Monday, September 19, 2005

Dark energy, quintessence

We mentioned dark energy and the cosmological constant just a few days ago in connection with "zero point energy". The distinction is that "dark energy" refers to an undefined energy of some sort that must exist in order for the universe to be geometrically "flat" in accordance with the equations of general relativity. In fact, there are now several lines of evidence that this dark energy must exist, even though its nature and origins are quite unknown.

The cosmological constant (denoted by the greek letter Λ), on the other hand, is a very specific candidate for dark energy. While it would do the job admirably, there is no independent evidence for its existence. And it has conceptual problems, namely that if (as generally supposed) the cosmological constant can be explained as zero point energy, then its apparent density is a factor of about 10120 smaller than straightforward calculations suggest it ought to be. In other words, Λ must be nonzero but very finely tuned to a very implausibly small number. Situations like that make physicists very nervous.

However, there are many other forms that the dark energy could take, which are generally referred to as "quintessence". The idea of quintessence was proposed in 1998 by R. R. Caldwell, R. Dave, and P. J. Steinhardt. See this overview article by Caldwell and Steinhardt for more details.

This article: Dark Energy by Caldwell is an excellent recent (2004) summary of our present knowledge of dark energy, including the evidence for its existence. The evidence is:

  1. Many measurements of the apparent luminosity of distant "standard candle" Type Ia supernovae show that the expansion of the universe is now accelerating, instead of decelerating as would be the case if there were no dark energy.

  2. The existence of dark energy predicts a phenomenon known as the integrated Sachs-Wolfe effect, which represents a slowing of the collapse of overdense regions of the universe. This prediction has been confirmed by combining data from detailed measurements of the cosmic microwave background (CMB) and the large-scale distribution of galaxies. (The article by Caldwell has a good explanation.)

  3. There is weaker circumstantial evidence from CMB and galaxy distribution data.

Given that we can now be fairly confident dark energy exists, the big question is: What is it? In particular, can we tell whether it is a result of a cosmological constant or, instead, of quintessence?

One characteristic of the cosmological constant Λ is that it is truly a constant. It is the same everywhere and for all time. Quintesence, on the other hand, gives an energy density which varies spatially (i. e., isn't homogeneous) and with time (it decreases). These differences should make it possible to distinguish the alternatives, with very sophisticated measurements of the acceleration of the universe at different time periods. The instruments and space missions that could make the measurements are under design and development.

Quintessence itself can come in many possible types, and a recent technical paper, The Limits of Quintessence, by R. R. Caldwell and Eric V. Linder distinguishes two subtypes of quintessence in some detail, and both of those from Λ. If dark energy is actually quintessence, the measurements which are being developed should be able to distinguish between the subtypes. A less technical discussion of quintessence and ways to test for it appears in this news article: Finding A Way To Test For Dark Energy.

But if you're willing to tolerate a few equations (and just a pinch of calculus), we can show the essence of the difference. Your reward for following along here is that you will be able to understand the technical articles just mentioned a little better.

The important thing is a number that's conventionally written as "w". w is simply a constant of proportionality between pressure and energy density. For any given type of matter or energy the relation is this: P = wε, where P is pressure and ε is energy (or mass) density. This equation is from the theory of gases and is known as the "equation of state".

One other equation we need is called the "fluid equation". It describes how energy density, pressure, and a third quantity called the "scale factor", denoted by "a", are related in an expanding (or contracting) universe. The scale factor can be thought of as a variable yardstick that expands or contracts in proportion as the universe does. (For much more about the scale factor and equations involving it, see this.) Here is the equation:
ε&prime + 3(&epsilon+P)a′/a = 0
The prime symbol (′) in there denotes derivative with respect to time. The derivative is zero just in case the quantity is a constant. So ε′ = 0 just in case we have the equation of state &epsilon=-P, which means w=-1.

Suppose the cosmological constant Λ is the dominant form of dark energy. Since Λ is a constant, the corresponding energy density ε is constant, so ε′=0 and w=-1. In other words, dark energy being entirely the result of a cosmological constant corresponds to the parameter w=-1.

But there's no a priori reason that w couldn't be just about any varying function of time. What would it be if the dark energy were solely the result of quintessence? To answer that we need one more equation, called the "acceleration equation":
a′′/a = -(4πG/3c2)(ε+3P)
The double prime denotes the second derivative, which is interpreted as acceleration. π is the constant 3.14159..., G is Newton's gravitational constant, c is the speed of light, and a, ε, and P are as before.

What this equation says is that the acceleration of the expansion of the universe is a negative number times &epsilon+3P. Since we now know observationally that the acceleration is positive, we must have &epsilon+3P<0. And since ε=wP by definition, we must have ε<-3wε, hence w<-1/3. To be consistent with observations, we must have w<-1/3 if ε is the energy density corresponding to quintessence. (This also assumes Λ=0. If dark energy consists of both quintessence and a cosmological constant, which isn't impossible, things would be much more complicated.)

The bottom line of all this is that we can distinguish between quintessence and a cosmological constant as the source of dark energy (if both are not present) just by measuring accurately enough how the universe is expanding, which will tell us what w is. If w=-1, we have a cosmological constant. If -1<w<-1/3, we have some form of quintessence. (It is also conceivable that w<-1, in which case things are really weird.)

In fact, we can put slightly tighter bounds on w, since we know roughly how much dark energy there is, and this is because we know the universe is spatially flat. Let εt stand for the total energy density in the universe, and let εm be the energy density due exclusively to matter (most of which is dark matter). Careful measurements of the motions of stars in galaxies and of galaxies gives us a value for εm. Knowing in addition that the universe is flat tells us what εt has to be, and hence that εm is about (1/3)εt. (Actually it's a little less, but that's close enough.) Since εd, the energy density of dark energy, accounts for all the rest, we have εd = (2/3)εt.

Since the expansion of the universe is accelerating, the acceleration equation implies εt+3P<0. But P=wεd, since matter does not contribute to pressure (it has its own effective w=0), and hence P=w(2/3)εt. Plugging that in, we have εt+2wεt<0, and so w<-1/2 (instead of w<-1/3).

Finally, we can indicate what the two subtypes of quintessence are that Caldwell and Linder identified in their paper. The types are distinguished according to whether the first time derivative of w, i. e. w′, is positive or negative. If w′<0, then w is decreasing with time and in the limit is -1, so in some sense the quintessence is "freezing" into a cosmological constant, which means that the acceleration of expansion will continue forever. On the other hand, if w′>0, then w will gradually increase from near -1, away from behaving like a cosmological constant, and this case is called "thawing". As the universe expands, both εd and εm decrease (the total amounts of quintessence and matter don't change, but the volume is increasing). The quantity εt+3P = εm + εd + 3wεd = εm + (1+3w)εd. Now -2<1+3w<0 since -1<w<-1/3, so εt+3P approaches 0 as the densities decrease, and so acceleration gradually goes to 0 and stops.

Both subtypes of quintessence can be modeled using a variety of different types of "scalar fields", but not any fields that are part of the current standard model of particle physics.

What about the case w<-1? In that scenario, acceleration increases very rapidly, leading to what is called the "big rip", in which not only the universe itself expands, but in the distant future even stars and eventually subatomic particles are torn apart. This would correspond to yet another type of quintessence, called "phantom energy". But that's a story for another time.

Related:


How Are We to Make Progress With w?

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