**7.3. Cosmological constant as a stochastic variable**

Current cosmological observations can be interpreted as showing that
the *effective value* value of
(which will pick up contributions from all vacuum energy densities of
matter fields) has been reduced from the natural value of
*L*_{P}^{-2} to
*L*_{P}^{-2}(*L*_{P}
*H*_{0})^{2} where *H*_{0}
is the current value of the Hubble constant.
One possible way of thinking about this issue is the following
[261]:
Let us assume that the quantum micro structure of spacetime at Planck
scale is capable of readjusting itself, soaking up any
vacuum energy density which is introduced - like a sponge soaking up water.
If this process is fully deterministic and exact, all vacuum energy
densities will cease to have
macroscopic gravitational effects. However, since this process is
inherently quantum gravitational, it is subject to quantum fluctuations
at Planck scales. Hence, a tiny part of the vacuum energy will survive the
process and will lead to observable effects.
One may conjecture that the cosmological constant we measure corresponds
to this small residual fluctuation which will depend on the volume of
the spacetime region that is probed.
It is small, in the sense that it has been reduced from
*L*_{P}^{-2} to
*L*_{P}^{-2}(*L*_{P}
*H*_{0})^{2},
which indicates the fact that fluctuations - when measured over a large
volume - is small compared to the bulk value. It is the wetness of the
sponge we notice, not the water content inside.

This is particularly relevant in the context of standard discussions of
the contribution of
zero-point energies to cosmological constant. The correct theory is likely
to regularise the divergences and make the zero point energy finite and
about *L*_{P}^{-4}. This contribution is most
likely to modify the microscopic structure of spacetime (e.g if the
spacetime is naively thought of as due to stacking of Planck scale
volumes, this will modify the stacking or shapes of the volume elements)
and will not affect the bulk gravitational field when measured at scales
coarse grained over sizes much bigger than the Planck scales.

Given a large 4-volume
of the spacetime, we
will divide it into *M* cubes of size
(
*x*)^{4} and label the cubes by *n* = 1, 2,.....,
*M*. The contribution to the path integral amplitude
, describing long
wavelength limit of conventional Einstein gravity, can be expressed in
the form

(107) |

where we have indicated the standard continuum limit. (In conventional units
*c*_{1} = (16)^{-1}.)
Let us now ask how one could modify this result to
describe the ability of spacetime micro structure to readjust
itself and absorb vacuum energy densities.
This would require some additional dynamical
degree of freedom that will appear in the path integral amplitude and
survive in the classical limit. It can be shown that
[261]
the simplest implementation of this feature is by modifying the standard
path integral amplitude
[exp(*c*_{1}(*R L*_{P}^{2}) + ^{
... })] by a factor [(*x*_{n}) /
_{0}]
where (*x*)
is a scalar degree of freedom and
_{0} is
a pure number introduced to keep this factor
dimensionless. In other words, we modify the path integral amplitude to
the form:

(108) |

In the long wavelength limit, the extra factor in (108) will lead to the term of the form

(109) |

Thus, the net effect of our assumption is to introduce a `scalar field
potential'
*V*() = -
*L*_{P}^{-4}
ln( /
_{0})
in the semi classical limit. It is obvious that
the rescaling of such a scalar field by
*q* is
equivalent to adding a cosmological constant with vacuum energy
- *L*_{P}^{-4}ln *q*. Alternatively, any
vacuum energy can be re absorbed by such a rescaling.
The fact that the scalar degree of freedom occurs as a potential in (109)
without a corresponding kinetic energy term shows that its dynamics is
unconventional and non classical.

The above description in terms of macroscopic scalar degree of freedom can, of course, be only approximate. Treated as a vestige of a quantum gravitational degrees of freedom, the cancellation cannot be precise because of fluctuations in the elementary spacetime volumes. These fluctuations will reappear as a "small" cosmological constant because of two key ingredients: (i) discrete spacetime structure at Planck length and (ii) quantum gravitational uncertainty principle.

To show this, we use the fact noted earlier in section 7.1 that the net cosmological constant can be thought of as a Lagrange multiplier for proper volume of spacetime in the action functional for gravity. In any quantum cosmological models which leads to large volumes for the universe, phase of the wave function will pick up a factor of the form

(110) |

from (104), where
is the four volume.
Treating
(_{eff} /
8
*L*_{P}^{2},
) as conjugate
variables (*q*, *p*), we can invoke the standard uncertainty
principle to predict
8
*L*_{P}^{2} /
. Now we use the
earlier assumption regarding the microscopic structure of the spacetime:
Assume that there is a zero point length of the order of
*L*_{P}
so that the volume of the universe is made of a large number (*N*)
of cells, each of volume
(
*L*_{P})^{4} where
is a numerical
constant. Then
=
*N*(
*L*_{P})^{4}, implying a Poisson fluctuation
(
*L*_{P})^{2} and leading to

(111) |

This will give
_{} =
(8 /
3^{2}) which
will - for example - lead to
_{} = (2/3) if
=
2
. Thus
Planck length cutoff (UV limit) and volume of the universe (IR limit)
combine to give the correct
.
(A similar result was obtained earlier in
[260]
based on a different model.) The key idea, in this approach, is that
is a stochastic variable with a zero mean and fluctuations. It is the
rms fluctuation which is being observed in the cosmological context.

