Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

88 Observational Properties of the Universe

4.5.2 Galaxies

Let us now explain in a little more detail how we arrive at the estimate Ωg given in Equation (4.5.2). We proceed by calculating the mean luminosity per unit volume produced by galaxies, together with the mean value of M/L, the mass-to-light ratio, of the galaxies. Thus,

The value Lg can be obtained from the luminosity function of the galaxies, Φ(L). This function is defined such that the number of galaxies per unit volume with luminosity in the range L to L + dL is given by

The best fit to the observed properties of galaxies is a orded by the Schechter function

where the parameters are, approximately, Φ 10−2h3 Mpc−3, L 1010h−2L and α 1. The value of Lg that results is therefore

To derive the mass-to-light ratio M/L we must somehow measure the value of M. One can calculate the mass of a spiral galaxy if one knows the behaviour of the orbital rotation velocity of stars with distance from the centre of the galaxy, the rotation curve. One compares the observed curve with a theoretical model in which the rotation curve is produced by a distribution of gravitating material. There is strong evidence from 21 cm radio and optical observations that the rotation curves of spiral galaxies remains flat well outside the region in which most of the luminous material resides. This demonstrates that spiral galaxies possess large ‘haloes’ of dark matter, concerning the nature of which there is a huge debate. Some of the possibilities are neutral hydrogen gas, white dwarfs, massive planets, black holes, massive neutrinos and exotic particles, like for instance photinos. The mass of these haloes is thought to be between 3 and 10 times the mass of the luminous component of the galaxy.

Elliptical and S0 galaxies do not have such ordered orbital motions as spiral galaxies, so one cannot use rotation curves. One uses instead the virial theorem:

where the mean kinetic energy Ek is estimated from the velocity dispersion of the stars and the potential energy U is estimated from the size and shape of the galaxy. The typical value of M/L one obtains is

This should probably be regarded as a lower limit on the contribution due to galaxies because it refers only to the luminous part and does not take account of the full extent of the dark haloes.

4.5.3 Clusters of galaxies

Using the virial theorem we can also estimate the mass of groups and clusters of galaxies. This method is particularly useful for rich clusters of galaxies like the Coma and Virgo clusters. The kinetic energy can be estimated from the velocity dispersion of the galaxies in the cluster

Mcl is the total mass of the cluster and vr2 1/2 is the line-of-sight velocity dispersion of the galaxies. The potential energy is given by

GM2

U − cl , (4.5.20)

Rcl

where Rcl is the radius of the cluster which can be estimated from a model of its density profile. One typically obtains from this type of analysis values of order

A more sophisticated approach involves more detailed modelling of the velocities within the cluster:

This gives the mass contained within a radius r in terms of the density profile ρ(r) and the two independent velocity dispersions in the radial and tangential directions σr2 and σt2; the quantity

90 Observational Properties of the Universe

is a measure of the anisotropy of the radial velocity dispersion. In order to use this equation, one needs to know the profile of galaxies and velocity dispersion as a function of radius from the centre of the cluster. In reality, one can only measure the projected versions of these quantities, so the problem is formally indeterminate. One can, however, use a modelling procedure to perform an inversion of the projected profiles. For the Coma cluster, the result is a total dynamically inferred mass within an Abell radius of

which corresponds to a value of M/L 320h. Galaxies themselves therefore contribute only about 15% of the mass of the Coma cluster.

This value can be compared with two alternative determinations of cluster masses. One of these takes account of the fact that rich clusters of galaxies are permeated by a tenuous gaseous atmosphere of X-ray emitting gas. Since the temperature and density profiles of the gas can be obtained with X-ray telescopes such as ROSAT and data on the X-ray spectrum of these objects is also often available, one can break the indeterminacy of the modelling method. The X-ray data also have the advantage that they are not susceptible to Poisson errors coming from the relatively small number of galaxies that exist at a given radius. Assuming the cluster is spherically symmetric and considering only the gaseous component, for simplicity, the equation of hydrostatic equilibrium becomes

µ is the mean molecular weight of the gas. The procedure adopted is generally to use trial functions for M(r) in order to obtain consistency with T(r) and the spectrum data.

Good X-ray data from ROSAT have been used to model the gas distribution in the Coma cluster (Briel et al. 1992) with the result that

for the mass inside the Abell radius. The gas contributes more than the galaxies, but is still less than the total mass.

The third method for obtaining cluster masses is to use gravitational lensing. We discuss this later, in Chapter 19. Generally speaking, all three of these methods give cluster masses of the same order of magnitude, although they do not agree in all details.

