9.6 Delayed recombination and the finite thickness effect 373smaller than the horizon scale. Therefore, one can try to use the result (7.128),
obtained in the imperfect fluid approximation, to estimate the extra dissipation
during non-instantaneous recombination.
Problem 9.7Using (3.202) for the ionization fraction (which is valid when the
ionization fraction drops below unity), calculate the dissipation scale and show
that
(kDη)−r^2 4. 9 × 10 −^3 c^2 s√
(^) mh^275
η 10
+
12
5
c^2 sσ^2. (9.62)The first term here is what was obtained for instantaneous recombination (equa-
tion (7.132)), where we have expressed (^) mh^2 in terms ofη 10 ≡ 1010 nb/nγ(see
(3.121)). This term accounts for the dissipation before recombination starts. The
second term accounts for the additional dissipation during recombination.
Note that the second term in (9.62) corresponds to a scale which is smaller than
the mean free pathτγatηr, and so the imperfect fluid approximation cannot be
trusted. However, within the time intervalη∼ηrσ, when the visibility function is
different from zero, free photons have only enough time to propagate the comoving
distanceλ∼ηrσ, which is roughly the second term in (9.62). The inhomogeneities
in the radiation can be smeared (mainly because of free streaming) only within these
scales but not on larger scales. Therefore, the result (9.62) can be still used as a
reasonable rough estimate for the scale below which the inhomogeneities will be
suppressed.
At very low baryon density the first term in (9.62) dominates, and most of the
dissipation happens before ionization significantly drops. However, for realistic
values of the dark matter and baryon densities, (^) mh^275 0 .3 andη 10 5, the sec-
ond term can be nearly twice the first term. Thus, the corrections to Silk dissipation
due to noninstantaneous recombination can be important.
Problem 9.8If the baryon density is too low the approximations used above are
invalid. What is the minimal value ofη 10 (or (^) bh^275 ) for which the derived results
can still be trusted? For smaller values, how would the results be modified?
In summary we have found that the finite duration of recombination produces
two effects. First, the damping scale can be essentially greater than if recombination
were instantaneous. Second, the uncertainty in the time of decoupling results in an
extra suppression of temperature fluctuations on small angular scales. Although
both effects are interconnected, they are distinct.
The key formulae derived in the instanteneous recombination approximation are
easily modified for the case of delayed recombination. Namely, for the damping