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

318 Nonlinear Evolution

as well as the form (14.8.8). This approach is the one most frequently adopted in practice, but the community is becoming increasingly aware of its limitations. A simple model of this kind simply cannot hope to describe realistically the relationship between galaxy formation and environment (Dekel and Lahav 1999).

One should say, however, that there is no compelling reason a priori to believe that galaxy formation should be restricted to peaks of particularly high initial density. It is true that peaks collapsing later might produce objects with a lower final density than peaks collapsing earlier, but these could (and perhaps should) still correspond to galaxies. Some astrophysical mechanism must be introduced which will inhibit galaxy formation in the lower peaks. Many mechanisms have been suggested, such as the possibility that star formation may produce strong winds capable of blowing the gas out of shallow potential wells, thus suppressing star formation, but none of these are particularly compelling. We discuss briefly how such a mechanism might also explain the morphological di erence between elliptical and spiral galaxies in the next section. It is even possible that some large-scale modulation of the e ciency of galaxy formation might be achieved, perhaps by cosmic explosions or photoionisation due to quasars. Such a modulation would not be local in the sense discussed above and may well lead to a nonlinear bias parameter on large scales. We shall see later, however, in Chapter 17 that the latest clustering observations and the COBE microwave background fluctuations do not seem to support the idea of a strong bias, at least not in a CDM model.

At the present time b has a somewhat dubious status in the field of structure formation. The best way to think of b is not as describing some specific way of relating galaxies to mass, such as in (14.8.10), but as a way of parametrising our ignorance of galaxy formation in much the same way as one should interpret the mixing-length parameter in the theory of stellar convection. As we have mentioned already, to understand how this occurs we need to understand not only gravitational clustering but also star formation and gas dynamics. All this complicated physics is supposed to be contained in the parameter b.

14.9 Galaxy Formation

As we mentioned in Chapter 4, galaxies possess angular momentum. Its amount depends on the morphological type: it is maximum for spirals and S0 galaxies, and minimum for ellipticals. The angular momentum of our Galaxy, a fairly typical spiral galaxy of mass M 1011M , is J 1.4 × 1074 cm2 g s−1. The conventional parametrisation of galactic angular momenta is in terms of the ratio between the observed angular velocity, ω, and the angular velocity which would be required to support the galaxy by rotation alone, ω0:

where the dimensionless angular momentum parameter λ is typically as high as λ 0.4 for spirals, but only λ 0.05 for ellipticals. It is also probable that clusters of galaxies have some kind of rotation, large for the irregular open clusters like Virgo and smaller for the compact rich clusters like Coma.

The Kelvin circulation theorem guarantees that, in the absence of dissipative processes, an initially irrotational velocity field must remain so. The gravitational force can only create velocity fields in the form of potential flows which have zero curl. For a long time, therefore, the idea was held that the vorticity one appears to see now in galaxies must have been present in the early universe. This idea was developed much further in the theory of galaxy formation by cosmic turbulence which was at its most popular in 1970; this theory, however, predicted very high fluctuations in the temperature of the cosmic microwave background and some additional implausible assumptions were made. For this reason this scenario was rapidly abandoned and we mention it now only out of historical interest.

The origin of the rotation of galaxies within the framework of the theory of gravitational instability is described by a model, the first version of which was actually created by Hoyle (1949) and which has been subsequently modified by various authors and adapted to the various cosmogonical scenarios in fashion over the years (e.g. Efstathiou and Jones 1979). This model attributes the acquisition of angular momentum by a galaxy to the tidal action of protogalactic objects around it, at the epoch when the protogalaxy is just about to form a galaxy. At this epoch, protogalaxies have relatively large size (they will be close to their maximum expansion scale) and have a relatively small spatial separation compared with their size. Analytic calculations and N-body experiments show that this mechanism does indeed give a plausible account of the distribution of angular momentum observed in galactic systems.

This theory is valid in both top-down and bottom-up scenarios of structure formation. There is also another possibility: the circulation theorem is not valid in the presence of dissipative processes such as those accompanying the formation and propagation of a shock wave after the collapse of a pancake; the potential motion of the gas can become rotational after the gas has been compressed by a shock wave. This mechanism has not yet been analysed in great detail partly because of the di culty in dealing with nonlinear hydrodynamics and partly because of the apparent success of the alternative, simpler scenario based on tidal forces.

