Dark energy as a modification of the Friedmann equation Page: 4 of 12
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it follows that the first term must scale as H2. A calculation leads to Friedmann equation
for arbitrary 4-d brane-localized matter source pM(t) [11, 12]:
H 8 S7pM
H2 + H(5)
These general features persist for the arbitrary number of dimensions.
The basic point is that the higher-dimensional action is suppressed relative to 4-dimensional
one by an inverse power of the crossover scale. As a result, the scaling arguments suggest
that for the maximally symmetric ansatz the higher-dimensional contribution should scale
as a lower power of H2, and thus be important at late times (small H). This argument sug-
gests that infinite-volume extra-dimensional theories have the potential to explain cosmic
acceleration with dark energy.
Before proceeding, let us make a few important points. Naively, it appears that as far
as dark energy is concerned the self-accelerated solutions of higher-dimensional theory have
the same number of new parameters as the simplest model of dark energy, a cosmological
constant. The cosmological constant is replaced by the crossover scale rc. However, from
the quantum field theory point of view there is a crucial difference, the crossover scale r0 is
stable under quantum corrections.
There is an interesting coincidence, which also motivates the Hubble-scale value of r0
and is unrelated to dark energy. The scale r0 also sets the distance at which corrections
(coming from higher-dimensional gravity) to the usual metric for a gravitating source become
important. In particular, there are corrections to the Schwarzschild metric , which effect
planetary motions. The existing phenomenological bounds on such deviations demand that
the crossover scale be large. For instance, for Eq. (1) the most stringent bound comes from
lunar laser ranging experiments that monitor the moon's perihelion precession with a great
accuracy and imply a lower bound for r0 which is close to the present cosmological horizon
In the present paper we shall take a more radical and generic attitude. We shall use the
notion of infinite extra dimensions to motivate the modification of Friedmann equation at
late times. In that spirit, we shall assume that the physics that modifies Friedmann equation
satisfies the following simple requirements:
" There is a single crossover scale r0
" To leading order, the corrections to Friedmann equation can be parameterized a single
These two assumptions fix the form of the modified Friedmann equation:
z H a 8 r G pur
H - = (6)
In order to eliminate the need for dark energy, this term must specifically be: (1-QM)H /Hp -2
which implies that rc = (1 - QM)- Ho1.
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Dvali, Gia; U., /New York; Turner, Michael S. & /Chicago U., Astron. Astrophys. Ctr. /KICP, Chicago /Chicago U., EFI /Fermilab. Dark energy as a modification of the Friedmann equation, article, January 1, 2003; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc1409385/m1/4/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.