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

The window function WV must be chosen to model the way the sample is constructed. This is not completely straightforward because the observational selection criteria are not always well controlled and the results are quite sensitive to the shape of the window function. Top hat (13.3.14) and Gaussian (13.3.15) are the usual choices in this case, as for the density field.

Because the integral in Equation (18.3.4) is weighted towards lower k than the definition of σM2 given by Equation (13.3.8), which has an extra factor of k2, bulk flows are potentially useful for probing the linear regime of P(k) beyond what can be reached using properties of the spatial clustering of galaxies. The problem is that one typically has one measurement of the bulk flow on a scale R and this does not provide a strong constraint on σV or P(k), as is obvious from Equation (18.3.5): if a theory predicts an RMS bulk flow of 300 km s−1 on some scale, then a randomly selected sphere on that scale can have a velocity between 100 and 480 km s−1 with 90% probability, an allowed error range of a factor of almost five. Until much more data become available, therefore, such measurements can only be used as a consistency check on models and do not strongly discriminate between them. Velocities can, however, place constraints on the possible existence of bias since σV is simply proportional to b (in the linear bias model). For example, the standard CDM model predicts a bulk flow on the scale of 40h−1 Mpc of around 180 km s−1 if b = 1. This reduces to 72 km s−1 if b = 2.5, which was, at one time, the favoured value. The observation of a velocity of 388 km s−1 on this scale is clearly incompatible with SCDM with this level of bias; it is, however, compatible with a b = 1 CDM model.

It is also pertinent to mention that the factor f in Equation (18.3.4) means that high values of V tend to favour higher values of f and therefore higher values of Ω, remembering that f Ω0.6. We return to this in Section 18.6.

There is an interesting way to combine large-scale bulk flow information with small-scale velocity data. Let us consider the unsmoothed velocity field V(x; 0). In fact, some smoothing of the velocity field is always necessary because of the sparseness of the velocity field data, but we can assume that this scale, RS, is so much less than R that its value is e ectively zero. Consider the quantity

where the average is taken over a single smoothing window centred at x0. Clearly this represents the variance of the unsmoothed velocity field calculated with respect to the mean value of the velocity in the window, V(x0; R). The ratio

measures, in some sense, the ‘temperature’ of the velocity field on a scale R. If M2 > 1, then the systematic bulk flow in the smoothing volume exceeds the random motions. If, on the other hand, M2 < 1, these small-scale random ‘thermal’ motions are larger than the systematic flow. It is appropriate therefore to regard the spatial average of the quantity M2,

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as defining a kind of cosmic Mach number as a function of scale, M(R) (Ostriker and Suto 1990). In fact, the usual definition of the cosmic Mach number is slightly di erent from that given in Equation (18.3.8) and is more straightforward to calculate:

σ2(R)

M2(R) = V , (18.3.9)

ΣV2 (R)

where ΣV2 (R) is the spatial average of ΣV2 (x0; R) taken over all positions x0, by analogy with Equation (18.3.8).

The cosmic Mach number has the advantage that it probes the shape of the primordial power spectrum in a much more sensitive manner than the bulk flow statistics. Its main disadvantage is that M2 is defined in terms of the ratio of two quantities which are both subject to substantial observational uncertainties. Until the available peculiar velocity data improve, this statistic is therefore unlikely to provide a powerful test of structure-formation theories.

18.4 Velocity–Density Reconstruction

A more sophisticated approach to the use of velocity information is provided by a relatively new and extremely ingenious approach developed primarily by Bertschinger et al. (1990) which is now known as POTENT; see also Dekel et al. (1993). This makes use of the fact that in the linear theory of gravitational instability the velocity field is curl-free and can therefore be expressed as the gradient of a potential. We saw in Section 18.1, Equation (18.1.8), that this velocity potential turns out to be simply proportional to the linear theory value of the gravitational potential. Because the velocity field is the gradient of a potential ΦV , one can use the purely radial motions, Vr, revealed by redshift and distance information to map ΦV in three dimensions:

It is not required that paths of integration be radial, but they are in practice easier to deal with.

Once the potential has been mapped, one can solve for the density field using the Poisson equation in the form (18.1.7). This means therefore that one can compare the density field as reconstructed from the velocities with the density field measured directly from the counts of galaxies. This, in principle, enables one to determine directly the level of bias present in the data. The only other parameter involved in the relation between V and δ is then f, which, in turn, is a simple function of Ω. POTENT holds out the prospect, therefore, of supplying a measurement of Ω which is independent of b, unlike that discussed in Section 17.3 for example. We return to the estimation of Ω from velocity data in Section 18.6.

