3.2.1. Cosmic shear simulations for SKA

We create forecasts for the SKA1 Medium-Deep Band 2 Survey. This survey is very similar to the optimal observing configuration found from catalogue-level simulations in Bonaldi et al. (Reference Bonaldi, Harrison, Camera and Brown2016). We assume the survey will use the lower $1/3$ of Band 2 and the weak lensing data will be weighted to give an image plane point spread function (PSF) width of $0.55\,{\text{arcsec}}$ , with the source population cut to include all sources which have flux $>10\sigma$ and a size $>$ 1.5 $\times$ the PSF size. These source populations are also rescaled, as in Bonaldi et al. (Reference Bonaldi, Harrison, Camera and Brown2016), to more closely match more recent data and the T-RECS simulation (Bonaldi et al. Reference Bonaldi, Bonato, Galluzzi, Harrison, Massardi, Kay, De Zotti and Brown2018). For comparison to a similar Stage III optical weak lensing experiment, and for use in shear cross-correlations, we take the DES with expectations for the full 5-yr survey. The assumed parameters of the two surveys are fully specified in Table 4. For the Medium-Deep Band 2 Survey, we assume a sensitivity corresponding to baseline weighting resulting in an image plane PSF with a best-fitting Gaussian FWHM of $0.55\,{\text{arcsec}}$ .

Table 4. Parameters used in the creation of simulated weak lensing data sets for SKA1 Medium-Deep Band 2 Survey and DES 5-yr survey considered in this section.

We assume redshift distributions for weak lensing galaxies follow a distribution for the number density of the form

(9) \begin{equation}\frac{{\text{d}} n}{{\text{d}} z} \propto z^{2} \exp\left( -(z/z_0)^{\gamma} \right),\end{equation}

where $z_0 = z_{m} / \sqrt{2}$ and $z_{m}$ is the median redshift of sources using best fitting parameters for the SKA1-MID Medium-Deep Band 2 Survey population and DES survey given in Table 4. Sources are split into ten tomographic redshift bins, with equal numbers of sources in each bin and each source is attributed an error as follows. A fraction of sources $f_{{\text{spec-}}z}$ out to a redshift of $z_{{\text{spec-max}}}$ are assumed to have spectroscopic errors, in line with the predictions of Yahya et al. (Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015); Harrison et al. (Reference Harrison, Lochner and Brown2017). The remainder of sources are given photometric redshift errors with a Gaussian distribution (constrained with the physical prior $z>0$ ) of width ${(1+z)\sigma_{{\text{photo-}}z}}$ out to a redshift of $z_{{\text{photo-max}}}$ . Beyond $z_{{\text{photo-max}}}$ , we assume very poor redshift information, with ${(1+z)\sigma_{{\text{no-}}z}}$ .

Of crucial importance to weak lensing cosmology is precise, accurate measurement of source shapes in order to infer the shear transformation resulting from gravitational lensing. For our forecasts, we assume systematic errors due to shear measurement will be sub-dominant to statistical ones. For the Medium-Deep Band 2 Survey, the formulae of Amara & Réfrégier (Reference Amara and Réfrégier2008) allow us to calculate requirements on the multiplicative shear bias of $\sigma_{m} < 6.4\times10^{-3}$ and additive shear bias of $\sigma_{c} < 8.0\times10^{-4}$ . These requirements are of the same order of magnitude as those achieved in current optical weak lensing surveys such as DES and the Kilo-Degree Survey,Footnote ^j but tighter (by an order of magnitude in the case of multiplicative bias) than current methods for radio interferometer to date (Rivi & Miller Reference Rivi and Miller2018; Rivi et al. Reference Rivi, Lochner, Balan, Harrison and Abdalla2018). We assume that in the period to 2028, when observations are currently expected to begin, sufficient progress will be made in radio shear measurement methods such that biases are comparable to those achievable in optical surveys today. Previous work has shown that this is highly unlikely to be possible with images created with the CLEAN algorithm (Högbom Reference Högbom1974) meaning access to lower level data products such as gridded visibilities (or equivalently dirty images) will be essential (see also Patel et al. Reference Patel2015; Harrison & Brown Reference Harrison and Brown2015). For the intrinsic ellipticity distribution of galaxies, we use a shape dispersion of $\sigma_{g_i} = 0.3$ .

Figure 3. Forecast constraints for weak lensing with the SKA1 Medium-Deep Band 2 Survey as specified in the text, compared to the Stage III optical weak lensing DES and including cross-correlation constraints.

There are significant advantages to forming cosmic shear power spectra by cross-correlating shear maps made using two different experiments. In such power spectra, wavelength-dependent additive and multiplicative systematics can be removed (Camera et al. Reference Camera, Harrison, Bonaldi and Brown2017) and almost all of the statistical constraining power on cosmological parameters is retained (Harrison et al. Reference Harrison, Camera, Zuntz and Brown2016). Care must be taken in identifying the noise power spectra in the case of cross-power spectra; it will be affected by the overlap in shape information between cross-experiment bins. We note that constraints are relatively insensitive to the number of galaxies which are present in both bins, being degraded by only 4% when the fraction of overlap is varied between zero and one (see Harrison et al. Reference Harrison, Camera, Zuntz and Brown2016, Figure 1).

Table 5. One-dimensional marginalised constraints, from weak lensing alone and in combination with Planck CMB (Planck CMB2015 + BAO + lensing as described in Section 2.6), on the parameters considered, where all pairs (indicated by brackets) are also marginalised over the base ΛCDM parameter set.

Figure 4. The effect of including a prior from the Planck satellite (Planck 2015 CMB + BAO + lensing as described in Section 2.6) on the forecast Dark Energy constraints for the specified cross-correlation weak lensing experiment (note that constraints in the other two parameter spaces are not significantly affected).

3.2.2. Results from autocorrelation

We show forecast constraints in three cosmological parameter spaces in Figure 3: matter ( $\Omega_\textit{m}\hbox{-}\sigma_8$ ), Dark Energy equation of state in the CPL parameterisation ( $w_0\hbox{-}w_a$ ), and modified gravity modifications to the Poisson equation and Gravitational slip ( $\mu_0\hbox{-}\gamma_0$ ). Our results show that the SKA1 Medium-Deep Band 2 Survey will be capable of comparable constraints to other DETF Stage III surveys such as DES and also, powerfully, that cross-correlation constraints (which are free of wavelength-dependent systematics) retain almost all of the statistical power of the individual experiments. In Figure 4, we also present forecast constraints in the Dark Energy parameter space including priors from the Planck CMB experiment, specifically a Gaussian approximation to the Planck 2015 CMB + BAO + lensing likelihood as described in Section 2.6 with constraints on the other parameters considered not significantly affected by application of the Planck prior. We note that future CMB experiments may improve their constraining power, the lowering the impact of the SKA measurements on this particular parameter space, however as outlined below, a major motivation for weak lensing in the radio is the independence of the systematics compared to measurements in the optical.

We also display tabulated summaries of the one-dimensional marginalised uncertainties on these parameters in Table 5.

3.2.3. Results for mixed-stage surveys

The current SKA timeline expects large surveys such as the Medium-Deep Band 2 Survey specified here to begin in 2027, by which time Stage III optical surveys such as DES will have been completed and analysed (DES data have been taken up to year 6 and the year 3 data release is currently being prepared. One may expect the 5-yr release to be in 2021). Stage IV optical surveys (LSST and the Euclid satellite) are currently scheduled to begin taking data in the middle years of the next decade, with the full data sets becoming available around 2030, possibly concurrent with those from SKA phase 1. We therefore also consider forecasts for mixed-stage cosmic shear surveys, with the radio data coming from SKA phase 1 Medium-Deep Band 2 Survey as described above, and optical data from the Stage IV LSST survey. Figure 5 shows the relevant contours for the $\Omega_\textit m\hbox{-}\sigma_8$ parameters, with the expected significant gain when going from a Stage III to Stage IV survey. The contours from the SKA1-Medium-Deep Band 2 Survey $\times$ LSST combination show degradation of constraints with respect to the LSST case, but will be significantly less susceptible to systematics, as discussed above and below in this section. For LSST, we assume a galaxy number density of $n = 37\,$ arcmin^–2 and a sky area of $18\,000\,$ deg² and photometric redshifts only out to $z=3$ . For the cross-correlation, we consider only the $5\,000\,$ deg² SKA Medium-Deep Band 2 Survey area.

Figure 5. Forecast constraints for weak lensing with the SKA1 Medium-Deep Band 2 Survey as specified in the text, compared to the Stage IV optical weak lensing LSST survey and including cross-correlation. constraints.

3.2.4. Results from radio-optical cosmic shear cross-correlations

A key consideration in weak lensing surveys are the systematics induced by the instrument on galaxy shape measurements, which must be controlled to high levels in order to ensure unbiased constraints on cosmological parameters. In contrast with the optical weak lensing surveys conducted to date, radio weak lensing surveys will measure galaxy shapes from uv-data, allowing for direct Fourier plane measurement, as well as measurement in images reconstructed by deconvolving the interferometer PSF. The systematics from these shape measurements will be very different, and uncorrelated with, those from measuring shapes from CCD images. In Rivi & Miller (Reference Rivi and Miller2018), the authors adapted the optical method lensfit to shape measurement on Fourier-domain interferometer data which is capable of satisfying the requirements for the SKA1 Medium-Deep Band 2 Survey on sources with ${\text{SNR}}>18$ . Residual systematics are typically modelled as linear in the shear and shear power spectrum, with an additive and multiplicative component. In Figure 3 (and Harrison et al. Reference Harrison, Camera, Zuntz and Brown2016), the unfilled black contours show the constraints from cross-correlating radio and optical weak lensing experiments, demonstrating that nearly all of the statistical constraining power remains.

We explictly show this removal of systematics through cross-correlations in Figure 6 (and Camera et al. Reference Camera, Harrison, Bonaldi and Brown2017). Both panels show forecasts (made using Fisher matrices validated on the MCMC chains described above) for constraints on the $\lbrace w_0, w_{a}\rbrace$ dark energy parameters. The upper panel shows the effect of systematics which are additive in the power spectrum, for a given choice of additive systematics power spectrum of fixed slope and varying amplitudes (see Camera et al. Reference Camera, Harrison, Bonaldi and Brown2017, for a full description of both this and the multiplicative power spectrum systematics models). As can be seen, such systematics significantly bias the recovered values of $\lbrace w_0, w_{a}\rbrace$ away from the input cosmology shown by the dashed cross. By construction, additive systematics are removed for the Radio $\times$ Optical combination and the correct input cosmology is recovered. The lower panel shows the effect of systematics which are multiplicative in the power spectrum (i.e. are calibration systematics). Here, whilst the combined Radio $\times$ Optical contour remains biased away from the input cosmology, the three separate contours available allow a self-calibration procedure to be applied; each contour has different systematics, but all are measuring the same cosmology, meaning a correction can be found which makes all three consistent with each other, and the input cosmology. Mitigation of such multiplicative systematics is expected to be extremely important even at the level of Stage III surveys and represents a powerful argument for performing weak lensing in the radio band.

