# Can Einstein's Field equation explain inflation?

• andrewkirk

#### andrewkirk

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I was looking at the wikipedia article about cosmic inflation (you know, the first 10-32 seconds).

It says that an inflation hypothesis is a neat way to explain a number of otherwise puzzling observations, but it doesn't say much about what would make inflation occur and it doesn't mention either Einstein's or Friedman's equation. Does that mean that there is no known explanation of inflation in terms of just GR? Does it need something else? If so, what?

## Answers and Replies

One of the tricky aspects of learning about inflation is that their is a huge number of viable models that can explain inflation. The exact cause or which model is correct is unknown. In the earliest model inflation was explained by "false vacuum" by Allen Guth. this model however was later replaced. In this model inflation is a process of quantum tunnelling between the true vacuum (normal vacuum state) and the false vacuum (lowest energy state). This model had a problem called runaway inflation and was later replaced by chaotic eternal inflation. As well as the slow roll approximation model.

This article uses the slow roll approximation which is known good fit to observational data to compare the numerous (over 60) different models. Each model has its own formulas and is covered individually in this article.

http://arxiv.org/abs/1303.3787

be forewarned its extremely lengthy and technical but it will give you a decent listing of the different and viable models involved today in regards to inflation

1 person
Thanks Mordred. It sounds like the Guth theory relies on a theory outside GR - quantum mechanics. The few theories I've looked at about inflation all seem to rely on quantum mechanics or some speculative hypothesis derived from that. To your knowledge are there any theories that rely solely on GR?

strictly in terms of GR. I've never heard of any. A lot of the models require some form of scalar field ie the inflaton, Higg's field etc.

First comes a big bang,or at least a 'bang' without known cause; then we hypothesize inflation, also without known cause; then we have the FLRW cosmological model together with Einstein's field equations. The "old' cosmological model, without inflation, could not reproduce a cosmos with the parameters we observe. GR alone does not explain how the universe started.

The basis of modern cosmology are the FLRW solutions of the Einstein field equations. The Universe is assumed homogeneous (space has the same metric properties (measures) at all points) and isotropic (space has the same measures in all directions). The present consensus is that the isotropic model gives an adequate [approximate] description of the the large scale Universe. The Friedmann equations are derived by inserting the metric for a homogeneous and isotropic universe into Einstein’s field equations for a fluid with a given density and pressure. The FLRW metric starts after the initial inflationary epoch; the LCDM is a ‘fine-tuned version’ of the general FLRW model where parameters are chosen to get the best possible fit to our universe.

Here are brief snippets describing inflation from THE ROAD TO REALITY by Roger Penrose, pages section 28.4.
I am paraphrasing here:
INFLATION:

The commonly described history of the universe begins with conventional causal expansion. This means, for example, that thermodynamic energy had enough time to create rather uniform temperatures...to 'communicate' causal effects. Then from about 10-35 second to about 10-32 seconds an arbitrary scalar [inflation] field is introduced [sometimes referred to a 'Higgs field']. Some models require more than one inflationary phase, in which case there would have to be a different scalar field for each'.
...'During the inflation stage we have a region of a false vacuum which represents a quantum mechanical phase transition to a vacuum different from the one we are familiar with today'...Lambda [the cosmological constant of expansion] was about 10110 times it's current density. This transition is the 'slow roll' version
described in Wikipedia

http://en.wikipedia.org/wiki/Cosmic_...roll_inflation [Broken]

Note:
[Penrose seems to think such an arbitrary introduction of a field to 'fix' problems of the 'old model' 'casts doubt on the entire idea' and 'few will be as negative as I am' but he acknowledges 'inflationary cosmology, as described, has become a major part of the body of modern cosmological thinking'..

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I checked my notes further...here is a description of the inflation field I like.
You may have seen it already...

INFLATON: cause of Inflation

http://en.wikipedia.org/wiki/Inflaton

The inflaton is a hypothetical scalar field (and its associated particle) that may be responsible for the hypothetical inflation in the very early universe. According to inflation theory, the inflaton field provided the mechanism to drive a period of rapid expansion from 10−35 to 10−34 seconds after the initial expansion that formed the universe…. Random quantum fluctuations triggered a phase transition whereby the inflaton field released its potential energy as matter and radiation as it settled to its lowest-energy state. [At the end of inﬂation, the scalar ﬁeld will die away and get converted to matter and radiation.] ...

I see there is no mention of 'spontaneous symmetry breaking' here...

for a nice visual interpretation of energy during the brief period of inflation see the illustration here:

http://en.wikipedia.org/wiki/Mexican_hat_potential#A_pedagogical_example:_the_Mexican_hat_potential

Shortly after the bang, a high energy and 'symmetrical', but unstable state ensued, picture the top of the Mexican hat...it 'spontaneously' decayed, maybe by tunneling, down the crown of the hat and ended up in a lower energy, stable, non symmetrical state...the very early universe around the brim of the hat. That's the brief period of inflation Roger Penrose refers to from 10-35 sec to about 10-32 seconds.

The early 'symmetrical state' means, for example, all the fundamental forces were unified...
the initial quantum fluctuations which may have initiated the initial bang are magnified and expanded during inflation, becoming particles and radiation at the end...and planets, stars, and galaxies now. Until about 380,000 years after the bang, radiation [energy] could not get through the charged plasma...then the cosmic microwave background got through...that's what we observe still...but now it has cooled way down...

