Exploring an Alternative to Hubble's Law

[gptejms]

Garth,Chronos:
I don't think you(Garth) are reading my posts carefully.I asked for your comments on 'my explanation' for the equivalence of the two statements for non-gravitating bodies moving away from one another--you repeat your own explanation!
Anyway,continuing from my post no. 33,I now switch on the gravitation between the bodies moving away from one another.Before gravitation is switched on,the 'two statements' are equivalent in the manner pointed out in post 33.Once gravitation is switched on,there is an additional source of red shift due to changing clock rates(due to diminishing(over time) gravitation).Now are the two statements equivalent?Unless the relative velocity changes(by switching on gravitation)I don't see the two statements to be equivalent.Does the relative velocity change..it should.

So Garth,you could be right that the two statements are equivalent,though I am not 100 per cent sure!What do you have to say,Chronos?

 

[Garth]

I don't have enough time or background in cosmology or GTR to take this up in a serious way.

I am talking about cosmology, i.e. GR, if you are talking about the empty (zero density) cosmological model than I have commented on it - it expands and mutual red shift would be observed by 'test' (no mass) observers within it. Read up some GR cosmology textbooks, or perhaps here.

Ouch. Observational evidence suggests the universe includes gravitating matter. Empty universe models are intriguing, but to put it delicately: unphysical.

I did not intend to be rude but in light of that first quote of gptejms I wanted to set this discussion on the basis of recognised physics and not speculation.

In order to seriously make other hypotheses you do need "to have enough time or background in cosmology or GTR to take this up in a serious way".

Chronos , of course I agree that the universe is not empty, however the Milne model is a recognised cosmological model of the limit of a universe that asymptotically tends to zero density.

In the discussion about non-gravitating observers in an expanding universe the Milne model is the correct one to use, or otherwise, if there is no space-time curvature the discussion is one of SR in which mutally receding observers just observe (relativistic) doppler red shift.

I have located the detail for Dennis A Wright, he has a website where he has some of the math posted, obviously not everything because he wants you to buy the books. It’s not what normally attracts serious science but has some interesting concepts, he took up the study of physics because he is clairvoyant and wanted to understand how time functions. (Don’t we all?)

If this is turning into a discussion of clairvoyance I am out of here...

Garth

 

[Tzemach]

How do you measure the 'acceleration' of time? Which two clocks are you comparing to detect the acceleration of one clock relative to the other?

(Note: I am not objecting to the idea, it is part of my own SCC model, however that statement d) does not make sense on its own - you need to specify how you measure it.)

Garth

Each unit of time is $$T-t_p or 10^-43$$ smaller than the unit that preceeded it. So instead of comparing clocks we are examining the behaviour of a single clock.

 

[Garth]

Each unit of time is $T-t_p or 10^-43$ smaller than the unit that preceeded it. So instead of comparing clocks we are examining the behaviour of a single clock.

How do you propose to do that?
How is the comparison made experimentally of a clock and itself at a later time?

Garth

 

[gptejms]

In order to seriously make other hypotheses you do need "to have enough time or background in cosmology or GTR to take this up in a serious way".

Yours is a valid objection.In fact,I did read about 60% of Dirac's thin book on GR a few years back,though I don't remember all the details--may be I need to re-read it to refresh my knowledge.GR is not my main field, so I read just enough to get a fair idea of the field.

In the discussion about non-gravitating observers in an expanding universe the Milne model is the correct one to use, or otherwise, if there is no space-time curvature the discussion is one of SR in which mutally receding observers just observe (relativistic) doppler red shift.

Milne model must be the correct one to use for non-gravitating observers--even in this model I presume the 'two statements'('red shift caused by expansion'&. 'red shift caused by changing gravitational field') are equivalent--right?

 

[Garth]

Milne model must be the correct one to use for non-gravitating observers--even in this model I presume the 'two statements'('red shift caused by expansion'&. 'red shift caused by changing gravitational field') are equivalent--right?

Yes, and the statements "cosmological red shift is caused by a) the expansion of space, and b) an evolving cosmological (i.e. not local) gravitational field, are equivalent in GR with massive objects and observers.

The questions are:
Precisely what are you measuring?
How is the measurement made?
What standard units of length, time and mass are you comparing them with? and
How is that comparison made over cosmological space and time?

I hope this helps

Garth

 

[gptejms]

Garth:It will be nice if you can do a calculation and show that the two statements are equivalent(for Milne metric or any other metric).

