Probing non-gaussian features in the H I distribution at the epoch of re-ionization

Mon. Not. R. Astron. Soc. 358, 968 976 (25) doi:./j.365-2966.25.8836.x Probing non-gaussian features in the H I distribution at the epoch of re-ionization Somnath Bharadwaj Sanjay K. Pey 2 Department of Physics Meteorology & Centre for Theoretical Studies, IIT Kharagpur, Pin: 72 32, India 2 Department of Mathematics, LBS College, Gonda 27, India Accepted 25 January 2. Received 24 December 23; in original form 24 October 5 ABSTRACT The H I distribution at the epoch of re-ionization (EOR) is largely determined by the sizes distribution of the ionized regions. In the scenario where the ionized regions have comoving sizes of the order of a few Mpc, the large-scale statistical properties of the H I distribution are dominated by the Poisson noise of the discrete ionized regions it is highly non-gaussian. We investigate the possibility of probing re-ionization by studying these non-gaussian features using future radio-interferometric observations of redshifted 2-cm H I radiation. We develop a formalism relating correlations between the visibilities measured at three different baselines frequencies to the bispectrum of H I fluctuations. For visibilities at the same frequency, this signal is found to be of the same order as the two visibility correlation, which probes the H I power spectrum. For visibilities at different frequencies, we find that the correlations decay within a frequency difference of MHz. This implies that it is, in principle, straightforward to extract this H I signal from various contaminants, which are believed to have a continuum spectrum are expected to be correlated even at large frequency separations. Keywords: intergalactic medium cosmology: theory diffuse radiation large-scale structure of Universe. INTRODUCTION There has recently been a lot of interest in understing exactly how when the Universe was re-ionized. There now are significant observational constraints mainly from three different kinds of observations. The observation of quasars at redshift z 6, which show strong H I absorption (Becker et al. 2), indicates that at least per cent of the total hydrogen mass at z 6isneutral (Fan et al. 22) the neutral mass fraction decreases rapidly at lower redshifts. This is a strong indication that the epoch of re-ionization (EOR) ended at z 6. Observations of the cosmic microwave background radiation (CMBR) polarization, generated through Thomson scattering of CMBR photons by free electrons along the line of sight, indicates that the re-ionization began at a redshift z > 4. On the other h, the observed anisotropies of the CMBR indicate that the total optical depth of the Thomson scattering is not extremely high, suggesting that re-ionization could not have started at redshift much higher than approximately 3 (Kogut et al. 23; Spergel et al. 23). A third constraint comes from determinations of the IGM temperature from observations of the Lyα forest in the z range 2 to 4, which indicates a complex re-ionization history with there possibly being an order unity change in the neutral hydrogen fraction at z (Theuns et al. 22; Hui & Haiman 23). Mapping the H I distribution at high redshifts using radio observations of the redshifted 2-cm radiation (Scott & Rees 99; Kumar, Padmanabhan & Subramanian 995; Madau, Meiksin & Rees 997) holds the possibility of probing the transition from a largely neutral to a largely ionized Universe at a level of detail surpassing any other techniques. Zaldarriaga, Furlanetto & Hernquist (23, hereafter ZFH) have developed a statistical technique based on the angular power spectrum, on lines similar to the analysis of CMBR anisotropies, for analysing the H I signal from the EOR in radio-interferometric observations. Extracting the H I signal from various Galactic extragalactic contaminants (e.g. Shaver et al. 999; DiMatteo et al. 22; Oh & Mack 23; Cooray & Furlanetto 24; DiMatteo, Ciardi & Miniati 24; Gnedin & Shaver 24) is one of the most important challenges. Most of the known contaminants are expected to have continuum spectra ZFH show that it should in principle be possible to extract the H I signal using the fact that, unlike the contaminants, it will be uncorrelated at two slightly different frequencies. The frequency dependence of the angular power spectrum of the H I signal foregrounds has recently been analysed in detail by Santos, Cooray & Knox (24). E-mail: somnathb@iitkgp.ac.in (SB); spey@iucaa.ernet.in (SKP) C 25 RAS

