LATE-TIME COSMOLOGY WITH 21 cm INTENSITY MAPPING EXPERIMENTS

Radio telescopes measure flux density—the integral of the source intensity, ${{I}_{\nu }},$ over the solid angle of the telescope beam. We derive an expression for the H i line intensity in Appendix A. In the Rayleigh–Jeans limit, this can be converted into an effective H i brightness temperature, ${{T}_{b}}={{c}^{2}}{{I}_{\nu }}/2{{k}_{B}}{{\nu }^{2}},$ that can be split into a homogeneous part and a fluctuating part, ${{T}_{b}}={{\bar{T}}_{b}}(1+{{\delta }_{{\rm HI}}}),$ where (from Appendix A)

$$\begin{eqnarray}&&{{\bar{T}}_{b}}=\frac{3}{32\pi }\frac{h{{c}^{3}}{{A}_{10}}}{{{k}_{B}}{{m}_{p}}\nu _{21}^{2}}\frac{{{(1+z)}^{2}}}{H(z)}{{{\Omega}}_{{\rm HI}}}(z){{\rho }_{c,0}}.\end{eqnarray} \tag{ 2 }$$

The fluctuations are the quantity of interest here, and so we identify the cosmological signal as

$$\begin{eqnarray*}&&\delta {{T}^{S}}({{{\boldsymbol{ \theta }} }_{p}},{{\nu }_{p}})={{\bar{T}}_{b}}(z){{\delta }_{{\rm HI}}}({{{\boldsymbol{r}} }_{p}},z).\end{eqnarray*}$$

At late times, most of the neutral hydrogen content of the universe is expected to be localized to dense gas clouds within galaxies, where it is shielded from ionizing photons. We therefore expect HI to be a biased tracer of the dark matter distribution, just as galaxies are. This allows us to write the H i density contrast as ${{\delta }_{{\rm HI}}}={{b}_{{\rm HI}}}\star {{\delta }_{M}}$ (where ${{\delta }_{M}}$ is the total matter density perturbation, and ⋆ denotes convolution, accounting for the possibility of scale- and time-dependent biasing).

Because the H i intensity is measured as a function of frequency (and thus redshift) rather than comoving distance, we must also account for RSDs caused by the peculiar velocities of the clouds and the galaxies in which they reside. Following Kaiser (1987), we write the (Fourier-transformed) redshift-space H i contrast as

$$\begin{eqnarray}&&{{\delta }_{{\rm HI}}}({\boldsymbol{k}} )=({{b}_{{\rm HI}}}+f{{\mu }^{2}}){\rm exp} \left( -{{k}^{2}}{{\mu }^{2}}\sigma _{{\rm NL}}^{2}/2 \right){{\delta }_{M}}({\boldsymbol{k}} ),\end{eqnarray} \tag{ 3 }$$

where $\mu \equiv {{k}_{\parallel }}/k$ and the flat-sky approximation has been used again. We have assumed that the H i velocities are unbiased. The linear growth factor, f, is a key observable, telling us much about the growth of structure on linear scales; we will study it in detail in Section 4.3. The exponential term accounts for the "Fingers of God" effect due to uncorrelated velocities on small scales,⁸ which washes out structure in the radial direction past a cutoff scale parameterized by the non-linear dispersion, ${{\sigma }_{{\rm NL}}}.$

Substituting (3) into (1) and making use of the definition of the isotropic matter power spectrum,

$$\begin{eqnarray*}&&\langle \delta _{M}^{*}({\boldsymbol{k}} ){{\delta }_{M}}({\boldsymbol{k}} ^{\prime} )\rangle \equiv {{(2\pi )}^{2}}P(k){{\delta }^{3}}({\boldsymbol{k}} -{\boldsymbol{k}} ^{\prime} ),\end{eqnarray*}$$

we can write the signal covariance as

$$\begin{eqnarray}&&{{C}^{S}}({\boldsymbol{q}} ,y)=T_{b}^{2}({{z}_{i}})\frac{{{P}_{{\rm tot}}}({{z}_{i}},\frac{{\boldsymbol{q}} }{r},\frac{y}{{{r}_{\nu }}})}{{{r}^{2}}{{r}_{\nu }}},\end{eqnarray} \tag{ 4 }$$

where the factor of ${{r}^{2}}{{r}_{\nu }}$ is from the conversion into observational Fourier coordinates, $({\boldsymbol{q}} ,y),$ and we have defined

$$\begin{eqnarray*}\begin{array}{rcl} {{P}_{{\rm tot}}}({{z}_{i}},{{{\boldsymbol{k}} }_{\bot }},{{k}_{\parallel }}) & = & {{F}_{{\rm RSD}}}({{{\boldsymbol{k}} }_{\bot }},{{k}_{\parallel }}){{D}^{2}}(z)P(k,z=0) \\ {{F}_{{\rm RSD}}}({{{\boldsymbol{k}} }_{\bot }},{{k}_{\parallel }}) & = & {{({{b}_{{\rm HI}}}+f{{\mu }^{2}})}^{2}}{\rm exp} (-{{k}^{2}}{{\mu }^{2}}\sigma _{{\rm NL}}^{2}). \\ \end{array}\end{eqnarray*}$$

The redshift dependence of the matter power spectrum has been factored out into the linear growth factor, D, which is normalized to $D(z=0)=1.$ This is related to the growth rate by $f=d{\rm log} D/d{\rm log} a.$ Strictly, the growth factor should be scale-dependent on small scales, but for simplicity we neglect this possibility here as we will mostly be concerned with larger scales.

It is straightforward to calculate fiducial values for the cosmological functions in (4). We use CAMB (Lewis et al. 2000) to calculate P(k) at z = 0 for our chosen fiducial cosmological parameters, and use the simple parameterization $f(z)={\Omega}_{M}^{\gamma }(z)$ for the linear growth rate (Peebles 1980; Linder 2005), where ${{{\Omega}}_{M}}(z)={{{\Omega}}_{M}}{{(1+z)}^{3}}H_{0}^{2}/{{H}^{2}}(z),$ and $\gamma \approx 0.55$ for ΛCDM. For the other functions in (4), however, there is considerably more uncertainty in the choice of fiducial model.

One key uncertainty is the behavior of the H i bias, ${{b}_{{\rm HI}}}.$ The bias depends on the size of host dark matter halos; if a halo is too small, gas clouds would be unable to gain sufficient density to shield themselves and keep the hydrogen neutral. The halo dependence can be modeled using the halo mass function with an appropriate lower mass cutoff (or lower rotation velocity; see, e.g., Bagla et al. 2010). There are a few candidate models for the evolution of the bias as a function of redshift that fit the current constraints from observations (Switzer et al. 2013) or are calibrated against simulations (Wilman et al. 2008), but there is considerable disagreement between them. In Section 6.1 and Appendix B, we discuss the impact of the uncertain bias evolution. Unless stated otherwise, we will use a linear bias model for the rest of the paper and—rather conservatively—marginalize over the value of b_HI separately in each redshift bin.

Another major uncertainty is in the H i density fraction, ${{{\Omega}}_{{\rm HI}}}={{\rho }_{{\rm HI}}}/{{\rho }_{c,0}}.$ This enters the signal covariance through ${{\bar{T}}_{b}}(z),$ since ${{\bar{T}}_{b}}\propto {{{\Omega}}_{{\rm HI}}}.$ The current best constraints on the H i fraction come from Switzer et al. (2013), who find

$$\begin{eqnarray*}&&{{{\Omega}}_{{\rm HI}}}{{b}_{{\rm HI}}}=4.3\pm 1.1\times {{10}^{-4}}\end{eqnarray*}$$

at the 68% confidence level at z = 0.8. This constitutes a relatively large uncertainty in the overall amplitude of the H i signal and, correspondingly, the signal-to-noise ratio that can be achieved by a given experiment. We investigate the impact of this uncertainty in Section 6.1, but for the rest of the paper we will adopt a fiducial value of ${{{\Omega}}_{{\rm HI},0}}=4.86\times {{10}^{-4}}.$

The non-linear dispersion scale, ${{\sigma }_{{\rm NL}}},$ is yet another source of uncertainty. Recent values from the literature vary between ${{\sigma }_{{\rm NL}}}\approx 4\;{\rm and}\;10$ Mpc (e.g., Li et al. 2007; Reid & White 2011; Reid et al. 2012; Contreras et al. 2013); for our fiducial model, we choose a middling value of ${{\sigma }_{{\rm NL}}}=7$ Mpc (Li et al. 2007), which corresponds to a non-linear scale of ${{k}_{{\rm NL}}}\sim 0.14$ Mpc⁻¹ (or a velocity dispersion of ∼500 km s⁻¹). We check the sensitivity of our results to this choice in Section 6.2. IM experiments can independently constrain ${{\sigma }_{{\rm NL}}},$ so we leave it free, marginalizing over it as a nuisance parameter in all forecasts.

The noise covariance models the instrumental and sky noise for a given experiment, but we will also use it to include the effects of instrumental beams. For radio telescopes, assuming uncorrelated Gaussian noise, the noise covariance has the standard form

$$\begin{eqnarray}&&{{C}^{N}}({\boldsymbol{q}} ,y)=\frac{T_{{\rm sys}}^{2}}{{{t}_{{\rm tot}}}{\Delta}\nu }{{U}_{{\rm bin}}}\;\mathcal{I}B_{\bot }^{-2}B_{\parallel }^{-1},\end{eqnarray} \tag{ 5 }$$

where ${{T}_{{\rm sys}}}$ is the system temperature, ${{t}_{{\rm tot}}}$ is the total integration time, ${{U}_{{\rm bin}}}={{S}_{{\rm area}}}\;{\Delta}\tilde{\nu }$ is the volume of an individual redshift bin, and ${{S}_{{\rm area}}}$ and ${\Delta}\tilde{\nu }$ are the survey area and (dimensionless) bandwidth within the redshift bin, respectively. The factors of $\mathcal{I}$ and B describe the number (or number density) of receivers and their corresponding frequency and angular responses.

The system temperature has two main components: the instrument temperature, T_inst, which depends on the hardware design, and ${{T}_{{\rm sky}}}\approx 60\;{\rm K}\times {{(\nu /300\;{\rm MHz})}^{-2.5}},$ which accounts for atmospheric and background radio emission, to give a total ${{T}_{{\rm sys}}}={{T}_{{\rm inst}}}+{{T}_{{\rm sky}}}.$ Values for T_inst are quoted in the design specifications for a given experiment and are typically a few tens of kelvin.

The survey area and total integration time are not intrinsic to the design of the instrument, but are instead chosen as part of the survey strategy. We will systematically examine the effects of varying these parameters in Section 7. One of the advantages of IM with radio telescopes is that substantial fractions of the sky can be surveyed to a useful depth over the course of only a year or so. This is thanks in part to the relative cheapness of low-noise multiple-receiver systems and the ∼degree-scale primary beams of dishes in most arrays, both of which act to substantially improve survey speed. For much of what follows, we will assume that all experiments can perform between 10 and 25,000 deg² surveys over 10,000 hr total observing time, which is reasonable for a dedicated survey telescope.

We now turn to the effective beam terms. The instrumental resolution in the radial direction is limited by the bandwidth of an individual frequency channel, $\delta \nu .$ We approximate the channel bandpass by a Gaussian, which gives an effective beam in the parallel direction,

$$\begin{eqnarray*}&&{{B}_{\parallel }}(y)={\rm exp} \left( -\frac{{{(y\delta \nu /{{\nu }_{21}})}^{2}}}{16{\rm ln} 2} \right);\end{eqnarray*}$$

the numerical factor comes from the definition of the FWHM, $\sigma ={{\theta }_{{\rm FWHM}}}\sqrt{8{\rm ln} 2}.$ Modern radio receivers can typically be built with narrow channel bandwidths of around 100 kHz or less. Narrow channels allow for more precise removal of artificial radio interference (RFI), for example, but also increase the data rate, which may require expensive increases in correlator performance for interferometers. In practice, the channel bandwidth is not the limiting factor in the radial resolution, as for realistic $\delta \nu$ this is instead determined by the non-linear dispersion scale, ${{\sigma }_{{\rm NL}}}.$

The expressions for the dish multiplicity factor, $\mathcal{I},$ and transverse effective beam, ${{B}_{\bot }},$ depend on whether the array is used as an interferometer or a collection of independent single dishes, i.e., whether the signals from individual dishes are correlated with one another or not. For single-dish experiments that only use the autocorrelation of the signal from each dish, we have

$$\begin{eqnarray*}\begin{array}{rcl} \mathcal{I} & = & \frac{f(\nu )}{{{N}_{{\rm b}}}{{N}_{{\rm d}}}} \\ {{B}_{\bot }}({\boldsymbol{q}} ) & = & {\rm exp} \left( -\frac{{{(q{{\theta }_{{\rm B}}})}^{2}}}{16{\rm ln} 2} \right), \\ \end{array}\end{eqnarray*}$$

where ${{\theta }_{{\rm b}}}\approx \lambda /{{D}_{{\rm dish}}}$ is the FWHM of the beam of an individual dish of diameter D_dish at some wavelength λ, and ${{N}_{{\rm d}}}$ is the number of dishes in the array. ${{N}_{{\rm b}}}$ is the number of beams, which differs from unity if the experiment has multiple pixels or phased array feeds (PAFs) per dish. Each dish/beam will typically survey a different patch of the sky, thus increasing the survey speed. Any additional frequency dependence of the sensitivity (e.g., due to beam overlap for PAF receivers) can be accounted for by the $f(\nu )$ factor; specific forms of the noise expression appropriate for different types of receiver are given in Appendix D.

For interferometers, in the case where we assume multiple pointings, i.e., ${{S}_{{\rm area}}}\gt {\rm FOV},$ we have

$$\begin{eqnarray*}&&\mathcal{I}B_{\bot }^{-2}=\frac{{\rm FOV}}{n(u=q/2\pi )},\end{eqnarray*}$$

where n(u) is the number density of samples in the uv plane as a function of $|u|,$ and ${\rm FOV}\approx {{(\lambda /{{D}_{{\rm dish}}})}^{2}}$ is the field of view. Each configuration of baselines will lead to a different n(u), although if one assumes constant sampling in uv, it can be approximated by

$$\begin{eqnarray}&&n(u)=\frac{{{N}_{d}}({{N}_{d}}-1)}{2\pi (u_{{\rm max} }^{2}-u_{{\rm min} }^{2})},\end{eqnarray} \tag{ 6 }$$

where ${{u}_{{\rm max} }}={{D}_{{\rm max} }}/\lambda ,$ ${{u}_{{\rm min} }}={{D}_{{\rm min} }}/\lambda ,$ and D_max, D_min are the lengths of the longest/shortest baselines. The effective beam in the transverse direction is determined by n(u) and so does not need to be defined separately.

