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Parameterized descriptions of the galaxy–halo connection provide an effective way to synthesize diverse datasets, and have wide-ranging application in astrophysics and cosmology. Below, we highlight three of the major areas in which they have been applied: understanding the physics of galaxy formation (Section 7.1), inferring cosmological parameters (Section 7.2), and probing the properties and distribution of dark matter (Section 7.3).

7.1. Understanding the Physics of Galaxy Formation

We have discussed several of the key insights into galaxy formation that have been informed by studies of the galaxy–halo connection, as well as the interplay between physical and empirical models. We summarize a few of the most interesting aspects here.

We expect that as we become more able to empirically constrain the relationship between multi-variate properties of the galaxy–halo connection, constraints on these and other aspects of galaxy formation physics will improve significantly.

7.2. Inferring Cosmological Parameters

Future galaxy surveys will provide tremendous power for high precision cosmological constraints, especially if clustering measurements can be pushed to smaller scales, within the trans-linear or non-linear regime where the galaxy–halo connection is of increasing importance. For many statistics, the spatial scale at which the minimum fractional error is achieved is in the range of 1 ≤ r ≤ 10 Mpc (see discussion of Figure 13). This is true even for surveys that are specifically designed to probe structure on linear scales, such as measurements of baryon acoustic oscillations. However, galaxy bias is highly complex at these scales. Higher-order perturbation theory generally breaks down at scales around 10–20 Mpc, or at larger scales for redshift-space clustering (Carlson, White & Padmanabhan 2009, Wang, Reid & White 2014). Thus, extracting information out of Mpc scales requires a model that is fully non-linear. The galaxy–halo connection provides such a model, provided the model is flexible enough to incorporate any systematic uncertainties, including the accuracy of predictions for scale-dependent halo clustering, galaxy assembly bias, and the impact of baryonic physics on the abundance and clustering of dark matter halos.

Figure 13

Figure 13. Top Left: Projected galaxy clustering for five cosmological models that vary σ8. Each model is able to match the same two-point galaxy clustering but with different halo occupations. Top Middle: Mean halo occupation for the five cosmological models. Higher values of σ8 require fewer galaxies in massive halos, because they have stronger matter clustering. Top Right: Cosmological constraints from the combination of three different two-point correlations (3x2pt): galaxy–galaxy lensing, shear two-point, and angular galaxy–galaxy clustering using an LSST-like photometric survey. The different colors indicate the minimum scale used in the measurements. Bottom panels show three observables that can break the degeneracy between matter clustering and halo occupation. From left to right, the panels show redshift-space distortions (RSDs), the mass-to-number ratio in clusters, and galaxy–galaxy lensing for each of the values of σ8.

Clustering measurements at small scales are sensitive both to growth of structure and to the universal expansion rate, thus providing complementary information to test competing models of cosmic acceleration (see, e.g., the comprehensive review by Weinberg et al. 2013). In general, the pathway to cosmological constraints using halo occupation methods starts with measurements of projected galaxy clustering, either through wp(rp) or the angular correlation function w(θ). Using projected quantities is key because they eliminate the effect of redshift-space distortions (RSDs). The top-left panel in Figure 13 shows measurements of the projected clustering of galaxies in the BOSS survey from DR10. The figure shows halo occupation fits using the analytic model of Tinker et al. (2005) with five different values of σ8, as listed in the panel (the other cosmological parameters are held fixed). For reference, the clustering of dark matter is shown for each of these cosmologies. The amplitude of matter clustering, and thus the bias of the galaxies in the halo occupation model, varies significantly with σ8, but for each cosmology a good fit can be found to the real-space two-point galaxy clustering. Thus real-space clustering at these scales provides limited information on the amplitude and growth of structure when considered alone. Even though the real-space clustering is the same, each cosmology requires that galaxies occupy different halos (as shown in the top-middle panel). Thus, the occupation functions constrained by wp(rp) can be used to make predictions for statistics that contain more cosmologically sensitive information. These include RSDs, the mass-to-number ratio of galaxy clusters, and galaxy–galaxy lensing, which are shown in the bottom panels of Figure 13 and discussed in more detail below.