 

[Garth]

Garth:It will be nice if you can do a calculation and show that the two statements are equivalent(for Milne metric or any other metric).

I'll try, my tex/itex get screwed up for some reason since the new server.

The Robertson Walker metric for a homogeneous and isotropic universe is:

$$d\tau ^2 = dt^2-\frac{R(t)^2}{c^2 }[dr^2+\bar{r}^2 d\Omega^2]$$

where
$$d\Omega^2 = d\theta^2+\sin^2\theta d\phi^2$$

and $$R(t)$$ is the scale factor describing the expansion of the universe. This is derived from the Field Equation of the gravitational theory (normally GR).

$$\bar{r} =\begin{cases} R_C \sinh(r/R_C), &\mbox{for negative curvature} \\ r, &\mbox{for zero curvature} \\R_C \sin(r/R_C), &\mbox{for positive curvature} \end{cases}$$

[EDIT: here my tex and itex get screwed up again ( - help?), I hope you can understand the following.]

R_C the absolute value of the radius of curvature

r gives the comoving distance from the observer and

\bar{r} gives the proper distance distance.

The cosmological time dilation is given by
dtau/dt and the red shift of an object at time t_1 observed at time t_0 is given by

1 + z = R(t_0)/R(t_1)

now, given a fixed k and R_C, R(t) describes both the 'expansion of the universe' and the 'evolving cosmological gravitational field', they are one and the same.

Garth

 

[Tzemach]

How do you propose to do that?
How is the comparison made experimentally of a clock and itself at a later time?

Garth

If I remember from the first book there was a thought experiment which looked at a hypotheical spaceship which could exceed the speed of light (suspend your relativistic paradigms for a moment) in this spaceship one could not tell whether you were moving in space or time. It made sense but I will have to find and reread it for the full explanation.

I want to wait for the next book to come out because his website says that dark matter, dark energy and the Pioneer Anomaly are all proof that the theory is correct and time accelerates. He also says that gravity is due to the inertia or reluctance of mass to accelerate in time, which is why mass distorts time. (Time moves more slowly in a gravitational Field).

 

[Garth]

If I remember from the first book there was a thought experiment which looked at a hypotheical spaceship which could exceed the speed of light (suspend your relativistic paradigms for a moment) in this spaceship one could not tell whether you were moving in space or time. It made sense but I will have to find and reread it for the full explanation.

I want to wait for the next book to come out because his website says that dark matter, dark energy and the Pioneer Anomaly are all proof that the theory is correct and time accelerates. He also says that gravity is due to the inertia or reluctance of mass to accelerate in time, which is why mass distorts time. (Time moves more slowly in a gravitational Field).

First, here, let us discuss recognised and published physics. Private theories have to pass the criteria and be published in the IR Forum.

You can certainly ask questions, such as: "Does time move more slowly in a gravitational field?" but you then have to respect the answer if given in a clear and logical way.

I'm not against the concept of red shift being an artifact of "time acceleration", indeed it is a feature of my own, published, theory. However you do have to be clear about what you are saying; to detect time acceleration you have to compare one clock with another one and define everything consistently.

All we can do here on Earth is compare our observations of the distant universe with clocks, rulers and standard masses here in the laboratory. One such anomaly that might be indictating the standard understanding requires modification is indeed the Pioneer Anomaly and there has been much discussion on the subject in these Forums.

Garth

 

[Tzemach]

I'm not against the concept of red shift being an artifact of "time acceleration", indeed it is a feature of my own, published, theory.
Garth

This is something I am interested in following further, is there a link to your published theory somewhere? I apologise for being a little too enthusiastic and not observing the protocols properly but I was just a little keen to go further as there seemed to be several people thinking along the same lines.

 

[Garth]

This is something I am interested in following further, is there a link to your published theory somewhere? I apologise for being a little too enthusiastic and not observing the protocols properly but I was just a little keen to go further as there seemed to be several people thinking along the same lines.

The theory is called Self Creation Cosmology, which can be found in a Wikipedia http://en.wikipedia.org/wiki/Self_creation_cosmology .

If you want to discuss the theory please do so there, I don't want it to 'hog' other threads.

[Note: I see the links etc. are all screwed up as well so please find the links to my papers from the bottom of that Wikipedia article if you want to.]
Garth

 

[gptejms]

The cosmological time dilation is given by
dtau/dt and the red shift of an object at time t_1 observed at time t_0 is given by

1 + z = R(t_0)/R(t_1) ...(1)

now, given a fixed k and R_C, R(t) describes both the 'expansion of the universe' and the 'evolving cosmological gravitational field', they are one and the same.

