Fisher Matrix Based Predictions for Measuring the z = 3.35 Binned 21-cm Power Spectrum using the Ooty Wide Field Array (OWFA)

TL;DR

This paper predicts the ability of the OWFA telescope to measure the 21-cm power spectrum at redshift 3.35 using Fisher matrix analysis, highlighting detection prospects and optimal observation times.

Contribution

It provides the first detailed Fisher matrix-based forecast for 21-cm power spectrum detection with OWFA at z=3.35, including optimal observation strategies.

Findings

5 sigma detection possible with 1,000 hours in certain k-bins

Peak SNR of 10 sigma in 0.1-0.2 Mpc^{-1} range with 2,000 hours

Limited gains beyond 1,000 hours of observation in a single field

Abstract

We use the Fisher matrix formalism to predict the prospects of measuring the redshifted 21-cm power spectrum in different $k$-bins using observations with the upcoming Ooty Wide Field Array (OWFA) which will operate at $326.5 {\rm MHZ}$. This corresponds to neutral hydrogen (HI) at $z=3.35$, and a measurement of the 21-cm power spectrum provides an unique method to probe the large-scale structures at this redshift. Our analysis indicates that a $5 \sigma$ detection of the binned power spectrum is possible in the $k$ range $0.05 \leq k \leq 0.3 \, {\rm Mpc}^{-1}$ with $1,000$ hours of observation. We find that the Signal-to-Noise ratio (${\rm SNR}$) peaks in the $k$ range $0.1- 0.2\, {\rm Mpc}^{-1}$ where a $10 \sigma$ detection is possible with $2,000$ hours of observations. Our analysis also indicates that it is not very advantageous to observe much beyond $1,000$ hours in a single field of view as the ${\rm SNR}$ increases rather slowly beyond this in many of the small $k$-bins. The entire analysis reported here assumes that the foregrounds have been completely removed.

Equations

$${\mathbf{U}}_a=a\left(\frac{d}{\lambda }\right)\hat{i}(1\le a\le 263)$$

$$\mathcal{V}({\mathbf{U}}_a,{\nu }_n)=\mathcal{S}({\mathbf{U}}_a,{\nu }_n)+\mathcal{N}({\mathbf{U}}_a,{\nu }_n)$$

$$v({\mathbf{U}}_a,{\tau }_m)=\Delta {\nu }_c\sum _{n=1}^{N_c}{e}^{2\pi i{\tau }_m{\nu }_n}\mathcal{V}({\mathbf{U}}_a,{\nu }_n)$$

$${\tau }_m=\frac{m}{B}\text{with}\frac{N_c}{2}<m\le \frac{N_c}{2}$$

$$C_{ab}(m)=⟨v({\mathbf{U}}_a,{\tau }_m)v^{\ast }({\mathbf{U}}_b,{\tau }_m)⟩.$$

$$C_{ab}(m)=\frac{B}{r_{\nu }^2{r}_{\nu }^{{}^{\prime }}}{\left(\frac{2k_B}{{\lambda }^2}\right)}^2\int d^2{U}^{{}^{\prime }}\tilde{A}({\mathbf{U}}_a-{\mathbf{U}}^{{}^{\prime }}){\tilde{A}}^{\ast }({\mathbf{U}}_b-{\mathbf{U}}^{{}^{\prime }})$$

$$\times P_T({\mathbf{k}}_{\perp },k_{\parallel })+\frac{2\Delta {\nu }_{c}B{\sigma }_N^2}{(264-a)}{\delta }_{a,b}$$

$$F_{\alpha \gamma }=\frac{1}{2}\sum _m{C}_{ab}^{-1}(m)[C_{bc}(m){]}_{,\alpha }{C}_{cd}^{-1}(m)[C_{da}(m){]}_{,\gamma }$$

$$P_T(\mathbf{k})=(1+\beta {\mu }^2{)}^2{P}_T^r(k)$$

$$P_T^r(k)=({\bar{x}}_{\mathrm{HI}}{b}_{\mathrm{HI}}\bar{T}{)}^{2}P(k)$$

$$\bar{T}(z)=4.0\text{mK}(1+z{)}^2\left(\frac{1-Y_P}{0.75}\right)\left(\frac{{\Omega }_b{h}^2}{0.020}\right)\left(\frac{0.7}{h}\right)\left(\frac{H_0}{H(z)}\right)$$

$$\text{SNR}=\frac{1}{{\sigma }_{ic}}=\frac{[P_T^r{]}^i}{\Delta [P_T^r{]}^i},$$

Full text

1 Introduction

The redshifted 21-cm emission from the discrete, unresolved neutral hydrogen (HI) sources in the post-reionization era ($z<6$) appears as a faint diffuse background radiation in low frequency observations below 1420 MHz. This provides us a useful tool to explore the large scale structure of the universe in the post-reionization era, using the fluctuations in the diffuse background radiation to trace the HI power spectrum (Bharadwaj, Nath & Sethi 2001; Bharadwaj & Sethi 2001). In addition to probing the HI power spectrum (Bharadwaj & Pandey 2003; Bharadwaj & Srikant 2004), fluctuations in the diffuse background radiation also probe of the bispectrum (Ali et al. 2006; Guha Sarkar & Hazra 2013). In recent years, a considerable amount of work has been done to explore the prospects of detecting the 21-cm HI signal from the post-reionization era (Visbal et al. 2009; Bharadwaj et al. 2009; Wyithe & Loeb 2009; Seo et al. 2010; Mao 2012; Ansari et al. 2012; Bull et al. 2014).