Figure 6. Weak lensing marginal joint 1 $\sigma$ error contours in the dark energy equation-of-state parameter plane with additive (top) and multiplicative (bottom) systematics on the shear power spectrum measurement. The black cross indicates the ΛCDM fiducial values for dark energy parameters. Blue, red, and green ellipses are for radio and optical/near-IR surveys and their cross-correlation, respectively. (Details in the text.)

3.3.1. Forecasting

In order to estimate the effectiveness of the surveys and make predictions for the constraints on the cosmological parameters, we simulate the auto- and cross-correlation galaxy clustering angular power spectra, including the effects of cosmic magnification and the ISW. As only the observed galaxy distributions (which are affected by gravitational lensing) can be measured, it is impossible to measure the galaxy angular power spectrum decoupled from magnification. Hence, the galaxy clustering angular power spectrum contains both the density and magnification perturbations.

We use the simulated source count and galaxy bias model from Section 3.1 to simulate the angular correlation and cross-correlation functions $C_\ell$ , and the relevant measurement covariance matrices, for the Wide Band 1 Survey and Medium-Deep Band 2 Survey. In the case of galaxy clustering and ISW, we limit the analysis to the multipoles $\ell_{{\text{min}}}\leq \ell\leq 200$ , where $\ell_{{\text{min}}} = \pi/(2f_{{\text{sky}}})$ and $f_{{\text{sky}}}$ is the fraction of sky surveyed.

When making our forecasts, we also compare to and combine with current constraints from Planck CMB 2015, BAO, and RSD observations, as described in Section 2.6 (with additional relevant information for the extension parameters under consideration). We also assume that the overall bias for a particular redshift bin to be unknown, and so marginalised over. As such there are five (or nine, depending on the number of photometric bins for the given survey) extra parameters being considered in the Fisher matrix, which will degrade the performance of these cosmological probes.

Table 6. Predicted constraints from continuum galaxy clustering measurements using the two different survey strategies (Wide Band 1 Survey and Medium-Deep Band 2 Survey). These are 68% confidence levels on each of the parameters of the four different cosmological models we tested. The three main columns show results of galaxy clustering (GC) by itself (left), GC combined with ISW constraints (centre), and when Planck priors from Planck CMB 2015 + BAO are added to GC + ISW (right). Note that these cases assume that the overall bias in each of the photometric redshift bins is unknown and needs to be marginalised over.

3.3.2. Results

The 68% confidence level constraints on the different parameters described in Section 2.6 for the Wide Band 1 Survey and the Medium-Deep Band 2 Survey are given Table 6.

We show the predicted 68% and 95% confidence level constraints as a 2D contour, for the dark energy parameters $w_0$ and $w_a$ in Figure 7, and the modified gravity parameters $\mu_0$ and $\gamma_0$ in Figure 8. These constraints are shown for the Wide Band 1 Survey and the Medium-Deep Band 2 Survey in red, combining measurements from all photometric redshift bins, and including constraints from the ISW. In the dark energy case, we also show current constraints from Planck in blue, but for the modified gravity case, the Planck MCMC chains for these models are not public.

The predicted constraints on the dark energy parameters do not improve significantly on those presently available. This is also somewhat the case for the modified gravity parameters and the curvature, in the case of the Medium-Deep Band 2 Survey, though the Wide Band 1 Survey does improve on current knowledge. However, such constraints will improve with a better knowledge of the bias (decreasing the number of extra parameters to be marginalised over) and with a larger number of photometric redshift bins.

Constraints on $f_{{\text{NL}}}$ from the Medium-Deep Band 2 Survey will not be significantly better than those currently made by the Planck surveyor, $f_{{\text{NL}}} = 2.5 \pm 5.7$ (Planck Collaboration et al. Reference Collaboration2016Reference Planckb). In contrast, the Wide Band 1 Survey is capable of improving the constraint, with further potential gain from an increased number of redshift bins (Raccanelli et al. Reference Raccanelli2017). Finally, more competitive constraints on all parameters, but especially for $f_{{\text{NL}}}$ , may be achievable through the use of different radio galaxy populations as tracers of different mass halos, as described in Ferramacho et al. (Reference Ferramacho, Santos, Jarvis and Camera2014).

Figure 7. 68% and 95% confidence level forecast constraints on the deviation of the dark energy parameters $w_0,w_a$ from their fiducial values for the Wide Band 1 Survey (top) and Medium-Deep Band 2 Survey (bottom), using galaxy clustering data, including the effects of cosmic magnification. We show constraints from Planck CMB 2015 and BAO and RSD observations, as described in Section 2.6 in blue, SKA1 forecasts in red and the constraints for the combination of both experiments in green. We show here that for the dark energy parameters, the continuum data adds little to the existing constraints, owing to the uncertainty in the bias in each redshift bin. As such the blue Planck + BAO ellipse is only slightly bigger than the SKA + Planck + BAO for the continuum data. For the modified gravity parameters on the right, the Planck + BAO only chains were not available, and so the blue ellipse was left out of the figure.

Figure 8. 68% and 95% confidence level forecast constraints on the deviation of the modified gravity parameters $\mu_0,\gamma_0$ from their fiducial values for the Wide Band 1 Survey (top) and Medium-Deep Band 2 Survey (bottom), using galaxy clustering data, including the effects of cosmic magnification. We show SKA forecasts constraints in red and the constraints for the combination of SKA1 with Planck CMB 2016 and BAO in green.

3.4.1. Forecasting

In this section, we estimate the ability of SKA1 continuum surveys to measure the cosmic radio dipole using realistic mock catalogues, which include the effects of LSS and the kinematic dipole. Details of that study will be published elsewhere. Briefly, the mock catalogues assumed an angular power spectrum of the radio galaxies generated by CAMB sources (Challinor & Lewis Reference Challinor and Lewis2011), assuming the Planck best-fit flat $\Lambda$ CDM model (Planck Collaboration et al. Reference Collaboration2016Reference Plancka). The redshift distribution N(z) is shown in Figure 1, and the bias b(z) follows Alonso et al. (Reference Alonso, Bull, Ferreira, Maartens and Santos2015b). The available sky area is $f_{{\text{sky}}} \approx 0.52$ due to the removal of the galactic plane on low latitudes ( $|b| \leq 10^{\circ}$ ). Using the lognormal code flask (Xavier, Abdalla, & Joachimi Reference Xavier, Abdalla and Joachimi2016), we produced ensembles of 100 catalogues each, where the radio source positions follow the expected clustering distribution.

The effect of the kinetic dipole is implemented by boosting the maps of galaxy number densities according to the theoretical expectation (Ellis & Baldwin Reference Ellis and Baldwin1984),

(10) \begin{align}\left[\frac{{\text{d}} N}{{\text{d}}\Omega} (S, \mathbf{n})\right]_{{\text{obs}}} = \left[\frac{{\text{d}} N}{{\text{d}}\Omega} (S, \mathbf{n})\right]_{{\text{rest}}} \left( 1 + [2 + \beta (1+\alpha)] \frac{\mathbf{n}\cdot\mathbf{v}} {c}\right),\end{align}

where S denotes the flux density threshold of the survey, $\mathbf{n}$ is the direction on the sky and $\mathbf{v}$ is the Sun’s proper motion. This expression assumes that radio sources follow a power-law spectral energy distribution, $S \propto \nu^{-\beta}$ with $\beta = 0.75$ . The source counts are assumed to scale with S as $dn/dS\propto S^{-\alpha}$ , and we assume $\alpha = 1$ (which is very similar to the values of $\alpha$ for the individual redshift bins from simulations given in Table 3).

Here, we show results from estimations of the radio dipole direction and amplitude, $A = [2+ \beta(1+\alpha)]|\mathbf{v}|/c$ , of the generated mock catalogues by means of a quadratic estimator in pixel space on a HEALPixFootnote ^k grid with Nside = 64. Using pixel space has the advantage that incomplete sky coverage does not bias the results.

3.4.2. Results

Figure 9 shows an example of a simulated sky for a flux density threshold of 22.8 $\mu$ Jy at a central frequency of 700 MHz (Band 1), demonstrating the effect of the dipole on the source counts, as the southern sky appears to be slightly more dominated by blue than the northern hemisphere. The results from a set of 100 such simulations is shown in Figure 10. Given the assumptions, we would expect our mocks to produce a kinematic radio dipole amplitude of $A = 0.0046$ , pointing to the CMB dipole direction. The LSS contributes a dipole with a mean amplitude of $A = 0.0031 \pm 0.0016$ . This prediction depends on the assumed luminosity functions, spectral energy distributions, bias, redshift, and luminosity evolution of radio sources, see e.g. Tiwari & Nusser (Reference Tiwari and Nusser2016).

Figure 9. Simulated source count per pixel for the SKA1-MID Wide Band 1 Survey at a central frequency of 700 MHz and a flux threshold of 22.8 $\mu$ Jy in galactic coordinates and Mollweide projection at HEALPix resolution $N_{{\text{side}}}=64$ including the kinematic dipole and cosmic structure up to multipole moment $\ell_{{\text{max}}}=128$ . This shows the effect of the dipole on the source counts, as the southern sky appears here slightly bluer than the northern hemisphere.

Figure 10 shows the expected total radio dipole, which comprises contributions from LSS and the proper motion of the solar system. The expected kinematic contribution dominates the structure contribution and the measured amplitude is $A = 0.0056 \pm 0.0017$ in direction $(l,b) = (263.5\pm 28.0, 38.8\pm 19.7)$ deg. The distribution of dipole directions from the mocks is centred on the CMB dipole direction, but with some scatter due to the LSS.

The structure dipole is in fact dominated by contributions from local structure. Removing the low-redshift structure dipole ( $z < 0.5$ ), which might be possible using optical or infra-red catalogues, or by means of the HI redshift measured by the SKA, we measure the dipole direction $(l,b) = (265.3 \pm 4.9, 46.4 \pm 4.3)$ deg, in excellent agreement with the simulated dipole direction, with an amplitude of $A = 0.0047 \pm 0.0004$ , also agreeing with the input value. The distributions of dipole amplitudes are shown in the right panel of Figure 10.