Probing non-gaussian features in H I at re-ionization 969 An alternative statistical technique for analysing the H I signal is to study the correlations between the complex visibilities measured at different baselines frequencies in radio-interferometric observations. This has been developed in the context of observing H I from z < 6 (Bharadwaj & Sethi 2; Bharadwaj & Pey 23; Bharadwaj & Srikant 24) later generalized to the EOR signal in Bharadwaj & Ali (25, hereafter BA). The possibility of using visibility correlations to quantify the EOR signal has also been proposed by Morales & Hewitt (24), who further discuss how the different frequency signatures of the contaminants the H I signal can be used to distinguish between the two. Recently, Morales (25) has addressed the issue of the power spectrum sensitivity of the EOR H I signal. Various investigations (e.g. ZFH, BA) show that the power spectrum of H I fluctuations at the EOR has contributions from mainly two distinct effects, the clustering of the hydrogen, which, on large scales, is assumed to follow the dark matter distribution the fluctuations arising from the presence of discrete regions of ionized gas surrounding the sources responsible for re-ionizing the Universe. The details of the re-ionization process are not very well understood (e.g. Barkana & Loeb 2), the shape, size distribution of these ionized regions is one of the very important issues that will be probed by 2-cm H I observations. There has recently been progress in analytically modeling the growth of the ionized regions (Furlanetto, Zaldarriaga & Hernquist 24a, hereafter FZH) based on the findings of simulations (Gnedin 2; Benson et al. 2; Nusser et al. 22; Ciardi, Stoehr & White 23; Sokasian et al. 23a,b), which show that there will not be a large number of small H II regions around individual ionizing sources, rather there will be a few large ionized regions centred on places where the ionized sources are clustered. The size of these ionized regions is expected to be around a few Mpc (comoving) or possibly larger at the EOR. In such a scenario, on scales larger than the size of the individual ionized regions, the H I signal will be dominated by the Poisson noise arising from the discrete nature of the ionized regions (e.g. ZFH, BA, FZH, Furlanetto, Zaldarriaga & Hernquist 24b). Further, the H I signal is expected to be highly non-gaussian. Nearly all of the work on quantifying the EOR H I signal expected in radio-interferometric observations has focused on the two-point statistics namely the angular power spectrum the correlations between pairs of visibilities. Both these quantities are actually equivalent they basically probe the power spectrum of H I fluctuations at the EOR. The power spectrum completely quantifies a Gaussian rom field, but the higher order statistics would contain independent information if the H I fluctuations at EOR were not a Gaussian rom field. FZH have used the pixel distribution function, a one-point statistic, to quantify non-gaussian features in the H I distribution. He et al. (24) have studied the non-gaussian features that arise in the H I distribution in the lognormal model. In this paper, we address the issue of quantifying the non-gaussian features of the H I signal expected in radio-interferometric observations. In particular, we focus on the correlation between three visibilities. This is expected to be zero if the signal were a Gaussian rom field deviations from zero are a clear signature of the non-gaussian properties of the H I distribution. Here, we derive the relation between the three visibility correlation the bispectrum of the H I fluctuations. The bispectrum quantifies correlations between three Fourier modes this is non-zero only when there are phase correlations between different modes. The three visibility correlation, as we show, is comparable to the correlations between two visibilities this leads us to speculate that this will play an important role in detecting the H I signal. Further, the higher order correlations contain independent information, observing these would throw independent light on the topology morphology of the H I distribution at the EOR. Finally, an outline of the paper. In Section 2, we present the formalism relating the three visibility correlation to the H I bispectrum. In Section 3, we introduce a simple model for the H I distribution at re-ionization calculate its bispectrum. In Section 4, we present results for the three visibility correlation expected from H I at re-ionization discuss some consequences. It may also be noted that we use the values ( m, λ, b h 2, h) = (.3,.7,.2,.7) for the cosmological parameters throughout. 