Where Does the 10^-35 s^-2 Cosmological Constant Value Come From?

[Thomas Moore]

Hi.

I was just curious about the current value of the cosmological constant. My astrophysics class lecture notes say on the order of 10^-122, but the Wikipedia article says 10^-35 s^-2.

Could someone explain where the 10^-35 s^-2 value comes from?

Thanks!

 

[mathman]

From:

http://en.wikipedia.org/wiki/Cosmological_constantObservations announced in 1998 of distance–redshift relation for Type Ia supernovae[7][8] indicated that the expansion of the universe is accelerating. When combined with measurements of the cosmic microwave background radiation these implied a value of

,[9] a result which has been supported and refined by more recent measurements. There are other possible causes of an accelerating universe, such as quintessence, but the cosmological constant is in most respects the simplest solution. Thus, the current standard model of cosmology, the Lambda-CDM model, includes the cosmological constant, which is measured to be on the order of 10−52 m−2, in metric units. Multiplied by other constants that appear in the equations, it is often expressed as 10−52 m−2, 10−35 s−2, 10−47 GeV4, 10−29 g/cm3.[10] In terms of Planck units, and as a natural dimensionless value, the cosmological constant, λ, is on the order of 10−122.[11]

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Hope this helps.

 

[Thomas Moore]

Hi. Yes, I saw this Wikipedia article. I was curious where the 10^-35 s^-2 comes from. Thanks.

 

[Chalnoth]

Hi. Yes, I saw this Wikipedia article. I was curious where the 10^-35 s^-2 comes from. Thanks.

That's the square of the rate of expansion that would occur if we only had a cosmological constant and no matter or radiation.

 

[yogi]

That's the square of the rate of expansion that would occur if we only had a cosmological constant and no matter or radiation.

To balance gravity, Lambda needs to be on the order of 3H^2 which is about 10^35 sec^-2 as pointed out in the quote. There is, however, another very interesting solution to Einstein's equation where density is real and G is balanced by Lambda as intended by Einstein. Specifically, when negative pressure equals [rho(c)^2]/3. In such a case the universe has density rho = (3/R) kgm/m^2, (compatible with the real universe), expansion is exponential, and negative pressure (rather than density) remains constant during expansion.

 

[PeterDonis]

To balance gravity, Lambda needs to be on the order of 3H^2 which is about 10^35 sec^-2 as pointed out in the quote.

No, that's not what the quote says. The quote is talking about the expansion rate in a universe that is empty except for a cosmological constant; there's no "balancing of gravity" because there's nothing else to balance.

There is, however, another very interesting solution to Einstein's equation where density is real and G is balanced by Lambda as intended by Einstein. Specifically, when negative pressure equals [rho(c)^2]/3. In such a case the universe has density rho = (3/R) kgm/m^2, (compatible with the real universe), expansion is exponential, and negative pressure (rather than density) remains constant during expansion.

Do you have a reference for this? It doesn't look like any solution I'm aware of.

 

[Chronos]

I concur, Peter.

 

[yogi]

Response to post 6 and 7:

Einstein introduced Lambda to prevent gravitational collapse - he didn't specify what lambda was or how it had to work, but what he was looking for was a value that balanced gravity. Total force = - 4(pi)G(rho)/3 + Lambda(R)/3, therefore lambda = 4(pi)G(rho) and net force is zero. "This was just what Einstein wanted to achieve a static universe" The quote is From Harrison (in my 2003 edition its on page 331).

Secondly, there is more than one solution of Einstein's equation that leads to exponential expansion. Three are listed below, there may be more.

1)The empty universe,
2) negative pressure = (rho(c^2) ...this is McCrea's scenario for the production of matter from negatively expanding space, first seized upon by Hoyle and company to support steady state theory - and later by Guth as the inflationary epoch.(the same math leads to constant density during expansion of negative pressure).
3) negative pressure = rho(c^2)/3 -- When negative pressure equals positive matter - the solution is the same as the empty universe

 

[Chalnoth]

Response to post 6 and 7:

Einstein introduced Lambda to prevent gravitational collapse - he didn't specify what lambda was or how it had to work, but what he was looking for was a value that balanced gravity. Total force = - 4(pi)G(rho)/3 + Lambda(R)/3, therefore lambda = 4(pi)G(rho) and net force is zero. "This was just what Einstein wanted to achieve a static universe" The quote is From Harrison (in my 2003 edition its on page 331).

