Chapter 10

Just to be clear from the get-go, inflation is an idea more than a testable scientific theory. It solves "problems" in cosmology, but does so by inventing a new field that conveniently exists only at early times (when it's needed) and does not exist today. The problems it solves could also be solved by simply stating that these are the initial conditions of the universe. Is that answer unsatisfying? Yes! Is it simpler than postulating an entirely new thing with no other evidence to support its existence? Also yes! By Occam's razor, the idea of inflation should be rejected. Does this mean inflation can't be correct? No! What we need is a crucial experiment or observation, that when conducted would either confirm inflation or rule it out. Does such an experiment exist? No. There are observations that could provide evidence for certain types of inflation, but none I am aware of that could rule it out. This is good if you're a theorist who specializes in inflation physics, but bad if you want predictive theories, which is arguably what science is. All of that said, inflation is part of the modern conception of modern cosmology, so it's good to understand what it's all about.

Flatness Problem

The universe is close to, if not exactly, flat: $\kappa =0$ or $R_0$ is very, very large. Writing the Friedmann equation in terms of $\kappa$:

where $\Omega(t)$ is the total energy density relative to the critical energy density. Observations constrain $|1-\Omega_0| \leq 0.005$, so $R_0 > 14c/H_0$. We don't know if the universe is exactly flat, but it is very close to being flat today, which means that it had to be even flatter in the past if it's only close to, but not exactly, flat. The generic form of the Friedmann equation is

In the past, as $a\ll 1$, the radiation and matter terms blow up, while the cosmological constant and curvature terms stay small, so when radiation dominates ($a_{\rm rm} < 3\times10^{-4}$), we can neglect those last two terms and substitute this expression for $H(t)$ into the equation for $1-\Omega(t)$ above, also swapping out $H_0$ using $1-\Omega_0 = -\kappa c^2 / (H_0^2 R_0^2)$ from the same equation set to the present day:

As the scale factor gets smaller and smaller, $1-\Omega(t) = (1-\Omega_0)a^2 / \Omega_{r,0} \propto a^2$, and thus $1-\Omega(t) \ll 1-\Omega_0 < 0.005$. The universe must be flatter in the past than it is today. Extrapolating back as far as we dare, to the Planck time, we have $|1-\Omega_P| \leq 2\times10^{-62}$. This is pretty small, and of all possible values of energy density/curvature, it seems unlikely it would end up this small by chance. However, we have no idea how likely various values of the energy density are, so perhaps this is totally fine. In any case, it would be nice to explain why the universe is so close to being flat.

Horizon Problem

In Chapter 8, we calculated that the observed angular size of the horizon distance at the time of last scattering, i.e., how large a patch of the universe was that could theoretically be in causal contact, is

Yet, the temperature of the CMB is nearly the same in any direction you look. This is the horizon problem, that there are nearly 40,000 causally disconnected patches of the CMB we can see ($4\pi~{\rm rad}^2 (180^\circ /\pi~{\rm rad})^2 \approx 41,000~{\rm deg}^2$ on the surface of a sphere). Because their light cones don't intersect, there's no physical way for them to have come into the same equilibrium state — equilibrium occurs when particles can interact and exchange energy. That is not possible here.

Could it be the case that the initial state of the universe was just the same everywhere? Of course! But that's not terribly satisfying, and it would be nice if there were an explanation for why that was the case.

Monopole Problem

This problem is not yet a problem, in the sense that it results in (some) unproven high energy physics theories, and so the solution could simply be that those theories are wrong. In the event you like theories that produce monopoles, however, you would want to explain why we don't see their effects, i.e., you need to get rid of them somehow.

But first, what are monopoles? Technically, they are 0-dimensional (i.e., point like) topological defects. What, you need more explanation? Imagine a bucket of water undergoing a phase transition from a liquid to solid state — ice crystals will form in various different places at about the same time and grow outward, adding molecules to their crystalline structure. That structure most likely won't have the same orientation in different patches, so where they meet, there will be a defect where the crystalline structures don't match up. This particular 2D type of defect is called a domain wall; a 1D defect is referred to as a cosmic string (like in HW1). A monopole is just the point-like version of a defect, and it acts in much the same way as a magnetic monopole would (a magnetic monopole is the equivalent in electromagnetic theory of a particle with a magnetic charge, which as far as we can tell, doesn't exist).

