4 Cosmological Inflation
The standard BB theory described in the previous two chapters is incomplete. Key features of the observed universe, such as its large-scale homogeneity and flatness, seem to require that the universe started with very special, finely tuned initial conditions. These initial conditions must be imposed by hand. In this chapter, how inflation-an early period of accelerated expansion-drives the primordial universe towards the special state, even if it started from more generic initial conditions.
To specify the initial condition, we define the position and velocities of all particles on an initial time slice, or in the fluid approximation, we specify the density and pressure as a function of position. The laws of gravity and fluid dynamics are then used to evolve the system forward in time. We assumed the distribution of matter to be homogeneous and isotropic. But, why is this good assumption? How do we explain the extraordinary smoothness of the early universe? This is particularly surprising since most of universe appears not be have been in causal contact and there is no dynamical reason why these causally-disconnected regions have such similar physical properties. This homogeneity problem is often called horizon problem.
Additionally in order for the universe to remain homogeneous at late times the initial fluid velocities must take very precise value. If the initial velocities are just slightly small, the universe recollapses within a fraction of a second. If they are just slightly too big, the universe expands too rapidly and quickly becomes nearly empty. The fine-tuning of the initial velocities is made even more dramatic by considering it in the combination with the horizon problem, since the fluid velocities need to be fine-tuned across causally-disconnected regions of space. In Section 2.3.5, we see that local curvature of a region in space is determined by the sum of the particles and kinetic energies in that region. The fine-tuning of the initial velocities is therefore often called the flatness problem and can be phrased as the question why the spatial curvature of the universe is so small.
Maybe it could be argued on the basis of symmetry that perfect homogeneity and exact flatness are preferred set of initial conditions and appeal to an unknown theory of quantum gravity to pick out this special state of the early universe. Even taking this attitude, however, one then faces the problem that the universe is not perfectly homogeneous, but has small fluctuations in the distribution of mater. Moreover, these fluctuations are not random, but display correlations over very large distances. For example, the correlations observed in the CMB extend over distances that are larger than the distance that light could have traveled since the beginning of the hot Big Bang. What created these superhorizon correlation?
One of the amazing feature of inflation is that it contains a mechanism to produce the primordial density fluctuations and its superhorizon correlations. Small quantum fluctuation get stretched by the inflationary expansion and becomes the seeds for the formation of the large-scale structure of the universe. Chapter 8 is devoted to a detailed account of this. In this chapter, the basic idea of cosmic inflation and how it solves the problem of the standard hot Big Bang will be presented.
4.1 Problems of the Hot Big Bang
In this section, the mentioned three problems-the horizon problem, the flatness problem and the problem of superhorizon correlations- will be described in more detail and make them quantitative.
4.1.1 The Horizon Problem
The size of a causally-connected patch of space is determined by the maximal distance from which light can be received. This is best studied in comoving coordinate, where null geodesics are straight lines and the distance between two points equals the corresponding difference in conformal time, ∆𝜂. hence, if the Big Bang "started" with the singularity at 𝑡𝑖 ≡ 0,1 then the greatest comoving distance from which an observer at time 𝑡 will be able to receive signals traveling at the speed of light is the (comoving) particle horizon:
(4.1) 𝑑𝘩(𝜂) = 𝜂 - 𝜂𝑖 = ∫𝑡𝑖𝑡 𝑑𝑡/𝑎(𝑡).
The term "particle horizon" is used to make a distinction with "event horizon" which is the maximal distance that we can influence future events, rather than the maximal distance from which we can be influenced by past events. In conventional Big Bang cosmology, we can set 𝜂𝑖 ≡ 0 and the comoving particle horizon is simply equal to the conformal time. The size of the horizon at time 𝜂 may be visualized by the intersection of past lightcone of an observer with the spacelike surface at 𝜂𝑖 (see Fig. 4.1). Causal influence have to come within this region.
Equation (4.1) can be written in the following illuminating way
(4.2) 𝑑𝘩(𝜂) = ∫𝑡𝑖𝑡 𝑑𝑡/𝑎(𝑡) = ∫𝑎𝑖𝑎 𝑑𝑎/𝑎ȧ = ∫ln 𝑎𝑖ln 𝑎 (𝑎𝐻)-1 𝑑(ln 𝑎), [= ∫ln 𝑎𝑖ln 𝑎 (𝑎𝐻)-1 𝑑(ln 𝑎)/𝑑𝑎 𝑑𝑎 = ∫𝑎𝑖𝑎 (𝑎2𝐻)-1 𝑑𝑎]
where 𝑎𝑖 ≡ 0 corresponding the BB singularity. The causal structure of the spacetime is hence related to the evolution of the comoving Hubble radius, (𝑎𝐻)-1. For ordinary matter sources, the comoving Hubble radius is a monotonically increasing function of time (or scale factor), and the integral in (4.2) is dominated by the contribution from late times. This implies that in the standard cosmology 𝑑𝘩 ~ (𝑎𝐻)-1, which has led to the confusing practice of referring to both the particle horizon and the Hubble radius as the "horizon."
