Theoretical Approaches to Cosmic Acceleration

Mark Trodden, Syracuse University
trodden-at-physics.syr.edu

Less than a decade ago, observations of the lightcurves of type Ia supernovae first suggested that the expansion of the universe is accelerating. In the intervening years, a range of further observations [1,2] have provided firm support for this result, to the extent that, even if we were to ignore the supernova data entirely, the accelerating universe would remain unavoidable.

At the level of cosmological models, described by perfect fluids with phenomenological equations of state, the accelerating universe just requires new parameters to fit the known data. Augmenting the general Cold-Dark-Matter (CDM) cosmology with an extra fluid component, X, with present-day energy density and
constant equation of state parameter , satisfying
, the Friedmann equation becomes

(1) |

It is to this parametrization (with other parameters determining the initial spectrum of perturbations) that cosmological datasets are fit, perhaps the best-known being the WMAP data.

That such a small number of parameters can provide such a tremendous fit to the
evolution of the universe, including its large-scale structure, over its entire
history, is a triumph of modern cosmology comparable to the broad successes of the expansion, the discovery of the CMB and the agreement of the abundances of the light elements. However, such an approach, while remarkably useful, does not provide an *explanation* for the origin of cosmic acceleration. Indeed, the biggest impact of the accelerating universe is in its implications for fundamental physics.

Clearly, one possibility is that cosmic acceleration is due to the cosmological constant (with ). The cosmological constant problem itself - why is the vacuum energy so much smaller than we expect from effective-field-theory considerations? - requires a solution even in the absence of cosmic acceleration, and perhaps the final answer to this problem will yield a value appropriate to lead to late-time acceleration of the universe. Despite continuous theoretical pressure, the status of dynamical solutions to this conundrum has changed little since Weinberg's review article of 1988 [3]. Historically, this has led some researchers to consider an anthropic solution to the problem, although without a specific fundamental framework in which to investigate it.

However, in the context of string theory, the possibility of a *landscape*,
containing at least discrete vacua with vacuum energy densities ranging up to the Planck scale, coupled with the mechanism of eternal inflation to populate the landscape, has recently led to a specific implementation of the anthropic argument.

While such a conclusion would seem to limit the testability of the proposal, one hope is that it might be possible for the statistics of the distribution of vacua [4,5,6,7,8,9], to allow statistical predictions for other observable quantities, such as the fundamental coupling constants. Should increasingly accurate cosmological observations reveal a dark energy equation of state not equal to -1, or evidence for temporal or spatial variation of the dark energy density, then we will know that a cosmological constant is not the answer and it will be harder to imagine anthropic arguments from the string landscape being the correct answer.

If the cosmological constant is not responsible for dark energy (because it is zero, or much smaller than the dark energy scale), then several possibilities have been suggested for a dynamical origin for cosmic acceleration.

The first of these - *dark energy* [10,11,12] - seeks to find an underlying microscopic description of the perfect fluid. The most popular approach to this new dynamical component of the cosmic energy budget is to invoke a new scalar field driving late-time inflation (but without the need for an end, as in the reheating that takes place after early universe inflation). In such a *quintessence* model, the instantaneous effective dark energy equation of state is

(2) |

If one assumes that cosmic acceleration is due to such a field, with potential-dominated dynamics, then generally one finds that the scale of the potential () should be of order eV, and that the mass of the associated particle be of order the Hubble scale. These scales present obstacles to finding a sensible particle physics model of quintessence. One that does seem to work, with such technically natural parameter values, is if the quintessence field is realized as the pseudo-Nambu-Goldstone boson of some broken symmetry. In this case the unusually small values of parameters required are protected from quantum corrections by the symmetry.

One advantage of a subset of quintessence models is that they exhibit *tracking*. This means that there exist attractors of the dynamical system for which the scalar field tracks the equation of state of the background fluid. It can then be arranged that the field follows the evolution of the universe during radiation domination and then transitions to an accelerating attractor during matter domination. This allows a partial explanation of the coincidence problem, since acceleration is triggered by the onset of matter domination.

