Could an entropic force help explain size of the cosmological constant?

[marcus]

When we measure the cosmological constant Λ what we are really doing is measuring the longterm value of the Hubble rate, namely H_∞, the distance expansion rate that the present rate H(t) is seen to be declining towards and leveling out at.

It's convenient to write distance expansion rates as percentage growth per million years
H_now = 1/144 percent per My
H(t) → H_∞ = 1/173 percent per My

In c=1 units, Λ/3 = H_∞² = (1/173 percent per My)²
or if you want to put c into the formula
c²Λ/3 = H_∞²

Recent analysis shows that volume has entropy---which increases with volume. Essentially, up to a logarithmic term, entropy increases in proportion to V (or slightly faster due to the log V term).

It's my understanding that at least one QG researcher has been studying the question: can the tendency of entropy to increase help explain the inherent tendency of large-scale volumes to increase?
It would at least make sense to consider the possibility.

We are used to associating entropy with AREAS.

Here is a recent QG paper that computes the entropy associated (in the absence of other contributions) with any VOLUME:
http://arxiv.org/abs/1603.01561

 

[marcus]

Obviously the longterm baseline expansion rate for volumes is (1/173 percent)³ per My
numerically much smaller than the rate for distance.
According to the result given in http://arxiv.org/abs/1603.01561 this is would essentially correspond to a small fractional increase in the volume entropy of the given bulk region.
One would have to see if this could be associated with an "entropic force", a natural tendency for the volume entropy to increase over time.

 

[marcus]

People aren't so used to seeing and dealing with the cosmo constant Λ in Einstein's original GR form, as curvature---for instance as reciprocal area (e.g. meters^-2) or as reciprocal time squared (e.g. seconds^-2)
Either m^-2 or s^-2 are good units for Λ---the quantities only differing by a factor of c²)

As a check put this expression for Λ = 3H_∞² into google:

3((1/173) percent per million years)^2

The google calculator will give back
3 * ((((1 / 173) percent) per (million years))^2) = 1.00656 × 10^-35 s^-2

Λ just happens to be approximately a "round number" expressed in s^-2 terms, namely 10^-35 s^-2
You can check that this agrees with other values for the cosmological constant in the literature given in other terms such as "critical energy density" and fractions thereof.

Or if you want to see the curvature constant in reciprocal square meters, paste this into google
3((1/173) percent per million years)^2/c^2 in m^-2
and the calculator will give back something like 1.12... × 10^-52 m^-2

 

[Drakkith]

Here is a recent QG paper that computes the entropy associated (in the absence of other contributions) with any VOLUME:
http://arxiv.org/abs/1603.01561

Interesting. I'll have to give this a read when I get the chance.

 

[marcus]

Glad to know you find it interesting. Whether or not Volume Entropy contributes to cosmo constant (that highly speculative for now) there is a good chance that it helps to explain the EARLY UNIVERSE LOW ENTROPY --- which is otherwise something of a puzzle given that matter was in high entropy state of nearly uniform hot gas.

Here's an excerpt from that paper;
==quote from "conclusions" section==
For the second principle of thermodynamics to hold, the initial state of the universe must have had low entropy. On the other hand, from cosmic background radiation observations, the initial state of matter must have been close to having maximal entropy. Penrose addresses this discrepancy by taking into consideration the entropy associated to gravitational degrees of freedom. His hypothesis is that the degrees of freedom which have been activated to bring the increase in entropy from the initial state are the ones associated to the Weyl curvature tensor, which in his hypothesis was null in the initial state of the universe. A definition of the bulk entropy of space, which, as would be expected, grows with the volume, could perhaps perform the same role as the Weyl curvature degrees of freedom do in Penrose’s hypothesis: the universe had a much smaller volume close to its initial state, so the total available entropy was low - regardless of the matter entropy content - and has increased since, just because for a space of larger volume we have a greater number of states describing its geometry.

We close with a very speculative remark. Does the fact that entropy is large for larger volumes imply the existence of an entropic force driving to larger volumes? That is, could there be a statistical bias for transitions to geometries of greater volume? ...
...
==endquote==

 

[osotou]

I'm admittedly a lurker and a layperson. So much of this is way above my head. I did skim the paper and this stood out to me at the conclusion:

We close with a very speculative remark. Does the
fact that entropy is large for larger volumes im-
ply the existence of an entropic force driving to
larger volumes? That is, could there be a statistical
bias for transitions to geometries of greater volume?

