Problem set X: From CMB to super-cluster due Monday 11 Apr, 12pm
believe or not, we are now at the end! I hope you have all become wiser from all the hard works.
The first peak on CMB was originally detected by the COBE satellite in 1992. But it was predicted by a fundamental and
foresightful paper, Peebles & Yu (ApJ, 1970, 162, 815), based on the observed structure in the present universe. This
paper (and others) led Peebles to win the Nobel Physics prize (2019). Here, we follow his logic (at least in spirit) to
connect the dots from CMB to super-clusters.
For the universe, adopt the following parameters: our universe is flat; at the current moment, the universe is mostly made
of two components, matter and dark energy, with ⌦M,0 ⇠ 0.3 and ⌦⇤,0 ⇠ 0.7; but at higher redshift, the impact of
dark energy becomes less important, and we can consider a universe that is matter dominated. At the time of CMB, the
universe is roughly 370, 000 yrs old.
In the following questions, you are asked to perform a series of estimates.
- CMB shows that the dimensionless amplitude for the primordial temperature fluctuation is T /T ⇠ 3 ⇥ 105.
Assume a similar amplitude for the density fluctuations (an assumption we revisit in the last question). Compared to
the average universe, the denser patch should expand a little slower, and as a result, it should become even denser
relative to the average density. The density di↵erence should go as eq. (18.1) in the Lecture Notes. Follow the
discussion in CO §30 to derive this expression, and estimate at which time t the denser spot become about twice
the universal average – this is when structure formation will begin in earnest. (Here is a link to the relevant pages
in CO.) - You may find a strange answer above. But let’s march on regardless and connect the CMB spots with the
super-clusters. The largest structure in the universe today is the so-called ’super-clusters’, clusters of galaxies
that are marginally bound by their own gravity against the universal expansion. The Milky Way is on the outskirts
of one such super-cluster (’Laniakea’) with a radius of ⇠ 0.3 Glys (or ⇠ 100 Mega-parsecs). How high the mass of
our super-cluster has to be (express in solar mass) to resist the universal expansion today, with a Hubble constant
of H0 = 72 km/s/Mpc? For comparison, the Milky Way (a typical galaxy) has a mass of ⇠ 1012M - The universe today has a mean density of ⇢mean = ⌦M,0⇢crit, where ⇢crit is the critical density. How high is the
mean density of this cluster, compared to ⇢mean? Given the mean density of the cluster, what is the free-fall time?
And given this, can you explain why structures larger than super-cluster haven’t yet appeared in the universe? (Hint:
in our universe, larger structures are more flu↵y) - Such a super-cluster is called a ’collapsed’ structure, meaning, even though it is now much denser than the mean
density of the universe, the collapsed material were sourced from an original patch that has pretty much the mean
density of the universe. Calculate the following two distances: the comoving distance for the original patch; the
physical size of the original patch at redshift z = 1100. Let the latter be lpatch. - For a patch that has the physical radius of lpatch but placed at z = 1100, how large an angle does it extend to an
observer on Earth? How does this compare with the CMB first peak at an angular size of ⇠ 1 deg? (Hint: Fig. 14.1
can be used) What conclusion can you draw from this calculation? - We now return to the discrepancy between your answer in the first question and the fact that super-clusters have
collapsed in our current universe (age 13.7 Gyrs). That can be explained by the presence of dark matter. Dark
matter decouples from radiation earlier and they are allowed to collapse under their gravity earlier, while baryons are
tied to radiation and are only liberated till recombination. For super-clusters to have started collapse just now, how
large the density fluctuations have to be at the time of CMB?