We conclude that the amount of conformal time between the initial singularity and the emission of the CMB2 was much smaller than the conformal age of the universe today, 𝜂rec ≪ 𝜂0. It means that most parts of the CMB have non-overlapping past lightcones and hence never were in causal contact. This is illustrated by the spacetime diagram in Fig. 4.2. Consider two opposite directions on the sky. The CMB photons that we receive from these directions were emitted at the point labeled 𝑝 and 𝑞 in Fig. 4.2. Wee see that the photons were released sufficiently close to the BB singularity that the past lightcones of 𝑝 and 𝑞 don't overlap. How do the photons coming from 𝑝 and 𝑞 "know" that they should be at almost exactly the same temperature? There was simply not enough time for differences in the initial temperatures to be erased by heat transfer. The same question applies to any two points in the CMB that are separated by more than 2 degrees on the sky. (anout four times the angular size of the Moon, seen from Earth). The homogeneity of the CMB spans scales that are much larger than the particle horizon when the CMB was emitted. In fact, in the standard cosmology, the CMB consists of over 40 000 causally-disconnected patches off space. Why do they looks so similar? This is the horizon problem.
To make this more explicit, consider a flat universe filled with only matter and radiation. The Friedmann equation gives us the evolution of the Hubble rate
(4.3) 𝐻2 = 𝐻02 [𝛺𝑚,0𝑎-3 + 𝛺𝑟,0𝑎-4], with 𝛺𝑟,0 = 𝑎eq𝛺𝑚,0
The comoving Hubble radius can then be written as
(4.4) (𝑎𝐻)-1 = 1/√𝛺𝑚,0 𝐻0-1 𝑎/√(𝑎 + 𝑎eq).
The conformal time as a function of scale factor then is
(4.5) 𝜂 = ∫0𝑎 𝑑(ln 𝑎)/𝑎𝐻 = 2/√𝛺𝑚,0 𝐻0-1 (√(𝑎 + 𝑎eq) - √𝑎eq). [∫0𝑎 𝑑(ln 𝑎)/𝑎𝐻 = ∫0𝑎 𝑑(ln 𝑎)/𝑑𝑎 𝑑𝑎 1/𝑎𝐻 = ∫0𝑎 1/𝑎 𝑑𝑎 1/𝑎𝐻 = ∫0𝑎 𝑑𝑎/𝑎2𝐻]
Notice that the result has the correct limits: at early times (𝑎 ≪ 𝑎eq ≈ 3400-1), we get 𝜂 ∝ 𝑎 (as expected for the correct for the radiation-dominated era), while at late time (𝑎 ≫ 𝑎eq), we have 𝜂 ∝ 𝑎1/2 (as expected for the matter-dominated era). The conformal times today (𝑎0 = 1) and at recombination (𝑎rec ≈ 𝑎dec ≈ 1100-1) are
(4.6) 𝜂0 ≈ 2/√𝛺𝑚,0 𝐻0-1,
(4.7) 𝜂rec = 2/√𝛺𝑚,0 𝐻0-1 [√(1000-1 + 3400-1) - √3400-1] ≈ 0.00175 𝜂0.
Using the observed value of the Hubble constant, the comoving horizon at recombination is 𝑑𝘩(𝜂rec) = 𝜂rec ≈ 265 Mpc. This should be compared to the comoving distance to the surface of last-scattering, which in a flat universe is3 𝑑𝛢(𝜂rec) = 𝜂0 - 𝜂rec ≈ 𝜂0 ≈ 15.1 Gpc. The angle subtended by the horizon at recombination then is
(4.8) 𝜃𝘩 = 2𝑑𝘩(𝜂rec)/(𝜂0 - 𝜂rec) = (2 × 0.0175)/(1 - 0.0175) = 0.036 rad ≈ 2.0∘.
