Constraining large scale HI bias using redshifted 21-cm signal from the post-reionization epoch

0.56 MB·English

by  Tapomoy Guha Sarkar

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed20January2012 (MNLATEXstylefilev2.2)
Constraining large scale HI bias using redshifted 21-cm signal from
the post-reionization epoch
2
Tapomoy Guha Sarkar1⋆, Sourav Mitra1†, Suman Majumdar2,3‡
1
0
Tirthankar Roy Choudhury1§
2
1Harish-ChandraResearchInstitute,ChhatnagRoad,Jhusi,Allahabad,211019India.
n
2DepartmentofPhysics&Meteorology,IIT,Kharagpur721302,India.
a
J 3CentreforTheoreticalStudies,IIT,Kharagpur721302,India.

9
1
20January2012

]
O
C ABSTRACT
Intheabsenceofcomplexastrophysicalprocessesthatcharacterizethereionizationera,the
.
h 21-cmemission fromneutralhydrogen(HI)in the post-reionizationepochis believedto be
p an excellent tracer of the underlyingdark matter distribution. Assuming a backgroundcos-
-
o mology,itismodelledthrough(i)abiasfunctionb(k,z),whichrelatesHItothedarkmatter
r distributionand(ii)ameanneutralfraction(x¯HI)whichsetsitsamplitude.Inthispaper,we
st investigatethenatureoflargescaleHIbias.Thepost-reionizationHIismodelledusinggrav-
a ityonlyN-Bodysimulationsanda suitableprescriptionforassigninggastothedarkmatter
[ halos. Using the simulated bias as the fiducialmodelfor HI distribution at z ≤ 4, we have

2 generated a hypotheticaldata set for the 21-cm angular power spectrum (Cℓ) using a noise
v modelbasedonparametersofanextendedversionoftheGMRT.ThebinnedCℓ isassumed
2 tobemeasuredwithSNR&4intherange400≤ℓ≤8000atafiducialredshiftz =2.5.We
5 explorethe possibility of constrainingb(k) using the PrincipalComponentAnalysis (PCA)
5 onthissimulateddata.Ouranalysisshowsthatintherange0.2 < k < 2Mpc−1,thesimu-
5 lateddatasetcannotdistinguishbetweenmodelsexhibitingdifferentkdependences,provided
9. 1 . b(k) . 2 whichsets the 2-σ limits. Thisjustifies theuse of linearbiasmodelon large
0 scales. The largely uncertain x¯HI is treated as a free parameter resulting in degradation of
1 thebiasreconstruction.Thegivensimulateddataisfoundto constrainthefiducialx¯HI with
1 an accuracy of ∼ 4% (2-σ error). The method outlined here, could be successfully imple-
v: mentedonfutureobservationaldatasetstoconstrainb(k,z)andx¯HIandtherebyenhanceour
understandingofthelowredshiftUniverse.
i
X
Keywords: cosmology:theory–large-scalestructureofUniverse-cosmology:diffusera-
r
a diation
1 INTRODUCTION constant with Ω ∼ 0.001 (Lanzetta,Wolfe&Turnshek1995;
HI
Storrie-Lombardi,McMahon&Irwin1996;
Following the epoch of reionization (z ∼ 6), the low den-
Rao&Turnshek2000;Pe´rouxetal.2003).
sity gas gets completely ionized (Becker,Fan&White2001;
Fan,Carilli&Keating2006).However,asmallfractionofneutral
The distribution and clustering properties of DLAs suggest
hydrogen(HI)survives,andisconfinedtotheover-denseregions
thattheyareassociatedwithgalaxies,whichrepresenthighlynon-
oftheIGM.Atthisredshiftsthebulkoftheneutralgasiscontained
linear matter over densities (Haehnelt,Steinmetz&Rauch2000).
incloudswithcolumndensitygreaterthan2×1020atoms/cm2.Ob-
These clumped HI regions saturate the Gunn-Peterson optical
servationsindicatethattheseregionscanbeidentifiedasDamped
depth (Gunn&Peterson1965) and hence cannot be probed
Ly-α (DLA) systems (Wolfe,Gawiser&Prochaska2005), which
using Ly-α absorption. They are, however the dominant source
areself-shieldedfromfurtherionizationandhouse∼ 80%ofthe
for the 21-cm radiation. In the post reionization epoch, Ly-α
HIat1<z <4.Inthisredshiftrangetheneutralfractionremains
scatteringandtheWouthuysen-Fieldcoupling(Wouthuysen1952;
Purcell&Field1956; Furlanetto,Oh&Briggs2006) increases
the population of the hyperfine triplet state of HI. This makes
⋆ E-mail:[email protected]
the spin temperature T much greater than the CMB tem-
† E-mail:[email protected] s
‡ E-mail:[email protected] perature Tγ, whereby the 21-cm radiation is seen in emis-
§ E-mail:[email protected] sion (Madau,Meiksin&Rees1997; Bharadwaj&Ali2004;
2 GuhaSarkar,Mitra,Majumdar& Choudhury
Loeb&Zaldarriaga2004). The 21-cm flux from individual HI ofHIasatraceroftheunderlyingdarkmatterfield.Thiswouldbe
cloudsistooweak(< 10µJy)fordetectioninradioobservations usefulfor bothanalyticalandnumerical workinvolvingthepost-
withexistingfacilities,unlesstheeffectofgravitationallensingby reionizationHIdistribution.
intervening matterenhances theimage oftheclouds significantly The paper is organized as follows – in the next section we
(Saini,Bharadwaj&Sethi2001). The redshifted 21-cm signal discuss the simulation of HI distribution and the general features
however forms a diffuse background in all radio observations at of the bias function. Following that, we discuss the HI multi-
z < 6 (frequencies > 203 MHz). Several radio telescopes, like frequency angular power spectrum (MAPS), a statistical quanti-
thepresentlyfunctioningGMRT1,andfutureinstrumentsMWA2 fier directly measurable from radio-interferometric experiments.
andSKA3aimtodetectthisweakcosmologicalsignalsubmerged Finallyweusetheprincipalcomponentanalysistoinvestigatethe
inlargeastrophysical foregrounds (Santos,Cooray&Knox2005; possibility of constraining the bias model with simulated MAPS
McQuinnetal.2006;Ali,Bharadwaj&Chengalur2008). (Datta,Choudhury&Bharadwaj2007)data.
The study of large scale structures in redshift surveys and
numericalsimulationsrevealthatthegalaxies(forthatmatterany
non linearstructure) tracetheunderlying dark matterdistribution
2 SIMULATIONRESULTS-THEBIASMODEL
with a possible bias (Mo&White1996; Dekel&Lahav1999).
