I have read somewhere that CMB (cosmic microwave background radiation) fluctuations in temperature are linked to mass distribution fluctuations in the early universe (at ~350000 years after Big Bang, which is of course when the cosmic radiation was emitted), and that is used to explain the formation of large structures (galaxies, clusters of galaxies..). Why is that so?
[Physics] the origin of CMB fluctuations
astrophysicsbig-bangcosmic-microwave-background
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The cosmic microwave background does not originate with the big bang itself. It originates roughly 380,000 years after the big bang, when the temperature dropped far enough to allow electrons and protons to form atoms. When it was released, the cosmic microwave background wasn't microwave at all- the photons had higher energies. Since that time, they have been redshifted due to the expansion of the universe, and are presently in the microwave band.
The universe is opaque from 380,000 years and earlier. The galaxies that we can see only formed after that time. Before that, all that is observable is the CMB.
You have asked a lot of questions here so my apologies if I do not cover them all. Let us start with a misconception which you may (or may not) hold:
The CMB does not originate from the big bang it originates from what is called the surface of last scattering. This occurred about $380,000$ years after the big bang when the universe had a temperature of about $0.23-0.25eV$ ($\sim 3000K$). Consider the reaction: $$p+e^-\leftrightarrow H+\gamma \tag{(1)}$$ above $3000K$ the photons have enough energy to cause this reaction to go into reverse once the universe as cooled (due to the expansion of the universe) the energy of the photons is to small for the reverse reaction to occur. The consequence of this is that all the protons and electrons get taken up to form hydrogen atoms. With no free ions in the universe Thomson scattering becomes less probable ans the universe becomes optically thin. I.e. Before this time photons would travel a very small distance before scattering but after they will travel a very large distance - so large that the photons we see as the CMB are the photons from this time which are yet to scatter.
Now this surface of last scattering will occur at $3000K$ no matter what happened before it (this is determined by something known as the Saha equation). So only the rate of expansion after this time will effect the redshift of the CMB. Now inflation is thought to have happened within approximately $10^{-32}$ seconds after the big bang. Clearly $10^{-32}$ seconds $\lt $ $380,000$ years so inflation will not affect the redshift of the CMB.
Now you actually pose another query in your question - whether the cosmic horizon will change during inflation. The answer to this is yes. And no there is no contradiction here - the cosmic horizon is different from the surface of last scattering. The Cosmic Horizon is the furtherest point we could possible see. I.e. if a photon started traveling at the big bang singularity and has carried on in a straight line (without scattering) to reach us today - the distance it has traveled is the cosmic horizon. This is affected by inflation because the photon is assumed to be traveling through this period.
All that said inflation does have an effect on the CMB. It explains why it looks so homogeneous (something called the horizon problem). Inflation explains this since it means a very large are of the universe now was once all in causal contact since before inflation is was much smaller - allowing photons from one side to the other.
EDIT
Concerning the question of evidence of the age of the CMB. We can only measure the CMB as it appears to us now. This includes things like the wavelength, blackbody spectrum and the angular power spectrum of the anisotropies of the CMB. To work out the age of the CMB however, you need to know the temperature of the CMB when it was created and compare that to its value today (which we can measure). There is no way to measure the temperature when it was created (we simply weren't there) and thus for this we need the theoretical calculation involving the Saha equation.
Best Answer
The anisotropies are very small,
One needs a model and the dominant one is the Big Bang model
This light stopped interacting with electrons and nucleons at 380.000 years after the BB, before gravitational attraction formed stars and galaxies. The great uniformity of temperatures, at the level of 10^-5, cannot be explained thermodynamically because there was no uniform communication over the then universe ( due to special relativity) of the particles and energy forms before the photon separation, to homogenize the soup. This uniformity forced the inflation period in the very beginning of the Big Bang model. The rapid inflation homogenized the primordial soup so that we end up with the observed CMB spectrum.
The CMB has been used to model a homogeneous early inflation period to explain the CMB observation. Once the CMB homogeneity was modeled by the inflationary period, the homogeneity in the density of cluster of galaxies and galaxies also follows.
The CMB does not explain the formation of large structures, those are explained by statistical mechanics and gravitational forces as the universe cooled from the BB . It explains the homogeneity , which is still under study.