For interferometric experiments where the baseline distribution is available, we calculate n(u) specifically for that distribution; otherwise, we use the approximation in Equation (6). For the former, one has to assume a declination of observation as well as a baseline distribution. As the sky drifts, a set of tracks will be mapped out onto the uv plane. These are typically split into bins of size ${{({\Delta}u)}^{2}}\sim 1/{\rm FOV},$ which can be taken to be independent. It is then possible to construct a simple model for n(u) that is good enough for our forecasts (see Appendix C).

Foreground contamination from the galaxy and extragalactic point sources dwarfs the cosmological H i signal. A number of different methods have been proposed for modeling and subtracting the foregrounds (Oh & Mack 2003; Barkana & Loeb 2005; Santos et al. 2005; Morales et al. 2006; Wang et al. 2006; Gleser et al. 2008; Jelić et al. 2008; Liu et al. 2009; Petrovic & Oh 2011), but in this paper we will simply assume that some sort of method has been applied that removes them, leaving behind some residual contamination whose variance can be modeled as a sum of smooth power spectra. (One could also model instrumental calibration residuals in this way.) Our model for the residual foreground is

$$\begin{eqnarray}&&{{C}^{F}}({\boldsymbol{q}} ,y)=\epsilon _{{\rm FG}}^{2}\mathop{\sum }\limits_{X}\;{{A}_{X}}{{\left( \frac{{{l}_{{\rm p}}}}{2\pi q} \right)}^{{{n}_{X}}}}{{\left( \frac{{{\nu }_{p}}}{{{\nu }_{{\rm i}}}} \right)}^{{{m}_{X}}}}.\end{eqnarray} \tag{ 7 }$$

The amplitude and index parameters for four representative foregrounds are given in Table 1, following Santos et al. (2005).

Subtraction algorithms also introduce a minimum wave number below which cosmological information cannot be extracted. This is because the smooth variation of the foregrounds in frequency is difficult to separate from cosmological modes on scales comparable to the total survey bandwidth, ${{k}_{{\rm FG}}}\sim 1/({{r}_{\nu }}{\Delta}{{\tilde{\nu }}_{{\rm tot}}}).$ A subtraction method that relies on fitting and subtracting smoothly varying functions will necessarily remove some power from radial cosmological modes larger than this scale as well.

We have introduced an overall scaling, ${{\epsilon }_{{\rm FG}}},$ to parameterize the efficiency of the foreground removal process: ${{\epsilon }_{{\rm FG}}}=1$ corresponds to no foreground removal, while we will probably need ${{\epsilon }_{{\rm FG}}}\lesssim {{10}^{-5}}$ to be able to extract the cosmological signal. One can also interpret ${{\epsilon }_{{\rm FG}}}$ as a measure of how smooth (or correlated) foregrounds are in frequency, and thus how well they can be modeled by smooth deterministic functions. For example, if we assume a Gaussian correlation function along the frequency direction, we have that ${{\epsilon }_{{\rm FG}}}\sim {\rm exp} [-{{({\Delta}\nu /\xi )}^{2}}],$ where ${\Delta}\nu$ is the bandwidth of the redshift bin we are probing and ξ is the correlation length in frequency. We have neglected cross-correlations between different frequencies here, although including them would not be difficult.

This treatment of foreground subtraction is necessarily simplified. For example, we are assuming that the various contributions to the total signal remain uncorrelated, yet all of the subtraction methods proposed so far remove modes that receive contributions from all components. Although the foregrounds will be the dominant part of the subtracted signal by far, this process will nevertheless induce cross-correlations between whatever is left in the residual. Practical experience of foreground subtraction from real data is quite limited so far (see Chang et al. 2010; Masui et al. 2013; Switzer et al. 2013 for some initial attempts), and so we do not have a good picture of how important various aspects of the foreground problem are yet. For the time being, we believe that our model captures the essential elements of the foregrounds sufficiently well to be of use. We will return to the issue of foreground modeling in Section 6.3.

We are now in a position to construct the full Fisher matrix. To do this, we need to sum over the number of independent modes in ${\boldsymbol{q}}$ and y. If ${{S}_{{\rm area}}}$ is the survey area, ${\rm FOV}$ is the field of view of a single array element, and ${\Delta}\tilde{\nu }$ is the (dimensionless) bandwidth in the redshift bin, we have ${\Delta}y=2\pi /{\Delta}\tilde{\nu },$ and ${\Delta}q=2\pi /\sqrt{{{S}_{{\rm area}}}}$ (single-dish) or ${\Delta}q=2\pi /\sqrt{{\rm FOV}}$ (interferometer). The sum over modes is then given by

$$\begin{eqnarray*}&&\int \frac{{{d}^{2}}qdy}{{{({\Delta}q)}^{2}}{\Delta}y}\to {{U}_{{\rm bin}}}\int \frac{{{d}^{2}}qdy}{{{(2\pi )}^{3}}},\end{eqnarray*}$$

where ${{U}_{{\rm bin}}}={{S}_{{\rm area}}}\times {\Delta}\tilde{\nu }.$ If we define ${{C}^{T}}={{C}^{S}}+{{C}^{N}}+{{C}^{F}},$ the Fisher matrix for a set of cosmological parameters $\{{{p}_{i}}\}$ is given by

$$\begin{eqnarray}&&F_{ij}^{{\rm IM}}=\frac{1}{2}{{U}_{{\rm bin}}}\int \frac{{{d}^{2}}qdy}{{{(2\pi )}^{3}}}[{{\partial }_{i}}{\rm ln} {{C}^{T}}({\boldsymbol{q}} ,y){{\partial }_{j}}{\rm ln} {{C}^{T}}({\boldsymbol{q}} ,y)],\end{eqnarray} \tag{ 8 }$$

where the derivatives will only act on C^S, since that is the only term containing parameters of interest.

We can rewrite (8) in terms of physical wave numbers by using the following dictionary: ${{V}_{{\rm bin}}}={{U}_{{\rm bin}}}{{r}^{2}}{{r}_{\nu }},$ ${\boldsymbol{q}} ={{{\boldsymbol{k}} }_{\bot }}r,$ and $y={{k}_{\parallel }}{{r}_{\nu }},$ where ${{k}_{\bot }}=k\sqrt{1-{{\mu }^{2}}}$ and ${{k}_{\parallel }}=k\mu .$ We then apply the substitutions

• 1.  
  ${{U}_{{\rm bin}}}\to {{V}_{{\rm bin}}}$
• 2.  
  $\int d{{q}^{2}}dy\to 2\pi \int _{-1}^{+1}d\mu \int _{{{k}_{{\rm min} }}}^{\infty }{{k}^{2}}dk$
• 3.  
  $q=kr\sqrt{1-{{\mu }^{2}}}$ and $y=k{{r}_{\nu }}\mu .$

We can now express the Fisher matrix in a familiar form to those working on galaxy redshift surveys—a comparison we will pursue in Section 3.

Our focus in this paper is on the lead-up to Phase I of the SKA. We consider a portfolio of planned experimental configurations, with the aim of exploring how they will impact constraints on cosmological parameters. Our approach is ecumenical—we try to include as many proposed configurations as possible, although we are limited by what information has been made publicly available.

We will first of all consider three illustrative experimental setups, roughly corresponding to successive "generations" of planned IM experiments. These are:

• 1.  
  Stage I—Small, specialized H i experiment focused on a relatively narrow redshift range, intended to provide initial detections of the BAO and other first cosmological results. Stage I experiments are envisaged as either surveys on existing general-purpose arrays or relatively cheap purpose-built telescopes using multi-feed receivers to improve sensitivity.
• 2.  
  Stage II—Larger interferometric experiment with enhanced sensitivity, covering a wider range of redshifts. Stage II experiments are intended to cover a substantial survey volume, with the aim of producing constraints on cosmological parameters that are competitive with contemporary (DETF⁹ Stage II/III) LSS surveys. They are likely to be either purpose-built H i arrays with a large number of receivers or surveys on forthcoming "SKA-precursor" arrays such as MeerKAT and ASKAP.
• 3.  
  Facility—Survey on a future large array, covering a wide redshift range over most of the sky. Facility-type surveys will compete with other large (DETF Stage IV) experiments to produce the most precise cosmological parameter estimates and will be able to probe novel H i-only effects for the first time. The only planned experiments of this type so far are the Phase I SKA arrays, although the full CHIME and Tianlai configurations could also fall into this class.

We have chosen representative configurations for each of these classes (see Table 2) that will be used to illustrate the expected progress of H i IM experiments over the next decade. We have also forecasted for the following real (existing, proposed, or plausible) experiments:

• An SKA pathfinder consisting of thirty-six 12 m dishes, each with 36-element PAFs, located at the eventual site of SKA1-SUR in Australia (Johnston et al. 2008).
• Proposed compact array of 128 1.6 m tiles with four dipoles per tile, co-located with GBT or SKA1-MID (Pober et al. 2013).
• A proposed 40 m (25 m illuminated) multi-receiver single-dish telescope in South America (Battye et al. 2013).
• A proposed array made up of 20 × 100 m cylinders (20 × 80 m illuminated), based in British Columbia, Canada. There is a pathfinder with two half-length cylinders, and a planned full experiment with five (CHIME Collaboration 2012).
• A proposed multi-beam system on the 500 m single-dish telescope currently under construction in southwest China (Smoot & Debono 2014).
• A 100 m single-dish telescope in West Virginia (USA). GBT has already been used for preliminary detections of the H i signal (Chang et al. 2010; Masui et al. 2013; Switzer et al. 2013).
• A planned seven-beam receiver system on GBT (Chang & GBT-HIM Team 2014).
• Array of 30 45 m dishes in Pune, India (Swarup et al. 1991).
• An array of 27 25 m dishes, based in New Mexico, USA (NRAO 2014).
• An SKA pathfinder made up of seven 12 m dishes, on the planned site of SKA1-MID (SKA South Africa 2014).
• An SKA pathfinder with 64 13.5 m dishes, on the site of SKA1-MID (Jonas 2009). Has a choice of two frequency bands.
• A proposal for a mid-frequency aperture array component of Phase II of the SKA (MFAA 2014).
• A single 64 m dish (with 13 beams) in NSW, Australia (ATNF 2014).
• A planned SKA Phase I configuration with 190 15 m dishes, based in the Northern Cape, South Africa (Dewdney et al. 2013). Can be extended to incorporate the 64 MeerKAT dishes.
• A planned SKA Phase I configuration with 60 15 m dishes, each with 36-element PAFs, based in Western Australia (Dewdney et al. 2013). Can be extended to incorporate the ASKAP dishes.
• A proposed array of eight 15 × 120 m cylinders to be built in northwest China (Chen 2012).
• An array of 10 25 m dishes distributed across North America (Napier et al. 1994).
• A proposed upgrade to WSRT that uses a PAF in the focal plane to produce multiple beams on the sky (Oosterloo et al. 2010).

Table 2.  Telescope Configurations Used in This Paper

		Experiments	Tinst	Nd × Nb	Ddish	Dmin	Dmax	${{\nu }_{{\rm crit}}}$	$\nu _{{\rm max} }^{{\rm IM}}$	$\nu _{{\rm min} }^{{\rm IM}}$	${\Delta}{{\nu }^{{\rm IM}}}$	zmin	zmax	Sarea
			$[{\rm K}]$		$[{\rm m}]$	$[{\rm m}]$	$[{\rm m}]$	$[{\rm MHz}]$	$[{\rm MHz}]$	$[{\rm MHz}]$	$[{\rm MHz}]$			[deg2]
Ref.		Stage I	50	1 × 50	30.0	⋯	⋯	⋯	1100	800	300	0.29	0.77	5,000
	$\bullet$	Stage II	35	160 × 1	4.0	4.0	53.0	1000	1000	600	400	0.42	1.37	2,000
		Facility	20	250 × 1	15.0	⋯	⋯	⋯	1100	400	700	0.29	2.55	25,000
Existing Facility		GBT	29	1 × 1	100.0	⋯	⋯	⋯	920	680	240	0.54	1.09	100
		GBT-HIM	33	1 × 7	100.0	⋯	⋯	⋯	900	700	200	0.58	1.03	1,000
		GMRT	70	30 × 1	45.0	⋯	⋯	⋯	1420	1000	420	0.00	0.42	1,000
		JVLA	70	27 × 1	25.0	⋯	⋯	⋯	1420	1000	420	0.00	0.42	1,000
		Parkes	23	1 × 13	64.0	⋯	⋯	⋯	1420	1155	265	0.00	0.23	5,000
		VLBA	27	10 × 1	25.0	⋯	⋯	⋯	1420	1200	220	0.00	0.18	5,000
	$\blacktriangle$	WSRT + APERTIF	52	14 × 37	25.0	⋯	⋯	⋯	1300	1000	300	0.09	0.42	25,000
Targeted IM	$\bullet$	BAOBAB-128	40	128 × 1	1.6	1.6	26.0	⋯	900	600	300	0.58	1.37	1,000
		BINGO	50	1 × 50	25.0	⋯	⋯	⋯	1260	960	300	0.13	0.48	5,000
	$\diamond$	CHIME	50	1280 × 1	20.0	⋯	⋯	⋯	800	400	400	0.77	2.55	25,000
		FAST	20	1 × 20	500.0	⋯	⋯	⋯	1000	400	600	0.42	2.55	2,000
	$\bullet$	MFAA	50	3100 × 1	2.4	0.1	250.0	950	950	450	500	0.49	2.16	5,000
	$\diamond$	Tianlai	50	2048 × 1	15.0	⋯	⋯	⋯	950	550	400	0.49	1.58	25,000
Pre-SKA	$\blacktriangle$	ASKAP	50	36 × 36	12.0	⋯	⋯	⋯	1000	700	300	0.42	1.03	25,000
		KAT7	30	7 × 1	13.5	⋯	⋯	⋯	1420	1200	220	0.00	0.18	2,000
		MeerKAT (B1)	29	64 × 1	13.5	⋯	⋯	⋯	1015	580	435	0.40	1.45	25,000
		MeerKAT (B2)	20	64 × 1	13.5	⋯	⋯	⋯	1420	900	520	0.00	0.58	25,000
SKA Phase I		SKA1-MID (B1) Autocorr.	28	190 × 1	15.0	⋯	⋯	⋯	1050	350	700	0.35	3.06	25,000
	$\unicode{x25E6}$	SKA1-MID (B1) Interferom.	28	190 × 1	15.0	⋯	⋯	⋯	1050	350	700	0.35	3.06	100
		SKA1-MID (B2) Autocorr.	20	190 × 1	15.0	⋯	⋯	⋯	1420	900	520	0.00	0.58	25,000
	$\unicode{x25E6}$	SKA1-MID (B2) Interferom.	20	190 × 1	15.0	⋯	⋯	⋯	1420	900	520	0.00	0.58	50
	$\blacktriangle$	SKA1-SUR (B1)	50	60 × 36	15.0	⋯	⋯	710	900	400	500	0.58	2.55	25,000
	$\blacktriangle$	SKA1-SUR (B2)	30	96 × 36	15.0	⋯	⋯	1300	1150	650	500	0.23	1.18	25,000
		SKA1-MID + MeerKAT(B1)a	⋯	⋯	⋯	⋯	⋯	⋯	1050	350	700	0.35	3.06	25,000
		SKA1-MID + MeerKAT (B2)a	⋯	⋯	⋯	⋯	⋯	⋯	1420	900	520	0.00	0.58	25,000

Notes. The assumed observing mode of each telescope is denoted by: ( ) Single-dish; ($\blacktriangle$) Single-dish with Phased Array Feed; ($\unicode{x25E6}$) Dish Interferometer; ($\bullet$) Dense Aperture Array; ($\diamond$) Cylinder Interferometer. Some Instruments can operate over a wider frequency range than shown here; where this is the case, our values correspond to the most appropriate ${{\nu }_{{\rm max} }}$ for IM, or we include multiple bands.