Associating the post-reionization HI with dark matter halos We have obtained the dark matter distribution using the PM N-
implies that the gas traces the underlying dark matter distribu- bodycodedevelopedbyBharadwaj&Srikant(2004),assuminga
tion with a possible bias function b(k) = [PHI(k)/P(k)]1/2, fiducial cosmological model (used throughout the paper) Ωm =
where PHI(k) and P(k) denote the power spectra of HI and 0.2726,ΩΛ =0.726,Ωb =0.0456,h=0.705,Tcmb =2.728K
dark matter density fluctuations respectively. This function is σ = 0.809, n = 0.96 (all parameters from WMAP 7 year
8 s
believed to quantify the clustering property of the neutral gas. It data (Komatsu&etal.2011; Jarosik&etal.2011)). We simulate
isbelievedthatonsmallscales(belowtheJean’slength),thebias 6083 particles in 12163 grids with grid spacing 0.1Mpc in a
is a scale dependent function. However, it is reasonably scale- 121.6Mpc3 box. The mass assigned to each dark matter parti-
independent on large scales (Fangetal.1993). Further, the bias cleis m = 2.12×108M h−1. The initial particle distribu-
part ⊙
depends on the redshift. The use of the post-reionization 21-cm tion and velocity field generated using Zel’dovich approximation
signal (Bharadwaj&Sethi2001; Bharadwaj,Nath&Sethi2001; (at z ∼ 25) are evolved only under gravity. The particle posi-
Wyithe&Loeb2007;Loeb&Wyithe2008;Wyithe&Loeb2008; tion and velocities are then obtained as output at different red-
Visbal,Loeb&Wyithe2009) as a tracer of dark matter shifts 1.5 ≤ z ≤ 4 at intervals of δz = 0.5. Wehave used the
opens up new avenues towards various cosmological in- Friends-of-Friends algorithm (Davisetal.1985) to identify dark
vestigations (Wyithe,Loeb&Geil2007; Changetal.2008; matter over-densities as halos, taking linking length b = 0.2 (in
Bharadwaj,Sethi&Saini2009; Maoetal.2008) and cross- unitsofmeaninter-particledistance).Thisgivesareasonablygood
correlation studies (GuhaSarkar,Datta&Bharadwaj2009; matchwiththetheoreticalhalomassfunction(Jenkinsetal.2001;
GuhaSarkar2010; GuhaSarkaretal.2011). The underlying bias Sheth&Tormen2002) for masses as small as = 10m . The
part
model is crucial while forecasting or interpreting some of these halomassfunctionobtainedfromsimulationisfoundtobeinex-
results. cellentagreementwiththeSheth-Tormenmassfunctioninthemass
In this paper we have investigated the nature of HI bias in range109 ≤M ≤1013h−1M .
⊙
the post-reionization epoch. The HI fluctuations are simulated at WefollowtheprescriptionofBagla,Khandai&Datta(2010),
redshifts z < 6 and HI bias is obtained at various redshifts to populate the halos with neutral hydrogen and thereby identify
from the simulated dark matter and HI power spectra. This is themasDLAs.Equation(3)ofBagla,Khandai&Datta(2010)re-
similartotheearlierworkbyBagla,Khandai&Datta(2010)and lates the virial mass of halos, M with its circular velocity v .
circ
Mar´ınetal.(2010). Thesimulatedbiasfunction isassumed tobe The neutral gas in halos can self shield itself from ionizing radi-
our fiducial model for HI distribution at low redshifts. We have ationonly if thecircular velocity isabove athreshold of v =
circ
studiedthefeasibilityofconstrainingthisfiducialmodelwithob- 30km/sec at z ∼ 3. This sets a lower cutoff for the halo mass
serveddata.Herewehavefocusedonthemultifrequencyangular M .Further,halosarepopulatedwithgasinaway,suchthatthe
min
power spectrum(MAPS)(Datta,Choudhury&Bharadwaj2007)– verymassivehalosdonotcontainanyHI.Anupper cut-offscale
measurable directly from observed radio data and dependent to halo mass M is chosen using v = 200km/sec, above
max circ
on the bias model. Assuming a standard cosmological model which we do not assign any HI to halos. This is consistent with
and a known dark matter power spectrum we have used the theobservationthatverymassivehalosdonotcontainanygasin
Principal Component Analysis (PCA) on simulated MAPS data neutralform(Pontzenetal.2008).Thetotalneutralgasisthendis-
for a hypothetical radio-interferometric experiment to put con- tributed such that the mass of the gas assigned is proportional to
straints on the bias model. The method is similar to the themassofthehalobetweenthesetwocut-offlimits.Wenotethat
one used for power spectrum estimation using the CMB there is nothing canonical about this scheme. However, with the
data (Efstathiou&Bond1999; Hu&Holder2003; Leach2006) basicphysicalpictureinthebackgroundthisisthesimplestmodel.
and constraining reionization (Mitra,Choudhury&Ferrara2011; ResultsobtainedusingalternativeHIassignment schemesarenot
Mitra,Choudhury&Ferrara2012). Stringent constraints on the expectedtobedrasticallydifferent(Bagla,Khandai&Datta2010).
bias function withfuture data sets would be crucial inmodelling Figure 1 shows the simulated power spectra of dark matter
the distributionof neutral gas at low redshifts and justifythe use andHIdistributionatafiducialredshiftz = 2.5.Thedarkmatter
powerspectrumisseentobeconsistentwiththetransferfunction
givenbyEisenstein&Hu(1998)andthescaleinvariantprimordial
1 http://www.gmrt.ncra.tifr.res.in/ power spectrum (Harrison1970; Zeldovich1972). The HI power
2 http://www.mwatelescope.org/ spectrumhasagreateramplitudethanitsdarkmattercounterpart
3 http://www.skatelescope.org/ in the entire k-range allowed by the simulation parameters. Fig-
ConstraininglargescaleHIbiasusingredshifted21-cm signalfromthepost-reionizationepoch 3
100 25
z = 2.5 6 k=0.2Mpc−1h
20 z) 4
10 (
2
) b 2
2
π 15
2
( ) 0
/ 1 k 1 2 3 4
k) 2( z
( b 10
P
3
k
0.1
5
0.01 0
0.1 1 10 0.1 1 10
k(Mpc−1h) k(Mpc−1h)
Figure1.Thesimulatedpowerspectrafordarkmatterdistribution(solid Figure2.Thesimulatedbiasfunctionforz=1.5,2.0,2.5,3.0,3.5and4.0
line)andtheHIdensityfield(dashedline)atredshiftz=2.5. (bottomtotop)showingthescaledependence.Theinsetshowsthevariation
ofthelarge-scalelinearbiasasafunctionofredshift.
