d
beenmeasuredforclustersofgalaxies.Therearetwocontri- y =y e e . (4)
h iφ 0 cosθ 2πL′
butions:thethermalcomponent(TSZ)duetothemotionof
the electrons in the potential wells and the kinematic com- Ifthesizeoffilamentsatredshift z issmallcomparedtoits
ponent (KSZ) due to the motion of the cluster as a whole. angular diameter distance, L DA(z), then the projected
The induced temperatureanisotropies are comovinglength L′ can bewr≪itten as L′ =Lcosθ αD(z)
∼
where D(z) is the co-moving distance and α is the angle
δT δT v
=G(ν)y, = τ cl, (1) subtendedby thefilament on the sky.Then,
T0 TSZ T0 KSZ − c
(cid:16) (cid:17) (cid:16) (cid:17) n T Ld2
whereG(x)=xcoth(x/2)−4,x=hν/kBT istheCMBfre- hyiφ=y02πeα2eD2(z). (5)
quencyindimensionlessunits,v isthevelocityoftheclus-
cl
terprojected along thelineof sight (los)and cthespeed of The probability that a given los crosses the filament is
light.TheComptonizationparameterandtheopticaldepth L/(πD(z)); if d L, then the average over all possible
≪
τ are defined as valuesof α is
n T L
y=y0 Tenedl, τ =σT nedl, (2) hyi=y0 2eπ2eD(z)d. (6)
Z Z
The WHIM and SZ effect 3
In their analysis of the properties of filaments, consider corrections due to the non-linear evolution of the
Klar & Mu¨cket(2012)obtainedd=ǫL,withǫ [0.01 0.1]. matter density field on scales about and below 8h−1Mpc.
∈ −
The mean distortion along the los can be obtained by In the linear regime, baryons follow the Dark Matter
adding the effect of all filaments of different masses at dif- (DM) distribution, but this is not longer true once struc-
ferent redshifts: turesbecomenon-linear.Inregionswherelarge-scalepertur-
1 dn ǫL2(M)dV(z) bations develop shocks, Klar & Mu¨cket (2010) showed that
hyi= 2π dMf dMy0neTe D(z) dz dz, (7) thegasperturbationsonsmallscalesweresuppressedbythe
ZzZM enhanced pressure and would be nearly erased. Then, the
where nf is thenumberof filaments perunit of volume. To distribution of the baryon WHIM filaments that are form-
proceed further we assume that filaments connect clusters ingandevolvingwithinthelarge-scaledensityperturbations
and groups of galaxies. Then, theirlength is approximately would be different from that of the DM. The linear density
thedistancebetweenclusters.Ifnclisthenumberofclusters contrastofbaryonsintheIGMcanbeobtainedfromthatof
ofagivenmassperunitofvolume,thenthemeanseparation theDMbysmoothingoverscalesbelowL ,thelargestscale
0
ofthefilamentswithinavolumeV isl (V/ncl)1/3 andits erased by shock-heating. According to Fanget al. (1993)
mean length L n−1/3. Thus, the me∼an Comptonization
parameter along∼thecllos is δB(k,z)= δ1D+M(Lk2,kz2). (13)
0
2 dn D(z) dD(z)
hyi= π ZzZM dMf dMy0neTeǫncl(M)−2/3 dz dz, (8) pTohseincgomthoavtinitgisletnhgethsmsaclalleestLp0oscsainblebelendgetthersmcainleedatbwyhiimch-
thelinear peculiar velocity v equals thesound speed v (z)
with p s
of the baryon fluid. The sound speed is determined by the
dD(z)=cH0−1[ΩΛ+Ωm(1+z)3+Ωk(1+z)2]1/2dz, (9) mean temperature TIGM(z) of the IGM at any given z:
