7 Introducing Inflation

The classical big bang picture can’t be correct, or at least it can’t be complete. As the scalefactor $a\to 0$, we hit a singularity in the solution where the Einstein equations lose their predictivity. What happened before the big bang? Sometimes people say “nothing” or “space and time didn’t exist”, but that’s not really justified: Einstein’s equations simply cannot tell you the answer, because the mathematical singularity gets in the way of any kind of sensible answer.

One possibility is that we need a fundamentally quantum theory of gravity to complete the picture. This is, for example, the vision of the Hartle-Hawking no-boundary proposal¹⁸¹⁸ 18 https://en.wikipedia.org/wiki/Hartle-Hawking_state. We might term these “solutions outside GR” as it requires a fundamental modification to the equations at early times.

But another possibility is to imagine some new physics that works within a classical (or semi-classical) regime to prevent the singularity from being reached. We might term these “solutions inside GR” since they maintain GR throughout but add in some extra energy-momentum content to modify our original expectations.

Perhaps, for example, one could have some previous steady state of the universe followed by a gigantic “explosion” that generates the expansion we see today? Not so – or, at least, not easily so. Recall the second Friedmann equation,

$$\ddot{a}=-\frac{4\pi Ga}{3}(\rho +3P).$$

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Since ordinary matter (satisfying the “strong energy condition”, $\rho +3P>0$, or equivalently $w>-1/3$) can only cause deceleration () the big explosion idea doesn’t quite cut it. But park the idea, because we’ll come back to it…

Why are the spatial sections of the the universe so closely approximated by flat Euclidean space? Even if homogeneous and isotropic, the cosmos could have a curved space (Section 1).

To understand the severity of the problem in more detail consider the Friedmann equation in the form of (141), reproduced here for completeness:

$$\mathrm{\Omega }(a)-1=\frac{\kappa }{{(aH)}^2}.$$

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In the standard Big Bang cosmology containing just matter and radiation, the comoving Hubble radius, ${(aH)}^{-1}$, grows with time and from (141), the quantity $|\mathrm{\Omega }-1|$ must diverge with time. $\mathrm{\Omega }=1$ is called an “unstable fixed point” – it’s a bit like balancing a ball on top of a hill and expecting it to stay there for 13.7 billion years. It’d have to be terribly well balanced at the start.

In standard Big Bang cosmology, the near-flatness observed today (${\mathrm{\Omega }}_0\sim 1$) requires an extreme fine-tuning of $\mathrm{\Omega }$ close to $1$ in the early universe. More specifically, one finds that the deviation from flatness at Big Bang nucleosynthesis (BBN, $\sim 0.1$ MeV ), during the GUT era ($\sim {10}^{15}$ GeV) and at the Planck scale ($\sim {10}^{19}$ GeV), respectively has to satisfy the following conditions

	$|\mathrm{\Omega }(a_{\mathrm{BBN}})-1|$	$\le$	$𝒪({10}^{-16})$		(203)
	$|\mathrm{\Omega }(a_{\mathrm{GUT}})-1|$	$\le$	$𝒪({10}^{-55})$		(204)
	$|\mathrm{\Omega }(a_{\mathrm{Pl}})-1|$	$\le$	$𝒪({10}^{-61}).$		(205)

But it turns out that this instability of $\mathrm{\Omega }=1$ is only true if the strong energy condition $w>-1/3$ is satisfied, as the next exercise demonstrates.

📝 Exercise 7A Differentiate (141) to calculate $\mathrm{d}\ln |\mathrm{\Omega }-1|/\mathrm{d}\ln a$ as a function of $\mathrm{\Omega }$ and $w$; show that $$\frac{\mathrm{d}\ln |\mathrm{\Omega }-1|}{\mathrm{d}\ln a}=(1+3w)\mathrm{\Omega }.$$ (206) Use this equation to determine under what circumstances can $|\mathrm{\Omega }-1|$ shrink (i.e. the universe become closer to flat) over time.

It’s now tempting to link problem (1) and (2): working fully within GR we could solve both if we have some content that violates the strong energy condition. Keep the idea parked…

We previously derived the FRW metric assuming the homogeneity and isotropy of the universe. Why is this a good assumption? It’s particularly surprising given that inhomogeneities are typically gravitationally unstable and therefore grow with time (as we’ll see in the second part of the course). To put it another way, whatever the inhomogeneities are like today, we expect they were even smaller in the past. Observations of the CMB confirm this expectation: the density of the universe at decoupling was homogeneous to the accuracy of one part in ten-thousand. Going to even earlier times, the universe would have been even more accurately homogeneous. Why would the big bang generate such a beautifully homogeneous universe?

