# The Marcus Conjecture

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I have chosen to honor PF by publishing my conjecture first here .

The Marcus Conjecture, which I formulated a few days ago on one of the cosmological redshift threads, is that the energy lost from the CMB by expansion has gone into the form of dark energy.

I shall now show plausibility by calculating the amount of CMB energy in a given volume of space. The amount which the photons in that volume have already lost we know to be about 1100 times their current energy.

It will turn out that the amount lost by the photons in the volume is within an order of magnitude of the dark energy in that volume.

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The CMB energy density

Natural units c=G=hbar=k=1 are used to facilitate the calculation.

The temperature of the CMB has been measured with exquisite accuracy by COBE and WMAP. It is 1.93E-32

The density of blackbody radiation at temperature T is
(pi2/15) T4

It is a simple matter to raise 1.93E-32 to the fourth and to see that the density of energy is

9E-128

The photons in any give volume have lost 1100/1101 of their
original energy by redshift therefore the density of the LOST
CMB energy is 1100 times 9E-128

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I would think rather that the photons get redshifted. As the frequency gets smaller (because the wavelength gets larger, c = fw with w being wavelength) the corresponding energy goes down (E = hf).

Though it would be interesting to see the mechanism you propose.

And should this not be in Theory Development?

FZ+
It will turn out that the amount lost by the photons in the volume is of the same order of magnitude as the dark energy in that volume.
But that doesn't prove anything... It could just be an artifact of the scale you use, or just a complete co-incidence...

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Comparison with currently estimated dark energy density

The density of dark energy is currently estimated to be 1.3E-123.

This corresponds to a Hubble parameter of 71 km/s per Mpc
which determines the critical density rhocrit at 1.8E-123, and to the WMAP estimate that dark energy comprises
73 percent of rhocrit.

Let us compare this figure for dark energy with the expansion loss in the CMB. Multiplying 9E-128 by 1100----1.0E-124

It seems that the CMB expansion loss could have supplied a substantial portion (about one tenth) of the dark energy created by the expansion of space.

This is assuming that dark energy is, as some believe, a vacuum energy or cosmological constant which is created along with space as space expands.

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Hmm... Isn't "conjecture" a mathematical term ?
While "theory" is the relevant word for this case ?
I mean conjecture is an abolute within an
abstract system while a theory is just
a likely explanation about the real world that
we use in physics, chemistry, biology and so on.

Just thought this may be worth mentioning...

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The number of CMB photons

Hello Brad and FZ, I do not have much time to anser but Brad that is just what I am saying! You say "I would think rather that the photons get redshifted." Yes, that is the cosmological redshift---it is an effect of the expansion of space and it follows the rule you mentioned, that E= hbar w, where w is the angular frequency-----except you put it in cyclic frequency terms, which is equivalent.

FZ you are mistaken, this is not an artifact of the scale. You could do the same calculation in metric and get that the energy densities are in the relation I say they are-----the CMB loss is about 20 percent of the existing dark, on a per volume basis.
The only reason I use the natural units scale is that metric is messy and time-consuming to use. (And natural units are a bit more up-to-date and fashionable )

**************

To continue: I am not concerned with "theory development" here but with calculation of two widely accepted mainstream things to show that they have the same order of magnitude. This is sometimes a clue to there being some connection. There may be one and there may not. In General Relativity there is no set rule of energy conservation. The CMB energy can simply have gone out of existence as space expands, lengthening or "redshifting" the wavelengths. And meanwhile, without any connection, the dark energy that is assumed to be at least approximately constant per volume, can have been being created "out of nothing" as space expanded. It is perfectly legal for energy to be created and/or destroyed. However in this case there may have been a flow----what was detroyed by the expansion of space (in the CMB case) may have somehow reappeared as dark energy was created by the expansion of space.