ure2 shows thebehaviour of thebias function b(k,z) .Wehave
obtainedthescaledependenceoftheHIbiasforvariousredshifts
z c3 c2 c1 c0
intherange 1.5 ≤ z ≤ 4. At theseredshifts, the biasisseento
begreaterthanunity,afeaturethatisobservedintheclusteringof 1.5 0.0029 0.0365 -0.1561 1.1402
highredshiftgalaxies(Mo&White1996;Wyithe&Brown2009). 2.0 0.0052 0.0177 0.0176 1.5837
2.5 0.0101 -0.0245 0.3951 2.1672
Onlargecosmologicalscalesthebiasremainsconstantandgrows
3.0 0.0160 -0.0884 1.0835 2.9287
monotonicallyatsmallscales,wherenon-lineareffectsareatplay.
3.5 0.0234 -0.1537 2.1854 3.8050
This is a generic feature seen at all redshifts. The k-range over
4.0 0.0248 -0.1655 3.6684 4.9061
whichthebiasfunctionremainsscaleindependent islargeratthe
lowerredshifts.Thelinearbiasmodelishenceseentoholdreason-
ablywellonlargescales.Thescaledependenceofbiasforagiven Table1.Thefitparametersforbiasfunctionoftheformb2(k)=c3k3+
redshiftisfittedusingacubicpolynomialwithparameterssumma- c2k2+c1k+c0forvariousredshifts1.5≤z≤4.0.
rized in Table 1. The inset inFigure 2 shows the redshift depen-
denceofthelinearbiaswhichindicatesamonotonicincrease.This MAPS, defined as C (∆z) = ha (z)a∗ (z + ∆z)i, where
ℓ ℓm ℓm
isalsoconsistentwiththeexpectedz-dependenceofhighredshift aℓm(z) = dΩnˆYℓ∗m(nˆ)T(nˆ,z). This measures the correlation
galaxybias.Thebehaviourofthelinearbiasforsmallk-valuesas of the spherical harmonic components of the temperature fieldat
afunctionofredshiftisnon-linearandcanbefittedbyanapproxi- two redshiftRslices separated by ∆z. In the flat-sky approxima-
matepowerlawoftheform∼z2.Thisscalingrelationshipofbias tionandincorporatingtheredshiftspacedistortioneffectwehave
with is found to be sensitive to the mass resolution of the simu- (Datta,Choudhury&Bharadwaj2007)
lation. The similardependence of HI bias withk and z hasbeen
∞
observedearlierbyBagla,Khandai&Datta(2010)withacompu- T¯2
C = dk cos(k ∆r)Ps (k) (1)
tationallyrobustTreeN-body code.Hereweshow that,thesame ℓ πr2 k k HI
Z
generic features and similar scaling relations for bias can be ob- 0
tainedbyusingasimplerandcomputationallylessexpensivePM for correlationbetween HI at comoving distances r and r+∆r,
N-bodycode.Ouraimistousethisscaleandredshiftdependence T¯ = 4mK(1+z)2 Ωbh2 H0 0.7 , k = ℓ 2+k2 and
ofbias,obtainedfromoursimulationasthefiducialmodelforthe 0.02 H(z) h r k
postreionizationHIdistribution.Weshallsubsequentlyinvestigate PHsIdenotestheredsh(cid:16)iftspac(cid:17)eHIpo(cid:0)wer(cid:1)spectrumqg(cid:0)ive(cid:1)nby
thefeasibilityof constraining thismodel using Principal Compo- k 2 2
nentAnalysis(PCA)onsimulatedMAPSdata. PHsI(k)=x¯2HIb2(k,z)D+2 "1+β(cid:18) kk(cid:19) # P(k) (2)
wherethemeanneutralfractionx¯ isassumedtohaveafiducial
HI
value 2.45×10−2, β = f/b(k,z), f = dlnD /dlna where,
3 HI21-CMANGULARPOWERSPECTRUM- +
D representsthegrowingmodeofdensityperturbations,aisthe
SIMULATEDDATA +
cosmologicalscalefactorandP(k)denotesthepresentdaymatter
Redshifted 21-cm observations have an unique advantage over powerspectrum.
other cosmological probes sinceit maps the3D density fieldand WeuseMAPSasanalternativetothemorecommonlyused
gives a tomographic image of the Universe. In this paper we 3Dpowerspectrumsinceithasafewfeaturesthatmakesitsmea-
have quantified the statistical properties of the fluctuations in the surement more convenient. Firstly we note that as a function of
redshifted 21-cm brightness temperature T(nˆ,z) on the sky is ℓ (angular scales) and ∆z (radial separations) the MAPS encap-
quantified through the multi frequency angular power spectrum sulate the entire three dimensional information regarding the HI
4 GuhaSarkar,Mitra,Majumdar& Choudhury
distribution.Inthisapproach,thefluctuationsinthetransversedi- poratethecorrelationfor∆ν < ∆ν .Weplantotakethisupina
c
rectionare Fouriertransformed, whiletheradial direction iskept futurework.
unchangedintherealfrequencyspace.Nocosmologicalinforma- Figure3showsthe3DHIpowerspectrumatthefiducialred-
tionishowever lost.Secondly,21-cmsignalisdeeplysubmerged shift z = 2.5 obtained using the dark matter power spectrum of
inastrophysicalforegrounds.Theseforegroundsareknowntohave Eisenstein&Hu(1998).WehaveusedtheWMAP7yearcosmo-
a smooth and slow variation with frequency, whereas the signal logicalmodelthroughout.Figure4showsthecorrespondingHIan-
is more localized along the frequency axis. The distinct spectral gularpowerspectrum.TheshapeofC isdictatedbytheshapeof
ℓ
(∆z)behaviourhasbeenproposedtobeanusefulmethodtosep- thematterpowerspectrum,thebiasfunction,andthebackground
arate the cosmological signal from foreground contaminants. In- cosmologicalmodel.Theamplitudeissetbyvariousquantitiesthat
factithasbeenshownthatforegroundscanbecompletelyremoved dependonthecosmologicalmodelandthegrowthoflinearpertur-
by subtracting out a suitable polynomial in ∆ν from C (∆ν) bations.Theglobal meanneutralfractionalsoappearsintheam-
ℓ
(Ghoshetal.2011).ItishenceadvantageoustouseMAPSwhich plitudeandplaysacrucial roleindeterminingthemeanlevelfor
maintainsthedifferencebetweenthefrequencyandangularinfor- 21-cmemission. Hence, for afixedcosmological model, thebias
mationinanobservation.The3Dpowerspectrumonthecontrary and the neutral fraction, solely determine the fluctuations of the
mixes up frequency and transverse information through the full post-reionizationHIdensityfield.Wehaveusedthebiasmodelob-
3D Fourier transform. Further, for a large band width radio ob- tained from numerical simulations in the last section to evaluate
servation,coveringlargeradialseparationslightconeeffectisex- theC .Weassumethatthebinnedangularpowerspectrumismea-
ℓ
pectedtoaffectthesignal.Thiscanalsobeeasilyincorporatedinto suredatsevenℓbins−thedatabeinggeneratedusingEquation1
MAPS unlike the 3D power spectrum which mixes up the infor- usingthefiducialbiasmodel.