Given that there are approximately 4 × 103 large clusters of galaxies within a distance of 6 ×102h−1 Mpc from the Local Group, the density of matter produced by such clusters is roughly

which is of the same order as ρ0g given by Equation (4.5.17). The reason for this is not that virtually all galaxies reside in such clusters, which they certainly do

not, but that the ratio M/L for the matter in clusters is much higher than that for individual galaxies. In fact this ratio is of order 300M /L , roughly a factor of ten greater than that of galaxies. This discrepancy is the origin of the so-called ‘hidden mass problem’ in galaxy clusters, namely that there seems to be matter there in some unknown form.

If the value of M/L for galaxies were to be reconciled with the galactic value, one would have to have systematically overestimated the virial mass of the cluster. This might happen if the cluster were not gravitationally bound and virialised, but instead were still freely expanding with the background cosmology. In such a case we would have

and, therefore, a smaller total mass. However, we would expect the cluster to disperse on a characteristic timescale tc lc/ v2 1/2, where lc is a representative length scale for the cluster and v2 1/2 is the root-mean-square peculiar velocity of the galaxies in the cluster; for the Coma cluster tc 1/16H0 and it is generally the case that tc for clusters is much less than a Hubble time. If the clusters we observe were formed in a continuous fashion during the expansion of the Universe, many such clusters must have already dispersed in this way. The space between clusters should therefore contain galaxies of the type usually found in clusters, i.e. elliptical and lenticular galaxies, and they might be expected to have large peculiar motions. One observes, however, that ‘field’ galaxies are usually spirals and they do not have particularly large peculiar velocities. It seems reasonable therefore to conclude that clusters must be bound objects.

In light of this, it is necessary to postulate the existence of some component of dark matter (matter with a large value of M/L) to explain the virial masses of galaxy clusters. It is known from X-ray observations of clusters that a large fraction of the mass is in the form of hot gas. In particular, an analysis by White et al. (1993b) of the ubiquitous Coma cluster, in conjunction with Equation (4.5.9), indicates that, if the ratio of baryonic matter to total gravitating matter in Coma is representative of the global ratio, then one can constrain Ω to be

which is less than unity for most sensible values of h. It seems, however, that this hot gas component is not su cient to explain the dynamical mass; another component is needed. This component is probably collisionless and could in principle be in the form of cometary or asteroidal material, large planets (Jupiter-like objects), low-mass stars (brown dwarfs), or even black holes. There are problems, however, in reconciling the value of Ωdyn with nucleosynthesis predictions if all the cluster mass were baryonic. A favoured option is that at least some of this material is in the form of weakly interacting non-baryonic particles (photinos, axions, neutrinos, etc.) left over after the Big Bang. It is even possible, as we shall explain in Section 4.7, that these particles actually constitute the dominant contribution to Ω globally, not just in cluster cores. This is an attractive notion because,

92 Observational Properties of the Universe

as we shall see, a universe with Ω 1 dominated by non-baryonic matter has some advantages when it comes to explaining the formation of galaxies and large-scale structure. The existence of such a high density of non-baryonic matter would not contradict nucleosynthesis because the weakly interacting matter would not be involved in nuclear reactions in the early Universe. Modern inflationary cosmologies also favour Ω0 1 for theoretical reasons and it is often argued that if the Universe turned out to have Ω 1, this could be construed as evidence for inflation. There is not much evidence that Ω0 1, but we can say that it is (probably) at least Ω0 0.2.

4.6 Deviations from the Hubble Expansion

In the previous section we showed how one can use virial arguments relating velocities to gravitating mass in order to estimate masses from velocity data. The logical extension of this type of argument is to attempt to explain the peculiar motions of galaxies with respect to the Hubble expansion as being due to the cosmological distribution of mass. This idea is of great current interest but the arguments are more technical than we can accommodate in this introductory section; details are given in Chapter 18. We can nevertheless introduce some of the ideas here to whet the reader’s appetite.

The (radial) peculiar velocity of a galaxy is defined to be the di erence between the galaxy’s total measured radial velocity vr (obtained from the redshift) and the expected Hubble recession velocity for a galaxy at distance d from the observer:

Obviously, knowledge of vp requires both the redshift and an independent measurement of distance to the galaxy. The latter is not easy to acquire, so the construction of catalogues of peculiar motions is not a simple task. Nevertheless, some properties of the local flow pattern of galaxies are known. The motion of our Local Group of galaxies towards the Virgo cluster has been known for some time to be v 250±50 km s−1 and, as we shall see in Section 4.8, it is possible to estimate our velocity with respect to the reference frame in which the cosmic microwave background is at rest: v 550 ± 40 km s−1 in a direction α = 10.7 ± 0.3 h and δ = −22 ± 5◦, 44◦ away from the Virgo cluster. For reasons we shall explain later, one expects the resultant velocity of the Local Group to lie in the same direction as the net gravitational acceleration on it produced by the distribution of matter around it. Clearly then, our velocity with respect to the microwave background is not explained by the action of the Virgo cluster. In fact, studies of galaxy-peculiar motions show that the peculiar flow of galaxies is actually coherent over a large scale. A region of radius 50h−1 Mpc centred on the Local Group seems to be moving en masse in a direction corresponding to the Hydra and Centaurus clusters with a velocity of v 600 km s−1. It was thought that this bulk flow was due to the action of a huge concentration of mass at a distance of order 50h−1 Mpc from the Local Group, called the Great Attractor, but it is now generally accepted that

the pull is not due to a single mass but to the concerted e ort of a large number of clusters.

