Breaking of time diffeomorphisms in inflation

[Anne-Sylvie]

Hello here,

I am currently working on the topic of inflation.

It seems that at the stage of inflation, the universe can be described as a de Sitter space. In such a space, all spacetime diffeomorphisms are preserved. (That is something I don't really understand but I keep reading that so I admit it for now.)

Now, I read that, in order to give the FLRW universe that we know today, the time diffeomorphisms are broken and therefore there is a Goldstone boson associated with this symmetry breaking. I also read somewhere else that it is not the inflaton field that is concerned by this broken symmetry but rather the fluctuation field $\delta \phi$ defined as $\phi(x,t) = \phi_0(t) + \delta \phi(x,t)$.

• What is the Goldstone boson ? It is not the inflaton field...
• Does it makes sense to say that only time diffs are broken, since space and time coordinates are not so evidently separated ? I know that the background field $\phi_0$ gives a clock but I'm still confused.
• Why is FLRW not invariant under time diffs ? Because of the evolution of $a(t)$ ?
• I thought that in order to apply Goldstone's theorem, the broken symmetry had to be continuous. When we speak about broken time diffs, is it a continuous symmetry ? (Maybe because it's only shift in time ?)

Sorry if it is not clear, as you can see I'm confused about this topic. Thank to anyone who can provide some answer, even partial.

 

[AndyW]

Hello there,

Thank you for your post and for your interest in the topic of inflation. I can understand your confusion, as the concept of symmetry breaking in the context of inflation can be quite complex. I will try to address your questions to the best of my knowledge.

Firstly, let's start with the concept of a Goldstone boson. A Goldstone boson is a type of particle that arises when a continuous symmetry is spontaneously broken. In the case of inflation, the symmetry that is broken is the time diffeomorphism, which is a continuous symmetry of the de Sitter space. This symmetry breaking results in the creation of a Goldstone boson, which is associated with the fluctuations in the inflaton field. This is where the concept of the fluctuation field \delta \phi comes in. It is not the inflaton field itself that is concerned with the broken symmetry, but rather the fluctuations around it.

Now, you may wonder why only the time diffeomorphisms are broken and not the space diffeomorphisms. This is because in the context of inflation, the universe is expanding at an accelerated rate, which breaks the time symmetry but not the spatial symmetry. This is why we say that only the time diffeomorphisms are broken.

Regarding the FLRW universe, it is not invariant under time diffeomorphisms because of the evolution of the scale factor, a(t). As the universe expands, the scale factor changes with time, breaking the time symmetry. This is why the FLRW universe is not invariant under time diffeomorphisms.

Lastly, in order to apply Goldstone's theorem, the broken symmetry does not necessarily have to be continuous. In the case of broken time diffeomorphisms, it is considered a continuous symmetry because it is only a shift in time.

I hope this helps clarify some of your confusion. If you have any further questions, please feel free to ask. Thank you again for your interest in this topic.