Future limits detecting the Cosmic Microwave Background (CMB)

I am working on a part of my story where a long-lived civilization is deciphering information in the cosmic microwave background (CMB). I have read that in the future as distant CMB light reaches us, there will be a larger radius of early space to observe but as the CMB light will redshift further there will be a point that it may be too faint to observe.

One estimate for when the CMB will be undetectable with current technology is in 100 trillion years but as the civilization is highly advanced they may be able to detect much fainter signals, assuming the rate of expansion remains constant.

Is it likely that technology could detect waves far beyond our current capabilities? If so what estimate of time in the future could we put on the limits of detecting the CMB?

• CMB = Cosmic Backgriund Radiation? Jan 14 at 18:00
• @Trioxidane Yes. Jan 14 at 18:02

There are several ways to discuss CMB detectability. The simplest one is that, as you say, the CMB is constantly being redshifted by the expansion of the universe. The expansion of the universe at some time $$t$$ from the present day is described by a scale factor $$a(t)$$, with $$a=0$$ today. It turns out that the frequency of CMB photons scales as $$\nu\propto a^{-1}$$ and has a present-day value of roughly 282 GHz. Ground-based observations from Earth will eventually be limited by the plasma frequency of free electrons in the upper atmosphere, which is roughly 280 kHz at the lower end.$$^{\dagger}$$ So Earth-bound telescopes will no longer be able to detect the CMB at its peak frequency when the universe reaches a scale factor of, conveniently, about $$a\sim10^6$$.

This is a slight simplification, of course, as the CMB is a black body and therefore emits photons across all frequencies. From this argument, then, we will always be able to see the CMB, but we'll start seeing the Earth's atmosphere begin to cut it off significantly once we hit $$a\sim10^6$$.

You could also make an argument that the CMB will no longer be detectable when it becomes too faint - the expansion of the universe also means that its energy density is decreasing, as is the related quantity that radio astronomers refer to as flux density (since the CMB will be far in the radio regime at this point). In the low-frequency limit, the flux density of the CMB is proportional to temperature. In particular, if you're observing at a frequency $$\nu$$, the flux density scales like $$S_{\nu}\propto T_b\nu^2$$, with $$T_b$$ the brightness temperature of the source. This means that the flux density of the CMB should also approximately scale as $$S_{\nu}\propto a^{-1}$$.

Today, at frequencies of several hundred GHz, the CMB has a flux density per solid angle of $$S_{\nu}\simeq3.80\times10^5\;\text{Jy}$$. This means that, given a certain telescope's sensitivity threshold across its receivers' frequency range, you could detect when that telescope will no longer be able to detect the CMB. I'm not as keen on speculating on this, though, as it depends on the particular telescope, and I don't know to how low frequencies spaceborne radio telescopes might be able probe in the far future.

There's a third possible cutoff. If the universe continues to expand at an accelerating rate, there are only a finite number of CMB photons we can receive due to the behavior of the universe's particle horizon, and one day we will indeed no longer be able to observe the CMB. This is a fundamental limit that goes beyond the properties of Earth's atmosphere or how sensitive our telescopes may reach. I don't have numbers for this, either, as it does depend on precisely how the universe continues to expand - but if we do live in a forever-accelerating universe, one day the CMB will no longer be detectable.

$$^{\dagger}$$ The plasma frequency of electrons with number density $$n_e$$ is $$\nu_p=\sqrt{\frac{e^2n_e}{\pi m_e}}\simeq8.97\;\text{kHz}\sqrt{\frac{n_e}{\text{cm}^{-3}}}$$

• For #3, Wikipedia references a figure of 1 trillion years from now: Timeline of the far future Jan 14 at 19:35
• This might be worth its own question but do you know how much more new detail the CMB could contain in the future? by that i mean if cold spots could have far away densities show up in them or details of densities behind hot spots could be resolved? I am wondering if this part of the story needs to be so far in the future. Jan 14 at 20:20
• @Alexander Thanks for the reference; I hadn't seen it. I noticed that the paper cited gets that figure by comparing the wavelength of CMB photons with the particle horizon - a little different than what I had in mind - so I might hold off on citing it if I can find a paper taking a different tack on the problem (also . . . not overly keen on referencing a Loeb paper). Jan 14 at 23:31
• @AlanDavies My intuition says no - the CMB is just cooling and individual regions shouldn't increase or decrease the area they take up on the sky - but I should think about that a bit and get back to you. The other thing is that as the scale factor increases and the temperature of the CMB decreases, the difference in temperature between two regions will also decrease, making it harder to differentiate between hot and cold spots - I think. Jan 14 at 23:32
• Ok, thanks. I just remembered that Cosmic neutrino background and gravitational wave background could be more useful for this part of the story, although they will reveal earlier history. Jan 15 at 12:45