2 FORMALISM FOR THREE VISIBILITY CORRELATION In this section, we follow the notation used in BA, which also contains a more detailed discussion of the formalism for calculating the H I signal. The H I radiation at frequency 42 MHz in the rest frame of the hydrogen is redshifted to a frequency = 42/( + z) MHz for an observer at present. The expansion of the Universe the H I peculiar velocity both contribute to the redshift. Incorporating these effects, the specific intensity I (ˆn)ofredshifted 2-cm H I radiation at frequency direction ˆn can be written as I (ˆn) = Ī (z) η H I (ˆn, z) where Ī = 2.5 2 Jy ( )( ) b h 2.7 H () sr.2 h H(z) η H I (ˆn, z) = ρ H I ρ H ( T )[ γ T s ( + z) H(z) ] v. r It should be noted that the terms on the right-h side of equations () (2) refer to the epoch when the H I radiation originated. Here, H(z) the Hubble parameter, ρ H the mean cosmological density of hydrogen r (or r ) the comoving distance to the H I calculated ignoring peculiar velocities, depend only on z. The quantities ρ H I the H I density, T γ the CMBR temperature, T s the H I spin temperature v the radial component of the H I peculiar velocity also vary with position should be evaluated at x = r ˆn, i.e. the position where the radiation originated. It may be noted that η H I (x, z), the 2-cm radiation efficiency, was originally introduced by Madau et al. (997) who did not include peculiar velocities. As shown in BA, equation (2) includes an extra term, which arises when the effect of the H I peculiar velocities are included. The quantity η H I (ˆn, z) incorporates the details of the H I evolution including effects of heating, re-ionization density fluctuations as a result of structure formation. (2)

97 S. Bharadwaj S. K. Pey We next introduce η H I (k, z), the Fourier transform of η H I (y, z), d 3 k η H I (y, z) = e ik y η (2π) 3 H I (k, z), where y refers to an arbitrary comoving position. Using this we can express η H I (ˆn, z) as d 3 k η H I (ˆn, z) = e ik r ˆn η (2π) 3 H I (k, z), where it is understood that this refers to the position x = r ˆn. The ensemble average of various products of η H I (k, z) are used to quantify the statistical properties of the fluctuation in the H I distribution. We first consider the H I power spectrum P H I (k, z) defined through η H I (k, z) η H I (k 2, z) =(2π) 3 δ 3 D (k + k 2 ) P H I (k, z), (5) where δ 3 D is the three-dimensional Dirac delta function. The power spectrum completely quantifies all properties of the H I distribution if the fluctuations are a Gaussian rom field. The higher order statistics contain independent information if the fluctuations are not a Gaussian rom field. Here, we proceed one step beyond the power spectrum also consider the H I bispectrum B H I (k, k 2, k 3, z) defined through η H I (k, z) η H I (k 2, z) η H I (k 3, z) =(2π) 3 δ 3 D (k + k 2 + k 3 ) B H I (k, k 2, k 3, z). (6) We next mention a few well-known properties of the power spectrum bispectrum, which are relevant to the discussion. The fact that not all modes are correlated, reflected in the Dirac delta functions in equations (5) (6), is a consequence of the assumption that H I fluctuations are statistically homogeneous. Further, P H I (k)isisotropic, i.e. it does not depend on the direction of k,ifthe effects of the peculiar velocity are ignored. The redshift space distortion caused by the peculiar velocities breaks the isotropy of P H I (k), which now depends on the orientation of k with respect to the line of sight. Similarly, ignoring redshift space distortions, B H I (k, k 2, k 3 ) depends only on the triangle formed by the wave vectors k, k 2 k 3, this is completely specified by the magnitude of the three vectors (k, k 2, k 3 ). The bispectrum also depends on how the triangle is oriented with respect to the line of sight if redshift space distortions are included. Finally, we note that both the power spectrum the bispectrum are real quantities. While the power spectrum is necessarily positive, there is no such restriction on the bispectrum. We now shift our attention to radio-interferometric observations of redshifted H I using an array of low-frequency radio antennas distributed on a plane. The antennas all point in the same direction m, which we take to be vertically upwards. The beam pattern A(θ) quantifies how the individual antenna, pointing upwards, responds to signals from different directions in the sky. This is assumed to be a Gaussian A(θ)=e θ2 /θ 2 with θ, i.e. the beam width of the antennas (in radians) is small the part of the sky that contributes to the signal can be well approximated by a plane. In this