So how can the observed peculiar motions tell us about the distribution of mass and, in particular, the total density? The arguments rely on the theory of gravitational instability which we shall explain later, but a qualitative example can be given here based on the motion of the Local Group with respect to the Virgo cluster. One takes this motion to be the result of ‘infall’, which can be modelled by a simple linear model in which a ‘shell’ of galaxies containing the Local Group falls symmetrically onto the Virgo cluster, which is assumed to be spherical. If the density of galaxies in the Virgo cluster is a factor (1 + ∆g) higher than the cosmological average, the infall velocity is vLG, and the Virgocentric distance of the Local Group is rLG, then one can estimate

This type of argument leads one to a value of Ωdyn which is consistent with that obtained from virial arguments in clusters, i.e. Ωdyn 0.2–0.4. More recent analyses using data covering much larger scales give results apparently consistent with Ωdyn = 1 though with a great uncertainty.

One of the problems with analyses of this type is that one has to estimate the density fluctuation ∆g producing the peculiar motion. In the example this is estimated as the excess density of galaxies inside the cluster compared with the ‘field’. Given that much of the mass one detects is dark, there is no reason a priori why the fluctuation in mass density ∆m has to be the same as the fluctuation in number density of galaxies ∆g. If these di er by a factor b, then, according to Equation (4.6.2), one’s estimate of Ωdyn is wrong by a factor b1.7. The idea that galaxies might not trace the mass is usually called biased galaxy formation and it considerably complicates the analysis of galaxy clustering and peculiar motion studies; we discuss bias in detail in Section 14.8. Note that a value of b 2 can reconcile the Virgocentric flow with Ω = 1.

A more accurate determination of the anisotropy of the Hubble expansion on large scales allows the construction of a map of the peculiar velocity field, which, as we shall see in Chapter 18, is an important goal of modern observational cosmology. It is hoped that such a map will allow an accurate determination of the distribution of matter in the Universe, even if galaxies are biased tracers of the mass. The reason for this optimism is that all matter components exert gravity and react to it, not just the component of luminous matter which appears in galaxies. Regardless of how a galaxy forms and what it is made of, its motion is due to the action of all the gravitating mass around it. Modern theoretical developments, as well as new observational techniques for measuring distances to galaxies, give good grounds for believing that this is a reasonable task.

We should also take this opportunity to make some more formal comments about the nature of deviations from the Hubble flow in the context of the Cosmological Principle. Deviations of the type (4.6.1) can be regarded as being due to an

with respect to a coordinate origin at our Galaxy. We discussed this in the context of globally anisotropic models in Chapter 3. The tensor Hαβ is called the Hubble tensor and can be written in the form

where δαβ is the Kronecker symbol, ωαβ is an antisymmetric tensor which represents a rotation (ωαβ = −ωβα), and σαβ is a symmetric traceless tensor which represents shear (σαβ = σβα; σαα = 0). The constant H is the familiar Hubble constant.

The only observable quantity is the line-of-sight velocity vr

where the nα are the direction cosines of a distant galaxy at d. It is found that the contribution to the shear σαβ from massive distant clusters is of the order of 10%. In fact, by considering a large-redshift sample of distant clusters, one can find a coordinate system in which σαβ is diagonal; in this system one finds that

This provides some evidence for the Cosmological Principle.

4.7 Classical Cosmology

In the early days of observational cosmology, much emphasis was placed on the geometrical properties of expanding-universe models as tools for estimating parameters of the cosmological models. Indeed, famous articles by Sandage (1968, 1970) called ‘Cosmology: the search for two numbers’ reduced all cosmology to the task of determining H0 and q0, the deceleration parameter. Remember that, at a generic time t the deceleration parameter is defined by

as usual, the zero su x means that q0 is defined at the present time. Matterdominated models with vanishing Λ have

so the parameters q0 and Ω0 are essentially equivalent. If there is a cosmological constant contributing towards the spatial curvature, however, we have the general relation

In the case where ΩΛ + Ω0 = 1 (κ = 0) we have q0 < 0 for Ω0 < 23 .

