1. Introduction and Friedmann equation

We can start with Newton’s law of Gravitation which is universal. Objects with masses $M_1$ and $M_2$ a distance $r$ apart act on each other with the force

$$ F = - \frac{GM_1M_2}{r^2} .$$

This is an action at a distance.

Cosmological principle states that all points in the universe are identical, the universe is isotropic. This is kind of a dog in the manger policy.

According to Newton’s law, a finite-size universe will implode under gravity, hence the Newtonian universe must be infinite (in both space and time) in order to be stable. There cannot be any time evolution in this universe.

But there are problems in this picture:

1. This is an unstable picture. We are assuming that the universe is uniform, but if a small non-uniformity arises at any point, the whole universe will become unstable. When we study structure formation, we will use exactly this unstable picture; this is called Jean’s instability.

2. Olber’s paradox: why is the sky dark at night?

If the universe is infinite and the stars are infinitely long-lived, then the flux ($S$) of any star as observed by us will be proportional to the inverse of the square of the distance: $S\propto I/r^2$ where $I$ is the actual intensity of the star.

Density of stars $\sim n 4\pi r^2 dr$ where n is number of stars per unit volume.

So the amount of light received $\sim \int_0^\infty I/r^2 (n 4\pi r^2 dr) = 4\pi I \int_a^b dr=$ constant.

The only resolutions of Olber’s paradox in the Newtonian picture:

1. Stars are not infinitely log lived. Infinite lifetime does not salvage the situation by addition of interstellar medium. Because the interstellar medium itself then would glow and make the whole sky bright.
2. Gravity bends light.

The problems of the Newtonian picture could be solved only after the arrival of the general theory of relativity.

But the final equation of Friedmann can actually be derived using Newtonian mechanics, by simply using the shell theorem and the conservation of energy. We first need to define a comoving coordinate system.

Cosmological principle says that all points are identical. For example, if the universe is spherical, its shape will never change.

So the only evolution in this scenario is the size-scale of the universe. But if the universe is infinite, what is the meaning of size? To address this, we use comoving coordinates:

We draw a grid. If the distance between the nodes in the grid is 1 m at time $t=0$, this distance will increase when the grid expands. We say that the scale $a$ evolves with time and write the scale as $a(t)$.

In this picture, the dynamics of the universe is mapped into the motion of a point particle; a fantastic picture.

Let us assume the universe is filled up with a fluid (something that can expand and flow; not a solid). Think of a part of a universe of size, that is radius, 1. So after a time $t$, the radius will increase and become $a(t)$. Here we are assuming a time that is same for everyone; everyone has the same clock. This is called comoving time.

Now let us consider a point particle on the surface of the left sphere. Now we are looking at that particle from the centre of the sphere, and hence we see that the particle is receding away from us. Its kinetic energy will be

$$ \frac{1}{2}M \left(\frac{da}{dt}\right)^2 = \frac{M}{2} \dot{a}^2 $$

where $a$ is again the scale factor which we have defined above and hence $v=da/dt$. Here $\dot{a}$ always means the first derivative of $a$ with respect to time ($\ddot{a}$ will be the second derivative).

Now the shell theorem says that the mass within the sphere appears to be concentrated at its centre. So the particle has mass $M$ and, let us say, the mass of the spherical blob has is $\mathcal{M}$. Then the potential energy of the particle is $-GM\mathcal{M}/a$. Now according to the conservation of energy law,

$$ \frac{M\dot{a}^2}{2} - \frac{GM\mathcal{M}}{a} = K $$

where $K$ is the total energy which will remain constant. $K$ must be proportional to the mass of the particle, $K=Mk'$ and therefore

$$ \frac{M\dot{a}^2}{2} - \frac{GM\mathcal{M}}{a} = Mk' \Rightarrow \frac{\dot{a}^2}{2} - \frac{G\mathcal{M}}{a} = k' .$$

Now the mass of the universe $\mathcal{M}$ can be written in terms of its density as $\mathcal{M}=4\pi a^3 \rho / 3$. Put this in the above euqation and multiply by $2$ throughout and you get

$$ \frac{\dot{a}^2}{a^2} - \frac{8\pi}{3} \rho G = \frac{2k'}{a^2} = \frac{-k}{a^2}.$$

Note that $2k'$ can be written as $-k$ because it is just a constant. Here $\dot{a}/a$ is nothing but Hubble’s constant $H$. Hence,

$$ H^2 = \frac{8\pi G \rho}{3} - \frac{k}{a^2} $$

and this is called Friedmann equation. It describes the evolution of a fluid-like universe with a scale factor $a$ that is changing with time.

Hubble constant

$$ H = \frac{\dot{a}}{a} $$

means $\dot{a} \propto a$, the distance ($a$) of any point in the universe from an observer is directly proportional to the receding velocity of that point ($\dot{a}$).

Edwin Hubble calculated distances and velocities of nearby galaxies and proved this law observationally. We know the distance and velocity of galaxies much better now.

You can find the latest measurements of the velocity and distance of many galaxies here and check that their ratio $H$ is indeed a constant. A plot from this tool is shown below.

V_ls is the line-of-sight receding velocity. Each blob is the measurement of a galaxy. H0 is the Hubble constant at $t=0$, i. e. at present. Credit: Kourkchi et al. 2020, AJ, 159, 67.