8.3. The pancake theory of galaxy formation
In Sections 4 and 5 we described how photon viscosity and the decay of neutrino perturbations can imprint characteristic mass scales on an initial power-law spectrum of adiabatic perturbations. If neutrinos are massless, the photon viscosity damping mass scale is given by Eq. (4.12). For sufficiently low ( ~ 0.1) this corresponds roughly to a cluster of galaxies. Since perturbations on smaller mass-scales are erased, the first; objects to collapse will be larger than individual galaxies and will have masses far in excess of the Jeans mass after recombination. Pressure forces are, therefore, negligible and so the collapse may be described using the Zel'dovich (1970) approximate theory discussed in Section 5.2. This predicts the formation of pancakes. However, in contrast to the collisionless collapse discussed in Section 5.2, the gas will form a shock and the shocked gas will be heated to a high temperature. In this Section, we calculate the fraction of the shocked gas that can radiatively cool to temperatures 104 K and thus the fraction of gas that can fragment into galaxies. We also consider the characteristic masses of the first fragments to condense from the cooled gas. Analytic and semi-analytic discussions have been given by Sunyaev and Zel'dovich (1972), Zel'dovich (1978), Doroshkevich, Shandarin and Saar (1978) and Jones, Palmer and Wyse (1981).
If neutrinos have a non-zero rest mass, the relevant damping mass is given by Eq. (5.21). The same qualitative picture as outlined above applies in this case, except that only a small fraction of the total pancake mass is in baryonic form (b / T) which can shock and cool. The rest, ( / T, T = + b) is in massive neutrinos which are collisionless. In this case, neutrinos and baryons collapse together until the pancake forms. The baryons then shock while the neutrinos can propagate through the caustic, leading to a neutrino-baryon sandwich (Doroshkevich et al., 1980). In this scenario, one has a natural explanation of the missing-mass in clusters of galaxies - a problem that was never satisfactorily addressed in purely baryonic versions of the pancake theory. Detailed numerical calculations of the shock structure and the fraction of the gas which can cool in neutrino-baryon pancakes are described by Bond, Szalay and White (1983) and by Shapiro, Struck-Marcell and Melott (1983).
In the absence of any substantial primordial small-scale fluctuations, such as might be expected, for example, in the hybrid isothermal- adiabatic scenario envisaged by Dekel (1983), fragmentation is inhibited until a sufficiently dense quasi-equilibrium gas layer has accumulated in the mid-plane of the pancake. The pancake structure is determined as follows. The trajectory of a fluid element initially at comoving coordinate measured perpendicular to the pancake plane is given, for a one-dimensional pressure-free collapse, by
This non-linear solution combines the Hubble flow (in the first term) with the influence of a linearized plane wave density perturbation on the Hubble flow (in the second term) and can be shown to be consistent both with the equations of motion and Poisson's equation (Zel'dovich, 1970). Mass conservation then yields
where 0 is the background density. A more general formulation in terms of an arbitrary spectrum yields
where is the Lagrangian coordinate and S() replaces sin(kd) in Eq. (8.16). Evidently, at a local maximum of d2S / d2 infinite density is attained in the midplane ( = 0) at a redshift given by a(t) = ap. This signifies the formation of a caustic surface which arises from the intersection of particle trajectories.
We can regard as prescribing the mass-fraction and z as the instantaneous location of the particles initially in a plane-wave perturbation of wave number k. The time evolution of the particle density is given by
This represents matter falling into the mid-plane as a ap, and n0 is the initial density of the background. The velocity acquired in free-fall is sufficiently great ( 103 km s-1) that the infalling matter shocks to a temperature > 108 K, and the density is large enough near the midplane that the post-shock primordial gas radiatively cools to ~ 104 K (although thermal conductivity may maintain much of the gas at ~ 106 K). To evaluate the cooled mass-fraction, we note that the cooling time-scale is
where Aff< is the free-free cooling coefficient and subscript s denotes post-shock values, while the dynamical time scale of the newly shocked gas is td = 4z / . Also the immediate post-shock temperature is
where µ is the mean mass per H atom. Equating tc and td and letting a ~ ap yields the critical mass-fraction c that cools:
For a neutrino sandwich with ~ 1, only a mass fraction
enters the cooled baryonic layer at caustic formation. If b 0.1 this means that no more than three percent of the closure density (and about 30% of the total baryonic density) may be expected to participate in galaxy formation when the cooled, compressed baryonic pancake eventually fragments. Numerical simulations of one-dimensional shocks confirm these simple estimates (Bond et al., 1983).
Since only the cooled gas fragments and forms galaxies, this leads to a clear prediction of the pancake model: there should be at least a factor three more hot gas in galaxy clusters than baryonic mass in galaxies. Observationally, one finds M/L for rich clusters to be of order 400h(M/L), with ~ 10 - 20% of this accounted for by hot X-ray emitting gas. Since the luminous regions of galaxies have M/L ~ 10, the pancake theory prediction will be satisfied provided that both galaxy halos and the dark matter in galaxy clusters are non-baryonic. One would not easily be able to reconcile this theory with observation if halos and clusters were predominantly composed of dissipative material, such as compact stellar remnants, which had not undergone collapse prior to pancake formation.
The minimum fragment mass turns out to be relatively small. The cool gas layer is ram-pressure confined, and the minimum mass that is gravitationally unstable is given by the criterion (Sunyaev and Zel'dovich, 1972):
where vs is the sound velocity and p is the ram pressure. Now
and is independent of (for kd << 1). Consequently, one obtains
It seems unlikely that one could avoid fragmentation down to Mmin. This is much smaller than the mass of a typical galaxy, so galaxy formation must presumably occur by the aggregation of many smaller clouds of mass 107 - 109 M, much as in the primordial isothermal fluctuation model (Doroshkevich, Shandarin and Saar, 1978). The principal difference now is that the clouds originate by fragmentation at the relatively late epoch of pancaking, and so perhaps are more likely to be predominantly gaseous as galaxies form.