Inflation

This, and the following posts, are expansions on and clarifications of the concerns raised at Avoiding a Contravention of the Extended Consistency Principles.

During the inflationary epoch, between 10^-36s after the Big Bang and approximately 10^-32s later, it is understood that the universe expanded by a factor of at least e⁶⁰=~10²⁶.

In my hypothesis of universal expansion, as described in earlier posts and occasionally referred to as FUGE – Flat Uniform Granular Expansion, the radius of the universe expands by 1 unit of Planck length every unit of Planck time.  A FUGE universe, at 10^-36s after the Big Bang, would have had a radius of 10^-36 light seconds.  In that case, at the end of an inflationary epoch with expansion by a factor of 10²⁶, that universe would have a radius in the order of at least 10^-10 light seconds (which is about 30mm).

There are widely varying estimates for how large the universe was before and/or after inflation.  Some say the size of a grapefruit (after), Alan Guth says a marble (after), some say it was almost infinitesimally small at the beginning (in the order of a Planck length, making the post inflation size in the order of 10^-5 m – diameter or radius, it doesn’t really matter which), and a Forbes article says that the universe was about two AUs across at 10^-12s after the Big Bang (which would imply inflation by a factor of 10¹⁵).

Note that, for the universe to remain flat, this would result in the introduction of a quite substantial amount of mass-energy, equivalent to 10¹⁸ kg (or about 1/10³⁴ of the total mass today) in a period of about 10^-32s, or 10⁵⁰kg/s.  Compare this with the current rate, assuming an age of the universe of 13.77 billion years and H=71km/s/Mpc, of 2.02x10³⁵kg/s.

If, at the end of an inflationary epoch with an expansion by a factor of 10²⁶ in 10^-32s, the universe flipped back to FUGE – that is flat uniform granular expansion, then the universe today would be … pretty much indistinguishable from what we observe.  It would have a radius that is 3cm greater than we would expect in purely FUGE universe, which larger by a factor of about one part in 10²⁷ and the density would be greater by 10^-61kg/m³ (or by one part in 10³⁴), both of which are practically unmeasurable.

So, basically, perhaps inflation happened.  Perhaps it didn’t.  But either way, if the universe otherwise expanded in accordance with the FUGE hypothesis, it could be just as it is observed today.  We could enormously simplify the scenario, by imagining two straight lines with the combined equation r=±ct (t is along the horizontal axis, r is along the vertical axis, c is a conversion factor [and the speed of light]):

This is for values of t between 0 and 20.  Say that we introduce some “inflation” at t=2, then we get:

When we consider a range of t=[0,20], the inflation looks obvious, even with inflation in the order of only 10³.  But when we consider massively greater values of t, we begin to see that the curves are indistinguishable from r=±ct, with t=10⁶:

If we used equivalent scales, it would be inflation of 10²⁶ at t=2 and a final value of t=10⁴³, we get this:

 

This is totally indistinguishable from r=±ct, for sufficiently high values of t (anything above the square root of the order of magnitude of the inflation would be sufficient, assuming that the inflation happens sufficiently early).

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There are two broad options for what would have happened to the Hubble parameter during an inflationary period.  Either its value snapped to a single value for about 10^-32s, enough to expand the universe out by a factor of 10²⁶ and then relaxed back to some much lower value, or the equation of state parameter flipped below w=-1 for about 10^-32s and then back to above w=-1.

The single value of H required would be enough expand x=10^-36 light seconds into dx=10^-10 light seconds in dt=10^-32 seconds.  Recalling from the previous post that:

We find that H=~10⁵⁸/s=~10⁶¹km/s/Mpc.

If it is a question of the equation of state parameter flipping to below w=-1, then the value needed would be around w=-1.00003 because H is already incredibly high at 10^-36s. But it would have to flip back pretty much instantaneously or it would overshoot.  That is to say, the conditions in the universe that manifest an equation of state parameter of w=~-1.00003 would need to able to change, everywhere in the universe, simultaneously (because you get an inhomogeneity problem otherwise), within less than 10^-33s (or the universe is ripped apart).  Of course, the value of the equation of state parameter could ramp down and ramp back up again to get the amount of inflation that we need, but again you have a simultaneity problem which is precisely what inflation is trying to avoid (see Does Inflation have a Homogeneity Problem?).

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Note that while the above indicates that, were inflation to have happened, it would be unlikely that we would be able to observe any effect so far as the size or density of the universe goes, there are arguments out there that inflation is not necessary in the first place.  Recall that inflation was introduced as a concept to explain why the cosmic microwave background is so uniform.  There is a paper that argues that gravity is all that is needed to explain homogeneity of the universe and another that argues that an “anti-cosmos” would negate any requirement for inflation.  The latter was referenced at the end of Half a Problem Solved, where I quoted the statement "a CPT-respecting universe naturally expands and fills itself with particles, without the need for a long-theorized period of rapid expansion known as inflation".

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After the inflationary epoch, it is understood that there were two periods of decelerating expansion, the radiation-dominated era and the mass-dominated era, and then the current, dark-matter-dominated era in which the expansion of the universe is understood to be accelerating.  I’ll take a look at these soon, but first I need to introduce the equations for cosmological expansion and then I need to touch on the inflaverse (as raised in Does Inflation have a Homogeneity Problem?).