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This optical depth is unity at zdec = 1100. From the optical depth we can compute the visibility function P(t), the probability that a photon was last scattered in the interval (z, z + dz). This is given by dT / Z \11.75

and has a sharp maximum at z = zdec and a width of Az « 100 (see Figure 3.2). The finite thickness of the last scattering surface has important observational consequences for CMB temperature fluctuations on small scales.

### 3.2.3 Coupling of gas and radiation

The CMB plays a key role in the early evolution of structures. First, it sets the epoch of decoupling, when baryonic matter becomes free to move through the radiation field to form the first generation of gravitationally bound systems. Second, it fixes the matter temperature that in turn determines the Jeans scale of the minimum size of the first bound objects.

Following Peebles (1993), consider an electron e~ moving at non-relativistic speed v ^ c through the CMB. In the e~ rest-frame, the CMB temperature measured at angle 0 from the direction of motion is

The radiation energy density per unit volume per unit solid angle d^ = d^ dcos 0 around 0 is dw =[aBT 4(0)]^ 4n

(here aB is the radiation constant), and the net drag force (i.e. the component of the momentum transfer along the direction of motion, integrated over all directions of the radiation) felt by the electron is

4ffTaB T4

This force will be communicated to the protons through electrostatic coupling. The formation of the first gravitationally bound systems is limited by radiation drag, as the drag force per baryon is xeF, where xe is the fractional ionization. The mean force divided by the mass mp of a hydrogen atom gives the deceleration time of the streaming motion:

v dt 3 mpc

The product of the expansion rate H and the velocity-dissipation time ts is tsH = 7.6 x 105hx-1(1 + z)-5/2. (3.21)

Prior to decoupling tsH ^ 1. Since the characteristic time for the gravitational growth of mass-density fluctuations is of the order of the expansion timescale, baryonic density fluctuations become free to grow only after decoupling.

We can use the above results to find the rate of relaxation of the matter temperature Te to that of the radiation. The mean energy per electron in the plasma is E = 3kBTe/2 = me(v2}. The rate at which an electron is doing work against the radiation drag force is Fv, so the plasma transfers energy to the radiation at the mean rate, per electron, of dE IT? \ 4 aTaBT4 / 2\ rp (o

At thermal equilibrium Te = T, and this rate must be balanced by the rate at which photons scattering off electrons increase the matter energy:

Thus the rate of change of the matter temperature is dTe = 2 dE = xe 8ctt aBT4 dt 3kB dt 1 + xe 3mec

The factor xe/(1 + xe) accounts for the fact that the plasma energy-loss rate per unit volume is -ne dE/dt = -xenH dE/dt, while the total plasma energy density is (ne + np + nH¡)3kBTe/2 = nH(1 + xe)3kBTe/2. The expression above can be rewritten as dTe T - Te

dt tc where the "Compton cooling timescale" is

The characteristic condition for thermal coupling is then tcH = « < 1. <3.27)

Figure 3.3. Evolution of the radiation (dashed line, labeled CMB) and matter (solid line, labeled GAS) temperatures after recombination, in the absence of any reheating mechanism.

Figure 3.3. Evolution of the radiation (dashed line, labeled CMB) and matter (solid line, labeled GAS) temperatures after recombination, in the absence of any reheating mechanism.

For fully ionized gas xe = 1 and the Compton cooling timescale is shorter than the expansion time at all redshifts z > zc = 5. With increasing redshift above 5, it becomes increasingly difficult to keep optically thin ionized plasma hotter than the CMB (Figure 3.3). At redshift z > 100, well before the first energy sources (stars, accreting black holes) turn on, hydrogen will have the residual ionization xe = 2.5 x 10-4. With this ionization, the characteristic relation for the relaxation of the matter temperature is

The coefficient of the fractional temperature difference reaches unity at the "thermaliza-tion redshift" zth ~ 130. That is, the residual ionization is enough to keep the matter in temperature equilibrium with the CMB well after decoupling. At redshift lower than zth the temperature of intergalactic gas falls adiabatically faster than that of the radiation, Te <x a-2.

From the analysis above, the rate of change of the radiation energy density due to Compton scattering can be written as

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