# Where did the energy in the CMB go to?

1. Mar 2, 2005

### moving finger

This may seem like a silly question, but it has me puzzled.

Theory tells us that as space expands then photons (and I guess all form of energy?) in that space are red-shifted, ie their wavelength increases, ie they lose energy. The most obvious example of this is the CMB, which consists of maybe 10^88 photons (in the observable universe) "left-over" from the Big Bang. At the time of decoupling from matter (300,000 years after the Big Bang?) each of these photons was of very high energy (low wavelength), but as the universe (space) expanded each of these photons was red-shifted (lost energy), to end up as the microwave background that we see today.

If the first law of thermodynamics is to be obeyed (mass/energy must be conserved), what happened to the energy that these photons lost?

2. Mar 3, 2005

### Chronos

Not lost, just stretched out by expansion. Does a rubber band lose mass when you stretch it? Of course not, it just has less mass [energy] per inch.

3. Mar 3, 2005

### Janitor

But if the number of CMB photons remains pretty much constant over time, and if typical photon wavelengths are increasing with time, then the product of number of photons and average energy per photon has to be decreasing, doesn't it?

At any rate, I find this a neat question!

4. Mar 3, 2005

### moving finger

sorry Chronos, your analogy doesn't apply in this case.

the total energy in a photon is proportional to its frequency E=hv. (the reason why your analogy does not apply is because the total mass of the rubber band is independent of how much the rubber band is stretched - this is not the case when we think of the relationship between energy and photon wavelengths)

as space expanded, the wavlength of each CMB photon was increased (frequency reduced).

hence the energy of each CMB photon was reduced.

the number of CMB photons has not changed much since decoupling with matter, hence the total energy locked up in the CMB has reduced over time.

this energy must have gone somewhere..... but where?

5. Mar 3, 2005

### Chronos

The energy is used to power expansion.

6. Mar 3, 2005

### moving finger

what does that mean?

does that imply the expansion "needs energy" from the CMB in order to "power it" - in which case what happens if we have a universe with no CMB, does it stop expanding? i don't think so.

if anything, the CMB contributes to the total mass-energy of the universe and should slow down the expansion rather than "power it".

7. Mar 3, 2005

### Janitor

Even when the expansion is the result of space itself being created? Maybe so, I don't know.

8. Mar 3, 2005

### SpaceTiger

Staff Emeritus
Although I'm not sure of this, I think that it does the opposite (slow the universe down). In a matter-dominated universe, the kinetic energy of the matter is gradually converted into potential energy, slowing the expansion. A radiation-dominated universe will decelerate, just like a matter-dominated one, but the light must move at the speed of light, so you can't take its kinetic energy. Instead, you reduce its frequency.

9. Mar 3, 2005

### moving finger

Ahhh. OK, yes. The energy of the photons is converted into potential energy.... because the expansion is pulling all the photons further apart hence (since they all attract each other gravitationally) the total gravitational potential energy must be increasing as the universe expands, and the energy of each CMB photon is reduced accordingly, so that everything remains balanced. Is that it?

10. Mar 3, 2005

### SpaceTiger

Staff Emeritus
I think so. I've never directly worked with general relativity, so I'm not sure if my semi-classical understanding of the problem applies here.

11. Mar 3, 2005

### Garth

It doesn't!

It is important not to inconsistently mix up Newtonian and GR gravitational concepts as has happened in the above posts. GR does not in general conserve energy, in particular it does not do so in the cosmological solution of a non-static evolving universe.

The total energy of the CMB does decrease with time as the photon number (apart from the few absorbed by matter) is conserved.

Classically gravitational potential energy is that energy used in lifting a body against a gravitational force. In GR that force is replaced by curvature. In particular work has to be done to accelerate a body out of its freely falling inertial frame, and weight is the reaction to an inertial force. Therefore classical gravitational potential energy is radically re-interpreted in GR and should not be used to explain either cosmological or gravitational red shift.

The question of “Where does the energy of a CMB photon go?” is similar to the question of “Where does the energy of a gravitationally red shifted photon go?” The GR answer is, "into the field", whatever that means.

GR just loses the CMB energy into the cosmological field with no mechanism to explain how that happens or where it might be re-located.

Remember GR does not in general conserve energy, it conserves energy-momentum instead and that is different. It is the curvature of space-time that violates the conservation of energy, which is why the energy of a system can only be properly defined in GR in the absence of curvature, where the space-time is asymptotically flat.

Garth

12. Mar 4, 2005

### Chronos

Interesting. You have pointed out a problem in both models.

13. Mar 4, 2005

### chronon

Sorry, but the redshifting of CMB photons has nothing to do with gravity. You get the same redshifting in the Milne universe where the effects of gravity are ignored.

Of course if you ignore the idea of space stretching and say that the photons are redshifted because their source was moving rapidly away from us then there isn't a problem.