mationfromdifferenttimeslices.Thekeyadvantage,however,in The noise estimates are presented using the formalism
usingtheangularpowerspectrumisthatitcanbeobtaineddirectly used by Maoetal.(2008) for the 3D power spectrum and
from radio data. The quantity of interest in radio-interferometric Bharadwaj&Ali(2005) and Bagla,Khandai&Datta(2010) for
experimentsisthecomplexVisibilityV(U,ν)measuredforapair theangularpowerspectrum.Wehaveusedhypotheticaltelescope
of antennas separated by a distance d as a function of baseline parameters for these estimates. We consider radio telescope with
U = d/λ and frequency ν. The method of Visibility correla- 60GMRTlikeantennae(diameter45m)distributedrandomlyover
tiontoestimatetheangular power spectrum hasbeen wellestab- aregion1km×1km.WeassumeT ∼ 100K. Weconsider a
sys
lished(Bharadwaj&Sethi2001;Bharadwaj&Ali2005).Thisfol- aradio-observationatfrequencyν = 405MHzwithabandwidth
lows from the fact that hV(U,ν)V∗(U,ν + ∆ν)i ∝ Cℓ(∆ν). B=32MHzforanobservationtimeof1000hrs.
Here the angular multipole ℓ is identified with the baseline U as Inordertoattaindesiredsensitivitieswehaveassumedthatthe
ℓ=2πUandonehasassumedthattheantennaprimarybeamisei- dataisbinnedwherebyseveralnearbyℓ−modesarecombinedto
therde-convolvedorissufficientlypeakedsothatitmaybetreated incresasetheSNR.Furthur,intheradialdirection,thesignalisas-
as a Dirac delta function. Further the constant of proportionality sumedtodecorrelatefor∆ν >0.5MHz,sothatwehave64inde-
takescareoftheunitsanddependsonthevarioustelescopeparam- pendentmeasurementsofC forthegivenbandwidthof32MHz.
ℓ
eters. The7-ℓbinschosenhereallowsthebinnedpowerspectrumtobe
The angular power spectrum at a multipole ℓ is obtained by measured at a SNR & 4 in the entire range 400 ≤ ℓ ≤ 8000.
projectingthe3Dpowerspectrum.TheintegralinEquation1,sums Onewouldideallyexpecttomeasurethepowerspectrumatalarge
over the modes whose projection on the plane of the sky is ℓ/r. number of ℓvalues whichwould necessarilycompromise theob-
Hence,C hascontributionsfrommatterpowerspectrumonlyfor tainedsensitivities.Withthegivensetofobservationalparameters,
ℓ
k > ℓ/r.The shape of C isdictated by thematter power spec- onemay,inprinciplechooseafinerbinning. Itshallhowever de-
ℓ
trum P(k) and the bias b(k). The amplitude depends on quanti- gradetheSNRbelowthelevelofdetectability.Choosingarbitrarily
tiesdependent onthe background cosmological model as wellas fineℓ−binsandsimulataneouslymaintainingthesameSNRwould
theastrophysical propertiesoftheIGM.Weemphasize herethat, require improved observational parameters which may be unrea-
the mean neutral fraction and the HI bias are the only two non- sonableifnotimpossible.Thesamereasoningappliestonoisees-
cosmological parameters in our model for the HI distribution at timationfor the3D power spectrum where for agiven set of ob-
lowredshifts.PredictingthenatureofC inagivencosmological servational parameters, the choice of k− bins is dictated by the
ℓ
paradigmisthencruciallydependentontheunderlyingbiasmodel requirement of sensitivity. Inthefigure3,showing the3D power
andthevalueoftheneutralfraction. spectruma4−σdetectionofP (k)inthecentralbinrequiresthe
HI
The∆νdependanceoftheMAPSC (∆ν)measuresthecor- fullk−rangetobedividedinto18equallogarithmicbinsforthe
ℓ
relationbetweenthevarious2Dmodesasafunctionofradialsep- sameobservationalparameters.
aration∆r(∆ν).Thesignalisseentodecorrelateforlargeradial ThenoiseinCℓandPHI(k)isdominatedbycosmicvariance
separations, thedecorrelationbeing fasterfor largerℓvalues. For atsmallℓ/k(largescales),whereas,instrumentalnoisedominates
agivenℓ,onegetsindependent estimatesofC forradialsepara- atlargeℓ/kvalues(smallscales).Wepointoutthattheerroresti-
ℓ
tionsgreaterthanthecorrelationlength.Projectionofthe3Dpower matespredictedforahypotheticalobservationarebasedonreason-
spectrumleadstheavailabilityof fewerFouriermodes. However, abletelescopeparametersandfutureobservationsareexpectedto
for a given band width B, one may combine the signals emanat- reflectsimilarsensitivities.
ing from epechs separated by the correlation length ∆ν in the Wenoteherethatseveralcrucialobservationaldifficultieshin-
C
radial direction. Noting that the amplitude of the signal does not derC tobemeasuredatahighSNR.Separatingtheastrophysical
ℓ
change significantly over the radial separation corresponding to foregrounds,whichareseveralorderlargerinmagnitudethanthe
thebandwidth,onehas∼ B/∆ν independent measurementsof signal is a major challenge (Santos,Cooray&Knox2005;
c
C (∆z = 0). We have adopted the simplified picture where the McQuinnetal.2006; Ali,Bharadwaj&Chengalur2008;
ℓ
noise in C (∆z = 0) gets reduced owing to the combination of Ghoshetal.2010; Ghoshetal.2011). Several methods have
ℓ
theseB/∆ν realizations.Amorecompleteanalysiswouldincor- been suggested for the removal of foregrounds most of which
c
ConstraininglargescaleHIbiasusingredshifted21-cm signalfromthepost-reionizationepoch 5
104 10-5
z = 2.5 z = 2.5
⊙
103 ⊙⊙
⊙ ⊙
⊙
⊙⊙ 10-6 ⊙
⊙
102 ⊙ )
k) ⊙⊙ 2K ⊙
(PHI 101 ⊙⊙⊙⊙ C (ml ⊙
⊙⊙ 10-7 ⊙
⊙
⊙
⊙
100
10-1 10-8
10-1 100 103 104
k (Mpc-1) l
Figure3.Thetheoretical3DHIpowerspectrumPHsI(k)forz = 2.5as Figure4. Thetheoretical angular powerspectrum Cℓ forz = 2.5as a
afunctionofk,atµ = 0.5.Thepointswith2-σerror-barsrepresentthe functionofaℓ.Thepointswith2-σerror-barsrepresentthehypothetical
hypotheticalbinneddata. data.