In the post-reionization era, the bulk of the HI 21-cm emission originates from the dense pockets of self-shielded HI regions, which are identified as damped Ly$\alpha$ (DLA) systems in quasar observations. The fluctuations in the HI 21-cm emission which are in general quantified through HI power spectrum, is expected to trace the matter power spectrum with a possible bias (Bharadwaj, Nath & Sethi 2001; Bharadwaj & Sethi 2001). Wyithe & Loeb (2009) have shown that the complications in the HI power spectrum arising due to the modulation of the ionizing field are less than 1%. Bagla, Khandai & Datta (2010) have used semi-numerical simulations to predict the HI bias. They have used three different prescriptions to assign HI mass to the dark matter haloes and have found the HI bias to be scale-independent on large scales ($k\leq 1$ Mpc*-1*). Guha Sarkar et al. (2012) and Vilaescusa Navarro et al. (2014) have used simillar simulations and their results are also found to be consistent with a scale-independent HI bias on large scales. Recently, Sarkar, Bharadwaj & Anathpindika 2016 have used semi-numerical simulations to model the HI bias and have provided a fitting formula for the HI bias $b_{HI}(k,z)$ in both $k$ and $z$ across $0.01\leq k\leq 10$ Mpc*-1* and $0\leq z\leq 6$.

The measurement of the HI power spectrum holds the possibility of constraining the background cosmological model through the Baryon Acoustic Oscillations (BAO) (Wyithe et al. 2008; Chang et al. 2008). Seo et al. (2010) have studied the possibility of measuring BAO using the HI power spectrum with a ground-based radio telescope. The measurement of the HI power spectrum can also be used to constrain cosmological parameters independent of the BAO (Bharadwaj et al. 2009; Visbal, Loeb & Wyithe 2009). The measurement of the HI power spectrum can also be used to constrain the neutrino mass (Loeb & Wyithe 2008). Villaescusa Navarro et al. (2015) have used hydrodynamical simulations to study the signatures of massive neutrinos on the HI power spectrum and put constraints on the neutrino mass. In a recent work, Pal & Guha Sarkar (2016) have studied the prospects of measuring the neutrino mass using the HI 21-cm and the Ly-$\alpha$ forest cross-correlation power spectrum.

Measurements of the HI power spectrum are sensitive to both the mean neutral hydrogen fraction $\Omega_{HI}(z)$ and the HI bias $b_{HI}$. Several measurements have been carried out in past years to measure the value of the $\Omega_{HI}$ both at low and at high redshifts. Measurements of the $\Omega_{HI}$ at low redshifts ($z\leq 1$) come from HI galaxy surveys (Zwaan et al. 2005; Martin et al. 2010; Delhaize et al. 2013), DLAs observations (Rao et al. 2006; Meiring at al. 2011) and HI stacking (Lah et al. 2007; Rhee at al. 2013), while measurements of $\Omega_{HI}$ at high redshifts ($1<z<6$) are from DLAs studies (Prochaska & Wolfe 2009; Norterdaeme et al. 2012; Zafar et al. 2013). Measurement of the HI power spectrum can provide astrophysical information about the HI distribution.

Efforts have been made to measure the $\Omega_{HI}$ at redshifts $z<1$. Chang et al. (2010) and Masui et al. (2013) have studied the cross-correlation of the HI intensity and the galaxy surveys, while Switzer et al. (2013) have studied the autocorrelation of HI intensity to measure the $\Omega_{HI}$. Ghosh et al. (2012) have used the GMRT observations to place a upper limit on the value of $\Omega_{HI}$ at $z=1.33$.

Several low frequency radio interferrometric arrays (CHIME111http://chime.phas.ubc.ca/, Bandura et al. 2014; BAOBAB222http://bao.berkeley.edu/, Pober et al. 2013) are planned to measure the BAO using the 21-cm signal from $z\leq 2.55$. Shaw et al. (2014) present theoretical estimates for the sensitivity of CHIME to constrain the line-of-sight and angular scale of the BAO. The Giant Meterwave Radio Telescope (GMRT 333http://gmrt.ncra.tifr.res.in/, Swarup et al. 1991) which operates at frequencies corresponding to HI in the redshift range $0\leq z\leq 8.5$. The GMRT is currently being upgraded. The prospects of detecting the HI power spectrum from the post reionization era for the upgraded GMRT (uGMRT) has been studied in Chatterjee et al. (2016). The proposed future telescopes SKA1-mid and SKA1-low 444https://www.skatelescope.org/home/technicaldatainfo/key-documents/, both hold the prospect of measuring the post-reionization HI power spectrum at a high level of precision (Guha Sarkar & Datta 2015; Bull et al. 2015; Santos et al. 2015).