and Ωk =(1 Ωm ΩΛ). Equation (8) can be used to ob- vs = 2kBTIGM(z)/mp. The condition vp >vs gives
− −
ǫteal=einct0ra.o0nn5osarodnnedrctohofmempWaugHtneIiMttuhdefienlaoumfmtebhneetrsd.oifsIftfiowlratemiotenanktinsedaTucecceo≈drdbi1ny0g7tKhtoe, L0(z)p= (ΩΛ+2Ωπm(1(1++z)z2)3v)s1H/20−D1+(z)δ0 . (14)
Sheth & Tormen (2002) then y = 10−6, compatible with In eq. (14), the sound speed is determined by the IGM
c
theFIRASupperlimitofy¯c =1.5×10−5.TheCMBdistor- temperature TIGM at mean density. TIGM ≈ 104K and it
tion duetoa network of filamentsof ionized gas iscompat- is mainly determined by the evolution of the UV back-
ible with observations. The integration is performed from ground. For that temperature, at redshift z = 0 we have
Mi =1013M⊙ toMf =1015M⊙.Ourresultsareinsensitive L0 ≈1.7h−1 Mpc.Thisvalueofcut-offscaleagreeswiththe
totheexact valueofM whose numberdensityisexponen- results of Klar & Mu¨cket (2012) on the formation and evo-
f
tially suppressed but depend on M which we took to be lution of WHIM filaments. Tittley & Meiksin (2007) have
i
themass of rich groups of galaxies, limit where thenumber studied different ionization models and discussed the pos-
of objects is not longer well described by Sheth & Tormen sible range of variation of the mean IGM temperature. In
(2002).Usingthatformalism Shimon et al.(2012)havecal- our calculations, we assume that TIGM varies as given in
culated the radiation power spectrum. In the next section Theunset al. (2002). Below we shall see that at redshifts
wewillgiveamorerefinedpredictionbasedonalog-normal z>3theWHIMdoesnotgenerate significant anisotropies.
model of thebaryon distribution. At redshifts z 6 3, the mean IGM temperature variation
with redshift is small and can be roughly approximated as
log (T /103K) (A+0.1(1+z)),whereAvariesinthe
10 IGM ≈
rangeA=0.5 0.9, correspondingtoT =103.6 104K.
3 THE LOG-NORMAL BARYON DENSITY − IGM −
TheFouriertransformofeq.(13)givesthedensitycon-
DISTRIBUTION MODEL
trast in real co-moving space, δ(x,z). If the density field
In perturbation theory, thedensity contrast is defined as grows non-linearly butthevelocity field still remains in the
linear regime, then the initial Gaussian density field is de-
ρ(x) ρ
δ(x) −h i, (10) scribedbyalog-normaldensitydistribution(Coles & Jones
≡ ρ
h i 1991). This log-normal model has been shown to de-
where ρ(x) is the density at any given point and ρ the scribe the matter distribution in hydrodynamical simula-
h i
mean density at any given redshift. In Fourier space and in tions (Bi & Davidsen 1997) and to reproduce the observa-
thelinear regime, all modes evolveat thesame rate and tionsoftheIGM(Choudhuryet al.2001).Foralog-normal
density field,the baryon numberdensityis given by
δ(k,z)=D (z)δ(k,0), (11)
+
where D (z) = D (z,Ω ,Ω ) is the linear growth factor nB(x,z)=n0(z)e−∆2B(z)/2eδB(x,z) (15)
+ + Λ m
GnoarumssailaiznedrantodoDm+fi(e0l)d,=th1en. Itfhethlieneianritlyiaelxdtreanpsiotlyatfiedeldpoiwsear iwtshevraeriδaBncies,twhiethlinear baryon density field and ∆2B = hδB2i
spectrum P(k) is definedby
