Advances in perturbative thermal field theory - Ultra-relativistic ...

408 U Kraemmer and A Rebhan
where T αβ = P 0 (4δ 0 µ δ0 ν − η µν) and P 0 , the ideal-gas pressure ∝ T 4 . These identities reduce
the number of independent structure functions to three, which may be chosen as
1 (k) ≡ ˜ 0000 (k)
ρ
, 2 (k) ≡ ˜ 0µ
µ 0 (k)
ρ
, 3 (k) ≡ ˜ µν µν (k)
, (9.5)
ρ
where ρ = T 00 = 3P SB .
The HTL limit is in fact universal: the thermal matter may be composed of any form
of ultrarelativistic matter including gravitons, which are equally important as any other
thermalized matter if the graviton background has a comparable temperature. It reads [450]
ˆ 1 (k) = k0
2|k| ln k0 + |k|
k 0 −|k| − 5 4 , ˆ 2 =−1, ˆ 3 = 0. (9.6)
As in ordinary hot gauge theories, the entire HTL effective action is determined by the
Ward identities and has been constructed in [455, 456]. The three-graviton HTL vertex has
moreover been worked out explicitly in [457,458], and subleading corrections beyond the HTL
approximation have been considered in [459].
From (9.6), one can formally derive an HTL-dressed graviton propagator that contains
three independent transverse-traceless tensors. As the HTL propagators in gauge theories, these
describes collective phenomena in the form of non-trivial quasi-particle dispersion laws and
Landau damping (from the imaginary part of ˆ 1 ). In the context of cosmological perturbations,
the latter corresponds to the collisionless damping studied in [460], as we shall further discuss
below.
In the limit k 2 0 , k2 ≫ GT 4 , where G is the gravitational coupling constant, one can ignore
any background curvature terms Gρ ∝ GT 4 and use the momentum–space propagator to
read off the dispersion relations for the three branches of gravitational quasi-particles. One
of these, the spatially transverse-traceless branch, corresponds to the gravitons of the vacuum
theory. Denoting this branch by A, (9.6) implies an asymptotic thermal mass for the gravitons
according to [450]:
ω 2 A → k2 + m 2 A∞ = k2 + 5 × 16πGρ. (9.7)
9
While in a linear response theory branch A couples neither to perturbations in the energy
density δT 00 nor the energy flux δT 0i , the two additional branches, B and C, both couple to
energy flux, but only C to energy density. The additional branches correspond to excitations
that are purely collective phenomena, and like the longitudinal plasmon in gauge theories,
they disappear from the spectrum as k 2 /(GT 4 ) →∞. In contrast to ordinary gauge theories,
however, they do so by acquiring effective thermal masses that grow without bound, while at the
same time the residues of the corresponding poles disappear in a power-law behaviour (instead
of becoming massless and disappearing exponentially) [450]. (Some subleading corrections
to these asymptotic masses at one-loop order have been determined in [461], but these are
gauge-dependent and therefore evidently incomplete.)
For long wavelengths k 2 ∼ GT 4 , the momentum–space HTL graviton propagator exhibits
an instability, most prominently in mode C, which is reminiscent of the gravitational Jeans
instability and which is only to be expected since gravitational sources, unlike charges, cannot
be screened. However, this instability occurs at momentum scales that are comparable with
the necessarily non-vanishing curvature R ∼ GT 4 . The flat-space graviton propagator is
therefore no longer the appropriate quantity to analyse; metric perturbations have to be studied
in a curved time-dependent background.