Past Values of the Hubble Parameter

I have tried to look up past values of the Hubble parameter a number of times now, without luck.  Even finding an equation for past values of the Hubble parameter is challenging.  Below is an effort to provide such an equation.

The Hubble parameter is defined as:

And it is stated that:

 

Where the deceleration parameter is defined as:

 

If a=t, as in a FUGE universe and presumably the flat cosmology model as well, then we would have the following:

If a=t^½, as per the radiation-dominated era in the Standard Model, we would have:

 

If a=t^⅔, as per the matter-dominated era in the Standard Model, we would have:

In the Standard Model, the current era is dominated by Dark Energy and, for that reason, we could say that a=e^H.t, where the Hubble parameter no longer changes as is equal to H₀, in which case:

It is interesting that we have three eras where the equations work perfectly with nice tidy values for the deceleration parameter in those eras.  It would be madness to think of the eras as punctuated by sudden flips to a new value, even if Standard Model does otherwise have some identified issues (see The Problem(s) with the Standard Model).  For this reason, we could look at generalising.

Say that a=t^η, where η is a constant.  Then we would have:

Looking at the equation of state, we can see that it would follow (“(i)f the (perfect) fluid is the dominant form of matter in a flat (and isotropic) universe”) that η=2/(3(1+w).  For a matter-dominated era, w=0, so η=⅔.  For a radiation-dominated era, w=⅓, so η=½.  For a dark-energy-dominated era, w=-1, so there is no valid value of η.  For a FUGE universe, w=-⅓, so η=1.  So we don’t really need η, we can use the standard formulation with the equation of state value, w, except for when it doesn’t work of course (in a dark-energy-dominated era – where we would have a division by zero error).

Note that we are still left with one discontinuity point.  It is unclear and where the scale factor changed from the form a=t^η to a=(e^H_0.t)/B (where B is notionally a constant such that B=e^H_0.t_0, where t₀ notionally means "now", but the value of "now" could be applied to any time since that transition to obtain a(t₀)=1 unless, for some reason, we are in a privileged era).

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To work out the value of H(t) in the past then, we just need to know what t is.  I ask this because it makes sense that t is the age of the universe (and in this context it is for a FUGE universe), but this is not stated explicitly in the documents that I have looked at.  The closest to an explicit statement is that we can compare two values of a(t), one now at time t₀ and another at time t.  This implies that we can use now as the reference and the value of t in the equation is now minus how long ago an event at time t happened.  If we take “now” as being the age of the universe, then t will be the age of the universe at that time.

The most likely reason that people working in this area avoid just saying that t is the age of the universe is that they use the value t_H=1/H₀, which is reported as both 13.79 billion years and 14.4 billion years (with the latter being more common).  This is despite there being the Hubble tension, which means that the Hubble time should be presented with much less certainty, to match the lack of overall certainty associated with the Hubble parameter value (which is very close 76km/s/Mpc [Cepheids], somewhere close to 71km/s/Mpc [Standard Sirens] or very close to 67km/s/Mpc [CMB]).  It’s more accurate to say that the Hubble time, as defined as t_H=1/H₀, falls somewhere in the region of 12.8 to 14.6 billion years (and somewhere around 13.8 billion years according to the Standard Sirens value).

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We can bite the bullet and say that t in H(t) is the age of the universe where the current age of the universe is in the region of t≈14 billion years (13.8 if you like artificial certainty), so H₀≈70km/s/Mpc.

Going further to establish the values that I actually want (for an earlier but recently updated article on redshift), namely the Hubble parameter value for the decoupling/recombination event, we can say that for the Standard Model:

H(380,000 years)=2/3/(380,000 years)=1,720,000km/s/Mpc

And for a FUGE universe (see  FUGE and Redshift):

H(12.5 million years)=(1/12.5 million years)=78,200km/s/Mpc

FUGE and Redshift

I have written before about the OE Curve and redshift, but it was blended together with consideration of someone else’s theory about the universe so it’s not as clear as I would have liked.  So, I’ll have another go at it.