P (k) d3k
δ(k,0)δ(k′,0) =(2π)3P(k)δ(k k′). (12) ∆2B(z)=hδB2(x,z)i=D+2(z) [1+DMk2L2]2(2π)3 (16)
h i − Z 0
Ineq.12thepowerspectrumP(k)isthatofthebackground If δ is Gaussian distributed, then the spatial mean
B
ΛCDM model, computed using linear theory. We took the n (x,z) equals the cosmic baryonic background density
B
h i
spectral index at large scales to be n = 1. We did not n (z). Hereafter we shall denote by ξ =n (x,z)/n (z) the
S 0 B 0
4 I. F. Suarez-Vel´asquez, J. P. Mu¨cket, and F. Atrio-Barandela
non-linear baryon density in unitsof thebaryon mean den-
sity. 8
In Atrio-Barandela & Mu¨cket (2006),
Atrio-Barandela et al. (2008) and G´enova-Santos et al. 7
(2009) we assumed that the gas in the WHIM follows
6
a polytropic equation of state. High resolution hydro- K ]
simulations have shown that at the physical conditions T / 5
o1f05Kthe<WTHeI6M,10t7hKe sahnodckdehnesiattyedcongtarsasatst 1te6mpξer6atu1r0e0s log [ 4
is not polytropic. Kang et al. (2005) and Cen & Ostriker T
1
(2006) provide phase diagrams of electron temperature Te 3 T2
T
versus baryon density n in the range of interest. Since 3
B T
4
the IGM is highly ionized due photo-ionization by the UV 2
background, then n n . This is even more accurate for 0.1 1 10 100
e B
the WHIM as collisio≈nal ionization at high temperatures ξ
also contributes to maintain n = n . We have used these
e B
results to construct fits to the baryon temperature- density Figure 2. Various fit functions Te = Te(ne) to phase diagrams
distribution, Te =Te(ne).We used thefitting formula derived from hydro-numerical simulations. Curves T3,T4 corre-
spond to the results of Kang et al. (2004) with α = 1.5,4, re-
log10 T10e(8ξK) =−log (4+2ξα+1/ξ). (17) rseppercetsiveenltys;tTh2ecpoorwreesrploanwdsTt∝o nC3e/n2.& Ostriker (2006); finally, T1
(cid:18) (cid:19) 10
This expression fits the results of Kanget al. (2005) when
α=1 4. where the electron and temperature distribution are given
Th−e phase space derived from simulations are not sim- byeqs. (15) and (17), respectively.
plelinearrelations ofthetypeT =T (n );phasediagrams The average in eq. (18) must belimited to those scales
e e e
have a large scatter. This theoretical uncertainty is impor- (ξ1,ξ2) where the non-linear evolution is well described by
tant since the Comptonization parameter and temperature thelog-normalprobabilitydensityfunction.Thechoiceofξ1
anisotropies depend on how steeply the temperature rises issomewhatarbitrary.Sinceonlygaswithdensitycontrasts
withincreasingdensitycontrast.Wemodelthisuncertainty ξ > 2 will undergo shock-heating, this could be taken as a
by taking α = 1 4 in eq. (17), a range wide enough lowerlimit.Asacriteria,wetakeξ1=ξmedian+σ,whereσ=
to reproduce suffici−ently well the phase diagrams and the var(ξ)(var(ξ)=exp(∆2B)−1)isthestandarddeviationof
variation therein. We have also considered the results of thelog-normaldistribution.Toincludeonlythescalesthat
p
Cen & Ostriker(2006);aftercorrectingapossibletypotheir undergo shocks we also require that ξ1 >2 at all redshifts.