approximation, the unit vector ˆn can be represented by ˆn = m + θ, where θ is a two-dimensional vector in the plane of the sky. Using this, the angular fluctuations in the specific intensity δi can be expressed as d 3 k δi (ˆn) = Ī (2π) 3 e ir (k +k θ) η H I (k, z), (7) where k = k m k are respectively the components of k parallel perpendicular to m. The component k lies in the plane of the sky. The quantity measured in interferometric observations is the complex visibility V (U, ), which is recorded for every independent pair of antennas at every frequency channel in the b of observations. For any pair of antennas, U = d/λ quantifies the separation d in units of the wavelength λ, werefer to this dimensionless quantity U as a baseline. A typical radio-interferometric array simultaneously measures visibilities at a large number of baselines frequency channels, V (U,) = d 2 θ A(θ) I (θ)e i2πu θ. (8) (3) (4) The visibilities record only the angular fluctuations in I (θ) the visibilities arising from angular fluctuations in the H I radiation are ( d 3 k V (U,) = Ī (2π) a U r ) 3 2π k η H I (k, z)e ik r, (9) where a(u)isthe Fourier transform of the antenna beam pattern A(θ), which for a Gaussian beam A(θ) = e θ2 /θ 2 gives the Fourier transform also to be a Gaussian a(u) =π θ 2 exp [ π2 θ 2 U 2 ], which we use in the rest of this paper. In this paper, we quantify the statistical properties of the quantity measured in radio-interferometric observations, namely the visibilities at different baselines frequencies. Further, we study their relation to the statistical properties of the H I distribution. To this end, we introduce the notation S 2 (U, U 2, ) = V (U,+ )V (U 2,) S 3 (U, U 2, U 3,, 2 ) = V (U,+ )V (U 2,+ 2 )V (U 3,) () to denote the correlations between the visibilities at different baselines frequencies. It should be noted that although we have shown S 2 S 3 as explicit functions of only the frequency differences, all these correlations also depend on the the central value, which is not ()

Probing non-gaussian features in H I at re-ionization 97 shown as an explicit argument. Further, throughout our analysis, we assume that all frequency differences are much smaller than the central frequency, i.e. /. The correlation S 2 (U, U 2, ) between the visibilities at two baselines frequencies has been calculated earlier by Bharadwaj & Sethi (2), Bharadwaj & Pey (23) BA who find that S 2 ifu 2 U. This is a consequence of the statistical homogeneity of the H I fluctuations. It is sufficient to restrict the analysis to U = U 2 = U, which we denote as S 2 (U, ), we have S 2 (U, ) = Ī 2 θ 2 2r 2 dk P H I (k) cos(k r ), (2) where k = k m + (2π/r ) U r = dr /d. The vector k has components k (2π/r ) U respectively parallel perpendicular to the line of sight. The fact that P H I (k), which includes redshift distortion, is isotropic in the directions perpendicular to the line of sight implies that S 2 is isotropic in U we can write S 2 (U, ). We also note that S 2 is real for the H I signal. This follows from the fact that P H I (k) is real it is unchanged if k k. The three visibility correlation S 3 is the quantity of interest in this paper. This will be related to the H I bispectrum. Here, as for the power spectrum, we assume that /, whereby the only term in equation (9) for the visibility V (U, + ), which is affected by +, ise ik r +, which can be approximated as e ik (r +r ). We then have S 3 (U, U 2, U 3,, 2 ) = Ī 3 (2π) 6 ( d 3 k d 3 k 2 d 3 k 3 a U r ) ( 2π k a U 2 r ) 2π k 2 a(u 3 r 2π k 3 )e i(k +k 2 +k 3 )r e i(k +k 2 2 )r δ 3 D (k + k 2 + k 3 ) B H I (k, k 2, k 3 ). (3) It is convenient to write the d 3 k integrals as dk d 2 k integrate over d k 3, whereby the term e i(k +k 2 +k 3 )r drops out because of the Dirac delta function. Also, we introduce a new variable y = k (2π/r ) U use the explicit form for the function a(u), whereby we have S 3 (U, U 2, U 3,, 2 ) = Ī 3 dk (2π) 6 dk 2 e i(k +k 2 2 )r d 2 y d 2 y 2 d 2 y 3 δ 2 D [(2π/r )(U + U 2 + U 3 ) + y + y 2 + y 3 ] ( πθ 2 ) 3 exp [ (r θ /2) 2( y 2 + y2 2 + y2 3)] BH I, (4) where the arguments of the bispectrum change as we carry out the integrals, but we do not show them explicitly. Carrying out the d 2 y 3 integral, we have S 3 (U, U 2, U 3,, 2 ) = Ī 3 dk (2π) 6 dk 2 e i(k +k 2 2 )r d 2 y d 2 y 2 exp [ ( r θ /2 ) 2( y 2 + y 2 2)] exp { (r θ /2) 2 [y + y 2 + (2π/r )(U + U 2 + U 3 )] 2} B H I. (5) The point to note is the two Gaussian functions exp[ (r θ /2) 2 (y 2 + y 2 2)] exp{ (r θ /2) 2 [y + y 2 + (2π/r )(U + U 2 + U 3 )] 2 } are peaked around different values of y y 2. While the former is