^aFor Combined Arrays, in redshift bins where the bands overlap, we find ${{T}_{{\rm inst}}},$ ${{D}_{{\rm dish}}}$ by averaging the values for each sub-array, weighted by the number of dishes.

Download table as:  ASCIITypeset image

This is intended to be a relatively exhaustive list of current and planned H i IM experiments at $z\lesssim 3,$ but inevitably some have been omitted due to a lack of publicly available specifications. We have made our code publicly available,¹⁰ so forecasts can be performed when specifications become available.

The instrumental parameters used for each experiment are listed in Table 2. The survey area is chosen (between 10 and 25,000 ${{{\rm deg} }^{2}}$) to maximize the DE figure of merit (FOM) (see Section 5), and the survey time is assumed to be ${{t}_{{\rm tot}}}={{10}^{4}}$ hr for all experiments. For interferometers, we use either the baseline density calculated from the actual array layout (see appendix C) or the $n(u)\sim {\rm const}.$ approximation of Equation (6). For simplicity, we consider only single-dish mode for some telescope arrays, even though they are capable of interferometric measurements.

It is useful to be able to compare the performance of IM experiments with competing probes, such as galaxy redshift surveys. To this end, we also produce forecasts for a fiducial DETF Stage IV spectroscopic galaxy redshift survey, similar to DESI, Euclid, or WFIRST, with an expected yield of $\sim 6\times {{10}^{7}}$ spectroscopic redshifts. We take the survey to cover roughly a third of the sky (${{f}_{{\rm sky}}}=0.35$) over a redshift range of $0.65\leqslant z\leqslant 2.05,$ with redshift distribution taken from Amendola et al. (2013) (Euclid reference case, Table 1.3). The bias is taken to evolve as $b(z)=\sqrt{1+z}.$ We forecast for the same set of cosmological parameters as IM experiments, with the same fiducial values, including relevant nuisance parameters like ${{\sigma }_{{\rm NL}}}.$ We use the redshift scaling from Smith et al. (2003) to set ${{k}_{{\rm max} }}={{k}_{{\rm NL}}}{{(1+z)}^{2/(2+{{n}_{s}})}}$ and choose ${{k}_{{\rm min} }}=2\pi /{{({{V}_{{\rm bin}}})}^{\frac{1}{3}}}.$ Constraints are quite sensitive to the choice of k_max (White et al. 2008), but mostly insensitive to k_min (if chosen sufficiently small).

Intensity mapping experiments cannot constrain all cosmological quantities of interest on their own; information from other probes must be added in order to break degeneracies and improve precision. High-quality data are already available from a range of other sources, including CMB temperature and polarization anisotropies at high and low-ℓ, galaxy redshift surveys at high and low redshift, weak gravitational lensing, and supernova distance measurements. By the time of the first IM surveys with the SKA, however, the number of experiments for each of these probes, and their precision, will have risen sharply.

While the best constraints will ultimately be obtained by combining information from all relevant experiments, our intention here is not to provide an exhaustive account of the expected state of observational cosmology in 10 years' time. Instead, we will (conservatively) focus only on the CMB as an external probe in this paper.

CMB data provide a high-redshift distance measurement that is vital for anchoring the low-z distance measures that most effectively probe dark energy. It also yields information about the shape and normalization of the matter power spectrum, the matter content at $z\simeq 1090,$ and the physical scale of the BAO.

We use the DETF Planck prior from Albrecht et al. (2009), which assumes temperature and E-mode polarization data over 70% of the sky for the 70, 100, and 143 GHz channels out to $\ell =2000.$ We rescale the Fisher matrix to reflect our different fiducial cosmology and then project it out to the following variables (fixing all others):

$$\begin{eqnarray*}&&\{h,{{{\Omega}}_{b}}{{h}^{2}},{{{\Omega}}_{{\rm DE}}},{{{\Omega}}_{K}},{{w}_{0}},{{w}_{a}},{{n}_{s}},{{\sigma }_{8}}\}.\end{eqnarray*}$$

Note that the optical depth to last scattering, τ, which depends on the cosmic reionization history, has been marginalized over in the original DETF Fisher matrix. We ignore constraints from CMB lensing, high-ℓ CMB experiments, and B-mode polarization, although in principle these would provide additional information on h, ${{\sigma }_{8}},$ n_s, and the effective number of neutrino species.

The BAOs are a "statistical standard ruler" that forms the primary distance measure in surveys of large-scale structure. We can get an idea of the detectability of the BAOs by looking at the fractional errors on P(k), using Equation (11). These are shown for the reference surveys in Figure 4 and are overplotted on the BAO wiggle function (to be defined shortly) in Figure 5. All three IM surveys are capable of strongly detecting the BAO feature when the constraints are combined over their full redshift ranges. Facility approaches the cosmic variance limit (represented by the DETF Stage IV survey out to $k\sim 0.1\;{\rm Mp}{{{\rm c}}^{-1}}$) over a substantial fraction of the scales relevant to the BAO, mostly due to the sensitivity of its single-dish component. This also helps to put sub-10% level constraints on the power spectrum on scales slightly larger than the matter-radiation equality peak, ${{k}_{{\rm eq}}}\approx {{10}^{-2}}$ Mpc⁻¹. Its interferometric component provides constraints on smaller scales, achieving $\sim 10\%$ errors on P(k) out to $k\approx 1\;{\rm Mp}{{{\rm c}}^{-1}}.$

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The interferometric Stage II survey is sensitive to generally smaller scales, but still achieves good constraints on the BAO thanks to its coverage out to intermediate redshifts ($z\sim 1.4$). The Stage I survey can comfortably detect the BAO despite its significantly lower sensitivity than Facility, but leaves smaller scales unconstrained.

Alternatively, one can look at the detectability of the BAO feature as a whole. We follow a similar approach to Blake & Glazebrook (2003) and split the matter power spectrum, P(k), into a "smooth" part, ${{P}_{{\rm smooth}}}(k),$ and an oscillatory part,

$$\begin{eqnarray}&&{{f}_{{\rm bao}}}(k)=\frac{P(k)-{{P}_{{\rm smooth}}}(k)}{{{P}_{{\rm smooth}}}(k)}.\end{eqnarray} \tag{ 13 }$$

We then introduce an amplitude parameter, A, such that

$$\begin{eqnarray}&&P(k)=[1+A{{f}_{{\rm bao}}}(k)]{{P}_{{\rm smooth}}}(k).\end{eqnarray} \tag{ 14 }$$

Constraints on A therefore give a measure of the detectability of the BAO feature.

The splitting of P(k) between smooth and oscillatory parts is somewhat arbitrary. We attempt to construct a "purely oscillatory" ${{f}_{{\rm bao}}}(k)$—i.e., one that lacks a smooth overall trend in k—as follows. First, we use CAMB to calculate P(k) for the fiducial cosmological model over a range of sample points in k. We then choose two reference values of k that bound the region in which the oscillations are significant ($k\approx 0.02$ and 0.45 Mpc$^{-1}$ for our fiducial cosmology) and construct a cubic spline for ${\rm log} P(k)$ as a function of ${\rm log} k$ using all points outside that region. Next, we construct a preliminary oscillatory function by dividing the sampled P(k) by the splined function (not its logarithm), then fit another cubic spline to the result and find the zeros of its second derivative with respect to k. These are the points at which the first derivatives of the oscillatory function are maximal/minimal, and in some sense define "mid-points" of the function—its overall trend. We construct a cubic spline through these too, and then divide the preliminary oscillatory function by it to "de-trend." This leaves ${{f}_{{\rm bao}}}(k)$ as the final result (Figure 5). Unlike other methods, which look at ratios of the form $P(k,{{{\Omega}}_{b}}\ne 0)/P(k,{{{\Omega}}_{b}}=0)$ to pick out oscillations (Rassat et al. 2008), this method is essentially model independent for a given fiducial P(k).

The constraint on the overall amplitude of the BAO feature, A, is plotted as a function of redshift for the reference surveys in Figure 6. Facility is capable of $\gt 3\sigma$ detections of the BAO feature out to $z\approx 1.5,$ but makes progressively weaker detections at higher redshift, predominantly due to its limited angular resolution in single-dish mode. In comparison, the Stage II survey's constraints degrade much less rapidly with redshift, owing to its greater sensitivity to smaller angular scales (which translate to intermediate physical scales at higher z).

Zoom In Zoom Out Reset image size

Figure 7 plots the errors on P(k) for Facility as a function of both scale and redshift. For $k\gtrsim 0.1\;{\rm Mp}{{{\rm c}}^{-1}},$ most of the information comes from low redshifts, where the amplitude of the power spectrum is largest. At smaller k, however, the volume of the redshift bin begins to matter, as the increase in bin volume with z allows progressively larger scales to be probed. For Facility, the constraints on intermediate BAO scales ($k\sim 0.07\;{\rm Mp}{{{\rm c}}^{-1}}$) come from a mixture of low- and intermediate-redshift bins, with the high-redshift bins taking over on larger scales.

Zoom In Zoom Out Reset image size

The angular diameter distance, D_A(z), and the expansion rate, H(z), are measures of distance in the transverse and radial directions, respectively. To include them in our forecasts, we introduce (Blake & Glazebrook 2003)

$$\begin{eqnarray}&&{{\alpha }_{\bot }}\equiv \frac{{{r}^{{\rm fid}}}}{r}=\frac{D_{A}^{{\rm fid}}(z)}{{{D}_{A}}(z)}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{{\alpha }_{\parallel }}\equiv \frac{r_{\nu }^{{\rm fid}}}{{{r}_{\nu }}}=\frac{H(z)}{{{H}^{{\rm fid}}}(z)},\end{eqnarray} \tag{ 16 }$$

where r^fid and $r_{\nu }^{{\rm fid}}$ are the fiducial ΛCDM values of r and ${{r}_{\nu }}$ at a given redshift. We then replace $q\to {{\alpha }_{\bot }}q$ and $y\to {{\alpha }_{\parallel }}y$ in Equation (4) to get

$$\begin{eqnarray}\begin{array}{rcl} {{C}^{S}}({\boldsymbol{q}} ,y) & = & T_{b}^{2}\frac{\alpha _{\bot }^{2}{{\alpha }_{\parallel }}}{{{r}^{2}}{{r}_{\nu }}}{{F}_{{\rm RSD}}}\left( \frac{{{\alpha }_{\bot }}{\boldsymbol{q}} }{r},\frac{{{\alpha }_{\parallel }}y}{{{r}_{\nu }}} \right){{D}^{2}}(z) \\ {} & {} & \times P\left( k=\sqrt{{{\left( \frac{{{\alpha }_{\bot }}{\boldsymbol{q}} }{r} \right)}^{2}}+{{\left( \frac{{{\alpha }_{\parallel }}y}{{{r}_{\nu }}} \right)}^{2}}} \right). \\ \end{array}\end{eqnarray} \tag{ 17 }$$

The distance measures enter this expression in four places: (a) an overall factor of $\alpha _{\bot }^{2}{{\alpha }_{\parallel }},$ related to the physical volume of the survey; (b) a distortion of the angular (μ) dependence of the RSDs; (c) a shift in the non-linear cutoff scale of the RSDs; and (d) a shift of the whole isotropic power spectrum, P(k), that can be further subdivided into corresponding shifts in the BAO feature and smooth power spectrum. The latter two, (c) and (d), are both due to the remapping of k, shown explicitly in the argument of P(k) in Equation (17) (Samushia et al. 2011).

Due to modeling uncertainties and degeneracies with other parameters, it is not necessarily desirable to use all of these terms to measure distances from real data. The BAO feature is the standard choice of distance measure, owing to its robustness to systematic error; the acoustic scale shifts only slightly when non-linearities are introduced (Crocce & Scoccimarro 2008; Smith et al. 2008), and smooth variations such as scale-dependent bias also have a relatively minor impact (Zhang 2008) (although corrections must still be made for precision measurements). Anisotropies of the correlation function can also be used to measure distances (Alcock & Paczynski 1979; Kaiser 1987), through RSDs (b) and the Alcock–Paczynski effect (a, c, and d), although these are more sensitive to the detailed modeling of the power spectrum (Reid et al. 2012). Thus, a particularly conservative analysis might only derive distances from the BAO and discard information from the other terms.

Figure 8 shows the effect of neglecting some of the distance terms for the Facility experiment. By using only the BAO, one is discarding a substantial amount of useful information, as shown by the comparatively poor constraints on D_A and H. This is partially compensated by the reduced risk of systematic error and the improved growth rate constraints that are due to weaker degeneracies with other parameters when compared to the other distance terms. Including broadband information—i.e., the shift in (smooth) P(k)—reduces the error on D_A by a factor of 2–5 over the entire redshift range and is especially beneficial at higher z, where the BAO-only constraints degrade rapidly due to the limited angular sensitivity of the telescope. Adding the RSD terms helps to distinguish between the radial and transverse directions, which also reduces the error on H(z).

