Table Of ContentMon.Not.R.Astron.Soc.000,000–000 (0000) Printed21January2009 (MNLATEXstylefilev2.2)
Gravitational Lensing Effect on the Two-Point Correlation
Function of Hot Spots in 21 cm fluctuations
9
Hiroyuki Tashiro1, Toshifumi Futamase2
0
0 1 Institut d’Astrophysique Spatiale, Universit´e Paris-Sud XI and CNR, Orsay, F-91405, (France)
2 2 Astronomical Institute, Tohoku University, Sendai 980-8578, (Japan)
n
a
J 21January2009
5
1
ABSTRACT
]
O
We investigate the weak gravitational lensing effect on the two-point correlation
C
functionoflocalmaxima(hotspots)inthecosmic21cmfluctuationmap.Theintrinsic
.
h two-point function has a pronounced depression feature around the angular scale of
p θ ∼ 40’, which depends on the observed frequency and corresponds to the scale of
-
o the acoustic oscillation of cosmic plasma before the recombination. It is found that
r the weak lensing induces a large w-dependent smoothing at that scale where w is the
t
s equation of state parameter of dark energy and thus provides a useful constraints on
a
thedarkenergypropertycombinedwiththedepressionangularscalesonthetwo-point
[
correlationfunction.
1
v Key words: cosmology:theory – large-scale structure of universe
0
3
2
2 1 INTRODUCTION
.
1 There has been a growing interest to use the cosmic 21 cm background fluctuations for a useful tool to study the so-called
0 dark age of the universe. The emission and absorption lines of the 21 cm spin-flip transition of neutral hydrogen produce
9
Cosmic Microwave Background (CMB) brightness temperature fluctuations of order ±10 mK. By scanning through redshift
0
frequencies of 21 cm line, it is possible to observe the evolution of the neutral hydrogen density with time. Therefore, the
:
v observation of the 21 cm fluctuations is expected as a promising probe of the reionisation history (Zaldarriaga et al. 2004;
i
X McQuinn et al. 2006). In order to measure the fluctuations, there are planned low-frequency radio arrays, for example,
MileuraWidefieldArray1 (MWA),LowFrequencyArray2 (LOFAR)andSquareKilometerArray3 (SKA).Moreover,the21
r
a cm cross-correlation with other observations, e.g, CMB, large scale structures and galaxies, also studied because the cross-
correlation with complementary observation gives more information than their respective auto-correlations (Alvarezet al.
2006; Tashiro et al. 2008; Adshead & Furlanetto 2008; Lidz et al. 2009).
The 21 cm observations also have a potential to reveal the Universe at high redshifts before the reionisation epoch
(Madau et al. 1997; Tozzi et al. 2000). The physics of the 21 cm fluctuations at high redshifts is understood better than at
lowerredshiftsaroundthereionisationepoch(foradetailedreview,seeLewis & Challinor2007).The21cmfluctuationsathigh
redshiftstracethematterpowerspectrumandarecalculatedinthelineartheory.Theobservationsoftheacousticoscillations
inthe21cmfluctuationsareexpectedtobeaprobeofthecompositionandgeometryoftheUniverse(Barkana & Loeb2005;
Mao & Wu 2008).
Inadditiontotheseprimordialfluctuations,thereexistsecondary21cmfluctuations.Modificationbyweakgravitational
lensing is considered as one of the main sources of secondary fluctuations (Mandel & Zaldarriaga 2006). The path of the
emitted 21 cm photon is perturbed by weak gravitational lensing of large scale structures. Gravitational lensing modifies
the primordial 21 cm fluctuations as in the context of CMB. However, in contrast to the CMB case where there is only one
sourceplaneatthelastscatteringsurface,theredshifted21cmfluctuationsprovideexcellentsourcesforgravitationallensing
because of the existences of many different source planes.
1 http://www.haystack.mit.edu/array/MWA
2 http://www.lofar.org
3 http://www.skatelescope.org
2 Tashiro, H. et al.
In this paper, we investigate the weak lensing effect on the 21cm fluctuations. In order to reconstruct matter den-
sity fluctuations along the 21 cm photon path, some authors have been studied the effect (Mandel & Zaldarriaga 2006;
Zahn & Zaldarriaga 2006;Metcalf & White2007;Lu & Pen2008).Forexample,MandelandZeldarrigastudiedtheeffecton
theangularpowerspectraaswellastheintrinsic,three-dimensionalpowerspectraofthe21cm fluctuationsduringtheeraof
reionisationandtheeffectonbothspectraislessthanapercentintheinterestingscales(Mandel & Zaldarriaga2006).Metcalf
and White studied the lensed shear map power spectrum in 21cm fluctuations and pointed out the potential of producing
high resolution, high signal-to-noise images of thecosmic mass distribution (Metcalf & White 2007).
