Table Of ContentMon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Probing the bispectrum at high redshifts using 21 cm HI
observations
SK. Saiyad Ali1⋆, Somnath Bharadwaj1† and Sanjay K. Pandey2‡
1 Department of Physics and Meteorology & Centre for Theoretical Studies, IIT Kharagpur, Pin: 721 302 , India.
2 Deptt. of Mathematics ,L.B.S.College,Gonda 271001 , India & Visiting Associate, InterUniversityCentre for Astronomy and Astrophysics,
6
Pune , India.
0
0
2
5February2008
n
a
J
ABSTRACT
5
Observations of fluctuations in the redshifted 21 cm radiation from neutral hydrogen
2 (HI) are perceived to be an important future probe of the universe at high redshifts.
v Under the assumption that at redshifts z ≤ 6 (Post-Reionization Era) the HI traces
8 theunderlyingdarkmatterwithapossiblebias,weinvestigatethepossibilityofusing
1 observationsof redshifted 21 cm radiationto detect the bispectrum arising from non-
1 linear gravitational clustering and from non-linear bias. We find that the expected
0 signal is ∼ 0.1mJy at 325MHz (z = 3.4) for the small baselines at the GMRT, the
1 strengthbeingafewtimeslargerathigherfrequencies(610MHz,z =1.3).Further,the
5
magnitudeofthesignalfromthebispectrumispredictedtobecomparabletothatfrom
0
thepowerspectrum,allowingadetectionofbothinroughlythesameintegrationtime.
/
h The HI signal is found to be uncorrelatedbeyond frequency separations of ∼1.3MHz
p whereas the continuum sources of continuum are expected to be correlated across
- much largerfrequencies. This signature can in principle be used to distinguish the HI
o
signal from the contamination. We also consider the possibility of using observations
r
t of the bispectrum to determine the linear and quadratic bias parameters of the HI at
s
a high redshifts, this having possible implications for theories of galaxy formation.
:
v Key words: cosmology: large scale structure of universe - intergalactic medium -
Xi diffuse radiation
r
a
1 INTRODUCTION (1989), Jing, Mo & Borner (1991), Gaztanaga & Frieman
(1994), Frieman & Gaztanaga (1994), Jing & Borner
Quantifying non-Gaussian features in the large-scale mat-
(1998). The recently completed Two -degreeField Galaxies
ter distribution is an important issue in cosmology. It is
Redshift Survey(2dfGRS) and thecurrently ongoing SDSS
widely accepted that the large-scale structures originated
havebeenusedtomeasurethegalaxythree-pointcorrelation
from initiallysmalldensityfluctuationswhichwereaGaus-
function at an unprecedented level of accuracy (2dFGRS,
sianrandomfield.Non-Gaussianfeaturesariseasthefluctu-
Jing & Borner 2004; Kayoet al. 2004; Gaztanaga et al.
ations grow through the process of gravitational instability
2005;SDSS)allowingvariousissuesliketheluminosityand
(eg.Peebles1980),andquantifyingtheseatvariousstagesof
colour dependence to be addressed. There have recently
thegrowthisexpectedtoyieldasignificantamountofinfor-
been a few measurements of the bispectrum for the IRAS
mation.Thethree-pointcorrelationfunctionanditsFourier
galaxies (Feldman et al. 2001; Scoccimarro et al. 2001) and
transform, the bispectrum, are the lowest order statistics
for the 2dFGRS ( 2002Verde et al. 2002) which have (as
sensitive to non-Gaussian features.
discussed later) been used to estimate the galaxy bias.