Zoom In Zoom Out Reset image size

Disregarding any of the distance terms can also substantially alter the correlation structure of the Fisher matrix, which has a knock-on effect on constraints for other parameters. This can be seen clearly for f(z) in Figure 8, which has a substantial scatter in fractional error depending on which distance measures are used. To most accurately reflect the interdependencies of the various cosmological parameters, we will use all of the distance measure terms in what follows. The increased uncertainty that comes from marginalizing over nuisance parameters such as ${{\sigma }_{{\rm NL}}}$ helps compensate for the increased risk of systematic error that attends some of the measures, which means that this is not too optimistic of us.

Figure 6 shows the fractional constraints that can be achieved on D_A(z) and H(z) for the full set of reference surveys. Facility measures the expansion rate to better than 1% out to $z\approx 1.7$ and stays within 2% as far out as z = 2.5. This is roughly a factor of 2 worse than the DETF Stage IV reference survey, although the redshift range covered by Facility is significantly larger. Stage II also obtains ∼2% constraints across its full redshift range, and Stage I hovers around 3%.

The picture is somewhat different for D_A(z). The fractional errors for Facility increase from $\sim 1.5\%$ at $z\approx 0.4$ up to 4% at z = 2, compared with the relatively flat errors for the galaxy survey that remain below 1% for most of the redshift range. Stage II's errors are also relatively flat as a function of z, staying at around the $2.5\%$ mark.

The limited angular resolution of the single-dish experiments is the cause of this behavior. H(z) is most sensitive to the resolution in the radial (frequency) direction, which is essentially the same for all experiments and does not evolve appreciably with redshift (being set by the non-linear scale rather than the channel bandwidth, as discussed in Section 3). The D_A(z) constraints depend more on the sensitivity to transverse physical scales, however, which differs between interferometers and single-dish experiments. For single-dish, $k_{\bot }^{{\rm max} }$ tends to be relatively small even at z = 0 for moderately sized dishes, and it continues to decrease (shift to larger scales) as z increases, as shown in Figure 3. As this happens, useful distance information from smaller scales is lost. The same happens for interferometers, but $k_{\bot }^{{\rm max} }$ is typically much larger, so the most useful transverse scales remain resolved. In fact, since $k_{\bot }^{{\rm min} }$ is also decreasing, additional distance information becomes available from larger scales.

It is worth noting that the effect would be different if only the BAO were being used as distance measures—both the D_A(z) and H(z) constraints would be affected by the loss of resolution in the transverse direction (Sánchez et al. 2013). To see this, consider a simplified model of the correlation function consisting of the sum of a smooth component and a feature, ${{\xi }_{{\rm BAO}}}(r)=A{\rm exp} [-{{(r-{{r}_{{\rm BAO}}})}^{2}}/2{{\sigma }^{2}}].$ The response to the loss of resolution along the transverse direction corresponds to convolving the correlation function with a window function such that

$$\begin{eqnarray*}&&{{\tilde{\xi }}_{{\rm BAO}}}({{r}_{\parallel }},{{{\boldsymbol{r}} }_{\bot }})=\int {{d}^{2}}r_{\bot }^{^{\prime} }W({{{\boldsymbol{r}} }_{\bot }}-{\boldsymbol{r}} _{\bot }^{^{\prime} }){{\xi }_{{\rm BAO}}}\left( \sqrt{r_{\parallel }^{2}+r_{\bot }^{^{\prime} 2}} \right),\end{eqnarray*}$$

which can be rearranged to have the form

$$\begin{eqnarray*}&&{{\tilde{\xi }}_{{\rm BAO}}}({{r}_{\parallel }},{{{\boldsymbol{r}} }_{\bot }})=\int {{d}^{2}}{\Delta}W({\Delta}){{\xi }_{{\rm BAO}}}\left( \sqrt{{{r}^{2}}+2{\bf \Delta }\cdot {{{\boldsymbol{r}} }_{\bot }}+{{{\Delta}}^{2}}} \right).\end{eqnarray*}$$

The smoothed correlation function remains a function of r and ${{r}_{\bot }},$ which means that the degradation of signal due to the smoothing is almost democratically taken up by both the transverse and parallel directions. This can be seen explicitly in Figure 8 for the BAO-only curves for D_A(z) and H(z), which both show a much stronger evolution with redshift than any other combination of distance measures. The inclusion of additional distance measures is therefore necessary to help limit the effect of the poor angular resolution of single-dish experiments.

The various distance measures for LSS surveys depend on combinations of D_A and H rather than constraining them individually. For example, moments of the correlation function (Reid et al. 2012) give the volume distance and Alcock–Paczynski terms,

$$\begin{eqnarray}&&{{D}_{V}}(z)={{\left[ {{(1+z)}^{2}}D_{A}^{2}\frac{cz}{H(z)} \right]}^{\frac{1}{3}}}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&F(z)=(1+z){{D}_{A}}(z)H(z)/c.\end{eqnarray} \tag{ 19 }$$

In some sense, these quantities define redshift-dependent figures of merit—many other studies forecast directly in terms of D_V, for example, and surveys are compared in terms of the errors that they can achieve on this parameter at a given redshift. We have therefore presented results for both D_A and H (Figure 6) and D_V and F (Figure 9) to facilitate comparison with previous studies.

Zoom In Zoom Out Reset image size

The other key observable provided by LSS surveys is the growth rate, which can be measured from the anisotropy of the correlation function in redshift space.

The growth rate has two major roles: first, as another measure of distance, since it can be related to the expansion rate (albeit in a model-dependent way); and second, as a non-geometric probe of gravity over cosmic time. The former is useful for helping to break parameter degeneracies that can crop up when only geometric distance measures are used. The latter is of importance in distinguishing theories of dark energy and modified gravity (Masui et al. 2010; Hall et al. 2013); it is often the case that theories can be made to have the same expansion history, for example, but differ in growth rate. (We will return to this in Section 5.4.)

In what follows, we will concentrate on the linear growth rate, f(z). H i IM experiments are also capable of probing non-linear growth, but modeling uncertainties on small scales reduces their usefulness for constraining cosmological parameters. Non-linear effects are discussed in more detail in Sections 5.4 and 6.2.

The linear growth rate appears in two places in Equation (17); explicitly, in the angle-dependent RSD factor, and implicitly, through the linear growth factor, D(z), that gives the redshift evolution of the power spectrum. The two are related by $f(z)=d{\rm log} D/d{\rm log} a.$ Because D(z) appears as an overall factor of the signal covariance, it is degenerate with other parameters, making it hard to measure in an unbiased way. We will assume that D(z) provides no new information on the growth rate here (i.e., we neglect its derivative w.r.t. f in the Fisher matrix).

Various other degeneracies crop up when measuring the growth rate from RSDs. The signal covariance is proportional to ${{C}^{S}}\propto {{\left[ {{b}_{{\rm HI}}}(z)+f(z){{\mu }^{2}} \right]}^{2}}{{D}^{2}}(z)T_{b}^{2}(z)\sigma _{8}^{2}.$ If no functional form is assumed for any of these terms (i.e., they are left free in each redshift bin), there are only two quantities that can be uniquely distinguished from this expression: an overall amplitude, and a factor of the angle dependence, e.g., ${{C}^{S}}\propto {{\kappa }^{2}}(z){{\left[ 1+\beta (z){{\mu }^{2}} \right]}^{2}},$ where $\beta =f(z)/{{b}_{{\rm HI}}}(z)$ and $\kappa ={{\sigma }_{8}}D(z){{T}_{b}}(z){{b}_{{\rm HI}}}(z).$ Alternatively, one can merge the linear growth factor with the overall normalization to give a redshift-dependent normalization, ${{\sigma }_{8}}(z)={{\sigma }_{8}}D(z),$ and write the two RSD functions as ${{T}_{b}}f{{\sigma }_{8}}$ and ${{T}_{b}}{{b}_{{\rm HI}}}{{\sigma }_{8}}.$

Either way, there are at least three unknowns to be determined from two functions, so it is clear that more information is needed to unpick the degeneracy. The CMB gives a prior on ${{\sigma }_{8}}(z\simeq 1090),$ D(z) can in principle be determined from f(z), and several models for the bias and brightness temperature exist, although there is significant disagreement between them (see Section 6.1). The brightness temperature will be measurable from the non-fluctuating part of the H i signal when future IM experiments come online, though, so this can reasonably be taken as a given quantity—T_b(z) is fixed to its fiducial form throughout this paper. We resolve the remaining degeneracy by treating the combinations $f(z){{\sigma }_{8}}(z)$ and ${{b}_{{\rm HI}}}(z){{\sigma }_{8}}(z)$ as independent parameters, with both being free functions of redshift. Figure 10 shows the joint constraints on them as a function of z for the Facility survey.

Zoom In Zoom Out Reset image size

Figure 6 shows the constraints on $f{{\sigma }_{8}}(z)$ that can be achieved by our set of reference surveys. Sub-2% errors are possible for the Facility and Stage II experiments out to a redshift of $z\sim 1.2,$ despite (pessimistically) taking the bias to be a completely free function of redshift. As shown in Figure 8, constraints on the growth rate are sensitive to the choice of other distance measures; for Facility at least, using only BAO to measure distances would result in a $\sim 50\%$ reduction in the error on $f{{\sigma }_{8}}(z)$ across the whole redshift range, albeit at the cost of significantly degraded H(z) and D_A(z) measurements.

As shown in Figure 6, the errors on $f{{\sigma }_{8}}(z)$ for the Stage I and Facility surveys increase significantly with redshift, while the evolution is less severe for Stage II. As with the angular diameter distance (see previous section), this is mostly due to the limited angular resolution of the dish-based surveys.

The current consensus is that cosmological data are well described by a flat ΛCDM model of structure formation that can be characterized in terms of six parameters: the Hubble parameter, ${{H}_{0}}=100$ h km s⁻¹ Mpc⁻¹, the density of dark energy (or cosmological constant), ${{{\Omega}}_{{\rm DE}}},$ the physical density of baryons, ${{\omega }_{b}},$ the linear amplitude of density fluctuations, parameterized by ${{\sigma }_{8}},$ the spectral index of primordial density perturbations, n_s, and the optical depth to last scattering, τ. In this section, we examine the constraints that IM experiments will be able to put on this model when combined with CMB data from Planck. Parameters that extend the "vanilla" ΛCDM model (${{w}_{0}},{{w}_{a}},{{{\Omega}}_{K}},\gamma$) are fixed to their fiducial values in this section, and ${{{\Omega}}_{{\rm DE}}}={{{\Omega}}_{{\Lambda}}}.$

Figure 11 presents forecasts for five of the six parameters for the Facility experiment, compared with Planck-only and the DETF Stage IV galaxy redshift survey. Although the reionization history will have a significant role in the evolution of the H i density and bias, we are focusing on sufficiently late times that our constraints will essentially be insensitive to variations of τ, within current constraints, and so we leave it out of the plot. (In fact, it has already been marginalized over in the Planck prior Fisher matrix.) We do not directly constrain ${{\omega }_{b}}$ with IM experiments either, but as it is strongly correlated with other parameters in the Planck prior, we leave it in.

Zoom In Zoom Out Reset image size

As expected, there is a modest improvement over Planck alone by a factor of a few. The Planck-only constraints are mostly limited by strong correlations between parameters, so the role of the IM survey is primarily to break degeneracies. Future high-resolution experiments such as ACTPol (Niemack et al. 2010) and SPTpol (Austermann et al. 2012) will be able to measure weak lensing of the CMB to sufficient accuracy that constraints from the CMB alone should be competitive with IM (and redshift surveys). This is contingent on the assumption of a fixed w = −1, however.

The biggest effect of adding IM data (or indeed any LSS data) is to substantially improve the constraints on h and ${{{\Omega}}_{{\rm DE}}}.$ As shown in Figure 11, the two are strongly correlated for Planck alone, as the CMB only measures the combination ${{{\Omega}}_{M}}{{h}^{3}}=(1-{{{\Omega}}_{{\rm DE}}}){{h}^{3}}$ (Planck Collaboration 2014b). The distance measures at late times depend on different combinations of these parameters and so help to break this degeneracy. This has a knock-on effect on other parameters that are correlated with them, especially n_s.

Table 3 summarizes the vanilla ΛCDM constraints for the full set of reference experiments. The majority of future H i surveys are capable of improving on constraints from Planck plus existing LSS data sets; notably, H₀ and ${{{\Omega}}_{{\rm DE}}}$ are determined at the sub-1% level by all but the smallest IM experiments.

Table 3.  Forecast 1σ Marginal Errors on Vanilla ΛCDM Model Parameters for the Set of Reference Surveys, Compared with Current Constraints from Planck (Temperature-only) and WMAP (Planck Collaboration 2014b)

Experiments	h	${{\omega }_{b}}$	ΩDE	ns	σ8
	/ 10−3	/ 10−4	/ 10−3	/ 10−4	/ 10−3
Planck + Stage I	5.8	1.5	6.1	36.9	7.0
Planck + Stage II	5.1	1.4	5.3	32.8	6.0
Planck + Facility	2.7	1.2	2.7	21.9	3.3
Planck + DETF IV	2.2	1.0	2.2	16.0	2.3
Planck + WMAP	12	2.8	17	73.0	12
Planck+WP+BAO	7.8	2.5	10	57.0	11

Download table as:  ASCIITypeset image

High-end Facility-class experiments should even be competitive with a DETF Stage IV galaxy survey (cf. Euclid or LSST), although this is only likely to be the case for parameters constrained by the distance measures, such as ${{{\Omega}}_{{\rm DE}}}.$ Those that depend more on the power spectrum at smaller scales will not be quite as close, because the galaxy surveys measure P(k) significantly better at relatively high wavenumbers of $k\sim 0.1$ Mpc⁻¹ (Figure 4); our IM experiments fall behind on these scales due to limited (single-dish) angular resolution.

The driver for the majority of the cosmological surveys currently under development is to find precision constraints on the dark energy equation of state, w(z), and in doing so to infer the physical nature of the substance that appears to be driving the accelerated expansion of the universe. H i IM provides a way of constraining w(z) with considerable precision, using the full combination of D_A(z), H(z), and f(z) reconstructed over a broad range of redshifts.