peaked around y = y 2 =, the latter is peaked around y + y 2 = ( 2π/r )(U + U 2 + U 3 ). The peaks of the two functions have very little overlap if U + U 2 + U 3 > the visibility correlations are exponentially suppressed if the vector sum of the baselines differs from zero. There are substantial correlations only for the sets of baselines for which U + U 2 + U 3 (πθ ).Inthe rest of our analysis, we only consider combinations of baselines for which U + U 2 + U 3 = the product of the two Gaussian functions becomes exp[ 2(r θ /2) 2 (y 2 + y 2 2 + y y 2 )]. This can be further simplified if the baselines we are dealing with are much larger than /(πθ ). We can then approximate this function by a product of two Dirac delta functions (6π/3) (r θ ) 4 δ 2 D (y + y 2 /2) δ 2 D (y 2). Using this in equation (5), we have S 3 (U, U 2, U 3,, 2 ) = Ī 3 θ 2 dk 2 π r 4 dk 2 e i(k +k 2 2 )r B H I (k, k 2, k 3 ), (6) where k = k m + (2π/r ) U, k 2 = k 2 m + (2π/r ) U 2 k 3 = (k + k 2 ) m + (2π/r ) U 3. Further, it can be verified that S 3 is real S 3 (U, U 2, U 3,, 2 ) = Ī 3 θ 2 dk 2 π r 4 dk 2 cos[(k + k 2 2 )r ] B H I(k, k 2, k 3 ), (7) where we have also incorporated the fact that S 3 depends only the triangle formed by U, U 2 U 3, which is completely specified by just the magnitudes (U, U 2, U 3 ). We use equations (2) (7) to calculate the visibility correlations expected during the EOR.

972 S. Bharadwaj S. K. Pey 3 A MODEL FOR THE H I DISTRIBUTION The re-ionization of H I in the Universe started, possibly at a redshift z 3, when the first luminous objects were formed. The radiation from these luminous objects from the subsequently formed luminous objects ionized the low-density H I in the Universe. The re-ionization commences in small spherical regions (Stromgren sphere) surrounding the luminous objects. These spheres are filled with ionized H II gas, the rest of the Universe being filled with H I. Gradually these ionized regions grow until they finally overlap, filling up the whole of space, all the low-density gas in the Universe is ionized. The H I distribution during re-ionization is largely determined by the ionized regions. This is expected to be highly non-gaussian carrying signatures of the size, shape distribution of the discrete ionized regions. Here, we adopt a simple model for the ionized regions. Though simple, this model suffices to illustrate the non-gaussian nature of the H I distribution allows us to calculate some of the salient observable consequences. We assume that the H I gas is heated well before it is re-ionized that the spin temperature is coupled to the gas temperature with T s T γ so that ( T γ /T s ). It then follows that η H I > (equation 2), i.e. H I will be seen in emission. At any epoch, a fraction of the volume f V is completely ionized, the ionized gas being in non-overlapping spheres of comoving radius R, the centres of the spheres being romly distributed. This model is similar to that used by ZFH in the context of H I emission, Gruzinov & Hu (998) Knox, Scoccimarro & Dodelson (998) in the context of the effect of patchy re-ionization on the CMBR. One would expect the centres of the ionized spheres to be clustered, given the fact that we identify them with the locations of the first luminous objects, which are believed to have formed at the peaks of the density fluctuations. This effect, included in BA, has not been taken into account here. Following ZFH, we assume that the mean neutral fraction x H I at any epoch is given by x H I (z) = (8) + exp[(z z )/ z] with z = z =.5 so that 5 per cent of the hydrogen is neutral at a redshift z =. The mean comoving number density of ionized spheres n H I is related to the quantities defined earlier as f V = x H I = (4πR 3 /3) n H I.WehavekeptR as a free parameter have used this to determine n H I. We assume that the total hydrogen density traces the dark matter hence it is ρ H ( + δ), where δ refers to the fluctuations in the dark matter distribution. Then, in our model, the H I density is ρ H I (x, z) = ρ H ( + δ)[ θ( x x a a /R)], where a refers to the different ionized spheres with centres at x a, θ(y) isthe Heaviside step function defined such that θ(y) = for y zero otherwise. We then have [ η H I (x, z) = + δ + z ] [ v ( ) ] x xa θ, (9) H(z) r R a where v refers to the peculiar velocity caused by δ. The point to note is that η H I (x, z) has contributions from two distinct effects, namely the fluctuations arising from the gravitational clustering of the hydrogen that follows the dark matter distribution the discrete ionized regions. Earlier studies (ZFH) have shown that the