We also simulated catalogues with $S = 5, 10$ and 16 $\mu$ Jy, which show that the structure dipole depends on the flux density threshold, providing an extra handle to separate them from the kinematic dipole. In none of our simulations was shot noise a limitation, in contrast to contemporary radio continuum surveys (Schwarz et al. Reference Schwarz2015).

4.2.1. Baryon Acoustic Oscillations and RSDs

The BAO feature is a preferred scale in the clustering of galaxies, set by sound waves emitted in the early Universe when photons and baryons were coupled. Since the true physical scale of the BAO is known from CMB observations, we can use the feature as a ‘standard ruler’ to measure the cosmological expansion rate and distance-redshift relation. This is achieved by separately measuring the apparent size of the BAO feature in the transverse and radial directions on the sky, and comparing with its known physical size [set by the size of the comoving sound horizon during the baryon drag epoch, $r_s(z_d)$ ]. The radial BAO scale is sensitive to the expansion rate, H(z), while the transverse BAO scale is sensitive to the angular diameter distance, $D_A(z)$ .

The HI galaxy Medium-Deep Band 2 Survey will be able to detect and measure the BAO feature at low redshift (Yahya et al. Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015; Abdalla et al. Reference Abdalla2015; Bull Reference Bull2016). This measurement has already been performed by optical spectroscopic experiments, such as BOSS and WiggleZ (Alam et al. Reference Alam2017; Kazin et al. Reference Kazin2014), but over different redshift ranges and patches of the sky. An SKA1 HI galaxy redshift survey will add independent data points at low redshift, $z \lesssim 0.3$ , which will help to better constrain the time evolution of the energy density of the various components of the Universe—particularly dark energy. The expected constraints on H(z) and $D_A(z)$ are shown in Figure 11 and are typically a few percent for the HI galaxy survey. While this is not competitive with the precision of forthcoming optical/near-IR spectroscopic surveys such as DESI and Euclid, it will be at lower redshift than these experiments can access, and so is complementary to them.

Figure 11. Forecast constraints on the cosmic expansion rate, H, (left panel) and angular diameter distance, $D_A(z)$ , (right panel) for several different experiments, following the forecasting methodology described in Bull (Reference Bull2016). The SKA1 Medium-Deep Band 2 Survey for HI galaxy redshifts is shown in light blue, HI IM are shown in red/pink (see Section 5 for details), and optical/NIR spectroscopic galaxy surveys are shown in black/grey.

Another feature that is present in the clustering pattern of galaxies is RSDs, a characteristic squashing of the 2D correlation function caused by the peculiar motions of galaxies (Kaiser Reference Kaiser1987; Scoccimarro Reference Scoccimarro2004; Percival et al. Reference Percival, Samushia, Ross, Shapiro and Raccanelli2011). Galaxies with a component of motion in the radial direction have their spectral line emission Doppler shifted, making them appear closer or further away than they actually are according to their observed redshifts. This results in an anisotropic clustering pattern as seen in redshift space. The degree of anisotropy is controlled by several factors, including the linear growth rate of structure, f(z), and the clustering bias of the galaxies with respect to the underlying CDM distribution, b(z). The growth rate in particular is valuable for testing alternative theories of gravity, which tend to enhance or suppress galaxy peculiar velocities with respect to the GR prediction (Jain & Zhang Reference Jain and Zhang2008; Baker, Ferreira, & Skordis Reference Baker, Ferreira and Skordis2014). RSDs not only occur on smaller scales than the BAO feature, but can also be detected by an HI galaxy redshift survey as long as the shot noise level is sufficiently low. The SKA1 HI galaxy survey will be able to measure the normalised linear growth rate, $f\sigma_8$ , to $\sim$ 3% at $z \approx 0.3$ (see Figure 12). This is roughly in line with what existing optical experiments can achieve at similar redshifts (see Macaulay et al. Reference Macaulay, Wehus and Eriksen2013 for a summary).

Figure 12. Forecast constraints on the linear growth rate of LSS, $f\sigma_8$ , for the same surveys as in Figure 11. Open circles show a compilation of current constraints on $f\sigma_8$ from Macaulay, Wehus, & Eriksen (Reference Macaulay, Wehus and Eriksen2013).

Figure 13 shows results for when the growth rate constraints are mapped onto the phenomenological modified gravity parametrisation defined in equations (3) and (4).Footnote ^l The constraints on both $\mu_0$ and $\gamma_0$ are improved by roughly a factor of two over Planck—comparable to what can be achieved with DES (galaxy clustering only). This is not competitive with bigger spectroscopic galaxy surveys like Euclid or DESI, but does provide an independent datapoint at low redshift.

Figure 13. Forecast constraints on phenomenological modified gravity parameters using the broadband shape of the power spectrum, detected using the HI galaxy sample of the Medium-Deep Band 2 Survey. Planck and DES (galaxy clustering only) constraints are included for comparison. The improvement from adding SKA1 is comparable to DES. Specifications for DES were taken from Lahav et al. (Reference Lahav, Kiakotou, Abdalla and Blake2010).

4.2.2. Doppler magnification

There is a contribution to the apparent magnification of galaxies due to their peculiar motion, as well as weak gravitational lensing (Bonvin Reference Bonvin2008). The motion of the galaxies causes a shift in their apparent radial position (as seen in redshift space), while their angular size depends only on the actual (real space) angular diameter distance. As such, a galaxy that is moving away from us will maintain fixed angular size while appearing to be further away than it really is (and thus ‘bigger’ than it should be for a galaxy at that apparent distance). This effect has been called Doppler magnification and dominates the weak lensing convergence at low redshift (Bacon et al. Reference Bacon, Andrianomena, Clarkson, Bolejko and Maartens2014; Borzyszkowski, Bertacca, & Porciani Reference Borzyszkowski, Bertacca and Porciani2017; Bonvin et al. Reference Bonvin, Andrianomena, Bacon, Clarkson, Maartens, Moloi and Bull2017; Andrianomena et al. Reference Andrianomena, Bonvin, Bacon, Bull, Clarkson, Maartens and Moloi2018). It can be detected statistically through the dipolar pattern it introduces in the density-convergence cross-correlation, $\langle \kappa \delta_g \rangle$ . The galaxy density, $\delta_g$ , can be measured from the 3D galaxy positions, while the convergence, $\kappa$ , can be estimated from the angular sizes of the galaxies.

As discussed above, an SKA1 HI galaxy redshift survey will yield high number densities of galaxies with spectroscopic redshifts at $z \lesssim 0.4$ , approximately covering the redshift range where Doppler magnification dominates the weak lensing convergence. If the HI-emitting galaxies can be resolved, their sizes can also be measured (e.g. from their surface brightness profile in continuum emission), making it possible to measure the Doppler magnification signal using a single survey. Galaxy size estimators often suffer from large scatter, and it remains an open question as to how well SKA1 will be able to measure sizes. This scatter has a significant effect on the expected SNR of the Doppler magnification signal. There is a known relation between the size of an HI disk and the HI mass (Wang et al. Reference Wang, Koribalski, Serra, van der Hulst, Roychowdhury, Kamphuis and Chengalur2016) that shows very little scatter over several orders of magnitude, however. For objects that are spatially resolved in HI, their expected sizes can be computed from their HI masses and compared with their apparent sizes.

Following the forecasting methodology of Bonvin et al. (Reference Bonvin, Andrianomena, Bacon, Clarkson, Maartens, Moloi and Bull2017), we expect SKA1 to achieve a signal-to-noise ratio of $\approx$ 8 on the Doppler magnification dipole for galaxies separated by $\sim$ 100 $h^{-1}$ Mpc (Figure 14), assuming a size scatter of $\sigma(\kappa) = 0.3$ (comparable to what optical surveys can achieve). The cumulative SNR over $0.1 \le z \le 0.5$ , for the full range of separations, is $\approx$ 40.

Figure 14. Signal-to-noise ratio of the Doppler magnification dipole for SKA1 as a function of separation d at $z=0.15$ (the redshift bin in which the SNR is largest). A pixel size of ${4h^{-1}\,{\text{Mpc}}}$ has been assumed. The upper bound and lower bounds are for convergence errors (size noise) of $\sigma_\kappa=0.3$ and $\sigma_\kappa=0.8$ , respectively.

4.2.3. Direct peculiar velocity measurements

The Tully–Fisher relation (Tully & Fisher Reference Tully and Fisher1977) can be used to infer the intrinsic luminosity of a galaxy from its 21-cm line width, which is a proxy for rotational velocity. Combined with the redshift of the line and a measurement of the galaxy inclination, this makes it possible to measure the galaxy’s peculiar velocity in the line-of-sight direction. The statistics of the peculiar velocity field, sampled by a large set of galaxies, can then be used to measure various combinations of cosmological quantities. Peculiar velocity statistics are particularly sensitive to the growth rate of structure and so can be used as powerful probes of modified gravitational physics (e.g. Hellwing et al. Reference Hellwing, Barreira, Frenk, Li and Cole2014; Koda et al. Reference Koda2014; Ivarsen et al. Reference Ivarsen, Bull, Llinares and Mota2016).

Measuring the width of the 21-cm line requires line detections with significantly better signal-to-noise that would be needed to measure redshift alone. Figure 15 shows the expected fractional error on the 21-cm line width of a galaxy as a function of the signal-to-noise ratio on the integrated flux of the line, assuming a simplified Gaussian line profile model. The $5\sigma$ and $8\sigma$ thresholds (on the peak per-channel SNR, not the integrated flux) from Table 7 are shown as red and yellow dashed lines, respectively. These are the thresholds we assumed for 21-cm line detection for a redshift-only survey. To measure the peculiar velocity to better than 20% of the speed of light (as required by the analysis in Koda et al. Reference Koda2014), a fractional measurement precision of $\sim$ 2.4% is required on the line width, which translates to a peak per-channel SNR of $\sim$ 110 $\sigma$ according to Figure 15. As such, the number density of galaxies for which peculiar velocity measurements are available will be significantly lower than for the redshift-only sample. Some way of measuring the inclination (e.g. from continuum or optical/NIR images) for all galaxies in the sample is also required. Nevertheless, direct measurements of the peculiar velocity field are sensitive cosmological probes, so the constraining power of even relatively small peculiar velocity samples can be substantial. Forecasts for SKA precursor experiments were presented in Koda et al. (Reference Koda2014) and showed that a $\sim$ 3% measurement of $f\sigma_8$ should be achievable at $z \simeq 0.025$ with a combined redshift + velocity survey, for example.