Here we are interested in the two-point correlation function before reionisation. If adopting the Gaussian assumption
for the primordial fluctuation field, it is known that the peak statistics can provide additional information about intrinsic
distribution of hot spots that those pairs have some characteristic separation angles (Heavens& Sheth 1999). In particular
there is a pronounced depression feature in the two-point function around the angular scale of θ ∼ 40’ depending on the
observed frequency.The weak lensing then redistributes hot spots in theobserved 21 cm fluctuation maps from the intrinsic
distribution. We found that the effect has large influence in the depression feature, typically several percent contrary to the
lensing effect on thepower spectra. Moreover, theeffect does not verymuchdependon thedetails of thenonlinearstructure
formation.Thusthedetailedobservationoftheredshiftdependenceoftheweaklensingeffectaswellasoftheangularposition
in the two-point function offer a very useful mean of investigating dark energy equation of state.
This paper is organised as follows. In Sec. II, we calculate and discuss the two-point correlation function of hot spots
in the 21 cm line fluctuations. In Sec. III, we give the formalisms the gravitational lensing effect on two-point correlation
function. In Sec. IV, we present the result of lensed two-point correlation function and the effect of dark energy equation of
state.Finally,weconcludeinSec.V.Asthefiducialcosmological modelinthispaper,weassumetheΛCDMcosmology with
cosmological parameters, h=0.7 (H =h×100 km s−1 Mpc−1), Ω =0.3, Ω =0.7 and σ =0.8.
0 m Λ 8
2 TWO-POINT CORRELATION FUNCTION OF HOT SPOTS IN THE 21 CM LINE FLUCTUATIONS
In the case of the CMB temperature anisotropy, two-point correlation functions of hot spots can be calculated from the
angularpowerspectrum oftheCMB temperatureanisotropy undertheGaussian assumption (Heavens& Sheth1999).Since
itisagoodassumptionthatthe21cmfluctuationsbeforethereionisationhaveaGaussianstatistic,wecancalculatetwo-pint
correlation functionsofhotspotsinthe21cmfluctuationsfromtheirangularpowerspectruminthesamewayasinthecase
of theCMB temperature anisotropy.
2.1 Angular power spectrum of the 21 cm fluctuations
The observed 21 cm fluctuations at λ=21(1+z ) cm can bewritten as (Zaldarriaga et al. 2004)
obs
η0
T(nˆ,z )=T (z ) dηW (η −η)ψ (nˆ,η), (1)
obs 21 obs 21 obs 21
Z
0
where η is the conformal time, the subscripts obs and 0 mean the values at the redshift z and the present time, respec-
obs
tively, and W (η −η) is a response function which characterises the bandwidth of an experiment and normalised as
21 obs
∞
dxW (x) = 1. In this paper, we assume that W (η −η) is the delta function for simplicity. In Eq. (1), T (z) is a
−∞ 21 21 obs 21
Rnormalisation constant which is given by
Ω h2 0.15 1+z 1/2
T (z)≃23 mK b , (2)
21 (cid:18)0.02 (cid:19)h(cid:16)Ωmh2(cid:17)(cid:16) 10 (cid:17)i
and ψ is thedimensionless brightness temperature written as
21
T (η)
ψ (nˆ,η)≡x (nˆ,η)[1+δ (nˆ,η)] 1− cmb , (3)
21 H b (cid:20) Ts(nˆ,η)(cid:21)
whereδ isthebaryondensitycontrast,x isthefractionofneutralhydrogen,T isthespintemperatureofneutralhydrogen
b H s
and T is the CMB temperature. For simplicity, we assume that the fraction of neutral hydrogen is x = 1 because the
cmb H
redshifts we are interested in are before the reionisation epoch. We also take another assumption that the gas temperature,
T ,isheatedbystarsorQSOs,andmuchhigherthanT .Inthiscase, T iswellcoupledtoT andT ≫T .Asaresult,
g cmb s g s cmb
Eq. (3) does not have the dependence on T . Under these assumptions, the 21 cm fluctuations are determined only by the
s
baryon density contrast.