The determination of the three-point correlation
function was pioneered by Peebles and Groth (1975) using All the measurements of the three-point correlation
the galaxies in the Lick and Zwicky angular catalogues. function or the bispectrum mentioned here are restricted
Later studies of the three point correlation function using tolowredshifts (z<1).This limitation arises becausethey
galaxy surveys include Peebles (1981), Bean et al. (1983), are based on galaxy surveys which do not extend to very
Efstathiou & Jedrzejewski (1984), Hale-Suttonet. al. highredshifts.Forexamplethe2dFGRSwhichisthelargest
completed redshift survey extends to a maximum redshift
of around z = 0.3. Further, a galaxy or quasar survey ex-
⋆ Email:[email protected]
Email:[email protected] tending out to large redshifts (z >1) would select only the
† Email:[email protected] brightest objects which would be sparse in number. These
‡
2 S. S. Ali, S. Bharadwaj and S. K. Pandey
objects would be distributed in the densest regions which bythesize,distribution andtopology oftheionized regions
are known to have a highly biased distribution and their and it is anticipated that this will not trace the underlying
distribution would not be representative of the underlying darkmatterdistribution.Itisthusexpectedthatthebispec-
matter distribution. In this paper we discuss how observa- trum of the HI during reionization will tell us more about
tionsoftheredshifted21cmemissionfromneutralhydrogen the ionization process and less about the non-Gaussianity
(HI) can be used to measure the bi-spectrum at high red- arising from the non-linear growth of density perturbation.
shifts (ie. z 1 and higher). Inthispaperweinvestigatehowthreevisibilitycorrelations
∼
The HI density in the redshift range 1.7 5.5 is can be used to study the HI bispectrum at redshifts z <6,
−
knownfromobservationsofLy-αabsorptionlinesalonglines where it is reasonable to assumed that the large scales HI
of sight to quasars (Storrie-Lombardi, McMahon & Irwin distribution traces thedark matter with a possible bias.
1996; Storrie-Lombardi, & Wolfe 2000; P´eroux et al. 2001; Wenextbrieflydiscusswhatwecanhopetolearnfrom
Prochaska, Herbert-Fort & Wolfe 2005). These observa- observations of non-Gaussianities in the HI. In perturba-
tions indicate that Ωg(z), the comoving density of neu- tion theory the lowest order at which there is a non-zero
tral gas expressed as a fraction of the current criti- three point correlation function is the second order (Fry
cal density, has a value 10−3. Recent observations 1984; Bharadwaj 1994). These predictions are expected to
∼
(Prochaska, Herbert-Fort& Wolfe 2005) show that Ωg(z) be valid on large scales which are weakly non-linear, and
drops by about 25% at z =2.2 2.5. Current observations it is in principle possible to compare these with observa-
−
ofΩg(z)areconsistentwithnoevolutionathigherredshifts tions and test the currently accepted scenario for struc-
(3 z 5.5). The bulk ( 70%) of the HI resides in high ture formation. The bispectrum has been perceived to be
colu≤mn-≤density regions ( ∼2. 1020cm−2) which are possi- a more effective statistics for quantifying non-Gaussianity
≥ ×
bly the progenitors of present day galaxies. The redshifted intheweakly non-linearregime anditstheoryhasbeende-
21 cm (1420MHz) emission from this HI is present as a veloped in Fry (1994), Hivon et al. (1995), Matarrese et al.
background radiation in radio observations at all frequen- (1997), 1998Verde et al. 1998 1998, Scoccimarro et al.
cies below 1420MHz. The fluctuations in this background (1998), Scoccimarro, Couchman & Frieman (1999) and
radiation arisefrom fluctuationsintheHIdensityandfrom Scoccimarro et al. (2001). These investigations show that
peculiar velocities at the redshift where the radiation orig- it possible to use observations of the bispectrum (or
inated (Bharadwaj, Nath and Sethi 2001, hereafter BNS). equivalently the three point correlation function) to de-
Thepossibilityofdetectingthisholdsthepotentialofbeing termine the linear and quadratic bias parameters. This
an important tool to probethelarge-scale structuresin the has been actually carried out using various galaxy sur-
redshift range z = 1 6. This is particularly significant in veys like the APM galaxies and the IRAS galaxies
