The Cosmological Constant

1. Aug 8, 2011

hylander4

So I was watching the show "Curiosity" on the Discovery channel last night. In the show, Stephen Hawking describes the birth of the Universe, and why our theory of inflation leaves no place for God.

I'm not going to comment on the God thing, but Hawking said something that got me thinking. Basically, he said that the entire Universe has zero energy because space is essentially equivalent to negative energy.

This gave me an idea--if virtual particles are constantly being created and annihilated everywhere, then positive energy is always being created (even if the two particles annihilate, the annihilation has to create some energy, right?). So if energy is being created from vacuum at a constant rate, then negative energy (space) must be created to balance out that constant rate of energy creation. This would cause a *constant* rate of expansion for the universe, possibly similar to the one we've been measuring in the past couple decades.

Does this make any sense at all? Is this what cosmologists mean when they say that vacuum energy is responsible for the expansion of the universe? I'm an undergraduate physics major, but I still haven't taken GR or Quantum Field Theory, so I could be way off.

2. Aug 8, 2011

Chronos

The annihilation of virtual particles does not actually create energy, they borrowed it from the vacuum to bootstrap themselves into existence, so they are just returning it. Most cosmologists consider gravity as the 'negative' energy that counteracts the positive energy of matter in arriving at the zero total energy of the universe. The dark energy responsible for expansion is not well understood. The vacuum energy idea is not so popular anymore since they calculated it should be absurdly high. See: http://math.ucr.edu/home/baez/vacuum.html

3. Aug 9, 2011

Chalnoth

Huh? The cosmological constant (which is equivalent to vacuum energy) is far and away the prevailing explanation for the accelerated expansion. The value of it is weird, which still makes some theorists squirm, but it is the prevailing explanation.

And it's not so much that dark energy isn't well-understood, but rather that the experimental evidence is so far pretty sketchy: we know it's accelerating, but it is difficult to distinguish between possible models.

4. Aug 9, 2011

5. Aug 9, 2011

Chalnoth

Yes, but it is the simplest, and the most strongly theoretically-motivated (which isn't saying much). In fact, there is such an utter lack of compelling alternative models that when trying to find a deviation from the cosmological constant, the order of the day is to use parameterizations which have little or no bearing on the physics (e.g. $w_0, w_a$).

6. Aug 9, 2011

hylander4

If this vacuum energy is fueling the expansion of the universe, what mechanism is it using to do this? Or do people think that space naturally provides a negative pressure on itself?

7. Aug 9, 2011

bapowell

Slip a cosmological constant into the Einstein field equations. You'll find that a homogeneous and isotropic universe accelerates.

8. Aug 10, 2011

hylander4

Although I'm not very familiar with Einstein's equations, I feel like this line of reasoning does absolutely nothing to explain a physical process which converts vacuum energy (predicted by QFT) to more space/an expanding metric/etc.

Are there theories about this? If there aren't, then I can't really see any strong theoretical link between the vacuum energy as defined by virtual particles, and the vacuum energy that is causing space to expand.

9. Aug 10, 2011

Chalnoth

There's nothing that "converts" it. This is simply how gravity reacts to vacuum energy. This can, perhaps, most easily be seen by looking at the Freidmann equations (constants omitted for clarity):

$$H^2 = \rho$$

Here $H$ is the Hubble parameter, $H = (1/a) da/dt$, and $\rho$ is the energy density of the universe (note that the Friedmann equations assume the universe is smooth).

If you have a cosmological constant, and only a cosmological constant, then $\rho$ is a constant. So you have a simple differential equation:

$${1 \over a}{da \over dt} = \sqrt{\rho} = H_0$$

If you know your simple differential equations, this is the equation for exponential growth: the rate of change is proportional to the value, which is similar to compounded interest or population growth without resource constraints.

This means that the solution is:

$$a(t) = a(t=0) e^{H_0 t}$$

...which is exponentially-accelerated growth.

10. Aug 10, 2011

bapowell

It can also be shown that stress-energy with the quantum numbers of the vacuum has negative pressure and constant energy density. So, the quantum vacuum does indeed have the stress-energy of a cosmological constant. Chalnoth has just shown above how the cosmological constant in turn leads to accelerated expansion. Does this address your concerns hylander4?

11. Aug 19, 2011

drollere

dark energy is a hypothesis hung on a number that has no physical explanation, except that the number is necessary to make cosmological expansion fit cosmological observations as cosmological theory is currently formulated.

recall that einstein first proposed the constant, because it was necessary to make a calculation come out correctly, then regretted it, because he felt it would show he fudged the work. he was right both ways.

put differently: dark energy (and for that matter, dark matter) have exactly the same scientific status in the 21st century that the ether had in the 19th century. a putative invisible cause to explain puzzling and challenging observations.

without putting on my tinfoil hat, i feel that science would do well to remember that once you put a name on something, the name takes on a life of its own and grows its own consequences.

at the limits of our knowledge, we are always stuck in the quandary that we lack an insight to explain what we know, but that we need a name or term with which to talk about it and under which to collect the important facts. the name seems to create an insight where none at present exists.

