 # Should we question the Big Bang? – amazing science

Today’s video deals with the “impossibly early” problem of galaxies discovered by the James Webb Telescope. Do they undermine the Big Bang Theory?

Here are always some comments and references for the most curious.

### halo mass function

Central to the argument for improbable galaxies are these curves, which give the expected abundance of dark matter halos according to their mass and redshift. This is called the mass function of halos. I modified this number from the post:

Steinhardt, CL, Capak, P., Masters, D., & Speagle, JS (2016). The Impossible Early Galaxy Problem. The Astrophysical Journal, 824(1), 21.

It tells us the number density of dark matter halos (how many halos are in a given volume expressed in $$\mathrm{MPc}^3$$.) as a function of mass and for different redshifts. Here it is a cumulative function, so it gives the number of halos that have at least X mass in a given volume.

We’re talking about the mass of dark matter halos here, because we know that galaxies form inside these halos, and the baryonic fraction is about 20%. Knowing baryonic matter, not everything becomes stellar mass, meaning that for a given halo, stellar mass is usually less than 10% of the halo’s mass.

If we take for example a halo of 1000 billion solar masses (which would probably give a galaxy the star mass of our galaxy), we see that it is very unlikely at redshift 10, but the probability increases by several orders of magnitude with time (and redshift decreases).

These curves are obtained from a formalism calculation that takes into account initial density fluctuations of matter and makes them evolve during the expansion of the universe to predict the mass function of halos. We can mention Formalities of Schechter Presswhich has since been improved That Sheth Turmin. This formalism is semi-analytical but has a number of free parameters that must be determined.

### the break by Balmer

I’m always a little annoyed when I use fairly specific scientific terms, because a lot of times they really only exist in English. I decided to use the term break Since it is generally found in publications (but I found a thesis in French that talked about it break).

Let’s go back to the mechanism in action. Here is one of the spectra I showed in the video And before we talk about the Palmer station wagon, let’s talk about the Lehman station wagon! It is a fact that below about 370 nm in the above figure, there is basically no signal. but why ?

When the hydrogen atom is in the ground state (n = 1), it can absorb light of different wavelengths. These are the Lyman lines found in UV rays (featured here in Angstroms) From right to left: we see that the alpha line is about 122 nm, the beta is about 103 nm, the gamma is about 97 nm, etc. On top of that, we see the gap between the lines narrowing because the high energy levels are close together. Asymptotically, the 91 nm boundary roughly corresponds to the ionization energy: the fact that the electron is completely shredded from level 1. It is with this value that the Lyman fracture is found. But why not a simple absorption line?

If the wavelength of a photon is 122nm then it can make the electron go from level 1 to level 2. But if its wavelength is 100nm then it is too energetic to reach level 2 but not enough for level 3. We can say that ‘too much energy’ should not be a problem because ‘he who can do more can do less’. Except that energy is conserved, the excess energy has to go somewhere, and there it has nowhere to go. So for energy level transitions we have absorption lines.

But when we get to ionization energy, it’s different! Once an electron is snatched away, all of the extra energy can go into its kinetic energy. This is the principle of Einstein’s equation for the photoelectric effect.

Thus all wavelengths beyond the Lyman limit are likely to be absorbed, hence the presence of this interval in the spectrum. In my spectrum above, since I have a redshift of 3, it’s about 4 times the wavelength of ionization, about 360 nanometers.

All this was Lehmann, that is, when we start from an atom in the ground state. But the same logic applies to an atom in the n = 2 state, giving rise to Balmer lines, which are more familiar to us because they are in the visible (Here they are represented by emission lines but the idea is the same). We find an asymptotic build-up of about 365 nm which corresponds to the initial electron ionization in the case n = 2. We therefore also expect a discontinuity for all wavelengths below 365 nm (multiplied by (1 + p) with the redshift). But be careful, the impact will be less violent than that of a Lehman station wagon!

In fact, for absorption to occur, the atom must be in the n = 2 level, which is potentially more rarefied than in the ground state. Thus, the existence and size of the separator will depend on the amount of hydrogen atoms present in this level, which itself is related to the temperature of the stars.

The effect turned out to be strongest for A-type stars, about 2 solar masses and a temperature of 8,000 to 10,000 K. For less bright stars, there are fewer atoms in level 2 because they are in level 1. Conversely, for brighter stars, hydrogen atoms will be more frequent in higher energy states.

And this is where the Balmer fracture intensity of a galaxy tells us about its star formation (and thus about how that distribution has evolved since the galaxy’s birth).

### Other solutions to the impossible early galaxies problem?

I touched on it briefly in the video, but one solution to the problem of early galaxies might lie in assuming that one takes a function of mass for stars, i.e. the distribution of mass during galaxy formation. The story is similar to what I was saying about the function of halomass. From the galaxies around us, we’ve inferred a kind of mass distribution that seems to be constant all the time.

There are many analytical variables (See Wikipedia) One of the recent publications you mentioned in the video examines the hypothesis that these distributions would not be as applicable as they are for very early galaxies, and in particular because at that time the fossil radiation had a temperature whose influence cannot be neglected. The ‘hot initial mass function’ hypothesis appears to resolve the tension.

Steinhardt CL, Kokorev F, Rusakov F, Garcia E, and Snebin A. (2023). Templates for photometric synthesis of ultraredshift galaxies. Astrophysical Journal Letters, 951(2), L40.

We can also cite a more recent simulation that leads to a similar conclusion

McCaffrey J, Hardin S, Wise J, and Regan J. (2023). Stressless: JWST galaxies at z>10 are consistent with cosmological simulations. arXiv primer arXiv reprint: 2304.13755.