# How long will it be until the cosmic microwave background is undetectable?

I've had a number of interesting conversations with people about how the cosmic microwave background (CMB) will evolve in the future. The CMB is the sea of photons left over from the epoch of recombination several hundred thousand years after the Big Bang. Right now, the CMB is quite cool, at a temperature of roughly 2.73 Kelvin, certainly detectable by radio telescopes. As the universe ages, however, it will become both cooler and more dilute - and therefore harder to detect.

I'm interested in just how long it will be before the cosmic microwave background is effectively undetectable, using today's technology. Let's assume that we, time-traveling radio astronomers, have access to any radio telescope or interferometer in the world, along with the receivers and backends that come with them. Let's also assume that the universe evolves in time as we would expect, undergoing accelerating expansion dominated by dark energy. **How long will it be before the CMB is undetectable?**

# Observable quantities

The cosmic microwave background is very nearly a black body. Its temperature $T$ evolves with the scale factor of the universe $a$ as $$T=T_0a^{-1}$$ with $T_0$ being the present-day temperature. By Wien's law, its peak emission takes place at a frequency $$\nu_p=\frac{c}{b}T=\frac{cT_0}{b}a^{-1}$$ where $b$ is Wien's displacement constant. As the CMB is isotropic, then by the Rayleigh-Jeans approximation, the flux density at a frequency $\nu$ is $$S_{\nu}=4\pi I_{\nu}\approx\frac{8\pi k_BT\nu^2}{c^2}=\frac{8\pi k_BT_0\nu^2}{c^2}a^{-1}$$ with $I_{\nu}$ the specific intensity and $k_B$ Boltzmann's constant. Therefore, the peak flux density is $$S_p=\frac{8\pi k_BT_0^3}{b^2}a^{-3}$$ We now have expressions for the three quantities most important to a radio astronomer: the temperature of the source, the frequency of peak emission, and the flux density.

# The scale factor

The above calculations depend on the scale factor, the quantity that dictates how the universe expands. In a flat universe like our own, with matter density $\Omega_M$ and dark energy density $\Omega_{\Lambda}$, the scale factor is $$a(t)=\left(\frac{\Omega_M}{\Omega_{\Lambda}}\right)^{1/3}\left(\sinh\left[\frac{3}{2}\Omega_{\Lambda}^{1/2}H_0t\right]\right)^{2/3}$$ where $H_0$ is the Hubble constant. As $H_0\gg1$, this becomes $$a(t)\approx\left(\frac{\Omega_M}{\Omega_{\Lambda}}\right)^{1/3}2^{-2/3}\exp\left(\Omega_{\Lambda}^{1/2}H_0t\right)$$ which is the scale factor for a dark energy-dominated universe.

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