Apparently the matter content of the Universe is dominated by an unknown form of dark matter (DM) without interactions with ordinary baryonic matter, perhaps not even with itself. It only interacts via the gravitational field, manifesting its effects on astrophysical and cosmological scales. The purpose of this review is to summarize the phenomenology of all such effects, that can serve as probes of dark matter. Regardless of the ultimate, correct explanation of its particle nature or field nature, theory needs to address all these effects.

This review does not cover the historical development, except by glimpses, because the rapid development of observational means tends to render all discoveries older than a decade unimportant.

Beginning from the first controversial conclusions from the motion of stars near the Galactic disk on missing matter in the Galactic disk (Section 2), and that of Fritz Zwicky in 1933 [1] of missing matter in the Coma cluster (Section 3), we describe the kinematics of virially bound systems (Section 3) and rotating spiral galaxies (Section 4). An increasingly important method to determine the weights of galaxies, clusters and gravitational fields at large, independently of electromagnetic radiation, is lensing, strong as well as weak (Section 5). Next follows a discussion of dark matter in elliptical galaxies (Section 6) and mass-to-light ratios which probe dark matter in all systems, notably in dwarf spheroidals (Section 7). Different ways to measure missing mass in groups and clusters derive from the comparison of visible light and X-rays (Section 8). Mass autocorrelation functions relate galaxy masses to dark halo masses (Section 9).

In radiation the most important tools are the temperature and polarization anisotropies in the Cosmic Microwave Background (CMB) (Section 10), which give information on the mean density of both dark and baryonic matter as well as on the geometry of the Universe. The large scale structures of matter exhibit similar fluctuations evident in the Baryonic Acoustic Oscillations (BAO) (Section 11). The amplitude of the temperature variations in the CMB prove, that galaxies could not have formed in a purely baryonic Universe (Section 12). Simulations of large scale structures also show that DM must be present (Section 13). The best quantitative estimates of the density of DM come from overall parametric fits to cosmological models, notably the Cold Dark Matter model “CDM” with a cosmological constant, of CMB data, BAO data, and redshifts of supernovae of type Ia (SNe Ia) (Section 14). A particularly impressive testimony comes from merging clusters (Section 15). We conclude this review with a brief summary (Section 16).

In 1922 the Dutch astronomer Jacobus Kapteyn [2] studied the vertical motions of all known stars near the Galactic plane and used these data to calculate the acceleration of matter. This amounts to treating the stars as members of a “star atmosphere”, a statistical ensemble in which the density of stars and their velocity dispersion defines a “temperature” from which one obtains the gravitational potential. This is analogous to how one obtains the gravitational potential of the Earth from a study of the atmosphere. Kapteyn found that the spatial density is sufficient to explain the vertical motions.

Later in the same year the British astronomer James Jeans [3] reanalyzed Kapteyn’s data and found a mass deficit: to each bright star two dark stars had to be present. The result contradicted grossly the expectations: if the potential provided by the known stars was not sufficient to keep the stars bound to the Galactic disk, the Galaxy should rapidly be losing stars. Since the Galaxy appeared to be stable there had to be some missing matter near the Galactic plane.

In 1932 the Dutch astronomer Jan Hendrik Oort [4] reanalyzed the vertical motions and came to the same conclusion as Jeans. There was indeed a mass deficit which Oort proposed to indicate the presence of some dark matter in our Galaxy. The possibility that this missing matter would be nonbaryonic could not even be thought of at that time. Note that the first neutral baryon, the neutron, was discovered by James Chadwick [5] only in the same year, in 1932.

However, it is nowadays considered, that this does not prove the existence of DM in the disk. The potential in which the stars are moving is not only due to the disk, but rather to the totality of matter in the Galaxy which is dominated by the Galactic halo. The advent of much more precise data in 1998 led Holmberg & Flynn [6] to conclude that no DM was present in the disk.

Oort determined the mass of the Galaxy to be , and thought that the nonluminous component was mainly gas. Still in 1969 he thought that intergalactic gas made up a large fraction of the mass of the universe [7]. The general recognition of the missing matter as a possibly new type of non-baryonic DM dates to the early eighties.