Including dark energy changes the distance to the surface of last-scattering by a factor of 0.9, so that 𝜃𝘩 ≈ 2.3∘.4
Although the total conformal time between the singularity and recombination is finite and small, we included times in the integral in (4.2) that were arbitrarily close to the initial singularity:
(4.9) ∆𝜂 = ∫0𝛿𝑡 𝑑𝑡/𝑎(𝑡) + ∫𝛿𝑡𝑡 𝑑𝑡/𝑎(𝑡).
However, in the first integral in (4.9) we have no reason to trust the classical geometry. We implicitly assumed that the breakdown of general relativity in the regime close to the singularity does not lead to a large contribution to the conformal time: i.e. we assumed 𝛿𝜂 ≪ ∆𝜂. This assumption may be incorrect and there may, in fact, be no horizon problem in a complete theory of quantum gravity. However, we will soon see that inflation provide a simple and computable solution to the horizon problem. Effectively, this is achieved by modifying the scale factor evolution in the second integral in (4.9), i.e. in the classical regime. We can choose this solution or a version of "quantum gravity magic".
4.1.2 The Flatness Problem
The flatness problem is closely related to the horizon problem, so that any solution to the horizon problem will also adress the flatness problem.
Let us define the time-dependent critical density of the universe as 𝜌crit(𝑡) = 3𝑀𝑃𝑙2𝐻2. [RE (2.141) and (3.2)] The time-dependent curvature parameter then is
(4.10) 𝛺𝑘 (𝑡) = (𝜌crit - 𝜌)/𝜌crit = (𝑎0𝐻0)2/(𝑎𝐻)2 𝛺 𝑘,0,
where we used that 𝜌crit ∝ 𝐻2 and (𝜌crit - 𝜌) ∝𝑎-2. [RE (2.144)] Recall that observations provide an upper bound on the size of the curvature parameter today, ∣𝛺𝑘,0∣ < 0.005. Because the comoving Hubble radius, (𝑎𝐻)-1 [= 𝑐/𝑎𝐻], is growing during the conventional hot BB, we expect that ∣𝛺𝑘∣ was even smaller in the past. Let us quantify this. Ignoring the shor period dark energy domination, the comoving Hubble radius is given by (4.4) and we can write (4.10) as
(4.11) 𝛺𝑘(𝑡) = 𝛺𝑘,0/𝛺𝑚,0 𝑎2/(𝑎 + 𝑎eq),
where we have set 𝑎0 ≡ 1. At matter-radiation equality, this implies
(4.12) ∣𝛺𝑘(𝑡eq)∣ = 𝛺𝑘,0/𝛺𝑚,0 𝑎eq/2 < 10-6.
At earlier time, the universe was dominated by radiation by radiation and we can use
(4.13) 𝐻2 = 𝐻eq2 𝛺𝑟,eq (𝑎eq/𝑎)4,
with 𝛺𝑟,eq = 0.5. The curavure parameter then becomes
(4.14) 𝛺𝑘(𝑡) = (𝑎eq𝐻eq)2/(𝑎𝐻)2 𝛺𝑘(𝑡eq) = 2𝛺𝑘(𝑡eq) (𝑎/𝑎eq)2.
Evaluating this at the time of BB nucleosynthesis, 𝑧𝛣𝛣𝑁 ≈ 4 × 108, or the the electroweak phase transition, 𝑧𝐸𝑊 ≈ 1015, we find
(4.15-6) ∣𝛺𝑘(𝑡𝛣𝛣𝑁)∣ < 10-16, ∣𝛺𝑘(𝑡𝐸𝑊)∣ < 10-29.
At even earlier times, the curvature parameter is constrained to be even smaller.
A useful way of rephrasing the problem is in terms of the curvature scale 𝑅(𝑡),
(4.17) 𝑅(𝑡) = 1/√𝛺𝑘(𝑡) 𝐻-1(𝑡).
We have seen that observations constrain the curvature scale today to be 𝑅(𝑡) > 14𝐻0-1. The above constrains on 𝛺𝑘(𝑡) then imply that the curvature scale in the early universe was many orders of magnitude larger than the Hubble radius at that time. Since in the standard BB cosmology, the Hubble radius is the same order as the particle horizon, this suggest a fine-tuning of many causally-disconnected patches of space. Section 2.3.5 [Friedmann Equation] relates the spatial curvature of a region of space to sum of kinetic and potential energies in the region. The flatness problem can therefore also be viewed as the problem of adjusting the initial velocities of all particles over distances that naively have never been in causal contact.
