#### 3.1.1. GR and modified gravity

To date, GR has passed every test with flying colours. The most stringent of these have been carried out in the solar system and with binary pulsars (Will Reference Will2014; Stairs Reference Stairs2003; Wex Reference Wex2014; Shao & Wex 2016; Kramer Reference Kramer2016), where a wide range of deviations from GR have been essentially ruled out with extremely high precision. The recent direct measurement of GWs by Advanced LIGO/Virgo has produced a new opportunity to validate GR in a very different physical situation, that is, a highly dynamical, strong field spacetime (Abbott et al. 2016c; Reference Abbott2017), and a growing variety of cosmological tests of gravity are beginning to be carried out with ever-increasing precision (Joyce et al. Reference Joyce, Jain, Khoury and Trodden2015; Bull et al. Reference Bull2016). These are just a few of the regimes in which new gravitational phenomena could be hiding, however (Baker et al. Reference Baker, Psaltis and Skordis2015), and most have not yet been tested with the high precision that is characteristic of solar system tests. Furthermore, intriguing clues of possible deviations from GR have been emerging (e.g., in recent studies of DM and dark energy) but are far from decisive and remain open to interpretation. Finally, GR may turn out to be the low-energy limit of a more fundamental quantum gravity theory, with hints of the true high-energy theory only arising in relatively extreme physical situations that we have yet to probe. As such, testing GR across a broader range of physical regimes, with increasing precision, stands out as one of the most important tasks in contemporary fundamental physics. The SKA will be a remarkably versatile instrument for such tests, as we will discuss throughout this section.

An important tool in extending tests of GR into new regimes has been the development of a variety of alternative gravity theories (Clifton et al. Reference Clifton, Ferreira, Padilla and Skordis2012b). These give some ideas of what possible deviations from GR could look like and help to structure and combine observational tests in a coherent way. While there are many so-called *modified gravity* theories in existence, it is possible to categorise them in a relatively simple way, according to how they break Lovelock’s theorem (Lovelock Reference Lovelock1971). This is a uniqueness theorem for GR; according to Lovelock’s theorem, GR is the only theory that is derived from a local, four-dimensional action that is at most second order in derivatives only of the spacetime metric. Any deviation from these conditions *breaks* the theorem, giving rise to an alternative non-GR theory that may or may not have a coherent structure. For example, one can add additional gravitational interactions that depend on new scalar or tensor degrees of freedom (e.g., Horndeski or bigravity models respectively), add extra dimensions (e.g., Randall-Sundrum models), introduce non-local operators (e.g., non-local gravity), higher-order derivative operators (e.g., *f*(*R*) theory), or even depart from an action-based formulation altogether (e.g., emergent spacetimes). Each of these theories tends to have a complex structure of its own, which is often necessary to avoid pathologies such as *ghost* degrees of freedom, derivative instabilities, and so forth. Viable theories are also saddled with the need to reduce to a theory very close to GR in the solar system, due to the extremely restrictive constraints on possible deviations in that regime. The result is that most viable modified gravity theories predict interesting new phenomena—for example, screening mechanisms that shield non-GR interactions on small scales as in Chameleon gravity (Khoury & Weltman Reference Khoury and Weltman2004a,b)—which in turn inform the development of new observational tests. Unsuccessful searches for these new phenomena can constrain and even rule out specific subsets of these theories and test GR in the process.

##### 3.1.1.1. Testing relativistic gravity with radio pulsar timing

Pulsar timing involves the use of large area radio telescopes or arrays to record the so-called times of arrival (TOAs) of pulsations from rotating radio pulsars. Millisecond pulsars (MSPs) are especially stable celestial clocks that allow timing precision at the nanosecond level (Taylor Reference Taylor1992; Stairs Reference Stairs2003). Such precision enables unprecedented studies of neutron star astronomy and fundamental physics, notably precision tests of gravity theories (Wex Reference Wex2014; Manchester:Reference Manchester2015; Kramer Reference Kramer2016).

The TOAs from pulsar timing depend on the physical parameters that describe the pulsar system. These include the astrometric and rotational parameters of the pulsar, velocity dispersion in the intervening interstellar medium, and the motion of the telescope in the solar system (including the movement and the rotation of the Earth). If the pulsar is in a binary system, the TOAs are also affected by the orbital motion of the binary, which in turn depend on the underlying gravity theory (Damour & Taylor Reference Damour and Taylor1992; Edwards et al. Reference Edwards, Hobbs and Manchester2006). Deviations from GR—if any—will manifest in TOAs, and different kinds of deviations predict different *residuals* from the GR template.

