Contribution of Kaluza-Klein modes to vacuum energy in models with large extra dimensions $.pdf
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Contribution of Kaluza-Klein modes to vacuum
energy in models with large extra dimensions
the Cosmological constant
Abhinav Gupta?
St. Stephen’s College,
University of Delhi, Delhi-110 007, India.
Abstract
In this paper, the generation of topological energy in models with large
extra dimensions is investigated. The origin of this energy is attributed
to a topological deformation of the standard Minkowski vacuum due to
compactification of extra dimensions. This deformation is seen to give rise
to an effective, finite energy density due to massive Kaluza-Klein modes of
gravitation. It’s renormalized value is seen to depend on the size of the
extra dimensions instead of the UV cut-off of the theory. It is shown that if
this energy density is to contribute to the observed cosmological constant,
there will be extremely stringent bounds on the number of extra dimensions
and their size.
Introduction
The Einstein equations with the cosmological constant are given by
Rμν ? 1
2
gμνR = gμνΛ? 8πGTμν (1)
where Λ is the cosmological constant, or vacuum energy. Modulo the factor of
8πG, this constant is identical to the stress tensor associated with vacuum. It can
?E–mail : abh@ducos.ernet.in, abhinav@physics.du.ac.in
1
be seen on grounds of covariance that the vacuum expectation value of the stress
tensor is of the form [1]
〈Tμν〉 = ρvac gμν (2)
No known symmetry forbids a cosmological term in Einstain equations. Based
on observations, it is seen that in suitable units, Λ is at most of the order of 10?47
(GeV)4. At the same time, the vacuum energy evaluated with an ultraviolet cut-off
at the Planck scale Mpl ～ 1019 GeV gives
〈Evac〉 ～ 1
2
∫ Mpl
0
d3k
(2π)3
√
k2 +m2 ～ M4pl (3)
which is about 120 orders of magnitude higher. Attempts have been made to
account for this discrepancy [2,3] and it has been realized that even if the cut-off
is taken at the Supersymmetry (SUSY) scale, the discrepancy is still about 60
orders o
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