This solution is unstable, which is why Einstein abandoned it. If some matter moves a bit from one region to another, then the region that the matter moved from will become less dense, leading to runaway expansion, while the region the matter moved to will collapse in on itself.

It's also not quite accurate to say that it was a "force to balance gravity", as the cosmological constant is part of gravity.

One way to understand this is to look at how the Einstein Field Equations are derived. One of the more intuitive ways of doing this is to make use of the action principle (see more here). It's possible to prove that the action has to be a scalar function of the Ricci curvature tensor. You can write a general function of the Ricci curvature tensor with a simple expansion:

f(R) = a_0 + a_1 R + a_2 R^2 + a_3 R^3 + ...

Here $a_0$, $a_1$, etc. are constants. The simplest possible action that is actually dependent upon R is to only take the first two terms:

f(R) = a_0 + a_1 R

With some appropriate choices for the constants $a_0$ and $a_1$ in terms of $\Lambda$ and $G$, this action gives General Relativity. You can create more complicated theories of gravity by adding additional terms. But the cosmological constant is a fundamental component that cannot be removed from the theory. It was long thought that there must be some symmetry that sets this value to zero, but no such symmetry has yet been found.

Secondly, there is more than one solution of Einstein's equation that leads to exponential expansion. Three are listed below, there may be more.

1)The empty universe,
2) negative pressure = (rho(c^2) ...this is McCrea's scenario for the production of matter from negatively expanding space, first seized upon by Hoyle and company to support steady state theory - and later by Guth as the inflationary epoch.(the same math leads to constant density during expansion of negative pressure).
3) negative pressure = rho(c^2)/3 -- When negative pressure equals positive matter - the solution is the same as the empty universe

(1) and (3) are not exponential expansion. In the case of (1), the expansion is meaningless (you can change your variables to get any expansion you want). (3) has linear expansion.

 

[yogi]

Einstein introduced Lambda as an ad hoc constant to avoid gravitational collapse - that was the 1916-17 edition. Critics pointed out that the balanced universe was unstable - but Einstein did not drop the CC until years later when the evidence begin to accumulate that the universe was expanding. His famous "away with it" was not because of potential cosmic instability, it was because a balancing factor was believed by Einstein to be no longer necessary. In the usual sense common associated with balancing, a balanced universe would be unstable. But there are other interpretations of lambda that are not unstable, but I can't raise those in this thread.

There is no difference in the expansion profile of an empty universe and a universe where negative energy is precisely balanced by positive matter energy - these are just two different ways to get to zero. The solution is exponential expansion in both cases. In the case of the empty universe, Howard Robertson showed early-on that sprinkling bits of matter throughout an empty universe would reveal the expansion as exponential (something more than an academic curiosity as the de Sitter universe was first viewed). The interesting aspect of the net zero energy universe, is that it emulates the real universe will all its ordinary matter and accompanying negative gravitational energy

Yogi

 

[Chalnoth]

Einstein introduced Lambda as an ad hoc constant to avoid gravitational collapse - that was the 1916-17 edition. Critics pointed out that the balanced universe was unstable - but Einstein did not drop the CC until years later when the evidence begin to accumulate that the universe was expanding. His famous "away with it" was not because of potential cosmic instability, it was because a balancing factor was believed by Einstein to be no longer necessary. In the usual sense common associated with balancing, a balanced universe would be unstable. But there are other interpretations of lambda that are not unstable, but I can't raise those in this thread.

It really is irrelevant what Einstein's motivation was. The cosmological constant is a necessary component of the theory. What Einstein did or thought is an interesting part of history to some, but it has precisely zero bearing on the status of the cosmological constant in the theory.

There is no difference in the expansion profile of an empty universe and a universe where negative energy is precisely balanced by positive matter energy - these are just two different ways to get to zero. The solution is exponential expansion in both cases. In the case of the empty universe, Howard Robertson showed early-on that sprinkling bits of matter throughout an empty universe would reveal the expansion as exponential (something more than an academic curiosity as the de Sitter universe was first viewed). The interesting aspect of the net zero energy universe, is that it emulates the real universe will all its ordinary matter and accompanying negative gravitational energy.