The double pulsar J0737–3039 (Kramer et al. Reference Kramer2006) represents the state-of-the-art in the field. Five independent tests have already been made possible with this system. GR passes all of them. When the SKA is operating, the double pulsar will provide completely new tests, for example, measuring the Lense–Thirring effect (Kehl et al. Reference Kehl, Wex, Kramer and Liu2017), which probe a different aspect of gravitation related to the spin.

What makes the field of testing gravity with pulsar timing interesting is that, although the double pulsar represents the state-of-the-art, other pulsars can outperform it in probing different aspects of gravity (Wex Reference Wex2014). For example, the recently discovered triple pulsar system (with one neutron star and two white dwarfs) is the best system to constrain the universality of free fall (UFF) for strongly self-gravitating bodies (Ransom et al. Reference Ransom2014; Shao Reference Shao2016; Archibald et al. Reference Archibald2018). UFF is one of the most important ingredients of the strong equivalence principle (SEP; Will Reference Will2014). When UFF is violated, objects with different self-gravitating energies could follow different geodesics (Damour & Schaefer Reference Damour and Schaefer1991). When the SEP is violated, for a binary composed of two objects with different self-gravitating energies, it is very likely that a new channel to radiate away orbital energy will open. If dipole radiation exists (in addition to the quadrupole radiation in GR), a binary will shrink faster, resulting in a new contribution to the time derivative of the orbital period (Damour & Taylor Reference Damour and Taylor1992). For example, this happens in a class of scalar-tensor theories (Damour & Esposito-Farese Reference Damour and Esposito-Farese1996), and in these theories, the dipole radiation might also be enhanced due to the strong field of neutron star interiors. Binary pulsars have provided the best constraints for this phenomenon (Freire et al. Reference Freire2012; Shao et al. Reference Shao, Sennett, Buonanno, Kramer and Wex2017).

Pulsars can be used to test the validity of theories (de Cesare & Sakellariadou Reference de Cesare and Sakellariadou2017; de Cesare et al. Reference de Cesare, Lizzi and Sakellariadou2016) that lead to time variation of Newton’s gravitational constant. A time-varying Newton’s constant will contribute to the decay of the binary orbit as (Damour et al. Reference Damour, Gibbons and Taylor1988; Nordtvedt Reference Nordtvedt1990)

where *P*, $m_c$, and *M* stand for the orbital period, the companion mass, and the sum of the masses of the pulsar and its companion, respectively, and *s* denotes a *sensitivity* parameter. Currently, the strongest constraint on the temporal variation of the gravitational constant results from lunar laser ranging analysis, which sets (Williams et al. Reference Williams, Turyshev and Boggs2004)

Pulsar timing of PSRs J1012+5307 (Lazaridis et al. Reference Lazaridis2009), J1738+0333 (Freire et al. Reference Freire2012), and J1713+0747 (Zhu et al. Reference Zhu2019) has achieved limits comparable to Equation (4).

Binary pulsars can also be used to test cosmological models that lead to local Lorentz invariance (LLI) violation. In particular, some modified gravity models, such as the TeVeS (Bekenstein Reference Bekenstein2004) or the D-material universe (a cosmological model motivated from string theory that includes a vector field; Elghozi et al. Reference Elghozi, Mavromatos, Sakellariadou and Yusaf2016) imply violation of LLI. Possible violation of LLI results in modifications of the orbital motion of binary pulsars (Damour & Esposito-Farese Reference Damour and Esposito-Farese1992; Shao & Wex Reference Shao and Wex2012; Shao Reference Shao2014), as well as leads to characteristic changes in the spin evolution of solitary pulsars (Nordtvedt Reference Nordtvedt1987; Shao et al. Reference Shao, Caballero, Kramer, Wex, Champion and Jessner2013); for the latter, LLI also leads to spin precession with respect to a fixed direction (Shao & Wex Reference Shao and Wex2012). Hence, LLI violation implies changes in the time derivative of the orbit eccentricity, of the projected semi-major axis, and of the longitude of the periastron, while it changes the time behaviour of the pulse profile. The strongest current constraints on LLI violation are set from pulsar experiments, using the timing of binary pulsars.