HOW MANY CMB PHOTONS ARE THERE

Since the temperature of the CMB is 1.93E-32, the average energy of a CMB photon is 2.701x1.93E-32

In blackbody radiation the average photon energy is always 2.701kT, which is 2.701 times the temperature. (Remember that Boltzmann's constant k =1)

thus the average quantum energy in the CMB is 5.2E-32.

the number of photons per unit volume, therefore, is
equal to the energy density divided by the average for an individual

9E-128/5.2E-32

1.7E-96

To get a picture of this at familiar scale, note that a mile is E38, so that a cubic mile is E114.

Therefore a cubic mile of space contains 1.7E18 CMB photons.

1.7 quintillion.

These are the photons which have lost all but 1/3000 of their original energy. An amount comparable to the dark energy believed to occupy the same cubic mile.

Originally posted by marcus
The density of dark energy is currently estimated to be 1.3E-123.

This corresponds to a Hubble parameter of 71 km/s per Mpc
which determines the critical density rhocrit at 1.8E-123, and to the WMAP estimate that dark energy comprises
73 percent of rhocrit.

Let us compare this figure for dark energy with the expansion loss in the CMB. Multiplying 9 by 3000----27000E-128----0.27E-123

It seems that the CMB expansion loss could have supplied a substantial portion (0.27 out of 1.3) of the dark energy created by the expansion of space.

This is assuming that dark energy is, as some believe, a vacuum energy or cosmological constant which is created along with space as space expands.

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Why it is merely a conjecture

Originally posted by drag
Hmm... Isn't "conjecture" a mathematical term ?
...

Hello drag, nice to hear from you. The reason I wouldnt
say theory is that I am not proposing a mechanism by which
work could be done by something on something else.
I am just pointing out that the energy lost in one place happens
to be 10 percent of the energy that has appeared in another place, during roughly the same period of time.

Since I am not suggesting a mechanism or explaining how something could happen I can't call it a theory.

But I am suggesting the possibility----conjecturing----that there might be some connection between the disappearance of some energy somewhere and its appearance somewhere else.

Anyway what you call it is not so crucial. BTW I'm skeptical that there is a physical connection, but the similarity in amounts is quite striking and needs to be pointed out.

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The amount which the photons in that volume have already lost we know to be about 3000 times their current energy.

Where do you arrive at that? Space could easily have grown 3000 times since they were created.
I am just pointing out that the energy lost in one place happens to be 20 percent of the energy that has appeared in another place, during roughly the same period of time.
If you're pointing it out, show it. Also, can you explain how the energy got from one place to another causally?

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I think the standard argument is that the energy 'lost' by CMBR goes into 'gravitational potential energy,' as much as those terms can be defined in GR -- I know I have seen this discussed without any mention of a cosmological constant being relevant, but I don't know any detials.

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Originally posted by damgo
I think the standard argument is that the energy 'lost' by CMBR goes into 'gravitational potential energy,' as much as those terms can be defined in GR -- I know I have seen this discussed without any mention of a cosmological constant being relevant, but I don't know any detials.

Hi damgo, I've been reading a lot recently on this in Usenet threads and Baez site. It is difficult and problematical to formulate global energy conservation in GR----requires
the dubious invention of "pseudotensors" and even then...----a fair amount of skepticism about the notion that it goes into
"gravitational potential energy" and the simplest thing to say is that it is simply lost. But people do argue back and forth.

In any case it has definitely not been PROVEN that the CMB energy is conserved and goes into some definite form and the speculation that it goes into gr. pot. is not proven but is speculative.

A lot of people say it is simply lost and this actually appeals to me because of Occam's razor sort of simplicity. In GR as it stands, the vanilla mainstream version, there is no global energy cons. law and so----why not? why shouldn't it just be lost.
To get anything else one must do dubious tinkering with GR.

However, I think it should be pointed out that the CMB loss over its lifetime is 10 percent of the energy now believed to reside in the vacuum or dark energy.

I'd appreciate your confirming (for Schwarzsch.'s benefit) the figure of 1100 for the CMB redshift. It is a standard figure people assign to the CMB, as I am pretty sure you know, but he has just questioned it. Please back me up on this. Indeed space has grown 1100-fold since "they were created" (in his words) and that is just the point.