4.1.3 Superhorizon correlations
A new theory of he initial condition of the universe might dismiss the horizon and flatness problem. What cannot be dismissed, however, is the fact that the universe contains density fluctuations that are correlated over apparently acausal distances. This modern version of horizon problem begs for a dynamical explanation.
To understand the severity of the problem, let us consider the fluctuations that w e observe in th CMB and in large-scale structure of the universe. Fig. 4.1 shows the evolution of a representative fluctuation of wavelength 𝜆 relative to the Hubble radius (𝑎𝐻)-1. In comoving coordinates, wavelength 𝜆 is fixed and the Hubble radius increase. This means that any fluctuation that is inside the Hubble radius today was outside the Hubble radius at sufficiently early times. For the standard hot BB, the Hubble radius is approximately equal to the particle horizon, so we call these regimes "subhorizon" and "superhorizon". Avobe we saw that the particle horizon at recombination was about 265 Mpc. Scales larger than this would not have been inside the horizon before the CMB was emitted. Yet, we find the CMB fluctuations to be correlated on scales that are larger than this apparent horizon. Not only is the CMB homogeneous on apparently acausal scale, this modern version of horizon problem also has correlated fluctuations on these scale.
4.2 Before the Hot Bing Bang
The causality issues described above suggested that there was a phase before the hot Big Bang during which the homogeneity of the univesr and its correlated fluctuations were generated.
4.2.1 A Shrinking Hubble Sphere
Our description of the horizon and flatness problems gas highlighted the fundamental role played by the growing comoving Hubble radius of the standard BB cosmology. A simple solution to these problem therefore suggests itself: let us conjecture a phase of decreasing comoving Hubble radius in the early universe.
(4.18) 𝑑/𝑑𝑡 (𝑎𝐻)-1 < 0.
The shrinking Hubble sphere as the fundamental definition of inflation is preferred since it relates most directly to the horizon problem and is also a key feature of the inflationary mechanism for generating fluctuations (see Chapter 8). Using 𝑎𝐻 = ȧ, it is easy to see that this is equivalent to ä > 0, leading to the familiar notion of inflation as a period of accelerated expansion. If inflation lasts long enough, then the horizon and flatness problems can be avoided.
It will now be important to distinguish clearly between particle horizon and the Hubble radius. The
Hubble radius is the distance that particles can travel in the course of one expansion time, 𝐻-1, which is roughly the time in which the scale factor doubles. The Hubble radius is another way of measuring whether particles are causally connected with each other: Particles that are separated by distances greater than the particle horizon could never have communicated with one another. Inflation is a mechanism to make the particle horizon much larger than the Hubble radius. This means that the particles can't communicate now, but were in causal contact early on,
4.2.2 Horizon Problem Revisited
Consider again the expression (4.2) for comoving particle horizon
(4.19) 𝑑𝘩(𝜂) = ∫ln 𝑎𝑖ln 𝑎 (𝑎𝐻)-1 𝑑(ln 𝑎).
In the standard BB cosmology, the integral is dominated by late times and the particle horizons is the same order as the Hubble radius, 𝑑𝘩(𝜂) ~ (𝑎𝐻)-1. However, if the comoving Hubble radius is instead a decreasing function of time, then the integral is dominated by early times and the particle horizon can be much larger than the Hubble radius, 𝑑𝘩(𝜂) ~ (𝑎𝑖𝐻𝑖)-1 ≫ (𝑎𝐻)-1. As we will see, that is precisely what the doctor ordered.
To illustrate this further, consider a flat universe dominated by a fluid with a consit equation of state 𝜔 ≡ 𝑃/𝜌. [= 𝑃/(𝜌𝑐2) (2.107)] From
the Friedmann equation, we infer that the evolution of the comoving Hubble radius is [RE (2.148)]
(4.20) (𝑎𝐻)-1 = 𝐻0-1 𝑎1/2 (1 + 3𝜔),
and the comoving particle horizon becomes
(4.21) 𝑑𝘩(𝑎) = 2𝐻0-1/(1 + 3𝜔) [𝑎1/2 (1 + 3𝜔) - 𝑎𝑖1/2 (1 + 3𝜔)] ≡ 𝜂 - 𝜂𝑖.
Ordinary matter satisfies the strong energy condition (SEC), 1 + 3𝜔 > 0, so that the Hubble radius grows and the particle horizon is dominated by late time, 𝑎 ≫ 𝑎𝑖 and get 𝑑𝘩 = 𝜂. Now, take 1 + 3𝜔 < 0 instead. The Hubble radius then shrunk and the particle horizon receives most of its contribution from early times. In fact, we now have
(4.22) 𝜂𝑖 ≡ 2𝐻0-1/(1 + 3𝜔) 𝑎𝑖1/2 (1 + 3𝜔) → [𝑎𝑖 → 0, 𝜔 <-1/3] → -∞.