There is also the potential for the SKA to search for the predicted effects of quantum gravity. Specifically, in a pulsar BH binary system, the disruption effect due to quantum correction can lead to a different gravitational time delay and interferometry of BH lensing. Recently, the discovery of PSR J1745–2900 (Eatough et al. Reference Eatough2013; Rea et al. Reference Rea2013; Shannon & Johnston Reference Shannon and Johnston2013) orbiting the galactic centre (GC) BH Sgr A* opens up the possibility for precision tests of gravity (Pen & Broderick Reference Pen and Broderick2014). The radio pulses emitted from the pulsar can be lensed by an intervening BH that is in between the pulsar and observer. Therefore, the gravitational time delay effect and interferometry between the two light rays can be used to investigate the possible quantum deviations from standard Einstein gravity (Pen & Broderick Reference Pen and Broderick2014). According to Pen & Broderick (Reference Pen and Broderick2014), the fractal structure of the BH surface due to quantum corrections can destroy any interference between the two light rays from the pulsars. In the future, the SKA will find a large number of pulsar BH binary systems, with which we will be able to perform stringent tests of gravity.

Finally, binary pulsars have been used to constrain a free parameter of a higher-derivative cosmological model, obtained as the gravitational sector of a microscopic model that offers a purely geometric interpretation for the standard model (Chamseddine et al. Reference Chamseddine, Connes and Marcolli2007). By studying the propagation of gravitons (Nelson et al. Reference Nelson, Ochoa and Sakellariadou2010b), constraints were placed on the parameter that relates coupling constants at unification, using either the quadrupole formula for GWs emitted from binary pulsars (Nelson et al. Reference Nelson, Ochoa and Sakellariadou2010a) or geodesic precession and frame-dragging effects (Lambiase et al. Reference Lambiase, Sakellariadou and Stabile2013). These constraints will be improved once more rapidly rotating pulsars close to the Earth are observed. Clearly, such an approach can be used for several other extended gravity models (Capozziello et al. Reference Capozziello, Lambiase, Sakellariadou and Stabile2015; Lambiase et al. Reference Lambiase, Sakellariadou, Stabile and Stabile2015).

Since the SKA will provide better timing precision and discover more pulsars, all the above tests will be improved significantly (Shao et al. Reference Shao2015).

##### 3.1.1.2. BH physics and Sgr A*

Testing BH physics is an intriguing and challenging task for modern astronomy. Relativity predicts that any astrophysical BH is described by the Kerr metric and depends solely on its mass and angular momentum (or equivalently spin). Sagittarius A* (Sgr A*), which is the closest example of a supermassive BH (SMBH), is an ideal laboratory with which the SKA can test gravity theories and the no-hair theorem (Kramer et al. Reference Kramer, Backer, Cordes, Lazio, Stappers and Johnston2004).

Pulsars are extremely precise natural clocks due to their tremendous rotational stability. Thus, a relativistic binary of a pulsar and Sgr A* would be a robust tool for testing relativity in stronger gravitational fields than is available from pulsar binaries with stellar mass companions. Such a test will be important since strong field predictions can be fundamentally different between GR and a number of alternative gravity theories (see Johannsen Reference Johannsen2016, for a review).

The GC hosts a large number of young and massive stars within the inner parsec, which can be the progenitors of pulsars (e.g., Paumard et al. Reference Paumard2006; Lu et al. Reference Lu, Do, Ghez, Morris, Yelda and Matthews2013). The population of normal pulsars can be hundreds within distance of $<\!4000AU$ from Sgr A* (e.g., Zhang et al. Reference Zhang, Lu and Yu2014; Pfahl & Loeb Reference Pfahl and Loeb2004; Chennamangalam & Lorimer Reference Chennamangalam and Lorimer2014). The orbits of the innermost ones could be as tight as $\sim\!100$–500 AU from the SMBH (Zhang et al. Reference Zhang, Lu and Yu2014). Furthermore, a magnetar recently discovered in this region (Rea et al. Reference Rea2013; Eatough et al. Reference Eatough2013) also suggests that a population of normal pulsars is likely to be present near the GC, since magnetars are rare pulsars.

To reveal pulsars in the GC region, a high-frequency (usually $>\!9$ GHz) radio survey is needed as there is severe radio scattering by the interstellar medium at low frequencies. Radio surveys so far have not found any normal pulsars in the innermost parsec of the GC (e.g., Deneva et al. Reference Deneva, Cordes and Lazio2009; Macquart et al. Reference Macquart, Kanekar, Frail and Ransom2010; Bates et al. Reference Bates2011). SKA1-Mid would be capable of revealing pulsars down to $2.4$ GHz with spin periods $\sim\! 0.5$ s in this region (Eatough et al. Reference Eatough2015). The timing accuracy of pulsars for SKA after $\sim1$ h integration can reach $\sigma_T\simeq100\,\mu$s (Liu et al. Reference Liu, Wex, Kramer, Cordes and Lazio2012) at a frequency of $\gtrsim15\,$GHz, and $\sigma_T\simeq0.1$–$10\,$ms if the frequency is between $\gtrsim\! 5\,$ and $\lesssim\! 15\,$GHz. Besides the timing measurements, proper motions would be measurable for these pulsars. Finally, the baselines of the SKA are expected to be up to $\sim 3000\,$km, and thus it can provide image resolution up to $2\,$mas at $10\,$GHz (Godfrey et al. Reference Godfrey2012) and astrometric precision reaching $\sim\!10\,\mu$ as (Fomalont & Reid Reference Fomalont and Reid2004).

