8.4. The nature of "dark energy"
It is beyond the scope of the present article to discuss dark energy in detail and the reader is referred to the excellent review by Peebles and Ratra (2002). However, since this is arguably the most dramatic discovery involving HST I would like to make a few points.
In general relativity, the stress-energy tensor of the vacuum, T^{}_{µ}, can be written as
(38) |
where _{} is constant (proportional to Einstein's cosmological constant) and g_{µ} is the metric. In a Minkowski flat spacetime this can be written simply as the "equation of state"
(39) |
More generally, if the equation of state is written as (for example, Canuto et al. 1977, Ratra and Peebles 1988, Sahni and Starobinsky 2000, Turner and White 1997),
(40) |
then the dark energy density behaves (for a constant w) like _{} ~ a^{-3(1+w)} (where a is the scale factor). Since the scale factor satisfies the equation (in units in which c = 1)
(41) |
the dark energy contribution will result in an accelerating universe if w < - 1/3.
It is interesting to note that from a purely fluid dynamical point of view, any equation of state of the type P = w is in fact dynamically unstable for any negative value of w except for w = - 1 (corresponding to a cosmological constant). This can be easily seen from perturbations on the equations for momentum and energy conservation (< >, < P > denote mean values)
(42) |
(43) |
Defining the speed of sound, c^{2}_{s} = dP / d, these two equations can be combined to form
(44) |
which is clearly unstable for negative c^{2}_{s} (except for w = - 1, when the combination < > + < P > vanishes, as expected from the fact that the vacuum is the same for all inertial observers).
Thus, formally speaking, from a fluid-dynamical point of view only a cosmological constant yields a stable solution.
One can certainly take the dark energy to be associated with a uniform scalar field (a "quintessence" field; see below), that rolls down a potential V() at a rate determined by
(45) |
where V'() = dV / d. In this case,
(46) |
For a sufficiently slowly varying dynamical component, such that the kinetic energy is much smaller than the potential, ^{2} << V(), one even obtains a field energy that mimics the cosmological constant (P_{} - _{}). Generally, quintessence solutions allow for different equations of state, and even values of w that are time variable. Nevertheless, I find the stability property expressed in eq. (44) sufficiently intriguing to warrant a deeper examination of the cosmological constant possibility, before that is abandoned in favor of other dynamical solutions. This point of view received some further support from the recent determination by WMAP that w < - 0.78 (Bennett et al. 2003).
There are two main problems with the implied energy density _{V} of the dark energy (e.g., Weinberg 2001). (i) Why is its value not ~ 120 orders of magnitude larger (as expected from fluctuations in the gravitational field up to the Planck scale)? (ii) Why now? (Namely, why _{} ~ _{M} now, even though _{} may be associated with a cosmological constant, while _{M} declined continuously from the initial singularity to its present value).
An interesting curiosity to note is that even though taking graviton energies up to Planck scale, M_{P}, misses the value of the dark energy density by ~ 10^{120}, and taking them up to the supersymmetry scale, M_{SUSY}, misses by ~ 10^{55}, a scale of M_{V} ~ (M_{SUSY} / M_{P})M_{SUSY} actually does give the right order of magnitude! While I am not aware at present of any theory that produces this scale in a natural way (although see, e.g., Arkani-Hamed et al. 2000), I find this coincidence worth following up.
Much of the efforts to resolve the above two problems revolved not around a cosmological constant, but rather around the behavior of quintessence fields. In particular, attention has concentrated on "tracker" solutions, in which the final value of the quintessence energy density is independent of fine-tuning of the initial conditions (e.g., Zlatev, Wang and Steinhardt 1998, Albrecht and Skordis 2000). For example if one takes a potential of the form
(47) |
where > 0 and M is an adjustable constant, and the field is initially much smaller than the Planck mass, then for _{M} >> V(), ^{2}, we find that _{M} decreases initially faster (~ t^{-2}) than (t)(~ t^{-2 / (2+)}).
Eventually, however, a transition to a _{}-dominated universe occurs (and _{} decreases as t^{-2 / (4+)}). In other words, the quintessence answer to the question: why is the dark energy density so small? is simply: because the universe is very old. Nevertheless, simple potentials of the form (47) do not offer a clear solution to the "why now?" problem. In fact, to have _{} ~ _{M} (and of order of the critical density) at the present time requires fine-tuning the parameters so that (for example, Weinberg 2001)
(48) |
with no simple explanation as to why this equality should hold.