The BB singularity has been pushed to negative conformal time, so that there was actually much more conformal time between the singularity and recombination than we had thought.
Fig. 4.4 shows that the new spacetime diagram. Let us denote the beginning and the end of inflation by 𝜂𝑖 and 𝜂𝑒, respectively. If ∣𝜂𝑒 - 𝜂𝑖∣ ≫ 𝜂0 - 𝜂rec, then the past lightcones of widely separated points in the CMB had enough time to intersect. The uniformity of the CMB is not a mystery anymore. In inflationary cosmology, 𝜂 = 0 is not the initial singularity, but instead is only a transition point between inflation and the hot BB. There is time before and after 𝜂 = 0.5
4.2.3 Flatness Problem Revisited
Next, we show how the shrinking Hubble sphere also solves the flatness problem. We have seen that the evolution of the curvature parameter is related to the evolution of the comoving Hubble radius as
(4.23) 𝛺𝑘(𝑡) = (𝑎𝑖𝐻𝑖)2/(𝑎𝐻)2 𝛺𝑘(𝑡𝑖).
Any initial curvature will therefore decrease, if the comoving Hubble radius decreases. If the inflation lasts long enough it therefore solves the flatness problem.
Let us consider the case of a perfect fluid with a constant equation of state 𝜔. The Friedmann equation then implies
(4.24) (𝑎𝐻)2 = (𝑎𝑖𝐻𝑖)2 [(1 - 𝛺𝑘,𝑖)(𝑎/𝑎𝑖)-(1 + 3𝜔) + 𝛺𝑘,𝑖],
and substituting this into (4.23), we get
(4.25) 𝛺𝑘(𝑁) = 𝛺𝑘,𝑖𝑒(1 + 3𝜔)𝑁/[(1 - 𝛺𝑘,𝑖) + 𝛺𝑘,𝑖𝑒(1 + 3𝜔)𝑁],
where 𝑁 ≡ ln(𝑎/𝑎𝑖) is the number of "e-folds" of expansion. This solution has two fixed points at 𝛺𝑘,𝑖 = 0 (a flat universe) and 𝛺𝑘,𝑖 = 1 (an empty Milne universe). Fig 4.5 shows the evolution of 𝛺𝑘 for a radiation-dominated universe. [RE (2.144) 𝛺𝑘 > 0 for 𝑘 < 0] With 1 + 3𝜔 > 0, any deviation from 𝛺𝑘 = 0 is unstable and grows with time. For a negative curved universe (with 𝛺𝑘 > 0), the growth then slows as 𝛺𝑘 approaches its second fixed point at 𝛺𝑘 = 1. This final state is an empty universe with negative curvature. For a positively curved universe (with 𝛺𝑘 < 0), the growth accelerates and the curvature parameter diverges at
(4.26) 𝑁* = 1/(1 + 3𝜔) ln ∣𝛺𝑘,𝑖 -1/𝛺𝑘,𝑖∣.
The divergence occurs when the Hubble parameter vanishes, cf. (4.24), and therefore corresponds to the turn-around point of the scale factor (see Fig. 2.12).
Exercise 4.1 Show that the curvature parameter in (4.23) satisfies the following evolution equation
(4.27) 𝑑𝛺𝑘/𝑑𝑁 = (1 + 3𝜔) 𝛺𝑘 (1 - 𝛺𝑘).
This makes it transparent that 𝛺𝑘 = 0 and 𝛺𝑘 =1 are fixed points, and that the behavior away from these points depends on the equation of state.
[Solution] (4.23) 𝛺𝑘(𝑡) = (𝑎𝑖𝐻𝑖)2/(𝑎𝐻)2 𝛺𝑘(𝑡𝑖); Setting 𝑁 = ln (𝑎𝑖𝐻𝑖)2𝑎 and 𝜔 = 𝑃/𝜌𝑐2
(a) 𝑑𝛺𝑘/𝑑𝑁 = 𝑑𝑡/𝑑[ln (𝑎𝑖𝐻𝑖)2𝑎] 𝑑𝛺𝑘/𝑑𝑡 = 1/[ln (𝑎𝑖𝐻𝑖)2𝐻] 𝛺𝑘 𝑑(ln 𝑑𝛺𝑘)/𝑑𝑡 = -2 𝛺𝑘/𝐻 𝑑(ln ȧ)/𝑑𝑡 = -2 𝛺𝑘/𝐻 ä/ȧ = -2𝛺𝑘 ä/𝑎𝐻2.