What happens if we think of time as a physical force? Decouple time and space, give them each a separate frame, what happens? Space retains the same density always and forever until something happens(a force, time) to the system that causes, what --entropy. Time collects entropy from an infinitely dense space, and that additional entropy drags more and more space along with it.

 

[Drakkith]

What happens if we think of time as a physical force? Decouple time and space, give them each a separate frame, what happens? Space retains the same density always and forever until something happens(a force, time) to the system that causes, what --entropy. Time collects entropy from an infinitely dense space, and that additional entropy drags more and more space along with it.

Since this is a violation of most of physics there is no answer to your question.

 

[marcus]

To be fair, the term "entropic force" is potentially confusing, especially as applied to gravity the way Erik Verlinde did around 2010 causing a brief fad.
The Wikipedia article on entropic force is in many ways very good IMHO. It talks about surface tension and the elastic tension in a polymer chain or stretched rubber band. Worth reading.
https://en.wikipedia.org/wiki/Entropic_force

Without explicitly doing so, the article refers to Ted Jacobson's great seminal article of 1995 on the thermodynamic origin of gravity. But it doesn't give a link and it doesn't explain that Jacobson was talking about gravity as changing geometry interacting with matter. He was not treating gravity as force. It's only a force in the Newtonian picture.

Also strictly speaking the cosmological constant Λ is not a force in any conventional sense. There is no work done as distance growth speeds up---no mysterious "dark energy" with its foot on the accelerator.
It's a feature of the spacetime geometry---that distances between otherwise stationary things increase at a certain small percentage rate of their own accord. Interesting to try to understand this and to explain the size of this baseline percentage increase.

It's understandable that layfolk would use the term "force" loosely and describe stuff in terms of force when that's not strictly appropriate because pros do that too to some extent. But they at least should be able to say what the mathematics is that corresponds to the words they are using---that's the key difference.

 

[marcus]

Let's look again at that excerpt from the Volume Entropy paper
==quote from "conclusions" section http://arxiv.org/abs/1603.01561 ==
...His hypothesis is that the degrees of freedom which have been activated to bring the increase in entropy from the initial state are the ones associated to the Weyl curvature tensor, which in his hypothesis was null ...
A definition of the bulk entropy of space, ..., could perhaps perform the same role as the Weyl curvature degrees of freedom do in Penrose’s hypothesis: the universe had a much smaller volume close to its initial state, so the total available entropy was low - regardless of the matter entropy content - and has increased since, just because for a space of larger volume we have a greater number of states describing its geometry...
==endquote==

Note that in a bounce cosmology it is no puzzle--no failure of the 2nd Law--that the entropy appears to be be RESET at the bounce. Entropy is observer dependent and the pre-bounce set of observers differ from the post-bounce. It is not continuously defined.

The puzzle is that from the standpoint of post-bounce observers the entropy must be LOW in spite of the fact that (as can be seen directly from the CMB map) the matter component of the entropy is HIGH. Nearly uniform temperature and density hot gas.

So there must be some other component of the entropy which in the post-bounce early universe outweighed the matter component, and made the total low compared with what it would be later---IOW able to increase as required by 2nd Law.

I like the way the Volume Entropy authors put it: "degrees of freedom which have been activated to bring the increase in entropy". Degrees of freedom which do not even exist for early post-bounce observers but which burgeon into existence as the universe expands.

But this raises a question for us: what is happening to the volume entropy in the pre-bounce contracting phase?

 

[Drakkith]

Note that in a bounce cosmology it is no puzzle--no failure of the 2nd Law--that the entropy appears to be be RESET at the bounce. Entropy is observer dependent and the pre-bounce set of observers differ from the post-bounce. It is not continuously defined.

Can you elaborate on this, Marcus? Entropy is observer dependent?

 

[marcus]

Robert Wald (GR expert) has an interesting paper emphasizing this in the GR context. Thanu Padmanabhan has also emphasized this dependency on the observer.
The simple explanation is that entropy depends on the macro state-microstate map. How many micro states (indistinguishable to the observer) are included in the particular observer's macro state. Macro and micro have no inherent meaning apart from some class of observers.

I don't have ready references for this, D. Hopefully someone else can provide some, or more elaboration on the topic.

 

[marcus]

Let's look again at that excerpt from the Volume Entropy paper
==quote from "conclusions" section http://arxiv.org/abs/1603.01561 ==
...His hypothesis is that the degrees of freedom which have been activated to bring the increase in entropy from the initial state are the ones associated to the Weyl curvature tensor, which in his hypothesis was null ...
A definition of the bulk entropy of space, ..., could perhaps perform the same role as the Weyl curvature degrees of freedom do in Penrose’s hypothesis: the universe had a much smaller volume close to its initial state, so the total available entropy was low - regardless of the matter entropy content - and has increased since, just because for a space of larger volume we have a greater number of states describing its geometry...
==endquote==
...
The puzzle is that from the standpoint of post-bounce observers the entropy must be LOW in spite of the fact that (as can be seen directly from the CMB map) the matter component of the entropy is HIGH. Nearly uniform temperature and density hot gas.