The planets move around the Sun along their orbits with orbital velocities balanced by the total gravity of the Solar system. Similarly, stars move in galaxies in orbits with orbital velocities determined by the gravitational field of the galaxy, or they move with velocity dispersion. Galaxies in turn move with velocity dispersion under the influence of the gravitational field of their environment, which may be a galaxy group, a cluster or a supercluster. In the simplest dynamical framework one treats massive systems (galaxies, groups and clusters) as statistically steady, spherical, self-gravitating systems of N objects with average mass m and average velocity v or velocity dispersion. The total kinetic energy E of such a system is then (we now use rather than v)

If the average separation is r, the potential energy of pairings is

The virial theorem states that for such a system

The total dynamic mass can then be estimated from and

This can also be written

where I is a surface luminosity, R is a scale, and is the mass-to-light ratio. Choosing the scale to be the half light radius, this implies a relationship between the observed central velocity dispersion, and called the Fundamental Plane. of the form

The virial theorem predicts the values, for the coefficients. This relationship is found in ellipticals [8,9] and in some other types of stellar populations, but with somewhat different coefficients.

Spiral galaxies are stable gravitationally bound systems in which visible matter is composed of stars and interstellar gas. Most of the observable matter is in a relatively thin disc, where stars and gas rotate around the galactic center on nearly circular orbits. The galaxy kinematics is measured by the Doppler shift of well-known emission lines of particular tracers of the gravitational potential: HI, CO and H_α.

If the circular velocity at radius r is v in a rotating galaxy with mass M(r) inside r, the condition for stability is that the centrifugal acceleration should equal the gravitational pull, and the radial dependence of v would then be expected to follow Kepler’s law

The surprising result for spiral galaxy rotation curves is, that the velocity does not follow Kepler’s inverse-root law, but stays rather constant after attaining a maximum. The most obvious solution to this is that the galaxies are embedded in extensive, diffuse halos of dark matter If the mass M(r) enclosed inside the radius r, is proportional to r it follows that constant.

The rotation curve of most galaxies can be fitted by the superposition of contributions from the stellar and gaseous disks, sometimes a bulge, and the dark halo, modeled by a quasi-isothermal sphere. The inner part is difficult to model because the density of stars is high, rendering observations of individual star velocities difficult. Thus the fits are not unique, the relative contributions of disk and dark matter halo is model-dependent, and it is sometimes not even sure whether galactic disks do contain dark matter. Typically, dark matter constitutes about half of the total mass.

In Figure 3 we show the rotation curves fitted for eleven well-measured galaxies [25] of increasing halo mass. One notes, that the central dark halo component is indeed much smaller than the luminous disk component.

At large radii, however, the need for a DM halo is obvious. On galactic scales, the contribution of DM generally dominates the total mass. Note the contribution of the baryonic component, negligible for light masses but increasingly important in the larger structures.

The mass discrepancy emerges also as a disagreement between light and mass distributions: light does not trace mass, the ratio

is not constant, but increases with radius [26].

Gentile et al. [27] have shown that cusped profiles are in clear conflict with data on spiral galaxies. Central densities are rather flat, scaling approximately as . The best-fit disk + NFW halo mass model fits the rotation curves poorly, it implies an implausibly low stellar mass-to-light ratio and an unphysically high halo mass. Clearly the actual profiles are of very uncertain origin.

One notes in Figure 3 that the shape of the rotation curve depends on the halo virial mass so that the distribution of gravitating matter, unlike luminous matter, is luminosity dependent. The old idea that the rotation curve stays constant after attaining a maximum is thus a simplification of the real situation. The rotation velocity can be expressed by a Universal Rotation Curve [25]. All spiral galaxies lie on a curve in the 4-dimensional space of luminosity, core radius, halo central density and fraction of DM, see Figure 4.