Using Friedmann equations,
(b) 𝐻2 = 8π𝐺/3 𝜌 - 𝑘𝑐2/𝑎2𝑅02 = 8π𝐺/3 𝜌 + 𝐻2𝛺𝑘.
(c) ä/𝑎 = -4π𝐺/3 (𝜌 + 3𝑃/𝑐2) = -4π𝐺/3 𝜌(1 + 3𝜔) = 𝐻2/2 (1 - 𝛺𝑘)(1 + 3𝜔).
Substituting (c) to (a), we get
(d) 𝑑𝛺𝑘/𝑑𝑁 = (1 + 3𝜔) 𝛺𝑘 (1 - 𝛺𝑘). ▮
The evolution of the curvature parameter is drastically different if the fluid violates the strong energy condition (SEC), 1 + 3𝜔 < 0. The solution in (4.25) still applies and shows that ∣𝛺𝑘∣ decreases as 𝑒-∣1 + 3𝜔∣𝑁, so that the fixed point 𝛺𝑘 = 0 is now an attractor (see Fig. 4.6). The flatness problem is solved if the inflationary expansion lasts long enough to reduce any initial curvature to such a low value at the beginning of the hot BB that the subsequent expansion doesn't increase it above the observational bound of ∣𝛺𝑘,0∣ < 0.005. In practice, that this means between 40 and 60 𝑒-folds of inflationary expansion (see Section 4.2.5).
4.2.4 Superhorizon Correlations
The shrinking Hubble sphere also explain how the fluctuations observed in the CMB can be correlated on apparently superhorizon scales. Fig. 4.7 shows the evolution of a fixed comoving scale relative to the Hubble radius. If inflaion lasted long enough, this scale started its life inside the Hubble radius, wherecausal processes were able to create nontrivial correlations. The fluctuations then exited the Hubble radius6 and are measured today as apparently superhorizon correlations in the CMB. Notice the symmetry of the evolution. Scales just entering the Hubble radius today, 60 𝑒-folds after the end of inflation, left the Hubble radius 60 𝑒-folds before the end of inflation.
4.2.5 Duration of Inflation
How much inflation is needed o solve the problems of the hot BB? Since the Hubble radius is easier to calculate than the particle horizon, it is common to use the Hubble radius as a means of judging the causality problem. The problems are solved if the entire observable universe is smaller than the comoving Hubble radius at the beginning of inflation (see Fig. 4.7).
(4.28) (𝑎0𝐻0)-1 < (𝑎𝑖𝐻𝑖)-1
Notice that this is more conservative than using the particle horizon, since 𝑑𝘩(𝑡𝑖) is bigger than (𝑎𝑖𝐻𝑖)-1. We don't have to assume anything about earlier times 𝑡 < 𝑡𝑖, while 𝑑𝘩(𝑡𝑖) depends on the entire history of time before inflation.
The physical Hubble rate is nearly constant during inflation, so that 𝐻𝑖 ≈ 𝐻𝑒. During infaltion the scale factor increase while the comoving Hubble radius is decrease. The latter is typically expressed in terms of the number of 𝑒-foldings
(4.29) 𝑁tot ≡ ln(𝑎𝑒/𝑎𝑖).
We would like to detertime the minimal number of 𝑒-foldings that is required to solve the problem of hot BB.
As in Fig. 4.7, the decrease of the comoving Hubble radius during inflation must compensate for its increase during the hot BB. The amount grown during the hot BB depends on the maximal temperature of the thermal plasma at the beginning of the hot BB. We will denote this so-called reheating temperature by 𝛵𝑅. We take the universe to be radiation-dominated throughout. Remembering that 𝐻 ∝ 𝑎-2 during the radiation-dominated era, we have
(4.30) 𝑎0𝐻0/𝑎𝑅𝐻𝑅 = 𝑎0/𝑎𝑅 (𝑎𝑅/𝑎0)2 = 𝑎𝑅/𝑎0 ~ 𝛵0/𝛵𝑅 ~ 10-28 (1015 GeV/𝛵𝑅),
where we have introduce a reference, 1015 GeV for the reheating temperature. We will furthermore assume that the energy density at the end of inflation was converted relatively quickly into the particles of thermal plasma, so that the Hubble radius didn't significantly grow between the end of inflation and the beginning of hot BB, (𝑎𝑒𝐻𝑒)-1 ~ (𝑎𝑅𝐻𝑅)-1. The condition can then written as
(4.31) (𝑎𝑖𝐻𝑖)-1 > (𝑎0𝐻0)-1 ~ 10-28 (𝛵𝑅/1015 GeV) 𝑎𝑒𝐻𝑒)-1,
and using 𝐻𝑖 ≈ 𝐻𝑒, we get
(4.32) 𝑁tot ≡ ln(𝑎𝑒/𝑎𝑖) > 64 + ln(𝛵𝑅/1015 GeV).