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There are (at least) two ways to think about redshift.

A photon which is emitted by a source that is moving away at a sufficiently high speed can be observed to be redshifted.  That’s not to say there isn’t redshift for slow (non-relativistic) light sources, it’s just that the redshift in that case is so small as to be unobservable.  This form of redshift is effectively the same as Doppler shift, but requires some extra thought due to relativistic effects (see relativistic Doppler shift).

A photon that is emitted by a source that is stationary (at rest relative to the Hubble flow) or “comoving” relative to the observer, if sufficiently distant from that observer, can also be observed to be redshifted.  Again, the redshift is there, irrespective of the distance under those conditions, but at relatively short distances it’s too small to be observed.

(There is a third source of redshift, which is due to gravity.  It’s actually similar to the redshift due to cosmological expansion in a way, but at a different scale, since both involve deformation of space along the path of the photon.  Redshift due to gravity occurs when the photon is moving out from a gravitational field.)

I want to address redshift due to cosmological expansion.  First think about a photon moving through space in a given direction.  It has a wavelength determined by the speed of light, since a photon is a wavelet oscillating with a frequency determined by its energy (which is related to its colour and associated temperature).  That wave traces a path through space like this:

Say we stretch the original space illustrated above by a factor of two (looking at only the first full wavelength), we now get this:

The speed of light remains constant, so what we have here is a photon with twice the wavelength and half the frequency.  It’s an extreme case of redshifting.

To determine the redshift, z, we take the second wavelength, λ_now, subtract the original wavelength, λ_then, and divide by the first wavelength:

z=(λ_now-λ_then)/λ_then=(λ_now/λ_then)-1

Or,

z+1=λ_now/λ_then

Note that this equation is the same as standard redshift, where now=obsv and then=emit.

In a FUGE universe, the radius of the universe at a given time t will be ct.  The wavelength of an arbitrary photon at that time will be some fraction of that radius, so we can consider a photon of wavelength λ_then=ct_then/B.  After a period of expansion that same photon (now) will have the wavelength λ_now=ct_now/B.  Substituting this into the above and we get:

z+1=t_now/t_then

Alternatively, we can think about scale factor over time.  In a FUGE universe, a(t)=ct/ct₀ where is a reference time (usually now, but it doesn’t have to be in the case).  Note that cosmological redshift is given by

z+1=a_now/a_then

So using the scale factor equation above:

z+1=(ct_now/ct₀)/(ct_then/ct₀)

z+1=t_now/t_then

Alternatively, we can think about the OE curve, for which the equation is x'=(ct₀-x).x/ct₀.  Note that this could be confusing, because in all the OE curve articles in which I clarify that x=ct, I use t to refer to the time elapsed between emission time of the photon (or when it was in a particular location) and the observer.  The t used above (and in the redshift literature more generally) is a reference to the age of the universe (for which I use ꬱ).  Also, in the OE curve equation, rather than being just a reference time, t₀ is specifically the current period of time since the beginning of the universe, or perhaps less confusingly t₀ is the current age of the universe, so t_now=t₀.

I do note, in Mathematics for Taking Another Look at the Universe, that in the OE curve equation it would be more accurate to say that it would be more accurate to use Δt and Δx, so Δx=c.Δt and thus

x'=(ct₀-Δx).Δx/ct₀=(ct₀-cΔt).cΔt/ct₀

I’ll use this notation for as much clarity as is possible under the circumstances, noting that Δt is referring to the delta between now and then, so Δt=t_now-t_then=t₀-t_then, and Δx=ct is the distance a photon in static space would travel in that time.

Consider, hypothetically, that redshift might be due to the difference between the actual separation crossed and the distance that the photon needed to travel to cross that separation, or:

z=Δx/x'

Note that what this is effectively doing is comparing the distance between two comoving locations (emitter and observer) at two times (relative to the observer), time of emission and time of observation.