Figure 15. Expected fractional error on the width of the 21 cm line, as a function of the signal-to-noise ratio on the integrated line flux. The vertical dashes lines show three different detection thresholds: (red) $5\sigma$ threshold on the peak per channel SNR; (yellow) $8\sigma$ threshold on the peak per-channel SNR; and (blue) threshold corresponding to $\sigma(v_{{\text{pec}}}) < 0.2 c$ .

4.2.4. Void statistics

Future large-area galaxy surveys will offer an unprecedented spectroscopic view of both large and small scales in the cosmic web of structure. Thanks to its high galaxy density and low bias, the SKA1 HI galaxy survey will allow unusually small voids and comoving scales to be probed compared to other spectroscopic surveys.

The number counts (Pisani et al. Reference Pisani, Sutter, Hamaus, Alizadeh, Biswas, Wandelt and Hirata2015; Sahlén, Zubeldia, & Silk Reference Sahlén, Zubeldia and Silk2016), shapes (Massara et al. Reference Massara, Villaescusa-Navarro, Viel and Sutter2015), RSDs (Sutter et al. Reference Sutter, Pisani, Wandelt and Weinberg2014), and lensing properties (Spolyar, Sahlén, & Silk Reference Spolyar, Sahlén and Silk2013) of voids are examples of sensitive void-based probes of cosmology. Voids are particularly sensitive to the normalisation and shape of the matter power spectrum, its growth rate, and the effects of screened theories of gravity which exhibit modifications to GR in low-density environments (Voivodic et al. Reference Voivodic, Lima, Llinares and Mota2017). This is because void distributions contain objects ranging from the linear to the non-linear regime, across scale, density, and redshift (Sahlén & Silk Reference Sahlén and Silk2016).

We forecast cosmological parameter constraints from the HI galaxy Medium-Deep Band 2 Survey in our fiducial cosmology, using a Fisher matrix method. The void distribution is modelled following (Sahlén et al. Reference Sahlén, Zubeldia and Silk2016; Sahlén & Silk Reference Sahlén and Silk2016) using an approximate modelling scheme to incorporate the effects of massive neutrinos on the void distribution (Sahlén Reference Sahlén2018). We also take into account the galaxy density and bias for the survey. Below $z \approx 0.18$ , the survey is limited by the void-in-cloud limit. Voids smaller than this limit tend to disappear due to collapse of the overdensity cloud within which they are situated.

We expect to find around $4 \times 10^4$ voids larger than $10\,h^{-1}$ Mpc. The marginalised constraints on $w_0$ and $w_a$ inferred from void abundances are shown in Figure 16 and Table 9. The SKA1 void counts and Planck + lensing + BAO parameter constraints offer similar but complementary constraining power, with their combination strengthening the $w_0-w_a$ figure of merit by a factor of $\sim$ 6–10. This effect will likely be increased with constraints of a future Stage IV CMB experiment. This is not directly competitive with future optical/NIR spectroscopic galaxy surveys at higher redshift, which are expected to provide $\sim$ few-percent constraints on $w_0$ for example, but demonstrates the usefulness of low-redshift void counts as an independent cross-check on these quantities. Also including the sum of neutrino masses as a free parameter only marginally weakens the void constraints (Sahlén Reference Sahlén2018). Recalling that additional cosmological information is also available in e.g. shapes/profiles, voids are therefore a promising application of an SKA1 HI galaxy survey.

Table 9. Forecast dark energy constraints for void counts.

Figure 16. Forecast marginalized parameter constraints for $w_0$ and $w_a$ from the void counts of the HI galaxy Medium-Deep Band 2 Survey (grey), Planck (blue), and both combined (yellow). Apart from the cosmological parameters, we have also marginalized over uncertainty in void radius (Sahlén & Silk Reference Sahlén and Silk2016), and in the theoretical void distribution function (Pisani et al. Reference Pisani, Sutter, Hamaus, Alizadeh, Biswas, Wandelt and Hirata2015).

4.2.5. Particle dark matter searches in cross-correlation with γ–ray maps

Camera et al. (Reference Camera, Fornasa, Fornengo and Regis2013) and subsequent studies (Fornengo & Regis Reference Fornengo and Regis2014; Camera et al. Reference Camera, Fornasa, Fornengo and Regis2015a) proposed a new technique for indirect particle dark matter detection, based on the cross-correlation of direct gravitational probes of dark matter, such as weak gravitational lensing or the clustering of galaxies. A cross-correlation between the unresolved $\gamma$ -ray background seen by the Fermi Large Area Telescope (LAT; Atwood et al. Reference Atwood2009) and various cosmological observables has already been detected (Fornengo et al. Reference Fornengo, Perotto, Regis and Camera2015; Xia et al. Reference Xia, Cuoco, Branchini and Viel2015; Cuoco et al. Reference Cuoco, Xia, Regis, Branchini, Fornengo and Viel2015; Branchini et al. Reference Branchini, Camera, Cuoco, Fornengo, Regis, Viel and Xia2017). Currently, the vast majority of the $\gamma$ -ray sky is unresolved and only a few thousand $\gamma$ -ray sources are known. On large scales, non-thermal emission mechanisms are expected to greatly exceed any other process in the low-frequency radio band and the $\gamma$ -ray range. Thus, radio data are expected to correlate with the $\gamma$ -ray sky and can be exploited to filter out the information concerning the composition of the $\gamma$ -ray background contained in maps of the unresolved $\gamma$ -ray emission.

Here, we present forecasts for the cross-correlation of SKA1 HI galaxies and the $\gamma$ -ray sky from Fermi. A major added value of SKA1 HI galaxies is that is their redshift distribution peaks at low redshift and has an extremely low shot noise (see Yahya et al. Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015, Figure 4). This is the very regime where the non-gravitational dark matter signal is strongest. Specifically, we adopt an SKA1 HI galaxy survey with specifics given in Yahya et al. (Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015) for the baseline configuration. We consider only galaxies in the redshift range $0<z\le0.5$ , which we further subdivide into 10 narrow spectroscopic redshift bins. For the $\gamma$ -ray angular power spectrum, we employ the fitting formulæ found by Tröster et al. (Reference Tröster2017) for Pass-8 Fermi-LAT events gathered until September 2016 (i.e. over 8 yr of data taking). This is a conservative choice, as by the time the SKA1 HI galaxy catalogue will be available, a much larger amount of Fermi-LAT data will be available. Figure 17 shows the improvement on bounds on particle dark matter cross section (assuming a generic phenomenological annihilating dark matter model; see Camera et al. Reference Camera, Fornasa, Fornengo and Regis2015a) as a function of dark matter mass when SKA1 HI galaxies are used, compared with the two main probes studied in Camera et al. (Reference Camera, Fornasa, Fornengo and Regis2015a), i.e. cosmic shear from DES (Year 1 data only) and Euclid. The high density of spectroscopically detected HI galaxies from the HI galaxy survey provides constraints on particle dark matter properties that are 10–60% tighter than with state-of-the-art and even future experiments.

Figure 17. Improvement factor in constraints on the velocity-averaged dark matter annihilation cross-section, $\langle \sigma_a v\rangle$ , as a function of particle dark matter mass, when an SKA1 HI galaxy survey is used for the cross-correlation with Fermi-LAT data, instead of DES year 1 (blue) or Euclid (red/orange).

4.2.6. Cross-correlation with gravitational wave sources

Gravitational wave (GW) experiments are expected to directly detect tens to thousands of binary black holes (BBHs) and neutron star (NS) coalescence events per year over the coming decade (e.g. Ng et al. Reference Ng, Wong, Broadhurst and Li2018), depending on the natural rate of mergers and how detector sensitivity improves with time. As the number of known events increases, and the accuracy of source localisation improves, large GW source catalogues numbering in the thousands to tens of thousands of events will be constructed. These can then be cross-correlated with galaxy surveys such as the SKA1 Medium-Deep Band 2 Survey to constrain cosmological models and determine properties of the BBH and NS host galaxy/halo populations.

GWs are lensed by intervening LSS just as light is, and so by cross-correlating foreground galaxies (that act as lenses) with a background of GWs, one can perform tests of GR and dark energy models in a way that is independent from current tests using galaxy surveys alone (Raccanelli Reference Raccanelli2017). Forecasts are not currently available for an SKA1 HI galaxy survey, but $\mathcal{O}(10\%)$ constraints on $w_0$ , $w_a$ , $\mu_0$ , and $\gamma_0$ are expected to be achievable with an SKA2 HI galaxy survey (Raccanelli Reference Raccanelli2017).

The angular correlation of GW sources with different types of galaxies can also be used to understand if merging high-mass BBHs preferentially trace star-rich galaxies (as would be the case if they form from objects at the endpoint of stellar evolution), or the dark matter distribution (as would be the case if they are primordial black holes). The most star-rich galaxies are typically found in halos of mass $\sim$ $10^{11-12}$ ${\text{M}}_\odot$ , while almost all mergers of primordial BBHs would happen in halos of $\lesssim$ $10^6$ ${\text{M}}_\odot$ , as shown in Bird et al. (Reference Bird, Cholis, Muñoz, Kamionkowski, Kovetz, Raccanelli and Riess2016). Other models (e.g. where high-mass BBHs are the relics of Population III stars) also predict different host halo populations. The range of host halo masses determines the mean bias of the host population. This can be measured through the cross-correlation of galaxy populations of known bias with the GW source catalogue, therefore determining the nature of BBH progenitors (Raccanelli et al. Reference Raccanelli, Kovetz, Bird, Cholis and Muñoz2016; Scelfo et al. Reference Scelfo, Bellomo, Raccanelli, Verde and Matarrese2018). HI galaxies are present across a wide range of halo masses, but there is expected to be a cut-off below $\sim$ 10 $^8\,{\text{M}}_\odot$ , where self-shielding of the HI from the ionising UV background fails (e.g. Bagla, Khandai, & Datta Reference Bagla, Khandai and Datta2010). The cross-correlation between HI galaxies and GW sources can therefore be expected to strongly constrain the primordial black hole scenario.

4.2.7. HI model uncertainties

Cosmological constraints from the HI galaxy survey are also subject to uncertainties in the abundance and spatial distribution of neutral hydrogen. The combination of current astrophysical uncertainties on the neutral hydrogen density and bias parameters, $\Omega_{{\text{HI}}}$ and $b_{{\text{HI}}}$ , can be shown to lead to about a 60–100% uncertainty in current models of the HI power spectrum (Padmanabhan, Choudhury, & Refregier Reference Padmanabhan, Choudhury and Refregier2015).