There is a rich literature on the sensistivity estimates for various low frequency radio telescopes. Morales et al. (2005) present a general method to calculate the Epoch of Reionization (EOR) power spectrum sensitivity for any radio-interferrometric array. Harkar et al. (2010) have made sensitivity estimates for the Low Frequency Array (LOFAR555http://www.lofar.org/, van Haarlem et al. 2013). Simillar estimates have been made in the context of the Murchison Wide-field Array (MWA666http://www.mwatelescope.org, Bowman et al. 2013; Tingay et al. 2013). Beardsley et al. (2013) have estimated the sensitivity of the MWA to the EOR 21-cm power spectrum and McQuinn et al. (2006) present estimates for cosmological parameter estimation using MWA5000, a hypothetical extended version of MWA. Parsons et al. (2012) have explored redundancy calibration in the context of the Donald C. Backer Precision Array to Probe the Epoch of Reionization (PAPER777http://astro.berkeley.edu/dbacker/eor, Parsons et al. 2010). Pober et al. (2014) have studied the constraints acheiveable with the Hydrogen Epoch of Reionization Array (HERA888http://reionization.org/, DeBoer et al. 2016; Neben et al. 2016) and have found that a very high significance (\leavevmode\nobreak\ \hbox to0.0pt{>\hss}{\lower 4.30554pt\hbox{\sim}}\,30\sigma) detection of the reionization power spectrum is possible in even the most pessimistic scenarios. Aaron Ewall-Wice et al. (2016) have used the Fisher matrix formalism to predict the sensitivity with which it will be possible to constrain reionization and X-ray heating models with the future HERA and SKA phase I.

Here, we discuss the prospects of measuring the HI power spectrum at $z\sim 3$, using the upgraded Ooty radio Telescope (ORT). The ORT consists of a 530 m long and 30 m wide parabolic cyllindrical reflector, which is placed in the north-south direction on a hill having the same slope as the latitude (11o) of the station (Swarup et al. 1971; Sarma et al. 1975). It is possible to observe the same part of the sky through a single rotation of the long axis, which is aligned with earth’s rotation axis. The entire feed system of the ORT has 1056 dipoles, spaced 0.47 m apart from each other, which are placed along the focal line of the telescope. The cyllindrical Ooty Radio Telescope (ORT) is currently being upgraded (Prasad & Subrahmanya 2011a, b; Marthi & Chengalur 2014, Subrahmanya 2017a, b) to function as a linear radio interferrometric array the Ooty Wide Field Array (OWFA). The OWFA works at a nominal frequency of $\nu_{0}=326.5$ MHz, which corresponds to HI radiation from the redshift $z=3.35$. The OWFA can operate in two independenet interferrometric modes

• Phase I and Phase II. In this work, we have considered Phase II only. The Phase II has 264 antenna elements, where each antenna element consists of 4 dipoles. Each antenna has a rectangular aperture of dimension 1.92 m $\times$ 30 m. The field-of-view of OWFA Phase II is highly assymetric in dimension, $27.4^{o}\times 1.75^{o}$. The operating bandwidth for the Phase II is 40 MHz. The field-of-view and the observing bandwidth of the OWFA Phase II allow to observe the universe over a real space volume $\sim 0.3\,{\rm Gpc}^{3}$.

We now report some recent works related to OWFA. Calibration is an important issue for OWFA and it has been addressed in Marthi & Chengalur (2014). Gehlot & Bagla (2017) have followed the approach of Ali & Bharadwaj (2014) to predict the HI signal expected at OWFA. Marthi (2017) presents a programmable emulator for simulating OWFA observations for which foreground modelling and predictions are presented in Marthi et al. (2017). Chatterjee, Bharadwaj & Marthi (2017) present simulations of the HI signal expected at OWFA.

Ali & Bharadwaj (2014) (hereafter, Paper I) have studied the prospects of detecting the 21-cm signal using OWFA. In this paper, We have also made detailed foreground predictions for OWFA. In a recent study, Bharadwaj, Sarkar & Ali (2015)(hereafter, Paper II) have used the Fisher matrix analysis to make predictions for hours of observations to measure the HI power spectrum. We showed that the dominant contribution to the OWFA HI signal is from the $k$-range $0.02\leq k\leq 0.2$ Mpc*-1*. It was found that a $5\sigma$ detection of the HI power spectrum is possible with $\sim 150$ hours of observations using the Phase II. In this study, we have also explored the possibility of measuring the redshift space distortion parameter $\beta$. We found that the non-uniform sampling of the $\mathbf{k}$-modes does not make OWFA suitable for measuring $\beta$.

The predictions for OWFA, mentioned earlier, have all assumed that the HI power spectrum is related to the $\Lambda$CDM power spectrum with a scale-independent linear HI bias. All of these studies have focussed on measuring the amplitude of the HI power spectrum assuming that the shape of the matter power spectrum (Eisenstein & Hu 1998) is precisely known. It is interesting and worthwhile to consider a situation where both amplitude and the shape of the HI power spectrum is unknown. There are several astrophysical processes which could in principle, change the shape of the HI power spectrum without affecting the matter power spectrum. Further, uncertainties in the background cosmological model would also be reflected as changes in the observed HI power spectrum through various effects like redshift space distortion and Alcock-Paczynski (AP) effect. In this paper, we have considered the possibility of measuring the HI power spectrum using OWFA. For this purpose, we have divided the $k$-range into several bins and employed the Fisher matrix analysis to make predictions for measuring the HI power spectrum in each of these $k$-bins. Throughout our analysis, we have used the $\Lambda{\rm CDM}$ cosmology with PLANCK+WMAP9 best-fit cosmological parameters (Ade at al. 2014).