12. Aug 19, 2011

bapowell

Adding a constant to the field equations is not a fudge per se. The guiding principle of general relativity is general covariance, and the only theoretical constraints one has when building the gravitational lagrangian is to adhere to this symmetry. Theoretically speaking, the honest and correct thing to do is to include all terms that are consistent with this symmetry in the equations of motion. Then, observations will enable you to constrain the likely values of these terms. So, adding a CC a priori is not a fudge -- its inclusion is mandated by the symmetry of the theory.

Strongly disagree with regards to dark matter. Don't get fooled by the word 'dark' here: just because both dark energy and dark matter contain the word 'dark', does not mean they are equally 'dark' in terms of our understanding. There's mounting evidence from various cosmological sectors (CMB, large scale structure, lensing) that support the hypothesis of particulate dark matter. The postulation of the ether was largely driven by aesthetics -- it was by no means a requirement of electromagnetic theory that waves needed a medium in which to travel. The postulation of dark matter was largely driven by observations that could not be explained with current theory. This is, by and large, how science is done. And it's not like the fact that its "invisible" is somehow a cop out or a weakness of the explanation -- the darn stuff literally is invisible!

I don't quite see the relevance of this statement to the discussion. True, the CC as calculated from QFT is off by a hundred orders of magnitude. But, we have good reason to believe that vacuum energy manifests itself gravitationally, and we have good reason to believe that QFT is not UV complete. We know that vacuum energy causes just the kind of accelerated expansion observed in supernova surveys. This makes vacuum energy a blatantly obvious candidate hypothesis if you ask me. Nobody should be claiming we understand it -- in fact, understanding the nature of dark energy -- and the gravitational behavior of the quantum vacuum in general -- should be one of the top 5 things on any serious cosmologists to-do list.

13. Aug 19, 2011

marcus

Good points. Putting Lambda into the equation was playing according to the rules, not a fudge. It should be there by rights. And then you try to determine is it zero? is it negative? Well it happens to turn out positive.

You also make the excellent point that we shouldn't necessarily call it an "energy" and think of it as arising from QFT. It is a constant (with dimensions of inverse area or curvature) which appears naturally in the Einst. field eqn.

Have to go, back later.

Yes, good point about dark matter. Vast amount of evidence. People make contour maps of density of various clouds. One can in effect see it by its lensing effect on background galaxies. I don't think there is any question any more. Predicting it was a triumph.

"Dark energy" is somewhat of a misnomer, like "big bang". Catchy but potentially misleading. Better, for now, to simply say cosmological constant. And what is all this fuss about a simple constant in an equation? Why all the drama? Until and if we learn different it is just a constant which belongs naturally in the 1915 equation of gravity.

Bianchi and Rovelli have a paper making that point.
http://arxiv.org/abs/1002.3966
Why all these prejudices against a constant?
Eugenio Bianchi, Carlo Rovelli
(Submitted on 21 Feb 2010)
The expansion of the observed universe appears to be accelerating. A simple explanation of this phenomenon is provided by the non-vanishing of the cosmological constant in the Einstein equations. Arguments are commonly presented to the effect that this simple explanation is not viable or not sufficient, and therefore we are facing the "great mystery" of the "nature of a dark energy". We argue that these arguments are unconvincing, or ill-founded.
9 pages, 4 figures

Last edited: Aug 19, 2011
14. Sep 5, 2011

keepit

could the universe be self flattening in the way entropy moves or some other such analogy?

15. Sep 6, 2011

Chalnoth

Well, that's sort of the idea behind inflation. Essentially, the effect of the overall curvature of the universe scales as $1/a^2$. Any time you have a universe which is dominated by stuff that dilutes more slowly than $1/a^2$, it gets flatter and flatter with time. This is the case at the present time, for example, with a universe dominated by something that is at least very much like a cosmological constant (that doesn't dilute at all with the expansion).

However, for much of the history of our universe, it has been dominated by matter, which scales in energy density as $1/a^3$, and before that by radiation, which scales in energy density as $1/a^4$. During this period of time, the effect of the curvature increased with time, so much so that for the curvature to be less than 1% right now, back when the CMB was emitted, the curvature had to be 0.003%. Back when Big Bang Nucleosynthesis started at (roughly) $z=10^8$, if my calculations are accurate, the curvature would have been a mere one part in $10^{13}$.

So what you need is some mechanism to get started before we have any matter that produces extreme flatness. Inflation does this, for one.