I think you also see the figure of 20 percent relating the two densities------as 0.10 is to the total 1.3----
as 0.1E-123 is to 1.3E-123---which I will have to make clearer, I see.

If anyone would like links to the speculation about energy conservation in GR, let me know. The main book on the cosmological redshift is, I believe, by Harrison, who I am told
takes the simple view that the energy just goes away. I have
not looked at the book, just seen people cite it. Maybe I will
edit this and put in the title.

found an article by Harrison Astrophys. Journal
Harrison E., 1993, The redshift-distance and velocity-distance laws, ApJ 403 28-31

caveat---I haven't seen this book---just seen Harrison cited as
an authority on cosmological redshift and related issues. here is his book:
Edward R. Harrison
"Cosmology, the Science of the Universe",
Cambridge University Press 1981

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Where do you arrive at that? Space could easily have grown 3000 times since they were created.

If you're pointing it out, show it. Also, can you explain how the energy got from one place to another causally?

Hi Schwarzschild, most of what I'm saying is just standard mainsteam stuff----I will flag stuff for you that is not (big example: this conjecture).

In reply, space HAS grown 1100 fold since CMB emitted and that is just the point. It reduces the energy of each photon by a factor of 1100.

It also thins out the CMB photons---by factor of 1100^3.
this is why you see people say that, while matter density falls off with the cube of the expansion factor the CMB radiation density falls off as the fourth power.

there are 1100^3 fewer photons in a vol of space and each photon is 1100 times less energetic.

think of it as expansion stretching out waves and longer wavelength radiation being less energetic.

I thought the 10 percent was clear. I calculated 0.1E-123
and 1.3E-123 for the two densities----these two figures are ordinary routine astronomy save for the fact that they are stated in c=G=hbar=1 units. You see that 0.1 is roughly 10 percent of 1.3.

The thing that's out of the ordinary run of the mill is that I am speculating that a mechanism might be discovered---when we understand a little better what dark energy is---whereby the energy lost from CMB ends up in the form of dark.
And I certainly don't FAVOR that hypothesis! It seems more reasonable to me, as to a lot of people, simply to view the lost energy as lost (GR has no global conservation law). But the similarity of magnitudes is suspicious and should be pointed out.

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Staff Emeritus
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Hmm...I'm on the fence about whether to keep this topic here or move it to T.D. Another gray area in astronomy I suppose. I'll just keep an eye on this for now. Carry on.

Tyger
So what is Dark Energy

supposed to be doing in the scheme of things? What is it needed to account for?

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Originally posted by Tyger
supposed to be doing in the scheme of things? What is it needed to account for?

Cosmologists like to wring their hands and say they didnt ask for it, it was just forced on them in 1998 to balance a couple of accounts.

the idea had been around since 1916
a uniformly distributed constant-density energy
but the modelers had been mostly assuming it was zero

then in 1998 there were surprising supernova (type Ia) observations by two independent teams which showed
that expansion is accelerating

only assuming a dark energy term could this be explained within standard GR

PLUS (and this is the real killer) if you include just the right amount of dark energy to provide the observed acceleration of expansion then it happens that there turns out to be enough total density to make the universe flat

inflation models and other theoretical reasoning had long suggested to people that it was flat. PLUS there were CMB observations coming in which said it was flat.

1. they weren't planning on having it (the idea had died)
2. it turned out to explain the supernova data
3. it turned out to agree with the CMB data that showed flatness
4. it agreed with the inflation scenario which implied flatness or very near to it.
5. there wasnt enough other stuff, ordinary matter and dark matter to make the U flat----they needed a WHOLE LOT more and suddenly here it was--eureka

So suddenly, in about 1998, there was a revolution in cosmology and this thing appeared with overwhelming evidence, a thing no body had been sponsoring or pushing, it was nobodys pet idea, it just came in because it made peoples equations balance

And the figure now (after this years CMB data from WMAP) is
that it is 73 percent of rho crit.