Now the question is, why should monopoles exist? Today, we have 4 fundamental forces: gravity, electromagnetism, weak, and strong. But it was not always this way. In the high energy universe, around $10^{-12}$ s after the Big Bang, the electromagnetic and weak forces were indistinguishable — there was simply a unified electroweak force. The breaking of that force into 2 separate forces can be thought of as a phase transition, like water freezing into ice. Now, at even higher energies, it is thought that this electroweak force should unify under a Grand Unified Theory, or GUT, with the strong force. There isn't a single definitive GUT; many have been proposed, I think largely incompletely, and in any case no practical experimental or observational predictions have been made to confirm or rule out most GUTs. But, in some GUTs, the separation of the strong and electroweak forces should produce monopoles, which would be hugely massive: $m_{\rm M}c^2 \sim E_{\rm GUT} \sim 10^{12}~{\rm TeV}$. The number density produced can also be estimated, and the resulting energy density would not be insignificant — shortly after being produced, they would become non-relativistic and would soon take over for radiation as the dominate source of energy density. We know that didn't happen because nucleosynthesis (which would happen long after monopoles dominated the universe) occurs in a predictable way — assuming the universe evolves as if dominated by radiation — which matches observations. Also, no evidence today exists that magnetic monopoles are present at all, let alone dominating the energy density of the universe. So, presuming they were created, where did they all go? This is the problem.

How Inflation Solves These Problems

Although I might call these curiosities, as opposed to problems, one appealing aspect of inflation is that it can solve all 3 of them at the same time. So what is inflation, exactly? It just means that at some very early time, the universe expands expands exponentially, or as $a(t) \propto e^{H_i t}$, where $H_i$ is the Hubble parameter or expansion rate at that time. You may recall that this is exactly the expression for $a(t)$ when the universe only contains a cosmological constant, and you'd be right. So the cause of inflation would be like a cosmological constant, but much larger (dominating over radiation at very very early times), and also it needs to disappear, since the universe was dominated by radiation and matter until recently.

The idea then is that the universe is expanding as we expect, with radiation dominating, until inflation "turns on" at time $t_i$. It lasts for awhile and "turns off" at time $t_f$, after which the universe is again radiation-dominated and expands thusly. In other words,

Between $t_i$ and $t_f$, the scale factor would have increased by factor of $a(t_f)/a(t_i) = \exp(H_i[t_f-t_i])$, or $N$ e-foldings, where $N$ is given by the expression inside the exponential. How does this idea solve the problems?

1) flatness

Remembering that exponential expansion occurs when the Hubble parameter is a constant, the curvature is then

Substituting times $t_i$ and $t_f$ into the equation, we find $|1-\Omega(t_f)| = e^{-2N}|1-\Omega(t_i)|$. Thus, the more the universe expands, the flatter it gets — if the universe had been strongly curved, such that $|1-\Omega(t_i)| \sim 1$, after inflation ends $|1-\Omega(t_f)| = e^{-2N} \sim 0$, close to flat as long as $N$ is sufficiently large. For the observed flatness we see, $N \gtrsim 60$. Inflation predicts a flat universe.

2) horizon

The horizon distance is just the proper distance between 2 points, where a photon left at $t=0$ and arrived at that time:

Using the expression for $a(t)$ above, we find $d_{\rm hor}(t_f) = e^N c (2t_i+H^{-1}_i) = d_{\rm hor}(t_i)e^N+c H_i^{-1} e^N$. We don't know what $H_i$ is, but regardless the horizon distance increases in size by at least a factor of $e^N \sim e^{60} = 10^{26}$. Before, we calculated the horizon distance assuming radiation dominated the entire time, and thus $a(t) \propto t^{1/2}$. Now, with this much faster expansion rate, there could have been a small patch of the universe in causal contact that got "inflated" to a size $10^{26}\times$ larger in practically an instant. So really, the entire observable universe was in causal contact due to this exponential expansion that we didn't account for earlier.