−
the context of the GMRT(Swarup et al. 1991) which oper- (Feldman et al. 2001; Scoccimarro et al. 2001). In a recent
ates in several bands in the frequency range of interest. It analysis 2002Verde et al. 2002 2002 have analysed the bis-
is also interesting to note that the fluctuationsin theback- pectrum of the 2dFGRS to conclude b = 1.04 0.11
1
±
ground HI radiation is expected to exceed the fluctuations for the linear bias parameter and b = 0.054 0.08 for
2
− ±
in the CMBR by a factor of 10 100 over the frequency the quadratic bias parameter, indicating that the 2dFGRS
−
range of interest (BNS). galaxies are anunbiased tracerof theunderlyingdarkmat-
Complex visibilities at different baselines and frequen- ter. Further, the linear bias parameter was used in combi-
cies are the primary quantity measured in radio interfer- nation with the redshift distortion parameter β ≈ Ω0m.60/b1
ometric observations. Correlations between pairs of visi- measuredfromthesamesurvey(Peacock et al.2001)tode-
bilities directly probe the three dimensional power spec- termine the density of the universe Ωm0 = 0.27 0.06. A
±
trum of HI fluctuation at the redshift where the radia- new analysis of the three point correlation function of the
tion originated. This has been studied in a series of papers 2dFGRSgalaxies (Gaztanaga et al.2005)confirmsthatthe
(Bharadwaj and Sethi 2001; Bharadwaj and Pandey 2003, large-scalestructuresformedthroughgravitationalinstabil-
Bharadwaj & Srikant 2004) which develop the relation be- itystartingfromGaussianinitialconditions.Thestudycon-
tweenvisibilitycorrelationsandtheHIpowerspectrum,and cludes that the galaxies do not trace the underlying dark-
estimate the expected signal in the redshift range z < 6 matter distribution with b1 = 0.930−+0..1008 and c2 = b2/B1 =
both analytically and using N-body simulations. This for- 0.34+0.11. Observations of theHI bispectrum would allow
− −0.08
malism has later been extended(Bharadwaj and Ali 2005) ustocarryoutsuchstudiesatredshiftsz>1,providingfur-
to quantify the expected HI visibility correlation signal thertestsofthegravitationalinstabilitypicture.Further,it
over a large redshift range extending from the dark ages would be possible to determine the bias of the HI at high
(z 100) tothepresentepoch.Thepossibility ofusingvis- redshifts, this being a potential probe of galaxy formation.
∼
ibility correlations to probe the 3D HI power spectrum has The cosmological evolution of the neutral hydrogen
also been considered by Morales and Hewitt (2004) while duringtheEpoch of Reionization (z=6 20) and thepre-
−
Zaldarriaga, Furlanetto and Hernquist (2004) propose the Reionizationera(z>20),andtheexpectedredshifted21cm
use of the angular power spectrum, both in the context of signalaretopicsofveryintenseresearch(eg.Scott and Rees
detecting theEpoch of Reionization (EOR) signal. 1990; Madau, Meiksin and Rees 1997; Gnedin & Ostriker
Bharadwaj and Pandey (2005) ( hereafter BP05) have, 1997; Shaveret al. 1999; Tozzi et al. 2000; Iliev et al. 2003;
inarecentpaper,investigatedthepossibilityofusingcorre- Furlanetto,Sokasian. & Hernquist 2004; Miralda-Escude
lations between visibilities observed at three different base- 2003; Zaldarriaga, Furlanetto and Hernquist 2004;
lines and frequencies to measure the bispectrum of HI fluc- Loeb & Zaldarriaga 2004; Bharadwaj & Ali 2004;
tuations at the epoch of reionization. The HI distribution Zaroubi & Silk 2004; Bharadwaj and Ali 2005;
during the epoch of reionization will be determined more Bharadwaj and Pandey 2005; Barkana & Loeb 2005a;
Probing the bispectrum at high redshifts. 3
Barkana & Loeb 2005b; Barkana & Loeb 2005c; on the central valueν which is not shown as an explicit ar-
Bowman ,Morales & Hewitt 2005). We note that most gument.Thevalueofν willbeclearfromthecontextofthe
of these papers have no direct bearing on the cur- discussion.Further,throughoutouranalysisweassumethat
rent work which is restricted to the Post-Reionization all frequency differences are much smaller than the central
Era (z < 6) where it is reasonable to assume that frequencyie.∆ν/ν 1.Thedefinitionofthethreevisibility
≪
the HI traces the dark matter, and the process of correlation S closely follows that of S .