contribution from the discrete ionized regions dominates the H I power spectrum on length-scales larger than the size of the individual ionized bubbles at redshifts z when f V.5 the H I signal is expected to be maximum. In the stard scenario, the initial dark matter fluctuation δ is assumed to be a Gaussian rom field for which the bispectrum is zero. Non-Gaussian features of order δ 2 arise from non-linear effects as the density fluctuation grows, but these effects are expected to be very small on the length-scales of our interest at redshifts z. The bispectrum B H I too will be dominated by the non-gaussian features arising from the discrete ionized regions. Further, we expect the gravitational clustering of the hydrogen to make a smaller contribution to the bispectrum than it does to the power spectrum. The aim here being to investigate the non-gaussian effects through a study of the bispectrum, it is justified to focus on just the contribution arising from the individual ionized regions, ignoring the effects of gravitation clustering. Under this assumption, [ ( ) ] x xa η H I (x, z) = θ (2) R a its Fourier transform for k > is η H I (k, z) = f V W (kr) e ik xa, (2) n H I a where W (y) = (3/y 3 )[sin(y) y cos(y)] is the spherical top hat window function. Using these, we have P H I (k) = f V 2W 2 (kr) (22) n H I B H I (k, k 2, k 3 ) = f V 3 W (k R) W (k 2 R) W (k 3 R) (23) n 2 H I respectively for the power spectrum the bispectrum. We use these to calculate the visibility correlations expected in this model. Our model has a limitation that it cannot be used when a large fraction of the volume is ionized as the ionized spheres start to overlap the H I density becomes negative in the overlapping regions. Calculating the fraction of the total volume where the H I density is negative, we find this to be f 2 V /2. We use this to assess the range of validity of our model. We restrict the model to z >, where f V <.5, the H I density is negative in less than 2.5 per cent of the total volume.

4 RESULTS AND DISCUSSION Probing non-gaussian features in H I at re-ionization 973 In this section, we present results for the visibility correlations expected from H I during the EOR. Our aim being to illustrate the non-gaussian nature of the expected signal its dependence on the ionized regions, we show results centred on only at a single frequency namely 25 MHz. This corresponds to a redshift z =.4 when the mean neutral fraction is x H I =.67 (i.e. f V =.33). We choose this particular frequency as x H I is quite close to.5 where the H I signal is expected to be maximum, simultaneously ensuring that the volume fraction where the H I density predicted by our model becomes negative is small ( 5 per cent). Further, the H I signal is expected to be dominated by discrete ionized regions hence we anticipate significant non-gaussian features. We have used equations (2) (7) to calculate the expected correlations between two three visibilities, respectively. For this, it is necessary to specify a value for θ, the beam size of the individual antennas in the array. Further, it may be noted that θ.6 θ FWHM. The value of θ will depend on the physical dimensions of the antennas the wavelength of observation. For the Giant Meterwave Radio Telescope (GMRT), θ = at 325 MHz. We scale this using θ λ to obtain θ = 2. 6at25 MHz, which we use here. The H I signal predicted here is for observations using the GMRT they can be directly compared to those in BA. Both S 2 S 3 scale as θ 2, it is straightforward to scale the results presented here to make visibility correlation predictions for other radio telescopes. The comoving radius of the ionized spheres R is a free parameter in our model. Investigations on the growth of the ionized spheres (FZH) show that these will be at least a few Mpc in radius (possibly larger) at the redshift of interest. We have considered three possible values R = (, 3, 5) h Mpc for which the respective values of n H I are (78, 2.9,.63) 3 h 3 Mpc 3. For ease of graphical presentation, we have restricted our analysis of S 3 to equilateral triangles for which the size of the baseline U completely specifies the triangle we have S 3 (U, ). Further, we first consider the correlations at the same frequency, i.e δ =. Figs 2 show the results [S 2 (U)] /2 [ S 3 (U)] /3, respectively. ] /2 mjy [S (U) 2.. R = 5 Mpc/h R = 3 Mpc/h R = Mpc/h. Figure. This shows the visibility correlation S 2 (U, ) /2 as a function of U for =, for different values of R, the comoving radius of the ionized spheres. These predictions are for observations centred at 25 MHz. U /3 mjy.. R = 5 Mpc/h R = 3 Mpc/h R = Mpc/h [ S (U)] 3.. e 5 Figure 2. This shows the visibility correlation [ S 3 (U, )] /3 as a function of U for =, for different values of R, the comoving radius of the ionized spheres. These predictions are for observations centred at 25 MHz. U

