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According to the lambda CDM model, $H_0$ is around 67.7. If this value was calculated from a value for $H$ at the emission of the CMB, what was it, or do we have constraints on its value at that time?

Qmechanic
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user141183
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2 Answers2

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No, the current CMB based value for $H_0$ is not directly calculated from the value for $H_\mathrm{ls}$ at the time of last scattering when the CMB was emitted. As discussed in the answer to "How is Hubble's constant (the expansion rate) predicted from LCDM and the CMB?", the value depends on comparing the currently observed angular distribution of CMB intensity fluctuations to Lambda-CDM model predictions for ther physical size of these fluctuations at the time of last scattering. The longer the time (which depends on $H_0$), the bigger the angles.

As discussed in the answers to

$H$ does change with time. The value at the time of last scattering can be calculated in the standard cosmological model to be:

$$H_\mathrm{ls} = 100\,\mathrm{km/sec/Mpc}\;\omega_r^{1/2}(1+z_\mathrm{ls})^2\sqrt{1+\frac{\omega_m}{\omega_r}\frac{1}{1+z_\mathrm{ls}}}\sim 2.3\times 10^4 H_0$$

where

  • $\omega_r$ is the total physical radiation density, $$\omega_r=\left[1+\frac{7}{8} N_\mathrm{eff} \left(\frac{4}{11}\right)^{4/3}\right]\omega_\gamma$$
  • $\omega_\gamma= 2.47\times 10^{−5}$ is the physical photon energy density,
  • $N_\mathrm{eff}=3.06$ is the effective number of light neutrinos,
  • $\omega_m=0.142$ is the physical nonrelativistic-matter density, and
  • $z_\mathrm{ls}\approx 1080$ is the redshift corresponding to the time of last scattering.

For estimated constraints on $H_\mathrm{ls}$, I leave it as an exercise to look up the uncertainties in all the imput values and estimate the total uncertainty. (Model uncertainties and possible correlations between the input values may be an issue.)

David Bailey
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The CMB was emitted when

$\rm a=1/(z+1)=1/1090$,

so with

$\rm H(a)=H_0 \ \sqrt{\Omega_R/a^4+\Omega_M/a^3+\Omega_{\Lambda}}$

and

$\rm \Omega_R=9.2136e-5, \ \Omega_M=0.315, \ \Omega_{\Lambda}=1-\Omega_R-\Omega_M$

we get

$\rm H(a)=23194 \cdot H_0$

Yukterez
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