Are the Friedmann Equations Applicable to the Pre-Inflationary Epoch?

[timmdeeg]

I wonder whether the Friedmann equations are applicable to the pre-inflationary epoch and if there are any theoretical expectations regarding the spatial curvature at that early times.

I suspect the curvature parameter $k$ should have a certain value at that time already and which never will change in the future, correct?

Does - as the inflation has not yet started - this mean that the universe is static and if yes, static due to the cancelling of repelling and attractive gravity?

Thanks for any help.

 

[skydivephil]

The universe is not necesarily static prior to infation, here is one possibility:
http://arxiv.org/abs/1302.0254

 

[marcus]

The universe is not necesarily static prior to inflation, here is one possibility:
http://arxiv.org/abs/1302.0254

Good paper to cite! Great example of recent research in pre-inflation cosmology.
There is a moment when H crosses from negative to positive (from contraction to expansion) So there is a moment when H = 0 and the universe is neither contracting nor expanding. It is not "static" in the usual sense---the bounce is extremely dynamic---but it is a moment when the opposite tendencies cancel---that could be what Timmdeeg was asking about.

The paper derives a quantum-modified Friedmann equation which makes it clear why there is a bounce at near-Planckian density and allows the critical turnaround density to be calculated. Timmdeeg asked about the Friedmann equation so he might be interested in that. I forget what page it's on.

 

[timmdeeg]

.
There is a moment when H crosses from negative to positive (from contraction to expansion) So there is a moment when H = 0 and the universe is neither contracting nor expanding. It is not "static" in the usual sense---the bounce is extremely dynamic---but it is a moment when the opposite tendencies cancel---that could be what Timmdeeg was asking about.

No marcus, I didn't have this moment in my mind, but the universe prior to inflation being in a high energy state due to the inflaton field.
According to my impression the bounce models are very speculative and apart from this said moment isn't a static state per se.

 

[bapowell]

No marcus, I didn't have this moment in my mind, but the universe prior to inflation being in a high energy state due to the inflaton field.

Can you elaborate on what you mean by this? The inflaton would have been a subdominant component of the total energy of the pre-inflationary universe.

 

[timmdeeg]

Can you elaborate on what you mean by this? The inflaton would have been a subdominant component of the total energy of the pre-inflationary universe.

My question refers to the state of the universe before the scalar field started to roll down the potential energy hill. How would we describe this state? It seems obvious to describe it as a static universe, as long as it doesn't expand. But on the other side it seems nonsense to think of a static universe which contains vacuum energy only.

According to Friedmann any kind of energy density creates a time dependence on the scale factor, unless repelling gravity is balanced by attractive gravity. So, in case there exists a static state prior to inflation which kind of a vacuum energy wouldn't force the universe to expand?

 

[bapowell]

My question refers to the state of the universe before the scalar field started to roll down the potential energy hill. How would we describe this state? It seems obvious to describe it as a static universe, as long as it doesn't expand. But on the other side it seems nonsense to think of a static universe which contains vacuum energy only.

If it contains only the inflaton (i.e. vacuum energy only) then of course it inflates. In order to not inflate, you must have other stuff in the universe that dominates the energy density. For example, prior to inflation, it's entirely possible to have had a radiation-dominated universe, with matter and inflaton subdominant. Remember -- to get inflation going, the inflaton must come to dominate the local energy density. Lastly, in order to not expand, you must have the right balance of matter/radiation and vacuum energy.

According to Friedmann any kind of energy density creates a time dependence on the scale factor, unless repelling gravity is balanced by attractive gravity. So, in case there exists a static state prior to inflation which kind of a vacuum energy wouldn't force the universe to expand?

Regular old vacuum energy will work: recall Einstein's employment of the cosmological constant for just this purpose. However, I don't see a good reason to postulate a static universe prior to inflation, since the conditions for regular Friedmann expansion are much easier to achieve and understand.

 

[timmdeeg]

However, I don't see a good reason to postulate a static universe prior to inflation, since the conditions for regular Friedmann expansion are much easier to achieve and understand.

Would you agree that a non-expanding universe containing nothing else than vacuum energy exists prior to inflation? But if so, how would you describe it?

Or perhaps do we simply have to admit that we don't know why it doesn't expand, although it contains vacuum energy only.

 

[bapowell]

Would you agree that a non-expanding universe containing nothing else than vacuum energy exists prior to inflation? But if so, how would you describe it?

No, I'm trying to say that this is impossible. Vacuum energy contributes a constant energy density which leads to accelerated expansion in the Friedmann universe. The most generic pre-inflationary universe would be one containing matter, radiation and inflaton field (vacuum energy) with the latter subdominant so that the universe undergoes non-inflationary, radiation- or matter-dominated expansion.

