84
5 General-Relativistic Matrix Kinetic Theory
is what we have already implicitly assumed in the construction of f s (x,
p) and the
Liouville equation, since these are defined on the mass shells E p,s =
m 2
s + +
p 2 of
the free particles. Thus, f s (x,
p) typically varies in x
μ over scales ∼ L free and obeys
an equation of the form
L f s (x,
p) = C s (x,
p) ,
(5.1.1)
where C s is the “collision term” of the s species capturing deviations from free
motion.
In the opposite case of a “dense” gas L free ∼ L coll , the interactions are an integral
part of the dynamics and therefore alter its description qualitatively. In particular,
the degrees of freedom are no longer the ones of free particles, but rather collective
excitations, whose precise structure is in general hard to obtain. More precisely,
the spectral distribution of the system is not of the form ∼ δ
(4)
( p a p
a
+ m
2
), as we
have used until now (implicitly or explicitly), but rather a generic function of the
4-momentum norm A( p
2
). Moreover, since free motion is no longer the typical
behavior of the particles, we cannot simply replace δ
(4)
( p
2
+ m
2
) → A( p
2
) in our
equations at the level of the 8-dimensional phase space LM, i.e. we cannot treat the
gas as a collection of particle species with a continuous mass spectrum.
1 Thus, the
case of dense gases, such as in the very early universe, cannot be modeled using some
f s (x,
p) and Eq. (5.1.1), i.e. as free motion that is perturbed by sporadic collisions.
Instead, one must consider a more fundamental non-equilibrium QFT description
[1–3].
Here we therefore focus on the case of “dilute” gases, which is a valid assumption
in cosmology way after the reheating era. We will refer to L free and L coll as the
“macroscopic” and “microscopic” scales, respectively. In the absence of unstable
particles, the collision term is dominated by 2 ↔ 2 scattering and is given by the
“Boltzmann-Uehling-Uhlenbeck equation” (BUU)
C s (x,
p) =
1
2
d 3 p 1
(2π) 3 2E p1,1
d 3 p 2
(2π) 3 2E p2,2
d 3 p 3
(2π) 3 2E p3,3
(2π)
4 δ
(4) ( p 1 + p 2 − p 3 − p s ) (5.1.2)
× |A|
2 (
p 1 ,
p 2 → →
p 3 ,
p) [ f 1 f 2 (1 ± f 3 ) (1 ± f s ) − (1 ± f 1 ) (1 ± f 2 ) f 3 f s ] ,
where f k := f k (x,
p k ). Here A is the “matrix element”, or “amplitude”, associated
with the scattering event
2 and it is related to the S-matrix of the QFT through
p 3 ,
p|S − I| |
p 1 ,
p 2 ≡ (2π)
4
δ
(4)
( p 1 + p 2 − p 3 − p s ) iA(
p 1 ,
p 2 → →
p 3 ,
p) .
(5.1.3)
1 There is an exception to this conclusion, i.e. there are cases where interactions are frequent but
where the dilute gas machinery can still be applied. This occurs when A( p 2 ) exhibits sharp enough
maxima around some p 2 = −m 2 value, in which case the degrees of freedom are effective particles
(“quasi-particles”) with effective mass m. More specifically, we need the width of A( p 2 ) around
−m 2 to be small compared to both L
−2
coll and L
−2
free .
2 The matrix element is usually denoted by “M”, but here this already denotes the space-time
manifold.
5 General-Relativistic Matrix Kinetic Theory
is what we have already implicitly assumed in the construction of f s (x,
p) and the
Liouville equation, since these are defined on the mass shells E p,s =
m 2
s + +
p 2 of
the free particles. Thus, f s (x,
p) typically varies in x
μ over scales ∼ L free and obeys
an equation of the form
L f s (x,
p) = C s (x,
p) ,
(5.1.1)
where C s is the “collision term” of the s species capturing deviations from free
motion.
In the opposite case of a “dense” gas L free ∼ L coll , the interactions are an integral
part of the dynamics and therefore alter its description qualitatively. In particular,
the degrees of freedom are no longer the ones of free particles, but rather collective
excitations, whose precise structure is in general hard to obtain. More precisely,
the spectral distribution of the system is not of the form ∼ δ
(4)
( p a p
a
+ m
2
), as we
have used until now (implicitly or explicitly), but rather a generic function of the
4-momentum norm A( p
2
). Moreover, since free motion is no longer the typical
behavior of the particles, we cannot simply replace δ
(4)
( p
2
+ m
2
) → A( p
2
) in our
equations at the level of the 8-dimensional phase space LM, i.e. we cannot treat the
gas as a collection of particle species with a continuous mass spectrum.
1 Thus, the
case of dense gases, such as in the very early universe, cannot be modeled using some
f s (x,
p) and Eq. (5.1.1), i.e. as free motion that is perturbed by sporadic collisions.
Instead, one must consider a more fundamental non-equilibrium QFT description
[1–3].
Here we therefore focus on the case of “dilute” gases, which is a valid assumption
in cosmology way after the reheating era. We will refer to L free and L coll as the
“macroscopic” and “microscopic” scales, respectively. In the absence of unstable
particles, the collision term is dominated by 2 ↔ 2 scattering and is given by the
“Boltzmann-Uehling-Uhlenbeck equation” (BUU)
C s (x,
p) =
1
2
d 3 p 1
(2π) 3 2E p1,1
d 3 p 2
(2π) 3 2E p2,2
d 3 p 3
(2π) 3 2E p3,3
(2π)
4 δ
(4) ( p 1 + p 2 − p 3 − p s ) (5.1.2)
× |A|
2 (
p 1 ,
p 2 → →
p 3 ,
p) [ f 1 f 2 (1 ± f 3 ) (1 ± f s ) − (1 ± f 1 ) (1 ± f 2 ) f 3 f s ] ,
where f k := f k (x,
p k ). Here A is the “matrix element”, or “amplitude”, associated
with the scattering event
2 and it is related to the S-matrix of the QFT through
p 3 ,
p|S − I| |
p 1 ,
p 2 ≡ (2π)
4
δ
(4)
( p 1 + p 2 − p 3 − p s ) iA(
p 1 ,
p 2 → →
p 3 ,
p) .
(5.1.3)
1 There is an exception to this conclusion, i.e. there are cases where interactions are frequent but
where the dilute gas machinery can still be applied. This occurs when A( p 2 ) exhibits sharp enough
maxima around some p 2 = −m 2 value, in which case the degrees of freedom are effective particles
(“quasi-particles”) with effective mass m. More specifically, we need the width of A( p 2 ) around
−m 2 to be small compared to both L
−2
coll and L
−2
free .
2 The matrix element is usually denoted by “M”, but here this already denotes the space-time
manifold.