Beyond the standard Hot big bang model

“Our whole universe was in a hot dense state,
Then nearly fourteen billion years ago expansion started. Wait …
That all started with the big bang (Bang)!”

is how the theme song of the popular TV show “The Big Bang Theory” starts.

” “Since the dawn of man” is really not that long,
As every galaxy was formed in less time than it takes to sing this song.
A fraction of a second and the elements were made …
Set in motion by the same big bang!”

“It’s expanding ever outward but one day
It will pause and start to go the other way.
Collapsing ever inward, we won’t be here, it won’t be heard
Our best and brightest figure that it’ll make an even bigger bang!”

Well, for all intents and purposes, that’s a pretty good introduction to what the postulates of the standard Hot big bang model are. That the universe started from a hot dense state and that it started expanding outwards. As it expanded, the temperature started to cool down. After a certain point matter and radiation, which were previously in equilibrium with one another, could now exist independently. The point in the expansion of the universe is referred to as the Epoch of decoupling. And we’ve come a long way since then, roughly 14 billion years, to get to where we are now. Also, that last bit in the song about a collapsing universe isn’t scientific fact any more because we are living in a flat universe, which will forever keep expanding.

(I am not going to give the balloon analogy because with the analogy comes the question what are we expanding into. And people aren’t satisfied when I say that we are expanding into nothing or that what we are expanding into doesn’t really matter (yet).)

I am not going to talk about how the Hot big bang model was triumphant in answering the questions of the day, namely the presence of a Cosmic Microwave Background radiation, the matter content of the universe and the expansion of the universe. Well, maybe later. For now, I would like to talk about some of the drawbacks of the Hot big bang model. Of the major drawbacks, I am going to focus on the Flatness (which I shall discuss here) and the Horizon (which I shall discuss later) problems and how the theory of Inflation solves them.

Before we talk about the flatness problem, let’s lay a mathematical foundation upon which we can formulate the question. For now, take my word that one of the equations that govern the expansion of the universe, referred to as the Friedmann equations, is

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

Note that $\dot{a}/a$ is also referred as the Hubble parameter $H$, $\rho$ is the density of our universe and $\kappa$ is the curvature parameter. If we manipulate the equations a bit, we get

$$\rho_c a^2 -\rho a^2= - \frac{3\kappa c^2}{8\pi G}$$

$\rho_c$ is referred to as the critical density of the universe and it is defined as

$$\rho_c = \frac{3H^2}{8\pi G}$$

Manipulating the equation further, we get that

$$(\Omega^{-1}-1)\rho a^2= - \frac{3\kappa c^2}{8\pi G} ,$$

where $\Omega$ is the ratio $\rho/\rho_c$.

Note that the right hand side of the above equation is a constant and therefore the left hand side too has to be a constant as the universe evolves and expands.

Conventionally, $\rho$ has three components; matter, radiation and vacuum (cosmological constant). The matter density behaves as $1/a^3$, radiation density as $1/a^4$ and the vacuum density is constant. The expansion of the universe is currently being driven by the cosmological constant but earlier, it used to be driven by radiation and matter. Therefore, during the early stages of the expansion of the universe, we can say that $\rho a^2$ decreases and to keep the left hand side constant, $(\Omega^{-1}-1)$ has to increase. The equation can also be solved trivially if $\Omega = 1$ i.e if the density of the universe we live in, $\rho$, is equal to the critical density $\rho_c$, meaning that we live in a flat universe i.e $\kappa = 0$. Astronomers have in fact observed that in the universe we live in, the value of $\Omega$ is very close to 1.

If you think about it, there is no logical reason as to why the value of $\Omega$ needs to be confined to $1$. This fine tuning of the $\Omega$ parameter and confining it’s value close to $1$ in order to satisfy the above equation is referred to as the Flatness problem. Richard Dicke was one of the first to identify this consequence of the Hot big bang model.

On the other hand, if during a certain stage of the expansion of the universe, the density of the universe driving the expansion was constant, then $\rho a^2$ increases drastically meaning that $(\Omega^{-1}-1)$ will decrease. Therefore, in such a scenario, no matter what value of $\Omega$ the universe starts with, it will eventually become $1$. This epoch of rapid expansion of the universe is referred to as Inflation.

[1] Theme song
[2] Wiki
[3] with numbers plugged in
[4] for the more theoretically inclined

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