An Introduction to Newtonian Cosmology

In the words of Carl Segan

The answer to the inquisitiveness of what there was? What there is? And what there will be? Is cosmology. 

Cosmology is the study of a harmonious universe, ranging from its formation and evolution to its future. It’s a science of exploration of the universe with the right amount of imagination and skepticism. By its very nature, cosmology is based on the collaboration of speculative theories and cosmological observation. And today we'll be overviewing it from a Newtonian perspective.

The only force of the four fundamental force of the nature that affects the large scale object at a cosmological level is gravity. Predominantly in physics there have been two leading theories explaining Gravity. The first one being by Newton which led the laws of physics for about 200 years. But with the proposal of special relativity, its inconsistency became quite evident. In the aftermath of this Einstein proposed a new theory of gravity widely known as General theory of relativity. Even though the modern-day fundamentals of cosmology are forged upon the relativistic framework, Newton's theory gives no serious anomalies at small distances (i.e., within our galaxy). Moreover, it is much simpler to grasp, hence we'll be taking look at cosmology from Newtonian framework.

Homogeneity and comoving coordinates

In the concern of a large-scale universe, cosmology puts forward a case of Homogenous and Isotropic universe. In simple terms it proposes that macrocosm is smooth at large scales. On their own the property of Homogeneity means that the universe looks same at all points and the property of isotropy states that the universe looks same in all direction.

One thing that needs to be kept in the mind is the existence of one property is not evidence for the existence of other. For example, a universe with uniform magnetic-filed would be homogenous but not isotropic similarly a spherically symmetric distribution would be isotropic but not homogenous.

Even though more recently homogeneity has become a matter of suspect in the eye of modern cosmologists but in this section, we'll consider the universe to be spatially homogenous and isotropic.

Now to understand the homogeneity and isotropy better we'll introduce another concept called comoving coordinates.

Let's consider a homogeneous sphere which is radially expanding (or contracting); however, density $\rho(t)$ remains spatially homogeneous. Now let's choose a point $t=t_{0}$ in time and introduce a coordinate system $x$ at this instant with the origin coinciding with the center of the sphere. Since the expansion the velocity vector of a particle at position $\boldsymbol{r}(t)$ is parallel to $\boldsymbol{r}$, the direction of $\boldsymbol{r}(t)$ is constant. Because $\boldsymbol{r}\left(t_{0}\right)=\boldsymbol{x}$, this implies

$$
\boldsymbol{r}(t)=a(t) \boldsymbol{x} .
$$

The function $a(t)$ can depend only on time. Here since the spatial density is constant, the ' $\mathrm{a}$ ' will also be spatially constant. This function $a(t)$ is called the cosmic scale factor; due to $\boldsymbol{r}\left(t_{0}\right)=\boldsymbol{x}$, it obeys

$$
a\left(t_{0}\right)=1
$$

The value of $t_{0}$ is arbitrary; lets choose $t_{0}=$ today. The observer which moves according to our above mention equation are called comoving observers, and $\boldsymbol{x}$ is the comoving coordinate. The world line $(\boldsymbol{r}, t)$ of a comoving observer is unambiguously determined by $\boldsymbol{x},(\boldsymbol{r}, t)=[a(t) \boldsymbol{x}, t]$

Expansion Rate. The velocity of such a comoving observer can be obtained from the time derivative of its position

$$
\boldsymbol{v}(\boldsymbol{r}, t)=\frac{\mathrm{d}}{\mathrm{d} t} \boldsymbol{r}(t)=\frac{\mathrm{d} a}{\mathrm{~d} t} \boldsymbol{x} \equiv \dot{a} \boldsymbol{x}=\frac{\dot{a}}{a} \boldsymbol{r} \equiv H(t) \boldsymbol{r}
$$

From the above-mentioned equation, we can infer $\mathrm{H}(\mathrm{t})$ as:

$$
H(t):=\frac{\dot{a}}{a}
$$
 

Redshift

In 1928, While observing Edwin Hubble discovered that the escape velocity of the galaxies is increasing with the distance. He came up with the following relation between distance and velocity

$$
v=H_{0} D
$$

The linear increase in velocity with distance were interpreted as the expansion of universe. This concept of expansion of universe was explained in terms of redshift.