Redshift-space distortions. Galaxy redshifts are a function of not just the smooth Hubble flow but also ‘peculiar' motions caused by the local gravitational potential, which causes galaxies to move toward overdensities and away from underdensities. Thus, the spatial distribution of galaxies in redshift space will have anisotropies due to the amplitude of the velocity field. The velocity field is, in turn, determined by the amount of matter in the universe, how clumpy that distribution is, and by the theory of gravity. Current analyses of RSD yield ∼ 10% measurements of the parameter combination fσ8, where f is the logarithmic growth of structure and σ8 is the amplitude of matter fluctuations (Alam et al., 2017). The bottom-left panel in Figure 13 shows the variation in the RSD monopole, ξ0(r), for the five cosmologies, based on the model by Tinker (2007). This panel also shows the expected measurement error for a full BOSS-like survey based on mock galaxy catalogs. As discussed above, the ‘sweet spot’ for optimal measurements is in the 1–10 Mpc regime, where sample variance is minimized but shot noise due to small number statistics is avoided. The most constraining power between models comes at r ∼ 1 Mpc, which represents the transition between pairs of galaxies between two distinct halos and pairs within a single halo. Galaxy pairs within a single halo have larger relative velocities, leading to significant suppression of clustering. As can be seen in the mean occupation functions, the fraction of galaxies that are satellites varies inversely with σ8, thus the model with the largest fsat has the largest pairwise velocity dispersion at r ∼ 1 Mpc, and the lowest ξ0(r) at this scale.

The mass-to-number ratio of galaxy clusters (M / N): This statistic is analogous to the mass-to-light ratios of galaxy clusters, but reduced the number of free model parameters by simply counting the number of galaxies, N, inside a halo. From the mean occupation functions shown in Figure 13, it is clear that measurements of the mean occupation themselves contain cosmological information. The bottom-middle panel of the figure shows predictions for M / N from the five cosmologies fit to the DR10 BOSS wp(rp). Here, the quantity M / N is normalized by the ratio ρcrit / bar{n}gal, where ρcrit is the cosmological critical density, and bar{n}gal is the mean space density of the galaxies in the sample. Points with errors represent estimates of the uncertainties achievable in a BOSS-like survey at z < 0.3. Errors in halo masses are taken from the weak lensing analysis of RedMaPPer clusters by Murata et al. (2018), which are added in quadrature with the expected Poisson noise from the number of clusters in the survey volume (although the mass estimates dominate the error bar). M / N and M / L have been effectively used to constrain cosmological parameters with low-redshift data (van den Bosch, Mo & Yang 2003, Tinker et al. 2012), and new large-scale redshift and lensing surveys make application to larger volumes imminent. Reddick et al. (2014) showed that with current constraining power, simple HODs are sufficient to obtain unbiased parameter constraints, but as the statistical power increases additional parameters may be needed.

Galaxy–galaxy lensing: Galaxy–galaxy lensing is a probe of the galaxy–matter cross correlation, and it is sensitive to both the matter density and amplitude of matter fluctuations. The observational quantity measured by galaxy–galaxy lensing is ΔΣ(Rp), the excess surface mass density at Rp, relative to the mean interior density, around a galaxy. The bottom-right panel of Figure 13 shows how ΔΣ(Rp) varies with σ8 for the models fit to the BOSS real-space clustering. Observational uncertainties in this quantity are shown from Leauthaud et al. (2017), which uses deep CFHTLS (Canada-France-Hawaii Telescope Legacy Survey) imaging in the Stripe 82 field of the BOSS spectroscopic survey. Note that this survey only covers ∼ 200 deg2, which is only 2% of the full spectroscopic BOSS catalog. Even this small area yields constraining power to distinguish these models, indicating the substantial potential of future combinations of large area spectroscopic and imaging surveys. Cacciato et al. (2013) and More et al. (2015) have demonstrated the efficacy of joint clustering and lensing analyses for constraining cosmological parameters in the halo occupation framework.

3x2pt: As discussed above, combinations of two-point statistics can break degeneracies in the galaxy–halo connection and provide powerful cosmological information. A recent study from the Dark Energy Survey (DES Collaboration et al., 2017) used a combination of galaxy–galaxy clustering, shear-shear clustering, and galaxy-shear clustering to put the tightest constraints yet on σ8 and Ωm in the local Universe (and see related work by Kilbinger et al. 2013 and van Uitert et al. 2018). To date, these analyses have assumed linear bias between the galaxy clustering and matter clustering and exclude small scales where this assumption is expected to fail from the analyses. However, substantially more constraining power may be available if the modeling can be extended to smaller scales (Krause & Eifler, 2017); a full comparison between constraints with a fully nonlinear modeling approach with a parameterized galaxy–halo connection and a quasi-linear approach with a smaller number of bias parameters has yet to be done.

