5.2. Formation of pancakes from primeval adiabatic perturbations
In Section 4, we discussed the damping of
adiabatic fluctuations
prior to and during recombination, with the result that linear
adiabatic ftuctuations are damped on mass scales
MD
[Eq. (4.12)]. Since the density fluctuations are assumed to be
everywhere small, the linear theory results of
Section 5.1 may be used
to discuss the early phase of gravitational instability.
From Eq. (5.7) we see that in linear theory, the peculiar velocity at any point q may be written as a separable function of the variables (q, t). On keeping just the dominant mode we have,
![]() |
(5.17) |
Integrating (5.17), it follows that the coordinate x of a fluid element may be written,
![]() |
(5.18) |
The coordinates q are taken to define the unperturbed background and
term
b(t)p(q) represents the initial irregularity
in the
matter distribution, assumed to be small. Equations (5.17) and (5.18) are
expected to be an accurate approximation while perturbations are in
the linear regime, but
Zel'dovich (1970)
has pointed out that they may
also be used as a guide to the evolution of irregularities when
/
>>
1. Consider the density of a fixed fluid element in the
neighbourhood of x at ti. The density at a
later time t is given by
the Jacobian of the transformation between the coordinate systems
(x, q), thus
![]() |
(5.19) |
Evaluating the Jacobian to first order,
![]() |
(5.20) |
in agreement with linear theory, Eq. (5.7a). By extrapolating the
result of (5.19), infinite density is achieved at some time t
when the
determinant in (5.19) vanishes. Thus the matter will in general pile
up into sheets (pancakes) along the surfaces defined by the condition
|xk /
qj| =
0. This is illustrated in Figure 5.1 where
we compare the
Zel'dovich approximate theory with a three-dimensional N-body
simulation (Davis, Efstathiou, Frenk and White, in preparation). Here
a particle is initially placed at the centre of each of 323
cells of a three dimensional lattice. The particle coordinates
qi are then displaced by a small amount
p(qi) evaluated from (5.17)
with Ak = const × exp(- ik
k)
for ki < kD and
Ak = 0 for ki >
kD where
D =
2
/ kD = 0.5
times the length of the periodic box. The cosmological
density parameter was set to
= 1 and the
amplitude of the initial
perturbations was adjusted so that pancakes would form when the system
had expanded by a factor of
4.5. The pictures in
the left-hand
panel of Figure 5.1 show an N-body
simulation in which forces were calculated using a Fast Fourier
transform potential solver on a 643 grid (see e.g.
Efstathiou and Eastwood
1981
and references
therein). The pictures in the right hand panel show the same initial
conditions evolved using the Zel'dovich approximation Eq. (5.18). The
Zel'dovich approximation compares very well with the N-body
simulation, especially at early times. At late times, after the
"caustics" have formed, the Zel'dovich approximation fails to reproduce
the dense knots which develop at the intersection of sheets and
filaments. In Figure 5.2 we show three
projections of the particle
positions in the N-body simulation after expansion by a factor of
8.2. The dominant visual impression is of long filaments, but careful
inspection reveals sheet-like structures as well. By the time the
system has expanded by a factor ~ 12, there remain only weak remnants
of filamentary and sheet-like structures.
![]() |
Figure 5.2. Three projections of the numerical simulation shown in Figure 5.1 after the system had expanded by a factor of 8.2. |
Two-dimensional simulations which include gravity have been done by Doroshkevich et al. (1980a) and by Melott (1983). Three-dimensional simulations have been presented by Klypin and Shandarin (1983) and by Frenk, White and Davis (1983).
These results show that if we have primordial adiabatic fluctuation
(and no significant contribution to the mass density from "exotic"
weakly interacting particles with masses >> 1 keV - see
Sections 9.3
and 10) the first objects to fragment out of
the background will be
"pancakes" or prolate structures with linear dimensions
~ 2 / kD.
However, since it is assumed that
/
is everywhere
small initially,
the matter will still be gaseous at the time of collapse and hence a
shock will form as fluid elements intersect with high relative
velocity. Now from Eq. (4.12) the characteristic damping mass scale
is MD
1.3 ×
1012(
h2)-3/2
M
and
is, therefore, much larger than a typical galactic mass
(1011
M
) if
< 1. In this
case, galaxy
formation must proceed by the fragmentation of the pancakes. Some
aspects of the fragmentation process will be discussed in
Section 8.