Note that the second assumption, T ≫T , might be invalid at high redshifts, if there are not sufficient heat sources to
s g
makeT enough high. In thiscase, T is set by thebalance between T and T , and theabsolute valueof theamplitudeof
g s g cmb
thebrightnesstemperaturefluctuationsdependsonT (Madau et al.1997).However,theabsolutevalueoftheamplitudedoes
s
notaffectthetwo-pointcorrelation functioninouranalysis,sincethethresholdforahotspotinthispaperisdefinedrelative
Gravitational Lensing Effect on 21 cm Fluctuations 3
to the dispersion of the fluctuations and the correlation function is determined by the gradient and the second derivative of
theangularpowerspectrumasshowninHeavens& Sheth(1999).Therefore, ourfinalconclusion isnotsignificantlychanged
in the case without thesecond assumption.
After expanding Eq. (3) in Fourier series and using Rayleigh’s formula, we can obtain spherical harmonic coefficients of
the21 cm fluctuations,
d3k
a2ℓm1(zobs)=4π(−i)ℓZ (2π)3(1+fµ2)δbkα2ℓ1(k,zobs)Yℓ∗m(k), (4)
where Y (k) is a spherical harmonic function and α21(k,z) is a transfer function for the21 cm line,
ℓm ℓ
α21(k,z)≡T (z)D(z)j [k(η −η)]. (5)
ℓ 21 ℓ 0
Here,j isthesphericalBesselfunctionandD(z)isthelineargrowthfactorofthedensityfluctuations.Weincludedthefactor
ℓ
(1+fµ2) in order to take into account of the redshift-space distortion by the bulk velocity fields, which is called “Kaiser
effect”,withµ≡kˆ·nˆ,andf ≡dlnD/dlna(Bharadwaj & Ali2004).Theangularpowerspectrumofthe21cmfluctuations
is obtained from
C (z ,ℓ)=h|a21(z )|2i. (6)
21 obs ℓm obs
2.2 Two-point correlation function
For calculating the two-point correlation function of hot spots in the 21 cm line fluctuations, we define the number density
fluctuationsof hot spots in the21 cm fluctuation map as
n (θ)−n¯
δn (θ)= pk pk, (7)
pk n¯
pk
where n and n¯ are the number density and the mean number density of hot spots above a certain threshold ν. The
pk pk
threshold ν is written as ν =∆T /σ with the21 cm fluctuations ∆T and theirdispersion,
21 21 21
1
σ ≡h|∆T |2i= dℓ2C (ℓ). (8)
21 21 (2π)2 Z 21
In what follows, we set ν =1 as thethreshold.
The two-point correlation function of hot spots is the ensembleaverage of the numberdensity fluctuations,
ξ (θ)≡hδn(θ )δn(θ )i, (9)
pk−pk 1 2
where|θ −θ |=θ.Thedetailed calculation of thecorrelation function of hot spotsfrom theangular spectrum iswritten in
1 2
Heavens& Sheth (1999). Therefore, we donot give thedetailed calculation and we only show some results here.
The left panel in Fig. 1 shows the two-point correlation functions of hot spots in the 21 cm fluctuations. In this figure,
wesetz =30.Thebaryonicoscillation bringstheoscillatory featurebetween20and50arcmininthecorrelation function.
obs
Since the 21 cm fluctuations do not have Silk damping, the correlation function does not damp below 10 arcmin unlike that
of theCMB temperature anisotropy.
Varyingtheobservational wavelength,we can obtain two-point correlation functions at differentz . Weplot thecorre-
obs
lation function for different z in the right panel of Fig. 1. When z are varied from high to low redshifts, the position of
obs obs
the oscillation shifts to large angle. This is because the apparent angular diameter of the baryonic oscillation becomes large
at low redshifts. In addition, decreasing the amplitude of the oscillation at low redshifts is also explained by the apparent
angular diameter. The increment of the apparent angular diameter makes the baryonic oscillation smooth in the angular
power spectrum of the 21 cm fluctuations. Since the two-point correlation function are related to the gradient and second
derivativeoftheangularpowerspectrum,thesmoothingofthebaryonicoscillation inlowz decreasestheamplitudeofthe
obs
two-point correlation function.