The homogeneity problem is sometimes restated as the horizon problem. If the universe is going to be homogeneous, the ‘right’ value of the density has to somehow be communicated across its entirety (or, at least, the entirety of what we can measure today). It might seem like “that won’t be a problem since the universe starts with zero size, so everything must be able to instantly communicate with everything else at that early time”, but at scalefactor zero is precisely the point where our predictivity breaks down, so the meaning of this statement is hazy at best.

Let’s remember our earlier distinction between solutions inside GR and solutions outside GR – and work firmly “inside”. That means doing a more careful job to see if the universe can sort itself out at very early times, but after the actual singularity. We can ask: what is the maximum comoving distance across which a signal could have propagated between the big bang and some time $a$? Remember that this is given by (176), the conformal time $\eta$, which is also the causal (or comoving) horizon, characterizing the distance travelled by light since $t=0$. Let’s look at equation (176) and integrate it for a general fluid.

$$\eta ={\int }_0^t\frac{\mathrm{d}{t}^{\prime }}{a(t^{\prime })}={\int }_0^a\frac{\mathrm{d}a}{a^2{H}_0{a}^{-3(1+w)/2}}\propto a^{(3w+1)/2}.$$

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For example, as we saw previously, for RD ($w=1/3$) and MD ($w=0$) universes we find

$\eta \propto \{\begin{array}{cc}a\hspace{1em}\hspace{1em} \mathrm{RD}\hfill & \\ a^{1/2}\hspace{1em}\mathrm{MD}\hfill & \end{array}.$

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This means that the comoving horizon grows monotonically with time (at least in an expanding universe) which implies that comoving scales entering the horizon today have been far outside the horizon at CMB decoupling. That is, unless . Is this message starting to seem familiar?

Park it again and let’s first assess how bad this problem is:

☞ Exercise 7B Consider the cosmic microwave background (CMB) which, as we’ll discuss in a later section, is formed at redshift $z\simeq 1100$. Calculate the physical particle horizon at this early time, assuming that the universe contains only radiation and matter. A good starting point is equation (182). Then calculate the angular diameter distance to this time, $d_A(z=1100)$. Use these results to calculate the maximum angular extent of patches on the CMB sky which have been able to communicate. Interpret your answer in terms of the homogeneity problem.

The flatness and horizon problems are certainly not logical inconsistencies in the standard cosmological model. If one assumes that the initial value of $\mathrm{\Omega }$ was extremely close to unity and that the universe began near-homogeneously then the universe will continue to evolve homogeneously in agreement with observations. The flatness and horizon problems are therefore really just severe shortcomings in the predictive power of the Big Bang model. The dramatic flatness of the early universe cannot be predicted by the standard model, but must instead be assumed in the initial conditions.

Perhaps this is a plausible state for a universe to start in, so why worry? For me, the real strangeness that demands an explanation is that the very near homogeneity of the universe is just slightly broken in such a way as to seed the structure that can then grow over cosmic time. A theory that can explain this structure is a major step forward over the raw ‘big bang’ picture, whatever way one choses to look at it. While the three problems above are used as ‘inspiration’ for inflation, really it is this fourth puzzle that can be construed as strong evidence in favour of something much like inflation actually having happened in the real universe. As we will see, inflationary theories predict about the right level and exactly the right spectrum of fluctuations – a prediction that was made before the measurements confirmed it.

We can start to see a resolution to the problems posed above by expressing them in terms of the comoving Hubble radius. To understand the Hubble radius, it’s quite useful to rewrite the comoving horizon/conformal time as an integral over $\ln a$ rather than $a$:

$$\eta ={\int }_{-\mathrm{\infty }}^{\ln a}d[\ln a^{\prime }]\frac{1}{a^{\prime }H(a^{\prime })}.$$

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Then the integrand ${(aH)}^{-1}$ – aka the comoving Hubble radius – tells us the comoving distance over which photons could have travelled in the last $e$-fold of the universe’s expansion. That’s a good way to quantify “recent communication” across the universe, as opposed $\eta$ itself which quantifies “all the communication there’s ever been”.

