16
2 Motivation
of gauge amounts to a choice of observer family, and in particular its dynamics.
For instance, we will see that we can choose a gauge where all observers are
in free-fall and their spatial frames are parallely transported along e 0 . Although
this is certainly a valid assumption at large enough scales, we will nevertheless
choose to work in a generic gauge in order to maintain generality. In particular,
we will be able to extend the diffeomorphism-invariant combinations of the metric
components in linear perturbation theory, the “Bardeen variables”, to the ones that
are invariant under both symmetries. Finally, the more the local symmetries of
our equations, the more ways we have to perform consistency checks, a tool that
becomes increasingly relevant given the complexity of higher-order perturbation
theory.
• Symmetry-based definition of cosmological observables from localized sources.
Just as a tensor is defined by its transformation properties, so can we now define a
cosmological observable associated with some localized source through its transformation properties under the present symmetries. Denoting by P the position of
the source, which is linked by a light-like geodesic to ˆ
P, an associated cosmological observable C(P, ˆ
P) is a function of P and ˆ
P, with the following properties:
– Under a coordinate transformation the observable transforms as a bi-scalar. This
means that C(P, ˆ
P) is invariant, because it is expressed as a function of points.
If we express it as a function of the coordinates of these points we rather have
˜
C( ˜
x, ˜ ˆ
x) = C(x, ˆ
x) .
(2.2.6)
A measurement cannot depend on the parametrization of space-time.
– Under a LLT, the variation of C(P, ˆ
P) depends only on
a
b (P) and
a
b ( ˆ
P),
because the only observers involved in the process are e a (P) and e a ( ˆ
P). The
geodesic path between them solely depends on the metric information g μν (x),
not on the choice of intermediate observers e
μ
a (x). This will not always be
explicit in our formalism, because we will have e
a
μ (x) appearing in the observables all along the line of sight, but its presence will be such that only
a
b (P)
and
a
b ( ˆ
P) will end up appearing under an LLT.
The fact that e 0 (P) is identified with a physical quantity, that is the source’s 4velocity, means that a boost at P changes the source to the one with a different
4-velocity ˜
e 0 (P), thus also modifying the corresponding observables. In the
case where the source has some non-uniform shape, one can also associate to
it a privileged spatial basis e i (P), say by picking three reference points that
are distinguished by the shape. Then, performing a local rotation at P would
mean that we change the source to one that is directed differently ˜
e i (P), thus
modifying again the corresponding observables. We therefore conclude that, in
the absence of such a privileged way of associating some e i (P) to the source,
the corresponding observables should also be invariant under local rotations at
P. Indeed, if they are not, it would mean that their value is ambiguous, as it
depends on a choice of e i (P) that has no physical interpretation.
2 Motivation
of gauge amounts to a choice of observer family, and in particular its dynamics.
For instance, we will see that we can choose a gauge where all observers are
in free-fall and their spatial frames are parallely transported along e 0 . Although
this is certainly a valid assumption at large enough scales, we will nevertheless
choose to work in a generic gauge in order to maintain generality. In particular,
we will be able to extend the diffeomorphism-invariant combinations of the metric
components in linear perturbation theory, the “Bardeen variables”, to the ones that
are invariant under both symmetries. Finally, the more the local symmetries of
our equations, the more ways we have to perform consistency checks, a tool that
becomes increasingly relevant given the complexity of higher-order perturbation
theory.
• Symmetry-based definition of cosmological observables from localized sources.
Just as a tensor is defined by its transformation properties, so can we now define a
cosmological observable associated with some localized source through its transformation properties under the present symmetries. Denoting by P the position of
the source, which is linked by a light-like geodesic to ˆ
P, an associated cosmological observable C(P, ˆ
P) is a function of P and ˆ
P, with the following properties:
– Under a coordinate transformation the observable transforms as a bi-scalar. This
means that C(P, ˆ
P) is invariant, because it is expressed as a function of points.
If we express it as a function of the coordinates of these points we rather have
˜
C( ˜
x, ˜ ˆ
x) = C(x, ˆ
x) .
(2.2.6)
A measurement cannot depend on the parametrization of space-time.
– Under a LLT, the variation of C(P, ˆ
P) depends only on
a
b (P) and
a
b ( ˆ
P),
because the only observers involved in the process are e a (P) and e a ( ˆ
P). The
geodesic path between them solely depends on the metric information g μν (x),
not on the choice of intermediate observers e
μ
a (x). This will not always be
explicit in our formalism, because we will have e
a
μ (x) appearing in the observables all along the line of sight, but its presence will be such that only
a
b (P)
and
a
b ( ˆ
P) will end up appearing under an LLT.
The fact that e 0 (P) is identified with a physical quantity, that is the source’s 4velocity, means that a boost at P changes the source to the one with a different
4-velocity ˜
e 0 (P), thus also modifying the corresponding observables. In the
case where the source has some non-uniform shape, one can also associate to
it a privileged spatial basis e i (P), say by picking three reference points that
are distinguished by the shape. Then, performing a local rotation at P would
mean that we change the source to one that is directed differently ˜
e i (P), thus
modifying again the corresponding observables. We therefore conclude that, in
the absence of such a privileged way of associating some e i (P) to the source,
the corresponding observables should also be invariant under local rotations at
P. Indeed, if they are not, it would mean that their value is ambiguous, as it
depends on a choice of e i (P) that has no physical interpretation.