There have been numerous efforts to build accurate halo models of the HI distribution (e.g. Bagla et al. Reference Bagla, Khandai and Datta2010; Davé et al. Reference Davé, Katz, Oppenheimer, Kollmeier and Weinberg2013; Villaescusa-Navarro et al. Reference Villaescusa-Navarro, Viel, Datta and Choudhury2014, Reference Villaescusa-Navarro2018a; Padmanabhan & Refregier Reference Padmanabhan and Refregier2017; Padmanabhan, Refregier, & Amara Reference Padmanabhan, Refregier and Amara2017, Reference Padmanabhan, Refregier and Amara2018), with free parameters typically constrained using some subset of currently available HI observables (galaxy number counts, IM observations, and Damped Lyman- $\alpha$ systems) across redshifts 0–5 in the post-reionisation universe. HI galaxy redshift and HI IM surveys with SKA1 will greatly expand the amount of data available to constrain these models, leading to significantly enhanced precision in our knowledge of the relevant parameter values (e.g. see Table 13 below), and allowing the models themselves to be distinguished from one another. Recent forecasts by Padmanabhan et al. (Reference Padmanabhan, Refregier and Amara2018) also suggest that, once priors on HI model parameters from existing observations are applied, the cosmological parameter constraints from an SKA1 HI IM survey will be generally insensitive to remaining uncertainties in the astrophysical model. The same conclusion is also expected to hold for the HI galaxy redshift survey, at least if we restrict our attention to linear scales, $k \lesssim 0.14\,{\text{Mpc}}^{-1}$ .

5.1.1. Temperature and bias

The total brightness temperature at a given redshift and in a unit direction n on the sky can be written as

(13) \begin{equation}T_{b}(z,{\bf n}) \approx \overline{T}_{b}(z) \Big[1+b_{{\text{HI}}}(z)\delta_m(z,{\bf n})-\frac{(1+z)}{H(z)}\,n^i\partial_i\big({\bf n}\cdot {\bf v} \big)\Big],\end{equation}

where $b_{{\text{HI}}}$ is the HI galaxy bias, $\delta_m$ is the matter density contrast, v is the peculiar velocity of emitters, and the average signal $\overline{T}_{b}$ is determined by the comoving HI density fraction $\Omega_{{\text{HI}}}$ . The last term in braces describes the effect of RSD. The signal will be completely specified once we have a prescription for the $\Omega_{{\text{HI}}}$ and $b_{{\text{HI}}}$ . This can be obtained by making use of the halo mass function, ${{\text{d}}}n/{{\text{d}}}M$ and halo bias, relying on a model for the amount of HI mass in a dark matter halo of mass M, i.e. $M_{{\text{HI}}}(M)$ (see Santos et al. Reference Santos2015, for details). Simulations have found that almost all HI in the post-reionisation Universe resides within dark matter halos (Villaescusa-Navarro et al. Reference Villaescusa-Navarro, Viel, Datta and Choudhury2014, Reference Villaescusa-Navarro2018b). This fact justifies the usage of halo models to study the spatial distribution of cosmic neutral hydrogen (Padmanabhan & Refregier Reference Padmanabhan and Refregier2017; Castorina & Villaescusa-Navarro Reference Castorina and Villaescusa-Navarro2017; Wolz et al. Reference Wolz, Murray, Blake and Wyithe2018; Villaescusa-Navarro et al. Reference Villaescusa-Navarro2018b).

5.1.2. Power spectrum

The first aim of the IM survey will be to measure the HI power spectrum (or its large sky equivalent, the angular power spectrum). In addition, we will take advantage of multi-wavelength coverage (e.g. BOSS, DES, Euclid, LSST see Section 2.5) to detect the signal in cross-correlation. The HI power spectrum signal (with RSDs) can be written as

(14) \begin{equation}P^{\text{HI}}(z,k) = \bar{T}_b(z)^2b_{{\text{HI}}}(z)^2[1+\beta_{{\text{HI}}}(z)\mu^2]^2P(z,k) \, ,\end{equation}

which allows to break the degeneracy between $\Omega_{{\text{HI}}}$ and $b_{{\text{HI}}}$ (Masui et al. Reference Masui2013b). The cross-correlation power spectrum will also depend on the galaxy bias, $b_{g}$ and the cross-correlation coefficient r of the two probes, and can be used, as mentioned, to mitigate systematic effects.

The following forecasts make use of the Fisher matrix formulation. Details of the noise calculation for both MID (single dish and interferometer) and LOW can be found in Bull et al. (Reference Bull, Ferreira, Patel and Santos2015b) and Santos et al. (Reference Santos2015). Details on SKA1-LOW EoR surveys can be found in Koopmans et al. (Reference Koopmans2015). Particular care must be taken when combining MeerKAT and SKA1-MID dishes due to different primary beams and bands. Note that, when considering measurements with the interferometer, we assume a strict non-linear cut-off to define the maximum wavevector in the Fisher matrix, $k_{{\text{max}}} = 0.2h\,{\text{Mpc}}^{-1}$ at all redshifts. This is a conservative choice, much smaller than the instrumental cut-off.

The finite number of HI samples in the intensity maps also results into a shot noise contribution on the power spectrum measurements. In hydrodynamic simulations, Villaescusa-Navarro et al. (Reference Villaescusa-Navarro2018b) found that the amplitude of the HI shot-noise is negligible at $z\leq5$ (see also Castorina & Villaescusa-Navarro Reference Castorina and Villaescusa-Navarro2017) and therefore BAO measurements through HI IM will barely be affected by this. They also found values of the linear HI bias equal to 0.84, 1.49, 2.03, 2.56, 2.82, and 3.18 at redshifts 0, 1, 2, 3, 4, and 5, respectively. While the HI bias is essentially scale-independent down to $k\simeq$ 1h Mpc^–1 at $z=1$ , at redshifts $z\geq3$ the HI bias is scale-dependent already at $k=0.3h\,{\text{Mpc}}^{-1}$ . In the following, we forecast the constraints on the linear bias $b_{{\text{HI}}}$ by the SKA1 IM surveys, however, they will also be the first surveys to investigate the scale-dependence of the HI clustering signal for all redshifts $0<z<6$ .

SKA1-MID

The expected error on the measurement of the HI power spectrum from the Wide Band 1 Survey is shown in Figure 18 (top panel) for a redshift bin of width $\Delta z = 0.1$ centred at $z=0.6$ . Keeping the cosmological parameters fixed to the Planck 2015 cosmology (Ade et al. Reference Ade2016a), the only unknown in $P^{\text{HI}}$ is ( $\Omega_{{\text{HI}}}b_{{\text{HI}}}$ ). Employing a Fisher matrix analysis, we calculate the expected constraints on $\Omega_{{\text{HI}}}b_{{\text{HI}}}$ (Pourtsidou, Bacon, & Crittenden Reference Pourtsidou, Bacon and Crittenden2017), which are summarised in the first column of Table 10. Using RSDs, the degeneracy between $\Omega_{{\text{HI}}}$ and $b_{{\text{HI}}}$ can be broken and the resulting constraints are presented in the second column of Table 10.

Figure 18. Upper panel: HI detection with the SKA1-MID Wide Band 1 Survey, showing the expected signal power spectrum (black solid) and measurement errors (cyan) from the HI auto-correlation power spectrum. The assumed k binning is $\Delta k = 0.01\,{\text{Mpc}}^{-1}$ . Lower panel: HI detection with the Deep SKA1-LOW Survey, signal power spectrum (solid black line) and measurement errors (cyan band) at $z=4$ . We have used a k-binning $\Delta k = 0.01\,{\text{Mpc}}^{-1}$ and a redshift bin $\Delta z = 0.3$ .

Table 10. Forecasted fractional uncertainties on $\Omega_{{\text{HI}}}b_{{\text{HI}}}$ , and $\Omega_{{\text{HI}}}$ assuming the SKA1-MID Wide Band 1 Survey and following the methodology in Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017). For the $\Omega_{{\text{HI}}}$ constraints, we utilise the full HI power spectrum with RSDs. Note that the assumed redshift bin width is $\Delta z = 0.1$ , but we show the results for half of the bins for brevity. The cosmological constraints are reported in Figures 11 and 12.

SKA1-LOW

Here, we present predictions on the Deep SKA1-LOW Survey. Other possibilities (in terms of sky coverage and observation time) as well as an optimisation study will be presented in an upcoming publication.

In Figure 18 (bottom panel), we show the predicted HI signal power spectrum neglecting the effect of RSDs, together with the predicted measurement errors at $z=4$ for the Deep SKA1-LOW Survey. Performing a Fisher matrix analysis following the methodology in Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017) we can constrain $\Omega_{{\text{HI}}}$ and $b_{{\text{HI}}}$ . Our derived constraints are quoted in Table 11. As we can see, IM with the Deep SKA1-LOW Survey probes the largely unexplored ‘redshift desert’ era and can give us valuable information on the evolution of the HI abundance and bias across cosmic time.

Table 11. Forecast fractional uncertainties on HI parameters for IM with the Deep SKA1-LOW Survey, following the methodology in Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017).

Finally, in Figure 19, we show the derived constraints for both SKA IM surveys (i.e. Wide Band 1 Survey and Deep SKA1-LOW Survey) on $\Omega_{{\text{HI}}}$ compared to current measurements.

Figure 19. Forecasts for the HI density, $\Omega_{{\text{HI}}}$ , using the Wide Band 1 Survey and Deep SKA1-LOW Survey (black points), and comparison with measurements (see Crighton et al. Reference Crighton2015 and references therein), following the methodology in Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017). Note that we have used a very conservative non-linear $k_{{\text{max}}}$ cutoff for these results.

At this point, we note that our forecasts have ignored residual foreground contamination and other systematic effects. Assessing these effects using simulations and exploring the possibility of performing BAO measurements using this survey is the subject of ongoing work.

5.2.1. Baryon acoustic oscillations and RSDs

As already mentioned in Section 4.2.1, BAOs can provide robust measurements on the angular diameter distance and Hubble rate as a function of redshift. Such measurements can in turn be used to constrain dark energy models and the curvature of the Universe (Bull et al. Reference Bull, Ferreira, Patel and Santos2015b; Bull et al. Reference Bull, Camera, Raccanelli, Blake, Ferreira, Santos and Schwarz2015a; Witzemann et al. Reference Witzemann, Bull, Clarkson, Santos, Spinelli and Weltman2018). The same is true for RSDs, which can measure the growth rate, a crucial ingredient for instance in constraining modified gravity models. In this section, we focus on what can be achieved with the Wide Band 1 Survey. Exploring the same for Deep SKA1-LOW Survey is the subject of ongoing work.