The relativistic effects cause orbital precession of the pulsars orbiting Sgr A*, in both the argument of pericentre and the orbital plane. A number of previous studies have focused on the relativistic effects according to the orbital averaged precession over multiple orbits (e.g., Wex & Kopeikin Reference Wex and Kopeikin1999; Pfahl & Loeb Reference Pfahl and Loeb2004; Liu et al. Reference Liu, Wex, Kramer, Cordes and Lazio2012; Psaltis et al. Reference Psaltis, Wex and Kramer2016) or the resolved orbital precession within a few orbits (Angélil & Saha Reference Angélil and Saha2010; Angélil et al. Reference Angélil, Saha and Merritt2010). These studies implement post-Newtonian techniques based on Blandford & Teukolsky (Reference Blandford and Teukolsky1976), Damour & Deruelle (Reference Damour and Deruelle1986), and Hobbs et al. (Reference Hobbs, Edwards and Manchester2006), or a mixed perturbative and numerical approach (Angélil et al. Reference Angélil, Saha and Merritt2010). For a pulsar orbiting an SMBH, it is also feasible to implement full relativistic treatments (Zhang et al. Reference Zhang, Lu and Yu2015; Zhang & Saha Reference Zhang and Saha2017).

The TOAs of pulsars rotating around Sgr A* are affected by a number of relativistic effects, for example, Einstein delay and Shapiro delay (Damour & Deruelle Reference Damour and Deruelle1986; Taylor Reference Taylor1992). The orbital precession caused by frame-dragging and quadrupole moment effects also impact the TOAs. Recent studies have found that the frame-dragging effect in TOAs for a pulsar-Sgr A* binary are quite strong compared to the timing accuracies of the pulsar (Liu et al. Reference Liu, Wex, Kramer, Cordes and Lazio2012; Psaltis et al. Reference Psaltis, Wex and Kramer2016), that is, orders of 10–100 s per orbit while the timing accuracies are typically $\sim0.1$ ms (Zhang & Saha Reference Zhang and Saha2017). Current TOA modelling assumes that the orbital precession increases linearly with time. However, it is found to be inaccurate compared to the TOA accuracy; thus, more sophisticated modelling of TOAs are needed, for example, explicitly solving the geodesic equation of the pulsars and the propagation trajectories of the photons (Zhang & Saha Reference Zhang and Saha2017).

Frame-dragging and quadrupole momentum effects can be tightly constrained by observing relativistic pulsar-Sgr A* binaries. If the orbital period of a pulsar is $\sim\!0.3$ yr, the frame-dragging and the quadrupole moment effect of the SMBH can be constrained down to $\sim10^{-2}$–$10^{-3}$ and $\sim10^{-2}$, respectively, within a decade, providing timing accuracies of $\sigma_{\rm
T}\sim100\,\mu$s (Liu et al. Reference Liu, Wex, Kramer, Cordes and Lazio2012). By monitoring a normal pulsar with an orbital period of $\sim2.6$ yr and an eccentricity of $0.3$–$0.9$, and assuming a timing accuracy of 1–5 ms, the magnitude, the line-of-sight inclination, the position angle of the SMBH spin can be constrained with $2\sigma$ errors of $10^{-3}$–$10^{-2}$, $0.1^\circ$–$5^\circ$, and $0.1^\circ$–$10^\circ$, respectively, after $\sim$8 yr (Zhang & Saha Reference Zhang and Saha2017). Even for pulsars in orbits similar to the currently detected stars S2/S0-2 or S0-102, the spin of the SMBH can still be constrained within 4–$8\,$yr (Zhang & Saha Reference Zhang and Saha2017); see Figure 7. Thus, any pulsar located closer than $\sim\! 1000$ AU from the SMBH is plausible for GR spin measurements and tests of relativity.

Figure 7.
*Left:* Apparent trajectories on the sky: blue for the pulsar and cyan for the SMBH. *Right:* Accuracy on the recovered spin magnitude, with green showing results when TOAs on their own are used, and blue showing results from combining both timing and proper motion information. (Zhang & Saha Reference Zhang and Saha2017). The filled red and empty white circles mark the pericentre and apocentre, respectively, of the pulsar orbit. The curves are interpolated from the computed accuracies at the epochs labelled 1–7.