Let's assume a galaxy to be at $\mathbf{r}_{0}=\mathbf{a}_{0}$ from which we are receiving light, from $a_{0}$ to us.

by assuming that the velocity of from $\mathbf{a}_{0}$ to ourselves with the velocity of light is $c$ for every fundamental observer. Let light leave $\mathbf{r}_{0}=\mathbf{a}_{0}$ at $t=t_{a}$ to reach $\mathbf{r}_{0}=0$ at $t=t_{0}$.

Thus, the light travelling from $\mathbf{r}_{0}=\mathbf{a}_{0}$ to $\mathbf{r}_{0}=0$ will pass an intermediate observer at $\mathbf{r}_{0}=\lambda \mathbf{a}_{0}, 0<\lambda<1$, at time $t$ in the range $t_{a}<t<t_{0}$. Since the velocity of such a typical observer is $\mathbf{r}_{0} \dot{S}(t)$ away from us, the light has a radial velocity

$$
\frac{d r}{d t}=-c+\lambda a_{0} \dot{a}(t)
$$

where $r=r_{0} a(t)=\lambda a_{0} a(t)$.

Since $d r / d t=\dot{\lambda} a_{0} a+\lambda a_{0} \dot{a}$, we get

$$
\frac{d \lambda}{d t}=-\frac{c}{a_{0} a^{\prime}}
$$

i.e., $a_{0}=\int_{t_{a}}^{t_{0}} \frac{c d t}{a(t)}$ since at $t=t_{a}, \lambda=1$ and at $t=t_{0}, \lambda=0$. In deriving (14) we have added the velocity of light to the velocity of the intermediate observer as per the Newtonian formula for vectorial addition of velocities. Although our operation is inconsistent with special relativity, it is fully consistent within the Newtonian framework. the above galaxy. The first crest leaves at $t_{a}$ and arrives at $t_{0}$. The second one leaves at $t_{a}+\Delta t_{a}$ and reaches at $t_{0}+\Delta t_{0}$, thus a relation similar to (14) holds, i.e.,

$$
a_{0}=\int_{t_{a}+\Delta t_{a}}^{t_{0}+\Delta t_{0}} \frac{c d t}{a(t)} .
$$

Subtracting (14) from (15) and using the approximation that $\Delta t_{0}$ and $\Delta t_{a}$ are small enough intervals for treating $S(t)$ constant over them, we get

$$
\frac{c \Delta t_{0}}{a\left(t_{0}\right)}=\frac{c \Delta t_{a}}{a\left(t_{a}\right)}
$$

But if $\lambda$ is the wavelength received by us, then $c \Delta t_{0}=\lambda$ while $c \Delta t_{a}=\lambda_{0}$. Therefore

$$
1+z \equiv \frac{\lambda}{\lambda_{0}}=\frac{a\left(t_{0}\right)}{a\left(t_{a}\right)}
$$

This is the relationship between $z$, the redshift and the scale factor increase with time, implying that the universe is expanding.

Moreover, from the concept of redshift a relation between scale factor and distance can be deduced,

For small distances $t_{a} \approx t_{0}$ and a Taylor expansion near $t=t_{0}$ gives

$$
a\left(t_{a}\right) \cong a\left(t_{0}\right)-\left(t_{0}-t_{a}\right) \dot{a}\left(t_{0}\right)
$$

Hence,

$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{a}\right)}=\left\{1-\left(t_{0}-t_{a}\right) \frac{\dot{a}\left(t_{0}\right)}{a\left(t_{0}\right)}\right\}^{-1} \approx 1+\left.\left(t_{0}-t_{a}\right) \frac{\dot{d}}{a}\right|_{t_{0}} \ldots \ldots \ldots \ldots \ldots \ldots . \mathrm{Eq} 1$

But, from (14), under the same approximation,

$a_{0} \approx \frac{c\left(t_{0}-t_{a}\right)}{a\left(t_{0}\right)}$

From Eq1 and Eq2 we find that the distance of the galaxy at $t=t_{0}$ is $D=a_{0} a\left(t_{0}\right) \cong$ $c\left(t_{0}-t_{a}\right)$.