We have chosen n = 61 and a k−range 0.13 ≤ k ≤ 5.3
uses the distinct spectral property of the 21 cm signal as against bin
Mpc−1. Our choice is dictated by the fact that for k < 0.13
thatoftheforegroundcontaminants.Themultifrequencyangular
Mpc−1, the C corresponding to the smallest ℓ is insensitive to
power spectrum (MAPS) C (∆ν) is itself useful for this pur- ℓ
ℓ b(k)andfork > 5Mpc−1 thereisnodataprobingthosescales.
pose (Ghoshetal.2010; Ghoshetal.2011). Whereas this signal
This truncation is also justified as the Fisher information matrix,
C (∆ν) decorrelates over large ∆ν, the foregrounds remain
ℓ
weshallsee,tendstozerobeyondthisk−range.
correlated − a feature that maybe used to separate the two. In
TheFishermatrixisconstructedas
our subsequent discussions we assume that the foregrounds have
beenremoved. Asmentioned earlier,theangular power spectrum 1 ∂Cth ∂Cth
F = ℓobs ℓobs , (4)
candirectlybemeasuredfromrawvisibilitydata.Onerequiresto ij σ2 ∂bfid(k )∂bfid(k )
incorporate the primary beam of the antenna in establishing this ℓXobs ℓobs i j
connection(Bharadwaj&Ali2005).Inthispaperweassumethat whereCth isthetheoretical(Eq.1)C evaluatedatℓ=ℓ using
ℓobs ℓ obs
such difficultiesareovercome and theangular power spectrum is thefiducialbiasmodelbfid(k)andσ isthecorrespondingob-
ℓobs
measuredwithsufficientlyhighSNR. servationalerror.Thedataisassumedtobesuchthatthecovariance
In the next section we use the Cℓ data generated with these matrixisdiagonalwherebyonlythevarianceσℓobs suffices.
assumptions to perform the PCA. If the 3D HI power spectrum The fiducial model for bias is, in principle, expected to be
ismeasured at some(k,µ)itwould bepossible todeterminethe closetotheunderlying “true”model. Inthisworkwehavetaken
biasdirectlyfromaknowledgeofthedarkmatterpowerspectrum. bfid(k) to be the fitted polynomial obtained in the earlier section
The bias would be measured at the k− values where the data is whichmatchesthesimulatedbiasuptoanacceptableaccuracy.
available.Theresultsforthe3Danalysisissummarizedinsection InthemodelforHIdistributionatlowredshifts,themeanneu-
5. tral fraction crucially sets the amplitude for the power spectrum.
However, a lack of precise knowledge about this quantity makes
theoverall amplitudeof C largelyuncertain. Toincorporate this
ℓ
4 PRINCIPALCOMPONENTANALYSIS we have treated the quantity x¯HI as an additional free parameter
over which the Fisher matrixis marginalized. The corresponding
Inthissection,wediscusstheprincipalcomponentmethodtowards degradedFishermatrixisgivenby
constrainingthebiasfunctionusingC data.Weconsiderasetof
n observational data points labeledℓ by C where ℓ runs Fdeg =F−BF′−1BT (5)
obs ℓobs obs
overthedifferentℓvaluesforwhichCℓisobtained(Fig.4). whereFistheoriginalnbin×nbinFishermatrixcorrespondingto
Inour attempt toreconstruct b(k)intherange[kmin,kmax], theparametersbi,F′isa1×1Fishermatrixfortheadditionalpa-
we assume that the bias which isan unknown function of k, can rameterx¯ ,andBisan ×1-dimensionalmatrixcontainingthe
HI bin
berepresentedbyasetofnbindiscretefreeparametersbi =b(ki) cross-terms.WeshallhenceforthrefertoFdeg astheFishermatrix
wheretheentirek-rangeisbinnedsuchthatki correspondstothe andimplicitlyassumethatthemarginalizationhasbeenperformed.
ithbinofwidthgivenby TheFishermatrixobtainedusingEq.4andEq.5isillustrated
lnk −lnk inFigure5asashadedplotinthek−kplane.Thematrixshows
∆lnki= mnax −1 min (3) a band diagonal structure with most of the information accumu-
bin
6 GuhaSarkar,Mitra,Majumdar& Choudhury
Figure5.ThedegradedFishermatrixFdeginthek−kplane.
ij
latedindiscreteregionsespeciallycorrespondingtothek−modes structionwouldbepoorforwidediscrepanciesbetweenthetwo.In
forwhichthedataisavailable.Intheregionk > 2andk < 0.2 ouranalysis,thesimulatedbiasservesastheinput.Intheabsence
Mpc−1,thevalueofF isrelativelysmall,implyingthatonecan- ofmanyalternativemodelsforlargescaleHIbias,thisservesasa
ij
notconstrainb(k)inthosek−binsfromthedatasetwehavecon- reasonablefiducialmodel.