Combining timing and astrometric measurements of GC pulsars, the mass and distance of Sgr A* can be constrained with extremely high accuracy. If the proper motion of pulsars can be determined with an accuracy of $10\,\mu$ as along with timing measurements, the mass and the distance of the SMBH can be constrained to about $\sim 1\,{\rm M}_\odot$ and $\sim1\,$pc, respectively (Zhang & Saha Reference Zhang and Saha2017).

It is important to note, however, that GC pulsars would experience gravitational perturbations from other masses, such as stars or other stellar remnants. These (non-relativistic) perturbations may obscure the spin-induced signals outside $\gtrsim\!100$–$400\,\rm ~AU$ (Merritt et al. Reference Merritt, Alexander, Mikkola and Will2010; Zhang & Iorio Reference Zhang and Iorio2017). Outside this region, how to remove this Newtonian ‘foreground’ remains an unsolved problem. One possible filtering strategy may be to use wavelets (Angélil & Saha Reference Angélil and Saha2014).

##### 3.1.1.3. Cosmological tests of gravity

While GR has proven robust against all observational and experimental tests that have been carried out so far, most of these have been restricted to the solar system or binary pulsar systems—that is, firmly in the small-scale, weak field regime. The recent LIGO GW detection has added a valuable strong field test of GR to the roster, but it is the relatively poorly constrained cosmological regime that has perhaps the greatest chance of offering a serious challenge to Einstein’s theory. The application of GR to cosmology represents an extrapolation by many orders of magnitude from where the theory has been most stringently tested, out to distance scales where unexpected new gravitational phenomena—specifically, DM and dark energy—have been discovered to dominate the Universe’s evolution. While it may yet be found that these have ‘conventional’ explanations, perhaps in terms of extensions to the standard model of particle physics, the fact remains that they have so far *only* been detected through their gravitational influence. As such, it is of utmost importance to examine whether the extrapolation of GR out to cosmological distances could be to blame for the appearance of these effects—perhaps we are interpreting our observations in the context of the wrong gravitational theory.

Cosmological tests of GR are still in their infancy, however. While most ‘background’ cosmological parameters are now known to better than 1% precision, additional parameters that describe possible deviations from GR are considerably less well constrained. Recent measurements of the growth rate of LSS have been made at the 10% level, for example, while many alternative theories of gravity have never even been subjected to tests beyond a comparison with background parameter constraints from, e.g., the CMB. It is clear, then, that there is some way to go before constraints on GR in the cosmological regime approach the accuracy that has been achieved in the small-scale, weak field limit.

The SKA is expected to play a central role in a multitude of high-precision tests of GR in cosmological settings, often in synergy with other survey experiments in different wavebands. In this section, we consider several examples of how SKA1 and SKA2 will contribute to precision cosmological tests of GR, including: growth rate and slip relation measurements with galaxy clustering and weak lensing observations; tests of gravity and dark energy using the 21-cm IM technique; detecting relativistic effects on ultra-large scales; peculiar velocity surveys; and void statistics.

On linear sub-horizon scales, there are two main ways in which deviations from GR can affect cosmological observables: by modifying how light propagates, and by modifying how structures collapse under gravity (Amendola et al. Reference Amendola, Kunz, Motta, Saltas and Sawicki2013). Both effects can be probed using large statistical samples of galaxies, for example, by measuring the weak lensing shear and RSD signals. At optical wavelengths, these observations are the preserve of photometric (imaging) and spectroscopic redshift surveys, respectively, but radio observations offer several alternative possibilities for getting at this information.