The relatively poor angular resolution of SKA1-MID in single-dish mode at high redshifts/low frequencies will partially smear out the shape of the BAO peak along the angular direction. Nevertheless, SKA1-MID can still provide competitive constraints on BAO measurements and its derived quantities using the HI IM technique. Following the Fisher matrix forecasting method described in Bull et al. (Reference Bull, Ferreira, Patel and Santos2015b), Bull (Reference Bull2016), Figure 11 shows the expected constraints as a function of redshift on the angular diameter distance $D_A$ and Hubble rate H, while Figure 12 shows the same for the growth rate $f\sigma_8$ . We see that the constraints are still quite competitive when comparing to concurrent surveys (e.g. Euclid like). The high redshift resolution of the HI IM survey makes it particularly fit for line of sight measurements, such as H(z) and the growth rate.

However, at frequencies $\nu\leqslant800$ MHz, the angular smoothing is so large that the BAO feature might be hard to extract from the angular direction. This depends on how well we can deconvolve the beam given the signal to noise. Even in this worst case scenario, the frequency resolution will be good enough to allow for a detection of the radial BAO. By means of numerical simulations incorporating the cosmological signal, instrumental effects, and the presence of foregrounds, Villaescusa-Navarro, Alonso, & Viel (Reference Villaescusa-Navarro, Alonso and Viel2017) demonstrated that the position of the radial BAO peak can be measured with percent precision accuracy through single-dish observations in the Band 1 of SKA1-MID.

5.2.2. Ultra-large-scale effects

One of the ‘transformational’ measurements expected from HI IM with the Wide Band 1 Survey is the constraints on the power spectrum on ultra-large scales (past the equality peak). This is an area where a single dish survey with SKA1-MID can excel given its low resolution, but large survey speed (Alonso et al. Reference Alonso, Bull, Ferreira, Maartens and Santos2015b). Such measurements can provide hints on new physics that only materialise on this ultra-large scales.

One example of such an effect is PNG. In particular, PNG of the local type $f_{{\text{NL}}}$ introduces a scale-dependent correction to clustering bias (Dalal et al. Reference Dalal, Dore, Huterer and Shirokov2008; Matarrese & Verde Reference Matarrese and Verde2008) such that $b_{{\text{HI}}}\propto f_{{\text{NL}}}/k^2$ . The $1/k^2$ term makes this effect particularly relevant on very large scales (small k) where statistical detectability is severely limited due to cosmic variance and large-scale systematic effects. Using HI IM only we forecast $\sigma(\,f_{{\text{NL}}})=2.8$ , assuming Band 1 for SKA dishes and UHF band for the MeerKAT dishes. Note that our calculations take into account the telescope beams and marginalise over the biases as well as any other large-scale effects. Currently, the best measurements on PNG come from the Planck satellite (Planck Collaboration et al. Reference Collaboration2016Reference Planckb) with $\sigma(\,f_{{\text{NL}}})=5.0$ using the bispectrum. Current bounds from galaxy surveys are roughly one order of magnitude worse than Planck (see e.g. Ross et al. Reference Ross2013; Ho et al. Reference Ho2015). The proposed SKA survey should improve current bounds from galaxy surveys and Planck. The ultimate goal would be to achieve $\sigma(\,f_{{\text{NL}}}) < 1$ such that we can start distinguishing between simple inflationary models (see e.g. de Putter, Gleyzes, & Doré Reference de Putter, Gleyzes and Doré2017).

Another type of very large-scale signatures are the so-called General Relativistic (GR) effects. These GR effects introduce corrections to the tracers’ transfer function as leading to a set of terms which are usually gathered together as a single contribution. They are an important prediction of GR over the very largest distances that it is possible to probe observationally, and so constitute a valuable test of alternative gravitational theories (Hall, Bonvin, & Challinor Reference Hall, Bonvin and Challinor2013; Lombriser, Yoo, & Koyama Reference Lombriser, Yoo and Koyama2013; Baker & Bull Reference Baker and Bull2015). Alonso et al. (Reference Alonso, Bull, Ferreira, Maartens and Santos2015c) have shown that these effects are not detectable in the single tracer case due to cosmic variance. However, it will be crucial to correctly model these relativistic corrections in future LSS surveys, in order not to bias the estimation of other ultra-large-scale effects such as PNG (Camera, Maartens, & Santos Reference Camera, Maartens and Santos2015b). In fact these contributions can mimic in some ways the effect of PNG in the bias (see e.g. Bruni et al. Reference Bruni, Crittenden, Koyama, Maartens, Pitrou and Wands2012; Jeong, Schmidt, & Hirata Reference Jeong, Schmidt and Hirata2012) so have to be considered in any realistic forecast. Here, we marginalise over them to safely take the effect into account.

It is possible to overcome cosmic variance with the MT technique (Seljak Reference Seljak2009), where one combines two differently biased dark matter tracers in such a way that the fundamental statistical uncertainty coming from cosmic variance can be bypassed. We updated the forecasts of Alonso & Ferreira (Reference Alonso and Ferreira2015) and Fonseca et al. (Reference Fonseca, Camera, Santos and Maartens2015) for $f_{{\text{NL}}}$ and GR effects using the MT technique with HI IM with SKA1 in combination with an overlapping $10\,000\deg^2$ Euclid-like survey and $14\,000\deg^2$ LSST-like photometric surveys. In Table 12, we show the forecast marginal errors on $f_{{\text{NL}}}$ and GR effects for 3 different sets of cosmological parameters: Case 1—marginal errors on $f_{{\text{NL}}}$ without including GR effects; Case 2—marginal errors on $f_{{\text{NL}}}$ including Lensing and GR effects all together; Case 3—marginal errors on $f_{{\text{NL}}}$ including Lensing and each GR effect individually. Note that all of the $\epsilon$ parameters have a fiducial value of $\epsilon=1$ (see Fonseca, Maartens, & Santos Reference Fonseca, Maartens and Santos2018 for the definitions). In Figure 20, we show the degeneracy between $f_{{\text{NL}}}$ and lensing (top) and GR effects (bottom) for the two synergy surveys considered assuming Case 2. It can be seen that using the MT technique, we will be able to break the barrier $\sigma(\,f_{{\text{NL}}})<1$ and make a detection of some GR effects such as the Doppler term.

Table 12. Marginal errors on $f_{{\text{NL}}}$ , lensing ( $\varepsilon_{{\text{Lens}}}$ ), and GR effects ( $\varepsilon_{{\text{GR}}}$ ), which include the Doppler term ( $\varepsilon_{{\text{Doppler}}}$ ), Time Delay ( $\varepsilon_{{\text{TD}}}$ ), Sachs–Wolfe ( $\varepsilon_{{\text{SW}}}$ ), and Integrated Sachs–Wolfe $(\varepsilon_{{\text{ISW}}})$ , using the MT technique with HI IM with the SKA1 Wide Band 1 Survey in conjugation with the Euclid and LSST surveys for three prior assumptions.

Figure 20. The $1\sigma$ (thin) and $2\sigma$ (thick) contours for the forecasted marginal errors on $f_{{\text{NL}}}$ and Lensing (top), and GR effects (bottom) using the MT technique from HI IM with the SKA1 Wide Band 1 Survey in combination with Euclid data (solid blue line) and LSST data (dashed red line). These forecasts assume Case 2 as presented in Table 12. Combination with LSST will allow to probe $f_{{\text{NL}}}\sim 1$ as well as detect large scale GR effects.

5.2.3. HI detection via synergies with optical surveys

Cross-correlations between HI IM and optical galaxy surveys can also provide precise and robust cosmological measurements, as they have the advantage of mitigating major issues like systematics and foreground contaminants that are relevant for one type of survey but not for the other. For example, in Masui et al. (Reference Masui2013b), the intensity maps acquired at the GBT were combined with the WiggleZ galaxy survey to constrain the quantity $\Omega_{{\text{HI}}}b_{{\text{HI}}}r$ at $z\sim 0.8$ with a statistical fractional error $\sim$ 16%. r is the cross-correlation efficiency of the two observables ranging $0<r<1$ .

We start by looking at the IM cross-correlations with a spectroscopic optical galaxy survey, following Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017). Figure 21, top panel, shows the expected signal and errors for a Euclid-like spectroscopic sample (Majerotto et al. Reference Majerotto2012) for a redshift bin of width $\Delta z = 0.1$ centred at $z=1$ . The assumed sky overlap is 10 000 deg $^2$ with corresponding 5 800 h total observing time for the IM survey which can be approximately achieved with the suggested SKA1 Wide Band 1 Survey. The resulting constraints on $\Omega_{{\text{HI}}}b_{{\text{HI}}}r$ (keeping the cosmological parameters fixed to the Planck 2015 cosmology Ade et al. Reference Ade2016a) are summarised in Table 13. This table also shows constraints on $f\sigma_8$ , $D_A$ , and H from cross-correlations with Euclid, considering $\bar{T}_b$ is known.

Cross-correlations with photometric optical galaxy surveys can also be used to constrain HI properties and perform joint probes studies (Pourtsidou et al. Reference Pourtsidou, Bacon, Crittenden and Metcalf2016). Figure 21, bottom panel, shows the expected signal and errors for Stage III DES-like photometric sample for a redshift bin of width $\Delta z = 0.1$ centred at $z=0.5$ . The assumed sky overlap is 5 000 deg². We can also combine probes such as HI clustering and optical lensing, or HI clustering and the CMB, to constrain gravity (Pourtsidou Reference Pourtsidou2016b) and inflation (Pourtsidou Reference Pourtsidou2016a).

5.2.4. Neutrino masses

The impact of massive neutrinos on the abundance and clustering of cosmic neutral hydrogen has been studied in Villaescusa-Navarro, Bull, & Viel (Reference Villaescusa-Navarro, Bull and Viel2015) through hydrodynamic simulations. It was found that neutrino masses do not affect much the halo HI mass function,Footnote ^m $M_{{\text{HI}}}(M,z)$ . Therefore, neutrino effects on HI properties can easily be explained through simple HI halo models. Villaescusa-Navarro et al. (Reference Villaescusa-Navarro, Bull and Viel2015) used those ingredients to forecast the sensitivity of the phase 1 of SKA to neutrino masses, finding that observations by SKA1-MID plus SKA1-LOW alone can place a constrain of $\sigma(M_\nu)=0.18$ eV ( $2\sigma$ ), where $M_\nu=\sum_i m_{\nu_i}$ . By adding information from Planck CMB 2015 data alone that limit can shrink to $\sigma(M_\nu)=0.067$ eV ( $2\sigma$ ), while a combination of data from SKA1-MID, SKA1-LOW, Planck and a spectroscopic galaxy survey like Euclid can yield a very competitive constraint of $\sigma(M_\nu)=0.057$ eV ( $2\sigma$ ). Those constraints have been derived with the Wide Band 1 Survey assuming observations in Band 1 and 2, and 10 000 h of interferometry observations by SKA1-LOW over 20 ${{\text{deg}}}^2$ at frequencies $\nu\in[200,355]$ MHz. Figure 22 shows those constraints projected in the $M_\nu-\sigma_8$ plane. We emphasise that the aforementioned constraints have been derived assuming different survey strategies than the ones in the rest of this article, and we aim to update them in future work.

