l_p ~ \Omega_tot^-1/2

[Michal Frackowiak]

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?

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.

3. sonic speed in the relativistic plasma is c/sqrt{3}. right? ;-) BUT: 
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 ;-)

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.
>
>
>
>  
>
>------------------------------------------------------------------------
>
>LISTNAME: shape-univ
>HELP: send an email to sympa w astro.uni.torun.pl with "help" 
>WEB ARCHIVE: http://www.astro.uni.torun.pl/sympa/shape-univ/
>UNSUBSCRIBE: email to sympa w astro.uni.torun.pl with "unsubscribe shape-univ"
>
>
>  
>