I think the short answer to the OP's question is that you can produce an inflating universe with the Friedmann equations provided you plug in a substance with the right equation of state. For instance, the simplest form of dark energy has P = -rho, where P is pressure and rho is energy density. If this were the only contribution to the density term in the Friedmann equation (no ordinary/dark matter), then the change in scale factor with time would be exponential, just as it was during inflation. We are already heading in that direction as the dark energy begins to dominate over the other constituents.

During the inflationary epoch, a scalar field called the "inflaton" with similar properties to what I described above is usually invoked to explain inflation. However there is no particle/field in the standard model of particle physics that could do this. As others have pointed out above, the physics of this inflaton that supposedly drove inflation is completely unknown, since we don't even know at what energy scale inflation occurred. However it was definitely at much higher energy scales than can be probed in terrestrial laboratories.

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1 person
I think the short answer to the OP's question is that you can produce an inflating universe with the Friedmann equations provided you plug in a substance with the right equation of state. For instance, the simplest form of dark energy has P = -rho, where P is pressure and rho is energy density. If this were the only contribution to the density term in the Friedmann equation (no ordinary/dark matter), then the change in scale factor with time would be exponential, just as it was during inflation.
Does the problem then become one of "why did the inflation stop"? If the Friedmann equation generates continued exponential expansion, given the equation of state, then it seems there would need to be some non-GR intervention to slow that inflationary expansion to the current, relatively much slower, rate.
We are already heading in that direction as the dark energy begins to dominate over the other constituents.
But if I understand it correctly (and it's quite likely that I don't!), it's not possible under any mainstream estimates of the cosmological constant, and without intervention by non-GR factors, to return to a situation like in the inflationary era, in which two particles that are 1cm apart can be several light years apart a femtosecond later. Or is it?

'

I think the short answer to the OP's question is that you can produce an inflating universe with the Friedmann equations provided you plug in a substance with the right equation of state. For instance, the simplest form of dark energy has P = -rho, where P is pressure and rho is energy density. If this were the only contribution to the density term in the Friedmann equation (no ordinary/dark matter), then the change in scale factor with time would be exponential, just as it was during inflation.
Does the problem then become one of "why did the inflation stop"? If the Friedmann equation generates continued exponential expansion, given the equation of state, then it seems there would need to be some non-GR intervention to slow that inflationary expansion to the current, relatively much slower, rate.
We are already heading in that direction as the dark energy begins to dominate over the other constituents.
But if I understand it correctly (and it's quite likely that I don't!), it's not possible under any mainstream estimates of the cosmological constant, and without intervention by non-GR factors, to return to a situation like in the inflationary era, in which two particles that are 1cm apart can be several light years apart a femtosecond later. Or is it?

cepheid:
I think the short answer to the OP's question is that you can produce an inflating universe with the Friedmann equations provided you plug in a substance with the right equation of state. ... We are already heading in that direction as the dark energy begins to dominate over the other constituents.

A very interesting perspective: the universe expands exponentially from both a very compact and high energy state early in life as well as an expansive low energy state late in life.

No wonder there are ideas about a multiverse...although I don't remember this type of link.

An interesting addendum to cepheid's post:

Inflation for beginners
http://www.lifesci.sussex.ac.uk/home/John_Gribbin/cosmo.htm#Inflation

This article provides a nice historical overview as well as some understanding of inflation issues.

There are similarities between the idea of eternal inflation and a self-reproducing universe and the version of the Steady State hypothesis developed in England by Fred Hoyle and Jayant Narlikar, with their C-field playing the part of the scalar field that drives inflation. As Hoyle wryly pointed out at a meeting of the Royal Astronomical Society in London in December 1994, the relevant equations in inflation theory are exactly the same as in his version of the Steady State idea, but with the letter "C" replaced by the Greek "Ø". "This," said Hoyle (tongue firmly in cheek) "makes all the difference in the world".

andrewkirk:
Does the problem then become one of "why did the inflation stop"?

Well, the current idea is the 'Mexican hat potential" I posted about in #6 above...but stopping inflation was considered a problem during development of inflation theory. Some still consider it an issue..like Penrose.

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Thanks Mordred. It sounds like the Guth theory relies on a theory outside GR - quantum mechanics. The few theories I've looked at about inflation all seem to rely on quantum mechanics or some speculative hypothesis derived from that. To your knowledge are there any theories that rely solely on GR?
Yes. The Starobinsky R^2 model.

Does the problem then become one of "why did the inflation stop"? If the Friedmann equation generates continued exponential expansion, given the equation of state, then it seems there would need to be some non-GR intervention to slow that inflationary expansion to the current, relatively much slower, rate.
Inflation stops whenever the equation of state becomes non-inflationary. As has been mentioned, the specific dynamics depend on the scenario, but there is nothing non-GR about it (beyond the use of a scalar field in the first place). Inflation occurs only when the potential energy of the field dominates over the kinetic energy (so-called slow roll phase). As the scalar field evolves, the kinetic energy increases and usually reaches a point at which the equation of state becomes non-inflationary. It's of course possible to devise a scalar field potential for which inflation is eternal and never stops, but plenty of the standard approaches (using Mexican hat-type potentials, hybrid inflation, 'chaotic'-type models based on polynomial potentials, etc) do see inflation end simply because at some point the field begins to roll to fast.