2.2 Dark matter halo properties

Dark matter was first proposed by [993Jump To The Next Citation Point] to explain the anomalously high velocity of galaxies in galaxy clusters. Since then, evidence for dark matter has been accumulating on all scales. The velocities of individual stars in dwarf galaxies suggest that these are the most dark matter dominated systems in the universe [e.g., 650, 509, 834, 635, 934]. Low surface brightness (LSB) and giant spiral galaxies rotate too fast to be supported by their stars and gas alone, indicating the presence of dark matter [286, 833, 153, 512]. Gravitationally lensed giant elliptical galaxies and galaxy clusters require dark matter to explain their observed image distributions [e.g., 761, 156, 935, 851, 244]. Finally, the temperature fluctuations in the cosmic microwave background (CMB) radiation indicate the need for dark matter in about the same amount as that required in galaxy clusters [e.g., 845, 968, 855].

While the case for particle dark matter is compelling, until we find direct evidence for such a particle, astrophysics remains a unique dark matter probe. Many varieties of dark matter candidates produce a noticeable change in the growth of structure in the universe [482, 865]. Warm dark matter (WDM) suppresses the growth of structure in the early universe producing a measurable effect on the small-scale matter power spectrum [143, 67, 87]. Self-interacting dark matter (SIDM) changes the expected density distribution within bound dark matter structures [273, 440]. In both cases, the key information about dark matter is contained on very small scales. In this section, we discuss previous work that has attempted to measure the small scale matter distribution in the universe, and discuss how Euclid will revolutionize the field. We divide efforts into three main areas: measuring the halo mass function on large scales, but at high redshift; measuring the halo mass function on small scales through lens substructures; measuring the dark matter density profile within galaxies and galaxy clusters.

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Figure 32: The baryonic mass function of galaxies (data points). The dotted line shows a Schechter function fit to the data. The blue line shows the predicted mass function of dark matter haloes, assuming that dark matter is cold. The red line shows the same assuming that dark matter is warm with a (thermal relic) mass of mWDM = 1 keV.

2.2.1 The halo mass function as a function of redshift

Attempts have already been made to probe the small scale power in the universe through galaxy counts. Figure 32View Image shows the best measurement of the ‘baryonic mass function’ of galaxies to date [758]. This is the number of galaxies with a given total mass in baryons normalized to a volume of 1 Mpc. To achieve this measurement, [758] sewed together results from a wide range of surveys reaching a baryonic mass of just 6 ∼ 10 M ⊙ – some of the smallest galaxies observed to date.

The baryonic mass function already turns up an interesting result. Over-plotted in blue on Figure 32View Image is the dark matter mass function expected assuming that dark matter is ‘cold’ – i.e., that it has no preferred scale. Notice that this has a different shape. On large scales, there should be bound dark matter structures with masses as large as 14 10 M ⊙, yet the number of observed galaxies drops off exponentially above a baryonic mass of ∼ 1012M ⊙. This discrepancy is well-understood. Such large dark matter haloes have been observed, but they no longer host a single galaxy; rather they are bound collections of galaxies – galaxy clusters [see e.g. 993]. However, there is also a discrepancy at low masses that is not so well understood. There should be far more bound dark matter haloes than observed small galaxies. This is the well-known ‘missing satellite’ problem [662, 511Jump To The Next Citation Point].

The missing satellite problem could be telling us that dark matter is not cold. The red line on Figure 32View Image shows the expected dark matter mass function for WDM with a (thermal relic) mass of m = 1 keV WDM. Notice that this gives an excellent match to the observed slope of the baryonic mass function on small scales. However, there may be a less exotic solution. It is likely that star formation becomes inefficient in galaxies on small scales. A combination of supernovae feedback, reionization and ram-pressure stripping is sufficient to fully explain the observed distribution assuming pure CDM [529, 756, 603]. Such ‘baryon feedback’ solutions to the missing satellite problem are also supported by recent measurements of the orbits of the Milky Way’s dwarf galaxies [594]. Weak and strong lensing measurements of the halo mass function

To make further progress on WDM constraints from astrophysics, we must avoid the issue of baryonic physics by probing the halo mass function directly. The only tool for achieving this is gravitational lensing. In weak lensing this means stacking data for a very large number of galaxies to obtain an averaged mass function. In strong lensing, this means simply finding enough systems with ‘good data.’ Good data ideally means multiple sources with wide redshift separation [776]; combining independent data from dynamics with lensing may also prove a promising route [see e.g. 893].

