Observed Events Curve - Shifting About Redshift (Including an Alternate FUGE-like Universe)

I tried calculating redshift in a FUGE universe, and ran into what looked to be a problem.

My logic was as follows: if we are observing a photon from an event that occurred a time t ago, then that photon had a proper distance of x'=x.(ct₀-x)/ct₀ and the event location (at the time of observation) will be x=ct from the observation location.  So for any arbitrary wavelength of the photon:

1+z=λ_observed/λ_emitted=x/x'=ct₀/(ct₀-x)

z=ct₀/(ct₀-x)-1=(ct₀-(ct₀-x))/(ct₀-x)=x/(ct₀-x)

z=t/(t₀-t)

The problem is that using this equation, redshift for the CMB does not come out to be what is generally attributed to it (z_CMB=1100).  Recall that t₀ here is the age of the universe and t is the transit time for an observed photon (see most recent posts, particularly Mathematics for Taking Another Look at the Universe), so for a photon emitted during recombination, the event (also confusingly known as decoupling) that led to the CMB when the universe was 380,000 years old, t_CMB=(13.8×10⁹-380,000) years and:

z_CMB=t_CMB/(t₀-t_CMB)

z_CMB=(13.8×10⁹-380,000)/(13.8×10⁹-(13.8×10⁹-380,000))=36,300

This is 33 times higher than the standard answer.  We can reorganise the equation above to work out how old the universe must have been for photons from the CMB to have a redshift of z=1100.  Using æ_CMB=t₀-t_CMB (and thus t_CMB=t₀-æ_CMB):

z_CMB=(t₀-æ_CMB)/æ_CMB=t₀/æ_CMB-1

t₀/æ_CMB=z_CMB+1

æ_CMB=t₀/z_CMB+1

For z_CMB=1100, we get an æ_CMB=1.25×10⁶ years.  Hm, it appears that something is not right.

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The question that immediately arises, at least for me, is how do we know that the redshift for photons from recombination is 1100 and how do we know that recombination happened when the universe was 380,000 years old?  And what are the error bars associated with these values?

The redshift value is a little rubbery, but is usually quoted as simply 1100, although it’s probably a bit lower.  Rhodri Evans (astrophysicist and author of The Cosmic Microwave Background - How It Changed Our Understanding of the Universe) has a blog post on the CMB redshift which gives a bit of the history, indicating that the value comes from a comparison between the temperature of the CMB radiation today and that at the time of recombination, so:

z_CMB=T_recomb/T_now=3000/2.725≈1100

This equation is a slight approximation, since it should be z+1=T_recomb/T_now and I will be using the non-approximation from now on.

Note also that the equation does not explicitly rely on the timing of the event.  The recombination is thought to have happened when the universe got sufficiently cool, so the timing isn’t actually key.  While Evans does write that “as the Universe expands, the temperature (..) decreases in inverse proportion to its size. Double the size of the Universe, and the temperature will halve”, to know when the temperature of the universe was 3000K, we would have to know what the temperature was at some other time, what that time was and what the relevant expansion rate was.

Note that in a FUGE universe, the radius of the universe is directly proportional to its age (so currently r₀=ct₀).  So, noting the inverse relationship, we could simply replace T_recomb and T_now with 1/æ_recomb=1/380,000 years and 1/æ_now=1/t₀=1/13.8×10⁹ years.  Which gives us … z≈36,300.

So, I still have questions about timing and redshift and now also temperature.

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When I expanded my search to find out how the relevant temperatures are calculated, I stumbled on what looks to be a variant of the FUGE universe, described in papers that have been published in what appear to be legitimate journals.  This model is referred to more frequently as Flat Space Cosmology, but there are also references to “r_H=ct models” (presumably those similar to Melia's, where t is the age of the universe) and “growing black hole models” which seems to describe something akin to the FUGE universe (I disagree with the terminology but that may just be a matter of perspective).