In the tidal action model the acquisition of angular momentum by a galaxy takes place in two phases. The first phase commences at the moment a fluctuation begins to grow after recombination and ends when it reaches its maximum expansion, at tm; the second phase lasts from then until the present epoch. This second phase is thought to be when the galaxy acquires its own individuality beginning at the stage it collapses, undergoes violent relaxation and reaches virial equilibrium. It can be shown that in the first phase the angular momentum of the perturbation grows roughly like t5/3, due to the e ects of deviations from the Hubble flow caused by the various sub-condensations which make up the protostructure in question. In the second phase the protogalaxy, which will not in general be spherical, is subject to a torque due to other protogalaxies in its vicin-

320 Nonlinear Evolution

ity. One finds that this tidal e ect, due to all the surrounding objects, increases

because the expansion of the Universe carries the protogalaxies away from each other.

The question of the angular momentum of galaxies is intimately related to the origin of the morphological types, discussed in Chapter 4. A full theory of the formation of galaxies is complicated by gas pressure e ects, as outlined in Section 14.6, and is yet to be elucidated. Possible answers to both the angular momentum and morphology questions may, however, come from the idea that dissipation is important for spiral galaxies but not for ellipticals. One can connect this to the problem of angular momentum as follows. The tidal action model can generate a value of λ 0.05–0.1, not quite large enough to account for spiral galaxies but comfortable for ellipticals. It seems clear for spirals that dissipation must be important to explain why the luminous matter in a galaxy is concentrated in the middle of its dark halo. If the gas collapses through cooling, as described in Section 14.7, then its binding energy will increase while the mass and angular momentum are conserved. If the binding energy of a spherical cloud is E GM2/R, as usual, then E 1/R as the gas cools and shrinks. This means that λ R−1/2, so cooling can increase the λ parameter. The problem with this is that, if the galaxy is all baryonic, the rate of increase is rather slow. If, however, there is a dominant dark halo, one can get a much more rapid increase in λ and a value of 0.4–0.5 is reasonable.

The problem of formation of elliptical galaxies is less well understood. The value of their angular momentum seems to be accounted for by the tidal action model if there is no significant dissipation, but how can it be arranged for spirals and ellipticals to be thus separated? A possible explanation for this is that ellipticals formed earlier, when the Universe was denser and star formation (perhaps) more e cient. One might therefore be motivated towards an extension of the idea of biased galaxy formation (Section 14.8) in which the very highest density peaks, which collapse soonest, become ellipticals, while the smaller peaks become spirals. The detailed physics of the dividing line between these two morphologies, which we have supposed may be crudely delineated by the e ciency of dissipation, is still very unclear. An alternative idea is that perhaps all galaxies form like spiral galaxies, but that ellipticals are made from merging of spirals. This would seem to be plausible, given that ellipticals occur predominantly in dense regions. There are also problems with this picture. It is not clear whether ellipticals have the correct density profiles for them to be consistent with mergers of disc galaxies if the mergers are dissipationless. This aspect would have to be explored using numerical simulations.

The di culty of understanding the complex e ects of heating, dissipation and star formation within a continuously evolving clustering hierarchy has spawned the field of semi-analytic galaxy formation. This approach encodes the complex physics of galaxy formation in a set of relatively simple rules applied within a merger-tree description of the formation and merging of dark-matter haloes. The basic picture described in this model is that gas falls into the haloes whereupon it

Comments 321

is shock-heated up to the virial temperature of the halo. It then undergoes radiative cooling. The cold gas component thus formed collapses into a rotationally supported disc and provides a reservoir of material that forms stars. The stars thus formed inject energy into the gas through supernova explosions, which also add a sprinkling of heavy elements to the mix. Crucial to this scenario is the assumption that the basic galaxy unit is disc. Elliptical and spheroidal galaxies are made through ‘major mergers’ of discs as suggested above. See Baugh et al. (1998) for a view of the state of this particular art.

14.10 Comments

It is clear that this chapter leaves many questions unanswered. We have shown that, while it is possible to use analytical methods and numerical simulations to understand the behaviour of density perturbations in the nonlinear regime, the complications of gas pressure, dissipation and star formation are still not fully understood. This means that we do not have an entirely satisfactory way of identifying sites of galaxy formation and every attempt to compare calculations with observations must take account of this di culty. The semianalytic approach has been a major advance in this area but it is still not clear how fully it can account for the observed properties of galaxies of di erent types.

