22
3 Mathematical Framework
light, weakly interacting and bosonic.
2 Gravity survives because it is a long-range
force and also universally attractive, i.e. the field is excitable at all scales and there
are no opposite charges that could cancel out the effect. As for electromagnetism, it is
also long-range, but the universe is electrically neutral on average, so the long wavelength electric field modes are suppressed. Nevertheless, we do infer the existence of
magnetic fields on cosmological scales through observations (see [1] and references
therein), the origin of which is still a subject of speculation to this date, which is why
we include the electromagnetic space-time field.
On the other hand, we also have particle-like excitations which are very well
localized with respect to cosmological scales. If the corresponding particle fluids are
dilute enough, their dynamics can be described statistically through 1-particle phase
space distributions f (x, p) on T
∗
M. As in the case of the space-time fields, the x
μ -
dependence captures only long wave-lengths, i.e. the distribution fluctuates mildly
in space-time, while the “short” length scales associated with the individual particle
dynamics are captured by the momentum dependence p μ . For instance, the typical
wave-length of a CMB photon today is of the order of the millimeter. Note also that,
although such photons are produced abundantly, this is not the case for short wavelength gravitons, because their coupling is so weak that their production requires
energy densities that are not resolved at cosmological scales. Thus, in cosmology all
gravitational effects can be taken into account through the classical space-time field
alone.
The bottom-line is that we will take into account both “manifestations” of electromagnetism, i.e. the phase space distribution of photons as well as the coherent
long wave-length space-time field, the two being effectively treated as different noninteracting components of the universe. This is consistent because photons have no
self-interactions. As for the world-line fields, we will only focus on the light-like
geodesics that connect a localized source to the observer, along with related objects.
We choose to describe them in field-theoretical language as well, because this will
make transparent a lot of manipulations and facilitate the relation to the other two
manifolds.
Finally, since respecting symmetries is a central aspect of this work, in Appendix
6.2 we propose for the interested reader a comprehensive discussion about the diffeomorphism symmetry and, in particular, the distinction between its “active” (pullbacks) and “passive” (coordinate transformations) versions. The difference is purely
conceptual at the level of local equations, but has practical implications when integrals are involved. This is the case here since cosmological observables are defined
through integrals over L. In the appendix, however, we choose to illustrate our discussion with the space-time manifold M instead, in order to avoid specificities of
the 1-dimensional case. For the uninterested reader, we directly mention our terminology. We will use the acronyms PMD (resp. AMD) for the passive (resp. active)
space-time diffeomorphisms, PLD (resp. ALD) for the passive (resp. active) world2 Fermionic fields can only give rise to elementary quantum excitations (particles) and bound states
thereof. Indeed, because of Pauli’s exclusion principle, the large occupation number configurations
that are required in order to reach classical behavior do not exist.
3 Mathematical Framework
light, weakly interacting and bosonic.
2 Gravity survives because it is a long-range
force and also universally attractive, i.e. the field is excitable at all scales and there
are no opposite charges that could cancel out the effect. As for electromagnetism, it is
also long-range, but the universe is electrically neutral on average, so the long wavelength electric field modes are suppressed. Nevertheless, we do infer the existence of
magnetic fields on cosmological scales through observations (see [1] and references
therein), the origin of which is still a subject of speculation to this date, which is why
we include the electromagnetic space-time field.
On the other hand, we also have particle-like excitations which are very well
localized with respect to cosmological scales. If the corresponding particle fluids are
dilute enough, their dynamics can be described statistically through 1-particle phase
space distributions f (x, p) on T
∗
M. As in the case of the space-time fields, the x
μ -
dependence captures only long wave-lengths, i.e. the distribution fluctuates mildly
in space-time, while the “short” length scales associated with the individual particle
dynamics are captured by the momentum dependence p μ . For instance, the typical
wave-length of a CMB photon today is of the order of the millimeter. Note also that,
although such photons are produced abundantly, this is not the case for short wavelength gravitons, because their coupling is so weak that their production requires
energy densities that are not resolved at cosmological scales. Thus, in cosmology all
gravitational effects can be taken into account through the classical space-time field
alone.
The bottom-line is that we will take into account both “manifestations” of electromagnetism, i.e. the phase space distribution of photons as well as the coherent
long wave-length space-time field, the two being effectively treated as different noninteracting components of the universe. This is consistent because photons have no
self-interactions. As for the world-line fields, we will only focus on the light-like
geodesics that connect a localized source to the observer, along with related objects.
We choose to describe them in field-theoretical language as well, because this will
make transparent a lot of manipulations and facilitate the relation to the other two
manifolds.
Finally, since respecting symmetries is a central aspect of this work, in Appendix
6.2 we propose for the interested reader a comprehensive discussion about the diffeomorphism symmetry and, in particular, the distinction between its “active” (pullbacks) and “passive” (coordinate transformations) versions. The difference is purely
conceptual at the level of local equations, but has practical implications when integrals are involved. This is the case here since cosmological observables are defined
through integrals over L. In the appendix, however, we choose to illustrate our discussion with the space-time manifold M instead, in order to avoid specificities of
the 1-dimensional case. For the uninterested reader, we directly mention our terminology. We will use the acronyms PMD (resp. AMD) for the passive (resp. active)
space-time diffeomorphisms, PLD (resp. ALD) for the passive (resp. active) world2 Fermionic fields can only give rise to elementary quantum excitations (particles) and bound states
thereof. Indeed, because of Pauli’s exclusion principle, the large occupation number configurations
that are required in order to reach classical behavior do not exist.