This is the famous statement that the solution of the horizon problem requires about 60 𝑒-folds of inflation. Notice the fewer 𝑒-folds are needed if the reheating temperature is lower.
1 Notice that the Big Bang singularity is a moment in time, not a point in space. In Fig. 4.1-2 we depict the singularity by an extended (possibly infinite) spacelike hypersurface.
2 In this chapter we don't distinguish between recombination and photon decoupling and simply refer to boths as 'recommbination."
3 Notice that we used the same notation, 𝑑𝛢, for the physical angular diameter distance in Section 2.2.3.
4 As we will see in Chapters 6 and 7, fluctuations in primordial plasma actually propagate at the speed of sound, which is smaller than the speed of light, 𝑐𝑠 = 𝑐/√3. The angular size of the corresponding sound horizon is 𝜃𝑠 = 𝜃𝘩/√3 ≈ 1.3∘, which explains why the pattern of CMB fluctuations has special features n the scale of 1 degree.
5 Notice that the BB singularity is still at 𝑡 = 0. The time between the singularity and recombination is fixed, but the lightcones get stretched drastically by the inflationary expansion.
6 We will later refer the Hubble radius as "horizon", but it is important to keep in mind that the Hubble radius and the horizon are vastly different during inflation.
4.3 The physics of Inflation
A key characteristic of inflation is that all physical quantities are slowly varying, despite fact that the space is expanding rapidly. Let us write the time derivative of the comoving Hubble radius as
(4.33) d/d𝑡 (𝑎𝐻)-1 = -(𝑎̇𝐻 + 𝑎Ḣ)/(𝑎𝐻)2 = -1/𝑎(1 - 𝜀),
where we have introduced the slow-roll parameter
(4.34) 𝜀 ≡ -Ḣ/𝐻2 = -d ln 𝐻/d𝑁,
with d𝑁 ≡ d ln 𝑎 = 𝐻d𝑡. This shows that a shrinking Hubble radius, ∂𝑡(𝑎𝐻) < 0, is associated with 𝜀 < 1. In other words, inflation occurs if the fractional change of the Hubble parameter, ∆ln 𝐻 = ∆𝐻/𝐻, per 𝑒-folding of expansion, ∆𝑁, is small. Moreover, as we will see in Chapter 8, the near scale-invariance of the observed fluctuations in fact requires that 𝜀 ≪ 1. In the limit 𝜀 → 0, the dynamics becomes time-translation invariant, 𝐻 = const, and the spacetime is de Sitter space
(4.35) d𝑠2 = -d𝑡2 + 𝑒2𝐻𝑡d𝐱2.
Because inflation has to end, so that time-translation symmetry must be broken and the spacetime must deviate from a perfect de Sitter space. However, for small , but finite 𝜀 ≠ 0, the line element (4.35) is still a good approximation to the inflationary background. This is why inflation is also often referred to as a quasi-de Sitter period.
Finally, we want inflation to last for a sufficient long time (usually at least 40 to 60 𝑒-folds), which requires that 𝜀 remains small for a sufficiently large number of Hubble times. This condition is measured by a second slow-roll parameter7
(4.36) 𝜅 ≡ (d ln 𝜀)/d𝑁 = 𝜀̇/𝐻𝜀.
For ∣𝜅∣ < 1, the fractional change of 𝜀 per 𝑒-fold is small and inflation persists. In the next section, we will discuss what microscopic physics can lead to the conditions 𝜀 < 1 and ∣𝜅∣ < 1,
Exercise 4.2 Consider a perfect fluid with density 𝜌 and pressure 𝑃. Using the Friedmann and continuity equations, show that
(4.37-8) 𝜀 = 3/2 (1 + 𝑃/𝜌), 𝜀 = 1/2 (d ln 𝜌)/(d ln 𝑎).
This shows that inflation occurs if the pressure is negative and the density is nearby constant. Conventional matter sources all dilute with expansion, so we need to look for something more unusual.
