Dark energy and the Cosmological Constant

[Quds Akbar]

So the Universe expanded very rapidly in its very first moments (inflation). The Universe then slowed down and is speeding up again, and Dark energy is supposed to be responsible for this accelerated expansion.

The cosmological constant might as well be dark energy, but why is it still being considered when the Universe's expansion's rate has changed over time? And the cosmological constant is constant, so it should remain constant since it existed, right?

So I'm guessing my question is, why do physicists still consider the cosmological constant if the Universe's expansion rate has changed over time? Or was dark energy simply not responsible for inflation?

 

[phinds]

Dark Energy is not known to have been responsible for inflation, but we don't know what was responsible, so it's hard to say.

Expansion and acceleration of the expansion are not the same thing. Dark energy is only responsible for the acceleration.

 

[Quds Akbar]

Dark Energy is not known to have been responsible for inflation, but we don't know what was responsible, so it's hard to say.

Expansion and acceleration of the expansion are not the same thing. Dark energy is only responsible for the acceleration.

But why do we believe in the cosmological constant if the expansion since the beginning of the universe was not constant?

 

[Garth]

But why do we believe in the cosmological constant if the expansion since the beginning of the universe was not constant?

Because the Cosmological Constant, aka DE, has only become dominant in the latter history of the universe.

If it is DE, not $\Lambda$, then it might evolve over time and indeed it might possibly be the inflaton field at an earlier, higher energy stage. The trick will be getting the idea to work!

Garth

 

[Bandersnatch]

But why do we believe in the cosmological constant if the expansion since the beginning of the universe was not constant?

It kinda works like this:
You've got three 'things' that influence expansion: matter, radiation and the cosmological constant (which is like a property of space). As the universe expands (there's more space), matter and radiation get diluted: on average there's less of them per unit volume of space (radiation gets diluted faster due to being additionally redshifted). The cosmological constant doesn't get diluted - it remains the same per unit volume of space.

So, way back when everything was closer together, the matter and radiation density were dominant (i.e. pulled everything together slowing down the rate of expansion), but with time they got diluted so much that the influence of the cosmological constant took over and became dominant - driving the accelerated expansion.

The cosmological constant is (or at least appears to be) constant, because how much a unit of space 'pushes outwards' never changed. It's the other influences that went down.

 

[phinds]

Yeah, what he said

 

[marcus]

... And the cosmological constant is constant, so it should remain constant since it existed, right?

So I'm guessing my question is, why do physicists still consider the cosmological constant if the Universe's expansion rate has changed over time?...

Hi, Quds, I agree with Phinds that Bandersnatch's explanation was especially good---as verbal explanations go. There is a limit to how clearly you can think about the standard cosmic model if you think purely in words without ever considering definite measured quantities like RATES of distance expansion, numerically expressed. Words can get in the way of understanding because they come with a lot of other associations. Like the word "acceleration" is automatically associated with driving a car and brings up a whole batch of unconscious expectations.

It could be that you are content with the verbal understanding and don't want to look at the actual expansion RATES, expressed numerically. It's only optional. Only read what I'm writing if you are curious and like numbers. There are two important rates of distance expansion to learn: the present rate H₀ and the longterm constant future rate H_∞ which is being gradually approached over time.

To keep this post simple I will only describe the present rate and leave the other for another post if you say you want a more numerical understanding.

The distances we are talking about are long-range cosmic scale distances between disconnected things, things not bound together by gravity, e.g. in orbits. We aren't talking about distances within our own galaxy, or between us and our nearest neighbor galaxies (to which ours is gravitationally bound).

Those long-range distances are currently increasing by 1/144 of one percent per million years.

Another way to say the same thing, if you like scientific notation (powers of ten) is to say that the current fractional growth rate H₀ is 2.20 x 10^-18 per second.

 

[Quds Akbar]

If it is DE, not $\Lambda$ then it might evolve over time and indeed it might possibly be the inflaton field at an earlier, higher energy stage. The trick will be getting the idea to work!

Garth

What is the "A"?

 

[Bandersnatch]

What is the "A"?

It's a greek letter 'lambda', used to denote the cosmological constant. What Garth means there is that there are other than Λ theoretical possibilities for dark energy, where it would no be constant. The quoted statement can be a bit misleading, though, as Λ still is DE.