Figure 21. HI IM with SKA1 Wide Band 1 Survey in cross-correlation with optical surveys, showing the expected signal power spectrum (black solid) and measurement errors (cyan). Top: Cross-correlation with a Euclid-like spectroscopic optical galaxy survey with 10,000 deg $^2$ overlap Bottom: Cross-correlation with a DES-like photometric optical galaxy survey with 5 000 deg $^2$ overlap. The assumed k binning is $\Delta k = 0.01$ .

Table 13. Forecast fractional uncertainties on HI and cosmological parameters assuming HI IM with the SKA1 Wide Band 1 Survey and Euclid-like cross-correlation described in the main text, following the methodology in Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017). Note that the assumed redshift bin width is $\Delta z = 0.1$ , but we show the results for half of the bins for brevity.

Figure 22. This figure shows 1 $\sigma$ and 2 $\sigma$ constraints on the $M_\nu-\sigma_8$ plane from Planck CMB 2015 alone (grey), SKA1-LOW (green), SKA1-LOW plus Planck CMB 2015 (blue) and SKA1-LOW plus SKA1-MID plus Planck CMB 2015 plus a spectroscopic galaxy survey (magenta). The lower limit from neutrino oscillations, together with recent cosmological upper bounds are shown with dashed vertical lines.

Although future CMB-only constraints can result in tight limits on the total neutrino mass of about $\sigma(M_{\nu}) \sim 0.1$ eV Aguirre et al. (Reference Aguirre2019), it is expected that the limits could improve even by a factor 5–6 when the CMB is combined with the LSS data or by assuming a prior on the optical depth to reionisation, by exploiting the different degeneracies between the observables. Current terrestrial experiments like Katrin achieve a sensitivity of about 0.2 eV on the electron neutrino mass, which translates in an error of 0.6 eV on the total neutrino masses; this is a factor at least 4 worse than the current upper limits obtained from a combination of present cosmological data. Thereby, it is foreseen that combination of different LSS observables, including IM, could allow to discriminate between normal and inverted hierarchy, and at the same time provide limits much and/or detection that will be much tighter than laboratory constraints, at least in the standard scenario of structure formation.

5.2.5. Probing inflationary features

Possible anomalies observed in the CMB by WMAP (Peiris et al. Reference Peiris2003) and Planck (Ade et al. Reference Ade2014; Reference Ade2016b; Akrami et al. Reference Akrami2018) may be connected to features on ultra-large scales ( $10^{-3}<k\, {\text{Mpc}}/h<10^{-2}$ ) in the primordial power spectrum that are generated by a violation of slow-roll. Constraints on such primordial features from inflation are shown in Xu, Hamann, & Chen (Reference Xu, Hamann and Chen2016), Ballardini et al. (Reference Ballardini, Finelli, Maartens and Moscardini2018) to be significantly improved by using the ultra-large-scale HI IM and continuum surveys of SKA1-MID. The potential of such surveys for constraining the ‘resonant’ (Chen, Easther, & Lim Reference Chen, Easther and Lim2008), ‘kink’ (Starobinskii Reference Starobinskii1992), ‘step’ (Adams, Cresswell, & Easther Reference Adams, Cresswell and Easther2001; Adshead et al. Reference Adshead, Dvorkin, Hu and Lim2012), and ‘warp’ (Miranda, Hu, & Adshead Reference Miranda, Hu and Adshead2012) inflation models is illustrated in Figures 23 and 24.

Figure 23. The marginalized $1-\sigma$ error on the resonance parameter as a function of frequency in the resonant inflationary model, using HI IM power spectrum measurements (blue dashed line) and bispectrum measurements (black solid line) from the SKA1-MID Wide Band 1 Survey (fiducial value is $f^{{\text{res}}}=0$ ).

Figure 24. Marginalized 68% (shaded areas) and 95% (dashed lines) confidence level contours for the feature wave-number in the kink (top), step (middle) and warp (bottom) inflationary models, using the Planck CMB 2015 alone (which is similar to $Planck\ {\tilde 2}018)$ and combining IM and continuum data from the SKA1 Wide Band 1 Survey with the CMB.

Figure 23 shows constraints on the parameter of the resonant non-Gaussianity, $f^{{\text{res}}}$ , as a function of the resonance frequency $C_\omega$ , using either the scale-dependent bias of the power spectrum or the bispectrum, with the Wide Band 1 Survey of SKA1-MID (adding Band 2 IM observations for $z<0.4$ ), combining the single-dish observation mode with the interferometric mode. Note that the power spectrum measurement is the more informative probe to the inflationary features. Here, the parameter $C_\omega$ is the modulation frequency in the power spectrum, and models with lower $C_\omega$ could get tighter constraints partially because the amplitude of the oscillations in the power spectrum is proportional to $f^{{\text{res}}}/C^2_\omega$ . The results show that even in the presence of foreground contamination, the upcoming HI IM observations of the LSS with the SKA1-MID alone could put extremely tight constraints on the feature models, potentially achieving orders-of-magnitude improvements over the two-dimensional CMB measurements. For details on the parameterisation and forecasts see Xu et al. (Reference Xu, Hamann and Chen2016).

Figure 24 shows the Fisher forecast constraints on the amplitude of the feature versus the scale of the feature in Fourier space, using Wide Band 1 Survey in both IM and continuum on SKA1-MID. SKA1 can constrain parameters of the feature models at $>3\sigma$ (for details, see Ballardini et al. Reference Ballardini, Finelli, Maartens and Moscardini2018). We note that the constraining power of a Stage IV CMB experiment might be increased. The specific models investigated here are not generic within the inflationary scenario (which is itself still hypothetical), as well as we use these models as examples of how the SKA may be able to constrain the shape of the primordial power spectrum.

HI IM surveys could also be used in combination with CMB experiments to constrain the scalar spectral index ( $n_{s}$ ) and its runnings ( $\alpha_{s}, \beta_{s}$ ) and test the predictions of popular single-field slow-roll inflation models. Current constraints from Planck are $\sigma(n_{s})=0.006$ and $\sigma(\alpha_{s})=0.007$ . A Stage IV CORE-like CMB survey (Finelli et al. Reference Finelli2018) combined with an HI IM survey with SKA1 Wide Band 1 Survey could reach $\sigma(n_{s})=0.0011$ and $\sigma(\alpha_{s})=0.0019$ (Pourtsidou Reference Pourtsidou2016a).

5.2.6. Unveiling the nature of dark matter

IM offers the opportunity to measure the matter power spectrum also at intermediate and small scales. At such scales, there could be a signature of the so-called free streaming of dark matter particles (as in the case of Warm Dark Matter—WDM), which produces a suppression of power (Smith & Markovic Reference Smith and Markovic2011). It is thus natural to explore what could be the constraints achieved by looking at neutral hydrogen in emission as probed by IM surveys. In (Carucci et al. Reference Carucci, Villaescusa-Navarro, Viel and Lapi2015), the impact of WDM thermal relics is investigated on the 21-cm IM signal focusing on the high redshift, where structure formation is closer to the linear regime (Viel et al. Reference Viel, Markovič, Baldi and Weller2012); the authors find that there is no suppression of power but there is an increase of power in a redshift and scale-dependent way at mildly non-linear scales. In Figure 25, we show the difference for the HI IM power spectrum which is expected between the WDM model and a corresponding CDM model with the same cosmological parameters, assuming a deep and narrow IM survey with SKA1-LOW with an area of $\sim$ 3–6 deg $^2$ at $z=$ 3–5 with a range of observation times as described in the caption. It will be quite important to obtain independent constraints on $\Omega_{{\text{HI}}}$ since, as it is shown in Figure 26, it is evident that that there exists a relatively strong degeneracy between the HI cosmic density and the WDM mass.

Figure 25. Percentage difference for the HI IM power spectrum when the HI distribution is modeled using two different methods: the halo based method (dotted lines) and the particle based method (solid lines). Results are shown at z = 3 (left), z = 4 (middle) and z = 5 (right). The error on the HI power spectrum of the model with CDM, normalized to the amplitude of the 21 cm (CDM) power spectrum is shown in a shaded region for three different observation times: $t_0 = 1\,000$ h (grey), $t_0 = 3\,000$ h (blue) and $t_0 = 5\,000$ h (fuchsia). The field-of-view for this example corresponds to an area of between 2.7 and 6 deg $^2$ , at z = 3–5. For clarity, we show the error from one HI-assignment method only because both are very similar and overlap at the scale of the plot.

Figure 26. $1\sigma$ and $2\sigma$ contours (dark and light areas) of the values of $\Omega_{{\text{HI}}}$ and m $_{{\text{WDM}}}$ determined using the HI power spectrum measured by SKA1-LOW with three different observation times: 1 000, 3 000 and 5 000 h (red, green and violet) and using a field-of-view between 2.7 and 6 deg $^2$ . The Fisher matrix analysis is performed using information coming from redshift $z = 3, 4 \text{ and } 5$ .

The results indicate that we will be able to rule out a 4 keV WDM model with 5 000 h of observations at $z > 3$ , with a statistical significance larger than 3 $\sigma$ , while a smaller mass of 3 keV, comparable to present day constraints, can be ruled out at more than 2 $\sigma$ confidence level with 1 000 h of observations at $z > 5$ .

Figure 27. Photometric redshift estimation using HI intensity maps as calibrator for an optical survey such as the LSST. The pink shaded regions show the range in photometric redshift which galaxies are selected from. The orange line shows the distribution of these chosen galaxies according to their LSST-like photometric redshift (Ascaso, Mei, & Benítez 2015). The black-dashed line shows the true distribution, and the blue points show the HI clustering redshift estimate (Cunnington et al. Reference Cunnington, Harrison, Pourtsidou and Bacon2018).