1.R(t) describes 'expansion of the universe',the wavelength gets stretched when the scalefactor expands--this is clear to me.

2.What's not quite clear is the statement:-R(t) describes 'evolving cosmological gravitational field'.First, how do you define cosmological gravitational field?Second,how do you relate it to the scale factor?Why should a gravitational field 'cosmological' or otherwise lead to expansion?
(physical understanding is what I'm looking for)

3. I ask a stupid question here:-I understand your relation(1) (for the red shift)in the spirit of my point 1 above.But you seem to be relating it to the cosmological time dilation dtau/dt...which can be calculated from the RW metric.Is this what you are doing?

 

[gptejms]

He also says that gravity is due to the inertia or reluctance of mass to accelerate in time, which is why mass distorts time. (Time moves more slowly in a gravitational Field).

This seems like circular logic to me.Reluctance of mass to accelerate in time(where acceleration in time is caused by changing gravitational field)...which is why time moves more slowly in a gravitational field(i.e. mass distorts time).

 

[Garth]

1.R(t) describes 'expansion of the universe',the wavelength gets stretched when the scalefactor expands--this is clear to me.

2.What's not quite clear is the statement:-R(t) describes 'evolving cosmological gravitational field'.First, how do you define cosmological gravitational field?Second,how do you relate it to the scale factor?Why should a gravitational field 'cosmological' or otherwise lead to expansion?
(physical understanding is what I'm looking for)

3. I ask a stupid question here:-I understand your relation(1) (for the red shift)in the spirit of my point 1 above.But you seem to be relating it to the cosmological time dilation dtau/dt...which can be calculated from the RW metric.Is this what you are doing?

Consider co-moving galaxies, i.e. those that are carried along with the expansion of space and have no proper motions.

R(t) is the only thing that is evolving with time in the R-W metric.
We are used to the idea that in the Schwarzschild solution, with local static gravitational fields, the field changes with r and not t, in the cosmological solution it changes with t and not r.

It is the average gravitational field of the whole universe that we are talking about. Remember in the cosmological solution we are dealing with a homogeneous density smoothed out over space.

If the expression "evolving cosmological gravitational field" means anything, which is my way of translating your expression in the OP: "As we go back into the past, things become nearer and nearer--so gravitational fields become higher and higher", then it also refers to
R(t).

Garth

 

[gptejms]

It is the average gravitational field of the whole universe that we are talking about. Remember in the cosmological solution we are dealing with a homogeneous density smoothed out over space.

If the expression "evolving cosmological gravitational field" means anything, which is my way of translating your expression in the OP: "As we go back into the past, things become nearer and nearer--so gravitational fields become higher and higher", then it also refers to
R(t).

Garth

For non-gravitating bodies moving away from one another(described by Milne metric as you say) the average gravitational field is zero.But there must be an "evolving cosmological gravitational field" according to your definition.How do you explain this?

 

[Garth]

For non-gravitating bodies moving away from one another(described by Milne metric as you say) the average gravitational field is zero.But there must be an "evolving cosmological gravitational field" according to your definition.How do you explain this?

You must remember that space-time is static, there is no evolution.

What does evolve from one time coordinate to the next are the space-like foliations or 'slices' of space-time of all events defined at a particular moment of cosmological time. These can be clearly defined as those surfaces of homogeneity on which any observer would observe an isotropic universe.

Gravitational fields are described in GR as the curvature of space-time prescibed by the Riemannian tensor.

In the Milne universe every component of the Riemannian is zero.

Nevertheless, the homogeneous space-like foliations are hyperbolic and, with respect to the temporal-conscious experience of an observer within them, evolve, i.e. expand, linearly: R(t) = R₀t/t₀.

I agree with Chronos that as the Milne universe is empty it is unphysical, yet it is a valid solution of Einstein's field equation and describes the asymptotic limit as $\rho \rightarrow 0$.

Garth

 

[gptejms]

Gravitational fields are described in GR as the curvature of space-time prescibed by the Riemannian tensor.

In the Milne universe every component of the Riemannian is zero.

Nevertheless, the homogeneous space-like foliations are hyperbolic and, with respect to the temporal-conscious experience of an observer within them, evolve, i.e. expand, linearly: R(t) = R₀t/t₀.