If particles are separated by comoving distances greater than ${(aH)}^{-1}$, they cannot “currently” communicate. This is slightly vague, but nonetheless a helpful concept in what follows.

CAUTION: The “Hubble radius” ${(aH)}^{-1}$ and the “comoving horizon” $\eta$ are different, but in many circumstances can be used almost interchangeably if one just wants a rough indication of the scale over which causal connections have been made. This is because in standard big bang cosmology, one finds that $\eta \propto {(aH)}^{-1}$ up to numerical factors anyway. But beware, because when we introduce inflation below, we need to be much more careful to distinguish the two.

A possible approach to solving (or at least weakening) the problems listed in Section 1 is to invert the behavior of the comoving Hubble radius i.e. make it decrease sufficiently in the very early universe. As we will discuss below, the inversion of the hubble radius evolution has a variety of desirable consequences that patch up difficulties in the normal big bang scenario while remaining firmly “inside GR”. This is the essential idea behind inflation.

We can specify a decreasing comoving Hubble radius with any of the three equivalent conditions:

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Interpreted in words, these three equivalent conditions describing inflation are:

• •

  Decreasing comoving Hubble radius;

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  Accelerated expansion;

• •

  Strongly negative pressure, or more technically: violation of the strong energy condition.

The last of these is an important consequence of having a shrinking comoving horizon. Requiring negative pressure sounds concerning, but vacuum energy (Section 5) fits the bill; we will shortly discuss scalar field theories which show us how to arrange for a significant vacuum energy in the early universe.

The kind of arguments we’ll be using might seem abstract and can lead to skepticism. Being skeptical is healthy, but don’t forget that scalar fields are real things – we know from the discovery of the Higgs boson, for instance, that they crop up in nature. So it’s really not (any longer) such a jump to imagine them playing this kind of role in the early universe.

In inflation, for at least a brief time, ${(aH)}^{-1}$ decreased dramatically – or to put it another way, if you go back in time through inflation, ${(aH)}^{-1}$ was larger at the start than at the end. The comoving horizon $\eta$ gets a huge contribution from the early stages of inflation, but ${(aH)}^{-1}$ subsequently shrinks. At the end of inflation, ${(aH)}^{-1}$ starts growing as usual but it is now just “revealing” regions that had been in causal contact in the pre-inflationary era. Here’s the key point: if the universe underwent inflation, $\eta$ can actually be as large as you like, depending on how long inflation went on for.

We can put a lower bound on how long inflation must have lasted by assuming that the universe was RD since the end of inflation, and ignoring the MD epoch¹⁹¹⁹ 19 If ignoring the matter dominated epoch sounds weird, take a look at Figure 9 where the time axis is plotted as $\ln a$. From this perspective, which is quite a natural one when considering the overall dynamics of the universe, the matter domination epoch is a very recent and almost negligible phase.. Remembering that $H\propto a^{-2}$ during RD, the scale factor $a_e$ and Hubble parameter $H_e$ measured at the end of inflation is related to the current values $a_0$ and $H_0$ by

$$\frac{a_0{H}_0}{a_e{H}_e}=\frac{a_0}{a_e}{\left(\frac{a_e}{a_0}\right)}^2=a_e.$$

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How do we estimate $a_e$? We will see later that our current best fit inflationary models typically operate at energy scales ${10}^{15}$ GeV or larger. So, taking $T_e\sim {10}^{15}$ GeV $\simeq {10}^{28}$K,

$$a_e\sim \frac{T_0}{T_e}\sim \frac{T_0}{{10}^{28}\mathrm{K}}\sim {10}^{-28},$$

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i.e., ${(aH)}^{-1}$ at the end of inflation was ${10}^{28}$ times smaller than it is today.

For inflation to solve the horizon problem, ${(aH)}^{-1}$ at the start of inflation was larger than the current comoving Hubble radius, i.e. the largest scales observable today. Thus, during inflation, ${(aH)}^{-1}$ had to decrease by $\sim {10}^{28}$. The most common way to arrange this it to set up $H\sim \mathrm{const}$ during inflation:

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where $t_e$ is the time at the end of inflation. Thus, decreasing ${(aH)}^{-1}$ during inflation is solely due to the exponential increase in $a(t)$. For the scale factor to increase by ${10}^{28}$, $H(t_e-t)$ must be $\sim \ln ({10}^{28})\sim 64$.

