4.2 Diffusive and Hydrodynamic Processes
4.2.3 Heat Diffusion and Hydrodynamic Expansion
Heat transport between the high and low density regions, in particular regions with high and low baryonic pressure and concomitant high and low radiation pressure, can be accomplished by diffusing or free streaming neutrinos or photons. I will be inter- ested in the evolution of inhomogeneities generated by annihilation below tempera- tures of T
<
4.2. Diffusive and Hydrodynamic Processes 45
at the respective epoch and thus only phenomena relevant on these scales will be dis- cussed below. For a more detailed description, the interested reader is again referred to Jedamzik & Fuller (1994).
Neutrino Heat Conduction. The neutrino mean free path is given by
l 100 =G 2 F T 5 R 1 810 11 cm T MeV 5 (N l+ 3 N q) 1 R 1, (4.17)
whereGF is the Fermi constant, andN
l is the number of relativistic weakly interact-
ing leptons at temperatureT. These will includee , , , e, ¯ e, , ¯ , , and ¯ ,
whenever their masses satisfym l
< T. Below a temperature of 20 MeV,N
l = 8 since
onlye
and the neutrinos themselves are still relativistic. SimilarlyN
q is the number
of relativistic free quark flavours at temperatureT and may includeu,d, s,c,b, andt.
Thus N
q = 0 for T
<
100 MeV due to the confinement of quarks into hadrons at the
QCD phase transition. The dimensionless quantity depends on the flavours and on
the exact values of the weak couplings of the neutrinos. Jedamzik & Fuller (1994) have shown that neutrino heat conduction may play a role in the dissipation of large ampli- tude baryon inhomogeneities on small length scales. Of course, dissipation only occurs when the baryon inhomogeneities are accompanied by temperature inhomogeneities. The initial conditions envisioned here, constantjn
b(
r)jand thus constant temperature,
are not of this type. Only when significant annihilation occurs, temperature inhomo- geneities may develop. Only annihilations occuring close to or after weak freeze out are relevant for the light element abundances. At such late times the neutrino mean free path is much larger than the typical size of the fluctuations injn
b( r)j.
Neutrino heat conduction is however most effective when the neutrino mean free path is of the order of the size of a density fluctuation. The scale for the density fluctu- ations is set by the neutron diffusion length
d n 10010 4cm T MeV 9/4 R 1, (4.18)
which is always considerably smaller than the neutrino diffusion length at low tem- peratures. Therefore, neutrinos are not efficient heat transporters at this evolutionary stage of the early Universe, since the weak interactions are not effective any more in transfering energy from the neutrinos to the plasma.
Photon Heat Conduction. After neutrino decoupling and beforee
annihilation heat transport is not effective because the mean free path of the heat transporting photons is very short. During the time ofe
annihilation, the photon mean free path
l e 100 R 1 Tn e (4.19) increases enormously since it is inversely proportional to the pair number density,n
e
=
n e
+ +n
e . After the completion of e
electron number density is given by the sum of the charged nuclei times their electric charge. The increased photon mean free path may then affect the dissipation of fluctu- ations in the baryon-to-entropy ratio (Alcock et al. 1990). Since the photons, in contrast to the neutrinos, remain tightly coupled to the cosmic fluid until the recombination epoch, temperature fluctuations and thus pressure equilibrium cannot be maintained any more.
One may distinguish two limits: dissipation due to diffusive photon heat trans- port and dissipation due to hydrodynamic expansion, according to whether the photon mean free path is smaller or larger than the region of the fluctuations. Borrowing ter- minology from radiation transport studies, Jedamzik & Fuller (1994) labeled these by the ‘optically thick’ and ‘optically thin’ limit, respectively. In the optically thick limit, heat transport is described by the diffusion equation for photons, which is identical to Eq. (4.13), but with
i(
r) replaced by the temperature fluctuationsÆ(r). The diffusion
constant is now given approximately by
D gt g l e (4.20)
withgtthe statistical weight of the heat transporting particles (gt = 2 for photons) and g
is the statistical weight of the relativistic particles still coupled to the plasma ( g
= gt
aftere
annihilation, since the neutrinos are decoupled).
The time to double the size of an entropy fluctuation by photon heat advection may be estimated to be 1 1 510 5s l100 cm 2 T 10 keV 1 2 p+ 3 4 He p+ 2 4 He ! . (4.21)
In the opposite limit, when the photons are free streaming on the scale of the density fluctuations, the temperature will be the same in the high and low density regions and hydrodynamic expansion will effectively damp the density fluctuations. The high den- sity regions will expand towards the low density regions and thereby transport material towards the annihilation region. The motion of the charged particles, protons and light elements, is impeded by the Thomson drag force, which acts on the electrons dragged along by the charged nuclei (Peebles 1971). Balance between pressure forces and the Thomson drag force yields a terminal velocityv= dr100/dt,
v 3 4 T " n e 1 R 2 dP dr100 , (4.22)
where dP/dr100is the radial pressure gradient,r100 the radial coordinate as measured
on the comoving scale, "
the energy density in photons, and n e = n e n e + the net
electron density. One finds for the pressure exerted by baryons and electrons below
T 30 keV P T¯n net 0 B @ X i i+ 0 @ n pair2+ X i Z i i ! 21 A 1 2 n pair 1 C A , (4.23)