3 GRAVITATIONAL LENSING EFFECT ON TWO-POINT CORRELATION FUNCTIONS
During traveling to an observer, 21 cm photons are deflected by the gravitational potential of the density fluctuations along
thepath as CMB photons. Therefore, we measure thelensed 21 cm fluctuations. This means that thenumberdensity of hot
spots at thedirection θ on theobservation map corresponds to that at θ+δθ on thesource plane, where δθ is thedeflection
angle generated by the weak gravitational lensing effect. Therefore, the observed number density fluctuations of hot spots,
δnobs,are represented as
pk
δnobs(θ)=δn(z,θ+δθ). (10)
pk
4 Tashiro, H. et al.
0.09 0.016
0.08 0.014 zobs=40
0.07 0.012 zobs=30
z =20
ò) 0.06 ò) 0.01 obs
( 0.05 (
k k 0.008
àp 0.04 àp
k k 0.006
p 0.03 p
ø ø 0.004
0.02
0.01 0.002
0 0
-0.01 -0.002
10 20 30 40 50 60 70 80 90 100 30 40 50 60 70 80
ò(arcmin) ò(arcmin)
Figure 1. (Left) Thetwo-point correlation function of hot spots inthe 21 cm fluctuations. We setν =1, and zobs =30. (Right) The
correlationfunctionfordifferentzobs.Thedotted,solidanddashedlinesrepresentthecorrelationfunctionsforzobs=40,zobs=30and
zobs=20,respectively.
where δn(z,θ) denotethenumberdensity fluctuationsof hot spots at theredshift z.
The dispersion of δθ for theangular separation θ is written as (Seljak 1996)
σ2 (θ)≡2−1h|δθ −δθ |2i=σ2 (θ)+σ2 (θ), (11)
GL 1 2 GL,0 GL,2
where |θ −θ | = θ and h i denotes the ensemble average. In Eq. (11), σ and σ are the isotropic and anisotropic
1 2 GL,0 GL,2
contributions tothe lensing dispersion, respectively, and given by
1 dℓ
σ2 (θ) = C (ℓ)[1−J (ℓθ)],
GL,0 2π Z ℓ GL 0
1 dℓ
σ2 (θ) = C (ℓ)J (ℓθ), (12)
GL,2 2π Z ℓ GL 2
Here, C (ℓ) is the angular power spectrum of the deflection angle,
GL
ηobs ℓ
C (ℓ)=9H4Ω2 dηa−2(η)W2(η,η )P k= ,η . (13)
GL 0 mZ obs m(cid:18) χ(η) (cid:19)
η0
whereP isthematterdensitypowerspectrumandW(η,η )isrepresentedasW(η,η )=χ(η−η )/χ(η )whereχis
m obs obs obs obs
thecomoving angular diameter distance.
The effect of weak gravitational lensing on the correlation functions of hot spots was investigated by Takadaet al.
(2000)(seealsoTakada & Futamase2001).Thelensed two-pointcorrelation functionisexpressedwiththedispersionsofthe
deflection angle as
ξpobk−spk(θ) = hδnobs(θ1)δnobs(θ2)i|θ1−θ2|=θ =hδn(θ1+δθ1)δn(θ2+δθ2)i|θ1−θ2|=θ
= Z (2dπ2ℓ)2(d22πℓ)′2ei(ℓ·θ1−ℓ′·θ2)hδnℓ1nℓ2ihei(ℓ·δθ1−ℓ′·δθ2)i
∞ ℓdℓ ℓ2 ℓ2
= C (ℓ) 1− σ2 (θ) J (ℓθ)+ σ2 (θ)J (lθ) , (14)
Z 2π pk−pk (cid:20)(cid:18) 2 GL,0 (cid:19) 0 2 GL,2 2 (cid:21)
0
where, in order to obtain the finalexpression, we used the Gaussian assumption of δnℓ,
hδnℓδnℓ′i=(2π)2Cpk−pk(ℓ)δ2(ℓ−ℓ′), (15)
and thefollowing approximation,
heiℓ·(δθ1−δθ2)i|θ1−θ2|=θ ≃1− l22 σG2L,0(θ)+cos(2ϕl)σG2L,2(θ) . (16)
(cid:2) (cid:3)
The angular spectrum of the unlensed correlation function of hot spots C in Eq. (14) can be related with the unlensed
pk−pk
correlation function ξunlensed(θ) as
pk−pk
π
C (ℓ)=2π dθθξunlensed(θ)J (ℓθ). (17)
pk−pk pk−pk 0
Z
0
Gravitational Lensing Effect on 21 cm Fluctuations 5
4 WEAK LENSING AND DARK ENERGY
First,wecalculatetheweakgravitationallensingeffectontwo-point@correlationfunctionsinthefiducialcosmologicalmodel.