##### 3.1.1.4. Radio weak lensing

Effective weak lensing surveys can be performed using radio continuum observations (Brown et al. Reference Brown2015), where the total emission from each galaxy is integrated over the entire waveband to increase signal-to-noise. SKA1-Mid has excellent $u-v$ plane coverage, making it possible to image large numbers of galaxies and measure their shapes. It will perform a large continuum galaxy survey over an area of several thousand square degrees (Jarvis et al. Reference Jarvis, Bacon, Blake, Brown, Lindsay, Raccanelli, Santos and Schwarz2015a), achieving a sky density of suitable lensed sources of 2.7 arcmin^{–2} at a mean redshift of $\sim\! 1.1$ (Harrison et al. Reference Harrison, Camera, Zuntz and Brown2016). This is a substantially lower number density than contemporary optical surveys, for example, the Dark Energy Survey (DES) will yield $\sim\! 12$ arcmin^{–2} at a mean redshift of 0.6. However, forecasts suggest that the two surveys should constrain cosmological parameters with a similar level of accuracy—for example, both SKA1 and DES lensing surveys should produce $\mathcal{O}(10\%)$ constraints on the parameter $\Sigma_0$, which parametrises deviations of the lensing potential from its GR behaviour (Harrison et al. Reference Harrison, Camera, Zuntz and Brown2016). This is mainly due to the stronger lensing signal from a significant high-redshift tail of continuum sources that compensate for the lower source number density. Corresponding forecasts for SKA2 suggest that a number density of 10 arcmin^{–2} will be achievable at a mean redshift of 1.3, for a survey covering 30 000 deg$^2$, yielding $\sim\! 4\%$ constraints on $\Sigma_0$ (Harrison et al. Reference Harrison, Camera, Zuntz and Brown2016), surpassing what will be possible with Euclid. While SKA alone will produce strong constraints on modified gravity lensing parameters, the combination of SKA with optical lensing surveys should be the ultimate goal, as the two different methods have very different systematics that should mostly drop out in cross-correlation, producing much ‘cleaner’ lensing signals with enhanced signal-to-noise (Bonaldi et al. Reference Bonaldi, Harrison, Camera and Brown2016; Camera et al. Reference Camera, Harrison, Bonaldi and Brown2017).

##### 3.1.1.5. RSD and peculiar velocities from HI galaxies

SKA1 will have the sensitivity and spectral resolution to perform several different types of spectroscopic galaxy surveys, using the 21-cm emission line from HI. The simplest is a redshift survey, where the 21-cm line is detected for as many galaxies as possible, with a signal-to-noise ratio sufficient only to get a fix on each redshift. Both SKA1 and SKA2 will be able to perform very large redshift surveys; the SKA1 version will be restricted to quite low redshifts, due to the steepness of the sensitivity curve for HI (Yahya et al. Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015; Harrison et al. Reference Harrison, Lochner and Brown2017), while the SKA2 version will be essentially cosmic variance limited from redshift 0 to $\sim\! 1.4$ for a survey covering 30 000 deg$^2$ (Yahya et al. Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015; Bull Reference Bull2016). Precise spectroscopic redshifts allow the galaxy distribution to be reconstructed in 3D down to very small scales, where density fluctuations become non-linear, and galaxies have substantial peculiar velocities due to their infall into larger structures. These velocities distort the 3D clustering pattern of the galaxies into an anisotropic pattern, as seen in redshift-space. The shape of the anisotropy can then be used to infer the velocity distribution, and thus the rate of growth of LSS. HI redshift surveys with SKA1 and SKA2 will both be capable of precision measurements of these RSDs, with SKA1 yielding $\sim\! 10\%$ measurements of $f\sigma_8$ (the linear growth rate multiplied by the normalisation of the matter power spectrum) in several redshift bins out to $z \approx 0.5$, and SKA2 yielding $\lesssim\! 1\%$ measurements out to $z \approx 1.7$ (Bull Reference Bull2016). See Figure 8 for a comparison with other surveys.

Figure 8. Comparison of predicted constraints on the growth rate, $f\sigma_8$, from RSD measurements with various SKA and contemporary optical/NIR surveys. ‘GS’ denotes a spectroscopic galaxy survey, while ‘IM’ denotes an IM survey. The open circles show a compilation of recent RSD measurements. Taken from Bull (Reference Bull2016).

Note that redshift surveys are not the only possibility—one can also try to spectrally resolve the 21-cm lines of galaxies with high signal-to-noise ratios, and then measure the width of the line profile to obtain their rotation velocities. This can then be used in conjunction with the Tully–Fisher (TF) relation that connects rotation velocity to intrinsic luminosity to directly measure the distances to the galaxies, making it possible to separate the cosmological redshift from the Doppler shift due to the peculiar velocity of the galaxy. Direct measurements of the peculiar velocity are highly complementary to RSDs, as they measure the growth rate in combination with a different set of cosmological parameters (i.e., they are sensitive to $\alpha = f[z] H[z]$). The recovered velocity field can also be cross-correlated with the density field (traced by the galaxy positions), resulting in a significant enhancement in the achievable growth rate constraints if the source number density is high enough (Koda et al. Reference Koda2014). SKA1 will be able to perform a wide, highly over-sampled TF peculiar velocity measurement at low redshift (cf., the sensitivity curves of Yahya et al. Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015), potentially resulting in better constraints on the growth rate than achievable with RSDs. The peculiar velocity data would also be suitable for testing (environment-dependent) signatures of modified gravity due to screening, as discussed by Hellwing et al. (Reference Hellwing, Barreira, Frenk, Li and Cole2014) and Ivarsen et al. (Reference Ivarsen, Bull, Llinares and Mota2016).