So you relate this (R(t)) to a 'cosmological graviational field' even though the gravitational field or every component of the Riemannian is zero?

 

[Garth]

So you relate this (R(t)) to a 'cosmological graviational field' even though the gravitational field or every component of the Riemannian is zero?

Yes, the behaviour of geodesics, described by R(t), are determined by the solution of the Einstein field equation. A gravitational field where all the components of the Riemannian are zero has a characteristic behaviour, just as when it has non-zero components.

What it is saying is that, given the universe is expanding, if there is no source of gravitation i.e. no gravitational forces, then the universe expands linearly, i.e. there is 'nothing to slow it down'. (when the cosmological constant $\Lambda = 0$)

 

[Tzemach]

Garth thank you for your help and the links, this has helped me sort out a couple of questions. I think your work is great, when I digest it all and do some more work I might have a few more clues.

 

[gptejms]

Yes, the behaviour of geodesics, described by R(t), are determined by the solution of the Einstein field equation. A gravitational field where all the components of the Riemannian are zero has a characteristic behaviour, just as when it has non-zero components.

What it is saying is that, given the universe is expanding, if there is no source of gravitation i.e. no gravitational forces, then the universe expands linearly, i.e. there is 'nothing to slow it down'. (when the cosmological constant $\Lambda = 0$)

'Taking the limit $$\rho$$ tending to zero(empty universe) starting with a universe with matter' is not the same as 'starting with non-gravitational bodies and then doing the calculations'.It could well be that the universe needs a trigger to start expanding and when that trigger is withdrawn($$\rho\rightarrow 0$$) it doesen't stop expanding:---it has an inertia of motion...or let me say inertia of expansion.

I googled for Milne cosmology today and found an interesting paper 'An interpretation of Milne Cosmology' by Alasdair Macleod.You'll see from the paper that Milne didn't like the idea of spacetime curvature.He tried to give an explanation of Hubble's law from SR.It needs a very special(and unlikely) initial condition for it to work.$$\rho$$ tending to zero is also called as Milne cosmology though it's a misnomer and not the original idea.

This author also argues that Milne and empty universe models are not mathematically equivalent.

 

[Garth]

'Taking the limit $$\rho$$ tending to zero(empty universe) starting with a universe with matter' is not the same as 'starting with non-gravitational bodies and then doing the calculations'.It could well be that the universe needs a trigger to start expanding and when that trigger is withdrawn($$\rho\rightarrow 0$$) it doesen't stop expanding:---it has an inertia of motion...or let me say inertia of expansion.

I googled for Milne cosmology today and found an interesting paper 'An interpretation of Milne Cosmology' by Alasdair Macleod.You'll see from the paper that Milne didn't like the idea of spacetime curvature.He tried to give an explanation of Hubble's law from SR.It needs a very special(and unlikely) initial condition for it to work.$$\rho$$ tending to zero is also called as Milne cosmology though it's a misnomer and not the original idea.

This author also argues that Milne and empty universe models are not mathematically equivalent.

Milne had his own cosmological theory called "Kinematic Relativity" in which space-time is flat, do not confuse that with his name being attached to the empty GR curved space-time model, which is what we are talking about here.

As far as your first comment is concerned, you have to be clear about whether you are discussing SR or GR.

In SR if non-gravitating bodies are mutually receeding then the red shift observed is due to their motions through (Minkowski) flat space-time and is relativistic doppler shift.

In GR the bodies are normally assumed to be at rest in space-time (co-moving particles) and it is space-time itself that expands and carries the bodies with it. The red-shift observed is cosmological red shift as I have discussed above.

I have been considering GR cosmological red shift all along.

Garth

 

[gptejms]

If you consider non-gravitating bodies(to start with),then of course you are in the SR domain(i.e. the original Milne cosmology) and you have to use the Doppler effect(as was done by Milne).What I argued in the posts above was that even in this case the Doppler effect may be traced back(as an alternative explanation) to the 'acceleration phase'.So even though we are not talking of cosmological red shift here, the red shift may be thought of as either due to the 'expansion'(where this is not cosmological expansion but 'expansion' of the universe due to bodies moving away from one another) or due to a gravitational effect(by equivalence principle the 'acceleration phase' may be thought of as a phase of gravity coming into play).So the 'two statements' that you have been claiming are equivalent(for the cosmological case) are equivalent even for this case.