 

[Garth]

It's a greek letter 'lambda', used to denote the cosmological constant. What Garth means there is that there are other than Λ theoretical possibilities for dark energy, where it would no be constant. The quoted statement can be a bit misleading, though, as Λ still is DE.

I disagree, there is a difference between $\Lambda$ (on the left hand side of Einstein's Field Equation) and DE (on the right hand side of EFE) - see the discussion here.

Garth

 

[Bandersnatch]

Fair enough, but... He'll still find the the one referred to as the other in the less rigorous sources if he continues reading about the subject. Even your earlier post from which the statement was quoted can be confusing in this way, as you both equate the two and differentiate between them.

 

[wabbit]

Those long-range distances are currently increasing by 1/144 of one percent per million years.

Another way to say the same thing, if you like scientific notation (powers of ten) is to say that the current fractional growth rate H₀ is 2.20 x 10^-18 per second.

I sometimes wish this kind of units were used more often instead of the like of "km/s/Mpc", which combines units from at least two different systems and can make quick order-of-magnitude estimates tricky.

My favorite ? "About 7 % per billion years".

 

[PeterDonis]

there is a difference between $\Lambda$ (on the left hand side of Einstein's Field Equation) and DE (on the right hand side of EFE) - see the discussion here.

If I understand your argument in that other thread correctly, you are saying that if $\Lambda$ is on the LHS of the EFE, it must be a constant (not varying in either space or time), because otherwise it would violate the Bianchi identities; whereas if it is on the RHS, it can vary? That's not a valid argument; the RHS of the EFE has to obey the Bianchi identities just like the LHS does (since the EFE is just an equality between the LHS and RHS), so if $\Lambda$ must be a constant if it's on the LHS, it must be a constant if it's on the RHS as well.

 

[Garth]

If I understand your argument in that other thread correctly, you are saying that if $\Lambda$ is on the LHS of the EFE, it must be a constant (not varying in either space or time), because otherwise it would violate the Bianchi identities; whereas if it is on the RHS, it can vary? That's not a valid argument; the RHS of the EFE has to obey the Bianchi identities just like the LHS does (since the EFE is just an equality between the LHS and RHS), so if $\Lambda$ must be a constant if it's on the LHS, it must be a constant if it's on the RHS as well.

Hi Peter,

Well it first depends on whether you want to stay within GR or not. If we are to stay within GR then $\Lambda$ must be constant, otherwise we start again with some other non-metric theory. Of course it might just be that an observation of '$\Lambda$/DE' varying may force such consideration in future.

However the GR Field Equation (EFE) is given by: $R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu}$

The whole construction of GR and the Equivalence Principle revolves around the 'covariant conservation' of both sides of the EFE, see MHW chapters 15 (Bianchi Identities and the boundary of a boundary (is zero)), 17.2 (Automatic conservation of the source - a dynamic necessity), 17.3 (Cosmological Constant).

If we move $\Lambda$ over to the RHS to become DE then we have: $R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} = 8\pi [T_{\mu\nu} + E_{\mu\nu}]$, where $E_{\mu\nu}= -(\Lambda/(8\pi) )g_{\mu\nu}$

If the LHS of the EFE is conserved then the whole of the RHS: G, the stress-energy tensor of 'matter' $T_{\mu\nu}$and DE $E_{\mu\nu}$, must be 'covariantly conserved': 8$\pi (G[T^{\mu}_{\nu} + E^{\mu}_{\nu}]);_{\mu} = 0$. G may be allowed to vary in theories such as Brans Dicke and https://en.wikipedia.org/wiki/Self-creation_cosmology , however in general G is considered constant.

In which case we must have: $(T^{\mu}_{\nu} + E^{\mu}_{\nu});_{\mu} = 0$ and any variation in the DE stress-energy tensor $E_{\mu\nu}$ must be 'mopped up' by that of matter $T_{\mu\nu}$, this will violate the equivalence principle to some degree.

So if we allow variation of $E_{\mu\nu}$ i.e. DE but not $\Lambda$ then it is possible to consider an evolving DE and still stay within GR.

However, if we start varying $\Lambda$ then it would be very easy to stray into very strange territory.

