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Cosmology and the FRW model

1. Observations on our universe

The first step is to consider the general appearance of our universe. Up until fairly recently, the overall structure of our universe was fairly hard to guess at, but with the observations of the last 50 years, we can say that, within observable distances, the universe is approximatively isotropic around us (the distribution of matter around us is the same in all directions) and homogeneous (the distribution of matter is the same everywhere). We'll define more specifically what homogeneity, isotropy, and approximatively mean in this context later on.

There are two main elements contributing to those results : the cosmic microwave background and the search of structures at large scales.

There is a slight problem in using those observations to determine the spacetime metric, in that the measuring of distances at such scales requires prior knowledge of the spacetime metric.This leads to the issue that there could possibly be a metric that better describes our universe, but by our assumption, we cannot directly check if this is true. There are a few other possible models of the universe, but as they are less symmetric, they will usually require more parameters, and as far as we can tell, those parameters are usually close to being the limit in which we're back to the homogeneous model. We will see later on that the assumptions of homogeneity and isotropy are at least consistent with the observations we get.

1.1. The cosmological distance ladder

One of the basic tool used in cosmology is the measure of distances of celestial bodies. There's a wide variety of them, each of them appropriate at a certain range of scales, and each of them with their own assumptions, especially regarding the local structure of spacetime.

Parallax measurements

Parallax measurements are one of the oldest method to measure distances of celestial bodies,

1.2. The cosmic microwave background

The cosmic microwave background (or CMB) is a seemingly sourceless radiation that appears in all directions, and is fairly isotropic. That radiation has an isotropic measured temperature of $2.72548\ \textrm{K}$, within an uncertainty of $0.00057\ \textrm{K}$.

1.2. Lack of large scale structures

A variety of surveys of astronomical objects in the universe have shown the kind of structures we can expect from it.

The Two-degree-Field Galaxy Redshift Survey (or 2dFGRS) was one of the most comprehensive survey of far-off astronomical objects, spanning about $250,000$ galaxies, spanning an angle of about $80^\circ \times 15^\circ$ towards the southern galactic hemisphere and $75^\circ \times 10^\circ$ towards the northern galactic hemisphere, for a median depth of about $\bar z = 0.11$.

2. Homogeneity and isotropy

From those observations, a simple model of our universe will be a homogeneous and isotropic one. First, we need to discuss what these will mean in general relativity.

2.1. Homogeneity of a manifold

Homogeneity roughly means that space is the same if we move from one point to another. As the notion of two spaces being the space rely on isometries, this will mean some isometry between any two points, which brings us the following definition :

A Riemannian manifold $\Sigma$ is said to be homogeneous if the group of isometries on $\Sigma$ acts transitively on it, so that given $p, q \in \Sigma$, there exists an isometry $\phi$ such that $\phi(p) = q$. This is a pretty strong condition on the topology and geometry allowed, and given a dimension, there are only a handful of such manifolds to exist.

Theorem : If $\Sigma$ is a homogeneous Riemannian manifold, it is geodesically complete.

Proof : Suppose there exists a unit speed geodesic $\gamma : [a, 1) \to \Sigma$ that is not extendible to $\lambda = 1$. Take a point $p \in \Sigma$ and a length $\alpha > 0$ such that any unit speed geodesic starting at $p$ has a length $\ell \geq \alpha$, and consider the quantity $\delta = \min(\alpha / 2, (1-\alpha) / 2)$. Since isometries preserve geodesics,

Killing vector field

Theorem : If a manifold is homogeneous, then there exists a set of $n$ Killing vector fields.

Proof :

A slightly stronger condition than homogeneity is two-point homogeneity, which not only induces an isometry between any two points points, but also preserves distances

It is two-point homogeneous if given four points $p_1, p_2, q_1, q_2 \in \Sigma$ with $d(p_1, q_1) = d(p_2, q_2)$, there is an isometry $\phi$ such that $\phi(p_1) = q_1$ and $\phi(p_2) = q_2$.

2.2. Isotropy

A manifold is called isotropic at $p$ if, for two vectors $v, w \in T_p M$, such that $|v| = |w|$, there exists an isometry of the manifold $\phi \in I(M)$ such that

$$\phi_* (v) = w$$

For a Riemannian manifold in particular, this can be restated as the fact that the unit sphere of the tangent space at $p$, noted $S_p M$ (this is the set of all vectors of unit norm) obeys this property.