So there must be some other component of the entropy which in the post-bounce early universe outweighed the matter component, and made the total low compared with what it would be later---IOW able to increase as required by 2nd Law.

I like the way the Volume Entropy authors put it: "degrees of freedom which have been activated to bring the increase in entropy". Degrees of freedom which do not even exist for early post-bounce observers but which burgeon into existence as the universe expands.

But this raises a question for us: what is happening to the volume entropy in the pre-bounce contracting phase?

I think the answer to that one is that although V-entropy is decreasing in the pre-bounce contraction (as the sizes of bulk volumes decline), this is OFFSET by the rapid increase in MATTER entropy, which goes maximal for pre-bounce observers as they see all the matter falling into BLACK HOLES. The matter degrees of freedom become inaccessible to them, behind the growing area of horizons. So the TOTAL entropy is still increasing for pre-bounce observers, in agreement with the 2nd Law.

As I see it, this could in turn bring up other questions. The Volume Entropy paper is prolific in raising questions : - )

 

[eloheim]

I have a question about the early universe 'matter entropy.' Wasn't the matter in the early universe in a low entropy state because it was spread out so perfectly evenly? I get that when you normally think of stuff that is all spread out it is high entropy because it's generic, but in the early universe the matter has to be exactly spread out so no tiny density variation causes everything to start collapsing into black holes. Maybe I'm missing a consideration of 'maximum entropy' somewhere?

 

[marcus]

Its good to be asking questions like that. Let's be specific and say "early universe" means year 380,000 when CMB originated. We have a "snapshot" of matter at that time. It is nearly evenly spread out---to within a factor of 10^-5, we are told. Isn't that HIGH matter entropy as we normally think of it?---yes, and you say you GET that. But there is another aspect, the gravitational field associated with the matter distribution, that we normally do not include when we consider the matter entropy. You point out that a very uniform gravitational field is somehow LOW entropy because it is subject to structure formation--a tiny over density can trigger coagulation. Maybe if there is confusion it is partly just semantic. I am not considering that as part of the matter entropy--I'm thinking of that as a feature of the gravitational field.

I think that is what Penrose does, with the Weyl curvature hypothesis. The distribution of matter certainly goes into making the Weyl curvature low, or null---but it's a separate descriptor from the matter entropy.
Langauge a bit confused here but we can probably just put up with it. Hope you agree.

 

[eloheim]

Thanks I see what you mean. I should look into the bounce cosmology stuff more I think too.

 

[osotou]

.

It's understandable that layfolk would use the term "force" loosely and describe stuff in terms of force when that's not strictly appropriate because pros do that too to some extent. But they at least should be able to say what the mathematics is that corresponds to the words they are using---that's the key difference.

Curious onlookers and layfolk such as myself often approach these subjects without even a clear idea of Newtonian physics. I personally realize now that I have glossed over that stuff because it seems boring -- the deeper subject matter is mysterious and wonderful! The really great thing about making such even basic mistakes in thinking is that I can step back and think more deeply about what I'm not understanding! (useful on many levels, including avoiding the tendency toward dogma) I need to get my horse back in front of the cart. A good place to start is probably the Lineman Davis article on Misconceptions about The Big Bang that you link in your signature, which I am reading now!

 

[Berenices]

The final equation before the discussion reads cV ≤ S ≤ c ′VlogV. The paper only specifies the c and c' as constants, do the c's describe a non-random variable, or are they fine-tuned, or are they completely randomisable. Apologies if I misunderstood.

 

[marcus]

that's equation (27) on page 3 . I think it means just what it says:
==quote==
...
...so that the entropy satisfies
cV ≤ S ≤ c′V logV.
with c and c′ constants.
==endquote==

c and c' are just constants---not variable, not random. It's a preliminary paper, the constants haven't been determined yet.
at least one of them would depend on the value of the maximal spacing of the Volume eigenvalues which I don't think have been computed yet. If they do a followup paper they may get closer to some definite values.

 

[windy miller]

So would I be correct to say that basically in this scenario the entropy at the big bang is small because the volume of the ( observable) universe is small? Is it really that simple? Or have I misunderstood what you are saying?