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from IPython.display import Video, HTML, YouTubeVideo
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Cosmological principle

Of the fundamental forces of nature, the weak and strong forces act on very subatomic scales, the electromagnetic forces are long range, but because on large scales matter is neutral, the electromagnetic force gets self shielded on large scales. Gravity, thus remains the only interesting interaction for cosmological studies.

The equations of General relativity (GR) are extremely tedious and allow direct solutions only in very specific cases. You have studied one such case in GR, that of the Schwarzschild metric. There the metric turn out have a specific form which was motivated by looking at the symmetries in the problem and finding the weak field limit. Most importantly if we look at the Scwarzschild metric, you can note that it becomes the Minkowski metric as you go further and further away from the object.

To make any progress whatsover in cosmology we also need to make some simplifying assumptions. Could the Universe be explained by an asymptotically Minkowski metric? Well it is not guaranteed. After all, there is mass in the Universe everywhere we look. Here is a flythrough in a 3d survey of galaxies which was made with the Sloan Digital Sky Survey.

YouTubeVideo("08LBltePDZw")

Homogeneity and isotropy

Even though the Universe has clumpiness when you average on sufficiently large scales, you see that overall the Universe is homogeneous and isotropic. We can use these simplifying assumptions of homogeneity and isotropy in space to construct a metric for the Universe. Homogeneity implies invariance under translations, simply put, there is nothing special about our location in space (all observational properties including Hubble’s law should be seen from every other galaxy). Isotropy implies invariance under rotations, it implies that there is no special direction in space.

On one hand these seem to be very similar assumptions, but it is important to note that these two are independent assumptions. Let us look at some examples of surfaces which are homogeneous but not isotropic, or the other way round?

Exercise:

• Is the surface of the sphere homogeneous and isotropic?

• Is the surface of the cylinder homogeneous and isotropic?

• What about the surface of a cone? Is it homogeneous? Is there a point from which it looks isotropic?

Before constructing three dimensional spaces which are symmetric, let us think about some special examples.

Exercise:

• Can you think of a 1-d space which is homogeneous?

• Can you think of a 1-d space which is homogeneous but finite?

• What about a 2-d space which is homogeneous (what about finite)?

It is important to also note that these assumptions may appear strange at first sight. Given that we do not see exactly the same thing in every direction. As such isotropy and homogeneity would imply that we cannot exist in such a Universe. Thus these assumptions are approximate and so the world view we build from these assumptions can also account for the inhomogeneities that we observe in the Universe, and these inhomogeneities remain tiny enough to not affect the general inferences we draw from these models.

The metric of a Homogeneous and Isotropic Universe

Galaxy observations show that as we peer into the past, populations of galaxies undergo evolution. Therefore we cannot impose a time symmetry. Given that all galaxies seem to be moving away from each other, we can consider that there is a mean motion of matter in the Universe with respect to which all the observable properties remain isotropic. We can think about these galaxies which are moving away from each other as some sort of fundamental observers which experience the same history over time. We will write down the metric for a space such fundamental observers.

We can consider the metric to be given by \(ds^2 = g_{\mu\nu} dx^{\mu}dx^{\nu}\). For these fundamental observers, \(dx^{i}=0\). The time coordinate can be assigned to be the proper time of these fundamental observers. In that case, we obtain,

\(ds^2 = -c^2 dt^2 = g_{00} dt^2\), i.e., \(g_{00} = -c^2\).

Because of isotropy, \(g_{0i}\) has to be zero. Otherwise distances measured with opposite directions \(dx^{i}\), will result in opposite values for \(ds^2\). Thus the metric needs to have a form which looks like,

\(ds^2 = -c^2 dt^2 + \gamma_{ij}dx^{i}dx^{j}\)

This form of the metric implies that we can foliate space time, as homogeneous and isotropic slices of space at different instants of time. The metric elements of the space could in general be scaled with a time dependent factor without violating homogeneity or isotropy. Let us call this scaling \(a(t)\). Thus our metric could in general have the form:

\(ds^2 = -c^2 dt^2 + a^2(t) \gamma_{ij}dx^{i}dx^{j} = -c^2 dt^2 + a^2(t) dl^2\)

Flat, homogeneous and isotropic 3-dimensional space

The Minkowski metric is one example of the above form, where \(\gamma_{ij}= \delta_{ij}\) and \(a(t) =\) constant. We can generalize the Minkowski metric by including a scale factor \(a^2(t)\). Thus the metric

\(ds^2 = -c^2 dt^2 + a^2(t) [dx_1^2 + dx_2^2 + dx_3^2] = -c^2 dt^2 + a^2(t) \left[ dr^2 + r^2 \left( d\theta^2 + \sin^2\theta d\phi^2 \right) \right]\)

is an example of the metric with a homogeneous and isotropic 3-dimensional space.

However, there are two more ways of obtaining a homogeneous and isotropic 3-d space, by embedding it in a four dimensional hyper-space with one hidden dimension.

Positively curved homogeneous and isotropic 3-dimensional space

Given that imagining 4 dimensions is difficult, we will take the analogy of a two dimensional creature (let us call it an ant) sitting on the surface of a sphere. Unbeknownst to the ant, the surface has a simple description in 3 dimensional space, and the ant is restricted to a 2 dimensional subspace of this 3d space via a constraint equation, namely \(x^2+y^2+z^2 = R^2\).