Garth

 

[PeterDonis]

If we are to stay within GR then $\Lambda$ must be constant

it is possible to consider an evolving DE and still stay within GR.

These two statements are inconsistent. "Evolving DE" is just another name for "$\Lambda$ is not constant". It's either one or the other.

Let me try and clarify a bit. If we put $\Lambda$ on the LHS of the EFE, we have

$$
G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}
$$

Taking the covariant derivative of both sides (not assuming $\Lambda$ constant) gives

$$
\nabla^{\mu} G_{\mu \nu} + \Lambda \nabla^{\mu} g_{\mu \nu} + g_{\mu \nu} \nabla^{\mu} \Lambda = 8 \pi \nabla^{\mu} T_{\mu \nu}
$$

Now, if we want to say that the covariant derivative of the LHS must be zero, then we must have $\nabla^{\mu} \Lambda = 0$, since every other term on the LHS already has zero covariant derivative as an identity. This then makes the covariant derivative of the RHS equal to zero, which means that the energy and momentum of ordinary matter and radiation are locally conserved (see further comments below).

However, we could also shift the $\nabla^{\mu} \Lambda$ term to the RHS and write

$$
\nabla^{\mu} G_{\mu \nu} + \Lambda \nabla^{\mu} g_{\mu \nu} = 8 \pi \left( \nabla^{\mu} T_{\mu \nu} - \frac{1}{8 \pi} g_{\mu \nu} \nabla^{\mu} \Lambda \right)
$$

Now the LHS is identically zero, so the RHS must be as well, hence the two terms inside the parentheses must cancel, but neither one needs to vanish by itself. But that would still be true if we left the $\Lambda$ term on the LHS; the difference is just algebra, not anything physical. Just shifting a term from one side of an equation to the other doesn't somehow force the term to be constant, or allow it to vary.

The physical question is whether you want to allow the covariant derivative of $T_{\mu \nu}$, exclusive of any $\Lambda$ term, to be nonzero; in other words, whether you want to allow energy and momentum to be exchanged, locally, between ordinary matter and radiation, and "dark energy" (or "scalar field", or whatever you want to call the $\Lambda$ term now that it can vary in space and time). Both possible answers, "no" and "yes", can be consistently modeled within GR. Whether both kinds of models are physically reasonable is a different question, one on which I'm not sure there is a consensus among physicists.

 

[Garth]

Hi Peter,

Well I have already covered that point in my last post.

I would agree that $\Lambda$ or DE are most likely 'fixed', as you suggest, as its the simplest solution and there is no substantial evidence to say otherwise.

However that does not prevent others from investigating an evolving DE and I was just showing how that could happen within the EFE, so long as it is DE and not $\Lambda$ that you allow to 'vary'.

In your FE with $\Lambda$ moved across (I think it needs editing as you have left it on the LHS as well) you need to include G as well as that might vary is some Brans Dicke type modification of GR and the covariant differential operator $\nabla^{\mu}$ can operate on the whole of the RHS.
$$\nabla^{\mu}G_{\mu\nu} = 8\pi\nabla^{\mu}(G(T_{\mu\nu} - \frac{1}{8\pi}g_{\mu\nu}\Lambda))$$

However as $\Lambda$ is constant by definition there is not much point in doing this. Onthe other hand if we call it DE we can give it a more general energy-momentum tensor that approximates to $E_{\mu\nu}= -(\Lambda/(8\pi) )g_{\mu\nu}$ then it may be allowed to evolve as I describe in my post above #14.

Garth

 

[PeterDonis]

I was just showing how that could happen within the EFE, so long as it is DE and not $\Lambda$ that you allow to 'vary'.

And my point is that the distinction you are drawing between "DE" and $\Lambda$, which appears to be based on which side of the EFE the term is written on, is not valid. Physically it doesn't matter which side of the equation you write the $\Lambda$ term on, or whether you call it "DE". What matters physically is whether or not energy and momentum can be exchanged between ordinary matter/radiation and "dark energy" or whatever you want to call it. If the answer is "yes", then $\Lambda$ can vary. If the answer is "no", then it can't. Which side of the equation things are written on has nothing to do with that.

In your FE with $\Lambda$ moved across (I think it needs editing as you have left it on the LHS as well)

That's not the EFE; it's the covariant derivative of the EFE. The term in $\Lambda$ that I left on the LHS is identically zero (because $\nabla^{\mu} g_{\mu \nu}$ is identically zero). I left it on the LHS to underscore that fact.