While the evolution of the equation-of-state parameter depends on the underlying dark energy theory, and as such could take any number of functional forms, it is nevertheless useful to work with a simple expansion about z = 0,

$$\begin{eqnarray}&&w(a)\approx {{w}_{0}}+\frac{z}{1+z}{{w}_{a}}.\end{eqnarray} \tag{ 21 }$$

This commonly used parameterization should be reasonably accurate at late times, but will not capture more exotic behavior at $z\gg 1.$ The corresponding dark energy density evolves with redshift as

$$\begin{eqnarray}&&{{{\Omega}}_{{\rm DE}}}(z)={{{\Omega}}_{{\rm DE},0}}{\rm exp} [3{{w}_{a}}z/(1+z)]\;{{(1+z)}^{3(1+{{w}_{0}}+{{w}_{a}})}}.\end{eqnarray} \tag{ 22 }$$

The overall sensitivity of an experiment to a varying equation of state can be summarized (to some extent) by the dark energy FOM, defined by the Dark Energy Task Force as (Albrecht et al. 2009)

$$\begin{eqnarray}&&{\rm FOM}=1/\sqrt{{\rm det} ({{F}^{-1}}{{|}_{{{w}_{0}},{{w}_{a}}}})},\end{eqnarray} \tag{ 23 }$$

which is proportional to the reciprocal of the area enclosed by the 68% contour of the $({{w}_{0}},{{w}_{a}})$ error ellipse for Fisher matrix F.

The foremost task in understanding the nature of dark energy is to determine whether the equation of state differs from that of a cosmological constant, $w=-1.$ Current constraints on w₀ and w_a are relatively weak; the combination of Planck with SNLS supernova data does give values that are slightly in tension with a pure cosmological constant (Planck Collaboration 2014b), but the significance fades when other data sets are used instead. Figure 13 shows the improved constraints that can be expected on w₀, w_a, and ${{{\Omega}}_{{\rm DE}}}$ for the combination of our reference experiments with Planck, assuming flatness. Despite the addition of IM data, the parameters remain strongly correlated, so even substantial deviations from $w=-1$ will not necessarily be picked up. Nevertheless, a substantial fraction of the ${{w}_{0}}-{{w}_{a}}$ plane can be excluded by IM + Planck, so a successful detection is still possible if the real values lie orthogonal to the degeneracy direction. 1D marginal constraints for the full set of extensions to ΛCDM that we are considering here (including w₀ and w_a) are given in Table 4 for all of the experiments from Section 2.5.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

If one takes the possibility of a varying equation of state seriously, w₀ and w_a should be left free when deriving constraints on other cosmological parameters. Table 4 shows the effect of marginalizing over the equation of state on the vanilla ΛCDM model parameters. The parameters derived from the various distance measures are strongly affected—their 1D marginal uncertainty is typically increased by around an order of magnitude compared to the unmarginalized case shown in Table 3. This can be understood in terms of the degeneracies shown in Figure 13; adding new parameters always increases the overall uncertainty, but because ${{{\Omega}}_{{\rm DE}}}$ (and h) are highly correlated with w₀ and w_a, they are particularly strongly affected. Parameters that do not depend on distance measures, i.e., n_s and ${{\sigma }_{8}},$ are less affected by the equation-of-state parameters, and so their marginal uncertainties increase by only a modest amount.

As we have seen, even the addition of IM or other intermediate-redshift LSS data to the CMB constraints is insufficient to break all of the parameter degeneracies once w₀ and w_a are allowed to vary. In order to precisely determine these parameters, it is therefore necessary to add more data. Distance measurements from Type Ia supernovae are the obvious candidate, since they offer orthogonal constraints on ${{{\Omega}}_{{\rm DE}}}-{{{\Omega}}_{M}}$ (Efstathiou & Bond 1999). A local measurement of H₀ is also useful; as shown in Figure 11, h is strongly correlated with the dark energy density, so additional information about either parameter can substantially improve the constraints on both. Figure 12 shows the effect of adding H₀ data to Planck + Facility. We also consider the effect of allowing departures from spatial flatness; as we will see in the next section, the combination of CMB and IM data measures ${{{\Omega}}_{K}}$ well, mostly independent of dark energy, so marginalizing over curvature has a relatively minor effect on the ${{w}_{0}}-{{w}_{a}}$ ellipse.

Figure 14 shows the contribution to the dark energy FOM from each redshift bin. For our reference IM experiments, it is clear that the redshift range $z\lesssim 1.2$ is most critical; little improvement in FOM is seen above this redshift. The same cannot be said for the galaxy survey, however, which sees a roughly equal increase in FOM with each additional redshift bin across its whole z range. One way of understanding this behavior is to compare Figure 14 with the plots for D_A(z), H(z), and $f{{\sigma }_{8}}(z)$ in Figure 6. Above $z\sim 1.2,$ the angular diameter distance and growth rate constraints begin to worsen for Facility, but remain relatively flat for the galaxy survey. Since w₀ and w_a are obtained from projections of these functions, it is no surprise that little is gained on the FOM at redshifts where they are poorly constrained.

Zoom In Zoom Out Reset image size

The potential for H i IM experiments to span extremely wide redshift ranges—from $z\approx 0.1$ out to $z\gtrsim 2.5$ without too much difficulty—makes them an interesting prospect for unraveling the geometric degeneracy, i.e., the interplay between dark energy and curvature. Without strong assumptions on one or the other, it is difficult to separate the effects of ${{{\Omega}}_{K}}$ and w(z) using only a single type of distance measure (Mortonson 2009; Shafieloo & Linder 2011), and for the CMB power spectrum alone they are completely degenerate. As discussed in Section 4.2, IM provides a suite of distance measures. The combination of IM and, for example, CMB data should therefore be very useful in separating curvature from the evolution of dark energy in a precise and unambiguous manner.

A precision determination of spatial curvature on horizon scales would also provide a rare opportunity to test inflation. Current observations seem to point in the direction of flatness, with the most recent bounds from Planck finding $|{{{\Omega}}_{K}}|\lesssim {{10}^{-2}}$ (95% CL), consistent with the vast majority of inflation models, but if a detection of $|{{{\Omega}}_{K}}|\gtrsim {{10}^{-4}}$ were made, the whole class of eternally inflating models would be put under pressure (Guth & Nomura 2012; Kleban & Schillo 2012) .

Zoom In Zoom Out Reset image size

The minimum curvature that can be detected unambiguously also happens to be at around the 10⁻⁴ level (Vardanyan et al. 2009; Bull & Kamionkowski 2013). Future CMB experiments should be able to approach this order of magnitude if w is fixed to −1, and so too should a Facility-type IM experiment combined with Planck, as shown in Figure 16. There is little justification for putting such a strong prior on the equation of state, though—any rigorous constraint on ${{{\Omega}}_{K}}$ must confront the geometric degeneracy head-on and allow the full freedom of w(z). Figure 16 also shows the limits on curvature that can be achieved when the equation of state is left free. Though clearly worse than for fixed w = −1, the difference is relatively modest—the combination of a Facility survey with Planck should still be able to measure $|{{{\Omega}}_{K}}|$ to around 10⁻³ without any particularly strong assumptions about the form of w(z).

Zoom In Zoom Out Reset image size

The effect of the geometric degeneracy runs both ways—a lack of knowledge about ${{{\Omega}}_{K}}$ also degrades the reconstruction of the time evolution of the equation of state. Indeed, a percent level uncertainty in ${{{\Omega}}_{K}}$ can lead to a $\sim 100\%$ error on the recovery of more exotic forms for w(z) (Clarkson et al. 2007), although it has been argued that current constraints can already mitigate this (Okouma et al. 2013). As shown in Figure 12, the errors on the equation-of-state parameters do increase when we allow ${{{\Omega}}_{K}}$ to be free, albeit not substantially in the case of both Stage II and Facility; the combination of H(z), D_A(z), and f(z) measurements from IM, plus the Planck prior, is enough to prevent any strong degeneracies from completely killing the ${{w}_{0}}-{{w}_{a}}$ constraints. In fact, Figure 12 shows that they are more sensitive to assumptions about H₀ than to curvature.

It is improved knowledge of the late-time expansion that most helps separate the effects of curvature and dark energy. We see this clearly in Figure 15, where ${{[\sigma ({{{\Omega}}_{K}})]}^{-1}}$ is plotted as a function of the depth of each survey. There is an optimal point beyond which little new information is gained by the IM surveys, coinciding with the redshift at which the constraints on f(z) and D_A(z) start to degrade due to the limited angular resolution of the experiments (Figure 6). This happens at higher z_max for the galaxy survey, which makes up for its lack of low-redshift bins by having relatively flat fractional errors in D_A(z), H(z), and f(z) out to $z\approx 2.$ At even higher redshift, the dynamical effect of curvature is completely negligible, so little extra information could be gained anyway.

The growth history of the universe is a particularly powerful test of gravity. Modified theories of gravity generically alter the growth of structure from its GR behavior, typically enhancing clustering on non-linear scales, and increasing peculiar velocities. Signatures of modified gravity in the non-linear regime are difficult to disentangle from less exotic astrophysical effects, leaving the linear velocity field as, in some sense, the "cleanest" modified gravity observable from large-scale structure.

There is great variety in the effects that different modifications to gravity have on the linear growth history; the space of theories is complex and has proved difficult to parameterize in a simple way (Hu & Sawicki 2007; Battye & Pearson 2012; Baker et al. 2013). For the purposes of illustration, we will fall back on one of the simplest parameterizations of growth, using the growth index formulation of Peebles (1980): $f(z)={\Omega}_{M}^{\gamma }(z).$ Deviations from GR are captured, in part at least, by the difference in γ from its ΛCDM+GR value, ${{\gamma }_{{\rm GR}}}\approx 0.55.$

Two notes of caution are necessary: first, many modified gravity theories do not have growth histories that are well described by a constant γ. Allowing γ to be a function of redshift can help (Linder & Cahn 2007; Ishak & Dossett 2009), but even then there are many cases where the growth rate becomes scale dependent or is otherwise poorly described by this parameterization. Second, dark energy models that require no modifications to GR can also modify the growth history, often in a way that is well described by making the growth index a function of the equation-of-state parameter, $\gamma (w)$ (Linder 2005; Gong et al. 2011b). We neglect this possibility here and instead treat γ as being independent of w.

Figure 17 shows forecasts for the various dark energy parameters for the DETF Stage IV galaxy redshift survey and the Facility reference experiment. The 1D marginal constraints from the galaxy survey outperform the IM experiment for all parameters, except one—the growth index. Furthermore, the 2D constraints involving γ are roughly orthogonal between the two surveys, despite this not being the case for other combinations of parameters.

Zoom In Zoom Out Reset image size

At first, this may seem surprising. In Figure 6, the galaxy redshift survey constrains $f{{\sigma }_{8}}(z)$ to around 1% across most of its redshift range, while Facility's precision can only match this in the lowest-redshift bins, increasing to $\sim 4\%$ at higher z. For γ, though, it is the very lowest redshifts that make the most difference. At low z, the growth factor evolves most rapidly and is most sensitive to the value of γ (i.e., $|df/d\gamma |$ increases as $z\to 0$), whereas at higher redshifts, matter begins to dominate, growth is slower, and the dependence on γ is relatively weak. By virtue of its substantially lower z_min, then, the Facility experiment captures more of the redshift range most sensitive to γ and wins out over the galaxy survey.

Figure 18 shows the effect of the lowest-redshift bins on γ more clearly. Even Stage II outperforms the Stage IV survey for ${{z}_{{\rm max} }}\lesssim 1$—again thanks to its lower z_min—despite producing significantly worse constraints on almost every other parameter. This behavior is also related to the choice of distance measures and how the degeneracies between them get broken. As shown in Figure 8, the choice of measure can have a big effect on the f(z) errors, so one might expect the strength of the constraint on γ, and its orthogonality to the galaxy redshift survey, to change if a different subset of measures was used.

Zoom In Zoom Out Reset image size

Assuming the full set of distance measures, the complementarity between IM and galaxy redshift surveys can be used to significantly increase the precision of the constraint on γ, to the point where it becomes possible to clearly distinguish between many modified gravity models. Figure 19 shows the result of combining the two surveys on the errors for w₀ and γ, along with some example predictions from modified gravity theories. The marginal error on γ goes from ${{\sigma }_{\gamma }}=0.024$ for Facility + Planck to ${{\sigma }_{\gamma }}=0.015$ for Facility + DETF IV + Planck.

Zoom In Zoom Out Reset image size

We have assumed a fiducial value of ${{{\Omega}}_{{\rm HI},0}}=6.5\times {{10}^{-4}}$ throughout our analysis. Clearly our results will strongly depend on this value, as it is important in setting the "signal-to-noise" of the experiment—the more neutral hydrogen there is, the more easily the cosmological signal can be detected. There are, however, large uncertainties in ${{{\Omega}}_{{\rm HI}}}(z)$ from current observations (Figure 20). In particular, different tracers of the H i density give inconsistent results, so neither the normalization nor the redshift evolution of ${{{\Omega}}_{{\rm HI}}}(z)$ is well understood.

Zoom In Zoom Out Reset image size

The constraint of most relevance to us is from Masui et al. (2013), where IM measurements were cross-correlated with the WiggleZ galaxy redshift survey. It was found that

$$\begin{eqnarray*}&&{{{\Omega}}_{{\rm HI}}}{{b}_{{\rm HI}}}\bar{r}=4.3\pm 0.7({\rm stat}.)\pm 0.4({\rm sys}.)\times {{10}^{-4}}\end{eqnarray*}$$

at $z=0.8,$ where the best theoretical estimates for the cross-correlation coefficient are $\bar{r}$ = 0.9–0.95. This was obtained by restricting the analysis to $0.075\;h\ {\rm Mp}{{{\rm c}}^{-1}}\lt k\lt 0.3\;h\ {\rm Mp}{{{\rm c}}^{-1}}$; extending it to $0.04\;h\ {\rm Mp}{{{\rm c}}^{-1}}\lt k\lt 0.8\;h\ {\rm Mp}{{{\rm c}}^{-1}}$ lowers the constraint slightly to

$$\begin{eqnarray*}&&{{{\Omega}}_{{\rm HI}}}{{b}_{{\rm HI}}}\bar{r}=4.0\pm 0.5({\rm stat}.)\pm 0.4({\rm sys}.)\times {{10}^{-4}}.\end{eqnarray*}$$

The value of ${{{\Omega}}_{{\rm HI}}}$ is entangled with the bias, which, from semi-analytic models combined with N-body simulations (Khandai et al. 2011), is found to be consistently low, b_HI ≈ 0.55–0.66, although it can go up to unity for certain model choices. As a result, Masui et al. (2013) proposes that one should assume ${{{\Omega}}_{{\rm HI}}}=(4.5-7.5)\times {{10}^{-4}}$ at z = 0.8.