Substituting in the OE curve equation:

z=Δx/((ct₀-Δx).Δx/ct₀)

z=ct₀/(ct₀-Δx)

Noting that Δx=c.Δt:

z=t₀/(t₀-Δt)

And then noting that Δt=t₀-t_then and t₀=t_now:

z=t_now/t_then

This isn’t precisely the same as above, but for sufficiently high values of z, z+1≈z, so it’s a good approximation.

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The problem, of course, as identified in the earlier article about redshift, is that the value for CMB redshift calculated this way is in the order of z≈36,000 (assuming t_d=380,000 years).  The accepted value is z≈1100.

However, this value is based on the assumption that the universe expanded such that H(t)=2/3t since decoupling/recombination (at t=t_d) which in turn would mean that z=(t_d/t₀)^⅔.  Note that 36,000^⅔≈1100.

If the FUGE universe model were to be correct, the redshift value associated with the CMB would remain z=1100, because this is required to reach the temperature at decoupling/recombination, but the timing of decoupling/recombination would be different.  In a FUGE universe, H(t)=1/t, and therefore the 2/3 power factor disappears, leaving us with z=t_d/t₀, such that decoupling/recombination would have happened at t_d≈12.5 million years (leaving plenty of time for the development of the oldest stars – including Methuselah – which are calculated to be in the order of 12 to 13.7 billion years old).

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There’s another aspect of redshift, which I haven’t really mentioned above, which is the shift in colour.  The colour of a photon is related to its wavelength – red photons have a wavelength in the region 625-740 nm.  Blue photons have a wavelength in the region 450-495nm.  You can see, therefore, that as a photon’s wavelength is increased, its colour is changed in the red direction.

If the colour of a photon is pushed even further, it leaves the visible spectrum into infrared and eventually into the microwave spectrum (pushing it further you could even get radio waves).

We can think of the CMB as being a blackbody radiation spectrum associated 2.75K and indeed the correspondence is nigh on perfect with that (source):

The find the peak of this spectrum, we can use Wein’s displacement law, λ_max=b/T, where b(=2.897×10^-3 m.K) is the constant of proportionality and T is the temperature in kelvin.

Given the current temperature and the temperature at decoupling/recombination (t_d=3000K), we can work out the wavelengths:

λ_{max_CMB}(t₀)=2.897×10^-3/2.75=1.05×10^-3m

λ_{max_CMB}(t_d)=2.897×10^-3/3000=966×10^-9m

Note that this latter wavelength actually corresponds with infrared and the first, unsurprisingly, is in the microwave spectrum.

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There is another way to arrive at a value of redshift in a FUGE universe.  This is via equations in notes from Oxford university (3.c. (Solution)), where it is stated that H(t_d)=H₀.1000√1000 and that (t_d/t₀)^⅔=1/1000.  If so, then it follows that (H(t_d)/H₀)^⅔=1000=(t₀/t_d)^⅔, and thus H(t_d)/H₀=t₀/t_d.  Since t₀=t_now and, in this instance, t_d=t_then, we have:

z≈t_now/t_then=H(t_then)/H(t_now)

Again, this should come as no surprise as, in a FUGE universe, H(t)=1/t.  However, if there were any direct evidence that, at decoupling/recombination, the value of the Hubble parameter was in fact H(t_d)≈ 77,000, see Past Values of the Hubble Parameter, then we’d have another way to work out that z_CMB≈1100.

Note that in the Standard model, H(t_now)=H₀≈70km/s/Mpc (using the centre point of the values per the Hubble tension) and H(t_d)=2/3t_d.  Using the Standard Model value of 380,000 years, that is H(t)=1,720,000km/s/Mpc.

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In conclusion, the FUGE universe model is consistent with redshift as we know it.  There are a number of ways of looking at it with all results being consistent, although the timing of decoupling/recombination is different due to the different rate of expansion between then and now.