Nearby Supernova Factory: Étalonnage des données de SNIFS et ...

tel-00372504, version 1 - 1 Apr 2009
CHAPTER 1. BIG BANG COSMOLOGY
Imposing these symmetries 2 to the universe at large scale, we can thus study its evolution.
1.2 FLRW models in general relativity
1.2.1 Robertson-Walker (RW) metric
Under the assumption of the cosmological principle, our spacetime metric can be written in
terms of an invariant four dimensions geodesic metric gµν which follows
ds 2 = gµνdx µ dx ν , (1.1)
that is, relates length measurements ds to a chosen coordinate system dx (and thus is an intrinsic
description of space), as the so-called Robertson-Walker metric
ds 2 = dt 2 − a 2
dr
(t)
2
(1 − kr2 ) + r2 dθ 2 + sin 2 θdφ 2
. (1.2)
We are assuming the speed of light is equal to unity (c = 1) and: t is the proper time, r, θ, φ
are comoving (spherical) spatial coordinates, a(t) is the scale factor (normalized by a(t0) ≡ 1) 3 ,
and the constant k characterizes the spatial curvature of the universe. k = {−1, 0, 1} correspond
respectively to open (saddle-like), spatially flat and closed (3-sphere) universes.
1.2.2 Einstein equations
The cosmological equations of motion are derived from the Einstein (1917) field equations
Rµν − 1
2 gµνR = 8πGTµν − Λgµν , (1.3)
where G is Newton’s gravitational constant, and which relate the curvature, represented by the
Ricci tensor Rµν to the energy-momentum tensor Tµν 4 . The Λgµν term is the cosmological
constant (Λ) term and was added by Einstein since he was interested in finding static solutions,
both due to his personal beliefs and in accordance to the astronomical data available at the
time, but which his initial formulation did not allowed. This effort however was unsuccessful:
the static universe described by this theory was unstable, and both de Sitter’s solution of (1.3)
for empty universes with Λ > 0, and measurements of the radial velocities of distant galaxies
using the redshift (§ 1.3.1) of their spectra by Slipher (1924), showed that our universe is in fact
not static, which lead Einstein to abandon the cosmological constant.
Despite Einstein’s misguided motivation for introducing the cosmological constant term,
there is nothing wrong (i.e. inconsistent) with the presence of such a term in the equations. We
will see in § 1.2.4 what this term may represent in modern cosmology.
2
The homogeneity and isotropy are symmetries of space and not of spacetime. Homogeneous, isotropic spacetimes
have a family of preferred three-dimensional spatial slices on which the three-dimensional geometry is
homogeneous and isotropic.
3
Throughout this document, this normalization will be implicit and the 0 subscript will always represent the
present epoch.
4
Or in other words, matter and energy are the source of the curvature of spacetime.
8