The left panel in Fig. 2 shows the lensed and the unlensed two point correlation functions. The effect of weak gravitational
lensing is smoothing of the oscillatory feature of two-point correlation function as in the CMB case (Takada et al. 2000;
Takada & Futamase 2001). Therefore, theeffect of gravitational lensing arises prominently around 30-50 arcmin where there
are thetop and bottom of the oscillatory features.
In this calculation, we have not taken into account the nonlinear evolution of matter density fluctuations, although
this gives the amplification of gravitational lensing on small scales in the CMB case (Seljak 1996). As shown in Fig. 2, the
scale where the gravitational lensing effect arises prominently is about more than 30 arcmin. According to Eq. (12), most
contribution of the gravitational lensing effect at each θ comes from C (ℓ) where ℓ corresponds to ℓ ∼π/θ. Therefore, the
GL
gravitational lensing effect at about 30 arcmin is sensitive to C (ℓ) at ℓ ∼ 300. The discrepancy between the linear and
GL
nonlinear density power spectra on the power spectra of the deflection angle is made on small scales ℓ>1000 (see Fig. 1 in
Mandel & Zaldarriaga 2006). Thus,neglecting thenonlinear evolution is valid in this paper.
Next,westudytheeffectofthedarkenergyequationofstatewonweakgravitationallensinginthetwo-pointcorrelation
function.Fig.3showsthelensedtwo-pointcorrelationfunctionfordifferentw.Oneoftheeffectsofwistheshiftoftheposition
of the oscillatory feature in the unlensed correlation function. Decreasing w means that the acceleration of the universe in
the dark energy dominated epoch becomes high and the distance to z increases. As a result, the oscillatory feature in the
obs
correlation function shifts to small angle scale as in the case of the baryonic oscillation in CMB or galaxy redshift surveys
(Blake & Glazebrook 2003). Gravitational lensing does not affect the position of the oscillatory feature. Therefore, even in
lensed correlation functions, the w-dependence of the position of the oscillatory feature is same as in the case of unlensed
correlation functions.
Varying the observational wavelength, we can obtain two-point correlation functions at different z . The left panel of
obs
Fig. 4 shows the evolution of the position of the oscillatory trough at the redshift z for different w. As z decreases, the
obs obs
apparent angle of the baryonic acoustic oscillation becomes large. Therefore, the position of the oscillatory trough shifts to
large scale with z decreasing.
obs
Wepresenttheevolutionofthefractionalchangebygravitationallensingatthetrough∆ξ/ξunlensed with∆ξ=(ξobs −
pk−pk pk−pk
ξunlensed), as a function of z for different w in the right panel of Fig. 4. The deflection angle depends on the distance to
pk−pk obs
the redshift z which is the redshift of a ‘source plane’, as described in Eq. (13). Decreasing z means that the distance
obs obs
becomesshortandthedeflectionanglebygravitationallensingdecreases.Therefore,theintegratedvalueofEq.(13)issmaller
for low z than for high z .
obs obs
Theamplitudeof∆ξ/ξunlensed dependsonw asshownintherightpanelofFig.4.Decreasingwwithkeepingz makes
pk−pk obs
thecomovingdistancetoz large.Asaresult,integralrangeofEq.(13)becomelargeandthegravitational lensingeffectis
obs
enhancedforsmallw.Themodificationofthegrowthrateofdensityfluctuationsbythedarkenergyequationofstateaffects
weakgravitational lensing.Howeverwefoundthatthiseffectisminorinourparameterregion (−1.2<w<−0.8),compared
with theeffect of themodification of the distance to z .
obs
Atthelast,weinvestigatewhichredshiftmakesthemostcontributiontothelensingeffect.Sincethegravitationallensing
effectmainlyarisesatabout30arcmininthe21cmfluctuationsbetweenz =20and40,thedominantcontributiontothe
obs
lensingdispersion inEq.(12) comesfrom C (ℓ)at ℓ∼300. Weplottheredshift distributionofC (ℓ)at ℓ=300 inFig.5.