Also, once again, getting hung up on which side of the equation a term is on is pointless; the physics doesn't depend on that.

you need to include G as well as that might vary is some Brans Dicke type modification of GR

I was only considering standard GR, not modifications of it; my point was to show that both $\Lambda$ constant and $\Lambda$ varying in space and time can be modeled in standard GR.

It's worth noting, though, that when considering the possibility of $G$ varying, it's important to separate, conceptually, two different functions that $G$ serves in standard GR. First, it provides a conversion factor between geometric units (curvature) and stress-energy units (energy density). Second, it tells how much curvature is produced by a given amount of stress-energy. It's only the second function that can vary in Brans-Dicke type theories.

 

[Garth]

If the cosmological constant (LHS) and DE (RHS) are the same thing then they cannot vary - the cosmological constant is constant by the Bianchi identities, and consequently DE cannot evolve (except by it simply varying in its effect because of the decreasing densities of other contents of the universe.)

However the identical mathematical structure ($p = -\rho c^2$) belies the difference between CC and DE. On the LHS the cosmological constant is simply a modification of space-time curvature and therefore an addition to the behaviour of gravitation - that is repulsive at cosmological ranges and attractive at short ones. On the RHS the DE is seen as an extra content of the universe, a mysterious substance, and a source of gravitation.

I think we are saying the same thing, it is just that when others talk about DE evolving and its EoS departing from $\omega = -1$, I prefer to keep it on the RHS where it can exchange energy and momentum between ordinary matter/radiation, and because I like to reserve the LHS for a constant $\Lambda$.

Okay, I understand what you are doing with the $\nabla^{\mu}g_{\mu\nu}$, sorry for any confusion.

You make an interesting point about the two functions of G in the EFE - my question would be how do you measure geometric units, derived from 'stress-energy' units, apart from the amount of curvature produced by a given amount of stress energy? A similar question in the Post Newtonian approximation of the One Body Problem is whether the Robertson parameter $\alpha$ has to be unity? It does as it is a consequence of the empirical definition of mass M.

Garth

 

[PeterDonis]

If the cosmological constant (LHS) and DE (RHS) are the same thing then they cannot vary - the cosmological constant is constant by the Bianchi identities

No, this is not correct. You can have a varying $\Lambda$ term on the LHS or the RHS, it's just a matter of algebra. Whether or not the LHS or RHS of the equation has zero covariant divergence is a red herring; the physical question is whether $T_{\mu \nu}$, the stress-energy tensor exclusive of any "dark energy", has zero covariant divergence.

On the LHS the cosmological constant is simply a modification of space-time curvature and therefore an addition to the behaviour of gravitation - that is repulsive at cosmological ranges and attractive at short ones).

This is really a matter of interpretation, not physics. If $\Lambda$ is variable and you put it on the LHS, you're just saying that the behavior of gravitation is variable in space and time, instead of saying that "dark energy" is variable in space and time. You'll get the same predictions for observables either way.

when others talk about DE evolving and its EoS departing from $\omega = -1$, I prefer to keep it on the RHS where it can exchange energy and momentum between ordinary matter/radiation, and because I like to reserve the LHS for a constant $\Lambda$.

I have no problem with this as a preference; I tend to have the same preference. But you were making (or appearing to make) a much stronger (and false) claim: that $\Lambda$ can only be variable if it's put on the RHS of the equation.

 

[marcus]

I disagree, there is a difference between $\Lambda$ (on the left hand side of Einstein's Field Equation) and DE (on the right hand side of EFE) - see the discussion here.

Garth

If I understand correctly, Garth, I think I agree with you. For me, here's the important thing that distinguishes. We do not know that ALL spacetime curvature is caused by mass-energy density, pressure and associated material quantities.
To me, it seems important to acknowledge this incompleteness of our knowledge.
What we have observed and measured is a residual curvature, a property of spacetime geometry.

We cannot blithely move Lambda over to the other side, the "matter" side, because doing so involves mythology. When we do that we mythologize an imaginary energy density which "causes" the curvature in the same way we are used to having other energy density cause curvature.

