l_p ~ \Omega_tot^-1/2

[szajtan odwieczny]

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.