Our Galaxy is complicated because of what appears to be a noticeable density dip at 9 kpc and a smaller dip at 3 kpc, as is seen in Figure 5 [28]. To fit the measured rotation curve one needs at least three contributing components: a central bulge, the star disk + gas, and a DM halo [28-30]. For small radii there is a choice of empirical rotation curves, and no DM component appears to be needed until radii beyond 15 kpc.

A consequence of the Strong Equivalence Principle (SEP) is that a photon in a gravitational field moves as if it possessed mass, and light rays therefore bend around gravitating masses. Thus celestial bodies can serve as gravitational lenses probing the gravitational field, whether baryonic or dark without distinction.

Since photons are neither emitted nor absorbed in the process of gravitational light deflection, the surface brightness of lensed sources remains unchanged. Changing the size of the cross-section of a light bundle only changes the flux observed from a source and magnifies it at fixed surface-brightness level. If the mass of the lensing object is very small, one will merely observe a magnification of the brightness of the lensed object an effect called microlensing. Microlensing of distant quasars by

compact lensing objects (stars, planets) has also been observed and used for estimating the mass distribution of the lens-quasar systems.

In Strong Lensing the photons move along geodesics in a strong gravitational potential which distorts space as well as time, causing larger deflection angles and requiring the full theory of General Relativity. The images in the observer plane can then become quite complicated because there may be more than one null geodesic connecting source and observer; it may not even be possible to find a unique mapping onto the source plane cf Figure 6. Strong lensing is a tool for testing the distribution of mass in the lens rather than purely a tool for testing General Relativity. An illustration is seen in Figure 7 where the lens is an elliptical galaxy [32].

At cosmological distances one may observe lensing by composed objects such as galaxy groups which are ensembles of “point-like”, individual galaxies. Lensing effects are very model-dependent, so to learn the true magnification effect one needs very detailed information on the structure of the lens.

Weak Lensing refers to deflection through a small angle when the light ray can be treated as a straight line (Figure 6), and the deflection as if it occurred discontinuously at the point of closest approach (the thin-lens approximation in optics). One then only invokes SEP to account for the distortion of clock rates.

The large-scale distribution of matter in the Universe is inhomogeneous in every direction, so one can expect that everything we observe is displaced and distorted by weak lensing. Since the tidal gravitational field and the deflection angles depend neither on the nature of the matter nor on its physical state, light deflection probes the total projected mass distribution. Lensing in infrared light offers an additional advantage of being able to sense distant background galaxies, since their number density is higher than in the optical range.

Background galaxies would be ideal tracers of distortions if they were intrinsically circular, because lensing transforms circular sources into ellipses. Any measured ellipticity would then directly reflect the action of the gravitational tidal field of the interposed lensing matter, and the statistical properties of the distortions would reflect the properties of the matter distribution. But many galaxies are actually intrinsically elliptical, and the ellipses are randomly oriented. This introduces noise into the inference of the tidal field from observed ellipticities. A useful feature in the sky is a fine-grained pattern of faint and distant blue galaxies appearing as a “wall paper”. This makes statistical weak-lensing studies possible, because it allows the detection of the coherent distortions imprinted by gravitational lensing on the images of the galaxy population.

Thus weak lensing has become an important technique to map non-luminous matter. A reconstruction of one of the largest and most detailed weak lensing surveys undertaken with the Hubble Space Telescope is shown in Figure 8 [33]. This map covers a large enough area to see extended filamentary structures.

A very large review on lensing by R. Massey et al. [34] can be recommended. We show several examples of lensing by clusters in Section 15.

Elliptical galaxies are quite compact objects which mostly do not rotate so their mass cannot be derived from rotation curves. The total dynamical mass is then the virial mass as derived from the velocity dispersions of stars and the anisotropies of their orbits. However, to disentangle the total mass profile into its dark and its stellar components is not straightforward, because the dynamical mass decomposition of dispersions is not unique. The luminous matter in the form of visible stars is a crucial quantity, indispensable to infer the dark component. When available one also makes use of strong and

weak lensing data, and of the X-ray properties of the emitting hot gas. The gravity is then balanced by pressure gradients as given by Jeans’ Equation.

