# Does the cosmic microwave background change over time?

Does the cosmic microwave background pattern change over time? I would assume it is getting "cooler" as in more redshifted as time passes, but I am more interested if the pattern on the background, as depicted below changes. I would expect we would see something like the cosmic microwave background retreating one light year (or possibly further due to inflation) every year, and it would be neat if we could make a "3D map" of the background as time passes (e.g. take an image of it every year).

http://lambda.gsfc.nasa.gov/product/map/dr5/map_images/wmap_planck/wmap_planck_ilc_300uK_med.png

The CMB patterns do indeed change over time, although statistically they remain the same, and although it will not be noticeable on human timescales.

The CMB we observe now comes from a thin shell with us in the center, and with a radius equal to the distance that the light has traveled from the Universe was 379,000 years old and until now. As time passes, we will receive CMB from a shell with an increasingly larger radius. As that light has traveled farther through space, it will, as you say, be more redshifted, or "cooler". But it will also have been emitted from more distant regions in the early Universe that, although statistically equivalent, simply will be other regions and hence look different.

The patterns that change the fastest are the smallest patterns we can observe. The angular resolution of the Planck satellite is 5-10 arcmin. Since the CMB comes from a redshift of ~1100, the angular diameter distance defining the physical distance spanned by a given angle — is ~13 Mpc, so 5 arcmin corresponds to a physical scale of roughly 19 kpc in physical coordinates, or 21 Mpc in comoving coordinates (that is, a structure spanning 5 arcmin today were ~19 kpc across at the time of emission, but have now expanded to a size of ~21 Mpc, with 1 Mpc = 1000 kpc = 3261 · 10³ lightyears).

Assuming an isotropic Universe, if the smallest observable parcels of gas were 19 kpc across perpendicular to our line of sight, they are also on average 19 kpc across along our line of sight.

So the question of how fast the CMB changes comes down to how much time did it take light to travel 19 kpc when the Universe was 379,000 years old. This is not simply 19 kpc divided by the speed of light, since the Universe expands as the light travels, but it's pretty close. Hence, it took light roughly 62,000 years to traverse such a patch.

Since we see events at redshift $$z$$ time dilated by a factor $$1+z$$, we will have to wait $$62\,\mathrm{kyr}\times1100$$, or roughly 70 million years (assuming that Planck will not get replaced by better instruments within that time which is, um, dubious).

So you're right, you could make a 3D image of the CMB, but since the patterns are much larger than a light-year, you don't have to take a new picture every year.

• 60,000 years is many orders of magnitude too low for the universe to expand by a factor of 12/11. It should be ~1 billion years. I think the quantity you want is cd.lookback_time(0, (1+zCMBnow)/(1+zCMBfuture)-1, **cosmo). 19 kpc and 21 Mpc looks right to me. The easiest way to compute how long we have to wait for a difference of "1 pixel" is 21 Mpc / c ≈ 68 Myr. That would be inaccurate if there was significant spacetime curvature at that scale, but there isn't. Nov 12 '19 at 18:43
• @benrg Yes you're right. Thanks! Another way to see this is that the ~60 kyr (or more precisely 62 kyr) it took light to traverse the 19 kpc in the past, is seen time dilated by a factor ~1100 today, increasing the time to 1100×62 kyr = 68 Myr.
– pela
Nov 12 '19 at 23:45

When we observe the CMBR we are observing the surface of last scattering, however the comoving points that make up the surface of last scattering (which infact will actually have a comparitvely very small, but non-zero thickness rather than being a 2D surface) change with time, which in theory should correspond to a change in the observed pattern (of anistropy) over time. On human time scales and given the limitations on our ability to observe the minute details of the CMBR, the observable effect on the pattern will be negligible.