14. Mar 4, 2005

### Garth

chronon you have to be consistent, in this case with the principles of GR.
According to the principles of GR what is 'gravity' except the effect of 'curvature', and what is cosmological expansion except the effect of 'curvature'? The Milne universe may not have matter but it still has 'curvature', that is if you want to remain consistent to GR. Empty space has 'gravity', i.e. 'curvature'!
According to the principles of GR are the sources moving through space-time producing a Doppler effect, or is it space-time itself that is 'stretching'?

Garth

Last edited: Mar 4, 2005
15. Mar 4, 2005

### chronon

Gravity is the curvature of space-time. The space-time of the Milne universe is flat. You might consider the space to be curved, but that depends on the choice of the coordinate system.
Sources don't move through space-time. Space-time doesn't stretch. These are time-based verbs.

16. Mar 4, 2005

### marcus

I agree with Garth. No one, to my knowledge, has ever explained where the lost CMB energy went. there is no energy-conservation law in GR, but GR is currently the prevailing largescale theory of spacetime and other theories just apply locally or approximately. So as far as anyone knows ENERGY CONSERVATION IS NOT TRUE except in some restricted local or approximate sense.

The CMB is an excellent example to show that energy conservation is not true, since no one can find the lost energy.

The concept of energy is a growing, evolving idea. Feynman's "dennis the menace" story illustrates this (I dont have a link). So one can conjecture that someday the concept of energy will be extended to something we dont know about, that we can measure, and people will measure it and say ahah THAT is where the missing CMB energy went, so they will have "found" the missing energy and conservation will be vindicated.

but that is only a conjecture. there would need to be an improved theory of spacetime, to replace GR. Because in GR the CMB energy when the photons get stretched out just goes away. It is a wellknown fact that GR does not support the energy conservation law.

So moving finger question is not at all silly. It seems very sensible to be puzzled. I've quoted several posts, including Garth's to keep track of the main points in the thread.

Last edited: Mar 4, 2005
17. Mar 4, 2005

### marcus

It is fun to calculate how much the lost energy is. I did this a couple of years ago and found that (unless I made a mistake) the lost CMB energy is within a factor of ten or so of the estimated "dark energy" density.

that is, if you take a cubic kilometer it has estimated about 0.6 joules of "dark energy" in it, according to astronomers.

but if you look at all the CMB photons in that cubic kilometer and add up their energy it is something roughly on the order of 1/10000 of that, like
on order of 0.00006 joules.

now each one of those photons has lost 999/1000 of his energy by being stretched out by the expansion of space

so therefore, if you gave each photon, in that cubic kilometer, BACK the energy that he has lost by redshifting, then within a factor of ten or so there would be about the same CMB energy in the cubic kilometer as there is supposed to be dark energy.

but there is no theory that allows lost CMB energy to reappear as dark energy. indeed I am skeptical that dark energy exists. it is not proved.
also one can be skeptical of the estimate of how much dark energy there is, assuming it exists----the 0.6 joule per cubic km.

I will recalculate the energy per cubic kilometer of the CMB and see how it compares, however. this is just for fun and not serious.

Last edited: Mar 4, 2005
18. Mar 4, 2005

### marcus

calculation in natural units:
WMAP estimate of dark energy density is 0.85E-120
current CMB temperature 0.96E-31
current CMB energy density is (pi2/15) (0.96E-31)4

but redshift of CMB is 1100, so each photon started life 1100 times more energetic, so if you gave them back their lost energy the density would be E-121

so about a tenth of "dark energy" density

... comparable anyway, same ballpark

Last edited: Mar 4, 2005
19. Mar 4, 2005

### SpaceTiger

Staff Emeritus
Actually, I think this is a natural consequence of the standard model (sort of). In order for the universe to remain flat, the total energy density must be critical. In the standard model, the currently dominant form of energy is dark energy, while in the past it was matter and radiation, respectively. During its era, of course, radiation had an energy density comparable to critical, but since then the comoving energy density has dropped due to the redshift. If it weren't for that redshift, the radiation would still have an energy density comparable to critical density and, therefore, the dark energy density.

Now, it's no coincidence that the matter-radiation transition and decoupling occur at about the same time (within a factor of a few), because recombination is related to the fact that matter is becoming the dominant constituent. Thus, the comoving energy density lost since decoupling is roughly equivalent to that lost since matter-radiation equality which, by definition, is equivalent to that lost by redshift of photons. In light of this, your calculation is perhaps a bit less surprising.

That said, however, we have no idea what the dark energy is, so it's possible that you're right (redshift -> dark energy). I doubt it, though.

20. Mar 4, 2005

### Garth

The Milne universe is hyperbolic, k = -1, to my knowledge that means it isn't flat or even conformally flat! In the cosmological solution to Einsteins field equation the space-time of an empty universe has curvature.
True, I was reflecting your own choice of words back to you, again, be consistent!

However from our frame of reference and our temporal "time-based" existence we can ask whether these sources are moving away from us and cosmological red shift is doppler in nature or whether space-time "stretches", i.e. the world lines of distant observers co-moving with the cosmological frame of reference in which the universe (CMB) is globally isotropic and homogeneous diverge from each other.

Garth

Last edited: Mar 4, 2005