Temporal Variation of Cosmic Microwave Background Radiation

Question

For some time now I am wondering how fast the Cosmic Microwave Background (CMB) Radiation varies with time, in particular, how fast the primary CMB anisotropies are varying. These anisotropies were generated due to density variations etc. at the time when the radiation was "released" at ca. 380000 years after the Big Bang.

I could not find a single site even mentioning temporal variations, but I found a nice paper from the Particle Data Group that's not too technical. References below refer to sections in that paper.

• According to section 29.5.3, the smallest resolved features (by current technology) are for multipole moments with $\ell\approx 2000$ at the end of the "damping tail". This can also be seen in graphs like from figure 29.2.

• According to section 29.3.4, there is no 1:1 relation between moments $\ell$ and the feature size $\theta$, but there is the approx relation of an angular variation of $\theta\sim \pi/\ell$. Hence the smallest resolvable features have $\theta\approx 1.6\cdot10^{-3}$ radians.

• Without metric expansion of space, the above $\theta$ corresponds to a feature size of $d=\theta c t$ where $c$ is the speed of light and $t$ is time of travel. This assumes $\theta$ is small, which is the case, and that the angular size of a feature matches its size in the line of sight, which follows from isotropy. As the signal is conveyed by light, a feature of size $d$ takes time $d/c = \theta t$ to pass Earth at the speed of light. The time of travel is almost the same like the age of the universe, which is $13.8\cdot10^9$ years. Time scale on which variations are to be expected is then $$ \frac\pi{2000}\cdot\text{Age of Universe} \approx 2.1\cdot 10^7 \text{ years} $$ which isn't even close to being accessible by measurement.

The last point ignores expansion of space. If the universe was smaller by a factor of $z$ when the CMB was released, then expansion of space would zoom the signal by the same factor of $z\approx 1100$ while it was travelling towards us. Is that correct? I.e. does expansion of space really cancel out of the calculation? And if not, what are the correct calculation and time scale on which variations of CMB can be expected?

And of course I am not talking about noise or foreground effects, i.e. effects that affected CMB after it was released, like when it passed through clouds of gas.

Answer 1

The expansion does affect the result, but by a much smaller factor. If you use conformal coordinates, then the argument is almost unchanged, except that you need to replace the coordinate time since last scattering with the conformal time, which is about 46 Gyr.

Your calculation is off by a factor of 10: you should have gotten 21 Myr. With the conformal-time correction, it's about 70 Myr.