Inside the half light radius the contribution of the dark matter halo to the central velocity dispersion is often very small, km·s^–1, so that the dark matter profile is intrinsically unresolvable. The outer mass profile is compatible with NFW, Equation (7), and with Burkert, Equation (9), as well. Important information on the mass distribution can be obtained from the Fundamental Plane, Equation (6). which yields the coefficients,. Note that this is in some tension with the Virial Theorem, perhaps due to variations in the central dispersions, , of the stellar populations.

O. Tiret et al. [36] concluded from a study of 23 giant elliptical galaxies with central velocity dispersions km·s^–1, that the mass within 5 - 10 kpc is dominated by the stars, not by DM. On the average the dark matter component contributes less than 5% to the total velocity dispersions.

The ELIXR survey is a volume-limited (≤110 Mpc) study by P. J. Humphrey et al. [35], of optically selected, isolated, L^* elliptical galaxies in particular the NGC 1521, for which X-ray data from Chandra and XMM exist. The isolation condition selects the appropriate galaxy halo and reduces the influence of a possible group-scale or cluster-scale halo.

Most of the baryons are in a morphologically relaxed hot gas halo detectable out to kpc, that is well described by hydrostatic models. The baryons and the dark matter conspire to produce a total mass density profile that can be well-approximated by a power law, over a wide range (as has been noted before, see references in [35,36]).

The fitting method involves solving the equation of hydrostatic equilibrium to compute temperature and density profile models, given parametrized mass and entropy profiles. The models are then projected onto the sky and fitted to the projected temperature and density profiles. A fit ignoring DM was poor, but inclusion of DM improved the fit highly significantly: DM was required at. We show this fit in Figure 9. In several studies [36,37], for most of the radii the dark matter contribution is very small although statistically significant.

The mass-to-light ratio of an astronomical object is defined as. Stellar populations exhibit values in solar units, in the solar neighborhood, in the Galactic disk from C. Flynn et al. [38].

Dwarf spheroidal galaxies (dSph) are the smallest stellar systems containing dark matter and exhibit very high M/L ratios,. In Andromeda IX = 93 + 120/–50, in Draco. The dwarf spheroidals have radii of pc and central velocity dispersions km·s^–1 which is larger than expected for self-gravitating, equilibrium stellar populations. The generally accepted picture has been, that dwarf galaxies have slowly rising rotation curves and are dominated by dark matter at all radii.

However, R. A. Swaters et al. [39] have reported ob-

servations of H I rotation curves for a sample of 73 dwarf galaxies, among which eight galaxies have sufficiently extended rotation curves to permit reliable determination of the core radius and the central density. They found that dark matter only becomes important at radii larger than three or four disk scale lengths. Their conclusion is, that the stellar disk can explain the mass distribution over the optical parts of the galaxy, and dark matter only becomes relevant at large radii. However, the required stellar mass-to-light ratios are high, up to 15 in the Rband.

Comparing the properties of dwarf galaxies in both the core and outskirts of the Perseus Cluster, Penny and Conselice [40] found a clear correlation between massto-light ratio and the luminosity of the dwarfs, such that the faintest dwarfs require the largest fractions of dark matter to remain bound. This is to be expected, as the fainter a galaxy is, the less luminous mass it will contain, therefore the higher its dark matter content must be to prevent its disruption. Dwarfs are more easily influenced by their environment than more massive galaxies.

The distance to the Perseus Cluster prevents an easy determination of, so S. J. Penny and C. J. Conselice [40] instead determined the dark matter content of the dwarfs by calculating the minimum mass needed in order to prevent tidal disruption by the cluster potential, using their sizes, the projected distance from the cluster center to each dwarf and the mass of the cluster interior. Three of 15 dwarfs turned out to have mass-to-light ratios smaller than 3, indicating that they do not require dark matter.

Ultra-compact dwarf galaxies (UCDs) are stellar systems with masses of around 10⁷ - 10⁸ M_sun and half mass radii of 10 - 100 pc. A remarkable properties of UCDs is that their dynamical mass-to-light ratios are on average about twice as large as those of globular clusters of comparable metallicity, and also tend to be larger than what one would expect based on simple stellar evolution models. UCDs appear to contain very little or no dark matter.