Here we can infer that since our coordinates are comoving hence the framework is increasing proving the expansion of the universe.

Dynamics of Universe

In this section we'll be discussing about How Universe behaves in large on the motional front, As mentioned before the fundamental of cosmology is based on the Relativistic framework but for a simpler understanding, we'll take the classical mechanics and relate it the modern cosmological principles.

Now we know that the mass $(M)$ inside a shell of radius $x$ is

$$
M=\int_{M} d m=\frac{4 \pi}{3} \rho(t) a^{3}(t) x^{3}
$$

where $\rho_{0}$ is the mass density of the Universe today $\left(t=t_{0}\right)$.

Here density is inversely proportional to volume of the universe

$$
\rho(t)=\rho_{0} a^{-3}(t)
$$

Now the gravitational acceleration of a particle is $G M(x) / r^{2}$

Hence:

$$
\ddot{r}(t) \equiv \frac{\mathrm{d}^{2} r}{\mathrm{~d} t^{2}}=-\frac{G M(x)}{r^{2}}=-\frac{4 \pi G}{3} \frac{\rho_{0} x^{3}}{r^{2}}
$$

or, after substituting $r(t)=x a(t)$, an equation for $a$,

$$
\ddot{a}(t)=\frac{\ddot{r}(t)}{x}=-\frac{4 \pi G}{3} \frac{\rho_{0}}{a^{2}(t)}=-\frac{4 \pi G}{3} \rho(t) a(t) .
$$

Here it is interesting to note that the expansion is solely governed the mass density.

With the equation now we'll try to apply energy conservation principle hence we'll take the acceleration equation and integrate it trying to obtain expressions for the kinetic energy.

$$
\ddot{a}(t)=-G \frac{4 \pi}{3} \rho(t) a(t)=-G \frac{M_{i n t}}{a^{2}(t)}
$$

Now we multiply by $\dot{a}(t)$ and we integrate the equation.

$$
\begin{aligned}
a(t) \ddot{a}(t) & =G \frac{M_{i n t}}{a^{2}(t)} \dot{a}(t) \\
\frac{1}{2} \dot{a}^{2}(t) & =G \frac{M_{i n t}}{a(t)}+U
\end{aligned}
$$

Now we can clearly see that this equation has the form of the energy conservation equation. It is possible to distinguish the terms of the kinetic energy and the potential energy. After manipulating the above equation we'll get the following equation

$$
\left(\frac{\dot{a}(t)}{a(t)}\right)^{2}=\frac{8 \pi G}{3} \rho(t)+\frac{2 K}{a^{2} a^{2}(t)}
$$

Here it's interesting to note down the term one that contains $a^{-2}(t)$

We can easily infer here that the last term is related to the curvature term in the relativistic case, but in the Newtonian theory we cannot discuss about curvature since it's all a Euclidean metric. We can relate $K$ to mechanical energy and as a dynamic constant.

So, here with the curvature term, we can differentiate between different types of universes as a function of the $K$. If $K>0$ the equations are going to a universe that will recollapses. If $K<0$ the universe is will expand forever. And as an intermediate case, $\mathrm{K}=\mathrm{o}$, here the universe will keep expending but the acceleration will eventually lead to zero that means $\mathrm{d} a / \mathrm{d} t \rightarrow 0$.

Relativistic correction

After coming up with the concept of General relativity Einstein modified the Friedmann equation, framing them in the we see them today

$$
\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho-\frac{K c^{2}}{a^{2}}+\frac{\Lambda}{3}
$$

and

$$
\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+\frac{3 P}{c^{2}}\right)+\frac{\Lambda}{3}
$$

These two equations changed the course of cosmology forever. We'll discuss it in more detail in the future. For now, thank you for sticking out till the end.

- Sanskar (MS21234)

References: 

"An Introduction to Cosmology" by J.V. Narlikar

"An Introduction to Modern Cosmology" by Andrew R. Liddle

"Cosmological Physics" by Jhon A. Peacock