### 7.2 Cosmology

Cosmology also provides a way to test Lorentz violation. The most obvious connection is via inflation. If the
number of e-foldings of inflation is high enough, then the density fluctuations responsible for the observed
cosmic microwave background (CMB) spectrum have a size shorter than the Planck scale before inflation. It
might therefore be possible for trans-Planckian physics/quantum gravity to influence the currently observed
CMB spectrum. If Lorentz violation is present at or near the Planck scale (as is implicit in models that use
a modified dispersion relation at high energies [213]), then the microwave background may still carry an
imprint.
A number of authors have addressed the possible signatures of trans-Planckian physics in the CMB
(for a sampling see [94, 212, 214, 262, 101, 164, 66, 236] and references therein). While the
possibility of such constraints is obviously appealing, the CMB imprint (if any) of trans-Planckian
physics, much less Lorentz violation, is model dependent and currently the subject of much
debate.
In short, although such cosmological explorations are interesting and may provide an eventual method for
ultra-high energy tests of Lorentz invariance, for the purposes of this review we forego any more discussion
on this approach.
A simple low energy method to limit the coefficients in the aether model (47) that is less fraught with
ambiguities has been explored by Carroll and Lim [76]. They consider a simplified version of the
model (47) without the term and choose the potential to be of the form ,
where is a Lagrange multiplier. Without loss of generality, we can rescale the coefficients in
Equation (47) to set . In the Newtonian limit Carroll and Lim find that Newton’s constant as
measured on earth is rescaled to be
In comparison, the effective cosmological Newton’s constant is calculated to be
The difference between the cosmological and Newtonian regimes implies that we have to
adjust our measured Newton’s constant before we insert it into cosmological evolution equations.
Such an adjustment modifies the rate of expansion. A change in the expansion rate modifies
big bang nucleosynthesis and changes the ratio of the primordial abundance of to .
By comparing this effect with observed nucleosynthesis limits, Carroll and Lim are able to
constrain the size of , , and . In addition to the nucleosynthesis constraint, the authors
impose restrictions on the choice of coefficients such that in the preferred frame characterized by
the perturbations have a positive definite Hamiltonian, are non-tachyonic, and
propagate subluminally. With these assumptions Carroll and Lim find the following constraint:

where the dependence has been included for completeness.