1 Introduction and Summary
5
on the spectral observer space. In particular, this induces a non-trivial coordinate
transformation
{z, ϑ, ϕ, ˆ
ω} → {˜ z, ˜
ϑ, ˜
ϕ, ˜ ˆ
ω} ,
(1.0.3)
so that this space is actually endowed with an atlas. The different observable functions
associated with each chart correspond to the observations of all possible observers
and all possible sources in a given space-time geometry.
In the case of the CMB, which is a diffuse source, the corresponding observables
are obtained by evaluating the photon phase space distribution at the observer position
and pulling-back its tangent space dependence on the spectral observer sky S spec
S × R + . If one is only interested in relating these observables to the corresponding
“source”, which would be the last scattering surface, then one can use the geometrical
optics approach (eikonal approximation), whereby the photon intensity tensor is
parallely-transported from that surface along light-like geodesics up to the observer
point. For this task, one can again use the observer space-time formalism described
above, now applied to the case of a continuous collection of sources situated at
z = z last. scat. (ϑ, ϕ). However, this is only an approximative approach, neglecting
for instance non-gravitational interactions, or the fact that the photon decoupling
process is not instantaneous. In order to take into account all possible effects, one
needs to consider the full dynamics of the photon phase space distribution, and of
the ones it couples to, which is required anyways for the cosmological evolution of
matter in general. The second important part of this work therefore contains a detailed
description of matrix kinetic theory on curved space-time.
5 As also recognized in [52,
87–92], the tetrad formalism appears as the natural language for this task, especially
for relating the particle phase space distribution to the underlying quantum field
theory (QFT) quantities, which is required in order to obtain the collision term of
the Boltzmann equation for matrix distributions. We revisit the construction of this
theory, including in particular detailed discussions over subtle issues that are not
addressed in the literature, and also deviate from the latter in our definition of the
collision term, which leads to different results at higher order corrections in the
coupling constants. Finally, another original output of our treatment is that, since
we have access to the spin polarization of fermionic matter through the associated
matrix distribution, we can compute the intrinsic magnetization of the corresponding
fluid. In particular, this leads to an extra magnetic moment that sources the cosmic
magnetic fields and might therefore be relevant in studying their generation.
In summary, in this work we propose an ab initio derivation of the essential
equations regarding cosmological observables employing only exact definitions and
relations. In order to achieve this without specifying a coordinate system, one needs
to introduce tetrads, which is the core concept of our formalism and its distinguishing
feature with respect to the standard practice. The content is organized as follows. In
Chap. 2 we motivate the use of the tetrad formalism of GR in cosmology. In Chap. 3
5 Here by “matrix” is meant the fact that the phase space distributions take into account the possibility
of quantum superposition of particle polarizations and are therefore hermitian matrix functions of
phase space, instead of scalars.
5
on the spectral observer space. In particular, this induces a non-trivial coordinate
transformation
{z, ϑ, ϕ, ˆ
ω} → {˜ z, ˜
ϑ, ˜
ϕ, ˜ ˆ
ω} ,
(1.0.3)
so that this space is actually endowed with an atlas. The different observable functions
associated with each chart correspond to the observations of all possible observers
and all possible sources in a given space-time geometry.
In the case of the CMB, which is a diffuse source, the corresponding observables
are obtained by evaluating the photon phase space distribution at the observer position
and pulling-back its tangent space dependence on the spectral observer sky S spec
S × R + . If one is only interested in relating these observables to the corresponding
“source”, which would be the last scattering surface, then one can use the geometrical
optics approach (eikonal approximation), whereby the photon intensity tensor is
parallely-transported from that surface along light-like geodesics up to the observer
point. For this task, one can again use the observer space-time formalism described
above, now applied to the case of a continuous collection of sources situated at
z = z last. scat. (ϑ, ϕ). However, this is only an approximative approach, neglecting
for instance non-gravitational interactions, or the fact that the photon decoupling
process is not instantaneous. In order to take into account all possible effects, one
needs to consider the full dynamics of the photon phase space distribution, and of
the ones it couples to, which is required anyways for the cosmological evolution of
matter in general. The second important part of this work therefore contains a detailed
description of matrix kinetic theory on curved space-time.
5 As also recognized in [52,
87–92], the tetrad formalism appears as the natural language for this task, especially
for relating the particle phase space distribution to the underlying quantum field
theory (QFT) quantities, which is required in order to obtain the collision term of
the Boltzmann equation for matrix distributions. We revisit the construction of this
theory, including in particular detailed discussions over subtle issues that are not
addressed in the literature, and also deviate from the latter in our definition of the
collision term, which leads to different results at higher order corrections in the
coupling constants. Finally, another original output of our treatment is that, since
we have access to the spin polarization of fermionic matter through the associated
matrix distribution, we can compute the intrinsic magnetization of the corresponding
fluid. In particular, this leads to an extra magnetic moment that sources the cosmic
magnetic fields and might therefore be relevant in studying their generation.
In summary, in this work we propose an ab initio derivation of the essential
equations regarding cosmological observables employing only exact definitions and
relations. In order to achieve this without specifying a coordinate system, one needs
to introduce tetrads, which is the core concept of our formalism and its distinguishing
feature with respect to the standard practice. The content is organized as follows. In
Chap. 2 we motivate the use of the tetrad formalism of GR in cosmology. In Chap. 3
5 Here by “matrix” is meant the fact that the phase space distributions take into account the possibility
of quantum superposition of particle polarizations and are therefore hermitian matrix functions of
phase space, instead of scalars.