We also have the problem that, in order to run an N-body simulation or perform an analytical calculation, one needs to normalise the spectrum appropriately. In the past this was done by matching properties of the density fluctuation field to properties of galaxy counts. In more recent times, after the COBE result, the usual approach has become to normalise models to the microwave background anisotropy they predict. Even this latter method still carries some uncertainty, as we shall see in Chapter 17. To this one can add the problem of not knowing the form and quantity of any dark matter, which alters the primordial spectrum before the nonlinear phase is reached. Clearly there is an enormous parameter space to be explored and the tools we have to probe it theoretically are relatively crude.

Nevertheless, there has been substantial progress in recent years in the field of structure formation, and there is considerable cause to be optimistic about the future. Numerical techniques are being refined, the computational power available is steadily increasing and powerful analytical extensions of those we have discussed in this chapter have also been developed. On the observational side, tens of thousands of galaxy redshifts have been compiled over the last three decades. These allow us to probe the distribution of luminous matter on larger and larger scales; models for the bias are used to translate this into the mass distribution. New methods we shall describe in the following chapters have been devised to minimise the bias-dependence of tests of structure-formation scenarios. And finally, the microwave background fluctuations on small angular scales may allow us to test these theoretical ideas in a much more rigorous way than has hitherto been possible.

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Bibliographic Notes on Chapter 14

Analytic nonlinear methods for large-scale structure are reviewed by Shandarin and Zel’dovich (1989) and Sahni and Coles (1995). The Burgers equation is discussed by Gurbatov et al. (1989). The basics of N-body simulation are discussed by Hockney and Eastwood (1988) in a general context. Numerical N-body techniques in cosmology are discussed by Efstathiou et al. (1985) and Bertschinger and Gelb (1991), while SPH variants are covered by Evrard (1988). For a discussion of Eulerian hydrodynamics, see Cen (1992).

Problems

1. For a Universe with Ω0 ≠ 1, show that the generalisation of Equation (14.1.8) is

π2 χ(Ω0) = 4Ω0(H0t0)2 .

2.Show that the Zel’dovich approximation is an exact solution of the one-dimensional gravitational clustering problem provided no trajectories have crossed. (Hint: substitute the Zel’dovich trajectories into the Euler equation for the problem and show that the potential gradients implied are consistent with the Poisson equation.)

3.Find the Zel’dovich displacement field corresponding to a spherical ‘top-hat’ density perturbation like that discussed in Section 14.1. Show that the Zel’dovich approximation predicts the formation of a singularity (i.e. that δ → ∞ at a finite time).

4.Prove the relation (14.4.19).

5.The self-similar evolution described in Section 14.4.2 requires that very largeand very small-scale velocities give convergent contributions to the peculiar velocity field. What restriction does this place on the spectral index, n, of the density fluctuations?

6.Derive the approximate results (14.8.5) and (14.8.6).

15

Models of

Structure

Formation

15.1 Introduction

In the preceding four chapters we have laid out the basic ingredients of the theory of cosmological structure formation according to the standard paradigm. The essential components of this recipe are primordial density perturbations, gravitational instability and dark matter, but many variations on this basic theme are viable. Despite the great progress that has undoubtedly been made, further steps are di cult because of uncertainties in the cosmological parameters, in the modelling of relevant physical processes involved in galaxy formation, and in the uncertain relationship between galaxies and the underlying distribution of matter.

Our aim in this chapter is to explain how the various components we have described come together in ‘models’ of structure formation that can be tested against observations. This will involve taking stock, and reducing the rather detailed physical discussion we have followed so far to a few key ideas and model parameters. Our role is not to advocate one particular mix of ingredients over another, but to point out how these di erent ingredients might be constrained or ruled out.

For example, as we have seen in Chapter 10, the expansion of the Universe renders the cosmological version of gravitational instability very slow, a power law in time rather than the exponential growth that develops in a static background. This slow rate has the important consequence that the evolved distribution of mass still retains significant memory of the initial state. If the perturbations were to

324 Models of Structure Formation

grow exponentially, all memory of the initial conditions would be rapidly erased. This, in turn, has two consequences for theories of structure formation. One is that a detailed model must entail a complete prescription for the form of the initial conditions, and the other is that observations made at the present epoch allow us to probe the form of the primordial fluctuations and thus test the theory.

15.2 Historical Prelude

Progress in the field of structure formation during the 1970s was characterised by the construction of scenarios for the origin of cosmic protostructure in twocomponent models containing baryonic material and radiation. (As we shall see, the cosmological neutrino background does not greatly influence the evolution of perturbations in matter and radiation, as long as the neutrinos are massless.) There can exist two fundamental modes of perturbations in such a twocomponent system: adiabatic perturbations, in which the matter fluctuations, δm = δρm/ρm, and radiation fluctuations, δr = δρr/ρr, are coupled together so that 4δm = 3δr; and isothermal perturbations, which involve only fluctuations in the matter component, i.e. δr = 0. These two kinds of perturbation led to two distinct scenarios for galaxy formation.