Also, we must admit that it is totally bizarre that 73percent of the energy in the universe is something we didnt know about and have no idea what it is. (and something that Einstein thot of in 1916 and then took back and regreted as "his greatest blunder" and then came back with a vengeance in 1998. This is to roll on the floor laughing about. It shows how much we humans know.
(Maybe this is it's "purpose" if you must have a purpose for everything

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The density of blackbody radiation at temperature T is(pi2/15) pi2/15 T4
marcus, this is not a personal attack, but you might want to use SI units in the future, just because they are cool. OK, so are you fairly certain that the acceleration of expanding space is caused by the lost energy in redshifted photons? Or is it something inherent in space itself? If the former were true, energy would be creating space. The latter true, space would be creating space. Which do you think it is?

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marcus, this is not a personal attack, but you might want to use SI units in the future, just because they are cool. OK, so are you fairly certain that the acceleration of expanding space is caused by the lost energy in redshifted photons? Or is it something inherent in space itself? If the former were true, energy would be creating space. The latter true, space would be creating space. Which do you think it is?

Hello Schw. you mistake my position radically.
The mainstream view (to which I adhere) is that the dark energy term is responsible for accelerating expansion

My view of cosmology is essentially that presented in Michael Turner's survey article that I gave you the link for earlier

http://xxx.lanl.gov/PS_cache/astro-ph/pdf/0202/0202008.pdf

This is a fine thorough rigorous mainstream survey by an eminent guy that totally does not popularize and talk down. It is really worth downloading. But it will take some computer time.
There are alternative links so if this does not work tell me and I will supply others.

As I have said repeatedly here and elsewhere I am SKEPTICAL that the lost CMB energy went anywhere. The simplest thing to assume is that it was lost.

(no one has proved a global energy conservation theorem for GR, except under restrictive unrealistic assumptions, the burden of proof is on those who imaging that the CMB energy was conserved)

What I wish to do is to call attention to the fact that the lost CMB energy, on a per volume basis, is roughly 10 percent of the estimated dark energy.

Since no one knows what dark energy is or where it came from, and no one knows of any place the CMB energy went, there is a remote and tantalizing possibility of some connection. But since so little is understood at the present time about these things the proper attitude is to be skeptical and to merely point to the remarkable coincidence that the energies have the same order of magnitude.

One way to call attention to the similarity in energy is to say "prove or disprove: the CMB energy went into forming part of the dark energy". Hence my conjecture.

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My view of cosmology is essentially that presented in Michael Turner's survey article that I gave you the link for earlier
you better not call it the marcus conjecture then.

And I don't think you can avoid deciding between space making more space and matter/energy making more space.

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inflector
Marcus,

Finally, on the subject of redshifts, has anyone looked at the idea of whether or not the acceleration of the expansion itself is being caused by the energy that is given up during the process of redshifting itself? i.e. that dark energy is nothing more than energy lost to the metric of spacetime during photon travel? If this idea has been refuted, can anyone point me to the relevant papers?

Then after a few days more research here, I found the Marcus Conjecture.

Interestingly, Marcus, my question above is very much related to the conjecture idea you had seven years ago. You addressed the redshifting of the CMB, and got to about an order of magnitude of the value for dark energy. What would happen if you included all the energy lost from redshifting for all photons from all sources, including those that were emitted after the CMB?

I realize you haven't proposed any mechanism for the transfer of energy, but it seems that the energy values are close enough that it might pay off to look for one.

Have you done any further work on the Marcus Conjecture in the meantime? Or do you view it as a mere curiosity?

- Curtis

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inflector
I've done some more research and found out that the CMBR has more energy than all the photons ever emitted so adding the redshift from photons created after the CMBR would not help cross the order of magnitude gap in missing energy found by the Marcus Conjecture. You've already accounted for the bulk of the energy by using the CMBR.

However, what about the energy of photons that were created and absorbed before the decoupling? Is there any way to estimate the energy lost due to redshifting before this period while the universe was still opaque?