GL GL
Varyingz doesnotmakethedistributionchangeradically,becausetheradialdistancestothesourceplaneateachz are
obs obs
not different strongly. For all z , the distribution has a peak around z ∼1.5. Therefore, gravitational lensing effect on the
obs
21 cm fluctuationsis sensitive to thematter density fluctuationsat this redshift
5 CONCLUSION
We have calculated two-point correlation functions of hot spots in 21 cm fluctuations and studied the weak gravitational
lensing effect on thecorrelation function.Particularly, we haveexamined itspossibility as a probeof theequation of stateof
dark energy w.
The two-point correlation function of hot spots in the 21 cm fluctuations is more smoothed than that of the CMB
temperature anisotropy. On large scales, the correlation function is very flat. However, the amplitude of the correlation
function increases toward small scales due to the absence of Silk damping in the 21 cm fluctuations. The baryonic acoustic
oscillation produces the oscillatory feature around 30-50 arcmin. By decreasing the observation redshift z , the oscillatory
obs
featureshiftstolargescales.Therefore,thepositionoftheoscillatoryfeatureinmulti-frequencyobservationsplaysaimportant
role in thedecision of thecosmological parameter as thebaryonicoscillation in theCMB temperatureanisotropies and large
scale structures.
The weak gravitational lensing effect on thetwo-point correlation function appears on theoscillatory feature. The effect
is smoothing of the feature by smearing the baryonic oscillation without shifting its position. The advantage of the 21 cm
6 Tashiro, H. et al.
0.014
0.0072 lensed
0.012 lensed non lensed
non lensed
0.01 0.0068
ò) ò)
( 0.008 (
k k 0.0064
àp 0.006 àp
k k
øp 0.004 øp 0.006
0.002
0.0056
0
-0.002 0.0052
30 40 50 60 70 80 29 30 31 32 33 34 35
ò(arcmin) ò(arcmin)
Figure 2. The two-point correlation functions of hot spots in the 21 cm fluctuations. In the left panel, the solid line represents the
lensed correlation function and the dotted line indicates the unlensed correlation function. We set zobs = 30 and σ8 = 0.8. The right
panelshowstheresultaround30arcmininthesamecase.
0.007
w=-1:2
w=-1:0
0.0065 w=-0:8
)
ò
(
k
p
à 0.006
k
p
ø
0.0055
0.005
29 30 31 32 33 34 35
ò(arcmin)
Figure3.Thelensedtwo-pointcorrelationfunctionsofhotspotsfordifferentw.Thedotted,solid,anddashedlinesrepresentthelensed
correlationfunctionsforw=−1.2,w=−1.0,andw=−0.8,respectively.Forcomparison,wealsoplotunlensedcorrelationfunctionas
thethinlines.Wesetzobs=30andσ8=0.8.
fluctuationobservationisthatwecanobtainindependentlensedmapsatdifferentredshifts.Asz decreases,thedistanceto
obs
thesourceplanebecomessmallandthedeflectionanglebygravitationallensingdecreases.Asaresult,thedifferencebetween
thelensed and the unlensed correlation functions becomes small at low z .
obs
We have studied the sensitivity of the 21 cm two point correlation function to the dark energy equation of state w.
The effects of w on the two point correlation function appear on the shift of the position of the oscillatory features and the
smoothing by gravitational lensing. Since the distance to z depends on w, decreasing w makes the position shift to small
obs
scale and thesmoothing emphasised.
37 0.054
w=-0:8 w=-1:2
36 w=-1:0 0.052 w=-1:0
) w=-1:2 w=-0:8
n 35 k 0.05
mi àp
arc 34 =øpk 0.048
( 33 ø 0.046
ò É
32 0.044
31 0.042
30 0.04
15 20 25 30 35 40 15 20 25 30 35 40
z z
obs obs
Figure 4. (Left) The evolution of the position of the oscillatory trough for different w. The dotted, solid, and dashed lines are for
w =−1.2, w =−1.0, and w =−0.8, respectively. (Right) The evolution of fractional changes of the two-point correlation function for
differentw.Thedotted, solid,anddashedlinesrepresenttheevolutions forw=−1.2,w=−1.0,andw=−0.8,respectively.