H. Baumgardt and S. Mieske [41] have presented collisional N-body simulations which study the coevolution of a system composed of stars and dark matter. They find that DM gets removed from the central regions of such systems due to dynamical friction and mass segregation of stars. The friction timescale is significantly shorter than a Hubble time for typical globular clusters, while most UCDs have friction times much longer than a Hubble time. Therefore, a significant dark matter fraction remains within the half-mass radius of present-day UCDs, making dark matter a viable explanation for their elevated mass-to-light ratios.

A different type of systems are the ultra-faint dwarf galaxies (UFDs). When interpreted as steady state objects in virial equilibrium by V. Belokurov [42], would be the most DM dominated objects known in the Universe. Their half-light radii range from 70 pc to 320 pc.

A special case is the ultra-faint dwarf disk galaxy Segue 1 studied by M. Xiang-Gruess et al. [43] which has a baryon mass of only about 1000 solar masses. One interpretation is that this is a thin non-rotating stellar disk not accompanied by a gas disk, embedded in an axisymmetric DM halo and with a ratio. But if the disk rotates, f could be as high as 2000. If Segue 1 also has a magnetized gas disk, the dark matter halo has to confine the effective pressure in the stellar disk and the magnetic Lorentz force in the gas disk as well as possible rotation. Then f could be very large [43].

There are examples of groups formed by a small number of galaxies which are enveloped in a large cloud of hot gas (ICM), visible by its X-ray emission. One may assume that the electron density distribution associated with the X-ray brightness is in hydrostatic equilibrium, and one can extract the ICM radial density profiles by fits.

The amount of matter in the form of hot gas can be deduced from the intensity of this radiation. Adding the gas mass to the observed luminous matter, the total amount of baryonic matter, , can be estimated, see M. Markevitch et al. [44] and C. De Boni and G. Bertin [45]. In clusters studied, the gas fraction increases with the distance from the center; the dark matter appears more concentrated than the visible matter.

The temperature of the gas depends on the strength of the gravitational field, from which the total amount of gravitating matter, , in the system can be deduced. In many such small galaxy groups one finds , testifying to a dark halo present. An accurate estimate of requires that also dark energy is taken into account, because it reduces the strength of the gravitational potential. There are sometimes doubts whether all galaxies appearing near these groups are physical members. If not, they will artificially increase the velocity scatter and thus lead to larger virial masses.

On the scale of large clusters of galaxies like the Coma, it is generally observed that DM represents about 85% of the total mass and that the visible matter is mostly in the form of a hot ICM.

If galaxy formation is a local process, then on large scales galaxies must trace mass. This requires the study of how galaxies populate DM halos. In simulations one attempts to track galaxy and DM halo evolution across cosmic time in a physically consistent way, providing positions, velocities, star formation histories and other physical properties for the galaxy populations of interest.

Guo et al. [46] use abundance matching arguments to derive an accurate relation between galaxy stellar mass and DM halo mass. They combine a stellar mass function based on spectroscopic observations with a precise halo/ subhalo mass function obtained from simulations. Assuming this stellar mass-halo mass relation to be unique and monotonic, they compare it with direct observational estimates of the mean mass of halos surrounding galaxies of given stellar mass inferred from gravitational lensing and satellite galaxy dynamics data, and use it to populate halos in simulations. The stellar mass-halo mass relation is shown in Figure 10.

The implied spatial clustering of stellar mass turns out to be in remarkably good agreement with a direct and precise measurement. By comparing the galaxy autocorrelation function with the total mass autocorrelation function, as averaged over the Local Supercluster (LSC) volume, one concludes that a large amount of matter in the LSC is dark.

A similar study is that of Boyarsky et al. [47] who find a universal relation between DM column density and DM halo mass, satisfied by matter distributions at all observable scales in halo sizes from 10⁸ to 10¹⁶ M_sun, as shown in Figure 11. Such a universal property is difficult to explain without dark matter.