3 2
gravitational instability has progressed sufficiently to It has been shown that the two-visibility cor-
produce measurable departures from the Gaussian ini- relation probes the power spectrum of HI fluctu-
tial conditions. Subramanian & Padmanabhan (1993), ations (Bharadwaj and Sethi 2001, Bharadwaj and Ali
Kumar, Padmanabhan & Subramanian (1995) and 2005) and the three visibility correlation the bispectrum
Bagla, Nath & Padmanabhan (1997) have considered (Bharadwaj and Pandey2005).ConsideringS (U ,U ,∆ν)
2 1 2
the possibility of detecting HI at z < 6 focusing on the first, we note that it has a significant value only when
possibility of detectingindividualfeatures corresponding to U +U < (πθ )−1. Therefore we restrict the analysis
2 1 0
big HI clumps. Bagla and White (2003) have used N-body t|o U = U| = U which we denote as S (U,∆ν). This is
1 2 2
simulations toestimatethetwo-pointstatistics of thelow z related to−theredshift space HI power spectrum P (k) as
HI
HI signal.
We next present the structure of the paper. Section 2 S (U,∆ν)= I¯ν2θ02 ∞dk P (k)cos(k r′∆ν). (3)
reviews the formulae needed to calculate the visibility cor- 2 2rν2 Z0 k HI k ν
relation signal expected from HI at high z.InSection 3. we
wherer isthecomovingdistancetotheHIwhose1420MHz
ν
present the results and discuss their implications. radiation is observed at a frequency ν, r′ = dr /dν, k =
Finally we note that, unless stated otherwise, we use ν ν
tchosemvoallougeisca(lΩpma0r,aΩmλe0t,eΩrsb.hF2u,hrt)h=er,(w0.e3,u0s.e7,Ω0g.0=2,01.×7)1f0o−r3thaes Akkmpoi+nt(t2oπ/nroνt)eUisathnadtI¯PνH=I(k2).5in×cl1u0d2esJsyrth(cid:16)eΩe0bff.0he22c(cid:17)ts(cid:0)o0fh.7r(cid:1)edHsHh(0zi)ft.
afiducialvalueand presentresultsonlyforthisthroughout space distortions, and is therefore anisotropic along the the
thepaper. lineofsightm.WefurthernotethatS isrealandisotropic
2
ie it does not depend on the direction of the baseline U
whereby we can write S (U,∆ν).
2
Considering the three visibility correlation
2 CALCULATING THE HI VISIBILITY S (U ,U ,U ,∆ν ,∆ν ) next, this has a significant
CORRELATION SIGNAL va3lue1onl2y if3 U1+ U2 + U (πθ )−1. We therefore
1 2 3 0
| |≤
restrict ouranalysistoonlythosecombinationsofbaselines
Thequantitymeasuredinradio-interferometricobservations
isthecomplexvisibilityV(U,ν)whichrecordsonlythean- for which U1 +U2 +U3 = 0. Further S3 is real and it
dependsonly on the shape and size of thetriangle which is
gular fluctuations of the specific intensity I on the sky.
ν
completelyspecifiedbythemagnitudeofthethreebaselines
This is measured for every pair of antennas in a radio-
(U ,U ,U ). Wehave
interferometric array. For any pair of antennas at a sepa- 1 2 3
ration d, we refer to the two dimensional vector U = d/λ
as a baseline. In this paper it has been assumed that all I¯3θ2
theantennasin thearray arecoplanar, and that theanten- S3(U1,U2,U3,∆ν1,∆ν2) = 12νπr04 Z dk1kdk2kcos[(k1k∆ν1
nas all point vertically up wards in the direction m with ν
hmav·eUa=G0a.uFssuiartnhbere,atmhepiantdtievrinduAa(lθa)n=tene−nθa2s/aθ02rewaistshumθ0e≪dto1 wherek =k m+(2π+/r )kU2k∆,kν2)=rν′k] ×mB+HI((2kπ1/,rk2),Uk3,)(4)
(inradians)i.e.thebeamwidthθ oftheantennasissmall, 1 1k ν 1 2 2k ν 2
0 k = (k +k )m+(2π/r )U andB (k ,k ,k )isthe
and the part of the sky which contributes to the signal can 3 − 1k 2k ν 3 HI 1 2 3
redshift space HI bispectrum.
bewell approximated bya plane.
