## Sunday, 16 July 2017

### Inflation of the Hubble Constant

I've been chewing on an old bone recently, metaphorically that is.

In A Little Expansion on the Lightness of Fine-Tuning, I wrote about how I visualised the way two spaceships might approach each other, with both of them travelling at half the speed of light (relative to an implied third observer) and yet have a closing velocity of less than the speed of light.  The resultant model, for me at least, also managed to explain the spatial and temporal effects of special relativity.

A consequence of this model is that the universe is expanding at the speed of light and this expansion is time (see also On Time) - so that were you to be at rest in spatial terms, you would not be at rest in temporal terms, you would still be travelling "through" time by virtue of the universal expansion (at a rate equivalent to the speed of light).

The problem is that if the universe is expanding at the speed of light, per my model, then what about reports that the rate of expansion of the universe is increasing?  The speed of light is invariant, so the rate of expansion of the universe should also be invariant - if my model is valid.  What about inflation?  Well, I'll get to inflation in a moment.

I have previously (and rhetorically) asked the question Is the Universe Expanding at the Speed of Light?  My conclusion was that, if a single spatial Planck unit were added to the universe (in the direction we are looking) for every temporal Planck unit, then we would observe an expansion of the universe (today) at pretty much the rate that we observe the universe expanding at (today) - about 70 kilometres per second per megaparsec.

This was calculated, however, using a different model to that presented in A Little Expansion on the Lightness of Fine-Tuning.  It was if I were looking at a segmented ruler, with each segment being a Planck length long and I was adding a unit of Planck length somewhere in the middle for every unit of Planck time.  I then calculated the rate of the expansion of the ruler after 8.08x1060 units of Planck time (which is the age of the universe) and found that this matched the Hubble Constant.

So the question I have: is what happens if I use the onion like model to calculate the effect of the universe expanding at the speed of light?  This is taken from A Little Expansion on the Lightness of Fine-Tuning:

I was trying to explain something a little different there, so it's a bit more cluttered than it might need to be.  For our purposes at the moment, all we really need to consider is the difference between the value of the arc defined by xG at tE and its value at tG, noting the relevant angle θ.  My contention is that the universe expands at c, so therefore Δt = c.  Let us call the arc length x and refer to any change as Δx.

An arc length is calculated reasonably simply, x = θ.r (where θ is expressed in radians, and a full circle circumference is therefore given by x = 2π.r – see Hugging the World).  We therefore know that the difference in x would be given by Δx = θ.Δr (and in this model Δx = θ.Δt).  This gives us enough to work with.

Consider two moments in time:

x = θ.tnow

and

x + Δx = θ.(tnow + Δt)

And eliminate θ:

(x + Δx)/x = (tnow + Δt)/tnow

1 + Δx/x = 1 + Δt/tnow

Δx/x = Δt/tnow

Δx/Δt = x/tnow

So the rate of expansion of the universe is proportional to the age of the universe (tnow).  The Hubble Constant is presented as expansion over a given distance, so:

Ho = Δx/Δt/x = 1/tnow

Phew, my model stands up - because the value of the Hubble Constant is actually the reciprocal of the age of the universe - noting that the age of the universe is not uniquely calculated from the reciprocal of the Hubble Constant, there are other methods (see strong priors a little further down the page).  However, my model suggests that the universe is, in a sense, expanding at the speed of light - always has done so and always will do so - and the Hubble Constant should be decreasing with the age of the universe.  Nevertheless, we have people telling us that the rate of expansion of the universe is increasing.

This sounds like a potential worry because if the Hubble Constant were increasing rather than decreasing, then its current value at the reciprocal of the age of the universe would be coincidental.  This would be another, worrying example of fine-tuning that would have to be explained.

Fortunately, while there are observations that indicate that expansion of the universe might be increasing, the Hubble Constant is not.  This might seem counter-intuitive.  As Sean Carroll explains, the Hubble Constant gives us a scale by which to measure the velocity at which distant objects recede from us due to universal expansion, v = Ho.d, where d is the distance to the object receding away from us.  But note that this is caveated with "due to universal expansion".  Carroll is considering dark energy here.  If, in addition to universal expansion, things are being pushed apart even only by smidgen, then d will be increasing at rate greater than v (where v is "due to universal expansion").

Note that there is some room to doubt whether there actually is this acceleration in the rate of universal expansion, I myself remain a bit dubious maybe in part because if dark energy is real then my model may be fatally flawed, despite explaining so many things so well.

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But what about inflation, I hear you yell excitedly.

Well, there's two things.  Firstly, we need to remember Hugging the World.  The term tnow doesn't necessarily mean the time since any absolute beginning to the universe, it means time since some key event - what that event was may have been no more than a phase change from inflation to the current state of affairs.  (Or from an earlier aeon to this aeon.  This makes my model consistent with conformal cyclic cosmology, which can bypass inflation.)

Secondly, in my model, we currently have a nice, orderly, temporal expansion of the universe, with one layer (one moment) added "at a time".  This isn't necessarily the way things have to be.  Instead, there could have been a situation in which each Planck volume spawned a new Planck volume each unit of Planck time.  This would lead to an exponential cascade of expansion - and to get to the lower limit of inflation, an increase in size by a factor of 1026, it is only necessary to have about 86 doublings … if it is assumed that every Planck volume splits during each doubling.  However, there are about 2x1011 units of Planck time in the period during which inflation is thought to have occurred meaning that all that is required to achieve the minimum for inflation is that for each unit of Planck time, there would be an average one additional Planck volume for every existent 3x109 Planck volumes.  This is what would happen about 3x109 units of Planck time in, or alternatively, at 1.62x10-34s – noting that inflation is believed to have occurred sometime about 10-33s in.

In other words, something very like inflation happens anyway with my model.