There is a difference, however: in an empty universe, you can change the variables to have a faster expansion or a slower expansion or no expansion or even collapse instead of expansion and no observables will change. With the balanced universe, well, it's unstable so it won't stay balanced for long.

Now, if you had a universe dominated by matter with $w = -1/3$, then that would be a different story: you could have linear expansion in that case.

 

[yogi]

It really is irrelevant what Einstein's motivation was. The cosmological constant is a necessary component of the theory. What Einstein did or thought is an interesting part of history to some, but it has precisely zero bearing on the status of the cosmological constant in the theory.

I know you are a professional cosmologist, but I will still take issue with your statements. The OP's question was "where does the value of 10^-35 sec^-2 come from, and that answer comes from History, it had to have a value that counteracted the obvious potential for gravitational collapse. Prior to the ad hoc introduction if Lambda, it was not a part of the theory, and for many years it simply hung around popping up off-and-on as theories and knowledge changed. After 1998, it was re-invented as part of the LambdaCDM model, but if I remember correctly, that line of thinking only evolved after the 1a SN studies were interpreted as exponential expansion.

As an aside, it is an interesting curiosity that de Sitter argued early on with Einstien, that the universe could be better explained if it were expanding. Einstein had several opportunities to rethink GR and incorporate expansion, but never did [in particular his communications with Friedmann should have prompted a reconsideration of his static theory]. Nor did he embrace it after Robertson deduced the velocity-distance law after Hubble published his red shift data ... IMO, Einstein was chagrined by the lost opportunity to predict expansion, and instead of interpreting his constant as an expansion field of some sort, he discarded it. To the extent it is now part of Standard Theory should not be credited to Einstein, or to GR. Several prominent writers have expressed similar opinions, specifically, that his greatest mistake was not that he introduced Lambda, but rather, that he discarded it. The irony in the end is that lambda is consistent with q = -1 de Sitter expansion, or at least that appears to be where its headed.

 

[marcus]

Hi.

I was just curious about the current value of the cosmological constant. My astrophysics class lecture notes say on the order of 10^-122, but the Wikipedia article says 10^-35 s^-2.

Could someone explain where the 10^-35 s^-2 value comes from?

Thanks!

In a practical sense it comes from redshift-distance data the measures the asymptotic (longterm) expansion rate H_∞

H_∞² = Λ/3

To say that Lambda is about 10^-35 s^-2 = 10 x 10^-36 s^-2
is the same as saying that the longterm distance expansion rate is about (10/3)^1/2x 10^-18 s^-1
which happens to be roughly true. It is not a bad approximation.

That square root is about 1.826. Let's compare and see how close to the real estimate it is.

I think of the longterm expansion rate that H is settling down to as 1/17.3 ppb per year. currently expansion rate is about 20% larger but it is gradually declining and seems to be on track to level off at that 1/17.3.

If I type "1/17.3 per year in Hz" into google it gives me:
(1 / 17.3) per year = 1.8317205 × 10^-9 hertz
In other words 1/17.3 ppb per year is the same as 1.8317 × 10^-18 per second.
That is remarkably close to what your rough order of magnitude handle on Lambda implied which was 1.826...

 

[Chalnoth]

I know you are a professional cosmologist, but I will still take issue with your statements. The OP's question was "where does the value of 10^-35 sec^-2 come from, and that answer comes from History, it had to have a value that counteracted the obvious potential for gravitational collapse. Prior to the ad hoc introduction if Lambda, it was not a part of the theory, and for many years it simply hung around popping up off-and-on as theories and knowledge changed. After 1998, it was re-invented as part of the LambdaCDM model, but if I remember correctly, that line of thinking only evolved after the 1a SN studies were interpreted as exponential expansion.

I guarantee you that if you look at GR textbooks prior to 1998, nearly all of them will include the cosmological constant.

For example, in 1987 Stephen Weinberg argued on anthropic grounds that the cosmological constant should be within about an order of magnitude of the current matter density:
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.2607

It had long been thought that the cosmological constant should be identically zero, because it would have had to be so tiny to remain unobserved. But this doesn't mean that it was absent from the theory, or considered a non-issue.