The paper is structured as follows. In section 2, we present the theoretical HI model which was used for calculating the signal and noise covariance. Here we also show the Fisher matrix technique which was employed for estimating the binned HI power spectrum. In section 3, we use the results from the Fisher matrix analysis to make predictions for measuring the binned 21-cm power spectrum. We end with summary and conclusions in Section 4.

2 Visibility covariance & Fisher matrix

OWFA measures visibilities $\mathcal{V}(\mathbf{U}_{a},\nu_{n})$ at given baselines $\mathbf{U}_{a}$ and frequency channel $\nu_{n}$. The baseline configuration of the OWFA is one-dimensional. It consists of 264 antennas, arranged in a linear array along the length of the cylinder. Assuming the $\mathbf{x}$-axis to be along the length of the cylinder, the baselines of the OWFA can be written as follows, i.e,

[TABLE]

where $a$ denotes the baseline number, $d=1.92$ m, is the distance between two consecutive antennas and $\lambda$ is the wavelength corresponding to the central observing frequency $\nu_{0}$. OWFA has a high degree of redundancy in baselines. For OWFA, any given baseline $\mathbf{U}_{a}$ occurs $(264-a)$ times in the array. This can be used to both calibrate the antenna gains (independent of the sky model) as well as to estimate the true visibilities (marthi & Chengalur 2014).

In reality baselines $\mathbf{U}_{a}$ change as frequency varies across the observing bandwidth $(B)$. This is an extremely important factor that needs to be taken into account in the actual data analysis. The expected fractional variation in the baseline $\Delta U/U$, about the central frequency $\nu_{0}$ over the bandwidth of observation is $\Delta U/U=B/2\nu_{0}\sim 4.5\%$ for $B=30$ MHz. This is not significant enough to consider in our analysis and we have kept the baselines fixed at the value, corresponding to the central frequency $\nu_{0}$. The actual bandwidth may be somewhat larger than $B=30$ MHz.

We express the telescope’s observing frequency bandwidth as $B=N_{c}\Delta\nu_{c}$ where $N_{c}$ is the number of the frequency channels and $\Delta\nu_{c}$ is the channel-width. For our analysis, We have used $N_{c}=300$ with $\Delta\nu_{c}=0.1$ MHz.

The measured Visibilities $\mathcal{V}(\mathbf{U}_{a},\nu_{n})$ can be expressed as of sum of the HI signal $\mathcal{S}(\mathbf{U}_{a},\nu_{n})$ and the noise $\mathcal{N}(\mathbf{U}_{a},\nu_{n})$, i.e.,

[TABLE]

assuming that foregrounds have been completely removed from the data.

For the Fisher matrix analysis, it is of convenience to decompose the visibilities $\mathcal{V}(\mathbf{U}_{a},\nu_{n})$ into delay channels $\tau_{m}$ (Morales 2005) rather than frequency channels $\nu_{n}$, i.e.,

[TABLE]

where $\tau_{m}$ is the delay channel which is defined as follows, i.e.,

[TABLE]

The visibilities $v(\mathbf{U}_{a},\tau_{m})$ and $v(\mathbf{U}_{b},\tau_{n})$ are uncorrelated for $m\neq n$ (Paper II). It is therefore necessary to only consider the visibility correlations with $m=n$ for which we define the visibility covariance matrix

[TABLE]

The visibility covariance matrix $C_{ab}(m)$ can be expressed in terms of the redshifted HI 21-cm brightness temperature power spectrum $P_{T}(\mathbf{k}_{\perp},k_{\parallel})$ as (eq. (5) of Paper II)

[TABLE]

where the first and the second terms refer to the signal and noise covariance respectively. Here $r_{\nu}$ is the co-moving distance between the observer and the region of space from where the HI radiation originated, $r^{{}^{\prime}}_{\nu}=\frac{dr_{\nu}}{d\nu}$ gives the conversion factor from frequency to co-moving distance ($r_{\nu}=6.85$ Gpc and $r_{\nu}^{{}^{\prime}}=11.5$ Mpc MHz*-1* for OWFA) and $\tilde{A}(\mathbf{U})$ is the Fourier transform of the OWFA primary beam pattern (eq. (6) of Paper I)). The factor $\frac{2k_{B}}{\lambda^{2}}$ gives the conversion from brightness temperature to specific intensity (where $k_{B}$ is the Bultzmann constant), $P_{T}(\mathbf{k_{\perp}},k_{\parallel})$ is the redshifted HI 21-cm brightness temperature power spectrum, $\mathbf{k_{\perp}}=\pi(\mathbf{U}_{a}+\mathbf{U}_{b})/r_{\nu}$ and $k_{\parallel}=2\pi\tau_{m}/r_{\nu}^{{}^{\prime}}$ respectively refer to the perpendicular and parallel components of the wavevector $\mathbf{k}$ with $k=\sqrt{k_{\perp}^{2}+k_{\parallel}^{2}}$ (Bharadwaj, Sarkar & Ali 2015). The rms. noise of the measured visibilities has the contribution from the system noise and has a value $\sigma_{N}=6.69\,{\rm Jy}$ for $16\,{\rm s}$ integration time (Table 1 of Paper I) and the factor $(264-a)^{-1}$ in the noise contribution accounts for the redundancy in the baseline distribution for OWFA.