Now coming to Milne cosmology in the (GR) sense i.e. $$\rho\rightarrow 0$$(i.e. the sense that you have been talking),let me first say that this is not a model for non-gravitating bodies moving away from one another.This is a model for(to start with) gravitating bodies moving away from one another for which gravity has become extremely weak or negligible at some stage---such a universe has an inertia of expansion and keeps expanding.At the stage when gravity is switched off,I do not see the equivalence of the 'two statements' holding any longer---now the only way to explain the red shift is the cosmological expansion;there is no cosmological gravitational field.

 

[Garth]

You have to define your terms very carefully and state exactly how they are measured.

We were discussing the term "cosmological expansion", there is no such thing in SR, neither is there an "'acceleration phase'", these can only be discussed in GR.

As I have said, the surprising thing in GR is that the empty universe expands - linearly so for there is nothing to slow it down.

Test (infinitesimal mass) particles in such a universe are carried along with the linearly expanding space and exhibit mutual red shift, that is cosmological red shift, which depends on their position in space, rather than doppler red shift, which depends on their velocity through space.

To any observer the red shift of the other test particles may be attributed to the expansion of the universe or to the universe's (null) gravitational field, that is, the geometry of its space-time. They are two different ways of interpreting the same effect.

Perhaps where there is confusion is over the GR concept that co-moving objects in an expanding universe are not moving through space, but rather are been carried along by the expanding space in which they are embedded.

Garth

 

[gptejms]

We were discussing the term "cosmological expansion", there is no such thing in SR, neither is there an "'acceleration phase'", these can only be discussed in GR.

Agreed.The only thing I've been saying in the above posts is that 'the two statements' are equivalent even for the original Milne cosmology(in the sense that I've mentioned above(see post #58)).

As I have said, the surprising thing in GR is that the empty universe expands - linearly so for there is nothing to slow it down.

Test (infinitesimal mass) particles in such a universe are carried along with the linearly expanding space and exhibit mutual red shift, that is cosmological red shift, which depends on their position in space, rather than doppler red shift, which depends on their velocity through space.

Right.

To any observer the red shift of the other test particles may be attributed to the expansion of the universe or to the universe's (null) gravitational field, that is, the geometry of its space-time. They are two different ways of interpreting the same effect.

Here,there's a difference in our perspectives.If you look at my post no. 1,I say that there are two sources of red shift---cosmological expansion &. clocks getting faster due to decreasing (overall)gravitational field.You said these are one and the same thing.
Now coming to the $$\rho \rightarrow 0$$ case,clocks ain't getting any faster due to a (null) gravitational field(!)---so how are the two statements equivalent?I hope I am able to convey my point.When I talk of a gravitational field, I am looking for clocks getting faster(or slower)--there is no such thing in the case of a (null) gravitational field.So the two statements are not equivalent 'in the sense' that I have been talking.

The way you are making them equivalent is an equivalence that exists per se---it does no value addition.It's not the same as the equivalence that I have been talking about.

Perhaps where there is confusion is over the GR concept that co-moving objects in an expanding universe are not moving through space, but rather are been carried along by the expanding space in which they are embedded.

There is no such confusion.

 

[Garth]

There is no such confusion

BTW A Very Happy Birthday!

If there is no confusion and we agree that in cosmological red shift objects are treated as if they had no proper motion through space, only that of the expanding space in which they are embedded. Then the observed red shift can only come from the evolution of the scale factor with time.

That scale factor, R(t), and the curvature factor, $(1 - kr^2)^{-1/2}$, determined by the average density, are the only descriptions of the "cosmological gravitational field" in the cosmological R-W metric.

Therefore in the Milne universe (where k = -1), the observed red shift is due to the scale factor, R(t) ~ t, i.e. the null gravitational field. (See d'Inverno "Introducing Einstein's Relativity" pages 324-5 for a derivation of the cosmological red shift)

I think it is confusing that this cosmological red shift is also called doppler shift when the objects are not moving through space.

Garth

 

[gptejms]

Thanks, Garth!

 

[ratfink]

Isn't this a bit strong

If there is no confusion and we agree that in cosmological red shift objects are treated as if they had no proper motion through space, only that of the expanding space in which they are embedded. Then the observed red shift can only come from the evolution of the scale factor with time.