Another interesting suggestion has been that it may be possible to explain an accelerated universe by invoking the effects of inhomogeneities on the expansion rate - perturbations may induce an effective energy-momentum tensor with a nearly-constant magnitude. Kolb et. al. [13] have considered sub-horizon higher order corrections to the backreaction, going up to sixth order in a gradient expansion, and suggest that higher order corrections are large enough for the backreaction to generate dark energy like behavior. There have been a number of challenges to this idea (see e.g. [14]), but if a successful mechanism is found it would be an elegant and minimal explanation of acceleration.

A further possibility is that curvatures and length scales in the observable universe are only now reaching values at which an infrared modification of gravity can make itself apparent by driving self-acceleration. This possibility turns out to be incredibly difficult to implement.

Although, within the context of General Relativity (GR), one doesn't think about it too often, the metric tensor contains, in principle, more degrees of freedom than the usual spin-2 *graviton*. However, the Einstein-Hilbert action results in second-order equations of motion that constrain away the scalars and the
vectors, so that they are non-propagating. But this is not the case if one departs from the Einstein-Hilbert form for the action. When using any modified action (and the usual variational principle) one inevitably frees up some of the additional degrees of freedom. In fact, this can be a good thing, in that the dynamics of these new degrees of freedom may be precisely what one needs to drive the accelerated expansion of the universe. In many situations though, there is a price to pay.

The problems may be of several different kinds. First, there is the possibility that along with the desired deviations from GR on cosmological scales, one may also find similar deviations on solar system scales, at which GR is rather well-tested. Second is the possibility that the newly-activated degrees of freedom may be badly behaved in one way or another; either having the wrong sign kinetic terms (ghosts), and hence being unstable, or leading to superluminal propagation, which may lead to other problems. These constraints are surprisingly restrictive when one tries to create viable modified gravity models yielding cosmic acceleration.

As an example, one simple way to modify GR is to replace the Einstein-Hilbert Lagrangian density by a general function of the Ricci scalar . For appropriate choices of the function it is then possible to obtain late-time cosmic acceleration without the need for dark energy [15]. However, evading bounds from precision solar-system tests of gravity turns out to be a much trickier matter, since such simple models are equivalent to a Brans-Dicke theory with in the approximation in which one may neglect the potential, and are therefore inconsistent with experiment.To construct a realistic model requires at the very least a rather complicated function, with more than one adjustable parameter in order to fit the cosmological data and satisfy solar system bounds.

It is natural to consider generalizing such an action to include other curvature invariants [16], and it is straightforward to show these generically admit a maximally-symmetric solution: de Sitter space. Further, for a large number of such models (see e.g. [17]), solar system constraints, of the type I have described for models, can be evaded. However, in these cases another problem arises, namely that the extra degrees of freedom that arise are generically ghost-like.

An alternative, and particularly successful approach, is that employed by Dvali
and collaborators [18,19,20] in which an interesting modification to gravity arises from extra-dimensional models with
both five and four dimensional Einstein-Hilbert terms. These *Dvali-Gabadadze-Porrati (DGP) braneworlds* allow one to obtain cosmic acceleration from the gravitational sector because gravity deviates from the usual four-dimensional form at large distances. One may also ask whether ghosts plague these models. However, Dvali has claimed that this theory reaches the strong coupling regime before
a propagating ghost appears. In fact, Dvali has shown that theories that modify
gravity at cosmological distances must exhibit strong coupling phenomena, or else either possess ghosts or are ruled out by solar system constraints.

Current observational bounds are entirely consistent with a cosmological constant, but also with a range of dark energy models and the possibility that a modification to GR is the origin of cosmic acceleration. While it is often stated that one or other of these ideas is the simplest or most natural theoretical explanation, only increasingly accurate observations can settle the question and allow us to make progress. In preparation for these, much theoretical work is necessary to extract concrete predictions with which to distinguish between the various suggestions. A number of authors have already begin to tackle this problem, with one possible answer being that the cross-correlation of kinematical observables with tests involving the linear growth of structure as the universe expands [21]. Whatever the ultimate answer, the accelerating universe looks bound to teach us a deep truth about fundamental physics.