**3.2. Is general relativity breaking down?**

If we believe that we live in a universe which is homogeneous, isotropic, and accelerating, general relativity (GR) is unambiguous about the need for some sort of dark energy source. GR has been fantastically successful in passing classic experimental tests in the solar system, as well as at predicting the amount of gravitational radiation emitted from the binary pulsar (Will 2001). Nevertheless, the possibility remains open that gravitation might deviate from conventional GR on scales corresponding to the radius of the entire universe. For our present purposes, such deviations may either be relevant to the cosmological constant problem, or to the dark energy puzzle.

The idea behind modifying gravity to address the cosmological
constant problem is to somehow allow for the vacuum energy to be
large, but yet not lead to an appreciable spacetime curvature
(as manifested in a rapidly expanding universe). Of course we
still need to allow ordinary matter to warp spacetime, so
there has to be something special about vacuum energy. One
special thing is that vacuum energy comes with a negative
pressure *p*_{vac} =
- _{vac},
as in (1.4). We might therefore imagine a theory
which gave rise to a modified version of the Friedmann equation,
of the form

(1.20) |

With such an equation, ordinary matter (for which *p* vanishes)
leads to conventional expansion, while vacuum energy
decouples entirely. Such a theory has been studied
(Carroll & Mersini
2001),
and may even arise in "self-tuning" models of extra dimensions
(Arkani-Hamed,
Dimopoulos, Kaloper & Sundrum, 2000,
Kachru, Schulz &
Silverstein 2000).
Unfortunately, close examination
of self-tuning models reveals that there is a hidden fine-tuning,
expressed as a boundary condition chosen at a naked singularity in
the extra dimension. Furthermore, any alternative to the conventional
Friedmann equation is also constrained by
observations: any alternative must predict the right abundances
of light elements from Big Bang nucleosynthesis (BBN; see
Burles, Nollett, &
Turner 2001),
the correct evolution of a sensible spectrum of primordial density
fluctuations into the observed spectrum of temperature anisotropies in the
Cosmic Microwave Background and the power spectrum of large-scale
structure
(Tegmark 2002,
Zahn & Zaldarriaga
2003,
Lue, Scoccimarro &
Starkman 2003),
and that the age of the universe is approximately thirteen
billion years.
The most straightforward test comes from BBN
(Carroll &
Kaplinghat 2002,
Masso & Rota 2003),
since the light-element abundances depend on the expansion rate
during a relatively brief period (rather than on the behavior of
perturbations, or an an integral of the expansion rate over a long
period). Studies of BBN in alternate
cosmologies indicate that it is possible for modifications of
GR to remain consistent with observations, but only for a very
narrow set of possibilities. It seems likely that the
success of conventional BBN, including its agreement with the baryon
density as determined by CMB fluctuations
(Spergel et al. 2003),
is not a misleading accident, but rather an indication that
GR provides an accurate description of cosmology when the universe
was of the order of one minute old. The idea of modifying GR to
solve the cosmological constant problem is not completely dead,
but is evidently not promising.

Rather than trying to solve the cosmological constant problem, we
can put aside the issue of why the magnitude of the vacuum energy
is small and focus instead on whether the current period of
acceleration can be traced to a modification of GR. A necessary
feature of any such attempt is to include a new scale in the theory,
since acceleration has only begun relatively
recently. ^{(3)} From a purely
phenomenological point of view we can imagine
modifying the Friedmann equation (1.15) so that acceleration
kicks in when either the energy density approaches a certain
value _{*},

(1.21) |

or when the Hubble parameter approaches a certain value
*H*_{*},

(1.22) |

The former idea has been suggested by
Freese & Lewis 2002,
the latter by
Dvali & Turner 2003;
in both cases we can fit the data for
appropriate choices of the new parameters. It is possible that
equations of this type arise in brane-world models with large
extra spatial dimensions; it is less clear whether the appropriate
parameters can be derived. An even more dramatic mechanism also
takes advantage of extra dimensions, but allows for separate
gravitational dynamics on and off of our brane; in this case
gravity can be four-dimensional *below* a certain length
scale (which would obviously have to be very large), and appear
higher-dimensional at large distances
(Dvali, Gabadadze &
Porrati 2000,
Deffayet, Dvali, &
Gabadadze 2002,
Arkani-Hamed, Dimopoulos,
Dvali & Gabadadze 2002).
These scenarios can also make the universe accelerate at late times,
and may even lead to testable deviations from GR in the solar system
(Dvali, Gruzinov, &
Zaldarriaga 2003;
Lue and Starkman 2003).

As an alternative to extra dimensions, we may look for an
ordinary four-dimensional modification of GR.
This would be unusual behavior, as we are used to thinking of
effective field theories as breaking down at high energies and
small length scales, but being completely reliable in the
opposite regime. Nevertheless, it is worth exploring whether
a simple phenomenological model can easily accommodate the data.
Einstein's equation can be derived by minimizing an action given
by the spacetime integral of the curvature scalar *R*,

(1.23) |

A simple way to modify the theory when the curvature becomes
very small (at late times in the universe) is to simply add a
piece proportional to 1/*R*,

(1.24) |

where *µ* is a parameter with dimensions of mass
(Carroll, Duvvuri,
Trodden and Turner 2003).
It is straightforward to show
that this theory admits accelerating solutions; unfortunately, it
also brings to life a new scalar degree of freedom, which may ruin
the success of GR in the solar system
(Chiba 2003).
Investigations are still ongoing to see whether a simple modification of
this idea could explain the acceleration of the universe while remaining
consistent with experimental tests; in the meantime, the difficulty
in finding a simple extension of GR that does away with the
cosmological constant provides yet more support for the standard
scenario.

^{3} One way of characterizing this scale is
in terms of the Hubble parameter when the universe starts accelerating,
*H*_{0} ~ 10^{-18} sec^{-1}.
It is interesting in this context to recall the coincidence pointed
out by Milgrom (1983),
that dark *matter* only becomes important in galaxies when the
acceleration due to gravity dips below a fixed value,
*a*_{0} / *c*
10^{-18} sec^{-1}. Milgrom himself has
suggested that the explanation for this feature of galactic dynamics
can be explained by replacing dark matter by a modified dynamics, and
it is irresistible to speculate that both dark matter and dark energy
could be replaced by a single (as yet undiscovered)
modified theory of gravity. However, hope for this possibility seems to be
gradually becoming more difficult to maintain, as different methods
indicate the existence of gravitational forces which point in
directions other than where ordinary matter is
(Van Waerbeke et
al. 2000,
Dalal & Kochanek
2002,
Kneib et al. 2003)
- a phenomenon that
is easy to explain with dark matter, but difficult with modified
gravity - and explanations are offered for
*a*_{0} / *c* ~ *H*_{0} within
conventional cold dark matter
(Scott, White, Cohn,
& Pierpaoli 2001,
Kaplinghat & Turner
2002).
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