It is next necessary to specify the form of the redshift
Fluctuations in the high redshift HI density con-
space HI power spectrum P and bispectrum B . Fluctu-
tributes to the visibilities measured at all frequencies be- HI HI
ationsin theredshifted HIradiation arise from fluctuations
low1420MHz.FollowingBP05weconsiderthecontribution
in the HI density and from HI peculiar velocities, and it is
from the HI signal tothe correlations
necessary to include both these effects in P and B . We
HI HI
S2(U1,U2,∆ν)= V(U1,ν+∆ν)V(U2,ν) (1) assume that the HI traces the dark matter with a possible
h i
bias, and retain terms uptothequadratic order in the rela-
and
tion between thefluctuations in the HI density δ and the
HI
S (U ,U ,U ,∆ν ,∆ν ) = V(U ,ν+∆ν )V(U , fluctuationsin thedark matter density δ
3 1 2 3 1 2 1 1 2
h
ν+∆ν2)V(U3,ν)i (2) δ =b δ+ b2δ2 (5)
HI 1 2
expected between the visibilities at different baselines and
frequencies. Here the two visibility correlation function where b and b are the linear and quadratic bias param-
1 2
S (U ,U ,∆ν) denotes the correlation expected between eters respectively. A non-zero b would indicate nonlinear
2 1 2 2
the visibilities measured at two different baselines U and biasing of HI distribution with respect to underlying mass
1
U oneatthefrequencyν andtheotheratν+∆ν.Itshould distribution. It is also assumed that the peculiar velocities
2
benotedthatalthoughwehaveshownS asanexplicitfunc- aredeterminedbythedarkmatterwhichdominatesthedy-
2
tions of the frequency differences ∆ν only, it also depends namics.
4 S. S. Ali, S. Bharadwaj and S. K. Pandey
The HI power spectrum P (k) is related to the real .
HI
space dark matter power spectrum P(k) as
P (k)=b2x¯2 (1+βµ2)2P(k) (6) 1
HI 1 HI
700 MHz
µ being the cosine of the angle between k and the line of 610 MHz
sight m, and β = f(Ω )/b Ω0.6/b is the linear dis- 0.1 500 MHz
tortion parameter. Themterm1(1≈+βmµ2)21in equation (6) in- y 325 MHz .
mJ
corporates the effect of the peculiar velocities, and x¯HI is . 1/2 0.01
themeanhydrogenneutralfraction.ConvertingΩgas tothe U)]
mean neutral fraction x¯HI = ρ¯HI/ρ¯H = Ωgas/Ωb gives us S (2
x¯HI=50Ωgash2(0.02/Ωbh2) orx¯HI=2.45 10−2.Itshould [ 0.001
×
be noted that δ 1 and terms higher than δ2 have been
| |≪
dropped in thepower spectrum.
0.0001
For theHI bispectrum we assume 10 100 1000 10000
B (k ,k ,k ,z)=x¯3 B (k ,k ,k ,z) (7) ∆ν MHz
HI 1 2 3 HI s 1 2 3
where B is the redshift space bispectrum calculated by
s
1998Verde et al. 1998 1998. We have used the form of B
s
given in equation (11) of ( 1998Verde et al. 1998), the ex-
pression being quite lengthy we do not reproduce it here.
Wenotethat B (k ,k ,k ) hasa quadraticdependenceon .
s 1 2 3
thedarkmatterpowerspectrumP(k),itdependsonthebias Figure1.Thisshowstheexpectedtwovisibilitycorrelationsig-
parameters b1 and b2, and on the parameters of the back- nal[S2(U,0)]1/2 asafunctionofU forthedifferentvaluesofthe
ground cosmological model. The value of B also depends centralfrequencyshowninthefigure.
s
on theshape of the triangle formed by(k ,k ,k ).
1 2 3
Thefactthattheneutralhydrogenisindiscreteclouds
makes a contribution which we do not include here. This Figure 1 shows the [S (U,0)]1/2, the two visibility cor-
2
effect originates from the fact that the HI emission line relation signal expected when the two visibilities are at the
from individual clouds has a finite width, and the visibility samebaselineandfrequency.Thesignalisstrongestatsmall
correlation is enhanced when ∆ν is smaller than the line- baselines,anditisaround0.1mJyat325MHzie.(z=3.4).
width of the emission from the individual clouds. Another The signal falls with increasing U, and the U dependence
important effect not included here is that the fluctuations reflects the shape of the power spectrum. Further, the sig-
become significantly non-linear at low z. Both these effects nalincreaseswithincreasingcentralfrequencyordecreasing
havebeenstudiedforthepowerspectrumusingsimulations redshift,. This is a consequence of the fact that the matter
(Bharadwaj & Srikant2004). power spectrum grows with time.