The lack of agreement between observations makes it difficult to reconstruct the redshift evolution of ${{{\Omega}}_{{\rm HI}}}.$ At the upper end of the redshift range we are considering ($z\lesssim 3$), constraints from damped Lyα (DLA) systems are scattered in the range ${{{\Omega}}_{{\rm HI}}}\approx (4-9)\times {{10}^{-4}}.$ At $z\sim 1$ there are discrepant results between theoretical models that find ${{{\Omega}}_{{\rm HI}}}\simeq 3\times {{10}^{-4}}$ (Duffy et al. 2012) and observations of DLAs with Hubble Space Telescope that give ${{{\Omega}}_{{\rm HI}}}\simeq 9\times {{10}^{-4}}.$ At z = 0, the ALFAFA and HIPASS surveys find ${{{\Omega}}_{{\rm HI}}}\simeq 4\times {{10}^{-4}},$ which is slightly lower than our fiducial ${{{\Omega}}_{{\rm HI},0}}.$

For the forecasts in this paper, we tread the middle ground (solid line, Figure 20). The ${{{\Omega}}_{{\rm HI}}}(z)$ redshift evolution is derived from a simulated H i halo mass function, as described in Appendix B, and we choose its fiducial normalization to be consistent with the GBT/WiggleZ cross-correlation measurement at z = 0.8. The magnitude and redshift evolution of the H i bias, ${{b}_{{\rm HI}}}(z),$ are derived from the same mass function.

Figure 21 shows the effect of rescaling ${{{\Omega}}_{{\rm HI}}}$ by a constant factor on the constraints for several observables. If ${{{\Omega}}_{{\rm HI}}}(z)$ was halved, for example, the FOM for Facility would drop by a factor of 5. This highlights the sensitivity of cosmological constraints from IM to the H i density and gives some idea of the degradation/improvement in performance that would be expected if ${{{\Omega}}_{{\rm HI}}}(z)$ substantially differs from what we have assumed.

Zoom In Zoom Out Reset image size

We have marginalized over the non-linear scale, ${{\sigma }_{{\rm NL}}},$ in all of our forecasts. As described in Section 3, this parameter is responsible for setting the resolution in the radial direction; beyond this scale, non-linear peculiar velocities wash out all redshift information.

Figure 22 shows the effect of changing the fiducial non-linear scale. As one might expect, increasing ${{\sigma }_{{\rm NL}}}$ degrades the various constraints, as information is lost at progressively larger scales. For the Facility experiment, which uses a combined single-dish and interferometer mode, the change in non-linear scale has a similar, relatively mild effect on all of the figures of merit, which change by less than a factor of 2 for a doubling of ${{\sigma }_{{\rm NL}}}.$ This is not the case for the purely interferometric Stage II survey, which is more sensitive to smaller scales (especially at low z), so it is hit harder by the loss of information there.

Zoom In Zoom Out Reset image size

The viability of IM as a cosmological probe vitally depends on the availability of accurate foreground removal techniques, as the contaminating signals have an amplitude of between 4 and 6 orders of magnitude greater than the cosmological H i signal (e.g., Alonso et al. 2014). In the previous sections we adopted a fiducial value for the residual foreground contamination amplitude of ${{\epsilon }_{{\rm FG}}}={{10}^{-6}},$ which is a reasonable target value for current foreground subtraction methods. In this section we quantify the sensitivity of our forecasts to the assumed removal efficiency and discuss a number of potential problems surrounding foreground contamination.

Most of the proposals for how to subtract foregrounds from IM data rely on a simple qualitative assumption: that foregrounds have a smooth (coherent) frequency dependence over the observed frequency ranges.¹¹ This is generally true, apart from in the presence of polarization leakage, which we will discuss shortly. A simple commonsense approach is to assume that the foreground signal along the frequency direction is accurately modeled as a sum of low-order polynomials or similar, which capture what should essentially be a (very mildly) modulated power-law behavior. This is at the heart of the methods presented in Wang et al. (2006), Gleser et al. (2008), Jelić et al. (2008), and Liu et al. (2009).

Another possibility is to be agnostic about the frequency dependence of the foregrounds, but decompose the total signal in some form of signal-to-noise eigenbasis. Since the foregrounds have such large magnitudes, the hope is that they will be contained only in the very high signal-to-noise part, and thus will be suitably segregated from the cosmic signal. This is the logic behind the methods used in Chang et al. (2010), and Wolz et al. (2014). When applied to real data in Chang et al. (2010), it was found that the foregrounds were not as strongly segregated from the cosmic signal as expected, so it was necessary to subtract a larger number of modes than originally planned (inevitably throwing out some of the cosmological signal too). Even in the optimal case, this type of foreground removal method has been shown to leave a residual bias in, for example, the cosmological parameters that best-fit the recovered BAO (Wolz et al. 2014).

As described in Section 2.3, we model the effects of foregrounds in terms of a residual noise term with an overall relative amplitude, ${{\epsilon }_{{\rm FG}}},$ and a minimum cutoff wavenumber, k_FG, along the radial (frequency) direction. For any given foreground removal method, these two parameters are intertwined—the more large-scale modes one uses to estimate the shape of the foregrounds (i.e., the larger k_FG is made), the better the removal efficiency, and so the lower ${{\epsilon }_{{\rm FG}}}$ should be. We have chosen k_FG to be a fixed fraction of the total bandwidth across all the redshift slices of a survey and have implicitly absorbed the removal efficiency into ${{\epsilon }_{{\rm FG}}}.$ (Note that this model is purely stochastic and does not allow us to model the biases that were discussed in Wolz et al. 2014.)

In Figure 23, one can clearly see that the optimal level of foreground subtraction is around ${{\epsilon }_{{\rm FG}}}\simeq {{10}^{-6}}$ for the Facility configuration, but can be larger for other configurations (which have higher noise levels). The impact of changing k_FG is shown in Figure 24; while larger k_FG should yield better foreground subtraction, there is a trade-off involved in losing more data at small k.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

While our analysis has so far focused on unpolarized foregrounds, problems may arise if one considers instrumental leakage from polarized foregrounds into the total intensity mode. For a typical receiver, one expects a cross-leakage of order a few percent, so given the large amplitude of the foregrounds, this can have a significant effect on the total signal. Synchrotron emission is the main polarized foreground and has a non-trivial angular and frequency dependence due to Faraday rotation. To see this, consider the polarization angle, ϕ, which is rotated by the galactic magnetic field, ${\boldsymbol{B}} ,$ through

$$\begin{eqnarray}&&\phi (r)={{\phi }_{0}}(r)+{{c}^{2}}{{\nu }^{-2}}\int _{0}^{r}{{n}_{e}}({{r}^{^{\prime} }}){\boldsymbol{B}} ({{r}^{^{\prime} }})\cdot d{{r}^{^{\prime} }},\end{eqnarray} \tag{ 24 }$$

where ${{\phi }_{0}}(r)$ is the initial angle, n_e is the electron density, and r is the distance along the line of sight. The galactic magnetic field is a non-linear superposition of an overall coherent mode, tied to the spiral structure of the Galaxy, with a turbulent stochastic mode on small scales (Beck 2001). From (24), one can see that the rotation of the polarization vector depends on both frequency and the line of sight through the galaxy with some decoherence length (corresponding to the coherence of ${\boldsymbol{B}}$ along the line of sight). This leads to a more complex foreground signal that is considerably less smooth in frequency than unpolarized foregrounds—an effect that increases in severity at low frequencies (Alonso et al. 2014). While we have not explicitly included it in our foreground model, the effects of polarization leakage can be partially accounted for by a larger fiducial ${{\epsilon }_{{\rm FG}}}.$

In this paper, we have advocated using some instruments as collections of single-dish experiments, i.e., in autocorrelation mode. This is common practice in CMB mapping experiments and has been the leading method of producing large-scale, high-resolution maps. Its use with radio telescope arrays at lower frequencies is less common, however, and must be treated with some care because of a number of potentially serious systematics.

Nevertheless, there is some precedent for using autocorrelation mode to detect both individual H i sources and unresolved emission. In Braun et al. (2003), the WSRT array was used in this mode to perform a wide-field survey, yielding a sample of $\sim 150$ H i galaxies. A key difficulty of the analysis was in obtaining an accurate calibration of the autocorrelation mode—while the average gain over all receivers was relatively stable, there were variations of up to 10% for individual receivers. This was calibrated out by using cross-correlation data for known radio sources. For the first attempt at mapping the unresolved H i signal with the GBT telescope (Chang et al. 2010), the flux calibration was controlled by fixing an intermittent noise source at the feed point, and by periodically monitoring a known source.

Drifts in the gain (e.g., due to instrumental temperature variations) are just one type of autocorrelation systematic. Another is due to spillover and sidelobe pickup, which can arise from a poorly characterized beam and ground contamination. This is not an insurmountable problem, and ground-based and balloon-borne CMB experiments commonly incorporate design features to mitigate these effects. One approach for H i instruments is that proposed by BINGO (Battye et al. 2013). There, the idea is to use a partially illuminated aperture to reduce the effect of sidelobes, spillover, and RFI contamination. Another aspect of the BINGO design is the use of a fixed dish, which scans the sky simply by allowing it to drift through the beam as the Earth rotates. This bypasses various problems that arise with moving parts, and allows a more precise pointing calibration than is possible with dynamic "raster" scan strategies.¹²

Another key obstacle in the analysis of autocorrelation data is the presence of $1/f$ noise, which is a coherent (correlated) noise drift on long timescales. Ideally, one would be able to construct a receiver system such that the $1/f$ knee (i.e., the timescale beyond which correlations become important) is at very low frequencies—a few $\times {{10}^{-3}}$ Hz, for example. If this is possible, then the noise will drift over periods of several minutes, which, for a drift scan, corresponds to angular scales of a few degrees (i.e., larger than the BAO scale). This is the method used by BINGO.

More traditionally, the way to deal with noise drifts or biases in CMB experiments has been to devise a scan strategy such that the modes one is looking at are scanned at frequencies higher than the knee frequency. The signal in a given pixel will then be localized in particular (frequency) modes of the time series that are subject only to white (i.e., uncorrelated) noise. This requires that the instrument should have the ability to scan quickly across the sky, which can be challenging for large dishes. One can further mitigate the effect of $1/f$ noise by devising a scan strategy with multiple cross-linking, i.e., in which each pixel is revisited a number of times on different timescales, from different directions. The inversion process for extracting the map from the time series (and estimating the noise) is then well defined and numerically robust (Ferreira & Jaffe 2000). If the noise is smooth in frequency, it may also be possible to clean out any noise bias component, as one would a foreground component in the signal.

In summary, there are clearly a number of issues that must be considered carefully when working in autocorrelation mode, but as we have shown in our analysis throughout this paper, the scientific potential of single-dish IM experiments is tremendous. Ideally, one would design the dishes, receivers, and other hardware of a given array specifically to mitigate the problems discussed above, but in some cases (e.g., with the SKA) this is not possible due to competing design constraints enforced by the need to operate primarily in an interferometric mode—optimizing the hardware for autocorrelation mode as well is simply too expensive. Any chance of controlling these (potentially critical) systematic effects to a sufficient degree is then left down to the choice survey strategy and data analysis methodology.

Experience with autocorrelation CMB experiments suggests that reliance on non-hardware techniques to reduce important systematics is a risky strategy, but there are a number of features of forthcoming radio telescope arrays that may be helpful in making this feasible. For example, an experiment with hundreds of individual dishes can make use of the fact that some systematics will be uncorrelated between the dishes; by cross-correlating (detected) maps of the same volume produced by different dishes, many effects can therefore be expected to correlate out. Nevertheless, it remains to be demonstrated that an experiment as complex as Phase 1 of the SKA can successfully control its autocorrelation calibration down to the required level and how the survey/analysis strategy affects its overall sensitivity. This must therefore be seen as an important caveat of our analysis.

A particular limitation to H i mapping with interferometers is the difficulty of sampling modes on large angular scales. In the simplest model of an interferometer—as a collection of dishes—the minimum measurable wavenumber is set by the minimum baseline, which cannot be smaller than the diameter of the dishes. From Figure 3 we can see that arrays with large dishes will not adequately sample BAO scales at low redshift in interferometer mode, so most of the constraints must come from single-dish mode.

There are a few ways to mitigate this shortcoming. The simplest is the approach effectively taken by BAOBAB and dense aperture arrays—to just use smaller dishes and pack them closer together. This results in smaller baselines and a larger FOV for the interferometer, but reduces its total effective collecting area (and thus its sensitivity). One can add more dishes to compensate, although this can substantially increase the cost of correlation hardware, which scales roughly like $\sim N_{{\rm dish}}^{2}.$ GPU-based correlators, or correlating only a subset of receiver pairs, can reduce costs for large numbers of receivers.

Alternatively, one can use a more novel reflector design. For example, CHIME uses long cylindrical reflectors with many closely spaced receivers installed along the cylinder (Shaw et al. 2014). This provides a large number of short baselines and a primary beam that is $\sim 180{}^\circ$ in one direction but much narrower along the orthogonal direction. This leads to a very anisotropic sampling of transverse Fourier modes, and only the large angular modes that are exactly aligned with the cylinders will be properly sampled. Because the visibilities measured by the interferometer are convolved with a window function defined by the primary beam, however, modes larger than the shortest side of the FOV will be aliased, making them difficult to disentangle. This is only the case if the interferometer tracks a single patch of the sky, though; if one progressively scans over the patch with different pointing offsets and has precise knowledge of the primary beam pattern, it is possible to remove the aliasing effect of the primary beam and thus independently measure modes larger than the instantaneous FOV by mosaicing (Holdaway et al. 1999). In the case of CHIME, drift scanning provides a continuous range of pointing offsets, and in principle the array can see the whole sky over a 24 hr observation period.

One can also make interferometric measurements over a number of separate pointings without mosaicing, simply to survey a larger area of sky (White et al. 1999). Drift scanning can be seen as a continuous limit of this. The advantage of such a method is that one can greatly reduce the sample variance of the smallest-baseline modes, simply by observing them on several independent patches of the sky. A crucial point is that simply patching together multiple fields does not allow modes larger than those defined by the minimum baseline to be measured, and therefore does not change the range of modes sampled, but does increase the sensitivity within that range. We have implicitly assumed that interferometers can handle multiple pointings by allowing ${{S}_{{\rm area}}}\gt {\rm FOV}$ in our forecasts.

Another possibility is to operate some experiments in a "combined mode," where both autocorrelation and cross-correlation data are collected. The simplest way of doing this in practice is to split the total survey time into two chunks, using only one of the observing modes for each. From the previous two sections, we can see that each mode will have different systematics and hardware requirements, and it is likely that substantially different survey strategies would be needed for each mode.

