**2.3. The coincidence scandal**

The third issue mentioned above is the coincidence between the observed vacuum energy (1.11) and the current matter density. To understand this, we briefly review the dynamics of an expanding Robertson-Walker spacetime. The evolution of a homogeneous and isotropic universe is governed by the Friedmann equation,

(1.15) |

where *a*(*t*) is the scale factor,
*H* = / *a*
is the Hubble
parameter, is
the energy density, and
is the
spatial curvature parameter. The energy density is a sum of
different components,
=
_{i}
_{i},
which will in
general evolve differently as the universe expands. For matter
(non-relativistic particles) the energy density goes as
_{M}
*a*^{-3}, as the number density is diluted with
the expansion of the universe. For radiation the energy density goes as
_{R}
*a*^{-4}, since each particle loses energy
as it redshifts in addition to the decrease in number density.
Vacuum energy, meanwhile, is constant throughout spacetime, so that
_{}
*a*^{0}.

It is convenient to characterize the energy density of each component by its density parameter

(1.16) |

where the critical density

(1.17) |

is that required to make the spatial geometry of the universe be flat ( = 0). The "best-fit universe" or "concordance" model implied by numerous observations includes radiation, matter, and vacuum energy, with

(1.18) |

together implying a flat universe. We see that the densities in matter
and vacuum are of the same order of magnitude.
^{(2)} But the ratio of these
quantities changes rapidly as the universe expands:

(1.19) |

As a consequence,
at early times the vacuum energy was negligible in comparison to
matter and radiation, while at late times matter and radiation are
negligible. There is only a brief epoch of the universe's history
during which it would be possible to
witness the transition from domination by
one type of component to another. This is illustrated in
Figure 1, in which the various density parameters
_{i} are
plotted as a function of the scale factor. At early times
_{R} is
close to unity; the matter-radiation
transition happens relatively gradually, while the matter-vacuum
transition happens quite rapidly.

How finely-tuned is it that we exist in the era when vacuum and
matter are comparable? Between the Planck time and now, the universe
has expanded by a factor of approximately 10^{32}. To be fair,
we should consider an interval of logarithmic expansion which is
centered around the present time; this would describe a total expansion
by a factor of 10^{64}. If we take the transitional period
between matter and vacuum to include the time from
_{} /
_{M} = 0.1
to _{} /
_{M} = 10,
the universe expands by a factor of
100^{1/3}
10^{0.67}. Thus, there is an
approximately 1% chance that an
observer living in a randomly selected logarithmic expansion interval
in the history of our universe would be lucky enough to have
_{M} and
_{} be
the same order of magnitude.
Everyone will have their own favorite way of quantifying such
unnaturalness, but the calculation here
gives some idea of the fine-tuning involved; it is substantial,
but not completely ridiculous.

As we will discuss below, there is room to imagine that we are
actually not observing the effects of an ordinary cosmological
constant, but perhaps a dark energy source that varies gradually
as the universe expands, or even a breakdown of general relativity
on large scales. By itself, however, making dark energy dynamical
does not offer a solution to the coincidence scandal; purely on
the basis of observations, it seems clear that the universe has
begun to accelerate recently, which implies a scale at which something
new is kicking in. In particular, it is fruitless to try to
explain the matter/dark energy coincidence by invoking
mechanisms which make the dark energy density time-dependent in such a
way as to *always* be proportional to that in matter. Such a
scenario would either imply that the dark energy would redshift away
as _{dark}
*a*^{-3}, which from (1.15)
would lead to a non-accelerating universe, or require departures from
conventional general relativity of the type which (as discussed below)
are excluded by other measurements.

^{2} Of course
the "matter" contribution consists both of ordinary baryonic
matter and non-baryonic dark matter, with
_{b}
0.04 and
_{DM}
0.25. The similarity
between these apparently-independent quantities is another coincidence
problem, but at least one which is independent of time; we have
nothing to say about it here.
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