3.2 Density Inhomogeneities in the Early Universe

The density fluctuations inferred from COBE are remarkably small and we know that for the overdensities in our own galaxy / > 1; furthermore, gravitationally collapsed structures, such as quasi stellar objects, are known to exist at z 5, so by this time at least some perturbations must have grown past / = 1. The big questions for galaxy formation then are: What is the scale-size of the first collapsing overdensities? How fast can fluctuations grow in the early Universe? What sort of fluctuations must they be?

In 1902 James Jeans considered the criterion for gravitational collapse in an infinite uniform medium in which small density perturbations (d) give rise to adiabatic pressure changes or acoustic waves (dp), such that

(2)

where V²_s is the adiabatic sound speed. The criterion for collapse is satisfied by a mass (known as the Jeans Mass, M_J) just large enough that the sound crossing time is larger than the free-fall collapse time - thus rendering the overdensity unable to respond with pressure changes fast enough to halt gravitational collapse.

It is straight forward to apply the basic idea of Jeans to the problem of galaxy formation in the early Universe and calculate the minimum spherical mass which would stop expanding with the universal Hubble flow, turn around, and gravitationally collapse. At z = 1000, just after decoupling

(3)

where ₀ is the present mass density, and M is the unit of solar mass (1 M = 2 x 10³⁰ kg).

This value assumes that the energy density of the Universe is matter dominated - at epochs earlier than about the decoupling stage (z >1000 or t 10⁶ yrs), the energy density of the radiation field exceeds that of the matter content and the sound speed is then relativistic. During this period, known as the radiation dominated era, the minimum mass required to undergo gravitational collapse of baryonic overdensities rises to

(4)

This mass is so large that it is in fact comparable to the entire baryonic content of each causally connected volume of space at any time during the radiation dominated era. Furthermore, the characteristic timescale required for matter fluctuations to collapse is longer than the characteristic time for the expansion of the Universe during this era. A combination of the large Jeans mass and rapid expansion time of the Universe guarantee that no baryonic fluctuations gravitationally collapse for the first 10⁶ yrs after the Big Bang and galaxy formation really only begins after this time at the onset of the matter dominated era.

An estimate of the redshift at which galaxies collapse can be made using the Jeans criterion: an overdensity larger than M_J will stop expanding and collapse in a time t_c, given by

(5)

Taking the example of our own Milky Way, the total mass implied from dynamical studies is 5 x 10¹¹ M and from the distance of the furthest stars in the galactic halo a plausible size to the Galaxy before it collapsed is 100 kpc. This implies a density 3.6 x 10^-24 kg m^-3 and, using equation (5) a collapse time of 10⁹ yrs, which corresponds to a look-back time of 7.2 Gyr if _M = 0.2 and 5.4 Gyr if _M = 1.0 (all assuming = 0, h = 1). From Figure 1 this translates to redshifts of formation z_f 4-2 respectively. If h = 0.5 then formation redshifts range from z_f 6-4 for the same mass densities. Finally, bound objects the size of galaxies are unlikely to exist much above these redshifts for the simple reason that the age of the Universe is then short compared to the free-fall collapse time.