 

[timmdeeg]

The most generic pre-inflationary universe would be one containing matter, radiation and inflaton field (vacuum energy) with the latter subdominant so that the universe undergoes non-inflationary, radiation- or matter-dominated expansion.

Yes, but this possibility can be ruled out too (besides vacuum energy only), because matter was produced later (at the of inflation).

 

[bapowell]

Yes, but this possibility can be ruled out too (besides vacuum energy only), because matter was produced later (at the of inflation).

Yes, the inflaton decayed into the matter and radiation making up the observable universe; however, this does not rule out the possibility that matter and radiation existed prior to inflation. What observation could one make to rule out this possibility?

Remember, the inflaton, if it exists, is likely nothing more than a scalar matter field, not unlike the Higgs. There is no "vacuum energy only" solution that works because pure vacuum energy leads to eternal inflation -- it has no dynamics. The inflaton would have come into existence along with all the other kinds of particles at the big bang. In some part of the universe after the big bang, the inflaton field energy density was extraordinarily high, higher than the energy densities of all the other particles. This part of the universe underwent inflation, diluting all the non-inflaton energy density. Matter and energy is re-introduced as the inflaton decays towards the end of the inflationary epoch.

In fact, many cosmologists assumed that the inflaton existed in thermal equilibrium with other matter and radiation prior to inflation; these thermal interactions were perhaps key to the inflaton developing a VEV as the universe cooled.

 

[timmdeeg]

Thank you for these helpful explanations, you are describing exactly what I've been missing.

In fact, many cosmologists assumed that the inflaton existed in thermal equilibrium with other matter and radiation prior to inflation; these thermal interactions were perhaps key to the inflaton developing a VEV as the universe cooled.

Does "other matter and radiation prior to inflation" mean particles with and without rest mass of unknown nature, or how should I think of "..."?

I try in my own words, kindly correct if wrong:

The inflaton which is a scalar matter field has existed in thermal equilibrium with "other matter and radiation" prior to expansion. Due to inflation the "other matter ..." was diluted such that it can't be found in the observable universe. There is no causality between the "other matter and radiation" and the matter characterizing the scalar field in the sense that both co-existed without influencing each other.

The inflaton would have come into existence along with all the other kinds of particles at the big bang. In some part of the universe after the big bang, the inflaton field energy density was extraordinarily high, higher than the energy densities of all the other particles. This part of the universe underwent inflation, diluting all the non-inflaton energy density..

If I understand you correctly, regarding this part of the universe there exists no universe prior to inflation, because inflation started at the same time when the high energy density inflaton field came into existence.

 

[bapowell]

Does "other matter and radiation prior to inflation" mean particles with and without rest mass of unknown nature, or how should I think of "..."?

It would mean all particles (fields, really) that exist within the universe that couple to the inflaton. This is model-dependent.

The inflaton which is a scalar matter field has existed in thermal equilibrium with "other matter and radiation" prior to expansion. Due to inflation the "other matter ..." was diluted such that it can't be found in the observable universe. There is no causality between the "other matter and radiation" and the matter characterizing the scalar field in the sense that both co-existed without influencing each other.

I think you had it until the end: the inflaton and the other matter/radiation were in thermal equilibrium (by hypothesis) prior to inflation, so they certainly influenced each other.

If I understand you correctly, regarding this part of the universe there exists no universe prior to inflation, because inflation started at the same time when the high energy density inflaton field came into existence.

No, I suggested earlier that the inflaton pops out of the big bang along with all the other fields. There can be inflatons in the universe without inflation! Remember, the inflaton must have sufficiently high potential energy to get inflation going, and this depends on the inflaton potential energy function. Take Linde's "chaotic inflation" model, $V \propto \phi^2$ where $\phi$ is the inflaton field. It's quite possible that $\phi$ sits at the minimum of the potential across some regions of the universe. These regions evolve according to the dominant energy density, say, radiation. But since $\phi$ is susceptible to thermal and quantum fluctuations, some parts of the universe will have inflatons that find themselves very high up on their potentials. If inflaton energy is dominant, these regions will inflate.

 

[timmdeeg]

No, I suggested earlier that the inflaton pops out of the big bang along with all the other fields. There can be inflatons in the universe without inflation! Remember, the inflaton must have sufficiently high potential energy to get inflation going, and this depends on the inflaton potential energy function. Take Linde's "chaotic inflation" model, $V \propto \phi^2$ where $\phi$ is the inflaton field. It's quite possible that $\phi$ sits at the minimum of the potential across some regions of the universe. These regions evolve according to the dominant energy density, say, radiation. But since $\phi$ is susceptible to thermal and quantum fluctuations, some parts of the universe will have inflatons that find themselves very high up on their potentials. If inflaton energy is dominant, these regions will inflate.

Ok, thanks for this excellent comprehensive description of the chaotic inflation model.

