The evolution of baryonic fluctuations

In order to follow the evolution of fluctuations of baryonic matter, let us consider the second of eqs.(), with the inclusion of a pressure term:

$${\partial \mbox{\bf v}\over \partial t}+(\mbox{\bf v}\cdot {... ...1\over \bar \rho} {\mbox{\bf$\nabla$}}p+\nabla \Phi  = 0 .$$ (22)

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

$${\partial ^2\tilde \delta_{\mbox{\bf k}}\over \partial t^2}+... ...- 4\pi G \bar \rho\right) \tilde \delta_{\mbox{\bf k}} = 0 .$$ (23)

Here

$$v_s = \left({\partial p\over \partial \rho}\right)^{1/2}_{adiabatic}$$ (24)

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

$$\lambda_J = v_s \left({\pi \over G\bar \rho}\right)^{1/2} ,$$ (25)

which discriminates between two different regimes for the perturbation evolution. For fluctuation modes with wavelength $\lambda >\lambda_J$, the pressure contribution can be neglected and the linear solution of eq.(14) is recovered. Viceversa, for $\lambda <\lambda_J$ 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 $\lambda_J$ the fluctuations behave like oscillating sound waves.

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

$$M_J = {2\over 3}\pi \rho_b \lambda_J^3 ,$$ (26)

which is the mass of the smallest baryonic fluctuation that is able to grow. Before recombination, at a redshift $z_{rec}\simeq 10^3$, matter and radiation are tightly coupled by Thomson scattering. In this regime they behave like a single fluid with

$$v_s = {c\over \sqrt 3} \left({3\over 4} {\rho_m \over \rho_r}+1\right)^{- 1/2} .$$ (27)

Since matter-radiation equality occurs at $z_{eq}=4.2\times 10^4(\Omega h^2)$, the Jeans mass just before recombination is

$$M_J \simeq  9\times 10^{16}(\Omega_o h^2)^{-2}M_\odot ,$$ (28)

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

$$v_s = \left({5k_{\mbox{\tiny B}}T\over 3m_p}\right)^{1/2} ,$$ (29)

$m_p$ being the proton mass. At the recombination temperature $T_{rec}\simeq 3000 K$, it corresponds to a Jeans mass

$$M_J = 1.3\times 10^6(\Omega_o h^2)^{-1/2}M_\odot .$$ (30)

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 $M_{H,b} =(4\pi /3)\rho_b(ct^3)$. According to eq.(27), it is

$${M_J\over M_{H}} \simeq  26 \left({3\over 4} {\rho_m \over \rho_r}+1\right)^{-3/2} ,$$ (31)

so that, until radiation dominates, the Jeans mass exceeds the mass inside the horizon and all the subhorizon fluctuations are constrained not to grow.

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

$$M_D \simeq  2\times 10^{12}(\Omega_o/\Omega_b)^{3/2}(\Omega_oh^2)^{-5/4} M_\odot ,$$ (32)

which obviously depends on the baryon density parameter $\Omega_b$. The Silk damping increases by several orders of magnitude the mass-scale of the smallest fluctuation, which starts growing after recombination, smaller scale perturbations being heavily suppressed.

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, $\Omega_bh^2 \raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$} }0.1$, with both dynamical estimates of the mean cosmic density and the inflationary prejudice $\Omega_o =1$, the baryonic spectrum gives too large fluctuations at the scale of 10-20 $h^{-1}{\rm Mpc}$, 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.