It involves assuming something that we do not know to be the case, that all curvature arises from some material property. This is a superstition encouraged by past experience, but it is not science. We have to acknowledge that there may be no actual energy in "dark energy". Lambda may simply be a small constant intrinsic curvature---just the way spacetime geometry IS, before any material effects on geometry register.

Anyway, that's my point of view. It's encouraged by the fact that there is no evidence that Lambda varies, and so far the measurements of w are very close to -1, consistent with Lambda as a curvature constant. If it were to turn out after finer and finer measurements that w = -1 is excluded and that w < -1 then that would be evidence for an energy interpretation, and reason for me to change my viewpoint.

 

[PeterDonis]

We cannot blithely move Lambda over to the other side, the "matter" side, because doing so involves mythology...

It involves assuming something that we do not know to be the case, that all curvature arises from some material property.

But whether or not you are modeling the curvature as arising from "some material property" as opposed to "something intrinsic to spacetime" doesn't depend on which side of the equation you put $\Lambda$ on. It depends on the physics; the simplest indicator I can come up with is what I have said in previous posts, that what matters is whether or not energy and momentum can be exchanged, locally, between ordinary matter/radiation and dark energy, or whatever you are calling the $\Lambda$ term. The way you determine that is by looking at the covariant divergence of $T_{\mu \nu}$, the stress-energy tensor exclusive of any $\Lambda$ term. If it's zero, then energy/momentum exchange is not possible and you are treating $\Lambda$ as an "intrinsic property of spacetime", regardless of which side of the equation it is on. If it's nonzero, then energy/momentum exchange is possible and you are treating $\Lambda$ as a "material property", again, whichever side of the equation it is on.

I understand that it can be a very useful convention to put all the "spacetime" stuff on the LHS of the equation and all the "material" stuff on the RHS. But that's a convention, not physics, and breaking it, in and of itself, doesn't change the physics or introduce any new assumptions about the physics.

 

[marcus]

Until we have palpable evidence that some actual energy is involved, I think it would be better not to use the word "energy". I think it misleads people.
From a pedagogical perspective it's probably better to focus on what we actually measure:
the longterm expansion rate that H(t) appears to be tending towards.
$$H_\infty = \sqrt{\Lambda/3}$$

Here I'm using Lambda in the form 1.007 x 10^-35 second^-2
People who take Lambda as an inverse area would say
$$H_\infty = \sqrt{\Lambda c^2/3}$$

 

[PeterDonis]

Until we have palpable evidence that some actual energy is involved, I think it would be better not to use the word "energy".

Our current evidence says that there is no energy/momentum exchange between ordinary matter/radiation and $\Lambda$ (since $\Lambda$ appears to be constant everywhere), and given that, yes, the word "energy" could be seen as making a claim that the current evidence does not support.

 

[marcus]

...I understand that it can be a very useful convention to put all the "spacetime" stuff on the LHS of the equation and all the "material" stuff on the RHS. But that's a convention, not physics, and breaking it, in and of itself, doesn't change the physics or introduce any new assumptions about the physics.

Thanks, Peter. In that sense I am very conventional. I think there is an aesthetic and pedagogical aspect to the way important equations are written.
They influence how we think.

A great thing about the GR equation is that it describes the interplay of GEOMETRY on the lefthand with MATTER on the righthand. There is something amazing and beautiful about this.

 

[marcus]

Our current evidence says that there is no energy/momentum exchange between ordinary matter/radiation and $\Lambda$ (since $\Lambda$ appears to be constant everywhere), and given that, yes, the word "energy" could be seen as making a claim that the current evidence does not support.

That's a good criterion! You mentioned that earlier. It could be found that there IS exchange and then we would really have "dark energy" and not a cosmological curvature constant. I'm a slow typer (and not a quick thinker either ) so our posts have been crossing.

One problem with the "dark energy" term is something we see happening here at PF. People naturally enough associate energy with FORCE. So members come to Cosmo forum who seem to have the mental image of a force that is accelerating other galaxies faster and faster away from our galaxy. This can be a serious confusion, until someone makes the point that the "acceleration" they've heard or read about is not like the familiar acceleration of ordinary motion, accompanied by increase in kinetic energy and limited by the speed of light etc etc.