##### 3.1.1.6. 21-cm IM

Twenty-one centimetre IM (Battye et al. Reference Battye, Davies and Weller2004; Chang et al. Reference Chang, Pen, Peterson and McDonald2008) is an innovative technique that uses HI to map the three-dimensional LSS of the Universe. Instead of detecting individual galaxies like traditional optical or radio galaxy surveys, HI IM surveys measure the intensity of the redshifted 21-cm emission line in three dimensions (across the sky and along redshift).

The possibility of testing dark energy and gravity with the SKA using 21-cm IM has been studied extensively (Santos et al. Reference Santos2015). More specifically, it has been shown that an IM survey with SKA1-Mid can measure cosmological quantities like the Hubble rate *H*(*z*), the angular diameter distance $D_{\rm A}(z)$, and the growth rate of structure $f\sigma_8(z)$ across a wide range of redshifts (Bull et al. Reference Bull, Ferreira, Patel and Santos2015), at a level competitive with the expected results from Stage IV optical galaxy surveys like Euclid (Amendola et al. Reference Amendola2018). For example, a very large area SKA1-Mid IM survey can achieve sub-1% measurements of $f\sigma_8$ at $z<1$ (Bull Reference Bull2016).

However, the IM method is still in its infancy, with the major issue being foreground contamination (which is orders of magnitude larger than the cosmological signal) and systematic effects. These problems become much more tractable in cross-correlation with optical galaxy surveys, since systematics and noise that are relevant for one type of survey but not the other are expected to drop out (Masui et al. Reference Masui2013a; Pourtsidou et al. Reference Pourtsidou, Bacon, Crittenden and Metcalf2016; Wolz et al. Reference Wolz2017a). Therefore, cross-correlating the 21-cm data with optical galaxies is expected to alleviate various systematics and lead to more robust cosmological measurements.

As an example, we can consider cross-correlating an HI IM survey with SKA1-Mid with a Euclid-like optical galaxy clustering survey, as discussed by Pourtsidou et al. (Reference Pourtsidou, Bacon and Crittenden2017). Assuming an overlap $A_{\rm sky} = 7000 \, {\rm deg}^2$, it was found that very good constraints can be achieved in ($f\sigma_8, D_{\rm A}, H$) across a wide redshift range $0.7 \leq z \leq 1.4$, where dark energy or modified gravity effects are important (see Table 1). Furthermore, it was found that combining such a survey with CMB temperature maps can achieve an ISW detection with a signal-to-noise ratio $\sim\! 5$, which is similar to the results expected from future Stage IV galaxy surveys. Detecting the ISW effect in a flat universe provides direct evidence for dark energy or modified gravity.

Table 1. Forecasted fractional uncertainties on $\{f\sigma_8, D_{\rm A}, H\}$ assuming the SKA1-Mid IM and Euclid-like spectroscopic surveys.

##### 3.1.1.7. Relativistic effects on ultra-large scales

Thanks to the unmatched depth of continuum radio galaxy surveys, the large sky coverage, and the novel possibilities available with HI IM, the SKA will probe huge volumes of the Universe, thus allowing us to access the largest cosmic scales. Scales close to the cosmic horizon and beyond carry valuable information on both the primeval phases of the Universe’s evolution and on the law of gravity.

On the one hand, peculiar inflationary features such as primordial non-gaussian imprints are the strongest on the ultra-large scales. On the other hand, if we study cosmological perturbations with a fully relativistic approach, a plethora of terms appears in the power spectrum of number counts besides those due to Newtonian density fluctuations and RSDs (Challinor & Lewis Reference Challinor and Lewis2011; Bonvin & Durrer Reference Bonvin and Durrer2011; Yoo et al. Reference Yoo, Hamaus, Seljak and Zaldarriaga2012; Jeong et al. Reference Jeong, Schmidt and Hirata2012; Alonso et al. Reference Alonso, Bull, Ferreira, Maartens and Santos2015b). For instance, lensing is known to affect number counts through the so-called magnification bias; but other, yet-undetected effects like time delay, gravitational redshift and Sachs-Wolfe and ISW-like terms also contribute to the largest cosmic scales. To measure such relativistic corrections would mean to further thoroughly confirm Einstein’s gravity, in a regime far from where we have accurate tests of it. Otherwise, if we found departures from the well known and robust relativistic predictions, this would strongly hint at possible solutions of the DM/energy problems in terms of a modified gravity scenario (Lombriser et al. Reference Lombriser, Yoo and Koyama2013; Baker et al. Reference Baker, Ferreira, Leonard and Motta2014b; Baker & Bull Reference Baker and Bull2015).