Espen Haug and Eugene Tatum (and others) have, in the past half a year, published a number of papers, for the most part on open archive sites but sometimes in journals (for example the International Journal of Theoretical Physics).  The paper that most attracted my attention provides a method for calculating temperature, Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database (available from HAL open science, which is an open archive).

I’m not going to get into whether or not they have actually solved the Hubble Tension, instead I am going to look at equations in that paper that I have issues with.

The first appears late in the paper:

The complexity of this equation does not appear justified, since it resolves to:

Ʊ=2(4π/T_P)²/t_P

where T_P is Planck temperature and t_P is Planck time.  This follows from T_P=E_P/k_b where E_P=m_Pc² is Planck energy and k_b is the Boltzmann constant, noting that m_P=√(ħc/G), so that:

Ʊ=k_b²32π²G^1/2/c^5/2ħ^3/2=((E_P/T_P)².2.(4π)²/c²ħ).√(G/ħc)

Ʊ=((m_Pc²/T_P)².2.(4π)²/c²ħ)/m_P=(m_Pc²/ħ).2.(4π)²/T_P²

Ʊ=(√(ħc/G).c²/ħ).2.(4π/T_P)²=(√(c⁵/ħG).c²/ħ).2.(4π/T_P)²

And since t_P=(√(ħG/c⁵):

Ʊ=2.(4π/T_P)²/t_P=2.923×10^-19K^-2s^-1

This “composite constant” upsilon, which has no other apparent name than the Latinised Greek letter used to denote it, has no other apparent use than in the equation H₀=ƱT₀², where T₀ is the (current) temperature of the CMB.  In the paper in which upsilon is introduced, Upsilon Constants and Their Usefulness in Planck Scale Quantum Cosmology, it is derived purely from this simpler relationship.  (I should provide a warning here that SCIRP is considered to be a predatory publisher, meaning that there is no peer review for articles which are published after payment.  Tatum indicates that he consulted Dr. Rudolph Schild of Harvard-Smithsonian Center for Astrophysics.  Unfortunately, Schild apparently publishes in a fringe (and allegedly predatory) astronomy journal, Journal of Cosmology, of which he is the editor in chief.  Tatum has published in that journal at least twice.  That all said, if the mathematics is correct, it is correct irrespective of where it has been published, even if the author paid to have it published.)

Earlier in Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, there was something that really attracted my attention, an equation for the current CMP temperature T₀:

Once again however, this can be simplified.  Note that R_H=c/H₀ is the Hubble radius which, in a flat universe, is equal to R_H=æ.l_P, and that l_P=√(ħG/c³) so:

T₀=ħc/(4π.k_b.√(æ.l_P.2.l_P))=ħc/(4π.(m_Pc²/T_P).l_P.√(æ.2))

T₀=ħc/(4π.(√(ħc/G).c²/T_P).√(ħG/c³).√(2æ))

T₀=T_P/(4π.√(2æ))

Meaning that the only equation in which upsilon is used, H₀=ƱT₀², resolves to:

H₀=ƱT₀²=2.(4π/T_P)²/t_P.(T_P/(4π.√(2æ))²

H₀=1/(æ.t_P)

This is precisely what one would expect from a flat universe.

There is a slightly different approach, just using R_H=H_o.c, but it is messy. 
This messiness can be alleviated if we work with T₀², so that:

T₀²=(ħc/(4π.k_b.√((c/H_o).2.l_P)))²=ħ²c²/((4π)².k_b².((c/H_o).2.l_P))

T₀²=ħ²c²/((4π)².(m_Pc²/T_P)².((c/H_o).2.√(ħG/c³)))

T₀²=ħ²c²/((4π)².(m_P²c⁴/T_P²).((c/H_o).2.√(ħG/c³)))

T₀²=ħ²c²/((4π)².((ħc/G).c⁴/T_P²).((c/H_o).2.√(ħG/c³)))

Then rationalising and rearranging

T₀²=√(ħG/c⁵).H_o/(2.(4π)²/T_P²)=t_P.H_o/(2.(4π)²/T_P²)=H_o/Ʊ

So, of course, H₀=ƱT₀².