5.2.7. Photometric redshift calibration

With next-generation optical surveys such as Euclid and LSST promising to deliver unprecedented numbers of resolved galaxies, immense strain will be placed on the amount of spectroscopic follow-up required. Through a clustering-based redshift estimation method which utilities HI intensity maps with excellent redshift resolution, a well-constrained prediction can be made on the redshift distribution for an arbitrarily large optical population.

SKA1 HI IM would be capable of reducing uncertainties in photometric redshift measurements below the requirements of DES and LSST (Alonso et al. Reference Alonso, Ferreira, Jarvis and Moodley2017) assuming adequate foreground cleaning could be achieved. Tests have been carried out whereby attempts were made to recover the redshift distribution for a simulated optical galaxy catalogue by cross-correlating its clustering with HI intensity maps (Cunnington et al. Reference Cunnington, Harrison, Pourtsidou and Bacon2018).

This method relies on an estimate of $b_{g}$ , $b_{{\text{HI}}}$ (the bias for the optical galaxies and HI intensity maps, respectively) and a model for the measurement of the mean HI brightness temperature $\bar{T}_{{\text{HI}}}$ . Assuming these are in hand, Figure 27 presents a proof of concept example using a small survey (25 deg $^2$ ) with 1’ beam size, neglecting noise, where HI emission is estimated using a HI-halo mass relation. Cunnington et al. (Reference Cunnington, Harrison, Pourtsidou and Bacon2018) show that this result is comparable to that for the proposed IM experiment with the SKA1 Wide Band 1 Survey. We use the Ascaso et al. (Reference Ascaso, Mei and Benítez2015) catalogue in which accurate LSST photometric redshifts are simulated, with 2 pixels per arcminute resolution and 30 redshift bins over a range of 0 < z < 3. This gives the optical catalogue a low number density of 0.27 galaxies per voxel, but despite this, a clustering redshift recovery is still possible. This is at the expense of the simulations’ halo mass resolution for the galaxies which must increase for larger skies due to computational cost. This means that the wide results are on the conservative side, since including lower mass galaxies would mean a more complete representation of the underlying mass density, potentially improving the cross-correlation.

5.3.1. Foregrounds

IM observations suffer from contamination from Galactic and extra-Galactic foregrounds. The main components of the Galactic foregrounds in IM are synchrotron and free-free emission with amplitudes up to 4–5 orders of magnitudes higher than the redshifted HI signal. Current all-sky observations of the foregrounds are sparse in frequency (Reich & Reich Reference Reich and Reich1988) and suffer under high systematic contaminations (Remazeilles et al. Reference Remazeilles, Dickinson, Banday, Bigot-Sazy and Ghosh2015) which limit the possibility of template-fitting. However, the spectral smoothness of the foregrounds—each component approximately following a power law in frequency—allows one to separate the spectrally varying HI signal from the foregrounds. Results of Green Bank Telescope data analysis show that blind component separation techniques like Singular-Value Decomposition (Switzer et al. Reference Switzer, Chang, Masui, Pen and Voytek2015) and independent component analysis (fastICA, Wolz et al. Reference Wolz2017) had some success in separating signal and foregrounds. Studies of the performance of these methods on large sky areas (see Wolz et al. Reference Wolz, Abdalla, Blake, Shaw, Chapman and Rawlings2014; Reference Wolz2015; Alonso et al. Reference Alonso, Bull, Ferreira and Santos2015a) show that large angular scales $\ell <30$ suffer the most from foreground contamination. In addition, Alonso et al. (Reference Alonso, Bull, Ferreira and Santos2015a) demonstrate how the leakage of polarised foregrounds can affect the cosmological analysis. Alternative, promising separation techniques have been proposed by Chapman et al. (Reference Chapman2013), Olivari, Remazeilles, & Dickinson (Reference Olivari, Remazeilles and Dickinson2016), Zhang et al. (Reference Zhang, Bunn, Karakci, Korotkov, Sutter, Timbie, Tucker and Wandelt2016), Zuo et al. (Reference Zuo, Chen, Ansari and Lu2018), which provide a diverse collection of techniques to tackle the foreground subtraction of the SKA data. Moreover, the overall effects of foreground residuals on the cosmological interpretation is dramatically reduced by combining IM data with optical galaxy surveys. Wolz et al. (Reference Wolz2015) present a comparison of foreground removal in the context of IM with the SKA.

5.3.2. Red noise

Red noise, also termed $1/f$ noise, is a form of noise inherent to radio receivers which is correlated in time and manifests itself as gain fluctuations (see Harper et al. Reference Harper, Dickinson, Battye, Roychowdhury, Browne, Ma, Olivari and Chen2018 for a detailed exposition of the subject in the context of IM).

On timescales larger than $1/f_{k}$ , where is $f_{k}$ is called the knee frequency, the noise no longer behaves as ‘white noise’ and does not integrate down as square root of time. This behaviour typically leads to scan synchronous ‘stripes’ in maps. Techniques have been proposed to clean such effects directly on the time ordered data (Janssen et al. Reference Janssen1996; Maino et al. Reference Maino, Burigana, Górski, Mandolesi and Bersanelli2002), which should provide unbiased results even for timescales longer than the knee frequency, but this is traded for an overall increase of the noise variance which could ultimately prevent single dish observations from being useful for cosmology.

When scanning the sky with the telescope at a particular scan speed, the timescale $1/f_{k}$ will translate into an angular scale, which should be larger than the scale of the feature (e.g. the BAO scale) one is trying to measure in the thermal noise dominated regime. Hence, scanning as fast as possible can help some or all the effects of the red noise. This may not be sufficient with the SKA for the BAO scale, and a knee frequency is $\sim$ 1 Hz since the maximum scan speed will be $\sim$ 3 deg ${{\text{s}}}^{-1}$ . However, one would expect that the red noise is strongly correlated along the frequency direction. If that is the case, it might be possible to remove its effects as part of the foreground cleaning process. In Harper et al. (Reference Harper, Dickinson, Battye, Roychowdhury, Browne, Ma, Olivari and Chen2018), such frequency correlations were injected directly in the noise power spectrum density in a simulated spectroscopic receiver. For levels of correlation expected for a typically SKA receiver, it was found that the effects could be removed to a level where the noise is within a factor of two of the thermal noise.

An alternative would be to try to calibrate such fluctuations using a noise diode or the sky itself. To calibrate out $1/f$ noise using a noise diode signal, the uncertainties on the calibration measurement will need to be significantly better than the r.m.s. fluctuations of the $1/f$ noise ( $\sigma(1/f)$ ). This cannot be done on short timescales over which $\sigma(1/f)$ itself is very small (and therefore, not expected to be a problem), but it might be possible to calibrate the SKA receiver on 100 s or longer timescales, for feasible diode brightnesses. On 100-s timescales, for a bandwidth of $\Delta \nu = 50$ MHz and diode brightness of $25\,{\text{K}}$ , the diode signal stability needs to be better than 1 part in $10^{4}$ . It might be possible to use the noise diode in conjunction with component separation techniques described above to relax this requirement.

5.3.3. Bandpass calibration

Bandpass calibration errors are multiplicative with the total system temperature of the receiver. As the system temperature is typically many orders-of-magnitude greater than the HI intensity signal, even very small bandpass calibration errors can have a big impact on signal recovery. For the SKA receivers, the system temperature is approximately $T_{{\text{sys}}} = 22$ K (at 1200 MHz), while the expected HI fluctuation scale will be approximately $\sigma_{{\text{HI}}} = 0.1$ mK in a 10 MHz channel bandwidth. Assuming that at a minimum, the r.m.s. of the HI signal and bandpass calibration errors ( $\delta$ ) should be equal, then

(15) \begin{equation} \delta = \frac{\sigma_{{\text{HI}}}}{T_{{\text{sys}}}} \approx 5 \times 10^{-6}\,,\end{equation}

at the scale of interest for the HI signal (e.g. $\ell \approx 100$ corresponding to angular scales $\sim$ 2^°). This is the calibration error that should be aimed for in the final SKA HI IM survey per voxel with $4\,{{\text{deg}}}^2\times 10\,{\text{MHz}}$ .

Assuming that calibration will be performed N times throughout a survey, and assuming that the bandpass calibration uncertainties are Gaussian then the bandpass uncertainty per calibration should not exceed

(16) \begin{equation} \Delta = 5 \times 10^{-6} \sqrt{N_{c}N_{{\text{dishes}}}},\end{equation}

where $N_c$ is the number of bandpass calibrations per dish and $N_{{\text{dishes}}}$ is the number of dishes ( $N = N_c N_{{\text{dishes}}}$ ). No particular calibration procedure has been assumed here (e.g. calibration can be performed from a noise diode or astronomical source), and it assumed that there is no uncertainty in the calibration procedure being performed. This calculation also neglects many of the complexities expected of real calibration errors, such as possible non-Gaussianity, and correlations in frequency. On the other hand, there could also be the possibility of dealing with these uncalibrated uncertainties at the power spectrum level, depending on the behaviour of such fluctuations.

5.3.4. RFI from navigation satellites

Residual contamination from satellites can also pose a problem for HI IM measurements. Although the proposed IM survey is in band 1 where such contamination is expected to be smaller than band 2. Here, we review the recent study of the effect for band 2 (Harper & Dickinson Reference Harper and Dickinson2018) which will indicate some aspects of the problem.

Figure 28 shows the expected r.m.s. fluctuations of L-band emission from satellites when filtering all satellite within 5 degrees of the main beam. The figure shows that the satellite signal is comparable to the expected instantaneous sensitivity of the SKA receivers and greatly exceeds the SKA survey sensitivity if we consider a large survey in band 2 ( $20\,000\,{{\text{deg}}}^2$ , 200 dishes, 30 d, 1 MHz channel widths). In Harper & Dickinson (Reference Harper and Dickinson2018), it is shown that in general the satellite emission does not integrate down on the sky and the residual structure exceeds the expected HI signal fluctuations at all frequencies within SKA Band 2. However, some regions of the sky are clearer than others such as a 8 700 ${{\text{deg}}}^2$ patch around the South Celestial Pole ( $\delta < -65^{\circ}$ ) that might make for a good SKA HI IM survey location if done in band 2.

Figure 28. Expected r.m.s. of emission from global navigation satellite services within SKA1 Band 2 compared with the expected instantaneous receiver noise (black dashed line) for 1 MHz channel widths. The red dot-dashed line shows the sensitivity for an SKA HI IM survey in Band 2 with $30\,000 {{\text{deg}}}^2$ , 200 dishes and 30 d while the orange dot-dashed line shows the expected HI signal. Note however that the proposed wide HI IM survey is in Band 1.