In order to overcome this requirement for fine-tuning, some versions of the quintessence models choose potentials in which the universe has periodically been accelerating in the past (the dark energy has periodically dominated the energy density; e.g., Dodelson, Koplinghat & Stewart 2000). Dodelson et al. 2000 have shown, for example, that with a potential of the form
(49) |
one can obtain solutions in which the dark energy density tracks the ambient energy density of the universe and satisfies observational constraints.
A different approach to the problems associated with the dark energy density has been through anthropic considerations (e.g., Weinberg 2001, Vilenkin 1995, Kallosh and Linde 2002). The key assumption in this class of models is that some constants of nature, in particular the cosmological constant, and possibly the density contrast at the time of recombination, _{rec}, are in fact random variables, whose range of values and a priori probabilities are nevertheless determined by the laws of physics. In this picture, some values of the constants which are allowed in principle, may be incompatible with the very existence of observers. Assuming a principle of mediocrity - that we should expect to find ourselves in a universe typical of those that allow the emergence of intelligent life - Garriga, Livio and Vilenkin (2000) were able to show that the "why so small?" and "why now?" questions find a natural explanation. While I personally believe that anthropic considerations should only be used as a last resort, there is no denial of the fact that most versions of "eternal inflation," the notion that once inflation starts it never stops, unavoidably produce an ensemble of "pocket" universes (Guth 1981, Vilenkin 1983, Linde 1986, Steinhardt 1983). Once such an infinite ensemble is believed to exist, the problem of defining probabilities on it, and the concept of "mediocrity" become real (e.g., Linde, Linde and Mezhlumian 1995, Vilenkin 1998).
In spite of the apparent "success" of the anthropic argumentation in solving the two main problems associated with dark energy, the search for a fundamental explanation is not, and should not, be abandoned. An interesting, if speculative, new direction is provided by alternative theories of gravity. For example, models in which ordinary particles are localized on a three-dimensional surface (3-brane) embedded in infinite volume extradimensions to which gravity can spread, have been developed (Deffayet, Dvali, and Gabadadze 2002). In a particular version of these models, the Friedmann equation (eq. 24) is replaced by
(50) |
where = ± 1, and r_{c} represents a crossover scale. At distances shorter than r_{c} (which can be of astronomical size, e.g., r_{c} ~ cH_{0}^{-1}), observers on the brane discover the familiar Newtonian gravity. At large cosmological distances, however, the force-law of gravity becomes five-dimensional (as gravity spreads into the extra dimensions) and weaker. The dynamics are governed by whether / M_{p}^{2} is larger or smaller than 1 / r_{c}^{2}. While highly speculative at this stage, some aspects of these alternative theories of gravity can be experimentally tested (e.g., by lunar ranging experiments; Dvali, Gruzinov and Zaldarriaga 2002). Also, the relative lack of power on large scales (in particular, absence of any correlated signal on scales larger than 60 degrees) found by the Wilkinson Microwave Anisotropy Probe (Spergel et al. 2003), may indicate a breakdown of canonical gravity on large scales (or, more speculatively, a finite universe).
Luckily, observations that are already being performed with HST, and observations with the proposed Super Nova/Acceleration Probe (SNAP) will provide some crucial information on the nature of dark energy. In particular, if the equation of state parameter w is constant, then a transition from a decelerating phase to an accelerating phase occurs at
(51) |
For example, for _{M} = 0.27 (Bennett et al. 2003) and a cosmological constant (w = - 1), this gives ztr 0.76. Equation (51) shows that if _{M} can be determined independently to within a few percent (a goal that can perhaps be achieved by a conbination of cosmic microwave background and large scale structure measurements), then observations of a large sample of Type Ia supernovae in the redshift range 0.8-2 can place very meaningful constraints on the dark energy equation of state. The situation is more ambiguous if w is time-dependent, but even then, a sufficiently large sample of SNe Ia will give useful information. At the time of this writing, a search for SNe Ia conducted in the GOODS fields has already yielded 10 SNe Ia, with six of them in the redshift range z ~ 1-2 (Principal Investigator: Adam Riess). This, together with a few current ground-based searches at lower redshifts, is an excellent first step. The survey proposed by SNAP, of 15 square degrees, could yield light curves and spectra for over 2000 SNe Ia at redshifts up to z ~ 1.7. With this sample in hand, one could reach the elimination or at least control of the sources of systematic uncertainties needed to discriminate between different dark-energy models.