☞ Exercise 7D Recompute the number of “$e$–folds” of inflation needed to solve the horizon problem, following the argument above but now accounting for the RD/MD transition at $a_{\mathrm{EQ}}$.

Notice the symmetry of the inflationary solution (cf. Fig. 9). Scales just entering the horizon today, $\sim 60$ $e$-folds after the end of inflation, left the horizon $\sim 60$ $e$-folds before the end of inflation.

So far, we have discussed inflation in terms of comoving coordinates. But it is also profitable to think of exponential expansion in terms of physical coordinates. The physical size of a causally connected region exponentially increases during inflation:

$$d_{\mathrm{physical}}\equiv a(t_e)\left[\eta (t_e)-\eta (t_s)\right]=e^{H{t}_e}{\left[\frac{-e^{-Ht}}{H}\right]}_{t_s}^{t_e}\simeq \frac{1}{H}{e}^{H(t_e-t_s)}\text{,}$$

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where I have used the null geodesic condition ($\mathrm{d}r=\mathrm{d}t/a(t)$) and assume $t_e\gg t_s$, with $H\sim$ constant. The exponential growth with $t_e$ reflects the growth of the scalefactor (the comoving horizon is staying roughly constant).

Another way to look at the very large physical horizon from inflation is to see that regions observed to be cosmological today were actually microscopically small before inflation. They were allowed to be in causal contact because the scales that light needed to transverse were small and, crucially, because $H$ was also not too large. This second condition is a required part of the paradigm – even in the normal hot big bang model, any given cosmological region was once very small. During inflation the expansion rate $H$ must be kept sufficiently slow so as not to yank the universe apart before it has time for light to cross the relevant patch²⁰²⁰ 20 This contrasts with the popular science sound-bite that inflation is a “super-fast” expansion, which in my view is not a helpful description; see Section 3..

Recall the Friedmann equation in the form

$$|1-\mathrm{\Omega }(a)|=\frac{\kappa }{{(aH)}^2}.$$

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Inflation ($H\approx \mathrm{const}.$, $a=e^{Ht}$) is characterized by a decreasing comoving horizon which drives the universe toward flatness (rather than away from it – see Exercise 1),

$$|1-\mathrm{\Omega }(a)|\propto \frac{1}{a^2}=e^{-2Ht}\to 0\mathit{\hspace{1em}}\text{as}\mathit{\hspace{1em}}t\to \mathrm{\infty }.$$

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This solves the ‘flatness problem’ of why $\mathrm{\Omega }$ is so close to $1$: $\mathrm{\Omega }=1$ is actually an attractor during inflation.

📝 Exercise 7E Assume ${\mathrm{\Omega }}_0\sim 0.3$ today, and ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0$, so that $1-\mathrm{\Omega }={\mathrm{\Omega }}_k=0.7$. Extrapolate $1-\mathrm{\Omega }(t)$ back to the end of inflation, then through $60$ $e$-folds of inflation. What is $\mathrm{\Omega }(t)-1$ right before the $60$ $e$-folds of inflation? Suppose a future experiment detects residual curvature in the universe, ${\mathrm{\Omega }}_k\ne 0$. What could be concluded about the number of $e$-folds of inflation that our Universe went through?

We said it already, but let’s emphasise again: a decreasing comoving horizon means that large scales entering the present horizon were inside the horizon before inflation (see Fig. 9). We can assume that causal physics before inflation therefore established thermal equilibrium and spatial homogeneity. The uniformity of the CMB is not a mystery if there was an inflationary era.

Besides solving the Big Bang puzzles, the decreasing comoving horizon during inflation is the key feature required for the quantum generation of cosmological perturbations described in the second part of the course. During inflation, quantum fluctuations are generated on subhorizon scales, but exit the horizon once the Hubble radius becomes smaller than their comoving wavelength. In physical coordinates this looks like “superluminal expansion” stretching perturbations to acausal distances (though be very careful: nothing is actually moving superluminally any more than it does in a classical big bang picture). The fluctuations can eventually be thought of as classical superhorizon density perturbations which reenter the horizon in the subsequent Big Bang evolution and then undergo gravitational collapse to form the large scale structure in the universe. This process will be a focus of the last part of our course.