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A potential critique, if one were to only consider Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, is that all Haug and Tatum have done here is shuffle numbers around in order to hide H₀ in T₀, and then created a new constant (Ʊ) which does nothing more than reveal the previously hidden H₀.  While that may appear to be the case, such critique ignores the fact that T₀ is a measured value, specifically the temperature of the CMB today or 2.72548±0.00057 K (from 2009 but seemingly still most commonly used).  As Tatum showed in Upsilon Constants and Their Usefulness in Planck Scale Quantum Cosmology, upsilon can be used to extract H₀ from the CMB temperature:

ƱT₀²=(2.923×10^-19K^-2s^-1).(2.72548K)²=2.171×10^-18s^-1

Noting the km to Mpc ratio, 3.086×10¹⁹km/Mpc, we find:

ƱT₀²=66.99km/s/Mpc≈H₀

This would correspond to a FUGE universe which is 14.60 billion years old.

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The equation for T₀ was presented in a slightly different format in an earlier paper by Tatum, Seshavatharam and Lakshminarayana The Basics of Flat Space Cosmology:

Why the third term is there is beyond mjue, given the parallels between the second and the fourth.  The first two terms are a modification to the Hawking radiation temperature equation, as Tatum acknowledges in Upsilon Constants and Their Usefulness in Planck Scale Quantum Cosmology, (noting that k_b is the same thing as k_B).  The standard Hawking radiation temperature equation:

The implication here is that Tatum et al. are conceptually equating the temperature of the CMB to the temperature of a black hole with the same radius, but with a different value for mass – which doesn’t make sense because with a different mass you are no longer talking about a black hole.  Alternatively, they are effectively scaling that temperature (and the expression of mass in their equation as being less than that of an equivalent Schwarzschild black hole is misguided).

This leads to an intriguing notion and the potential for a different approach, in terms of establishing redshift.

Consider the Hawking radiation temperature for a black hole with the mass and radius of the prevailing Hubble sphere, now and at the time of recombination (when the CMB was generated).  Recall that, per Carroll, “a spatially flat universe remains spatially flat forever” and “the corresponding Schwarzschild radius … equals the Hubble length”.  As a consequence, the radius of the universe now is r_H-0=c/H₀ and, at recombination, it would have been r_H-recomb=c/H_recomb, where H_recomb is Hubble parameter for that time. 

According to Andrei Starinets’ General Relativity and Cosmology solution notes, the Hubble parameter at recombination (note that the event is referred to in the notes as “decoupling”) is H(t_d)=H_recomb= H₀.1000√1000.  Noting that Schwarzschild radius is directionally proportional to mass, and Hawking radiation temperature is inversely proportional to radius, we have (per the equation above from Rhodri Evans, without the approximation):

z_CMB+1=T_H-recomb/T_H-0=r_H-0/r_H-recomb=(c/H_now)/(c/ H_recomb)=H_recomb/H₀

z_CMB+1=1000√1000=31,600

Hm, still not right but it is closer to my answer but given that a(t₀)/a(t_d)=1000 should be almost certainly thought of as a(t₀)/a(t_d)=10³, rather than a(t₀)/a(t_d)=1.000×10³.   The tutorial notes go on to state that:

So, if we plug in the values t_d=380,000 years and t₀=13.8×10⁹ years, we get:

Curiously, if a(t₀)/a(t_d)=1100 is used, we get z=36,000.  It is interesting to note that the common rendering of is z_CMB=1100, but from the above this appears just be the ratio of scale factors, or there's an issue with Rhodri Evans' equations.  If not, is there some widespread blurring between the ratio of scale factors and redshift z or is Starinets making a mistake in his solution notes?