As these examples demonstrate, pushing to smaller scales has significant potential to improve cosmological constraints from current and upcoming datasets, but there are significant challenges to realize this potential, many of which are related to aspects of the galaxy–halo connection. These include the following:

Modeling in the non-linear regime: Historically, researchers have either used fitting functions for the properties of dark matter halos to model galaxy clustering, or have used simulations directly when modeling a range of galaxy clustering models within one cosmological model. Achieving the required accuracy for these fitting functions is especially challenging in the regime in which there is significant power in the data, 1–10 Mpc. Methods based on perturbation theory or effective field theory (Perko et al., 2016) may be effective in the mildly non-linear regime, but they will not be effective in modeling collapsed regions. The solution may be to emulate the statistics directly (e.g. using techniques similar to those that Heitmann et al. (2010) used for the dark matter power spectrum), using suites of simulations combined with flexible models of the galaxy–halo connection.

Assembly bias: As discussed in Section 4.4, our understanding of the detailed dependence of galaxy properties on halo properties other than their mass is still in the early stages, and improved modeling will be essential to take small-scale cosmology probes that depend on accurate galaxy clustering models to the next stage. In particular, what is needed is a modeling framework that is flexible enough to encompass the full range of physically plausible manifestations of the complexities of assembly bias for realistic galaxy populations, without losing substantial constraining power; this has yet to be demonstrated.

Impact of baryons: Precision cosmology on small-scales will also require understanding the possible range of impacts of galaxy formation and feedback on the matter distribution itself (Rudd, Zentner & Kravtsov, 2008, Semboloni et al., 2011, Schneider & Teyssier, 2015), including its implications for the mass function and clustering of dark matter halos and subhalos (van Daalen et al., 2011, Sawala et al., 2013, Martizzi et al., 2014). We are still far from being able to simulate the full range of possibilities over a range of cosmological models in order to directly emulate these effects using hydrodynamical simulations, so empirical modeling of the effects, informed by our best physical models, will likely remain the best path forward for the foreseeable future. The primary impact on the power spectrum can be characterized by a change in galaxy density profiles (Zentner, Rudd & Hu 2008), but this may not be sufficient for all observable statistics. Additional observables should be combined to put constraints on the possible amount of feedback; e.g. the small-scale shear power spectrum (Foreman, Becker & Wechsler, 2016) and the SZ profiles of groups and clusters (Battaglia et al., 2017).

Intrinsic alignments: Weak gravitational lensing (see Mandelbaum (2018) for a recent review) depends on the spatial correlations between small distortions in galaxy shapes; if galaxy shapes are aligned with their dark matter halos or with the tidal field, this creates a systematic error that has to be modeled. Different galaxy populations have been shown to have different intrinsic alignments, so in detail one would like to model not just the halo occupation as a function of galaxy properties but also the alignment of galaxies with their halos (Schneider & Bridle 2010, Blazek, Vlah & Seljak 2015).

Additional aspects: Accurate modeling of the galaxy–halo connection will continue to be an important feature of the next generation of cosmological studies, even for those studies that are not pushing to small scales or explicitly including a galaxy–halo connection model. Examples include the following:

7.3. Probing the Properties and Distribution of Dark Matter

The nature of the dark matter that makes up ∼ 83% of the mass in the Universe is still unclear, and an understanding of the galaxy–halo connection can facilitate astrophysical constraints on its nature as well as inform constraints from indirect and direct detection. Although the ΛCDM model has had remarkable success on large scales, especially e.g. at larger than the typical sizes of dark matter halos, it is less constrained on smaller scales, where alternative dark matter models can suppress the power spectrum or change the density profiles or dynamics of halos (Buckley & Peter, 2017). Understanding and marginalizing over the range of possibilities for the galaxy–halo connection can be critical to robust dark matter constraints in this regime. More generally, there are wide-ranging problems where understanding the properties of the dark matter halos of a specific galaxy or population of galaxies is important, and statistical modeling of the possible halo population of the galaxies provides a way forward. We give a few examples of these applications here.

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