Accelerated Expansion

Observations suggest that, during the dark-energy-dominated era, the expansion of the universe has been accelerating. 

This is acceleration is understood to have happened after two periods of deceleration during the radiation-dominated and the matter-dominated eras (see Decelerated Expansion).

Using the representations from the previous post about inflation, the acceleration could look something like this (somewhat exaggerated, or very much not so, depending on what the time scale is considered to be):

 

Note that the accelerated expansion is not thought to be of the same magnitude as per the inflationary era but, if the equation of state parameter w<-1 (and never rose above w=-1), it would nevertheless lead to a Big Rip eventually (see Equations for Cosmological Expansion).

In the description of the acceleration of expansion, it is stated that:

The most important property of dark energy is that it has negative pressure (repulsive action) which is distributed relatively homogeneously in space.

It is later stated that “Different theories of dark energy suggest different values of w”.  See Equations for Cosmological Expansion for the effects of various values of the equation of state parameter, w.  Note that there is some observational skepticism with regard to dark energy, including an experiment intended to detect associated forces, but which failed to do so (although the associated paper accepted that the expansion of the universe is accelerating).

The value of w established by the Planck Collaboration is consistent with w=-1, but they give the value as w=-1.03±0.03.   (The authors refer to w as the evolution parameter, but this usage may be idiosyncratic.  Note also that w=-1.03 and w=-1.06 are the values used in the charts in Equations for Cosmological Expansion to reflect this Planck Collaboration range.  They also find the universe to be flat.)

With an equation of state parameter of w=-1.03, we have a rate of change of the Hubble parameter of dH/dt=0.045.H² that leads to a Big Rip in about 300 billion years.  That is probably long enough to finish everything I need to get done, but a key question here is how exactly did the Planck collaboration arrive at an equation of state parameter value of w=-1.03?

My suspicion is that it went a bit like this:

At the end of the matter-dominated era about 4 billion years ago, as calculated at Decelerated Expansion, the value of the Hubble parameter would have been in the order of 66.5km/s/Mpc.  The measured value, by the Planck Collaboration, is about 67.4km/s/Mpc.

So, what value of the equation of state parameter do we need to get to today’s Hubble parameter value over a period of about 4 billion years?  First, we need to standardise our units.  That is, express our Hubble parameters in the same units as the inverse of the time delta (so Gy^-1).

So, 66.5km/s/Mpc = 0.0680Gy^-1 and 67.4km/s/Mpc = 0.0689Gy^-1.

Well, what do you know?  The current value of the equation of state parameter is precisely the right value to get the Hubble parameter value that the Planck Collaboration measures today.  It certainly seems like a fudge.  And there are other measurements of the Hubble parameter available, in fact there is a bit of a “crisis” associated with range of Hubble parameter value measurements.

Note that, notionally, at no time since shortly after the inflationary epoch began would the value of the Hubble parameter have been even close to the inverse age of the universe.  I say “notionally” because I am assuming that for the first 10^-36s, w=-1/3 so that dH/dt=-H².  But, conveniently, when we happen to be in the position to measure it, the Hubble parameter is, by some accounts, not just mine, but notably not the Planck Collaboration, exceedingly close to the inverse of the age of the universe, not only within an order of magnitude (which is commonly close enough with things cosmological), but well within the uncertainty of the most recent measurement using a distance ladder of flaring red giant stars and slap in the middle of the current range of authoritative measurements (which, given how disparate they are, may be no more than a coincidence).  See the image below from Scientific American and consider that the Hubble parameter consistent with the inverse age of the universe is 71km/s/Mpc.

If the Planck Collaboration were fudging their value of the equation of state parameter, the question arises, what value would be needed to end up with a Hubble parameter today of 71km/s/Mpc?  That would be about w=-1.13.  And about w=-1.25 to get a value today of 74km/s/Mpc.

In any of those cases, however, the application of the fudge is required to get a value of the Hubble parameter today that is consisted with the inverse age of the universe.  We could avoid this massive temporal coincidence entirely by considering instead that, just maybe, there was no acceleration, nor additional deceleration and that the value of Hubble parameter has always been and will always be given by H=1/t (and that there would be, given a sufficiently large value t, a tiny but “natural” deceleration given only by the inverse square of the age of the universe).

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It is probably massively arrogant of me to suggest that the expansion of the universe is not accelerating.  Extremely competent astronomers and cosmologists have observed the universe and collated credible evidence that indicates that the expansion of the universe is in fact accelerating.  The equations I have used may be invalid (despite their appearing on Wikipedia without correction – but I accept that errors are made and some are not corrected for many years).  Or I may have used them incorrectly, despite the fact that I arrive at the same value of the equation of state parameter as the Planck Collaboration using their own assumptions.  Or I’m just some random person with access to a computer and few ideas from left field rather than being a credentialed expert in the area.  So, perhaps it behoves me to reiterate and expand on the issues that I have with cosmological acceleration.

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Rather than make this a massively long post, I’ll do that in the next one.