1 Introduction and Summary
3
{ϑ, ϕ} therefore leads to an artificial “lensing” effect that has not been accounted for
in the quantitative studies of the literature yet.
2
At a more qualitative level, the necessity of introducing a full observer frame in
order to obtain the correct angular parametrization of observables is a recognized
fact [46–61]. Since the effect of this frame is to correct a parametrization of observables in the observer’s tangent space, its consideration leads to extra terms at the
observer position in the expressions for the cosmological observables, at linear order
in perturbation theory. These “observer terms” can therefore only affect the first few
multipoles of the corresponding angular spectra, so their effect is irrelevant in a multipole analysis at linear order, which is why they have been generically neglected in
the literature. However, they can no longer be ignored at the non-linear level, since
they couple with source and line-of-sight terms, thus affecting all multipoles.
3 Let
us also stress that, already at the linear level, these observer terms are clearly relevant conceptually, since they are necessary for the full expression of the observable
to be gauge-invariant and free of infrared divergences [50, 51, 54, 57, 58, 62–67].
Thus, in general, as one delves into the non-linear regime, it is important that the
observer frame is properly taken into account in order to avoid miscalculations and
misinterpretations.
From the viewpoint of the tetrad formulation of GR [68], where the metric information is contained and generalized in a tetrad field, the introduction of a tetrad at a
single (observer) point begs for a generalization to all the points of the manifold. Here
we will therefore reconsider the issue of cosmological observables with the tetrad
description of space-time as our starting point. This leads to several conceptual and
practical advantages at a remarkably negligible price: the introduction of three extra
non-dynamical fields. For instance, the tetrads at other points than the observer one
can now be interpreted as the frames of sources, thus unifying all reference frames
involved in cosmological observables in a single space-time field. The extra six components that the tetrad field has with respect to the metric can therefore be interpreted
as the information of 4-velocities and rest-frame orientations of an observer/source
family. Since one already considers velocity fields in cosmology, the truly new information one has to keep track of here are the three fields determining the orientation
of the spatial frames. Moreover, with this viewpoint the local Lorentz symmetry of
the tetrad formalism is now interpreted as frame transformations, thus allowing us to
access all the possible observer/source families. One can therefore select frames that
are convenient for performing computations and then have access to the ones that
2 To avoid confusion for the reader who is specialized in the field, let us stress that the usual
introduction of a Sachs basis does not resolve the issue we are pointing out here. An orthonormal
basis in the tangent space of a given point on the sky manifold allows one to obtain the observed
components of tensors at that point. It does not, however, provide the global observer parametrization
of the sky manifold itself {ϑ, ϕ}, which is necessary when computing angular correlation functions
or spectra. In particular, note that the parametrization {ϑ, ϕ} induces a privileged Sachs basis to each
point on the sky {∂ ϑ , ∂ ϕ / sin ϑ}, thus forming the two sky vector fields that the observer implicitly
uses in practice for decomposing tensors on the sky.
3 See [61] for a detailed discussion of this issue and, in particular, the impact on the statistics of
observables.
3
{ϑ, ϕ} therefore leads to an artificial “lensing” effect that has not been accounted for
in the quantitative studies of the literature yet.
2
At a more qualitative level, the necessity of introducing a full observer frame in
order to obtain the correct angular parametrization of observables is a recognized
fact [46–61]. Since the effect of this frame is to correct a parametrization of observables in the observer’s tangent space, its consideration leads to extra terms at the
observer position in the expressions for the cosmological observables, at linear order
in perturbation theory. These “observer terms” can therefore only affect the first few
multipoles of the corresponding angular spectra, so their effect is irrelevant in a multipole analysis at linear order, which is why they have been generically neglected in
the literature. However, they can no longer be ignored at the non-linear level, since
they couple with source and line-of-sight terms, thus affecting all multipoles.
3 Let
us also stress that, already at the linear level, these observer terms are clearly relevant conceptually, since they are necessary for the full expression of the observable
to be gauge-invariant and free of infrared divergences [50, 51, 54, 57, 58, 62–67].
Thus, in general, as one delves into the non-linear regime, it is important that the
observer frame is properly taken into account in order to avoid miscalculations and
misinterpretations.
From the viewpoint of the tetrad formulation of GR [68], where the metric information is contained and generalized in a tetrad field, the introduction of a tetrad at a
single (observer) point begs for a generalization to all the points of the manifold. Here
we will therefore reconsider the issue of cosmological observables with the tetrad
description of space-time as our starting point. This leads to several conceptual and
practical advantages at a remarkably negligible price: the introduction of three extra
non-dynamical fields. For instance, the tetrads at other points than the observer one
can now be interpreted as the frames of sources, thus unifying all reference frames
involved in cosmological observables in a single space-time field. The extra six components that the tetrad field has with respect to the metric can therefore be interpreted
as the information of 4-velocities and rest-frame orientations of an observer/source
family. Since one already considers velocity fields in cosmology, the truly new information one has to keep track of here are the three fields determining the orientation
of the spatial frames. Moreover, with this viewpoint the local Lorentz symmetry of
the tetrad formalism is now interpreted as frame transformations, thus allowing us to
access all the possible observer/source families. One can therefore select frames that
are convenient for performing computations and then have access to the ones that
2 To avoid confusion for the reader who is specialized in the field, let us stress that the usual
introduction of a Sachs basis does not resolve the issue we are pointing out here. An orthonormal
basis in the tangent space of a given point on the sky manifold allows one to obtain the observed
components of tensors at that point. It does not, however, provide the global observer parametrization
of the sky manifold itself {ϑ, ϕ}, which is necessary when computing angular correlation functions
or spectra. In particular, note that the parametrization {ϑ, ϕ} induces a privileged Sachs basis to each
point on the sky {∂ ϑ , ∂ ϕ / sin ϑ}, thus forming the two sky vector fields that the observer implicitly
uses in practice for decomposing tensors on the sky.
3 See [61] for a detailed discussion of this issue and, in particular, the impact on the statistics of
observables.