Note also that the solution notes do something akin to a pea and cup trick (similar to Tatum’s apparent hiding and revealing of H₀ above).  He presents three equations:

a(t_d)=a(t₀)/1000 … H(t_d)= H₀1000√1000 … (t_d/t₀)^2/3= a(t₀)/a(t_d)=1/1000

It is not difficult to see, when these equations are put side by side to see that:

H(t_d)/H₀=1000^3/2 and t₀/t_d=1000^3/2

So:

H(t_d)/H₀=t₀/t_d

It is unclear why this much simpler and, in retrospect, obvious relationship is not used.

We can go further to show that, since H(t_d)=H_recomb, t_d=t₀-t_CMB (where t_CMB as defined above is how long ago the CMB was generated) and z_CMB+1=H_recomb/H₀:

 z_CMB+1=t₀/(t₀-t_CMB)=t₀/(t₀-t_CMB)-(t₀-t_CMB)/(t₀-t_CMB)+1=t_CMB)/(t₀-t_CMB)+1

And generalising:

z=t/(t₀-t)

Which is the result that I arrived at, using a second approach

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Another approach is on the basis of a FUGE universe using Evan’s equation z_CMB=T_H-recomb/T_H-0.

In a FUGE universe, the current mass is M(t₀)=(m_P/2)t₀/t_P and at any time t ago, M(t)=(m_P/2).(t₀-t)/t_P.  Noting that mass is directly proportional to Schwarzschild radius and temperature is inversely proportional to radius, the equation above becomes:

z+1=T_recomb/T_now=M(t₀)/M(t_CMB)=((m_P/2)t₀/t_P)/((m_P/2).(t₀-t)/t_P)

z+1=t₀/(t₀-t)=t₀/(t₀-t)-(t₀-t)/(t₀-t)+1=t/(t₀-t)+1

This is precisely what I arrived from Starinets solution notes, so we have the same result using a third (slightly different) approach.

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In Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database, Haug and Tatum get a third value for z.  This is established from use of his (apparently) scaled temperatures.  This ends up with him comparing T₀=T_P/(4π.√(2æ₀)) with T_CMB=T_P/(4π.√(2æ_CMB)) so, noting that the subscript CMB here is equivalent to recomb used above:

z+1=T_CMB/T_now=T_recomb/T_now=(T_P/(4π.√(2æ₀)))/(T_P/(4π.√(2æ_recomb)))

z+1=√(æ₀/æ_recomb)=√(t₀/(t₀-t))=√(36,000)

It should be noted that they state that there was a choice between T_t=T₀√(1+z) and T_t=T₀(1+z).  They chose the latter, but if we choose the former, we get:

z+1=(T_recomb)²/(T_now)²=(T_P/(4π.√(2æ₀)))²/(T_P/(4π.√(2æ_recomb)))²

z+1=æ₀/æ_recomb=t₀/(t₀-t)=t₀/(t₀-t)-(t₀-t)/(t₀-t)+1

z=t/(t₀-t)

So we have a fourth approach to arrive at the same result.

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Finally, returning to Starinets’ solution notes, he notes that in the “matter-dominated era” a(t)=(t/t₀)^2/3.  This is a Standard Model thing.  In a FUGE universe, a(t)=(t/t₀) at all times, which is to say that there has been more stretching of the universe over the period between recombination and today and more redshift, so it’s precisely what we should expect.

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As I have managed to find five* methods for arriving at equations for redshift which indicate that, for an event that occurred a period t ago, z=t/(t₀-t), I am no longer convinced that there is a problem with my calculations.

There is however some confusion on the part of Tatum et al. associated with the introduction of their “composite constant” upsilon.  This will be touched on again in the next (shorter) post.

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* I understand that not all five methods are entirely distinct.  Also questions remain about Tatum's equations, not only with respect to his methods for getting the papers that contain them published but also with respect to whether the implied relationship he identifies via those equations is anything more than a rather startling coincidence.