Chapter 4
Observer Space-Time Formalism
Abstract In this chapter we use the tetrad field to develop a new formalism
for describing cosmological observables from localized sources in an exact and
coordinate-independent way. We construct a new manifold, the observer spacetime, whose coordinates are the proper time, redshift and angles an observer uses
to parametrize observations. This manifold is not the actual space-time manifold on
which fields live, but is related to the latter through a map constructed out of all the
possible light-rays reaching the observer’s path. This map is therefore not necessarily
bijective and its image is the observable universe, by definition. All other observables can then be defined on this observer space-time manifold, thus yielding the
functional relations between observables that one measures in practice. In particular,
we derive the differential equations determining the image deformation and number
count observables.
We start by defining the fundamental observables and express the equation of motion
of photons in terms of them. The boundary value of the photon 4-momentum at the
observer, and in the ˆ
e a frame, provides the observed frequency and angle parametrization, and we fix the reparametrization symmetry of the geodesic map to the (log)redshift parametrization. We then successively define the observer sky, observer
space, their spectral generalizations and their mapping to the observer’s light-cone,
paying attention to the involved symmetries, and also discuss the issue of caustics. The construction is completed by introducing a Sachs basis along each photon
path and fixing its rotational freedom by matching it to the natural dyad on the
observer sky, thus obtaining the Sachs basis that is actually used to build sky maps
of tensorial quantities. With this we can then describe the Jacobi and volume maps
within this formalism to obtain the fundamental observables associated with localized
sources. Finally, we also consider the possibility of describing the drift of cosmological observables by discussing the extension of the observer space to the observer
space-time and the relation of the latter to the observational coordinates.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_4
51
Observer Space-Time Formalism
Abstract In this chapter we use the tetrad field to develop a new formalism
for describing cosmological observables from localized sources in an exact and
coordinate-independent way. We construct a new manifold, the observer spacetime, whose coordinates are the proper time, redshift and angles an observer uses
to parametrize observations. This manifold is not the actual space-time manifold on
which fields live, but is related to the latter through a map constructed out of all the
possible light-rays reaching the observer’s path. This map is therefore not necessarily
bijective and its image is the observable universe, by definition. All other observables can then be defined on this observer space-time manifold, thus yielding the
functional relations between observables that one measures in practice. In particular,
we derive the differential equations determining the image deformation and number
count observables.
We start by defining the fundamental observables and express the equation of motion
of photons in terms of them. The boundary value of the photon 4-momentum at the
observer, and in the ˆ
e a frame, provides the observed frequency and angle parametrization, and we fix the reparametrization symmetry of the geodesic map to the (log)redshift parametrization. We then successively define the observer sky, observer
space, their spectral generalizations and their mapping to the observer’s light-cone,
paying attention to the involved symmetries, and also discuss the issue of caustics. The construction is completed by introducing a Sachs basis along each photon
path and fixing its rotational freedom by matching it to the natural dyad on the
observer sky, thus obtaining the Sachs basis that is actually used to build sky maps
of tensorial quantities. With this we can then describe the Jacobi and volume maps
within this formalism to obtain the fundamental observables associated with localized
sources. Finally, we also consider the possibility of describing the drift of cosmological observables by discussing the extension of the observer space to the observer
space-time and the relation of the latter to the observational coordinates.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_4
51