When will the CMB reach radio wavelengths?

Question

How long will it take for redshift to turn the microwave-frequency photons from the CMB into radio frequency? Into the CRB, if you will.

Answer 1

The CMB spectrum (per Hz) currently peaks at about 160 GHz (about 5 cm$^{-1}$). Some would already class that as being part of the high frequency radio spectrum. However, as an example, let's say you wanted the peak to be at 10 GHz.

Since frequencyis redshifted by the inverse of the scale factor, then we need $a/a_0 \simeq 16$ - i.e. the universe needs to get bigger by a factor of 16.

An analytic approximation for the scale factor as a function of time is given by $$ \frac{a}{a_0} \simeq \sqrt[3]{\frac{1}{\Omega_{\Lambda} }-1} \ \sinh ^{2/3}\left(3 \ H_{0} \ t \ \sqrt{\Omega_{\Lambda} }/{2} \right)$$ for a flat $\Lambda$CDM universe, where $\Omega_\Lambda$ is the dark energy density and $H_0$ the current Hubble parameter (see https://physics.stackexchange.com/a/712026/43351 ). This approximation is valid once the radiation density becomes negligible, which is the case from a few million years after the big bang.

Rearranging, we have $$ t \simeq \frac{2}{3H_0 \sqrt{\Omega_\Lambda}} \sinh^{-1}\left( \frac{a/a_0}{\sqrt[3]{\Omega_{\Lambda}^{-1} -1}}\right)^{3/2}\ . $$

If we assume $\Omega_\Lambda = 0.7$ and $H_0 = 69$ km/s/Mpc ($=0.0706\ $ Gyr$^{-1}$), we can check that when $a/a_0 =1$, we get $t \simeq 13.7$ Gyr. If we assume $a/a_0=16$, then we get your answer of $t = 59.5$ Gyr (i.e. 46 Gyr from now).

This of course assumes the $\Lambda$CDM model is correct and in particular, that $\Omega_\Lambda$ does not change.

To bring the frequency peak down to 1 GHz would require $a/a_0 = 160$ and $t=98.6$ Gyr (85 Gyr from now).

The main uncertainty in terms of precision is the current few percent uncertainty in $H_0$ (a.k.a. the Hubble tension) leading to a similar uncertainty in the time estimates.

Answer 2

For one, it depends on precisely what counts as radio. For an example answer, I'll rephrase as "How long will it take to shift the peak of the CMB spectrum down to $1\,\mathrm{GHz}$?"

Right now, at $t=t_0$, the CMB has a temperature of around $T_0\approx 2.725\,\mathrm{K}$. This corresponds to a peak frequency of $\nu_0\approx160\,\mathrm{GHz}$. As the universe expands, the wavelength of the CMB photons will be stretched in proportion to the scale factor $a(t)$. As such, in terms of frequency we will see a scaling $\nu\propto a^{-1}(t)$. Finding the time to cut our frequency by a factor of 160 to bring it down to the radio region is thus the same thing as finding the time when the scale factor becomes 160.

Because the universe is (increasingly) dominated by dark energy, we can roughly model the scale factor into the future exponentially: $$a(t)\propto e^{H_0(t-t_0)}$$

From this, we can simply take the logarithm of both sides and do a little rearranging to find a form for the time to reach any future scale factor: $$t-t_0=\frac{\log a}{H_0}$$

For our example of bringing our peak frequency down a factor of 160, this leads to

$$\begin{align*} \Delta t&=\frac{\log 160}{2.2\times10^{-18}\,\mathrm{s}^{-1}}\\ &\approx 73\,\mathrm{Gyr} \end{align*}$$

This is a very rough estimate. I would not interpret this any more precisely than as an order of magnitude estimate of $\sim10^{11}\,\mathrm{yr}$.