Locally isotropic manifold

2.3. Topologies of isotropic homogeneous manifolds

The classification of all two-point homogeneous manifolds has been performed by Wolf. They consist of :

  1. Euclidian space : $\mathbb{R}^n = \mathrm{E}(n) / \mathrm{O}(n)$
  2. The $n$-sphere : $S^n = \mathrm{SO}(n+1) / \mathrm{SO}(n)$
  3. The real projective space : $\mathbb{R}\mathrm{P}^n = \mathrm{SO}(n+1) / \mathrm{O}(n)$
  4. The complex projective space : $\mathbb{C}\mathrm{P}^n = \mathrm{SU}(n+1) / \mathrm{SU}(n)$
  5. The quaternionic projective space : $\mathbb{H}\mathrm{P}^n = \mathrm{Sp}(n+1) / \mathrm{Sp}(n) \times \mathrm{Sp}(1)$
  6. ...

In three dimensions, we are limited to the Euclidan space $\mathbb{R}^3$, the hyperbolic space $\mathbb{H}^3$, the $3$-sphere $S^3$ and the elliptic space $\mathbb{R}\mathrm{P}^3$, meaning that a two-point homogeneous universe can only have three possible topologies : $\mathbb{R}^3$ (Euclidian and hyperbolic space), $S^3$ or $\mathbb{R}\mathrm{P}^3$. The elliptic space is generally not considered, as its lack of orientability is strongly hinted to be impossible by particle experiments.

The metrics associated with those spaces are the following :

2.4. Tensor quantities

2. The FRW model

The simplest model that would roughly fit the observed data is to consider a spacetime with a spacelike hypersurface that is both homogeneous and isotropic. That is, it contains $3$ spacelike Killing vectors generating translations, as well as $3$ Killing vectors generating rotations. It's possible to weaken it, for instance by only taking homogeneity (which generates the Bianchi universes).

This means that the spacetime is foliated by spacelike hypersurfaces which are maximally symmetric (it will be one of the three classes of such $3$-manifolds).

$$ds^2 = -dt^2 + a(t) h$$

with $h$ a homogeneous isotropic $3$-metric. As is well known, maximally symmetric spaces have a coordinate patch in which the Riemann tensor obeys the relation

$$R_{\mu\nu\sigma\tau} = \frac{R}{n(n-1)} (g_{\mu\sigma} g_{\nu\tau} - g_{\mu\tau}g_{\nu\sigma}) $$

The topologies of spacetime

To make sense of the requirement of spatial homogeneity and isotropy, the FRW spacetime needs to be foliated by spacelike hypersurfaces. This will reduce the possible topologies to fiber bundles with typical fiber $\Sigma$, the spacelike hypersurface, over a one-dimensional base space , either $\mathbb{R}$ or $S^1$. As there is only the trivial fiber bundle over $\mathbb{R}$, this means that the only possible topologies will be $\mathbb{R} \times \Sigma$, $S^1 \times \Sigma$, and possibly some non-trivial bundle $\pi : \mathcal{M} \to S^1$. If the base space is $\mathbb{R}$, it is said to have an open time structure, while the base space $S$ is said to have a closed time structure. As a closed time structure leads to an acausal spacetime, as well as not fitting cosmological observations (the function $a$ does not seem to be periodic), the open time structure is the one generally studied.

The topology of $\Sigma$ is then just one of the homogeneous isotropic Riemannian manifold. Since we usually consider three dimensions, and that the elliptic space is generally considered to be unphysical and also has the same local geometry as $S^3$, we can restrict the topologies of interest to $\mathbb{R}^3$, $S^3$ and $\mathbb{H}^3$, which are generally called the flat, closed and open universes.

3. The Friedmann equations

Once given the topology and geometry of the FRW spacetime, we can apply it to the Einstein field equations.

\begin{eqnarray} {\Gamma^\tau}_{\mu\nu} &=& \frac{1}{2} g^{\sigma\tau} (\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}) \end{eqnarray} The only non-vanishing derivatives of the metric are $\partial_t g_{ab}$ $$\frac{\dot a^2 + kc^2}{a^2} = \frac{8\pi G \rho + \Lambda c^2}{3}$$ $$\frac{\ddot a}{a} = -\frac{4\pi G}{3} (\rho + \frac{3p}{c^2}) + \frac{\Lambda c^2}{3}$$

4. Perturbation on homogeneity

While the FRW model is quite appealing, proving that an approximately homogeneous universe will approximately obey the FRW metric is hard to show.

Last updated : 2021-06-07 12:36:14
Tags : general relativity , cosmology