This has three implications: First, FRW equations now need to be solved
with a stochastic term on the right hand side and one should check
whether the observations can still be explained.
The second feature is that stochastic properties of
need to be
described by a quantum cosmological model. If the quantum state of the
universe is expanded in terms of the eigenstates of some suitable
operator (which does not commute the total four volume operator), then
one should be able to characterize the fluctuations in each of these
states. Third, and most important, the idea of cosmological constant
arising as a *fluctuation* makes sense only if the bulk value is
rescaled away.

The non triviality of this result becomes clear when we compare it with
few other alternative ways of estimating the fluctuations - none of
which gives the correct result. The first alternative approach is based
on the assumption that one can associate an entropy
*S* = (*A*_{H} / 4*L*_{P}^{2})
with compact space
time horizons of area *A*_{H} (We will discuss this idea in
detail in section 10). A popular interpretation
of this result is that horizon areas are quantized
in units of *L*_{P}^{2} so that *S* is
proportional to the number of bits of information contained in the
horizon area. In this approach, horizon areas can be expressed in the
form *A*_{H} = *A*_{P}*N* where
*A*_{P}
*L*_{P}^{2}
is a quantum of area and *N* is an integer. Then the
*fluctuations* in the area will be
*A*_{H} = *A*_{P}
*N*
=(*A*_{P} *A*_{H})^{1/2}. Taking
*A*_{H}
^{-1}
for the de Sitter horizon, we find that
*H*^{2}(*H L*_{P}) which is a
lot smaller than what one needs. Further, taking
*A*_{H}
*r*_{H}^{2}, we find that
*r*_{H}
*L*_{P};
that is, this result essentially arises from the idea that the radius of
the horizon is uncertain within one Planck length. This is quite true,
of course, but does not lead to large enough fluctuations.

A more sophisticated way of getting this (wrong) result is to
relate the fluctuations in the cosmological constant
to that of the volume of the universe is by using a canonical ensemble
description for universes of proper Euclidean 4-volume
[262].
Writing *V*
/
8
*L*_{P}^{2} and treating
*V* and as
the relevant variables, one can write a partition function for the
4-volume as

(112) |

Taking the analogy with standard statistical mechanics (with the
correspondence *V*
and
*E*), we
can evaluate the fluctuations in the cosmological constant in exactly
the same way as energy fluctuations in canonical ensemble.
(This is done in several standard text books; see, for example,
[263]
p. 194.) This will give

(113) |

where *C* is the analogue of the specific heat.
Taking the 4-volume of the universe to be
=
*bH*^{-4} = 9*b*
^{-2}
where *b* is a numerical factor and using
*V* = ( /
8
*L*_{P}^{2}) we get
*L*_{P}^{-1} *V*^{-1/2}. It follows
from (113) that

(114) |

In other words
*H*^{2}(*HL*_{P}), which is the same result
from area quantization and is a lot smaller
than the cosmologically significant value.

Interestingly enough, one could do slightly better by assuming that
the horizon *radius* is quantized in units of Planck length, so
that *r*_{H} = *H*^{-1} =
*NL*_{P}. This will lead to the fluctuations
*r*_{H} = (*r*_{H}
*L*_{P})^{1/2} and using
*r*_{H} = *H*^{-1}
^{-1/2},
we get
*H*^{2}(*HL*_{P})^{1/2} - larger than
(114) but still inadequate.
In summary, the existence of two length scales *H*^{-1} and
*L*_{P} allows different results for
depending on how
exactly the fluctuations are characterized
(*V*
*N*,
*A*
*N* or
*r*_{H}
*N*). Hence the
result obtained above in (111) is non trivial.

These conclusions stress, among other things, the difference between
*fluctuations* and the *mean values*. For, if one assumes
that every patch of the universe with size *L*_{P}
contained an energy *E*_{P}, then a universe with
characteristic size *H*^{-1} will contain the energy
*E* = (*E*_{P} /
*L*_{P})*H*^{-1}. The corresponding energy
*density* will be
_{V} =
(*E*/*H*^{-3}) = (*H* /
*L*_{P})^{2} which leads to the correct
result. But, of course, we do not know why every length scale
*L*_{P} should contain an energy *E*_{P} and -
more importantly - contribute coherently to give the total energy.