The parameters H0 and q0 thus furnish a general description of the expansion of a cosmological model: these are Sandage’s famous ‘two numbers’. Their importance is demonstrated in standard cosmology textbooks (Weinberg 1972; Peebles 1993; Narlikar 1993; Peacock 1999), which show how the various observational relationships, such as the angular diameter–redshift and apparent magnitude– redshift relations for standard sources, can be expressed in simple forms using these parameters and the Robertson–Walker metric. In the standard Friedmann– Robertson–Walker models, the apparent flux density and angular size of a standard light source or standard rod depend in a relatively simple way on q0 (Hoyle 1959; Sandage 1961, 1968, 1970, 1988; Weinberg 1972), but the relationships are more complex if the cosmological constant term is included (e.g. Charlton and Turner 1987).

During the 1960s and early 1970s, a tremendous e ort was made to determine the deceleration parameter q0 from the magnitude–redshift diagram. For a while, the preferred value was q0 1 (Sandage 1968) but eventually the e ort died away when it was realised that evolutionary e ects dominated the observations; no adequate theory of galaxy evolution is available that could enable one to determine the true value of q0 from the observations. To a large extent this is the state of play now, although the use of the angular size–redshift and, in particular, the magnitude–redshift relation for Type Ia supernovae have seen something of a renaissance of this method. We shall therefore discuss only the recent developments in the subsequent sections.

4.7.1 Standard candles

The fundamental property required here is the luminosity distance of a source, which, for models with p = Λ = 0, is given by

this relationship is simply defined in terms of the intrinsic luminosity of the source L and the flux l received by an observer using the Euclidean relation

One usually seeks to exploit this dependence by plotting the so-called ‘Hubble diagram’ of apparent magnitude against redshift for objects of known intrinsic luminosity: this boils down to plotting log l against z, hence the dependence on dL.

The problem with exploiting such relations to prove the value of q0 directly is that one needs to have a standard ‘candle’: an object of known intrinsic luminosity. The dearth of classes of object suitable for this task is, of course, one of the reasons why the Hubble constant is so poorly known locally. If it were not for recent developments based on one particular type of object – Type Ia supernovae – we would have been inclined to have omitted this section entirely. As it is now,

96 Observational Properties of the Universe

Figure 4.8 The magnitude–redshift diagram for high-redshift supernovae measured by two independent groups. The data show a preference for models with a contribution from Λ. Picture courtesy of Bob Kirschner.

we consider that these sources o er the most exciting prospects for classical cosmology within the next few years.

The homogeneity and extremely high luminosity of the peak magnitudes of Type Ia supernovae, along with physical arguments as to why they should be standard sources, have made these attractive objects for observational cosmologists in recent years (e.g. Branch and Tammann 1992), though the use of supernovae has been discussed before, for example, by Sandage (1961). The current progress

stems from the realisation that these objects are not in fact identical, but form a family which can nevertheless be mapped onto a standard object by using independent observations. Correlations between peak magnitude and the shape of the light curve (Hamuy et al. 1995; Riess et al. 1995) or spectral features (Nugent et al. 1995) have reduced the systematic variations in peak brightness to about two-tenths of a magnitude. The great advantages of these objects are

1.because their behaviour depends only on the local physics, they are expected to be independent of environment and evolution and so are good candidates for standard candles, and

2.that they are bright enough to be seen at quite high redshifts, where the dependence on cosmological parameters (4.7.4) is appreciable.

Two teams are pursuing the goal of measuring cosmological parameters using Type Ia supernovae. Originally, results seemed to suggest a measurement of positive q0, but more recently it has become apparent that the high-redshift supernovae may be fainter, i.e. be at larger luminosity distance, for a given z than is compatible with q0 > 0. If these measurements are being interpreted correctly, and there is as yet no reason to believe they are not, this is compelling evidence for a cosmological constant.

4.7.2 Angular sizes

The angle subtended by a standard metric ‘rod’ behaves in an interesting fashion as its distance from the observer is increased in standard cosmologies. It first decreases, as expected, then reaches a minimum after which it increases again (Sandage 1961). The position of the minimum depends upon q0 (Ellis and Tivon 1985; Janis 1986). This somewhat paradoxical behaviour can be more easily understood by remembering that the light from very-high-redshift objects was emitted a long time ago when the proper distance to the object would have been much smaller than it is at the present epoch. Given appropriate dynamics, therefore, it is quite possible that distant objects appear larger than nearby ones with the same physical size.

For models with Λ = 0 the relationship between angular diameter θ and redshift z for objects moving with the Hubble expansion and with a fixed metric diameter d is simply

where DL(z) is the luminosity distance given by Equation (4.7.4).

As with the standard candles, astronomers are generally not equipped with standard sources they are able to place at arbitrarily large distances. To try to use this method, one must select galaxies or other sources and hope that the intrinsic properties of the objects selected do not change with their distance from the observer. Because light travels with a finite speed, more distant objects emitted their light further in the past than nearby objects. Lacking an explicit theory of