A similar scenario results if the mean-mass density of the universe is dominated by massive neutrinos. In this case, fluctuations in the neutrino density are erased on mass scales smaller than
![]() |
(5.21) |
where m30 is the neutrino mass in units of 30 eV
(Bond, Efstathiou and
Silk, 1980).
The damping is due to a combination of directional
dispersion while the neutrinos are relativistic and a type of Landau
damping when they are non-relativistic (this is discussed in more
detail in section 9). Detailed numerical
calculations of the damping of neutrino fluctuations have been presented by
Peebles (1982a) and
Bond and Szalay (1983).
In this picture, baryons will collapse
together with the neutrinos into pancakes of mass
~ Mmax and the
luminous parts of galaxies must form by the fragmentation of the
shocked gas.
It has been argued (e.g.
Peebles, 1974a;
Fall, 1980a)
that these theories for the formation of galaxies and clusters might be
inconsistent with the observed power-law slope of the two-point
correlation function
(r)
(Eq. 2.27) since the fluctuation spectra
possess a preferred scale. However, the numerical simulations of
Frenk et al. and Klypin and Shandarin produce power-law correlation
functions. The slope of the two-point function varies with time, but
there is a unique epoch at which the models yield a power law
(r)
r-1.8
in agreement with observations [cf. Eq. (2.27)]. [We have done
similar numerical simulations to those of Frenk et al. and find that
at late times, when the clustering is highly non-linear, the models
develop a three-point correlation function with a form similar to that
of Eq. (2.28)]. It is difficult to judge whether these detailed
results provide strong support for pancake theories. As we have
remarked above, in these theories galaxies can only form after the
dissipative collapse of gas. The small-scale clustering properties of
galaxies could be quite different to those inferred from purely
dissipationless N-body models. For example, one might expect galaxies
to form in predominantly high density regions and one could even
envisage gas dynamical effects which could inhibit galaxy formation in
very massive pancakes
(Bond et al., 1983).
The two-point correlation function may be used to set useful constraints on these theories. The observed lack of clustering on scales > r0 ~ 5h-1 Mpc suggests that any fluctuations on larger scales should still be in the linear regime of growth. Consider, for example, the spectrum used in the models shown in Figures 5.1 and 5.2. The two-point correlation function in the linear regime is,
![]() |
(5.22) |
where (0) is the
amplitude of the correlation function at zero lag.
Perturbations will enter the non-linear regime when
(0)
1. A
measure of the width of
(x) is
given by
![]() |
(5.23) |
(Peebles, 1981a, Peebles, 1982a) and it is reasonable to require r > xD at the present epoch. The detailed numerical simulations of Frenk et al. suggest that 2r0 > xD is a more accurate criterion. Applying this restriction we find the limits
![]() |
(5.24a) |
for adiabatic fluctuations with massless neutrinos, and
![]() |
(5.24b) |
for adiabatic fluctuations with massive neutrinos. A more detailed
discussion should properly take into account the dependence of the
shape of
(x) on the
spectral index n; in the massless neutrino case,
the broad spectrum (Figure 4.2) implies that
n
2
(Peebles, 1981a,
Silk, 1982)
and the massive neutrino case is discussed in
Section 9.2
(also Peebles,
1982a).
The constraints of Eqs. (5.24) agree qualitatively with the limits
deduced from the small-scale anisotropies of the microwave background
described in Section 4, i.e. the theories are
compatible with observational constraints only if
h ~ 1. This
may be difficult to reconcile with measurements of the cosmological density
parameter (Section 2.4) unless the dark
material is distributed more uniformly
than galaxies on scales < r0. The observed cosmic
abundances of helium
and especially deuterium require a low value of the baryon density,
B ~ 0.1
(see Pagel, 1982,
for an excellent review). The latter
constraint is partly responsible for the current popularity of models
with massive neutrinos. In these models it is possible that
T =
+
B
1 in order to satisfy
the microwave background limits and Eq. (5.24b) whilst
B
0.1 to give agreement
with primordial nucleosynthesis
(Schramm and Steigman,
1981).
Massive neutrinos also
offer an excellent alternative to the traditional forms of dark
material such as low mass stars.
As the numerical simulations show, it is unlikely that pancake
structure would survive to the present epoch unless pancake formation
occurred very recently, at redshifts z ~ 1. Even a redshift
z ~ 3, at which quasars have been observed, would seem to be too
early to allow
pancakes to survive to the present. The survival of pancakes may be
easier to arrange in a low density cosmological model since
gravitational instability effectively ceases at redshifts
zf ~ 1 /
0 - 1
but this would seem to be difficult to reconcile with the limits
deduced from Eqs. (5.24).