The tight coupling between radiation and matter density before decoupling caused the primordial adiabatic perturbations to oscillate in phase. Beginning from the time of last scattering, the receding horizon has been revealing these frozen density perturbations, setting up a pattern of standing acoustic waves in the baryon—photon fluid. After decoupling, this pattern is visible today as temperature anisotropies with a certain regularity across the sky.

The primordial photons are polarized by the anisotropic Thomson scattering process, but as long as the photons continue to meet free electrons their polarization is washed out, and no net polarization is produced. At a photon’s last scattering however, the induced polarization remains and the subsequently free-streaming photon possesses a quadrupole moment.

Temperature and polarization fluctuations are analyzed in terms of multipole components or powers. The resulting distribution of powers versus multipole, or multipole moment, is the power spectrum which exhibits conspicuous Doppler peaks. In Figure 12 we display the radiation temperature (TT) and temperature— E-polarization correlation (TE) power spectra from the 7-year data of WMAP as functions of multipole moments [49]. The spectra can then be compared to theory, and

theoretical parameters determined. Many experiments have determined the power spectra, so a wealth of data exists.

Baryonic matter feels attractive self-gravity and is pressure-supported, whereas dark matter only feels attractive self-gravity, but is pressureless. Thus the Doppler peaks in the CMBR power spectrum testify about baryonic and dark matter, whereas the troughs testify about rarefaction caused by the baryonic pressure. The position of the first peak determines. Combining the TT data with determinations of the Hubble constant h, the WMAP team can determine the total mass density parameter. The ratio of amplitudes of the

second-to-first Doppler peaks determines the baryonic density parameter to be and the dark matter component to be [49], thus.

Power spectra at higher multipole moments have been measured with the Atacama Cosmology Telescope (ACT) [50] at 148 GHz and 218 GHz, as well as the crossfrequency spectrum between these two channels. and found to be in agreement with the 7-year WMAP 94 GHz maps in the common range. The ACT has also been able to measure the lensing of the CMB signal at a significance of, which slightly smooths out the acoustic peaks, Figure 13.

In a fit of the flat CDM model to the data the dark matter density parameter comes out slightly higher than WMAP and the baryonic density slightly lower so the total density parameter for WMAP and ACT added is [51].

Information on the TE correlations comes from several measurements, among them WMAP [49], and on the E-mode polarization power spectrum alone (EE) from the QUAD collaboration [52], Figure 14.

The results show two surprises: Firstly, since, a large component is missing, of unknown nature, and termed dark energy. The second surprise is that ordinary baryonic matter is only a small fraction of the total matter budget. The remainder is then dark matter, of unknown composition. Of the 4.5% of baryons in

the Universe only about 1% is stars.

A cornerstone of cosmology is the Copernican principle, that matter in the Universe is distributed homogeneously, if only on the largest scales of superclusters separated by voids. On smaller scales we observe inhomogeneities in the forms of galaxies, galaxy groups, and clusters. The common approach to this situation is to turn to nonrelativistic hydrodynamics and treat matter in the Universe as an adiabatic, viscous, non-static fluid, in which random fluctuations around the mean density appear, manifested by compressions in some regions and rarefactions in other. The origin of these density fluctuations was the tight coupling established before decoupling between radiation and charged matter density, causing them to oscillate in phase. An ordinary fluid is dominated by the material pressure, but in the fluid of our Universe three effects are competing: gravitational attraction, density dilution due to the Hubble flow, and radiation pressure felt by charged particles only.

The inflationary fluctuations crossed the post-inflationary Hubble radius, to come back into vision with a wavelength corresponding to the size of the Hubble radius at that moment. At time the overdensities began to amplify and grow into larger inhomogeneities. In overdense regions where the gravitational forces dominate, matter contracts locally and attracts surrounding matter, becoming increasingly unstable until it eventually collapses into a gravitationally bound object. In regions where the pressure forces dominate, the fluctuations move with constant amplitude as sound waves in the fluid, transporting energy from one region of space to another.

