4
1 Introduction and Summary
are physically relevant by using these transformations. This freedom comes on top
of the freedom to choose the coordinate system of the space-time manifold, which
is also intact thanks to the fact that the observer/source frame information is now
encoded in tensors: the four vectors of a tetrad.
The original and central content of this work is the use of the tetrad field to
develop a new formalism for defining and computing the aforementioned cosmological observable functions. We will focus in particular on the most important ones: the
angular diameter distance, weak lensing and galaxy number counts (associated with a
given source 4-velocity field) and the cosmic microwave background (CMB). Being
sky maps, all these observables will be fields on the “observer sky” S parametrized
by the two angles {ϑ, ϕ} that the actual observer uses in practice. All the considered
definitions and equations will be given at the fully non-linear level and without any
reference to some “background” (homogeneous and isotropic) space-time, so that
coordinate and model-independence are manifest. In the case of localized sources,
our formalism is based on the introduction of a new manifold, the “observer space”
C, which is parametrized by the fundamental observables {z, ϑ, ϕ} and has the topology of a 3-cylinder C R + × S. It is then mapped to the observer light-cone in the
space-time manifold M through the bundle of light-like geodesics emanating from
the observer position. All other observables are directly defined as functions on that
observer space C, thus achieving a fully coordinate and model-independent definition
of the relations between physical quantities indeed. In particular, caustics of light
rays now correspond to the map C → M being non-injective, not to singularities, so
our observable maps are definable and computable in the presence of strong lensing
as well. This absence of obstruction in resolving caustics is another important feature of our formalism and stands in contrast to the observational coordinate [69, 70]
and geodesic light-cone coordinate [22, 42, 66, 70–81] approaches for cosmological
observables. At the practical level, one no longer needs to compute redshift fluctuations and angular deflections with respect to some reference parametrization (e.g.
affine parameter and unlensed angles), since now {z, ϑ, ϕ} are the parameters with
respect to which our equations are defined. Thus, the operator controlling evolution
down the light-cone will be the derivative with respect to z, while the geodesic deviation operator leading to the Jacobi map will be the derivative with respect to {ϑ, ϕ}.
4
In order to also describe the drift of observables with respect to a given observer
world-line and transport of her frame, one simply considers a specific integral line
of the observer 4-velocity field and repeats the observer space construction at each
point. This therefore leads to the “observer space-time” O := R × C parametrized
by {τ , z, ϑ, ϕ}, where τ is the observer proper time. By construction, the image of O
in the true space-time M is then the observable universe of the observer under consideration. Finally, each space S, C or O can be promoted to its “spectral” analogue
by including the observed frequency ˆ
ω dimension. We also pay special attention to
the effect of local Lorentz transformations, i.e. the change of observer/source family,
4 Higher-order angular derivatives would then allow one to go beyond the infinitesimal beam approximation and thus consider finite shapes on the sky (see [82–86] for works on finite beams).
1 Introduction and Summary
are physically relevant by using these transformations. This freedom comes on top
of the freedom to choose the coordinate system of the space-time manifold, which
is also intact thanks to the fact that the observer/source frame information is now
encoded in tensors: the four vectors of a tetrad.
The original and central content of this work is the use of the tetrad field to
develop a new formalism for defining and computing the aforementioned cosmological observable functions. We will focus in particular on the most important ones: the
angular diameter distance, weak lensing and galaxy number counts (associated with a
given source 4-velocity field) and the cosmic microwave background (CMB). Being
sky maps, all these observables will be fields on the “observer sky” S parametrized
by the two angles {ϑ, ϕ} that the actual observer uses in practice. All the considered
definitions and equations will be given at the fully non-linear level and without any
reference to some “background” (homogeneous and isotropic) space-time, so that
coordinate and model-independence are manifest. In the case of localized sources,
our formalism is based on the introduction of a new manifold, the “observer space”
C, which is parametrized by the fundamental observables {z, ϑ, ϕ} and has the topology of a 3-cylinder C R + × S. It is then mapped to the observer light-cone in the
space-time manifold M through the bundle of light-like geodesics emanating from
the observer position. All other observables are directly defined as functions on that
observer space C, thus achieving a fully coordinate and model-independent definition
of the relations between physical quantities indeed. In particular, caustics of light
rays now correspond to the map C → M being non-injective, not to singularities, so
our observable maps are definable and computable in the presence of strong lensing
as well. This absence of obstruction in resolving caustics is another important feature of our formalism and stands in contrast to the observational coordinate [69, 70]
and geodesic light-cone coordinate [22, 42, 66, 70–81] approaches for cosmological
observables. At the practical level, one no longer needs to compute redshift fluctuations and angular deflections with respect to some reference parametrization (e.g.
affine parameter and unlensed angles), since now {z, ϑ, ϕ} are the parameters with
respect to which our equations are defined. Thus, the operator controlling evolution
down the light-cone will be the derivative with respect to z, while the geodesic deviation operator leading to the Jacobi map will be the derivative with respect to {ϑ, ϕ}.
4
In order to also describe the drift of observables with respect to a given observer
world-line and transport of her frame, one simply considers a specific integral line
of the observer 4-velocity field and repeats the observer space construction at each
point. This therefore leads to the “observer space-time” O := R × C parametrized
by {τ , z, ϑ, ϕ}, where τ is the observer proper time. By construction, the image of O
in the true space-time M is then the observable universe of the observer under consideration. Finally, each space S, C or O can be promoted to its “spectral” analogue
by including the observed frequency ˆ
ω dimension. We also pay special attention to
the effect of local Lorentz transformations, i.e. the change of observer/source family,
4 Higher-order angular derivatives would then allow one to go beyond the infinitesimal beam approximation and thus consider finite shapes on the sky (see [82–86] for works on finite beams).