l_p ~ \Omega_tot^-1/2

[Szajtan Odwieczny]

> 1. PLEASE attach a .ps or a .pdf if you are going to present many 
> formulas!!!! hard to render them on-the-fly and... why?
ok

> 
> 2. "why is l_p proportional to \Omega_tot^-1/2"
> it is not. it is an old formula for a matter-dominated universe as I 
> remember. forget it. some people still use it (peacock). dig some ads or 
> astro-ph stuff.
hmm 
a) then what is the use of such mailing list, if everybody can find
and learn everything by himself ? I presume this stuff is been already
done by someone before and I do not claim to discover anything new (in
that case I'd post on cosmo-torun,poland or world ;) or stright to the
nobel foundation ;) ) but this is much faster to learn this way than when
you're on your own. I seek answers everywhere. btw the article by Ruth
Durrer astro-ph/0109522 is very teaching and usefull. thanks

b) if all old formulas were to be forgotten then astronomy would fall
apart. so the proportion is valid only for universe with \Omega_m =
\Omega_tot = 1 or so ? But why ? Sure the position of the peak is
related to the density of matter, but in LCDM models if we variate Omega_m
then Omega_tot also changes at given \Omega_l, so I see no obstacles for
the proportion to work even with nonzero cosmological constant. As I
remember well, k-split of the CMBFAST works this way, to speed up the
calcutations, that it just shifts (and compreses or stretches) the
precalculated spectrum left or right
just to satisfy the given curvature, so given \Omega-tot and thus \Omega_m
for arbitrary \Omega_l :)
but perhaps these are only projeciton effects in spaces of different
curvature
> 
> 3. sonic speed in the relativistic plasma is c/sqrt{3}. right? ;-) BUT: 
don't know but looks suspicious to me  (where are densities) :)
> c_s is NOT constant at that time. it is not that easy and I would rather 
> start with learning what people have already done than playing 
> hide-and-seek with numbers.

> THESE QUESTIONS ARE ALREADY ANSWERED ;-) - and answers were applied to 
> soft such as CMBfast which you use often ;-)

d) does it mean we should treat it as some kind of black box, into which
we put some parameters from one side , and from the other we see what
shape of the spectrum comes out, with none a scant of related mathmatics
whatsoever ? i like to know what is at least the general physical
explanation that the spectra responds in this way or the other for some
cosmological parameter variation.

I just noticed that the equation of state I used is like for baryonic gas,
not necessary as for CDM. ;) aha ! maeybe that's it ;) in fact barions do
not really shift the peak, they only change it's height.

anyway there are still many open threads to follow.

bart.

> 
> regards - michal f
> 
> szajtan odwieczny wrote:
 > 
> >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.
> >
> >
> >
> >  
> >
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