Accordingly, the linearized equation for the evolution of the field in Fourier space reads

Here

is the adiabatic sound speed in a medium with equation of state . In eq.(23) we can define the critical Jeans wavelength

which discriminates between two different regimes for the perturbation evolution. For fluctuation modes with wavelength , the pressure contribution can be neglected and the linear solution of eq.(14) is recovered. Viceversa, for the gravitational term becomes negligible and the solution oscillates. Thus, while fluctuations on a scale greater than the Jeans length are not pressure-supported and are able to grow by gravity, at scales below the fluctuations behave like oscillating sound waves.

If is the average baryon density, we can define
a baryon Jeans mass scale,

Since matter-radiation equality occurs at , the Jeans mass just before recombination is

of the same order of the mass of a supercluster. After recombination, however, photons are no longer coupled to matter, so that the equation of state rapidly changes and the baryonic component behaves like a monoatomic gas with

being the proton mass. At the recombination temperature , it corresponds to a Jeans mass

Thus, although before recombination the Jeans mass involves scales of superclusters, after matter and radiation decouple it drops by several orders of magnitude to the value of the mass of globular clusters, and fluctuations on small scales are able to start growing again.

It is worth comparing the Jeans mass before recombination with the baryon
mass contained inside the Hubble radius
.
According to eq.(27), it is

A further characteristic scale, which enters in the spectrum of baryon
fluctuations, is due to the collisional damping occurring just before
recombination. As recombination is approached, the coupling between
radiation and baryons becomes no longer perfect and the photon mean free
path starts increasing. Thus, photons can diffuse more easily from
overdensities carrying with them matter, to which they are however still
quite tightly coupled. The final effect is to damp fluctuations below the
scale which corresponds to the distance travelled by a photon in an expansion
time-scale. This is known as Silk damping [29] and an accurate
evaluation of the smoothing mass scale in the post-recombination baryon
spectrum [10] gives

Although the simplicity of a purely baryonic model is rather attractive, nevertheless it suffers from a number of serious problems, which makes it extremely unlikely. Even without referring to the difficulty of reconciling the predictions based on primordial nucleosynthesis, , with both dynamical estimates of the mean cosmic density and the inflationary prejudice , the baryonic spectrum gives too large fluctuations at the scale of 10-20 , with respect to what observed for the galaxy distribution. Even more, a purely baryonic model is ruled out since it predicts too high CMB temperature fluctuations with respect to current detections.