Thesimpleanalytictreatmentadoptedheresufficesfor To estimate the magnitude of the expected three visi-
the purposes of this paper where the main focus is to esti- bilitycorrelationsignalS (U ,U ,U ,∆ν ,∆ν ),wefirstre-
3 1 2 3 1 2
mate the magnitude and the nature of the HI signal, and strict ouranalysis toequilateral triangles for which thesize
to investigate the feasibility of using such observations to of the baseline U completely specifies the triangle, We fur-
probe structureformation at high redshifts. therrestrict ouranalysisto∆ν =∆ν =∆ν inwhichcase
1 2
we can use the notation S (U,∆ν). The ratio [S3(U,0)]1/3
3 [S2(U,0)]1/2
(Figure2)quantifiestherelativestrengthofS asafraction
3 RESULTS AND DISCUSSION 3
ofS .Thefirstpointtonoteisthatthethreevisibilitycorre-
2
We present results for the two and three visibility correla- lationsignaliscomparableinmagnitudetothetwovisibility
tion signal expected from HI at high z. In calculating the correlationsignal.Theratioofthesignalstrengthsisoforder
expected signal we have used the GMRT parameters, but unity for the small baselines (U 1000) at 325MHz and it
≤
it is straight forward to scale the results presented here issomewhathigherathigherfrequencies.Further,S1/3 falls
3
to make the visibility correlation predictions for other ra- faster than S1/2 with increasing U. It is possible to under-
2
dio telescopes. The only telescope parameter which enters standboththesefeaturesofS intermsofthefact thatthe
3
equations (3) and (4) is θ , which is the beam size of the bispectrum is quadratic in the power spectrum whereby it
0
individualantennain thearray.Further,it shouldbenoted growsfasterwithtimeandithasasteeperk dependenceat
that θ 0.6 θ .Thevalueofθ willdependonthe large k.
0 FWHM 0
physical≈dimen×sions of the antennas and the wavelength of The bispectrum B (k ,k ,k ) depends on the shape
s 1 2 3
observation.FortheGMRT θ =1◦ at 325MHz.Wescale and size of the triangle formed by k ,k and k . This de-
0 2 2 3
this using θ λ to obtain θ corresponding to different pendencevaries with the bias parameters (B ,b ), and this
0 0 1 2
∝
observation frequencies. We note that the smallest baseline can be used to observationally determine the values of the
at the GMRT is a little smaller than U = 100 while the bias parameters. To study how the shape and bias depen-
largest is around U 10,000, and we present results for U dence manifests itself in S , we have considered bilateral
3
∼
in therange 10 10,000. It maybenoted that unlessmen- triangles for which two sides are of length U with opening
−
tionedotherwise,weusethevalues(b ,b )=(1.0,0.5)for angle 120◦ and thethird side of length √3U. Westudy the
1 2
thebias parameters. ratio [S (U,∆ν)]1/3/[S (U,∆ν)]1/3 for ∆ν = 0 where the
3 E 3 B
Probing the bispectrum at high redshifts. 5
2.2 .
2
700 MHz
1.8 610 MHz 0.14
500 MHz
1/2()]SU2 111...246 325 MHz 0.12 610 MHz 120000
/1/3[()[]SU3 00 ..168 1/2)]mJy 0 0.0.18 534000000
0.4 . ν
∆ 0.06
0.2 U,
0 10 100 1000 10000 [S (2 0.04
U
0.02
Figure2.Thisshowstheratio [S3(U,0)]1/3 asafunctionofU for
[S2(U,0)]1/2 0
thedifferentvaluesofthecentralfrequencyshowninthefigure. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
∆ν MHz
1.2
610 MHz
1 .
/13()]SU3B 0.8 b1 b2 Fnailgu[Sr2e(U4,.∆Tνh)i]s1/s2hoawssathfuenecxtipoenctoefd∆twνofovristihbeilidtyiffceorrernetlavtaiolunessigo-f
/1/3()[]S[U3E 0.6 (((( ( 1 1111....5.0500 , ,,,−,−0000..0.505..5 5 ))) )) U. showninthefigureatacentralfrequencyof610MHz
0.4 .
