230 Cosmic Structures
Note that only baryonic matter experiences pressure forces. Recall that dark matter
is pressureless, feeling only gravitational forces.
When푡Gis shorter than푡s, the fluctuations are unstable and their amplitude will
grow by attracting surrounding matter, becoming increasingly unstable until the mat-
ter eventually collapses into a gravitationally bound object. The opposite case is stable:
the fluctuations will move with constant amplitude as sound waves. Setting푡G=푡s,
we find the limitingJeans wavelength휆=휆Jat theJeans instability, discovered bySir
James Jeans(1877–1946) in 1902,
휆J=√
휋
퐺휌
푐s. (10.21)Actually, the factor
√
휋was not present in the above Newtonian derivation; it comes
from an exact treatment of Equations (10.1)–(10.3); see for instance [3]. The mass
contained in a sphere of radius휆Jis called theJeans mass,
푀J=^4
3휋휆^3 J휌. (10.22)
In order for tiny density fluctuations to be able to grow to galactic size, there must
be enough time, or the expansion must be exponential in a brief time. The gravita-
tional collapse of a cloud exceeding the Jeans mass develops exponentially, so the
cloud halves its radius in equal successive time intervals. But galaxies and large-scale
structures do not condense out of the primordial medium by exponential collapse.
The structures grow only linearly with the scale푎or as some low power of푎.
For subhorizon modes, the distinction between the radiation-dominated and
matter-dominated eras is critical. During the radiation era, growth of perturbations is
suppressed. During the matter era, perturbations can grow. But during the matter era
the Jeans wavelength provides an important boundary. Large wavelength fluctuations
will grow with the expansion as long as they are in the linear regime. In an acceler-
ated expansion driven by dark energy, the condition for gravitational collapse becomes
extremely complicated. This happens rather late, only when matter domination ends
and dark energy becomes dynamically important(푧∼ 1 ).
For wavelengths less than the Jeans wavelength the pressure in the baryonic matter
can oppose the gravitational collapse and perturbations will oscillate in the linear
regime as sound waves, never reaching gravitational collapse. An alternative way of
stating this is to note that the radiation pressure and the tight coupling of photons,
protons and electrons causes the fluid to be viscous. On small scales, photon diffusion
and thermal conductivity inhibit the growth of perturbations as soon as they arise, and
on large scales there is no coherent energy transport.
Mass fluctuations at still shorter wavelength, with휆≈푟U≪푟H,canbreakaway
from the general expansion and collapse to bound systems of the size of galaxies or
clusters of galaxies. Fluctuations which enter in the nonlinear regime, where the ratio
in Equation (10.17) is large, collapse rapidly into black holes before pressure forces
have time to respond.
For baryonic matter before the recombination era, the baryonic Jeans mass is some
30 times larger than the mass푀Hof baryons within the Hubble radius푟H,soifthere