Gravitational Lensing Effect on 21 cm Fluctuations 7
0.7
`=300 zobs=40
z 0.6 zobs=30
g z =20
o 0.5 obs
l
d
= 0.4
L
G
C 0.3
g
o 0.2
l
d
0.1
0
0.1 1.0 10
z
Figure 5. The redshift distribution of CGL(ℓ) at ℓ = 300 as a function of zobs. The dotted, solid and dashed lines are for zobs = 40,
zobs=30andzobs=20,respectively.
Theevolutionofthefractionalchangebythegravitationallensingeffect,whichisobtainedfromobservationsofdifferent
redshift slices, is useful for the constraint on w. The 21 cm fluctuations with gravitational effect can be estimated from the
linear theory.Thus, it will be easy to compare these results with observational data and to obtain theconstraint on w.
We mention the comparison between the angular power spectrum and the two-point correlation function of the 21 cm
fluctuations for the detection of the gravitational lensing effect. According to Mandel & Zaldarriaga (2006), the fractional
change by gravitational lensing in the angular power spectrum is about 1 %. On the contrary, the fractional change is
enhanced to about 4 % in the correlation function. Therefore, the two-point correlation function is a better probe of the
gravitational lensing effect than the angular power spectrum.
Finally,wegivesomecommentsonobservationalaspects.Inthispaper,wefocusonweakgravitationallensingonthe21
cm fluctuations before the reionisation epoch (z > 15). Therefore, we neglect any effect of reionisation process on the 21
obs
cm fluctuations. Planned observational projects in the near future are aimed at the measurement of the 21 cm fluctuations
during reionisation (z ∼ 10 for LOFAR and z ∼ 15 for SKA). The 21 cm fluctuations from the reionisation epoch are
obs obs
studied with numerical simulations by many authors. The angular power spectrum of the21 cm fluctuations dependson the
reionisation process, (e.g. Baek et al. 2008). Therefore, in order to measure w by the weak gravitational lensing effect in the
near futureobservations, we must take intoaccount thereionisation process precisely with numerical simulations.
REFERENCES
Adshead P. J., FurlanettoS. R., 2008, MNRAS,384, 291
Alvarez M. A., Komatsu E., Dor´e O., Shapiro P. R.,2006, Astrophys.J., 647, 840
Baek S., Di Matteo P., Semelin B., Combes F., RevazY., 2008, arXiv:0808.0925
Barkana R., Loeb A., 2005, MNRAS,363, L36
Bharadwaj S., Ali S.S., 2004, MNRAS,352, 142
Blake C., Glazebrook K., 2003, Astrophys.J., 594, 665
HeavensA. F., Sheth R.K., 1999, MNRAS,310, 1062
Lewis A., Challinor A., 2007, Phys. Rev.D, 76, 083005
Lidz A., Zahn O., Furlanetto S.R., McQuinn M., Hernquist L., Zaldarriaga M., 2009, ApJ, 690, 252
Lu T., Pen U.-L.,2008, MNRAS,388, 1819
Madau P., Meiksin A., ReesM. J., 1997, Astrophys.J., 475, 429
Mandel K. S., Zaldarriaga M., 2006, Astrophys.J., 647, 719
Mao X.-C., Wu X.-P., 2008, Astrophys.J., 673, L107
McQuinn M., Zahn O., Zaldarriaga M., HernquistL., FurlanettoS. R., 2006, Astrophys.J., 653, 815
Metcalf R. B., WhiteS. D.M., 2007, MNRAS,381, 447
Seljak U., 1996, Astrophys.J., 463, 1
Takada M., Futamase T., 2001, Astrophys.J., 546, 620
Takada M., Komatsu E., Futamase T., 2000, Astrophys.J., 533, L83
Tashiro H., Aghanim N., Langer M., Douspis M., Zaroubi S., 2008, MNRAS,389, 469
Tozzi P., Madau P., Meiksin A., Rees M. J., 2000, Astrophys.J., 528, 597
Zahn O., Zaldarriaga M., 2006, AstroPhys. J., 653, 922
Zaldarriaga M., FurlanettoS. R., Hernquist L., 2004, Astrophys.J., 608, 622