sideredinthiswork. Weassumethatonecanthenreconstructthefunctionδb us-
i
A suitable choice of basis ensures that the parameters are ing only the first M ≤ n modes (see Eq. 6). Considering all
bin
not correlated. This amounts to writing the Fisher matrix in its then modesensuresthatnoinformationisthrownaway.How-
bin
eigen basis. Once the Fisher matrix is constructed, we determine everthisisachievedatthecostthaterrorsintherecoveredquan-
itseigenvaluesandcorrespondingeigenvectors.Theorthonormal- tities would be very large owing to the presence of the negligi-
ityandcompletenessoftheeigenfunctions,allowsustoexpandthe bly small eigenvalues. On the contrary, lowering the number of
deviationofb(ki)fromitsfiducialmodel,δbi =b(ki)−bfid(ki), modes canreduce theerror but may introduce largebiasesinthe
as recovered quantities. An important step in this analysis is there-
nbin fore,todecideonthenumberofmodesM tobeused.Inorderto
δbi= mpSp(ki) (6) test this we consider a constant bias model to represent the true
p=1 model as against the fiducial model. For a given data, figure 8
X
whereS (k )aretheprincipalcomponentsofb(k )andm arethe shows how the true model is reconstructed through the inclusion
p i i p
suitableexpansioncoefficients.Theadvantageisthat,unlikeb(k ), of more and more PCAmodes. Thereconstruction isdirectlyre-
i
thecoefficientsm areuncorrelated. lated to the quality of the data. In the k-range where data is not
p
Figure6showstheinverseofthelargesteigenvalues.Beyond available,thereconstructionispoorandthefiducialmodelisfol-
thefirstsix,alltheeigenvaluesareseentobenegligiblysmall.It lowed.Thereconstruction isalsopoor for largedeparturesof the
isknownthatthelargesteigenvaluecorrespondstominimumvari- truemodelfromthefiducialmodel.Weseethataresonablerecon-
ance set by the Cramer-Rao bound and vice versa. This implies structionisobtainedusingthefirst5modesfork < 1wherethe
that theerrorsinb(k) would increasedrasticallyif modes i > 6 dataisavailable. InordertofixthevalueofM,wehaveusedthe
areincluded.Hence,mostoftherelevantinformationisessentially Akaikeinformation criterion(Liddle2007) AIC = χ2min +2M,
containedinthefirstsixmodeswithlargereigenvalues.Thesenor- whosesmallervaluesareassumedtoimplyamorefavoredmodel.
malizedeigenmodes areshownintheFigure7.Onecanseethat, Following the strategy used by Clarkson&Zunckel(2010) and
all these modes almost tend to vanish for k > 2 and k < 0.2 Mitra,Choudhury&Ferrara(2012),wehaveuseddifferentvalues
Mpc−1,astheFishermatrixisvanishinglysmallintheseregions. ofM (2to6)forwhichtheAICisclosetoitsminimumandamal-
Thepositionsofthespikesandtroughsinthesemodesarerelated gamatedthemequallyattheMonteCarlostagewhenwecompute
tothepresenceofdatapointsandtheiramplitudesdependonthe theerrors.Inthisway,weensurethattheinherentbiaswhichexists
correspondingerror-bars(smallertheerror,largertheamplitude). inanyparticularchoiceofM isreduced.
The fiducial model adopted inour analysis maybe different We next perform the Monte-Carlo Markov Chain (MCMC)
from the true model which dictates the data. Clearly, the recon- analysisovertheparameterspaceoftheoptimumnumberofPCA
ConstraininglargescaleHIbiasusingredshifted21-cm signalfromthepost-reionizationepoch 7
105
4
104
fiducial
constant
103 M=3
3 M=5
M=7
102
1
-λi 101 ∙ ∙ )k(b 2
∙
100 ∙ ∙
∙ 1
10-1
10-2 0
2 4 6 8 0.2 1.0 5.0
mode i k(Mpc-1)
Figure6.Theinverseofeigenvalues ofthedegradedFishermatrixFdeg
ij
whichessentiallymeasuresthevarianceonthecorrespondingcoefficient.
Figure8.Thefiducialandconstant(true)biasmodelsareshown.There-
construction ofthetruemodelisshownforcaseswherenumberofPCA
modesconsideredareM =3,5,7
1
0
-1
1 4
0 mean
-1 fiducial
1 3 constant
0
-1 )
1 k( 2
b
0
-1
1 1
0
-1
1 0
0.2 1.0 5.0
0
k(Mpc-1)
-1
0.2 1.0 5.0
k (Mpc-1)
Figure9.Themarginalizedposterioridistributionofthebinnedbiasfunc-
Figure7.Thefirst6eigenmodesofthedegradedFishermatrix. tionobtainedfromtheMCMCanalysisusingtheAICcriterionuptofirst6
PCAeigenmodes.Thesolidlinesshowsthemeanvaluesofbiasparameters
whiletheshadedregionsrepresentthe2-σconfidencelimits.Inaddition,
weshowthefiducialandconstantbiasmodels.
amplitudes{m }andx¯ .Othercosmologicalparametersareheld
p HI
fixedtotheWMAP7best-fitvalues(seeSection2).Wecarryout
theanalysisbytakingM = 2toM = 6forwhichtheAICcrite- suitable thinning conditions for each chain to obtain statistically
rionissatisfied.ByequalchoiceofweightsforM andfoldingthe independentsamples.
correspondingerrorstogetherwereconstructb(k)andtherebyC
ℓ
alongwiththeireffectiveerrors.Wehavedevelopedacodebased
on the publicly available COSMOMC (Lewis&Bridle2002) for
5 RESULTSANDDISCUSSION
thispurpose.AnumberofdistinctchainsarerununtiltheGelman
andRubinconvergencestatisticssatisfiesR−1<0.001.Wehave The reconstructed bias function obtained using the analysis de-
alsousedtheconvergencediagnosticofRaftery&Lewistochoose scribed in the last section is shown in Figure 9. The solid line
8 GuhaSarkar,Mitra,Majumdar& Choudhury
Parameters 2-σerrors
10-5
x¯HI 1.06×10−3
blin 0.453
mean
⊙ fiducial Table2.The2-σerrorsforx¯HIandblin(k=0.3Mpc−1)obtainedfrom
thecurrentanalysisusingAICcriterion.
10-6 ⊙ constant
)
2K ⊙
m rors decrease drastically for k > 2 and k < 0.2 Mpc−1. This
( ⊙ isexpectedfromthenatureoftheFishermatrixwhichshowsthat
C there is practically no information in the PCA modes from these
10-7 ⊙ ⊙ k−regions.Therefore,allmodelsshowatendencytoconvergeto-
⊙ wardsthefiducialone.Thisisadirectmanifestationoflackofdata
pointsprobingthesescales.Thus,mostoftheinformationiscon-
centratedintherange0.2 < k < 2Mpc−1 withinwhichrecon-
structionofthebiasfunctionisrelevantwiththegivendataset.
