Today’s video talks about the issue of cosmic inflation: what it is, and why this puzzling phenomenon seems necessary to explain some of the strange features of our universe.

### Other Notes and Agreements

Every time I do an article or video about cosmology, it’s the same circus for me: I have to go back to equations and symbols to find my cosmology. It must be said that no one uses the same conventions, and I find there are many sources of confusion. We can mention for example:

- The fact of taking \(k\) sometimes as curvature (homogeneous with the inverse of length squared) and sometimes as a dimensionless number equal to 0, -1, or +1 (in which case, for +1 and -1, \(a\) becomes homogeneous with length, not a dimensionless number!)
- The fact of setting \(k = 0\) is sometimes straightforward and just handle this case.
- Using either matter or energy density (There is a factor \(c^2\) between the two.)
- Related: adjusting \(c = 1\) which makes it difficult to check the dimensions.
- Write equations that use both \(H\) and \(\point {a}\)
- ambiguity about the fact that the quantities are time dependent, or that we consider their current value (eg: critical density or \(\omega))
- Nature of the contributions of matter and energy (dust liquid, radiation, with or without dark energy/cosmological constant).

In short, I told myself I had to write myself a special reminder with the notes that seemed to me the least ambiguous and most relevant. I’ll probably make a PDF doc with that if I have the guts, meanwhile if I get lost in these notes I’ll put you a reminder of my handwriting! (There may be typos left but I couldn’t be farther away)

### about flatness

The last equations are important for our purposes. In particular: **Critical density is a function of time! **It depends on the Hubble coefficient \(H

\(\Omega

Particularly thanks to this, we can see that to obtain a value of \(\Omega = 1\) within 1% at present, a value equal to 1 in \(10^{-64 }\) closes at The first few moments.

The last equation looks a bit barbaric but useful: it shows very well that the flatness \(\omega = 1\) is **unstable balance**. If we are a little lower, the value will decrease, and if we exceed it a little, it will increase. This is because of the factor \((1 + 3w)\) which is always positive for ordinary forms of matter and energy (w = 0 for dust, w = 1 for radiation).

And see how the form of dark energy-type matter changes the game. If \(w < -1/3\), then we flip the trick and can have \(\Omega \) leaning towards 1 instead of away from it:** The universe will tend toward flatness** !

### about homogeneity

I didn’t really insist, but when we look at fossil radiation for example, we can identify the idea of a causally linked patch, which marks areas that would have been in contact without having to invoke inflation. The angular size of this patch is on the order of 2 degrees. This means that when you look at two directions separated by less than 2°, we can explain the homogeneity by saying that these regions were causally connected at the time of the cosmic microwave background emission. But after two degrees that is no longer the case.

In addition, on the “minimum” calculations of the amount of inflation needed to explain the homogeneity of the observable universe, you can for example go to see these accounts.

### slow cycle

I cut a bit in my text, initially elaborating a bit more on the idea of slow rolling of inflation models. Basically, if you have a standard field that plays the role of an inflaton, to define your model you have to choose a possibility for that field. Forms of voltage that work well are called “slow-rolling”, in which the field is “slow-rolled” in power to zero power (so that the inflation stops).

But there can be a whole host of capabilities and models that meet the conditions for slow rolling. Hence the idea of restricting the probabilities to the values of \(n_S\) and \(r\), both of which are dimensionless:

- \(n_S\) is the exponent that governs the oscillation spectrum, which is a powers law in \(\sim k^{n_S-1}\)
- \(r\) is a ratio between the scalar and tensor components of the primitive fluctuations. We can mainly relate this to the primordial forms of gravitational waves, the peculiarity being that all models of inflation predict an absolutely positive value.

To learn more, I wrote a long time ago Article on BICEP2 ! (Before Discovery collapsed, and I’m happy to point out I was somewhat wary at the time!)

Furthermore, in conclusion, we must do justice to Andre Linde. When I say in my video “All these little people have already seen each other with the Nobel Prize,” I’m thinking more about the people responsible for the experiment, and not about Linde, who was very careful. The proof is his extremely skeptical reaction when someone comes to tell him of the discovery.