Inflationary models predict that the primordial mass density fluctuations should be adiabatic, Gaussian, and exhibit the same scale invariance as the CMB fluctu-

ations. The baryonic acoustic oscillations can be treated similarly to CMB, they are specified by the dimensionless mass autocorrelation function which is the Fourier transform of the power spectrum of a spherical harmonic expansion. The power spectrum is shown in Figure 15 [53].

As the Universe approached decoupling, the photon mean free path increased and radiation could diffuse from overdense regions into underdense ones, thereby smoothing out any inhomogeneities in the plasma. The situation changed dramatically at recombination, at time 380,000 yr after Big Bang, when all the free electrons suddenly disappeared, captured into atomic Bohr orbits, and the radiation pressure almost vanished. Now the baryon acoustic waves and the CMB continued to oscillate independently, but adiabatically, and the density perturbations which had entered the Hubble radius since then could grow with full vigor into baryonic structures.

The scale of BAO depends on and on the Hubble constant, h, so one needs information on h to break the degeneracy. The result is then. In the ratio the h-dependence cancels out, so one can also quantify the amount of DM on very large scales by.

We have seen in Section 10 that the baryonic density parameter, , is very small. The critical density is determined by the expansion speed of the Universe, and the mean baryonic density of the Universe (stars, interstellar and intergalactic gas) is only [49].

The question arises whether the galaxies could have formed from primordial density fluctuations in a purely baryonic medium. We have also noted, that the fluctuations in CMB and BAO maintain adiabaticity. The amplitude of the primordial baryon density fluctuations

would have needed to be very large in order to form the observed number of galaxies. But then the amplitude of the CMB fluctuations would also have been very large, leading to intolerably large CMB anisotropies today. Thus galaxy formation in purely baryonic matter is ruled out by this argument alone.

Thus one concludes, that the galaxies could only have been formed in the presence of gravitating dark matter which started to fluctuate early, unhindered by radiation pressure. This conclusion is further strengthened in the next Section.

In the ΛCDM paradigm, the nonlinear growth of DM structure is a well-posed problem where both the initial conditions and the evolution equations are known (at least when the effects of the baryons can be neglected).

The Aquarius Project [54] is a Virgo Consortium program to carry out high-resolution DM simulations of Milky-Way—sized halos in the ΛCDM cosmology. This project seeks clues to the formation of galaxies and to the nature of the dark matter by designing strategies for exploring the formation of our Galaxy and its luminous and dark satellites.

The galaxy population on scales from 50 kpc to the size of the observable Universe has been predicted by hierarchical ΛCDM scenarios, and compared directly with a wide array of observations. So far, the ΛCDM paradigm has passed these tests successfully, particularly those that consider the large-scale matter distribution and has led to the discovery of a universal internal structure for DM halos. As was noted in Section 12, the observed structure of galaxies, clusters and superclusters, as illustrated by Figure 16, could not have formed in a baryonic medium devoid of dark matter.

Given this success, it is important to test ΛCDM predictions also on smaller scales, not least because these are sensitive to the nature of the dark matter. Indeed, a number of serious challenges to the paradigm have emerged on the scale of individual galaxies and their central structure. In particular, the abundance of small DM subhalos predicted within CDM halos is much larger than the number of known satellite galaxies surrounding the Milky Way (M. Boylan-Konchin et al. [48] and references therein).

In Section 10 we have seen that the WMAP 7-year CMB data together with the Hubble constant value testify about the existence of DM [49,51]. In Section 11 we addressed the BAO data [53] with the same conclusion. In overall fits one combines these with supernova data (SN Ia) which offer a constraint nearly orthogonal to that of CMB in the -plane. The Union compilation of 307 selected SN Ia includes the recent large samples of SNe Ia from the Supernova Legacy Survey, the ESSENCE Survey, the older data sets, as well as the recently extended data set of distant supernovae observed with HST. M. Kowalski et al. [55] present the latest results from this compilation and discuss the cosmological constraints and its combination with CMB and BAO measurements. The CMB constraint is close to the line, whereas the supernova constraint is close to the line. The BAO data constrain, but hardly at all. This is shown in Figure 17.