At this point, however, it is worth mentioning some of the possible problems with the POTENT analysis. As always, one is of course limited by the quality and

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order 12h−1 Mpc. If one uses clusters instead of individual galaxies, then σi can be reduced by a factor equal to the square root of the number of objects in the cluster, assuming the errors are random. One e ect of the heavy smoothing is that the volume probed by these studies consequently contains only a few independent smoothing volumes and the statistical significance of any reconstruction is bound to be poor.

Notice that the potential field one recovers then has to be di erentiated to produce the density field which will again exaggerate the level of noise. (It is possible to improve on the linear solution to the Poisson equation by using the Zel’dovich approximation (Section 14.2) to calculate the density perturbation δ from the velocity potential.) The scale of the noise problem can be gauged from the fact that a 20% distance error is of the same order as the typical peculiar velocity for distances beyond 30h−1 Mpc.

Apart from the problem of noise, there are also other sources of uncertainty in the applicability of this method. In any redshift survey one has to be careful to control selection biases, such as the Malmquist bias (Section 4.2), which can enter in a complicated and inhomogeneous way into this analysis. One also needs to believe that the distance indicators used are accurate. Most workers in this field claim that their distance indicators are accurate to, say, 10–20%. However, if the errors are not completely random, i.e. there is a systematic component which actually depends on the local density, then the results of this type of analysis can be seriously a ected. In this case the systematic error in V correlates with density in a similar way to that expected if the velocities were generated dynamically from density fluctuations. There are some suggestions that there is indeed such a systematic error in the commonly used Dn–σ indicator for elliptical galaxies (Guzman and Lucey 1993). What may happen is that old stellar populations produce a di erent response in the distance indicator compared with young ones. Since older galaxies formed earlier and in higher-density environments, the upshot is exactly the sort of systematic e ect that is so dangerous to methods like POTENT. Applying a corrected distance indicator to a sample of elliptical galaxies essentially eliminates all the observed peculiar motions, which means that the motions derived using the uncorrected indicator were completely spurious. Whether this type of error is su ciently widespread to a ect all peculiar motion studies is unclear but it suggests one should regard these results with some scepticism.

18.5 Redshift-Space Distortions

The methods we have discussed in Sections 18.2–18.4 of course require one to know peculiar motions for a sample of galaxies. There is an alternative approach, which does not need such information, and which may consequently be more reliable. This relies on the fact that peculiar motions a ect radial distances and not tangential ones. The distribution of galaxies in ‘redshift space’ is therefore a distorted representation of their distribution in real space. For example, dense clusters appear elongated along the line of sight because of the large radial-velocity

component of the peculiar velocities, an e ect known as the ‘fingers of God’. Similarly, the correlation functions and power spectra of galaxies should be expected to show a characteristic distortion when they are viewed in redshift space rather than in real space. This is the case even if the real-space distribution of matter is statistically homogeneous and isotropic.

Let us first consider the e ect of these distortions upon the two-point correlation function of galaxies. The conventional way to describe this phenomenon is to define coordinates as follows. Consider a pair of galaxies with measured redshifts corresponding to velocities v1 and v2. The separation in redshift space is then just

and the separations parallel and perpendicular to this direction are then just

When the correlation function is plotted in the π–rp plane, redshift distortions produce two e ects: a stretching of the contours of ξ along the π-axis on small scales (less than a few Mpc) due to nonlinear pairwise velocities, and compression along the π-axis on larger scale due to bulk (linear) motions.

Linear theory cannot be used to calculate the first of these contributions, so one has to use explicitly nonlinear methods. The usual approach is to use the equation

which expresses the conservation of particle pairs; x is a comoving coordinate and v12 = |s|. The Equation (18.5.5) is actually the first of an infinite set of equations known as the BBGKY hierarchy (Davis and Peebles 1977). To close the hierarchy one needs to make an assumption about higher moments. Assuming that the three-point correlation function has the hierarchical form (16.5.1) and that the real-space two-point correlation function is of the power-law form (16.4.5) leads to the so-called cosmic virial theorem:

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where Cγ 23.8 if γ = 1.8. Assuming that the radial anisotropy in ξ(rp, π) is due to the velocities v12, then one can, in principle, determine an estimate of Ω0 from the small-scale anisotropy. Notice, however, that there is an implicit assumption that the galaxy correlation function and the mass covariance function are identical, so this estimate will depend upon b in a non-trivial way.

On larger scales, the e ect of redshift-space distortions is in the opposite sense. One can understand this easily by realising that a large-scale overdensity will tend to be collapsing in real space. Matter will therefore be moving towards a cluster, thus flattening structures in the redshift direction. This both enhances the appearance of walls and filaments and changes their orientation, producing a series of ring-like structures around the observer called the ‘bull’s-eye e ect’ (Melott et al. 1998).