After some search I found one article only which mentions the curvature parameter:

http://www.colorado.edu/philosophy/vstenger/Nothing/CompCosmosInflation.pdf

Normally, when discussing inflation, one considers the completely geometrically flat universe, k = 0. However, in order to be consistent with some ideas that will be discussed in the following chapter, I will take k = 1, that is,
assume a closed universe. As we will see, inflation still can occur under that condition..

Obviously the author means the initial (pre-inflationary) value of $k$, as he writes " inflation still can occur under that condition."
So it seems that $k=0$ is prefered. Probably this assumption is based on the "right" amount of "other matter and radiation" that fits to this conclusion, correct?

 

[bapowell]

Obviously the author means the initial (pre-inflationary) value of $k$, as he writes " inflation still can occur under that condition."
So it seems that $k=0$ is prefered. Probably this assumption is based on the "right" amount of "other matter and radiation" that fits to this conclusion, correct?

One of the merits of inflation is that you actually don't need to postulate spatial flatness as an initial condition. As long as inflation gets going, the universe is driven towards flatness regardless of the initial curvature.

 

[timmdeeg]

One of the merits of inflation is that you actually don't need to postulate spatial flatness as an initial condition. As long as inflation gets going, the universe is driven towards flatness regardless of the initial curvature.

Yes, therefore I was quite astonished about this sentence: "Normally, when discussing inflation, one considers the completely geometrically flat universe, k = 0"

 

[bapowell]

Yes, therefore I was quite astonished about this sentence: "Normally, when discussing inflation, one considers the completely geometrically flat universe, k = 0"

Yes, because "when discussing inflation" usually we are focused on the observationally relevant part -- the last 60 e-folds or so. Many inflation models last for much longer than this, and so by the time the last 60 e-folds come around, there has been sufficient inflation to make k=0 a very safe assumption.

 

[timmdeeg]

Yes, because "when discussing inflation" usually we are focused on the observationally relevant part -- the last 60 e-folds or so. Many inflation models last for much longer than this, and so by the time the last 60 e-folds come around, there has been sufficient inflation to make k=0 a very safe assumption.

Ah this is the context, now it makes sense.

Thanks a lot for your patience.

 

[bapowell]

Ah this is the context, now it makes sense.

Thanks a lot for your patience.

My pleasure. Happy to help.

 

[timmdeeg]

However, I don't see a good reason to postulate a static universe prior to inflation, ...

I agree totally after reading about chaotic inflation. This theory seems widely accepted among cosmologists. In the instant of time the universe comes into existence it has a certain spatial curvature und topology and starts to inflate immediately.There is no need for an initial state of thermal equilibrium. So, all my questions seem meaningless from this point of view.

What however is responsible for a value for $k$ and what selects the topology? It seems purely arbitrary (or the secret of the quantum fluctuation) and not subject of current theories.

 

[bapowell]

I agree totally after reading about chaotic inflation. This theory seems widely accepted among cosmologists. In the instant of time the universe comes into existence it has a certain spatial curvature und topology and starts to inflate immediately.

Only parts of the universe inflate during chaotic inflation. Chaotic inflation is the prototypical example of an eternally inflating spacetime, meaning that the universe is inflating even now, only somewhere else. The idea is that there are always inflating and non-inflating patches, and since the inflating patches grow in volume exponentially the proportion of non-inflating space goes to zero as time goes to infinity. But sure, you can imagine that there are regions of the universe that undergo inflation immediately after the big bang.

What however is responsible for a value for $k$ and what selects the topology? It seems purely arbitrary (or the secret of the quantum fluctuation) and not subject of current theories.

If the universe originated as a quantum fluctuation then it must have a finite energy and hence be compact without boundary (like a sphere or torus). This requirement does not fix the global curvature, however(k=1 for the sphere, k=0 for the torus).

 

[timmdeeg]

If the universe originated as a quantum fluctuation then it must have a finite energy and hence be compact without boundary (like a sphere or torus). This requirement does not fix the global curvature, however(k=1 for the sphere, k=0 for the torus).

Ok, thanks. So, in order to create infinite flat or negatively curved universe bubbles quantum creation is ruled out. Therefore I suspect that there should be other (competing?) processes in the framework of chaotic inflation, which create spatially infinite universes and would appreciate any further information or reference about this.

 

[bapowell]

The idea of chaotic inflation does no presuppose any global geometry or topology of the universe, because it is not a theory of the origin of the universe. It is instead a model of the high energy, post-big bang universe that postulates that a scalar field has widely varying initial values in space.

 

[timmdeeg]

The idea of chaotic inflation does no presuppose any global geometry or topology of the universe, because it is not a theory of the origin of the universe. It is instead a model of the high energy, post-big bang universe that postulates that a scalar field has widely varying initial values in space.

Understand, thanks again.