Defining the vacuum energy density parameter by, a flat Universe corresponds to. For a non-flat Universe with a cosmological constant responsible for dark energy, a simultaneous fit to the data sets gives

where the first error is statistical and the second error systematic. Clearly one notes that the Universe is consistent with being flat. Subtracting from one obtains the density parameter for DM,. Assuming flatness, M. Kowalski et al. [55] find. This compares well

with the combined 7-year WMAP data and the ACT data, [51]. If one fits different models having more free parameters, one gets slightly different results, but all within these errors.

In isolated galaxies and galaxy clusters all matter components contributing to the common gravitational potential are more or less centrally-symmetrically coincident. This makes the dissection of DM from the baryonic components difficult and dependent on parametrization, as we have discussed in Section 3. In merging galaxy clusters however, the separate distributions of galaxies, intracluster gas and DM may become spatially segregated permitting separate observations. The visually observable galaxies behave as collisionless particles, the baryonic intracluster plasma is fluid-like, experiences ram pressure and emits X-rays, but non-interacting DM does not feel that pressure, it only makes itself felt by its contribution to the common gravitational potential.

Major cluster mergers are the most energetic events in the Universe since the Big Bang. Shock fronts in the intracluster gas are the key observational tools in the study of these systems. When a subcluster traverses a larger cluster it cannot be treated as a solid body with constant mass moving at constant velocity. During its passage through the gravitational potential of the main cluster it is shrinking over time, stripped of gas envelope and decelerating. Depending on the ratio of the cluster masses, the gas forms a bow shock in front of the main cluster, and this can even be reversed at the time when the potentials coincide.

We shall now meet several examples of galaxy cluster mergers where the presence of DM could be inferred from the separation of the gravitational potential from the position of the radiating plasma.

What we have termed “dark matter” is generic for observed gravitational effects on all scales: galaxies, small and large galaxy groups, clusters and superclusters, CMB anisotropies over the full horizon, baryonic oscillations over large scales, and cosmic shear in the large-scale matter distribution. The correct explanation or nature of dark matter is not known, whether it implies unconventional particles or modifications to gravitational theory. but gravitational effects have convincingly proved its existence in some form.

The few per cent of the mass of the Universe found as baryonic matter in stars and dust clouds is well accounted for by nucleosynthesis. If there exist particles which were very slow at time when galaxy formation started, they could be candidates for cold dark matter. They must have become non-relativistic much earlier than the leptons, and then decoupled from the hot plasma.

Whenever laboratory searches discover a new particle, it must pass several tests in order to be considered a

viable DM candidate: it must be neutral, compatible with constraints on self-interactions (essentially collisionless), consistent with Big Bang nucleosynthesis, and match the appropriate relic density. It must be consistent with direct DM searches and gamma-ray constraints, it must leave stellar evolution unchanged, and be compatible with other astrophysical bounds.

The total dynamical mass of an astronomical system is derivable from the velocity dispersions or the rotation velocities of its components via the use of the Virial Theorem or Kepler’s law, respectively. A most important probe is strong gravitational lensing which measures the total mass, but also weak lensing, the oscillations in the Cosmic Microwave Background and in the ambient baryonic medium. Probes separating dark matter from total matter require in addition observations of visible light, infrared radiation, X-rays, the Sunyaev-Zel’dovich effect, and supernovae. Depending on the system under study there are many ways to combine these tools using empirical halo models, simulating stellar population models and galaxy formation models, comparing mass-tolight ratios and mass autocorrelation functions. The most remarkable systems are merging galaxy clusters which, by their motion, separate non-collisional dark matter from optically visible galaxies and hot, radiating gas.

Regardless of the nature of dark matter, all theories attempting to explain it share the burden to explain the gravitational effects described in here. Thus there remains much to be done.

I am grateful to Sylvain Fouquet, Carmen RodriguezGonzalvez, and Will Dawson for clarifying comments and addenda.