(bolding etc is from me)
Just a thought, my understanding is that the Doppler interpretation is abandoned because if the redshift was due to proper motion then galaxies would distort as they approach the speed of light.
They don't ergo we say that it is space expanding carrying the galaxies along with it.
1) is this true?
2) Is there any test that we can carry out to differentiate between the two scenarios?
Ratfink typing with slapped wrists

 

[Garth]

Hi ratfink.
If you think "Then the observed red shift can only come from the evolution of the scale factor with time." is a bit 'strong', then where else might it come from?

Note: we are in the field of GR, which is a theory that may not be the last word on the subject and might in future be adapted or modified, however, while we are understanding cosmology under the standard GR paradigm then we are accepting that cosmological red shift is due to the expansion of space itself rather than the motions of galaxies etc. through space.

As in GR we understand the universe to be expanding and the proper distance to these distant objects increasing with time then it is understandable that the observed red shift is described as Doppler shift. All I am saying in my posts above is that this is the same phenomenon as that described by the expression "evolution of the gravitational field".

Garth

 

[gptejms]

I have a question on CMB.Due to the expansion of the universe there should be two sources of cooling of background radiation--1.)adiabatic expansion like that of any gas,2.)stretching of wavelengths due to cosmological expansion of space.Is the latter effect taken into account in calculations?

 

[gptejms]

Okay,I see from Ned Wright's cosmology tutorial that the latter effect is indeed taken care of.Now my question is about the first effect--is that taken care of?

 

[Garth]

Okay,I see from Ned Wright's cosmology tutorial that the latter effect is indeed taken care of.Now my question is about the first effect--is that taken care of?

There are two temperatures referred to in cosmology, the black body temperature of the CMB, which is at the present time equal to 2.76⁰K, and the temperature of matter, whcih can be anything today above 2.76⁰K. (There may be super-cooled gas around below this temperature in which case please let me have any links to published papers)

The temperature of the CMB is slowly decreasing because of the "stretching of wavelengths due to cosmological expansion of space". $T \propto R(t)^{-1}$.

When the universe was ionised the plasma within it was heated to the temperature of the then temperature of the CMB by photon-particle interactions.

Once the universe became transparent the radiation cooled adiabatically until other processes, such as it forming dense halos, changed the physical conditions, the matter temperature evolution then became a bit messy!

I think SpaceTiger is the expert in this regime!

Garth

 

[hellfire]

I have a question on CMB.Due to the expansion of the universe there should be two sources of cooling of background radiation--1.)adiabatic expansion like that of any gas,2.)stretching of wavelengths due to cosmological expansion of space.Is the latter effect taken into account in calculations?

When a (relativistic) photon gas cools adiabatically, the dependence of its energy with volume is $E \propto V^{-1/3}$. This is equivalent to $E \propto 1/a$ with a scale factor. Since $E = h \nu$ and $\nu \propto 1/a$ in an expanding space, this energy loss in an comoving (expanding) volume is due to the decrease of the frequency. As far as I know this accounts for all the energy loss and one has not to consider anything additionally.

 

[gptejms]

I think both the effects mentioned in post #65 can be taken into account.Consider an ideal bose gas(background radiation) undergoing cosmological expansion.Internal energy of a bose gas per unit volume, $$U/V \sim T^4$$,where T is the temperature.Now consider V increasing due to (cosmological) expansion.$$V\sim R^3$$,so it would seem that T goes as $$R^\frac{-3}{4}$$.

But since every photon in the background radiation is red shifted by an equal factor,the (total) internal energy U also goes down by the same factor i.e. U goes as 1/R.So one can see that the temperature T goes as 1/R rather than $$R^\frac{-3}{4}$$.

There could be loopholes in the argument(!) but because it's appealing I am reporting it as soon as I've thought of it.

 

[gptejms]

The above argument is obviously wrong.Temperature can not depend on volume,so all the relation means is that for a gas at temperature T,larger volume means larger internal energy.In fact it sheds no light on how the temperature varies with increasing size of the universe.

Now assuming that temperature T~1/R (due to cosmological expansion),how does internal energy scale as a function of R?This is the only valid question to ask.Does the above relation answer this question?What is the answer?

Well, the answer is U ~ 1/R(!).See,there is a difference in what 'exactly' was said in the last post and this one.If in post #69 we were to 'if U goes as 1/R due to cosmological expansion then T goes as 1/R too' then it would be a correct statement.The present post says the reverse('if T goes as 1/R then U goes as 1/R).

So we have said nothing so far about cooling of the bose gas due to the effect of adiabatic expansion.Hope the cosmo tigers here have at least something to say on this.