For OWFA, the visibilities at any two baselines $\mathbf{U}_{a}$ and $\mathbf{U}_{b}$ are uncorrelated ($C_{ab}(m)=0$) if $|a-b|>1$ ie. the visibility at a particular baseline $\mathbf{U}_{a}$ is only correlated with the visibilities at the same baselines or the adjacent baselines $\mathbf{U}_{a\pm 1}$. Thus, for a fixed $m$, $C_{ab}(m)$ is a symmetric, tridiagonal matrix where the diagonal represents the visibility correlation at the same baseline whereas the upper and lower diagonals represent the visibility correlation between the adjacent baselines. Figure 1 of Paper II shows the signal contribution for the diagonal and off-diagonal terms of $C_{ab}(m)$. The covariance at adjacent baselines is approximately one fourth of the covariance at the same baselines. Further, the noise contributes only to the diagonal terms and it does not figure in the off-diagonal terms.

We have used the Fisher matrix (eq. (8) of Paper II)

[TABLE]

to predict the accuracy with which it will be possible to constrain the value of various parameters using observations with OWFA. The indices $\alpha,\gamma$ here refer to the different parameters whose values we wish to constrain. The inverse of the Fisher matrix $F_{\alpha\gamma}$ provides an estimate of the error-covariance (Dodelson 2003) for these parameters. In eq. (7), the indices $a,b,c,d$ are to be summed over all baselines. We have used eq. (6) to calculate the data covariance matrix $C_{ab}(m)$ and also its derivatives $[C_{ab}(m)]_{,\alpha}$ with respect to the parameters whose values we wish to constrain. A discussion of the parameters considered for the present analysis follows in the next section.

3 Modelling and Binning the HI power spectrum

The redshifted HI 21-cm brightness temperature power spectrum $P_{T}(\mathbf{k})$ is the quantity that will be directly measured by any cosmological 21-cm experiment. This quantifies the fluctuations in the brightness temperature originating from two different sources -

1. the intrinsic fluctuations of the HI density in real co-moving space and 2. the peculiar velocities which introduce brightness temperature fluctuations through redshift space distortion. For the purpose of this work, we have modelled $P_{T}(\mathbf{k})$ as

[TABLE]

where $P_{T}^{r}(k)$ is the power spectrum of the HI 21-cm brightness temperature fluctuations in real space and the factor $(1+\beta\mu^{2})^{2}$ quantifies the the effect of linear redshift space distortion due to peculiar velocities (Kaiser 1987; Bharadwaj, Nath & Sethi 2001; Ali & Bharadwaj 2014). Here $\beta$ is the redshift space distortion parameter and $\mu=k_{\parallel}/k$.

OWFA will probe the $k_{\perp}$ and $k_{\parallel}$ - range $1.9\times 10^{-3}\leq k_{\perp}\leq 5\times 10^{-1}$ Mpc*-1* and $1.8\times 10^{-2}\leq k_{\parallel}\leq 2.73$ Mpc*-1* respectively, thereby covering the $k$-range $1.82\times 10^{-2}\leq k\leq 2.73$ Mpc*-1*. The present work focuses on making predictions for the accuracy with which it will be possible to measure $P_{T}^{r}(k)$ using OWFA. For this purpose, we have divided the entire $k$-range probed by OWFA into 20 equally spaced logarithmic $k$-bins, and we use $k^{i}$ and $[P^{r}_{T}]^{i}$ (with $1\leq i\leq 20)$ to refer respectively to the average $k$ and $P_{T}^{r}(k)$ value for each bin. We have used $\ln([P_{T}^{r}]^{i})$ and $\ln(\beta)$ as the parameters for the Fisher matrix analysis (eq. (7)) which gives an estimate of the precision with which it will be possible to measure these parameters.

4 Results and Discussions

We need a fiducial model for the $P_{T}^{r}(k)$ and $\beta$ to carry out the Fisher matrix analysis. We model the $P_{T}^{r}(k)$ assuming that it traces the underlying matter power spectrum $P(k)$ with a linear bias $b_{\rm HI}$ as also assumed by Bharadwaj, Nath & Sethi (2001), Bharadwaj & Sethi (2001), Wyithe & Loeb (2009),

[TABLE]

where $\bar{x}_{\rm HI}$ is the mean neutral hydrogen fraction. The characteristic HI brightness temperature $\bar{\rm T}$ is defined as (Bharadwaj & Ali 2005)

[TABLE]

where $Y_{P}$ is the helium mass fraction and the other symbols in the above equation have their usual meanings.

The parameter $\beta$ in the observed $P_{T}(k)$ (eq. 9) is defined as $\beta=f(\Omega)/b_{\rm HI}$ where $f(\Omega)$ quantifies the growth rate of the matter density perturbations, whose value is specified by the background $\Lambda$CDM cosmological model. Note that the various terms used in eq. (9) correspond to the redshift, $z=3.35$ where HI radiation originated.

We have used $\bar{x}_{\rm HI}=0.02$ for our analysis which corresponds to the neutral gas mass density parameter $\Omega_{\rm gas}=10^{-3}$. This value of $\Omega_{\rm gas}$ comes from DLA observations in the redshift range of our interest (Prochaska & Wolfe 2009; Noterdaeme et al. 2012; Zafar et al. 2013). Simulations (Bagla, Khandai & Datta 2010; Guha Sarkar et al. 2012; Sarkar, Bharadwaj & Anathpindika 2016) and analytical modelling (Marin et al. 2010) suggest the use of a constant, scale-independent bias at wave numbers $k\leq 1$ Mpc*-1*, and we use the value $b_{HI}=2.0$ for our entire analysis. We use these values and the cosmological parameters to calculate the fiducial values of $[P_{T}^{r}]^{i}$ (pointsin Figure 2) and $\beta=4.93\times 10^{-1}$.