The situation is considerably more difficult if one tries to collect data in both modes simultaneously. As discussed above, in single-dish mode one has to mitigate $1/f$ noise, typically by rapidly scanning across the sky. Conversely, interferometers require precisely measured baselines and pointings to allow accurate phase calibration and reconstruction of the beam pattern; as dishes accelerate while scanning, even small distortions of the mounts can make this difficult. The combination of all these issues can potentially be overcome by drift scanning or through the use of novel mounts, but neither option appears to have been tested yet.

For a single dish with effective collecting area A_e, the noise associated with the measured flux is assumed Gaussian with an rms (flux sensitivity) given by

$$\begin{eqnarray*}&&{{\sigma }_{S}}=\frac{2{{k}_{B}}{{T}_{{\rm sys}}}}{{{A}_{e}}\sqrt{\delta \nu \;{{t}_{{\rm p}}}}},\end{eqnarray*}$$

integrated over a frequency interval $\delta \nu$ and observation time per pointing t_p. Telescope sensitivities are often quoted in terms of the System Equivalent Flux Density¹⁶ , ${\rm SEFD}\equiv 2{{k}_{{\rm B}}}{{T}_{{\rm sys}}}/{{A}_{e}},$ or alternatively just ${{A}_{e}}/{{T}_{{\rm sys}}}.$ The effective area A_e is usually $\sim 70\%-80\%$ of the actual dish area, depending on the efficiency, η, of the system. The total system temperature is ${{T}_{{\rm sys}}}={{T}_{{\rm sky}}}+{{T}_{{\rm inst}}},$ where ${{T}_{{\rm sky}}}\approx 60{{(300\;{\rm MHz}/\nu )}^{2.55}}$ K is the sky temperature and ${{T}_{{\rm inst}}}$ is the instrument temperature (which is typically higher than the sky temperature above 300 MHz). For typical instrumental specifications, the single-dish noise rms can be written as

$$\begin{eqnarray*}&&{{\sigma }_{S}}=18\;{\rm mJy}\left( \frac{{{T}_{{\rm sys}}}}{25\;{\rm K}} \right)\left( \frac{200\;{{{\rm m}}^{2}}}{{{A}_{e}}} \right){{\left( \frac{0.1{\rm MHz}}{\delta \nu } \right)}^{1/2}}{{\left( \frac{1{\rm hr}}{{{t}_{{\rm p}}}} \right)}^{1/2}}.\end{eqnarray*}$$

Because we are interested in brightness temperature sensitivity (i.e., where the signal fills the primary beam), we need to look instead at the intensity of the signal. This is found by dividing the flux by the primary beam solid angle at FWHM, $\theta _{{\rm B}}^{2}.$ In the Rayleigh–Jeans limit, the conversion from intensity to temperature then gives ${{\sigma }_{T}}\approx {{\lambda }^{2}}{{T}_{{\rm sys}}}/(\theta _{{\rm B}}^{2}{{A}_{e}}\sqrt{\delta \nu \;{{t}_{p}}}).$ For a given survey area, $\sim {{S}_{{\rm area}}}/\theta _{{\rm B}}^{2}$ pointings are needed, so the single-dish rms noise temperature is

$$\begin{eqnarray*}&&{{\sigma }_{T}}\approx \frac{{{T}_{{\rm sys}}}}{\sqrt{\delta \nu \;{{t}_{{\rm tot}}}}}\frac{{{\lambda }^{2}}}{\theta _{{\rm B}}^{2}{{A}_{e}}}\sqrt{{{S}_{{\rm area}}}/\theta _{{\rm B}}^{2}},\end{eqnarray*}$$

where ${{t}_{{\rm tot}}}$ is the total observation time for the survey. For an array with N_d identical dishes, the signals from each dish can be added incoherently, reducing ${{\sigma }_{T}}$ by a factor of $1/\sqrt{{{N}_{{\rm d}}}}.$ Telescopes can also use focal plane arrays on each dish to increase the instantaneous FOV, effectively increasing the number of beams, ${{N}_{{\rm b}}},$ using multiple feeds or PAFs. Receivers can also support more than one polarization channel, ${{n}_{{\rm pol}}}\geqslant 1,$ and the channels can be added incoherently. Taking all of this into account, we can write

$$\begin{eqnarray*}&&{{\sigma }_{T}}\approx \frac{{{T}_{{\rm sys}}}}{\sqrt{{{n}_{{\rm pol}}}\;\delta \nu \;{{t}_{{\rm tot}}}}}\frac{{{\lambda }^{2}}}{\theta _{{\rm B}}^{2}{{A}_{e}}}\sqrt{{{S}_{{\rm area}}}/\theta _{{\rm B}}^{2}}\sqrt{\frac{1}{{{N}_{{\rm d}}}{{N}_{{\rm b}}}}},\end{eqnarray*}$$

with the constraint that ${{S}_{{\rm area}}}\geqslant {{N}_{{\rm b}}}\theta _{{\rm B}}^{2},$ since nothing is gained by pointing all the feeds in the same direction.

We are interested in a statistical detection of the H i signal. The 3D noise power spectrum associated with an autocorrelation measurement is just ${{P}_{N}}=\sigma _{T}^{2}\;{{V}_{{\rm pix}}},$ where ${{V}_{{\rm pix}}}={{(r{{\theta }_{{\rm B}}})}^{2}}\times ({{r}_{\nu }}\delta \nu /{{\nu }_{21}})$ is the 3D volume of each volume element. We can then obtain the expression for the noise covariance from Equation (5) (ignoring the beams),

$$\begin{eqnarray*}&&{{C}^{N}}({\boldsymbol{q}} ,y)\equiv \frac{{{P}_{N}}}{{{r}^{2}}{{r}_{\nu }}}=\frac{T_{{\rm sys}}^{2}{{U}_{{\rm bin}}}}{{{n}_{{\rm pol}}}\;{{t}_{{\rm tot}}}{\Delta}\nu }\left( \frac{{{\lambda }^{4}}}{A_{e}^{2}\theta _{{\rm B}}^{4}} \right)\mathcal{I},\end{eqnarray*}$$

where ${{U}_{{\rm bin}}}={{S}_{{\rm area}}}\;{\Delta}\tilde{\nu }$ and $\mathcal{I}=1/{{N}_{{\rm b}}}{{N}_{{\rm d}}}.$ For a dish reflector, ${{A}_{e}}\equiv \eta \pi {{({{D}_{{\rm dish}}}/2)}^{2}}$ and ${{\theta }_{{\rm B}}}\approx \lambda /{{D}_{{\rm dish}}},$ and so the factor in brackets in the expression above is $\mathcal{O}(1)/{{\eta }^{2}}.$ For a dish equipped with a PAF, the beams begin to overlap below a critical frequency, ${{\nu }_{{\rm crit}}},$ and so there is a resulting loss of sensitivity, such that

$$\begin{eqnarray}{{C}^{N}}({\boldsymbol{q}} ,y)\to {{C}^{N}}({\boldsymbol{q}} ,y)\times \left\{ \begin{array}{ccccccccccccccc} 1, & \nu \gt {{\nu }_{{\rm crit}}} \\ {{\left( {{\nu }_{{\rm crit}}}/\nu \right)}^{2}}, & \nu \leqslant {{\nu }_{{\rm crit}}}. \\ \end{array} \right.\end{eqnarray} \tag{ D.1 }$$

For an interferometer, a pair of elements separated a baseline of length d measures a visibility $V({\boldsymbol{u}} ,\nu ),$ where ${\boldsymbol{u}}$ is the vector in uv space corresponding to that baseline,¹⁷ and $u=|{\boldsymbol{u}} |=d/\lambda .$ The uv-space resolution is set by the interferometer FOV, which for an array of dishes is given by the beam solid angle of a single dish, $\delta u\delta v\sim 1/{\rm FOV}\sim {{A}_{e}}/{{\lambda }^{2}}.$ Visibilities separated by more than this distance in uv-space can be taken as independent.

The detector noise for a single complex visibility measurement, $N({\boldsymbol{u}} ,\nu ),$ is assumed Gaussian with variance

$$\begin{eqnarray*}&&\sigma _{T}^{2}\equiv \langle N({\boldsymbol{u}} ,{{\nu }_{1}}){{N}^{*}}({\boldsymbol{u}} ,{{\nu }_{2}})\rangle ={{\left( \frac{{{\lambda }^{2}}{{T}_{{\rm sys}}}}{{{A}_{e}}\sqrt{\delta \nu {{t}_{{\boldsymbol{u}} }}}} \right)}^{2}}{{\delta }_{1,2}},\end{eqnarray*}$$

where $\delta \nu$ is the channel bandwidth and ${{t}_{{\boldsymbol{u}} }}$ is the observing time for a given baseline. While each visibility measurement is independent in terms of instrumental noise, for the same sky signal the measurements will be strongly correlated for distances smaller than $\sqrt{{{A}_{e}}/{{\lambda }^{2}}},$ as explained above. One way of dealing with this is to average all visibilities falling into each uv-space resolution element of area $\delta u\delta v.$ The noise will then be reduced by the number of points in that element, while the sky visibility stays essentially the same (assuming sufficiently high uv resolution).

Let t_s be the integration time for one visibility, and ${{N}_{s}}({\boldsymbol{u}} )$ the corresponding total number of visibilities falling into a given element after one "snapshot." N_s will then be directly related to the baseline distribution (once we have factored in the observation angle), as explained in the previous appendix. The noise in each uv resolution element is then

$$\begin{eqnarray*}&&\sigma _{T}^{2}({\boldsymbol{u}} ,\nu )={{\left( \frac{{{\lambda }^{2}}{{T}_{{\rm sys}}}}{{{A}_{e}}\sqrt{\delta \nu \;{{t}_{s}}{{N}_{s}}({\boldsymbol{u}} )}} \right)}^{2}}.\end{eqnarray*}$$

Note that t_s is usually just a few seconds, as longer integration times generate a "smearing" of the visibility in the uv plane due to Earth's rotation. Smaller integration times allow more visibility points to be sampled, but each with larger noise.

The total observation time in a given patch of the sky, t_p, is usually more than the snapshot time t_s, and the telescope tracks the patch. This implies that the same baseline might produce different vectors in the uv plane as the observation angle changes. To allow for different choices of the pixel area and integration time per visibility, one usually refers to the average number density of baselines averaged over a 24 hr period,

$$\begin{eqnarray*}&&n({\boldsymbol{u}} )=N({\boldsymbol{u}} )/(\delta u\delta v)/(24\;{\rm hr}/{{t}_{s}}).\end{eqnarray*}$$

$N({\boldsymbol{u}} )$ now corresponds to the total number of baselines falling into a given resolution element $\delta u\delta v$ in a 24 hr period, so that sky rotation is taken into account. For a given (square) resolution element, ${{({\Delta}u)}^{2}},$ we can write

$$\begin{eqnarray*}&&\sigma _{T}^{2}({\boldsymbol{u}} ,\nu )={{\left( \frac{{{\lambda }^{2}}{{T}_{{\rm sys}}}}{{{A}_{e}}\sqrt{\delta \nu \;n({\boldsymbol{u}} ){{({\Delta}u)}^{2}}{{t}_{p}}}} \right)}^{2}},\end{eqnarray*}$$

where $n({\boldsymbol{u}} )$ is usually only a function of $u\equiv |{\boldsymbol{u}} |$ due to the symmetrizing effect of the rotation on the uv coverage.

If we assume that n(u) is constant on the uv plane between some ${{u}_{{\rm min} }}={{D}_{{\rm min} }}/\lambda$ and ${{u}_{{\rm max} }}={{D}_{{\rm max} }}/\lambda$ (which is not the same as assuming a uniform distribution of antennas), we can write (see Equation (6))

$$\begin{eqnarray*}&&(\pi u_{{\rm max} }^{2}-\pi u_{{\rm min} }^{2})n(u)={{N}_{{\rm d}}}({{N}_{{\rm d}}}-1)/2.\end{eqnarray*}$$

This follows from noting that the integration of n(u) over the uv plane should give the total number of baselines.

The variance of the noise in 3D Fourier space is related to the variance of the visibilities through

$$\begin{eqnarray*}&&\langle {{N}_{S}}({\boldsymbol{k}} )N_{S}^{*}({\boldsymbol{k}} )\rangle \approx (\delta \nu /{{\nu }_{21}}){\Delta}\tilde{\nu }{{\left[ {{r}^{2}}{{r}_{\nu }}{{\sigma }_{T}}({\boldsymbol{u}} ,\nu ) \right]}^{2}},\end{eqnarray*}$$

which is obtained by noting that the 3D Fourier component corresponds to a Fourier transform of the visibility along the frequency direction. The noise power spectrum for a single-pointing observation is then given by

$$\begin{eqnarray*}&&{{P}_{N}}=\frac{\langle {{N}_{S}}({\boldsymbol{k}} )N_{S}^{*}({\boldsymbol{k}} )\rangle }{({\rm FOV}\;{{r}^{2}})({{r}_{\nu }}{\Delta}\tilde{\nu })},\end{eqnarray*}$$

which, using ${{({\Delta}u)}^{2}}\sim 1/{\rm FOV},$ reduces to an expression similar to Equation (5),

$$\begin{eqnarray*}&&{{C}^{N}}({\boldsymbol{q}} ,y)\equiv \frac{{{P}_{N}}}{{{r}^{2}}{{r}_{\nu }}}=\frac{T_{{\rm sys}}^{2}}{{{\nu }_{21}}{{t}_{p}}}\frac{{{\lambda }^{4}}}{A_{e}^{2}\;n({\boldsymbol{u}} )}.\end{eqnarray*}$$

In the above, t_p is the observation time for a single pointing of the interferometer. Once a signal-to-noise ratio of unity is achieved on the scales of interest, one can gain by moving to another pointing (i.e., increasing the survey area). The time spent at each pointing is then ${{t}_{p}}={{t}_{{\rm tot}}}({\rm FOV}/{{S}_{{\rm area}}}),$ and the number of observed modes is increased by a factor of ${{S}_{{\rm area}}}/{\rm FOV}.$ Taking this into account, as well as the possibility of having multiple beams, ${{N}_{{\rm b}}}\geqslant 1,$ and polarization channels, ${{n}_{{\rm pol}}}\geqslant 1,$ we arrive at the expression

$$\begin{eqnarray}&&{{C}^{N}}({\boldsymbol{q}} ,y)=\frac{T_{{\rm sys}}^{2}}{{{\nu }_{21}}{{n}_{{\rm pol}}}\;{{t}_{{\rm tot}}}}\frac{{{\lambda }^{4}}{{S}_{{\rm area}}}}{A_{e}^{2}\cdot {\rm FOV}}\frac{1}{{{N}_{{\rm b}}}\;n({\boldsymbol{u}} )}.\end{eqnarray} \tag{ D.2 }$$