10-8 Themeanreconstructedbiassimplyfollowsthefiducialmodel
103 104 for0.2<k<2Mpc−1.ThisisexpectedasthesimulatedC data
ℓ
isgeneratedusingthefiducialbiasmodelitself(Section3).Inthe
caseofanalysisusingrealobserveddatathismatchingwouldhave
statisticalsignificance,whereasherethisjustservesasaninternal
Figure10.ThereconstructedC withits2-σconfidencelimits.Thepoints consistencycheck.Theshadedregiondepictingtheerrorsaround
ℓ
witherror-barsdenotetheobservationaldata.Thesolid,short-dashedand the mean is however meaningful and tells us how well the given
long-dashed lines represent C for the mean, fiducial and constant bias datacanconstrainthebias.Theoutlineofthe2-σconfidencelim-
ℓ
modelsrespectively. itsshowsajaggedfeaturewhichisdirectlyrelatedtothepresence
ofthedatapoints.Weobservethatapartfromthefiducialmodel,a
constantbiasmodelisalsoconsistentwiththedatawithinthe2-σ
limits.Infact,otherthanimposingroughbounds 1 . b(k) . 2,
thepresentdatacanhardlyconstrainthescale-dependenceofbias.
104
ItisalsonotpossiblefortheC datawithitserror-barstostatisti-
ℓ
callydistinguishbetweenthefiducialandtheconstantbiasmodel
mean
⊙ in 0.2 < k < 2 Mpc−1. Figure 10 illustrates the recovered an-
103 ⊙⊙ fiducial gularpowerspectrumwithits95%confidencelimits.Superposed
⊙
⊙
⊙ onitaretheoriginaldatapointswitherror-bars.Wealsoshowthe
⊙
⊙ angularpowerspectrumcalculatedforthefiducialandtheconstant
102 ⊙
) ⊙ biasmodels.The2-σcontourfollowsthepatternoftheerror-bars
k ⊙
(HI ⊙⊙ otinveth(weidthaitnaiptsoienrtrso.r-Ibtairss)evtoidtehnetdthifafetrtehnetbdiaatsamisoldaerlgse.lHyeinncseentshie-
P 101 ⊙⊙ k−dependence ofbiasonthesescalesdoesnotaffecttheobserv-
⊙
⊙ ablequantityC withintheboundsofstatisticalprecision.
⊙ ℓ
While constructing the Fisher matrix, we had marginalized
100 overthelargelyunknownparameterx¯ .Treatingitasanindepen-
HI
dent free parameter, we have investigated the possibility of con-
strainingtheneutral fractionusing thesimulated C data. The2-
ℓ
10-1 σ error in this parameter obtained from our analysis is shown in
10-1 100 Table 2. We had used the fiducial value x¯ = 2.45×10−2 in
HI
k (Mpc-1) calculatingC .Itisnot surprising thatour analysis givesamean
ℓ
x¯ =2.44×10−2whichisinexcellentagreementwiththefidu-
HI
cialvalue.Itishowevermoreimportanttonotethatthegivendata
Figure11.ThereconstructedPHI(k)withits2-σconfidencelimits.The actuallyconstrainsx¯HIreasonablewellat∼4%.
pointswitherror-barsdenotetheobservationaldata.Wehavetakenµ=0.5 Notingthat,onlargescales(k.0.3Mpc−1),onecannotdis-
andz=2.5 tinguishbetweenthemean,fiducialandtheconstantbiasmodels,
weuseb (=1.496)todenotethebiasvalueonthesescales.The
lin
2-σ erroronb isevaluatedatk = 0.3Mpc−1 (showninTable
lin
represents the mean model while the shaded region corresponds 2).
to 95% confidence limits(2-σ). We have also shown the fiducial Inthek−rangeofourinterest,thefiducialmodeldoesnotre-
model (short-dashed) aswellasthepopularly usedconstant bias, flectsignificantdeparturefromtheconstantbias.Further,thecon-
b ∼ 1.5 model (long-dashed) for comparison. We find that the fidence interval obtained from the data also reflects that the ob-
fiducial model is within the 95% confidence limits for the entire served C is insensitive to the form of bias function b(k) in this
ℓ
k−range considered, while the constant bias is within the same range - provided that it is bound between approximate cut-offs
confidence limitsonly uptok ≈ 2Mpc−1.Wenotethattheer- (1.b(k).2).Moreover,thebiaslargelyaffectstheamplitudeof
ConstraininglargescaleHIbiasusingredshifted21-cm signalfromthepost-reionizationepoch 9
theangular power spectrumandhasonlyaweakcontributionto- BharadwajS.,AliS.S.,2004,MNRAS,352,142
wardsdeterminingitsshape.Ascaleindependent large-scalebias BharadwajS.,AliS.S.,2005,MNRAS,356,1519
seemstobesufficientinmodellingthedata.Themeanneutralfrac- BharadwajS.,NathB.B.,SethiS.K.,2001,JournalofAstrophysicsand
tion which globally sets the amplitude of the power spectrum is Astronomy,22,21
hence weakly degenerate with the bias. Thisis manifested in the BharadwajS.,SethiS.K.,2001,JournalofAstrophysicsandAstronomy,
22,293
factthatthoughx¯ isratherwellconstrained,thebiasreconstruc-
HI
BharadwajS.,SethiS.K.,SainiT.D.,2009,PRD,79,083538
tion which uses the degraded Fisher information (after marginal-
BharadwajS.,SrikantP.S.,2004,JournalofAstrophysicsandAstronomy,
izing over x¯ ) is only weakly constrained from the same data.
HI 25,67
Apriorindependentknowledgeaboutthepostreionizationneutral
ChangT.,PenU.,PetersonJ.B.,McDonaldP.,2008,PhysicalReviewLet-
fraction would clearly ensure a more statistically significant bias
ters,100,091303
reconstructionwithsmallererrors. ClarksonC.,ZunckelC.,2010,PhysicalReviewLetters,104,211301
Figure 11 shows the reconstructed 3D HI power spectrum. DattaK.K.,ChoudhuryT.R.,BharadwajS.,2007,MNRAS,378,119
The direct algebraic relationship between the observable PHI(k) DavisM.,EfstathiouG.,FrenkC.S.,WhiteS.D.M.,1985,ApJ,292,371
andthebiasb(k)makesthe3Danalysisrelativelystraightforward. DekelA.,LahavO.,1999,ApJ,520,24
ThisisspecificallyevidentsincetheFishermatrixelementsinthis EfstathiouG.,BondJ.R.,1999,MNRAS,304,75
case are non-zero only along the diagonal at specific k− values EisensteinD.J.,HuW.,1998,ApJ,496,605
corresponding tothedatapoints.Theentireroutinerepeatedhere FanX.,CarilliC.L.,KeatingB.,2006,AnnualReview ofAstronomy&
yieldssimilargenericfeatures.However,thekeydifferenceisthat Astrophysics,44,415
FangL.,BiH.,XiangS.,BoernerG.,1993,ApJ,413,477
we have a larger number of bins with high sensitivity leading to
FurlanettoS.R.,OhS.P.,BriggsF.H.,2006,PhysicsReport,433,181
animprovedconstrainingof bias1.3 < b(k) < 1.7intherange
GhoshA.,BharadwajS.,AliS.S.,ChengalurJ.N.,2010,ArXive-prints
0.2<k<0.7Mpc−1.
