# I need help from a cosmologist big time. (questions about relic densities, etc.)

Short intro.: I'm a M.Sc. student finishing up a thesis on DM from a HEP aspect. My background in particle physics is ok, but I have had a lot of trouble with the cosmology aspects. Any help would be appreciated since I can't seem to find clear, concise answers for them with references (maybe the answers are necessarily long).

1) Why do people tend to use little h in place of H for Hubble's constant. And what's with the units? To make it of order unity? To embed error? (read that somewhere)

2) Why is the relic density expressed as omega h^2 instead of simply omega? And why as a fraction of the critical density as opposed to the actual total density? That last question is probably a result of a very weak knowledge of FLRW but I feel I should ask.

3) I've heard several times about the analysis of the CMB to obtain the matter-energy content of the universe using a multipole expansion on the power spectrum. The units seem quite confusing and perhaps I should review basic multipole expansion, but I can't seem to see a simple way to understand the basic analysis. If this necessitates a good reference, please recommend me one.

Thanks!
Shiro

Chronos
Gold Member
1. Unsure about 'h' for the hubble constant. 'h' is normally reserved for the Heisenberg uncertainty thing.
2. relic density is dependant on usage.
3. See http://arxiv.org/abs/1012.3191

1) The h is there to convey the error in the measurement of Hubble's constant. At least that's what I assume your talking about. E.g. you could have (nothing physical here): r = 100h km where h would conventionally be taken to be ~ 0.72 if we have a Hubble constant of:

$$H = 72 \pm ( {\rm{Error}} ) {\rm{km}} {\rm{s}}^{-1} {\rm{Mpc}}^{-1} .$$

This is somewhat more historical when people would not wish to make explicit assumptions about the value of Hubble's constant as it was somewhat poorly constrained in the past few decades. It's just a convenient parameterisation to account for that.

2) The energy density is defined as:

$$\Omega_i (t) = \frac{\rho}{\rho_c}$$

where

$$\rho_c = \frac{3 H^2}{8 \pi G} .$$

So the reason that energy densities can be defined in terms of some Hubble parameter, h, is due to the explicit dependence on Hubble's constant, H, and our wish to not make an explicit assumption about the value of H. Again, a convenient parameterisation to avoid explicit assumptions about the cosmological model used.

Forgot to mention: A reason we often calculate things relative to some critical density is that this critical density is something of a characteristic scale for the density in an FRW Universe. It gives us something convenient to compare densities with and acts, in this case, as something of a normalisation. If $$\rho > \rho_c$$ we have an overdense Universe -> Spherical geometry/Closed $$S^3$$, if $$\rho = \rho_c$$ we have critically dense Universe -> Flat geometry $$E^3$$ and if $$\rho < \rho_c$$ we have an underdense Universe -> Hyperbolic geomtry/Open $$H^3$$.

3) As above! Also check out WMAP stuff in general: http://lambda.gsfc.nasa.gov/product/map/current/map_bibliography.cfm

Last edited:
Chalnoth
1) Why do people tend to use little h in place of H for Hubble's constant. And what's with the units? To make it of order unity? To embed error? (read that somewhere)
The small h is a dimensionless parameter to encode the expansion. Many cosmological observations depend trivially upon the expansion rate, and so before we had a good handle on what the expansion rate was, astrophysicists/cosmologists embedded h in their units, with h defined as:
$$h = {H_0 \over 100km/s/Mpc}$$

As for why we use units of $km/s/Mpc[/tex], that, a with many things in physics, is down to history. Megaparsecs are a common unit used for large distances, such as the distances between galaxies, and km/s are a convenient unit for the motions of galaxies (typically galaxy motions are on the order of a few hundred km/s). 2) Why is the relic density expressed as omega h^2 instead of simply omega? And why as a fraction of the critical density as opposed to the actual total density? That last question is probably a result of a very weak knowledge of FLRW but I feel I should ask. [itex]\Omega$ is a density fraction, not a density. $\Omega h^2$ is a dimensionless matter density. You can see this by looking at the first Friedmann equation:
$$H^2 = {8 \pi G \over 3 \rho}$$

With matter and a cosmological constant, for instance, this can be written as:
$$H(a)^2 = H_0^2\left({\Omega_m \over a^3} + \Omega_\Lambda\right)$$

Here what I've done is expressed the matter densities in terms of the fraction of the total density today ($\Omega_m + \Omega_\Lambda = 1$, $a = 1$ today), and factored in the effect of the expansion on the density of each type of matter/energy.

Now, one thing to notice here is that the units work out so that multiplying a density fraction by the Hubble expansion rate squared give something that behaves like density. This is useful for observations where the observation is sensitive to the total density of a certain form of matter, but not to the density fraction (as is the case with WMAP, for instance).

3) I've heard several times about the analysis of the CMB to obtain the matter-energy content of the universe using a multipole expansion on the power spectrum. The units seem quite confusing and perhaps I should review basic multipole expansion, but I can't seem to see a simple way to understand the basic analysis. If this necessitates a good reference, please recommend me one.
Well, first we take a spherical harmonic transform of the map:
$$a_{\ell m} = \int_\Omega m(\theta, \phi) Y_\ell^{m*}(\theta, \phi)d\Omega$$
Since the spherical harmonics $Y_\ell^m$ are dimensionless, the spherical harmonic coefficients $a_{\ell m}$ have the same units as the map units (typically kelvin, millikelvin, or microkelvin). The power spectrum is then:

$$C_\ell = {1 \over 2\ell + 1} \sum_{m = -\ell}^{\ell} a_{\ell m} a^*_{\ell m}$$

Thus the power spectrum necessarily has units that are the square of the spherical harmonic coefficient units, which is the square of temperature.

The reason why the map units are in temperature, by the way, is because the CMB fluctuations are temperature fluctuations.