The value of $\beta$ can be estimated by sampling the Fourier modes $\mathbf{k}$ with a fixed magnitude $k$ which are, however, oriented at different directions to the line-of-sight. In other words, $\mu=k_{\parallel}/k$ should uniformly span over the entire range $-1\leq\mu\leq 1$. The minimum value of $k_{\parallel}$ probed by OWFA is approximately $10$ times larger than the minimum value of $k_{\perp}$. The maximum value of $k_{\parallel}$ is also $\sim 4$ times larger than the maximum value of $k_{\perp}$ (Table II of Paper II). In addition to this, the sampling width for $k_{\parallel}$ is roughly $\sim 20$ times larger than that of $k_{\perp}$. These disparities lead to a non-uniform distribution where the ${\bf k}$ modes are largely concentrated around $\mu=1$ (see Figure 3 of Paper II). This anisotropic distribution of the ${\bf k}$ modes does not make OWFA very suitable for measuring $\beta$, and we do not consider this in our analysis.

We have considered two different cases for error predictions. In the first situation, we consider the Conditional errors $\sigma_{ic}$ for the measurement of the binned HI power spectrum $[P_{T}^{r}]^{i}$. The conditional error $\sigma_{ic}$ represents the error on the measurement of $[P_{T}^{r}]^{i}$ in a situation where the values of all the other parameters are precisely known. Here, we calculate $\sigma_{ic}$ for the $i$-th bin by assuming that the values of $\beta$ and $[P_{T}^{r}]^{j}$ are precisely known for all the other bins. We use $\sigma_{ic}=1/\sqrt{F_{ii}}$ to compute the conditional error for the $i$-th bin.

In the second situation, we have considered the Marginalized errors $\sigma_{im}$ for the measurement of $[P_{T}^{r}]^{i}$. The marginalized error $\sigma_{im}$ gives the error on the measurement of $[P_{T}^{r}]^{i}$ without assuming any prior information about the other parameters. While estimating the error for the $i$-th bin, we have marginalized over the values of $\beta$ and $[P_{T}^{r}]^{j}$ in the other bins. In our previous work (Paper II), we have calculated the marginalized error on the measurement of the amplitude of the HI power spectrum with a prior on $\beta$ in the range $0.329\leq\beta\leq 0.986$. In the present work, we have not imposed any prior on $\beta$ and we have marginalized $\ln([P_{T}^{r}]^{i})$ and $\ln(\beta)$ over the entire range $-\infty$ to $+\infty$. We use $\sigma_{im}=\sqrt{[F^{-1}]_{ii}}$ to calculate the marginalized error for the $i$-th bin. The conditional and the marginalized errors here represent the two limiting cases, and the error estimates would lie somewhere in between $\sigma_{ic}$ and $\sigma_{im}$ if we impose priors on the value of $\beta$ or any of the other parameters.

In Paper II, we have shown that a $5\sigma$ detection of the amplitude of the $P_{T}^{r}$ is possible with $\sim 150$ hours of observations. We therefore need to consider an observing time $t>150$ hours for measuring the $[P_{T}^{r}]^{i}$ in different $k$-bins. Figure 1 shows both the conditional ($\sigma_{ic}$) and marginalized ($\sigma_{im}$) errors for $1000$ hours of observation. Here $\sigma_{ic}$ and $\sigma_{im}$ are respectively the conditional and marginalized errors for different $\ln([P^{r}_{T}]^{i})$ which are the parameters for the Fisher matrix analysis. Here $\sigma_{ic}$ and $\sigma_{im}$ represent the two limiting cases for the error estimates. We expect the error estimates to lie between these two limiting values in case we impose a prior on the value of $\beta$ (Paper II).

We find that the values of $\sigma_{ic}$ and $\sigma_{im}$ agree within $15\%$, except at the $k$-bins lying in the range $0.06\leq k\leq 0.3$ Mpc*-1* where the difference is $\sim 20-35\%$. This suggests that $\sigma_{ic}=1/\sqrt{F_{ii}}$ and $\sigma_{im}=\sqrt{[F^{-1}]_{ii}}$ are not significantly different, indicating that the contribution from the off-diagonal terms of $F_{ij}$ are small. We therefore conclude that the measurements of $P_{T}^{r}$ in different $k$-bins are by and large uncorrelated. In the subsequent analysis, we have used $\sigma_{ic}$ for predicting errors on the measurements of $[P_{T}^{r}]^{i}$ in different $k$-bins.

