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Has the $z\sim 1100$ redshift of the CMB been actually measured by comparing the fingerprint (absorption spectrum) of the CMB with the theoretical radiation pattern of a $2.725\,\mathrm{K}$ blackbody, or calculated by simply dividing the theoretical recombination temperature ($3000\,\mathrm{K}$) by $2.725\,\mathrm{K}$?

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2 Answers 2

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It is the latter. The redshift at which electrons combine with photons is deduced from a calculated temperature at which that (re)combination occurs and the temperature of the microwave background now (see for example How is the Redshift of Recombination Calculated?).

Unfortunately, in terms of its frequency spectrum, the microwave background is almost featureless (although its shape gives us the temperature). That is because even though we expect some photons from transitions between energy levels as the newly formed atoms head towards the ground state, these are swamped by the continuum because photons outnumbered baryons by billions to one in the early universe. Proposals have been made to try and detect this very weak signature though (see Rao et al. 2015 and Why isn't there a high spike visible in the CMBR, due to a massive recombining of electrons and protons).

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    $\begingroup$ @user13964273 - The CMB are the oldest photons we get. Their distribution and its uniformity is some strong evidence against steady steady state models of the Universe. Prior to the emission of those photons, we get plasma-filled space that cannot be directly observed using photons, a very significant limitation to our observations of the early Universe. However, the CMB parameters in combination with other post-plasma-filled observations let us model the quickly expanding plasma-filled epoch as some 400,000 years, thus putting the Big Bang on the timeline. $\endgroup$ Commented Apr 30 at 12:19
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    $\begingroup$ @user13964273 - CMB was, in its day, convincing to many (in relation to steady-state vs. Big Bang) because it was predicted by the Big Bang theory and only much later actually detected. However, CMB alone is just a part of a larger story. And if its minute anisotropies are much of an evidence of anything else, we are still only learning to read those. $\endgroup$ Commented Apr 30 at 12:23
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    $\begingroup$ The CMB IS a prediction of the big-bang model. The physics of at what temperature it occurs is very well-known and depends only weakly on adopted cosmological parameters. The temperature we measure today then tells us by what factor the universe has expanded. @user13964273 $\endgroup$
    – ProfRob
    Commented Apr 30 at 12:54
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    $\begingroup$ @user13964273 unless you can provide an alternative reference, my understanding is that McKellar measured the temperature of molecules in deep interstellar space to be 2.3 K. Nothing to do with the predictions of a steady state theory that doesn't predict a cosmic microwave background and is falsified by it. It would make an interesting Q as to why Alpher etc got the temperature so wrong - it's a simple bit of physics. $\endgroup$
    – ProfRob
    Commented May 1 at 12:26
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    $\begingroup$ @user13964273 as for the 1948 predictions, well I know I said the answer depends weakly on cosmological parameters, but $H_0$ was thought to be $\sim 500$ km/s/Mpc in 1948 and folks had little idea of what $\Omega_{\rm baryon}$ was. $\endgroup$
    – ProfRob
    Commented May 1 at 15:45
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The redshift is measured from the shift in the peak of the blackbody spectrum of the CMB as a function of expansion factor (I think that the original reference for this is Tolman 1934). Indeed the last scattering surface is assumed to occur at a temperature corresponding to the ionization energy of atomic hydrogen. So yes, it's basically a ratio of the temperatures.

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