for those who are not in the topic it is about the question why is l_p (which is the number of the multipole on which there is the first peak - so called acustic peak - in the angular power spectrum of CMB fluctuations) proportional to \Omega_tot^-1/2 (which is the unitless total density of the Universe) ?
Lately I was thinking about such explanation but don't know if this mae be correct.
l_p \approx EH_LSS / SH_LSS where EH_LSS is (say *) the event horizon at the time of last scattering and SH_LSS is the sonic horizon at the last scattering (t=t_LSS).
EH_LSS = c * \int_{t_LSS}^{t_0} dt/S(t) SH_LSS = c_s * \int_{0}^{t_LSS} dt/S(t)
where c - speed of light, c_s - speed of sound, S(t) - is scale factor. So it's just the angle under which today we see the sonic horizon as it was at the time of last scattering.
so from the above it would be that:
l_p ~ c_s^-1 (~ means "proportional" here)
the speed of sound is defined by:
c_s = \sqrt{ (P/\rho)_S } (while entropy - S is constant) (P- pressure, \rho - density)
for adiabatic transformation the entropy is or mae be conserved and the equation of state is:
P\rho^-\gamma = const where \gamma is the adiabatic index
So in short we have
c_s ~ \rho^{ (\gamma - 1)/2 }
and thus
l_p ~ \rho^{ (1-\gamma)/2 } ~ \Omega_{tot}^{ (1-\gamma)/2 }
so if the adiabatic index gamma for primordinal plasma is 2 then this consideration mae answer the quiestion, however gamma for nonrelatistic, in moderate temperatures, single-atom gases is something like 1.67 - not 2. Don't know how it is with hydrogen plasma in temperature of few thousand K.
Any comments much appreciated. like it might be ok. or this is complete nonsense. ------------ Boud:
* - this is the comment about the event horizon.
sure that the event horizon is the maximal distance from which, say some light, will ever reach us - and because of that I agree there should be infinity in the top limit of integration in the above formula for event horizon, but I guess if it comes about the event horizon at the time t=t_LSS then wheather there is \infty or just t=t_0 doesn't change the integral much, because from the t=t_LSS point of view the time like 14,2*10^9 y is like infinity. (Because the cones of light of some simultanous events at the time t_LSS drown in "normal" (proper ?) coordinates (not comoving) which are separated (the events) by the distance of event hirizon at the time t=t_LSS, become almost paralel.**) So in other words the size of the event horizon given by the above formula will probably be just a little smaller than the real event horizon at the time in case when integrated to infinity.
bart. -----
btw. It is interesting that:
If calculate l_p for dust universe where S(t) ~ t^2/3 it is easy to show that l_p \approx 35,2 * c/c_s and thus for l_p=220 where it is observed, we see that at the time of last scattering the sound of speed was about 6,25 times smaller than the speed of light which sounds resonable ;) (oh, this is for flat univ.)
** - this requires some more work to think, and mae be not precise explanation.