I wish it were possible to do away with the term "dark energy" (absent evidence of the sort you suggest)
and focus on the longterm residual expansion rate, and associated curvature constant. Maybe in time that will happen.

 

[Garth]

No, this is not correct. You can have a varying $\Lambda$ term on the LHS or the RHS, it's just a matter of algebra. Whether or not the LHS or RHS of the equation has zero covariant divergence is a red herring; the physical question is whether $T_{\mu \nu}$, the stress-energy tensor exclusive of any "dark energy", has zero covariant divergence.

I see this is being unnecessarily complicated, and confusing.

As traditionally $\Lambda$ has been used for Einstein's cosmological constant then it should be used only when it is constant and it belongs on the LHS where it is part of the description of space-time curvature.

If you are going to introduce DE, which may be constant or not, then, of course, whichever side of the EFE you put it on is only a matter of algebra. However if it does vary, requiring it to exchange energy/momentum with ordinary matter/radiation, it should be put on the RHS where it is not a description of curvature but of the source of curvature.

So 'it' either varies or it does not. For clarity if it does not vary call it $\Lambda$ and leave it on the LHS, if it does vary call it DE and put it on the RHS.

As at the moment there is no evidence of any such energy-momentum exchange I agree that it should be left on the LHS as a cosmological constant.

At least that's my 'pennyworth'.

Garth.

 

[wabbit]

Just stopping by to thank you all for thiis great discussion, watching it from the sidelines has helped me clarify my understanding of this question of lambda vs DE quite a lot.

 

[PeterDonis]

I see this is being unnecessarily complicated, and confusing.

I wasn't talking about how the equations should be written; I was talking about the physics that they represent. As I said in an earlier post, I agree that it's a very useful convention to put "spacetime" on the LHS and "material" on the RHS, which is basically what you're advocating (as did marcus). But once again, the convention is not the physics, and I think there is value in being very clear about which is which.

 

[Garth]

Sorry about the delay in responding, I haven't had the time.

I wasn't talking about how the equations should be written; I was talking about the physics that they represent. As I said in an earlier post, I agree that it's a very useful convention to put "spacetime" on the LHS and "material" on the RHS, which is basically what you're advocating (as did marcus). But once again, the convention is not the physics, and I think there is value in being very clear about which is which.

I agree, it is the physics they represent that is important, which is why I make the distinction between $\Lambda$ and DE, a new mysterious 'substance'.

Here's the rub.

The 'Cosmological Constant' $\Lambda$ is understood as a 'substance' which has density, in fact its density is the largest component of the universe's mass and with $\Omega_{\Lambda} = 0.69, \Omega_{DM+b} = 0.31$ (Plank 2015) it comprises over twice the density of DM and ordinary baryonic matter put together.

Giving it a density indicates that we have selected to treat it as the mysterious substance DE, and as a consequence I say it should be kept on the RHS of the EFE. One possibility is that it could be vacuum energy, although its value is of the order 10^-120 smaller than predicted by QM.

However if it is in fact a simple cosmological constant, Einstein's $\Lambda$, then there is no such density involved!

The effect of such a $\Lambda$ on space-time is to induce a hyperbolic curvature on empty space so that adding ~ $\Omega_{DM+b} = 0.31$ is enough to result in a spatially flat universe.

Garth

 

[PeterDonis]

the distinction between $\Lambda$ and DE, a new mysterious 'substance'.

This is a distinction of convention, not physics. The physics is the same whether you use the term "cosmological constant" or "dark energy"; it doesn't depend on the name. It doesn't depend on whether you conceptualize it as "energy density" or as "curvature of empty space". It doesn't depend on which units--curvature units or energy density units--you use to describe its magnitude. It doesn't depend on which side of the equation you put the term on. It only depends on whether whatever-you-call-it is constant or variable--whether it can or can't exchange energy and momentum with other stuff.

That question of physics is an empirical question to which, as best we can tell right now, the answer is "it can't". If calling it a "cosmological constant" rather than "dark energy", and using curvature units rather than energy density units to describe its magnitude, and putting it on the LHS of the equation rather than the RHS, matches up better for you with that empirical answer, that's fine; but all those other things are still just conventions, not physics.