Alas, measurements on horizon scales are plagued by cosmic variance. For instance, forecasts for next-generation surveys show that relativistic effects will not be detectable using a single tracer (Camera et al. Reference Camera2015e; Alonso & Ferreira Reference Alonso and Ferreira2015) and primordial non-gaussianity detection is limited to $\sigma(\,f_{\rm NL})\gtrsim 1$ (Camera et al. Reference Camera, Santos and Maartens2015a; Raccanelli et al. Reference Raccanelli2015). This calls for the multi-tracer technique (MT), developed for biased tracer of the large-scale cosmic structure and able to mitigate the effect of cosmic variance (Seljak Reference Seljak2009; Abramo & Leonard Reference Abramo and Leonard2013; Ferramacho et al. Reference Ferramacho, Santos, Jarvis and Camera2014). Fonseca et al. (Reference Fonseca, Camera, Santos and Maartens2015) showed that the combination of two contemporaneous surveys, a large HI IM survey with SKA1 and a Euclid-like optical/NIR photometric galaxy survey, will provide detection of relativistic effects, with a signal-to-noise of about 14. Forecasts for the detection of relativistic effects for other combinations of radio/optical surveys are discussed by Alonso & Ferreira (Reference Alonso and Ferreira2015).

##### 3.1.1.8. Void statistics

As a particular case for the SKA, we consider number counts of voids, and forecast cosmological parameter constraints from future SKA surveys in combination with Euclid, using the Fisher matrix method (see also 5.4.2). Considering that additional cosmological information is also available in, for example, shapes/profiles, accessible with the SKA, voids are a very promising new cosmological probe.

We consider a flat wCDM cosmology (i.e., a CDM cosmology with a constant equation of state, *w*) with a modified gravity model described by a growth index $\gamma(a) = \gamma_0 + \gamma_1(1-a)$ (Di Porto et al. Reference Di Porto, Amendola and Branchini2012). The void distribution is modelled following Sahlén et al. (Reference Sahlén, Zubeldia and Silk2016) and Sahlén & Silk (Reference Sahlén and Silk2018), here also taking into account the galaxy density and bias for each survey (Yahya et al. Reference Yahya, Bull, Santos, Silva, Maartens, Okouma and Bassett2015; Raccanelli et al. Reference Raccanelli, Montanari, Bertacca, Doré and Durrer2016c). The results are shown in Figure 9. The combined SKA1-Mid and Euclid void number counts could achieve a precision $\sigma(\gamma_0) = 0.16$ and $\sigma(\gamma_1) = 0.19$, marginalised over all other parameters. The SKA2 void number counts could improve on this, down to $\sigma(\gamma_0) = 0.07$, $\sigma(\gamma_1) = 0.15$. Using the powerful degeneracy-breaking complementarity between clusters of galaxies and voids (Sahlén et al. Reference Sahlén, Zubeldia and Silk2016; Sahlén & Silk Reference Sahlén and Silk2018; Sahlén Reference Sahlén2019), SKA2 voids + Euclid clusters number counts could reach $\sigma(\gamma_0) = 0.01$, $\sigma(\gamma_1) = 0.07$.

Figure 9. Forecast 68% parameter confidence constraints for a flat wCDM model with time-dependent growth index of matter perturbations. Note the considerable degeneracy breaking between the Euclid and SKA1 void samples, and between the SKA2 void and Euclid cluster samples. SKA1-Mid covers 5 000 deg$^2$, $z = 0-0.43$. SKA2 covers 30 000 deg$^2$, $z=0.1-2$. Euclid voids covers 15 000 deg$^2$, $z=0.7-2$. Euclid clusters covers 15 000 deg$^2$, $z=0.2-2$. The fiducial cosmological model is given by $\{\Omega_{\rm m} = 0.3, w = -1, \gamma_0 = 0.545, \gamma_1 = 0, \sigma_8 = 0.8, n_{\rm s} = 0.96, h = 0.7, \Omega_{\rm b} = 0.044\}$. We have also marginalised over uncertainty in void radius and cluster mass (Sahlén & Silk Reference Sahlén and Silk2018), and in the theoretical void distribution function (Pisani et al. Reference Pisani, Sutter, Hamaus, Alizadeh, Biswas, Wandelt and Hirata2015).