0.2
10 100 1000 10000 0.18
U 0.16 610 MHz 100
F∆iνgu=r0e,3w.hTehreisEshroewfesrsthteoreaqtuioila[Ste3r(aUl,t∆riνan)]g1El/e3s/a[Sn3d(UB,∆toνb)]i1Bla/t3efroarl . mJy 00..1124 534200000000 .
ttrhieantghliersd.iTsw√o3sUid.eTshoefcthenetbraillaftreerqauletnricayngisle6a1r0eMUHz(a.tT1h2e0◦re)suanltds 1/3∆ν)]U, 0 0.0.18
areforthe differentcombinations of thebias parameters (b1,b2) S (3 0.06
showninthefigure. [ 0.04
0.02
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
subscriptsEandBrefertoequilateralandbilateraltriangles ∆ν MHz
respectively.Figure3showstheresultsat610MHz(z=1.3),
the behaviour at other frequencies is similar. We find that
S doesnotshowanysignificantdependenceonthetriangle
3 .
shape at baselines smaller than U 1000. There is a shape
∼ Figure 5. This shows the expected three visibility correlation
dependenceatlargerbaselineswhereS decreases fasterfor
theequilateraltrianglesthanforthebil3ateraltriangles.Fur- signal[S3(U,∆ν)]1/3 as afunctionof ∆ν forthedifferent values
ofU showninthefigureatacentral frequencyof610MHz
ther,theratiohasadistinctdependenceonthevaluesofthe
bias parameters. The first point to note is that the ratio is .
independent of b for b =0. For other values of b we find
1 2 2
thattheU dependenceoftheratio[S (U,0)]1/3/[S (U,0)]1/3 [S (U,∆ν)]1/2 withincreasing∆ν.Itmaybenotedthatthe
3 E 3 B 2
atthelargebaselinesissensitivetobothb andb .Itshould, overall behaviorissimilar at othervaluesof thecentralfre-
1 2
inprinciple,bepossibletouseobservationsoftheshapede- quency,withthedifferencethatS andS decayfasterwith
2 3
pendenceof S to determine thebias parameters. increasing ∆ν at lower valuesof thecentral frequency.
3
Wenextconsider thecorrelation between visibilities at An important issue which we have not addressed here
different frequencies. The results are presented at fixed val- in any detail is theextraction of the HI signal from various
uesofthebaselines,andFigures4and5show [S (U,∆ν)]1/2 contaminants which are expected to dominate at low fre-
2
and [S (U,∆ν)]1/3 respectively, both at the central fre- quencies.Theissueofextractingtheredshifted21cmsignal
3
quency of 610MHz. We find that the correlations decay has received considerable attention (eg. Shaveret al. 1999;
rapidlyasthefrequencyseparation∆νisincreased.Wehave DiMatteo. et al. 2004 ; Oh et al. 2003;Morales and Hewitt
S(U,δν) = 0 at ∆ν 1.3MHz for U = 100, and there is a 2004 ; Zaldarriaga, Furlanetto and Hernquist
∼
weak anticorrelation S < 0 for larger ∆ν. The visibilities 2004 ; Cooray & Furlanetto 2004 ; Santos et al.
2
atlargerbaselinesgetdecorrelatedfasterwithincreasingU, 2005 ; Wang, Tegmark ,Santos & Knox 2005;
andthezerocrossingisat 0.7MHzatU =100.Thethree Morales ,Bowman & Hewitt 2005). The contribution
∼
visibilitycorrelation showsasimilar behavior,themaindif- from the continuum sources of contamination at two
ference being that [S (U,∆ν)]1/3 decays much faster than different frequencies is expected to be correlated even at
3
6 S. S. Ali, S. Bharadwaj and S. K. Pandey
large ∆ν,whereas theHIsignal isfound tobeuncorrelated also like to acknowledge BRNS, DAE, Govt. of India,for fi-
beyond something like 1.3MHz or even less depending on nancial support through sanction No. 2002/37/25/BRNS.
the value of U. It is then, in principle, straightforward to SKPwouldliketoacknowledgetheAssociateship Program,
fit the visibility correlations S and S at large ∆ν and IUCAA for supporting his visit to IIT, Kgp and CTS,IIT
2 3
remove any slowly varying component thereby separating Kgp for theuse of its facilities.
the contaminants from the HI signal. We also use this
opportunity to note that this is a major advantage of
using visibility correlations as compared to the angular
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