974 S. Bharadwaj S. K. Pey We find that at small U, [S 2 (U)] /2 is more or less constant with a value of the order of.2 mjy for R = 3 h Mpc. The signal is proportional to R 3/2 its magnitude increases as the ionized spheres become larger. Each baseline U can be associated with a comoving length-scale r /(2πU)atthe comoving distance where the H I radiation originated. The signal from the ionized spheres is constant across the baselines for which r /(2πU) islarger than the size of the spheres, the signal falls at baselines for which U > r /(2πR). Each baseline resolves out features larger than r /(2πU) the presence of discrete ionized regions make very little contribution to the signal at the large baselines. Comparing the results for [S 2 (U)] /2 presented here with those presented in BA, which also includes the effects of gravitational clustering, we note that the gravitational clustering signal is also of the order. mjy at small U. The gravitational clustering signal also falls with increasing U the combined signal would depend critically on the size of the bubbles. For example, the signal from discrete ionized sources would dominate over the gravitational clustering signal at baselines U > 5 if the ionized spheres had comoving radius R = 5 h Mpc, whereas the gravitational clustering signal would possibly dominate throughout for R = h Mpc. Turning our attention next to S 3 (Fig. 2), the first point to note is that this is negative. The shape of S 3 as a function of U is very similar to that of S 2 its magnitude is around [ S 3 (U)] /3.6 mjy at R = 3 h Mpc, which is around three times smaller than [S 2 (U)] /2. At small U,[ S 3 (U)] /3 is more or less constant. Although our results are restricted to equilateral triangles, we expect the correlations to be nearly constant for triangles of all shapes provided all the baselines satisfy U < r /(2πR). The signal is proportional to R 2 its magnitude increases a little faster than that of [S 2 (U)] /2 as R is increased. We expect the dark matter density fluctuations at z > to be well in the linear regime on comoving length-scales h Mpc or larger the contribution to S 3 from non-linear gravitational clustering is expected to be very small on these scales. It may be noted that the contribution to S 3 from linear gravitational clustering is exactly zero in the stard scenario where the initial density fluctuations are a Gaussian rom field. Further, the comoving length-scale h Mpc corresponds to the baseline U we expect the contribution from individual ionized spheres considered here to be the dominant signal at these baselines. We next consider the correlations between the visibilities at different frequencies (Figs 3 4). Again, for ease of graphical presentation, we have restricted our analysis of S 3 (U,, 2 )toequilateral triangles with the added restriction that = 2 =, sowehave S 3 (U, ). We have shown results only for R = 3 h Mpc, but a similar behaviour is expected for other values also. We find that both S 2 (U, ) S 3 (U, ) fall rapidly, in nearly the same fashion independent of U, are very close to zero by.5 MHz. One of the main challenges in observing cosmological H I is to extract it from various contaminants, which are expected to swamp this signal. The contaminants include Galactic synchrotron emission, free free emission from ionizing haloes (Oh & Mack 23), faint radio loud quasars (DiMatteo et al. 22) synchrotron emission from low-redshift galaxy clusters (DiMatteo et al. 24). Fortunately, all of these foregrounds have smooth continuum spectra we expect their contribution to the visibilities to be correlated over large, whereas the H I contribution is uncorrelated beyond MHz or less. It is, in principle, straightforward to fit the visibility correlations S 2 S 3 at large remove any slowly varying component thereby separating the contaminants from the H I signal. We also use this opportunity to note that this is a major advantage of using visibility correlations as compared with the angular power spectrum, which exhibits substantial correlations even at two frequencies separated by MHz (Santos et al. 24). An important fact that emerges from our analysis is that the H I signal in the correlation between three visibilities is of the same order as the correlation between two visibilities, the former being around three times