GhoshA.,Bharadwaj S.,SaiyadAliS.,Chengalur J.N.,2011,ArXive-
Intheabsenceofrealobserveddata,ourproposedmethodap-
prints
pliedonasimulateddataset,reflectsthepossibilityofconstrain-
GuhaSarkarT.,2010,JournalofCosmologyandAstro-ParticlePhysics,2,
inglarge-scaleHIbias.Themethodisexpectedtoyieldbetterre- 2
sultsifonehaspreciseknowledgeabouttheneutralcontentofthe GuhaSarkarT.,BharadwajS.,ChoudhuryT.R.,DattaK.K.,2011,MN-
IGMandtheunderlyingcosmologicalparadigm.Wenotethatthe RAS,410,1130
problemofconstraininganunknownfunctiongivenaknowndata GuhaSarkarT.,DattaK.K.,Bharadwaj S.,2009,JournalofCosmology
dealt in this work is fairly general and several alternative meth- andAstro-ParticlePhysics,8,19
odsmaybeused.Thechiefadvantageofthemethodadoptedhere, GunnJ.E.,PetersonB.A.,1965,ApJ,142,1633
apartfromitseffectivedatareduction,isitsmodelindependence. HaehneltM.G.,SteinmetzM.,RauchM.,2000,ApJ,534,594
HarrisonE.R.,1970,PRD,1,2726
Thenon-parametricnatureoftheanalysisisspeciallyusefulinthe
HuW.,HolderG.P.,2003,PRD,68,023001
absenceofanyspecificpriorinformation.Astraightforwardfitting
JarosikN.,etal.2011,TheAstrophysicalJournalSupplement,192,14
of a polynomial and estimating the coeffecients may turn out to
Jenkins A.,FrenkC.S.,WhiteS.D.M.,ColbergJ.M.,ColeS.,Evrard
beeffectivebutthereisnoapriorireasontobelievethatitwould
A.E.,CouchmanH.M.P.,YoshidaN.,2001,MNRAS,321,372
work.Itislogicallymorereasonablenottoimposeamodel(with
KomatsuE.,etal.2011,TheAstrophysicalJournalSupplement,192,18
itsparameters)uponthedata,andinstead,letthedatareconstruct
LanzettaK.M.,WolfeA.M.,TurnshekD.A.,1995,ApJ,440,435
themodel. LeachS.,2006,MNRAS,372,646
Withtheanticipationofupcomingradioobservationstowards LewisA.,BridleS.,2002,PRD,66,103511
measurementofHIpowerspectrum,ourmethodholdsthepromise LiddleA.R.,2007,MNRAS,377,L74
forpinningdownthenatureofHIbiastherebythrowingvaluable LoebA.,WyitheJ.S.B.,2008,PhysicalReviewLetters,100,161301
lightonourunderstandingoftheHIdistributioninthediffuseIGM. LoebA.,ZaldarriagaM.,2004,PhysicalReviewLetters,92,2
MadauP.,MeiksinA.,ReesM.J.,1997,ApJ,475,429
MaoY.,TegmarkM.,McQuinnM.,ZaldarriagaM.,ZahnO.,2008,PRD,
78,023529
ACKNOWLEDGEMENTS
Mar´ınF.A.,GnedinN.Y.,SeoH.-J.,VallinottoA.,2010,ApJ,718,972
McQuinnM.,ZahnO.,ZaldarriagaM.,HernquistL.,FurlanettoS.R.,2006,
Computational work for this study was carried out at the Centre
ApJ,653,815
forTheoreticalStudies,IIT,Kharagpurandtheclustercomputing
MitraS.,ChoudhuryT.R.,FerraraA.,2012,MNRAS,419,1480
facilityof Harish-Chandra Research Institute4.Suman Majumdar
MitraS.,ChoudhuryT.R.,FerraraA.,2011,MNRAS,413,1569
would like acknowledge Council of Scientific and Industrial Re-
MoH.J.,WhiteS.D.M.,1996,MNRAS,282,347
search (CSIR), India for providing financial assistance through a
Pe´rouxC.,McMahonR.G.,Storrie-LombardiL.J.,IrwinM.J.,2003,MN-
SeniorResearchFellowship(FileNo.9/81(1099)/10-EMR-I).The RAS,346,1103
authors would like to thank Prof. Somnath Bharadwaj for useful PontzenA.,GovernatoF.,PettiniM.,BoothC.M.,StinsonG.,WadsleyJ.,
discussionsandhelp. BrooksA.,QuinnT.,HaehneltM.,2008,MNRAS,390,1349
PurcellE.M.,FieldG.B.,1956,ApJ,124,542
RaoS.M.,TurnshekD.A.,2000,TheAstrophysicalJournalSupplement,
130,1
REFERENCES
SainiT.D.,BharadwajS.,SethiS.K.,2001,ApJ,557,421
AliS.S.,BharadwajS.,ChengalurJ.N.,2008,MNRAS,385,2166 SantosM.G.,CoorayA.,KnoxL.,2005,ApJ,625,575
BaglaJ.S.,KhandaiN.,DattaK.K.,2010,MNRAS,407,567 ShethR.K.,TormenG.,2002,MNRAS,329,61
BeckerR.H.,FanX.,WhiteR.L.e.a.,2001,ApJ,122,2850 Storrie-LombardiL.J.,McMahonR.G.,IrwinM.J.,1996,MNRAS,283,
L79
Visbal E., Loeb A., Wyithe S., 2009, Journal of Cosmology and Astro-
4 http://cluster.hri.res.in/index.html ParticlePhysics,10,30
10 GuhaSarkar,Mitra,Majumdar&Choudhury
WolfeA.M.,GawiserE.,ProchaskaJ.X.,2005,AnnualReviewofAstron-
omy&Astrophysics,43,861
WouthuysenS.A.,1952,ApJ,57,31
WyitheS.,BrownM.J.I.,2009,ArXive-prints
WyitheS.,LoebA.,2007,ArXive-prints
WyitheS.,LoebA.,2008,ArXive-prints
WyitheS.,LoebA.,GeilP.,2007,ArXive-prints
ZeldovichY.B.,1972,MNRAS,160,1P