3) monopole

As long as inflation occurs after the strong and electroweak forces become distinct (after monopoles form), then monopoles (along with every other particle) get severely diluted due to the $>10^{26}$ factor expansion of the scale factor (or $\Delta V/V > 10^{78}$). For $N\sim65$ e-foldings, the number density today of monopoles would be expected to be $\sim 5 \times 10^{-16}~{\rm Mpc}^{-3}$, and given the volume of the observable universe, it's unlikely even 1 monopole would be left within our horizon.

Physics Underlying Inflation

It's not worth getting into the details here because there is no one "theory" of inflation (there are infinitely many, as theorists are infinitely creative), but the basics are the following. There is a scalar field, call it the inflaton field, that has a value at every point in space just like the Higgs field (the only scalar field we have actual evidence for). For some reason, the potential of this field (with units of energy density) did not start out at its minimum value, but at some value much larger than the energy density of the rest of the universe. Why? Because otherwise, inflation wouldn't have happened! And that's the whole point, of course. Why is its energy density so large? Because otherwise it wouldn't cause exponential expansion for long enough to solve the 3 problems discussed above!

As with any potential, it will move toward its minimum. In order to produce exponential expansion, it needs to do so slowly, as a function of the field — this is termed "slow roll inflation." The exact nature of this function is otherwise unconstrained, so while science can rule out certain forms of inflation, it's pretty much impossible to rule out any conceivable form of inflation. As it reaches the minimum value of its potential, the energy density of the field (assuming this field couples to Standard Model physics in some, almost any, way) is transformed into high energy particles/photons/etc. This is good, because exponential expansion dilutes everything, and any particles (like monopoles) that existed before inflation would be separated by huge distances. In this scenario then, the quark soup with its annihilations into photons, etc., all originates at the end of inflation. The inflaton field effectively disappears at this time.

A measurable quantity from this model does arise, the nature of fluctuations that create large-scale in the universe, which leads to galaxies and clusters of galaxies. Inflation essentially takes a tiny piece of space-time and blows it up by many orders of magnitude, to macroscopic size. On these small scales, particles can appear out of nothing due to the uncertainty principle, and the number of particles that have randomly appeared in any one patch of the universe will be slightly different than in other patches. This "quantum foam," as it is sometimes called, thus seeds the inhomogeneities that lead to the giant structures we see today. The power spectrum of these structures, given by the "scalar spectral index" $n_s$, is predicted to be around 1, which is what we observe it to be. (It's actually a tad less than one, but I think various flavors of inflation can accommodate that). This agreement, to me, is the most compelling evidence for inflation, although it is also the most generic kind of prediction (what you would guess if you had no idea where structure came from).

A more interesting prediction is that there should be leftover gravitational waves from the epoch of inflation, which would impart a certain type of polarization signal on the CMB. This signal is within experimental reach, and one experiment (BICEP-2) claimed to detect it, but they jumped the gun and confused Galactic dust with the signal. However, if the signature is not detected, it does not prove inflation wrong, because there are versions of inflation that where you don't get this signal. In other words, the "theory" of inflation can accommodate almost any future experimental result; it cannot be proved wrong! This is bad, but it doesn't necessarily mean some version of inflation can't be right. It means the theory is too flexible to be properly scientific.

How much money would you spend on an experiment if you couldn't be sure it would provide a result you could learn something from? More concretely, imagine you wanted to detect a new, faint kind of star predicted by someone's theory. You would want to design the telescope and observing plan based on the theory's prediction of how luminous the stars are and how far away they are. Now, the theorists' favorite version of the theory just happens to tell them that if you build a slightly bigger telescope, you'll find them, but they admit that the theory doesn't prevent the stars from being just a little fainter than that new telescope could detect — or even a lot fainter. Suddenly, this theory isn't very useful for selling the idea of your slightly bigger telescope, because it can't guarantee that if the stars aren't there, you've learned something interesting. This is currently the situation with inflation, and was also the situation with supersymmetry predictions at the LHC.