Texhperepshsaiosnedrieaagdrsamlosgo1b0t(aTien/e1d08frKo)m=cos−m2o.l5o/gilcoagl10si(m4u+laξt)io0.n9s. dAespuepnpdeerntlimonittwheistvoaolkueξ.2 6100,buttheresultswereweakly
agreequalitativelywiththeresultsofKlar & Mu¨cket(2012) Even though this criteria is somewhat arbitrary, the
on the effect of shocks on WHIM filaments. In particular, mass fraction on theinterval(ξ1,ξ2) is veryclose tothat of
as time evolves shocks propagate to lower density regions, the undetected baryons: 40-50%. At any redshift, the mass
steepening the T n relation as shown in the phase dia- fraction ofthegas containedin filamentswith overdensities
grams of numericea−l simeulations. intherange[ξ1,ξ2]dependsontheIGMtemperature.Ifξ¯is
Forillustration,inFig.2weplotaseriesofdifferentfits. themeanofthedistributionthenM(ξ1,ξ2)=ξ¯−1 ξξ12ξF(ξ)
The lines with labels T3 and T4 correspond to Kang et al. is the fraction of mass in the integration range. At z = 0,
R
(2005) model with α = 1.5 and 4, respectively, T2 corre- the fraction of baryons in the WHIM is 48% for A=0.9
≈
spondstoCen & Ostriker(2006)withthecorrectedformula and 43% for A = 0.5. One would expect the mass frac-
≈
given above. For comparison, T1 corresponds to a simple tion to be largest for TIGM = 103.6K, since smaller scales
powerlawwithTe ∝n3e/2.FromFig.2,wecanimmediately would survive shock heating than for TIGM = 104K. The
conclude that since filaments of any density in the T3, T4 difference comes from the lower limit ξ1. For A = 0.5, the
models are hotter than in the T1,T2 models, temperature rms of the log-normal distribution σ is larger and so is ξ1;
anisotropies and distortions would be larger in the former the integration is carried over a smaller range of densities
than in thelatter cases. and low density regions that are included for A = 0.9 are
excluded for A = 0.5. However, even if in this aspect the
modelisnot fully satisfactory, thelow denseregions donot
contribute significantly to spectral distortions or tempera-
4 THE COMPTONIZATION PARAMETER
ture anisotropies and the exact value of the lower limit ξ ,
1
Thelog-normalformalismintroducedintheprevioussection while important for the baryon fraction on the WHIM, has
permits to compute the statistical properties of the TSZ little effect on thefinal results.
generated by the baryon population of WHIM filaments.
The average Comptonization parameter y along a los is
av
(Atrio-Barandela & Mu¨cket 2006) 5 THE ANGULAR POWER SPECTRUM
The correlation function of the CMB temperature
y =y n (z,x)T (z,x) dl, (18)
av 0 h e e i anisotropies due to the WHIM TSZ contribution along two
Z
The WHIM and SZ effect 5
different los l and l′ separated by an angle θ, is given by
(Atrio-Barandela & Mu¨cket 2006) 10
′
C(θ)=y2 zf zf S(x,z)S(x′,z′) dl dl′dzdz′, (19)
0 h idzdz′
Z0 Z0
1
whereS(x,z)=n (x,z)T(n (x,z)).The densityaverage of
e e 6
0
eq.(19)iscarriedoutoverthesamedensityintervalthanin 1
*
eq. (18). yc
Sincetheequationofstateofthegasisnotlongerpoly- 0.1
tropic, the correlation of the electron density at different
positions can not be factored out. The average in eq. (19)
differsfrom ourprevioustreatmentsincenowitneedstobe
computedusingthebi-variatelog-normalPDF.Remember-
0.01 0.1 1
ing that ξ =n (x,z)/n (z), then
B 0 redshift z
S(x,z)S(x′,z′) = ξT(ξ)ξ′T(ξ′)F(ξ,ξ′)dξdξ′, (20)
h i Figure 3.Comptonization parameter 106y asafunctionofred-
Z Z shift.From top to bottom solidand dashed lines corresponds to
where thebi-variate log-normal PDF F(ξ,ξ′) is given by model T4 and A = 0.5,0.9, dotted and dot-dashed lines to T3
andA=0.5,0.9,respectively.Finallythedoubledot-dashedline
F(ξ,ξ′)= 2πξξ′∆B1∆′B√1−rc2 exp − 2(1−1rc2)× correspondstoT2 withA=0.5.
h (21)
(log∆ξ2B−µ)2 −2rc(logξ−∆µB)(∆log′Bξ′−µ′) + (logξ∆′′B−2µ′)2!#. characterized byaparameter A 0.5 0.9 (seesec. 3) and
(cid:16) ≈ −
for the latter we shall consider three different relations: A
Itnhethmiseaenxporfestshieonl,og∆-nBorims gailvPenDbFyaenqd. (r1c6i)s, tµhe=c−or∆re2Bla/t2ioins (p2o0ly06tr)ompiocdeeqluaantdioKnaonfgsteattaelT(2e0∝04n)3em/2o,dtehlegCiveenn&byOeqst.r(i1k8e)r
coefficient rc =hlog(ξ)log(ξ′)i/(∆B∆′B)that inthispartic- with α=1.5, 4, denoted by byT1, T2, T3, T4,respectively.