The reason I go on about this is that so many treatments of science, even by scientists, focus on the words instead of on the science. They say, in effect, "we should call it cosmological constant, not dark energy" without ever explaining the actual physics involved--the question of whether it can or can't exchange energy and momentum with other stuff. The result is that lay people think the important thing is the words, and that if they just use the right words, they know the physics. And so we get interminable threads here on PF where people all agree on the actual physics, but never find that out because they spend hundreds of posts arguing over which words to use to describe it. (I'm not saying this is one of them, btw.)

Ok, I'll stop ranting now.

 

[Haelfix]

The cosmological constant business on the left hand side vs right hand side is quite silly to be honest. It's a complete triviality mathematically and there is no classical sense in which you can prefer one over the other, and in neither case are things any more or less mysterious with respect to the cosmological constant problem. (it is still 120 orders of magnitude too small on the left hand side if you want to think of it that way --curvature undergoes renormalization just like matter does when you promote the equations into the more fundamental quantum realm)

The only sense in which it might be important whether to group the term on the left vs the right is the ultimate prejudice about whether Einsteins equations holds through all energy scales. If for instance a modification occurs, then it is in fact important where you group things, b/c you may have a situation where physical quantities will differ.

 

[marcus]

Personally I find this physical interpretation of Lambda interesting (not one that is interchangeable with energy density however.)
The paper is only two pages so people who are interested in non-DE might want to take a look.
http://arxiv.org/abs/1105.1898

It goes back to some earlier papers by John Madore on the "fuzzy sphere". quantum uncertainty in measuring angle (as opposed to position, momentum...)

1. John Madore, “The Fuzzy sphere,” Class.Quant.Grav., 9, 69–88 (1992).
2. John Madore, “Gravity on fuzzy space-time,” (1997), dedicated to Walter Thirring on the occasion of his 70th birthday, arXiv:gr-qc/9709002 [gr-qc].
3. John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications., London Mathematical Society Lecture Note Series, Vol. 257 (Cambridge University Press, 2002).

 

[marcus]

Within a certain theoretical context all curvature can be converted back and forth with energy etc---and Lambda is simply a curvature interconvertible with an energy density, with no special physical meaning outside that context. Someone only familiar with that context may well think it is a mere formality how one treats the constant---doesn't make any difference which side, treat it as an energy density if you want, and so on.

But such a person might be interested in other research lines and open to giving them a hearing, because they are different from his or her own familiar context.
Another interpretation of the Lambda constant has to do with the compactness of the phase space of geometry. And the finiteness of the number of distinguishable states of geometry. This is not the phase space of a system particles moving in a fixed geometry but actually that of the geometry itself.
It is work that is just getting started, which gives what I think is a new physical meaning to Lambda. I'll get the link.
http://arxiv.org/abs/1502.00278
Compact phase space, cosmological constant, discrete time
Carlo Rovelli, Francesca Vidotto
(Submitted on 1 Feb 2015)
We study the quantization of geometry in the presence of a cosmological constant, using a discretization with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete, pointing to discreetness of time, in addition to space. We work in 2+1 dimensions, but these results may be relevant also for the physical 3+1 case.
6 pages

Again it is a short paper, only 6 pages. So not terribly burdensome to read.

 

[PeterDonis]

Another interpretation of the Lambda constant has to do with the compactness of the phase space of geometry.

Yes, in the context of quantum gravity, as I understand it, having a nonzero $\Lambda$ turns out to make a big difference to the phase space of the theory. AFAIK, at the level of ordinary classical GR, this would require $\Lambda$ to be constant, not variable, so it would not exchange energy or momentum with anything else (which is consistent with our best current observations).

 

[marcus]

You put it concisely! In addition to this incipient compact phase space idea (which needs to be worked out in 4d) there are a bunch of papers by various people which explore incorporating Lambda in QG by replacing SU(2) with the quantum group version SU(2)_q. where q is a version of Lambda on the unit circle of the complex plane. Some authors:
Fairbairn and Meusburger
M. Han
S. Major
I should mention that the while the Lambda constant occurring on the LHS of the Einstein GR equation is a reciprocal area (and is formally interconvertible with energy density) what these authors are finding to be physically meaningful is the square root of Lambda, the reciprocal of a length constant. So in that work it is the square root of the Lambda we are used to, which is more fundamentally significant. that's what I should have said when I was talking about the quantum group q constant being a version of Lambda.