Figure 2 shows the binned HI power spectrum $[P_{T}^{r}]^{i}$ with the $1\sigma$ errors $\Delta[P_{T}^{r}]^{i}=\sigma_{ic}\times[P_{T}^{r}]^{i}$ for $1,000$ hours of observation . The error $\Delta[P_{T}^{r}]^{i}$ on the measurement of the $P_{T}^{r}$ in a given $k$-bin is the combination of contributions from the system noise and the cosmic variance. The noise term in eq.(6) is suppressed by the factor $(264-a)^{-1}$ due to the redundancy of the OWFA baselines. We see that the noise contribution goes up as the baseline number $a$ is increased. The small $k$-bins which correspond to small baselines, have smaller noise contribution than the larger $k$-bins which correspond to large baselines. Here we have used logarithmic binning where the bin width and the number of $k$-modes in a bin increase with $k$. The cosmic variance in a given $k$-bin goes down with number of $k$-modes in that bin. We therefore expect the cosmic variance to be maximum at the smallest $k$-bin and decrease with increasing $k$. As a whole the errors at smaller $k$-bins are dominated by the cosmic variance whereas at larger $k$-bins, the errors are dominated by the system noise.

We can see from Figure 1 that $\sigma_{ic}=\Delta[P_{T}^{r}]^{i}/[P_{T}^{r}]^{i}$, which is the relative error on the binned power spectrum, is minimum in the range $k\sim 0.1-0.2\,{\rm Mpc}^{-1}$. The cosmic variance dominates the relative error at smaller values of $k$ $(<0.1\,{\rm Mpc}^{-1})$ whereas the system noise dominates at larger $k$ values $(>0.2\,{\rm Mpc}^{-1})$. We also see that the relative error is lower than $0.2$ in the range $0.05\leq k\leq 0.3\,{\rm Mpc}^{-1}$ where our results predict a $5\sigma$ detection of the binned power spectrum (Figure 2).

We have so far considered the errors on the measurement of $[P_{T}^{r}]^{i}$ with a given hours of observing time ($1,000$ hours). We shall now try to understand how the errors $\sigma_{ic}$ vary with observation time $t$. The time dependence of the visibility covariance $C_{ab}(m)$ (eq. (6)) comes in through the rms. noise of the measured visibilities $\sigma_{N}$ which scales inversely with $\sqrt{t}$, ie. $\sigma_{N}\sim 1/\sqrt{t}$. We expect the visibility covariance $C_{ab}(m)$ to vary inversely with $t$, ie. $C_{ab}(m)\sim 1/t$ for small observing times where the noise contribution is considerably larger than the signal, and we expect $C_{ab}(m)$ to have a constant value, independent of the observing time, for large values of $t$. The derivatives of the $C_{ab}(m)$ which appear in the Fisher Matrix (eq. (7)) are independent of $t$. It then follows that the Fisher matrix $F_{\alpha\gamma}$ scales as $F_{\alpha\gamma}\sim t^{2}$ for small observation times and $F_{\alpha\gamma}$ has a constant value for large $t$. We therefore expect the relative errors $\sigma_{ic}$ to vary as $\sigma_{ic}\sim 1/t$ for small observing times, and become independent of $t$ for large observation times where the error is dominated by the cosmic variance.

Figure 3 shows a contour plot of the Signal-to-Noise Ratio (SNR)

[TABLE]

as functions of the Fourier mode $k$ and observation time $t$. We see that a statistically significant measurement ( $3\sigma$) of the binned power spectrum is only possible for observations times greater than $200\,{\rm hours}$. A $3\sigma$ detection of $[P_{T}^{r}]^{i}$ is possible in the $k$-range $0.04\leq k\leq 0.2$ Mpc*-1* with $200-300$ hours of observation. Detection at a significance of $5\sigma$ is not possible with $t\leq 500$ hours of observation. We find that a $5\sigma$ detection of $[P_{T}^{r}]^{i}$ is possible for $0.05\leq k\leq 0.3$ Mpc*-1* with $1,000$ hours of observing time. Note that the SNR peaks in the range $k\sim 0.1-0.2$ Mpc*-1*. In this $k$ range the SNR continues to increase with $t$ for the entire $t$ range shown here and a $10\sigma$ detection is possible with $2,000$ hours of observation. At $k<0.1\,{\rm Mpc}^{-1}$, the SNR stops increasing with $t$ beyond a certain point. The SNR here becomes dominated by the cosmic variance as $t$ is increased, and the SNR contour becomes parallel to the $t$ axis. We see that irrespective of the observing time, a $5\sigma$ detection is not possible for $k<0.036\,{\rm Mpc}^{-1}$ if only one pointing is considered. For $k>0.2\,{\rm Mpc}^{-1}$ the error is system noise dominated, and the SNR continues to increase with increasing $t$. However, we see that a $5\sigma$ detection is not possible for $k>0.5\,{\rm Mpc}^{-1}$ within $2,000$ hours of observation.

As mentioned earlier, we expect SNR $\propto\,t$ for small observing times when the error is system noise dominated, and we expect the SNR to saturate at a fixed value for large observing times where the cosmic variance dominates. Figure 4 shows how the SNR changes with observing time $t$ for a few representative $k$-bins. The small $k$ bins have a relatively large cosmic variance. We see that the SNR at the smallest $k$ bin ($0.036\,{\rm Mpc}^{-1}$) shown in this figure is nearly saturated at a very small observing time ($t\sim 300$ hours), and increases very slowly for larger observing times. A $5\sigma$ detection in this bin requires $\sim 10,000$ hours of observation. The $k$-bin at $0.33\,{\rm Mpc}^{-1}$ shows the ${\rm SNR}\propto t$ scaling for $t\leq 700\,{\rm hours}$, beyond which the increase in SNR is slower. The two larger $k$-bins shown in the figure show the ${\rm SNR}\propto t$ behaviour over the entire $t$ range considered here. However, note that the largest $k$-bin with $k^{i}=1.16\,{\rm Mpc}^{-1}$ shown in the figure has a rather low SNR, and a $5\sigma$ detection is only possible with $10,000\,{\rm hours}$ of observation.