Euclid will measure the halo mass function down to ∼ 1013M ⊙ using weak lensing. It will simultaneously find 1000s of strong lensing systems. However, in both cases, the lowest mass scale is limited by the lensing critical density. This limits us to probing down to a halo mass of ∼ 1011M ⊙ which gives poor constraints on the nature of dark matter. However, if such measurements can be made as a function of redshift, the constraints improve dramatically. We discuss this in the next Section. The advantage of going to high redshift

Dark matter constraints from the halo mass function become much stronger if the halo mass function is measured as a function of redshift. This is because warm dark matter delays the growth of structure formation as well as suppressing small scale power. This is illustrated in Figure 33View Image, which shows the fraction of mass in bound structures as a function of redshift, normalized to a halo of Milky Way’s mass at redshift z = 0. Marked are different thermal relic WDM particle masses in keV (black solid lines). Notice that the differences between WDM models increase significantly towards higher redshift at a given mass scale. Thus we can obtain strong constraints on the nature of dark matter by moving to higher z’s, rather than lower halo mass.

The utility of redshift information was illustrated recently by observations of the Lyman-α absorption spectra from Quasars [927Jump To The Next Citation Point, 812Jump To The Next Citation Point]. Quasars act as cosmic ‘flashlights’ shining light from the very distant universe. Some of this light is absorbed by intervening neutral gas leading to absorption features in the Quasar spectra. Such features contain rich information about the matter distribution in the universe at high redshift. Thus, the Lyman-α forest measurements have been able to place a lower bound of mWDM > 4 keV probing scales of ∼ 1 Mpc. Key to the success of this measurement is that much of the neutral gas lies in-between galaxies in filaments. Thus, linear approximations for the growth of structures in WDM versus CDM remain acceptable, while assuming that the baryons are a good tracer of the underlying matter field is also a good approximation. However, improving on these early results means probing smaller scales where nonlinearities and baryon physics will creep in. For this reason, tighter bounds must come from techniques that either probe even higher redshifts, or even smaller scales. Lensing from Euclid is an excellent candidate since it will achieve both while measuring the halo mass function directly rather than through the visible baryons.

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Figure 33: The fraction of mass in bound structures as a function of redshift, normalized to a halo of Milky Way’s mass at redshift z = 0. Marked are different masses of thermal-relic WDM particles in keV (black solid lines). Notice that the differences between different WDM models increases towards higher redshift.

2.2.2 The dark matter density profile

An alternative approach to constraining dark matter models is to measure the distribution of dark matter within galaxies. Figure 34View Image shows the central log-slope of the density distribution for 9 galaxies/groups and 3 lensing clusters as a function of the enclosed lensing mass [777, 757, 776]. Over the visible region of galaxies, the dark matter distribution tends towards a single power law: α ρ ∝ r. Marked in red is the prediction from structure-formation simulations of the standard cosmological model, that assume non-relativistic CDM, and that do not include any baryonic matter. Notice that above an enclosed lensing mass of ∼ 1012M ⊙, the agreement between theory and observations is very good. This lends support to the idea that dark matter is cold and not strongly self-interacting. However, this result is based on only a handful of galaxy clusters with excellent data. Furthermore, lower mass galaxies and groups can, in principle, give tighter constraints. In these mass ranges, however (Menc < 1012M ⊙), the lensing mass is dominated by the visible stars. Determining the underlying dark matter distribution is then much more difficult. It is likely that the dark matter distribution is also altered from simple predictions by the dynamical interplay between the stars, gas and dark matter during galaxy formation [e.g., 296].

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Figure 34: The central log-slope α of the density distribution ρ ∝ rα for 9 galaxies/groups and 3 lensing clusters as a function of the enclosed lensing mass. Marked in red is the prediction from structure formation simulations of the standard cosmological model, that assume non-relativistic CDM, and that do not include any baryonic matter.

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