The e ect of these distortions upon the correlation function is actually quite complicated and depends upon the direction cosine µ between the line of sight l and the separation s. One can show, however, that the angle-averaged redshiftspace correlation function is given by the simple form

where ξr is the real-space correlation function (Kaiser 1987; Hamilton 1992). More instructively one can decompose ξ(rp, π) into spherical harmonics using

A robust diagnostic of the presence of redshift distortions is via the quadrupole- to-monopole ratio:

In principle, these ideas permit one to estimate Ω (through the f dependence), but this again requires that ξr for the matter should be known accurately. Fortunately, with the arrival of redshift surveys like the 2dF GRS such measurements can now be made with confidence (Peacock et al. 2001).

Another way to use redshift-space distortions in the linear regime is to study their e ect on the power spectrum, where the directional dependence is easier to calculate. In fact, one can show quite easily that

where Ps and Pr are the redshift space and real space power spectra, respectively (Kaiser 1987). If one can estimate the power spectrum in various directions of k, then one can fit the expected µ dependence to obtain an estimate of f and hence Ω. If galaxy formation is biased, then f in Equations (18.5.9) and (18.5.10) is replaced by β = f/b. Given the paucity of available peculiar velocity data, it seems that this type of analysis is the most promising approach to the use of cosmological velocity information to estimate Ω.

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much whether they are biased, except if the velocities of galaxies are systematically di erent from those of randomly selected points.

There are various ways to use the properties of peculiar motions in the estimation of Ω0. As we have seen, the small-scale anisotropy introduced into statistical measures like the correlation function and power spectrum can be used to estimate the magnitude of the radial component of the typical galaxy-peculiar velocity. The velocities obtained by such methods are around 300 km s−1. One can also use this information to infer the total amount of mass using the statistical mechanics of self-gravitating systems in the form of the cosmic virial theorem (18.5.6). These methods, when applied on small to intermediate scales, consistently yield estimates of Ω0 in the range 0.1–0.3. These estimates also agree with virial estimates of the masses of rich clusters of galaxies, in which the analysis is considerably simplified if one assumes the clusters are fully relaxed and gravitationally bound systems, as discussed in Chapter 4; as we mentioned there, this value is about an order of magnitude larger than naive estimates of Ω0 based on the mass-to-light ratios inferred for galaxy interiors. This discrepancy was one of the initial motivations for the introduction of a bias b into the models of galaxy clustering. Typically one compares some statistical measure of the clustering of galaxies with the observed velocity, so what emerges is a constraint on the combination β = f/b Ω0.6/b if there is a linear bias.

As we have seen in Chapter 17, the COBE detection of microwave background fluctuations casts doubt upon the existence of a bias su cient to explain the observed peculiar motions if Ω = 1, at least in the context of the CDM model. There is still an escape route for adherents of the critical density. Since direct determinations of Ω from dynamics have been restricted to relatively small volumes which may not be representative of the Universe at large, one can claim that we just live in an underdense part of the Universe. It is probably true that, if one simulates an Ω = 1 CDM model, one will find some places where the local distribution of mass is such as to produce, by the above analyses, a local value of

Ω0.2 by chance. This does not, however, constitute an argument against the alternative that Ω is actually less than unity.

Recent advances in the accumulation of galaxy redshifts have made it possible to attempt analyses of redshift-space distortions on large scales, which we also discussed in Section 18.5. The recent analysis of the 2dF GRS by Peacock et al. (2001) shows that β 0.4. If the APM galaxies upon which this survey is based are unbiased, then this means the matter density must be low; redshift distortions are insensitive to the presence of Λ. As we have explained, these measurements probably supply more robust methods for estimating Ω0 than the relatively local peculiar-motion studies that have always seemed to suggest a high value of

Ω0.6/b, consistent with an Einstein–de Sitter universe. In particular, because one can compare the reconstructed density field with the observed galaxy distribution, it is possible, at least in principle, to break the degeneracy between models with a low value of Ω and models having a higher density but a significant bias. This is a relatively new technique for measuring the density parameter, however, and it would be wise to suspend judgement upon it, at least until all possible sys-

tematic biases have been investigated. These methods are nevertheless extremely promising and we anticipate that, in the near future, relatively unambiguous determinations of Ω will be forthcoming.

Bibliographic Notes on Chapter 18

Historically interesting reviews of peculiar motions can be found in Rubin and Coyne (1988), Burstein (1990), Bertschinger (1992), Dekel (1994) and Strauss and Willick (1995). A wonderful recent review of linear redshift distortions is given by Hamilton (1998). Other useful references are Vittorio et al. (1986), Vittorio and Turner (1987) and Bertschinger and Juszkiewicz (1988).

Problems

1.Derive the cosmic virial theorem (18.5.6).

2.Derive Equations (18.5.7) and (18.5.8).

3.Derive the Kaiser formula (18.5.9).

4.Show that the Zel’dovich displacements in redshift space are a factor (1 + f ) larger in the line of sight than at right angles to it. Deduce that caustics form earlier in redshift space than in real space.