ular case is given by
D (z)D (z′) P (k)j (kx x′)k2dk
rc = 2+π2∆ +∆′ [1+DLM (z′)20k2|][1−+L|(z)2k2]. (22) 6.1 The mean Comptonization parameter
B B Z 0 0
In this expression, j denotes the spherical Bessel function In Fig. 3 we plot the contribution to the average Comp-
0
of zero order and x x′ the proper distance between two tonizationparameteryav fromz=0uptothegivenredshift
patches at positio|ns−x(z)| and x(z′) separated an angular for different models. In all cases, most of the contribution
distance θ. In theflat-sky approximation comes from z 6 1. In retrospect, this justifies restricting
in sec. 3 the IGM temperature parameter to the interval
x x′ l⊥(θ,z)2+[r(z) r(z′)]2, (23) A=0.5 0.9, valid for the interval z 63, since baryons at
| − |≈ − −
higher redshifts do not contributeto the TSZ effect.
withl⊥(θ,zp)thetransversedistanceoftwopointslocatedat
In the figure, the amplitude of the CMB distortion de-
thesameredshift.Therefore,theangulardependenceofthe
pends on the equation of state and the temperature of the
correlation function in eq.(19)entersonly througheq.(23)
IGM. When T = 103.6K smaller (and therefore more)
in eq.(22). IGM
scales contribute to the shock-heated WHIM and L is
The power spectrum can be obtained by the Fourier 0
smaller.Thosesmallscalescontributetothevarianceofthe
transform of thecorrelation function (eq.19)
baryon field at all redshifts (see eq. 16). These regions are
+1 of high density and at high temperature and the total ef-
C =2π C(θ)P (cos(θ))d(cos(θ)) (24)
ℓ ℓ fect increases, compensating for the baryon fraction being
Z−1 smaller (see section 4).
where P denotesthe Legendre polynomial of multipoles ℓ.
ℓ
6.2 The correlation function and the power
6 RESULTS spectrum
To compute the temperature anisotropies generated by the InFig.4weshowthepowerspectrumoftheradiation tem-
WHIM we take as a background cosmological model the perature anisotropies generated by the WHIM for different
concordanceΛCDMmodelwith densitiesdarkenergy,dark temperature models, obtained by numerical integration of
matter and baryon fractions Ω = 0.75, Ω = 0.25, Ω = eqs.(19-24).LinesfollowthesameconventionthaninFig.2.
Λ m b
0.043 and Hubble constant h = 0.73 and the amplitude of InFig.5weshowthedependencewithequationofstateand
the matter density perturbations at 8h−1Mpc, σ = 0.8. T .Theoverallshapeofthepowerspectraareverysim-
8 IGM
For simplicity, we fix the frequency dependence of the TSZ ple:thefunctionalformhasasinglemaximum.Fortherange
effecttounity,G(ν)=1.Ourresultsdependalsoonphysical of variation of the model parameters, the maximum occurs
parameterssuchasL andtheshapeoftheequationofstate. in the range ℓ = 200 500 and the maximum amplitude
0
The former is determined by the mean IGM temperature, is ℓ(ℓ+1)C /2π 0.07− 70(µK)2. The different equations
ℓ
≃ −
6 I. F. Suarez-Vel´asquez, J. P. Mu¨cket, and F. Atrio-Barandela
2
102 TT21 00..21<<zz<<00..32
T3 0.3<z<0.4
2K] 101 T4 2K] 1.5 00..45<<zz<<00..56
µ µ
πC /2 [l 100 πC/2 [l 1
+1) 10-1 +1)
l(l l(l 0.5
10-2
10-3
10 100 1000 1 10 100 1000
l l
Figure 4. Power spectrum of the radiation temperature Figure 6. Contribution to the power spectrum of different red-
anisotropies generated by the WHIM. Lines correspond to the shiftintervals,fortheT4 modelwithA=0.5.
modelsgiveninFig.2,withA=0.5.