For a standard dish reflector, ${\rm FOV}\approx \theta _{{\rm B}}^{2}.$ For an interferometer equipped with PAFs, the primary beam scales as

$$\begin{eqnarray*}{{\theta }_{B}}(\nu )={{\theta }_{B}}({{\nu }_{{\rm crit}}})\times \left\{ \begin{array}{ccccccccccccccc} \left( {{\nu }_{{\rm crit}}}/\nu \right), & \nu \gt {{\nu }_{{\rm crit}}} \\ 1, & \nu \leqslant {{\nu }_{{\rm crit}}}. \\ \end{array} \right.\end{eqnarray*}$$

For an aperture array, the primary beam scales as usual with frequency (${{\theta }_{B}}={{\theta }_{{\rm B}}}({{\nu }_{{\rm crit}}})\times ({{\nu }_{{\rm crit}}}/\nu )$), but now the effective area picks up a correction as the array becomes dense below the critical frequency,

$$\begin{eqnarray*}{{A}_{{\rm eff}}}(\nu )={{A}_{{\rm eff}}}({{\nu }_{{\rm crit}}})\times \left\{ \begin{array}{ccccccccccccccc} {{\left( {{\nu }_{{\rm crit}}}/\nu \right)}^{2}}, & \nu \gt {{\nu }_{{\rm crit}}} \\ 1, & \nu \leqslant {{\nu }_{{\rm crit}}}. \\ \end{array} \right.\end{eqnarray*}$$

The noise expression for cylinder interferometers (like CHIME) is more complicated. First of all, the primary beam is anisotropic; in the direction along the cylinder, the beam is $\sim 90{}^\circ ,$ while it is limited by the cylinder width, ${{w}_{{\rm cyl}}},$ in the perpendicular direction, giving ${\rm FOV}\approx 90{}^\circ \times \lambda /{{w}_{{\rm cyl}}}$ (Newburgh et al. 2014). second, many closely packed feeds illuminate each cylinder, complicating the relationship between number of receivers and collecting area assumed in Equation (D.2). To fit the cylinder noise expression into this form, we write ${{A}_{{\rm e}}}=\eta \;{{l}_{{\rm cyl}}}\;{{w}_{{\rm cyl}}}/{{N}_{{\rm feed}}},$ the effective area per feed, where ${{N}_{{\rm feed}}}$ is the number of feeds per cylinder and ${{l}_{{\rm cyl}}}$ is the cylinder length. There is also a restriction on the survey area; the beam cannot be steered and the telescope drift scans, fixing ${{S}_{{\rm area}}}\sim 30,000$ deg² (we choose 25,000 deg² as an effective area).

Finally, note that most of the quantities above depend on frequency. In particular, the FOV (and thus the minimum angular resolution) changes with frequency, so usually the maximum possible size for the uv-space resolution element is taken. Moreover, the final equation above is an approximation, and the middle of the frequency interval is taken in some of the expressions; otherwise, we would need to consider an integral over the frequency when calculating the noise power spectrum.

The derivatives for most of the terms in the signal model, Equation (4), are relatively straightforward:

$$\begin{eqnarray*}\begin{array}{rcl} {{\partial }_{A}}{{C}^{S}} & = & \frac{{{f}_{{\rm bao}}}(k)}{1+A{{f}_{{\rm bao}}}(k)}{{C}^{S}} \\ {{\partial }_{b}}{{C}^{S}} & = & \frac{2}{b+f{{\mu }^{2}}}{{C}^{S}} \\ {{\partial }_{f}}{{C}^{S}} & = & \frac{2{{\mu }^{2}}}{b+f{{\mu }^{2}}}{{C}^{S}} \\ {{\partial }_{{{\sigma }_{8}}}}{{C}^{S}} & = & (2/{{\sigma }_{8}})\;{{C}^{S}} \\ {{\partial }_{{{n}_{s}}}}{{C}^{S}} & = & {\rm log} (k/{{k}_{{\rm piv}}})\;{{C}^{S}} \\ {{\partial }_{\sigma _{{\rm NL}}^{2}}}{{C}^{S}} & = & -{{k}^{2}}{{\mu }^{2}}{{C}^{S}}. \\ \end{array}\end{eqnarray*}$$

We have used the splitting of P(k) into a smooth power spectrum plus a BAO feature from Equation (14) to calculate the derivative for the BAO amplitude, A. The derivative for ${{\sigma }_{8}}$ can be found by renormalizing the power spectrum, $P(k)\to {{({{\sigma }_{8}}/\sigma _{8}^{{\rm fid}})}^{2}}P(k),$ and the derivative for n_s by rewriting P(k) in terms of the primordial power spectrum, $P(k)\propto {{T}^{2}}(k)\;{{(k/{{k}_{{\rm piv}}})}^{{{n}_{s}}-1}},$ where ${{k}_{{\rm piv}}}=0.05\;{\rm Mp}{{{\rm c}}^{-1}}.$

The derivatives for the distance scales, $\{{{\alpha }_{\bot }},{{\alpha }_{\parallel }}\},$ are more complicated, but remain mostly analytic. For each α, the derivative is

$$\begin{eqnarray*}&&{{\partial }_{\alpha }}{{C}^{S}}=\left[ \frac{{{n}_{\alpha }}}{\alpha }+\frac{2f}{b+f{{\mu }^{2}}}{{\partial }_{\alpha }}{{\mu }^{2}}+{{\partial }_{k}}{\rm log} P(k)\;{{\partial }_{\alpha }}k \right]{{C}^{S}},\end{eqnarray*}$$

where ${{n}_{\bot }}=2,$ ${{n}_{\parallel }}=1.$ Only ${{\partial }_{k}}{\rm log} P(k)$ must be evaluated numerically; the other terms are given by

$$\begin{eqnarray*}\begin{array}{rcl} {{\partial }_{{{\alpha }_{\bot }}}}{{\mu }^{2}} & = & -2\;\alpha _{\bot }^{-1}{{\chi }^{2}}/{{\left( 1+{{\chi }^{2}} \right)}^{2}} \\ {{\partial }_{{{\alpha }_{\parallel }}}}{{\mu }^{2}} & = & 2\;\alpha _{\parallel }^{-1}{{\chi }^{2}}/{{\left( 1+{{\chi }^{2}} \right)}^{2}} \\ {{\partial }_{{{\alpha }_{\bot }}}}k & = & {{\left( \frac{{{\alpha }_{\bot }}q}{r} \right)}^{2}}/\left( k{{\alpha }_{\bot }} \right) \\ {{\partial }_{{{\alpha }_{\parallel }}}}k & = & {{\left( \frac{{{\alpha }_{\parallel }}y}{{{r}_{\nu }}} \right)}^{2}}/\left( k{{\alpha }_{\parallel }} \right), \\ \end{array}\end{eqnarray*}$$

where we have used the definitions

$$\begin{eqnarray*}\begin{array}{rcl} \chi & = & \frac{{{\alpha }_{\bot }}{{r}_{\nu }}q}{{{\alpha }_{\parallel }}ry} \\ {{k}^{2}} & = & {{\left( \frac{{{\alpha }_{\bot }}q}{r} \right)}^{2}}+{{\left( \frac{{{\alpha }_{\parallel }}y}{{{r}_{\nu }}} \right)}^{2}} \\ {{\mu }^{2}} & = & \frac{{{y}^{2}}}{{{y}^{2}}+{{\left( \frac{{{\alpha }_{\bot }}{{r}_{\nu }}}{{{\alpha }_{\parallel }}r} \right)}^{2}}{{q}^{2}}}. \\ \end{array}\end{eqnarray*}$$

Where we are interested in constraining cosmological parameters rather than functions of redshift, we project from $\{{{D}_{A}}(z),H(z),f(z)\}$ into the parameters $\{h,{{{\Omega}}_{K}},{{{\Omega}}_{{\rm DE}}},{{w}_{0}},{{w}_{a}},\gamma \}$ using Equation (20). To do this, we first assume the following forms for the functions of redshift, based on simple extensions to ΛCDM:

$$\begin{eqnarray*}\begin{array}{rcl} H(a) & = & {{H}_{0}}\sqrt{{{{\Omega}}_{M}}{{a}^{-3}}+{{{\Omega}}_{DE}}(a)+{{{\Omega}}_{K}}{{a}^{-2}}} \\ r(a) & = & \frac{c}{{{H}_{0}}}S\left( \int \frac{d{{a}^{^{\prime} }}}{{{a}^{^{\prime} }}^{2}E({{a}^{^{\prime} }})} \right) \\ f(a) & = & {\Omega}_{M}^{\gamma }(a), \\ \end{array}\end{eqnarray*}$$

where for ${{{\Omega}}_{K}}=\{-{\rm ve},\;0,\;+{\rm ve}\}$ we have defined $S(x)=\left\{ {\rm sin} (x\sqrt{|{{{\Omega}}_{K}}|})/\sqrt{|{{{\Omega}}_{K}}|},x,{\rm sinh} (x\sqrt{{{{\Omega}}_{K}}})/\sqrt{{{{\Omega}}_{K}}} \right\},$ and

$$\begin{eqnarray*}\begin{array}{rcl} w(a) & \approx & {{w}_{0}}+(1-a){{w}_{a}} \\ {{{\Omega}}_{M}}(a) & = & H_{0}^{2}{{{\Omega}}_{M}}{{a}^{-3}}/{{H}^{2}}(a) \\ {{{\Omega}}_{{\rm DE}}}(a) & = & {{{\Omega}}_{{\rm DE}}}\frac{{\rm exp} [3{{w}_{a}}(a-1)]}{{{a}^{3(1+{{w}_{0}}+{{w}_{a}})}}}, \\ \end{array}\end{eqnarray*}$$

$H(a)\equiv {{H}_{0}}E(a),$ ${{H}_{0}}=100\;{\rm hr}\;{\rm km}\ {{{\rm s}}^{-1}}\;{\rm Mp}{{{\rm c}}^{-1}},$ and ${{{\Omega}}_{M}}=1-{{{\Omega}}_{K}}-{{{\Omega}}_{{\rm DE}}}.$

Next, we need the derivatives of the original functions of redshift with respect to the new parameters. Most of them enter through the dimensionless Hubble rate, E(a), for which the relevant derivatives are

$$\begin{eqnarray*}\begin{array}{rcl} {{\partial }_{{{{\Omega}}_{k}}}}E(a) & = & ({{a}^{-2}}-{{a}^{-3}})/2E(a) \\ {{\partial }_{{{{\Omega}}_{{\rm DE}}}}}E(a) & = & (1-{{a}^{-3}})/2E(a) \\ {{\partial }_{{{w}_{0}}}}E(a) & = & -\frac{3}{2}\frac{{{{\Omega}}_{{\rm DE}}}}{E(a)}{\rm log} a \\ {{\partial }_{{{w}_{a}}}}E(a) & = & -\frac{3}{2}\frac{{{{\Omega}}_{{\rm DE}}}}{E(a)}[{\rm log} a+(1-a)] \\ {{\partial }_{\gamma }}E(a) & = & {{\partial }_{h}}E(a)=0. \\ \end{array}\end{eqnarray*}$$

For ${{\alpha }_{\parallel }},$ we then have (with β being any of the parameters except h, and evaluating on ΛCDM)

$$\begin{eqnarray*}&&{{\partial }_{\beta }}{{\alpha }_{\parallel }}=\frac{{{\partial }_{\beta }}E}{{{E}_{{\Lambda}}}}.\end{eqnarray*}$$

The expression for ${{\alpha }_{\bot }}$ is more complicated. For all but ${{{\Omega}}_{k}},$ the derivatives are given by

$$\begin{eqnarray*}\begin{array}{rcl} {{\partial }_{\beta }}{{\alpha }_{\bot }} & = & -\frac{{{\alpha }_{\bot }}}{{{r}_{{\Lambda}}}}\frac{\partial S(x)}{\partial x}{{\partial }_{\beta }}x \\ {{\partial }_{\beta }}x & = & -\int \frac{cd{{a}^{^{\prime} }}}{{{a}^{^{\prime} }}^{2}{{E}^{2}}({{a}^{^{\prime} }})}{{\partial }_{\beta }}E({{a}^{^{\prime} }}). \\ \end{array}\end{eqnarray*}$$

Additional terms appear in the derivative w.r.t. ${{{\Omega}}_{K}},$

$$\begin{eqnarray*}&&{{\partial }_{{{{\Omega}}_{K}}}}{{\alpha }_{\bot }}=-\frac{{{\alpha }_{\bot }}}{{{r}_{{\Lambda}}}}\left[ \frac{\partial S(x)}{\partial x}{{\partial }_{{{{\Omega}}_{K}}}}x+\frac{{\Theta}({{{\Omega}}_{K}})}{2{{{\Omega}}_{K}}}\left( x\frac{\partial S(x)}{\partial x}-S(x) \right) \right],\end{eqnarray*}$$

where ${\Theta}=0$ for ${{{\Omega}}_{K}}=0$ and unity elsewhere. For both ${{\alpha }_{\parallel }}$ and ${{\alpha }_{\bot }},$ the h derivative is ${{\partial }_{h}}\alpha =\alpha /h.$

For the growth function, the derivatives for all but $\{\gamma ,{{{\Omega}}_{K}},{{{\Omega}}_{{\rm DE}}}\}$ are given by

$$\begin{eqnarray*}&&{{\partial }_{\beta }}f(a)=-\gamma f(a)\frac{{{\partial }_{\beta }}E}{{{E}_{{\Lambda}}}}.\end{eqnarray*}$$

The derivatives with respect to the other parameters are

$$\begin{eqnarray*}\begin{array}{rcl} {{\partial }_{\gamma }}f & = & f(a){\rm log} {{{\Omega}}_{M}}(a) \\ {{\partial }_{{{{\Omega}}_{K}}}}f & = & -\gamma f\left[ {\Omega}_{M}^{-1}+{{\partial }_{{{{\Omega}}_{K}}}}{\rm log} E \right] \\ {{\partial }_{{{{\Omega}}_{{\rm DE}}}}}f & = & -\gamma f\left[ {\Omega}_{M}^{-1}+{{\partial }_{{{{\Omega}}_{{\rm DE}}}}}{\rm log} E \right]. \\ \end{array}\end{eqnarray*}$$