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.
On Sun, 5 Oct 2003, 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) ?
Maybe Ruth Durrer's paper "The theory of CMB anisotropies" might help?
http://arxiv.org/abs/astro-ph/0109522
e.g. p26
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@astro.uni.torun.pl with "help" WEB ARCHIVE: http://www.astro.uni.torun.pl/sympa/shape-univ/ UNSUBSCRIBE: email to sympa@astro.uni.torun.pl with "unsubscribe shape-univ"
- 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
- "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
- 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.
LISTNAME: shape-univ HELP: send an email to sympa@astro.uni.torun.pl with "help" WEB ARCHIVE: http://www.astro.uni.torun.pl/sympa/shape-univ/ UNSUBSCRIBE: email to sympa@astro.uni.torun.pl with "unsubscribe shape-univ"
cześć,
On Wed, 8 Oct 2003, Michal Frackowiak wrote:
- PLEASE attach a .ps or a .pdf if you are going to present many
formulas!!!! hard to render them on-the-fly and... why?
Or write them to a scratch TWiki page linked from
http://adjani.astro.uni.torun.pl/cosmo
with %$\int_{-\infty}^\infty e^{-\alpha x^2} dx$% i.e. using %$ latex expression $%
- "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.
- 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 ;-)
In other words, "dig some ads or astro-ph or CMBfast code". ;)
Certainly learning what other people have already done is a good idea - but being skeptical about it is also useful and helps in understanding.
E.g.: http://arxiv.org/abs/astro-ph/0310186 Cosmological parameters and the WMAP data Authors: Antony Lewis
which supports the claim that the WMAP data is not as easy to interpret as is claimed by the official team...
pozd boud
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
-
Boud Roukema -
Michal Frackowiak -
szajtan odwieczny -
Szajtan Odwieczny