100 ilarly to what occurs with the Comptonization parameter,
M2 the dominant contribution comes 0.3 < z < 0.5, and the
M1
contributionforstructureswithz>1isnegligible,inagree-
ment with the results of numerical simulations reported by
2]
K 10 Smith et al. (2011). Again, this justifies the use of a simple
µ
π [ parametrization of the IGM temperature that is only valid
2
C /l for redshifts z63.
l(l+1) 1 tempTehraeturarediaantiiosontrpoopwieesraslpsoecdtreupmendofstohnetWheHcIoMsmionldoguicceadl
parameters of the concordance model. As the amplitude of
theTSZeffectdependsonthenumberofelectronsprojected
0.1 along the los, the amplitude of the effect scales as ΩBh.
10 100 1000 Other parameters like Ω or Ω change the induced tem-
m Λ
l peratureanisotropiesbecausetheychangetheshapematter
power spectrum in eq .(16). But the variance of the linear
Figure 5. Effect of the cut-off length in the radiation power baryon density field ∆2 is dominated by the amplitude of
B
spectra.CurvesM1,M2 correspondtoA=0.5,A=0.9,respec- the matter power spectrum at small scales, and the largest
tively.Fromtoptobottom, the equationofstate corresponds to effect is produced by variations in the amplitude at those
theT4,T3 andT2 models. scales.Then,thelargestvariationsareproducedbychanges
in σ . These are shown in Fig. 7. A linear regression fit to
8
thepowerspectraofdifferentcosmological parametersgives
of state T T give power spectra that differ by three or-
1 4 thefollowing scaling relation at themaximum
−
dersofmagnitudeinamplitude.Thiswiderangereflectsour
theoreticaluncertaintyontheequationofstateoftheshock- [(ℓ+1)ℓC ] /2π σ2.6(Ω h)2. (25)
heatedgas.Usingapolytropicequationofstategaveevena ℓ max ∝ 8 b
larger range, and a better determination of the TSZ power Thisscalingrelationsdifferfromourpreviousresultsintwo
spectrum can only come through better understanding of very important aspects: the power spectrum does not de-
thephysicsoftheWHIM,thatis,throughcosmological hy- pend so strongly on the model parameters and the TSZ
drosimulations with larger dynamical range. contribution of the WHIM is always smaller than that of
The effect of the IGM temperature changes both the clusters.However,likeinAtrio-Barandela&Mu¨cket(2006),
amplitude and the location of the maximum. This was to the bulk of the contribution occurs well within the angu-
be expected since for smaller IGM temperature the cut-off lar range measured by WMAP and Planck. This opens the
length L is smaller and the contribution of small scales possibility of looking for the WHIM contribution in the ra-
0
is more important, increasing the amplitude reducing the diationpowerspectrumasinG´enova-Santosetal(2009).A
average angular size of the anisotropies. For A = 0.5 the more suitable data set is theforthcoming Planck data. The
power spectrum reaches a maximum at ℓ 450 with an high resolution and wide frequency coverage of this instru-
amplitude of ℓ(ℓ+1)C /2π 70(µK)2≃. For A = 0.9, ment makes it feasible to trace the WHIM not only by its
ℓ,max
≈
the maximum is at ℓ 300 and its amplitude is smaller by power spectrum, but also by the changing amplitude with
∼
a factor 3. frequency. In particular, the 217GHz channel, that corre-
∼
In Fig. 6 we show the contribution to the power spec- spondstotheTSZnull,willhelptoseparatethesignalfrom
trum due to the WHIM at different redshift intervals. Sim- foreground contributions and other larger systematics.
The WHIM and SZ effect 7
ACKNOWLEDGMENTS
150 ISV thanks the DAAD for the financial support, grant
0.70 A/08/73458. FAB acknowledges financial support from the
0.80
0.90 SpanishMinisteriodeEducaci´onyCiencia(grantsFIS2009-
2] 1.00 07238 andCSD2007-00050). Healso thanksthehospitality
K 100
µ of theLeibniz-Institut fu¨r Astrophysik Potsdam.
π [
2
C/l
+1) 50 References
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