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Have you done any further work on the Marcus Conjecture in the meantime? Or do you view it as a mere curiosity?
- Curtis

Thankyou for asking in fact I was mostly just having fun.
AFAIK it's just a coincidence that the two quantities are same rough order of magnitude.

There was no real conjecture, no idea of a physical connection. I think Phobos suspected that it was just half-serious (on my part at least) and let it pass.

inflector
I don't think any energy is 'lost' by redshifted photons. Redshifted photons are time dilated, so, in the end, all energy emitted by the source is recovered. See also Ted Bunn's recent comments: http://blog.richmond.edu/physicsbunn/2010/04/08/more-on-the-cosmological-redshift/

Thanks Chronos. Funny, I was just looking at a previous blog entry he referenced in that post:
http://blog.richmond.edu/physicsbunn/2009/12/02/interpreting-the-redshift/

Wherein Ted Bunn replies in a comment to another poster:
Ted Bunn said:
The “purist” position goes something like this. It only makes sense to talk about changes in energy if all measurements are made in the same inertial reference frame. When a photon travels over a cosmologically large distance, there is no inertial reference frame that accommodates both the emission and absorption events. Therefore, the question of whether the photon has or has not lost energy is simply meaningless.

When people use the “stretching-of-space” language, they’re implicitly assuming that comoving coordinates (i.e., a coordinate system in which both the emitting galaxy and the observer are at rest) are the most natural ones to use. But comoving coordinates are not an inertial system — that is, they’re not a system in which it makes sense to compare energies at widely different locations. So when people say that the photon has lost energy, they are committing at least a minor sin against general relativity.

Now what about my (and Hogg’s) position? We certainly agree that, if both the observer and emitter measure the photon’s energy (each in his own local inertial frame), the observer’s measurement will come out smaller than the emitter’s. But we think that that difference is more like a Doppler shift than anything else — that is, as you say, that it’s due to relative motion rather than to anything happening to the photon en route.

To expand on this a bit, suppose I throw a baseball to you, and you’re running away from me as you catch it. The baseball’s speed relative to you will be less than its speed relative to me. If each of us computes the ball’s kinetic energy, you’ll get a lower number than me. But it would be perverse to say that this means the ball has lost energy; rather, it’s just that we measured energy in different frames, which can’t reasonably be compared.

I understand his logic here but I'm not sure I buy it completely. He seems to be in conflict with a lot of other sources. It's hard for me to know who to trust here. It seems to be an area where there are some varied positions.

I've read some things that say during redshifting due to relative velocities you don't lose energy as it is just a matter the reference frame, but that cosmological redshifting is different. Many things I've read say that energy is lost but that this isn't a problem because there is no conservation of energy at cosmological scales; presumably, because as Bunn stated: "When a photon travels over a cosmologically large distance, there is no inertial reference frame that accommodates both the emission and absorption events. Therefore, the question of whether the photon has or has not lost energy is simply meaningless."

So I've read some sources that says it is lost but we don't know where it goes but that this violation of conservation of energy is not a problem at the cosmological scale for GR because there is no conservation of energy at the cosmological scale; and other sources saying that the energy is not lost. Is this just a matter of perspective or semantics?

The more I read, the more I seem to find differences of opinion. I'm trying to figure out if the differences exist in reality or only in my head because of my immature understanding.

[EDIT: This paper by Bunn and Hogg is also very relevant: http://arxiv.org/abs/0808.1081] [Broken]]

marcus said:
Thank you for asking in fact I was mostly just having fun.
AFAIK it's just a coincidence that the two quantities are same rough order of magnitude.

Thank you marcus. I thought that might be the case.

Just for kicks, do you have any idea how I might compute the energy "lost" (or seemingly lost anyway) from photons emitted from the start of the Big Bang through until the time of the decoupling? I'm trying to get a relative feel for the amount of energy involved in the Big Bang and subsequent developments. Since the CMBR represents more energy than all the photons emitted from that point forward, I am wondering if the photons that were emitted before that point (the decoupling and start of the CMBR) contained even more energy.

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