As mentioned earlier, we expect the SNR to increase with time $t$, ie. ${\rm SNR}\propto t$ for small observing times. The increase in the SNR slows down for larger observing times, and for even larger observing times $(t\geq t_{\rm CV})$ the SNR saturates at a fixed value which is determined by the cosmic variance in the particular $k$-bin. Here $t_{\rm CV}$ referes to the observing time beyond which the SNR is determined by the cosmic variance, and we have estimated this for the different $k$ bins such that the SNR increases slower than $t^{0.002}$ for $(t\geq t_{\rm CV})$. Figure 5 shows $t_{\rm CV}$ for the different $k$-bins. For any particular bin, it is not possible to increase the SNR any further by increasing the observing time beyond $t_{\rm CV}$. We find that $t_{\rm CV}$ increases approximately as $t_{\rm CV}\propto k^{0.63}$ for $k\leq 1.0\,{\rm Mpc}^{-1}$. The increase in the $t_{\rm CV}$ is rather slow for $k\geq 1\,{\rm Mpc}^{-1}$ and saturates at $t_{\rm CV}\sim 15000\,{\rm hr}$ beyond $k=2.0\,{\rm Mpc}^{-1}$. This behaviour is decided by a combination of several factors including the OWFA baseline redundancy, the sampling of the 3D Fourier modes and the logarithmic binning.

The discussion till now has entirely focused on observations in a single field of view. As already mentioned, the ${\rm SNR}$ ceases to increase with observing time once $t\sim t_{\rm CV}$. We see that $t_{\rm CV}\sim 1,000\,{\rm hr}$ at the smallest $k$ bin. The ${\rm SNR}$ in this bin will saturate for $t>1,000\,{\rm hr}$ and it is necessary to observe multiple fields of view to increase the ${\rm SNR}$ any further. The ${\rm SNR}$ scales as ${\rm SNR}\propto\sqrt{N}$, where $N$ is the number of fields of view. A possible observational strategy for OWFA would be to observe multiple fields of view, with each field being observed for a duration of $1,000-2,000\,{\rm hr}$. The $3\sigma$ contour in Figure 3 would correspond to $5.2\sigma$ for observations in $N=3$ fields of view. We see that a $5\sigma$ detection is possible in nearly all the $k$ bins if $3$ fields of view are observed for $1,500\,{\rm hr}$ each.

5 Summary and Conclusions

We have considered Phase II of OWFA to study the prospects of measuring the redshifted 21-cm power spectrum in different $k$-bins. The entire analysis is restricted to observations in a single field of view. We find that a $5\sigma$ detection of the binned power spectrum is possible in the $k$ range $0.05\leq k\leq 0.3\,{\rm Mpc}^{-1}$ with $1,000$ hours of observation. The ${\rm SNR}$ peaks in the $k$ range $0.1-0.2\,{\rm Mpc}^{-1}$ where a $10\sigma$ detection is possible in $2,000$ hours of observation. Our study reveals that it is not very advantageous to observe much beyond $1,000$ hours as the error in measuring the power spectrum become cosmic variance dominated in several of the small $k$-bins, and the ${\rm SNR}$ in these bins increase rather slowly with increasing $t$.

As discussed earlier, the variation of the baseline over the observing bandwidth is $\sim 5\%$. This makes both the diagonal and off-diagonal components of the Fisher matrix to change, which is expected to be not more than $5-10\%$.

The redshifted 21-cm signal provides an unique way to measure the BAO in the post-reioinization era ($z\leq 6$). This is perceived to be a sensitive probe of the dark energy. The BAO is a relatively small feature ($\sim 10-15\%$) that sits on the HI power spectrum. The five successive peaks of the BAO span the $k$ range $0.045\leq k\leq 0.3$ Mpc*-1*, which is well within the $k$-range probed by OWFA. The detection of the BAO requires measuring the HI power spectrum at a significance of $50\sigma$ or more. From Figure 3, we find that such a sensitivity cannot be achieved in the relevant $k$ range within $t\sim 2000$ hours of observation. It is also clear that the required sensitivity cannot be achieved by considering observations in a few fields of view. For detecting the BAO it is necessary to consider a different observational strategy covering the entire sky (e.g. Shaw et al. 2013). We plan to address this in future work.

Acknowledgement

The authors acknowledge Jayaram N. Chengalur, Jasjeet S. Bagla, Tirthankar Roy Choudhury, C.R. Subrahmanya, P.K. Manoharan and Visweshwar Ram Marthi for helpful discussions. AKS would like to acknowledge Rajesh Mondal and Suman Chatterjee for their help. AKS would like to specially mention Debanjan Sarkar for a detailed reading of the manuscript and providing important suggestions. AKS would also like to express thanks to the anonymous referees whose critical suggestions helped towards a major improvement of the language and presentation of the manuscript. SSA would like to acknowledge CTS, IIT Kharagpur for the use of its facilities and thank the authorities of IUCAA, Pune, India for providing the Visiting Associateship programme.