In the adiabatic scenario the first structures to form are on a large scale, M 1012–1014M , corresponding to clusters or superclusters of galaxies. Galaxies then form by successive processes of fragmentation of these large objects. For this reason the adiabatic scenario is also called a ‘top-down’ scenario.

On the other hand, in the isothermal scenario the first structures, protoclouds, are formed on a much smaller mass scale, M 105–106M , and then structure on larger scales is formed by the successive e ect of gravitational instability, a process known as hierarchical clustering. For this reason, the isothermal scenario is described as ‘bottom-up’.

The adiabatic and isothermal scenarios were in direct competition with each other during the 1970s. One aspect of this confrontation was that the adiabatic scenario was chiefly championed by the great school of Russian astrophysicists led by Zel’dovich in Moscow, and the isothermal model was primarily an American a air, advocated in particular by Peebles and the Princeton group. In fact, neither of these adversaries actually won the battle: because of several intrinsic di culties, the baryonic models were overtaken in the 1980s by models involving non-baryonic dark matter.

The main di culty of the adiabatic scenario was that it predicted rather large angular fluctuations in the temperature of the microwave background, which were in excess of the observational limits. We can illustrate the problem in a simple qualitative manner to bypass the complications of the kinetic approach described above. In a universe made only of baryons with Ωb 1, photons and massless neutrinos, the density fluctuation δm(zrec)M > MD(a)(zrec) must have amplitude greater than the growth factor between recombination and t0, which we called Ar0. From Section 11.4, one can see that, if Ω 1, then Ar0 zrec 103; if we are going to produce nonlinear structure by the present epoch, the density fluctuations

must have amplitude at least unity by now. Thus, one requires δm(zrec) 10−3 or higher. But these fluctuations in the matter are also accompanied in the adiabatic picture by fluctuations in the radiation which lead to fluctuations in the microwave background temperature δr 3δT/T 10−3, greater than the observational limits on the appropriate scale by more than two orders of magnitude. Moreover, if one recalls the calculations of primordial nucleosynthesis in the standard model, one cannot have Ωb as large as this, and a (generous) upper bound is given by Ω Ωb 0.1. This makes things even worse: in an open universe the growth factor is lower than a flat universe: Ar0 zrec/z(t ) 103Ω 102. In such a case the brightness fluctuations on the surface of last scattering exceed the observational limits by more than three orders of magnitude.

There is a possible escape from the limits on microwave background fluctuations provided by the possible existence of a period of reheating after zrec, perhaps caused by the energy liberated during pregalactic stellar evolution, which smooths out some of the fluctuations in the microwave background. There are problems with this escape route, however, as we shall see later in Chapter 19.

The isothermal scenario does not su er from the same di culties with the microwave background, chiefly because δr 0 for the isothermal fluctuations, and in any case the mass scale of the crucial first generation of clouds is so small. The major di culty in this case is that isothermal perturbations are ‘unnatural’: only very special processes can create primordial fluctuations in the matter component while leaving the radiation component undisturbed. One possibility we should mention is that inflation, which generically produces fluctuations of adiabatic type, can produce isocurvature fluctuations if the scalar field responsible for generating the fluctuations is not the same as the field – the inflaton – that drives the inflation. Isocurvature perturbations are, as we have mentioned, similar to isothermal perturbations but not identical. Indeed a variation of the old isothermal model has been advocated in recent years by Peebles (1987). His Primordial Isocurvature Baryon Model (PIB model) circumvents many of the problems of the old isothermal baryon model, but has di culties of its own.

Di culties with the adiabatic and isothermal pictures, chiefly the large-ampli- tude fluctuations they predicted in the cosmic microwave background, opened the way for the theories of the 1980s. These theories were built around the hypothesis that the Universe is dominated by non-baryonic dark matter, in the form of weakly interacting (collisionless) particles, perhaps neutrinos with mass mν 10 eV or some other ‘exotic’ particles (gravitinos, photinos, axions, etc.) predicted by some theories of high-energy particle physics. There are various possible models; the simplest is one of three components: baryonic material, non-baryonic material made of a single type of particle, and radiation (also in this case, the addition of a component of massless neutrinos does not have much e ect upon the evolution of perturbations). In this three-component system there are two fundamental perturbation modes again, similar to the two-component system mentioned above. These two modes are curvature perturbations (adiabatic modes) and isocurvature

326 Models of Structure Formation

perturbations. In the first mode, all three components are perturbed (δm δr δi, where i denotes the ‘exotic’ component); there is, therefore, a net perturbation in the energy-density and hence a perturbation in the curvature of space–time. In the second type of perturbation, however, the net energy-density is constant, so there is no perturbation to the spatial curvature.

