A very simple question: why should it be that the CMB power spectrum allows constraints to be placed on the combination of parameters $$\omega_c = \Omega_c h^2$$ $$\omega_b = \Omega_b h^2$$ as opposed to $\Omega_b$ and $\Omega_c$? I understand roughly where the constraints come from in terms of matter-radiation equality determining the scale at which modes have some extra radiation driving determining the total matter density in the early Universe and the Baryon density determining the ratio in peak heights but my question is why is it the physical densities that are constrained- why should the $h^2$ factor need to be included here?

  • $\begingroup$ For those of us who haven't seen these equations before, would you mind defining the variables and perhaps citing a source where we could get more information? $\endgroup$
    – Connor Garcia
    Mar 17, 2021 at 16:29
  • $\begingroup$ The variables $\Omega_i$ are defined as $\frac{\rho_i}{\rho_{crit}}$ where $\rho_i$ is the energy density of component i and $\rho_c$ is the critical energy density, $\omega_i$ parameters are defined in the original question. An example source which describes these parameters is: en.wikipedia.org/wiki/Lambda-CDM_model. My question is why should the CMB TT spectrum be sensitive to $\omega_i$ not $\Omega_i$. $\endgroup$ Mar 18, 2021 at 14:19
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    $\begingroup$ Well $\rho_{crit} = 3H_0^2/(8 \pi G)$, so if you can measure $\rho_i$ you still have the dependence on the unknown value of $H_0^2$. $h = H_0/100$ $\endgroup$
    – eshaya
    Mar 18, 2021 at 20:30
  • $\begingroup$ Yep just realised that! Thanks that’s exactly what I was looking for thanks- for some reason I had it in my head that $\Omega_i$ were the fundamental densities $\endgroup$ Mar 22, 2021 at 20:40


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