Exercise:

• Show that the metric for a sphere of radius R in terms of the (\(\theta, \phi\)) coordinate system \(dl^2 = R^2 [d\theta^2 + \sin^2\theta d\phi^2]\)

• Show that the metric for a flat plane in terms of the (\(\theta, \phi\)) coordinate system where \(\theta\) denotes the radial direction and \(\phi\) the azimuthal direction is given by \(dl^2 = d\theta^2 + \theta^2 d\phi^2\)

Now we can generalize this to the 3 dimensional surface of a 4-dimensional hypersphere to obtain a homogeneous and isotropic 3-dimensional space. The first one is a Euclidean space (\(dl^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2\)) where the geometry is Euclidean in 4 dimensions, but a constraint equation ensures that you stay on the surface of a hypersphere of curvature \(R\),

\(x_1^2 + x_2^2 + x_3^2 + x_4^2 = R^2\)

This way we are transforming to a new simpler coordinate system (\(\chi, \theta, \phi\)) based on this constraint equation.

• \(x_4 = R \cos\chi\)

• \(x_3 = R \sin\chi \cos\theta\)

• \(x_2 = R \sin\chi \sin\theta \sin\phi\)

• \(x_1 = R \sin\chi \sin\theta \cos\phi\)

The differentials are given by:

• \(dx_4 = - R \sin\chi d\chi\)

• \(dx_3 = R \cos\chi \cos\theta d\chi - R \sin\chi \sin\theta d\theta\)

• \(dx_2 = R \cos\chi \sin\theta \sin\phi d\chi + R \sin\chi \cos\theta \sin\phi d\theta + R \sin\chi \sin\theta \cos\phi d\phi\)

• \(dx_1 = R \cos\chi \sin\theta \cos\phi d\chi + R \sin\chi \cos\theta \cos\phi d\theta - R \sin\chi \sin\theta \sin\phi d\phi\)

Adding up the squared differentials we get the metric

\(dl^2 = R^2 [d\chi^2 + \sin^2\chi (d\theta^2 + \sin^2\theta d\phi^2 ) ]\)

Substituting, \(r=R\sin\chi\) we obtain the following metric for the 3-dimensional homogeneous and isotropic space

\(dl^2 = (1-r^2/R^2)^{-1}dr^{2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2 )\)

The entire metric will then be given by

\(ds^2 = -c^2dt^2 + a^2(t) \left[ (1-r^2/R^2)^{-1}dr^{2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2 ) \right]\)

Negatively curved homogeneous and isotropic 3-dimensional space

The third way of obtaining a homogeneous and isotropic 3 dimensional space, is by embedding it in a non-Euclidean space (\(dl^2 = dx_1^2 + dx_2^2 + dx_3^2 - dx_4^2\)) with the constraint equation,

\(x_1^2 + x_2^2 + x_3^2 - x_4^2 = -R^2\)

Exercise:

Using the following transformation

• \(x_4 = R \cosh\chi\)

• \(x_3 = R \sinh\chi \cos\theta\)

• \(x_2 = R \sinh\chi \sin\theta \sin\phi\)

• \(x_1 = R \sinh\chi \sin\theta \cos\phi\)

show that the metric in the above space is given by

\(dl^2 = R^2 [d\chi^2 + \sinh^2\chi (d\theta^2 + \sin^2\theta d\phi^2 ) ]\)

With the substitution, \(r=R\sinh\chi\), the metric for such a space is given by

\(dl^2 = (1+r^2/R^2)^{-1}dr^{2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2 )\)

The full metric in this case will be given by

\(ds^2 = -c^2dt^2 + a^2(t) \left[ (1+r^2/R^2)^{-1}dr^{2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2 ) \right]\)

Friedmann Robertson Walker metric

The 3 metrics above can be combined into a single one as

\(ds^2 = -c^2dt^2 + a^2(t) \left[ (1-k r^2)^{-1}dr^{2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2 ) \right]\)

with \(k=-1/R^2, 0, 1/R^2\) denoting negatively curved space, flat space and positively curved space, respectively. The above general metric is called the Friedmann Robertson Walker metric. You will see different forms of this metric used and they will be some coordinate transformations of each other.

For example,

\(ds^2 = -c^2dt^2 + a^2(t) \left[ d\chi^{2} + f^2(\chi) (d\theta^2 + \sin^2\theta d\phi^2 ) \right]\)

with

\(f(\chi) = R \sin \left(\chi/R \right)\) \(\,\quad\,\) (for \(k>0\))

\(f(\chi) = \chi\) \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\) (for \(k=0\))

\(f(\chi) = R \sinh \left(\chi/R\right)\) \(\,\,\,\,\,\) (for \(k<0\))

The assumptions for homogeneous and isotropic Universe has simplified the GR equations quite considerably. Now we just need to solve for the scale factor as a function of time, and figure out the signature of \(k\).

Exercise:

Please read the Feynman lectures on Curved space. It is an interesting read and helps you understand some of the analogies I have mentioned in the notes above but in much better and lucid language.