The fashionable models of the 1980s can also be divided into two categories along the lines of the top-down/bottom-up labels we mentioned above. Here the discriminating factor is not the type of initial perturbation, which is usually assumed to be adiabatic in each case, but the form of the dark matter, as we shall discuss in Chapter 13.

In the hot-dark-matter (HDM) scenario, which is similar in broad outline to the old adiabatic baryon picture, the Universe is dominated by collisionless particles with a very large velocity dispersion (hence the name ‘hot’), by virtue of it decoupling from the other components when it is still relativistic. A typical example is a neutrino with mass mν 10 eV.

The cold-dark-matter (CDM) scenario has certain similarities to the old isothermal picture. This is characterised by the assumption that the Universe is dominated again by collisionless particles, but this time with a very small velocity dispersion (hence the term ‘cold’). This can occur if the particles decouple when they are no longer relativistic (typical examples are supersymmetric particles such as gravitinos and photinos) or have never been in thermal equilibrium with the other components (e.g. the axion).

The rapid explosion in the quantity and quality of galaxy-clustering data (Chapters 16 and 18) and the discovery by the COBE team in 1992 of fluctuations in the temperature of the cosmic microwave background on the sky (Chapter 17) have placed strong constraints on these theories. Nevertheless, the general picture that Jeans instability produces galaxies and large-scale structure from small initial fluctuations seems to hold together extremely well. It remains to be seen whether the remaining questions can be resolved, or are symptomatic of a fundamental flaw in the model.

15.3 Gravitational Instability in Brief

In order to focus our attention on the various possible models, let us now recapitulate the essentials of the gravitational instability model.

In order to understand how structures form we need to consider the di cult problem of dealing with the evolution of inhomogeneities in the expanding Universe. We are helped in this task by the fact that we expect such inhomogeneities to be of very small amplitude early on so we can adopt a kind of perturbative approach, at least for the early stages of the problem. If the length scale of the perturbations is smaller than the e ective cosmological horizon dH = c/H0, a Newtonian treatment of the subject is expected to be valid. If the mean free path of a particle is small, matter can be treated as an ideal fluid and the Newtonian equations governing the motion of gravitating particles in an expanding universe that we used in Chapters 10–12 can be used.

From these equations the essential point is that, if one ignores pressure forces, one obtains a simple equation for the evolution of δ:

For a spatially flat universe dominated by pressureless matter, ρ0(t) = 16 πGt2 and Equation (15.3.1) admits two linearly independent power law solutions δ(x, t) = D±(t)δ(x), where D+(t) a(t) t2/3 is the growing mode and D−(t) t−1 is the decaying mode.

15.4 Primordial Density Fluctuations

The above considerations apply to the evolution of a single Fourier mode of the density field δ(x, t) = D+(t)δ(x). What is more likely to be relevant, however, is the case of a superposition of waves, resulting from some kind of stochastic process in which the density field consists of a superposition of such modes with di erent amplitudes. A statistical description of the initial perturbations is therefore required, and any comparison between theory and observations will also have to be statistical.

The spatial Fourier transform of δ(x) is

˜

It is useful to specify the properties of δ in terms of δ. We can define the power spectrum of the field to be (essentially) the variance of the amplitudes at a given value of k:

where δD is the Dirac delta function; this rather cumbersome definition takes account of the translation symmetry and reality requirements for P(k); isotropy is expressed by P(k) = P(k). The analogous quantity in real space is called the two-point correlation function, or, more correctly, the autocovariance function, of δ(x):

which is itself related to the power spectrum via a Fourier transform. The shape of the initial fluctuation spectrum is assumed to be imprinted on the universe at some arbitrarily early time. As we have explained, many versions of the inflationary scenario for the very early universe (Guth 1981; Guth and Pi 1982) produce a power-law form

with a preference in some cases for the Harrison–Zel’dovich form with n = 1 (Harrison 1970; Zel’dovich 1972). Even if inflation is not the origin of density fluctuations, the form (15.4.4) is a useful phenomenological model for the fluctuation spectrum.