smaller. This is a generic feature of the EOR H I signal, valid if the ionized regions are bubbles of the size R = h Mpc or larger. This signal arises from the Poisson noise of the discrete ionized regions it is enhanced if the size of the bubbles is increased. The fact that there is a substantial S 3 tells us that there are large phase correlations between the visibilities. This is a consequence of the fact that there are only a few coherent features (the ionized regions) that dominate the whole H I signal. Investigations on the growth of ionized regions (FZH) show that there will be a spread in the sizes of the ionized regions at any given epoch. This will smoothen some of the sharp features seen in Figs 2. The ringing seen in the these figures is an artefact of there being only a single value of R we do not expect this feature to be there if we have a spread in R. Further, the gravitational clustering signal not shown here may also dominate at large U. Despite all these limitations, we can still expect substantial correlations between three visibilities in a a more realistic analysis, this being a robust signature of the fact that re-ionization occurs through a few, large (R afew Mpc) bubbles of ionized gas the H I signal is dominated by Poisson noise. We next briefly discuss the noise levels the integration times required to observe the H I signal, particularly addressing the question of whether S 3 can be detected with integration times comparable to those needed for S 2.Weconsider an array of N antennas, the observations lasting a time duration t, with frequency channels of width δ spanning a total bwidth B. Itshould be noted that the effect of a finite channel width δ has not been included in our calculation, which assumes infinite frequency resolution. This effect can be easily included by convolving our results for the visibility correlation with the frequency response function of a single channel. Preferably, δ should be much smaller than the frequency separation at which the visibility correlation become uncorrelated. We use S to denote the frequency separation within which the visibilities are correlated beyond which they become uncorrelated. We use N 2 N 3 to denote the rms noise in S 2 S 3, respectively. It is well known that ( ) 2 2kB T SYS N 2 = A ef δ t (Thompson, Moran & Swenson 986) we have ( ) 3 2kB T SYS N 3 (δ t) 3/2 A ef

Probing non-gaussian features in H I at re-ionization 975.25 mjy /2 [S 2 (U, )].2.5..5 U = 2 U = 6 U =..2.3.4.5.6.7.8.9 MHz Figure 3. This shows the visibility correlation S 2 (U, ) /2 as a function of for the three different values of U shown in the figure. The comoving radius of the ionized spheres is assumed to be R = 3 h Mpc the predictions are for observations centred at 25 MHz. The topmost curve refers to U = 2..6 mjy /3 [S 3 (U, )].5.4.3.2 U = 2 U = 6 U =...2.3.4.5.6.7.8.9 MHz Figure 4. This shows the visibility correlation [ S 3 (U, )] /3 as a function of for the three different values of U shown in the figure. The comoving radius of the ionized spheres is assumed to be R = 3 h Mpc the predictions are for observations centred at 25 MHz. The topmost curve refers to U = 2. assuming that we have Gaussian rom noise, where T SYS is the system temperature A ef is the effective area of a single antenna. The noise contributions will be reduced by a factor / N o if we combine N o independent samples of the visibility correlation. A possible observational strategy for a preliminary detection of the H I signal would be to combine the visibility correlations at all baselines frequency separations where there is a reasonable amount of signal. This gives N o = [N(N )/2] (B/δ)(S/δ) for the two visibility correlation N o = [N(N )(N 2)/6] (B/δ) (S/δ) 2 for the three visibility correlations. It should be noted that we have used the fact that the S 3 is non-zero only for the baselines between triplets of antennas. Combining all of this, we have ( )[ ] /4 [N 2 ] /2 2kB T SYS 2 A ef N(N )BS t /2 ( )[ ] /6 [N 3 ] /3 2kB T SYS 6 A ef N(N )(N 2)BS 2 t. /2 The ratio [N 3 ] /3 /[N 2 ] /2 [N(N )B/(N 2) 2 S] /2 has a very weak dependence on N, B S for a reasonable choice of values, is of order unity. We thus see that, for a given integration time, we will achieve comparable noise levels in both the two three visibility correlations. Estimates of the integration time to detect S 2 (or equivalently the angular power spectrum; ZFH; BA) indicate this to be around afew hundred hours. We find that it should be possible to also detect S 3 in a comparable integration time.

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