[Solution] From the definition of the slow-roll parameter, we have
(a) 𝜀 ≡ -Ḣ/𝐻2 = -1/𝐻2 d/d𝑡 (𝑎̇/𝑎) = -1/𝐻2 (-𝑎̇2/𝑎2 + (d𝑎̇/d𝑡)/𝑎 = 1 - (d𝑎̇/d𝑡)/𝑎 1/𝐻2.
Using the Friedmann equations for a flat universe
(b) 𝐻2 = 8π𝐺/3 𝜌, (d𝑎̇/d𝑡)/𝑎 = -4π𝐺/3 (1 + 𝑃/𝜌), ⇒ (d𝑎̇/d𝑡)/𝑎 = -𝐻2/2 (1 + 3𝑃/𝜌),
(c) 𝜀 = 1 - [-𝐻2/2 (1 + 3𝑃/𝜌)] 1/𝐻2 = 3/2 (1 + 𝑃/𝜌).
Using the continuity equation (2.106), 𝜌̇ + 3𝐻(𝜌 + 𝑃) = 0 with 𝑐 ≡ 1, we can write
(d) 𝜀 = 3/2 [1 + (-𝜌̇/3𝐻 - 𝜌)/𝜌] = 3/2 (-1/3𝐻 𝜌̇/𝜌) = -1/2𝐻 𝜌̇/𝜌 = -1/2𝐻 (d ln 𝜌)/d𝑡 = -1/2 (d ln 𝜌)/(d ln 𝑎). ▮
4.3.1 Scalar Field Dynamics
The simplest models of inflation implement the time-dependent dynamics during inflation in terms of the evolution of a scalar field, 𝜙(𝑡, 𝐱), called the inflaton. [𝑡: time, 𝐱: position] Associated with each value is a potential energy density 𝑉(𝜙) (see Fig. 4.8) When the field changes with time then it it also carries a kinetic energy density 1/2 𝜙̇2. If the energy associated with the scalar field dominates the universe, then it sources the evolution of the FRW background. We want to determine under which conditions this can derive an inflationary expansion.
Let us begin with a scalar field in Minkowski space. Its action is
(4.39) 𝑆 = ∫ d𝑡 d3𝑥 [1/2 𝜙̇2 - 1/2 (∇𝜙)2 - 𝑉(𝜙)],
where we have also included the gradient energy, 1/2 (∇𝜙)2, associated with a spatially varying field. To determine the equation of the scalar field, we consider the variation 𝜙 → 𝜙 + 𝛿𝜙. Under this variation, the action changes a
(4.40) 𝛿𝑆 = ∫ d𝑡 d3𝑥 [𝜙̇𝛿𝜙̇ - ∇𝜙 ⋅ 𝛿∇𝜙 - d𝑉/d𝜙 𝛿𝜙]
{= ∫ d𝑡 3𝑥 [-d/d𝑡 (1/2 𝜙̇2) 𝛿𝜙 -(-∇ ⋅ ∇𝜙 𝛿𝜙) - d𝑉/d𝜙 𝛿𝜙]}
(4.41) = ∫ d𝑡 d3𝑥 [-d𝜙̇/d𝑡 + ∇2𝜙 - d𝑉/d𝜙]𝛿𝜙
where we have integrated by parts and dropped a boundary term. If the variation around the classical field configuration, then the principle of least-action states that 𝛿𝑆 = 0. For this to be valid for an arbitrary field variation 𝑑𝜙, the expression in the square brackets in (4.41) must vanish:
(4.42) d𝜙̇/d𝑡 - ∇2𝜙 = -d𝑉/d𝜙
This is the Klein-Gordon equation.
It is straightforward to extend this analysis to the evolution in an expanding FRW spacetime. For simplicity we will ignore spatial curvature.Physical coordinates are related to comoving coordinates by the scale factor, 𝑎(𝑡)𝐱.Taking this into account, it is easy to guess that the generalization of the action (4.49) to an FRW background is
(4.43) 𝑆 = ∫ d𝑡 d3𝑥 𝑎3(𝑡) [1/2 ᾡ2 - 1/2𝑎2(𝑡) (∇𝜙)2 - 𝑉(𝜙)],
We are interested in the evolution of a homogeneous field configurations, 𝜙 = 𝜙(𝑡), in which case the action reduces to
(4.44) 𝑆 = ∫ d𝑡 d3𝑥 𝑎3(𝑡) [1/2 𝜙̇2 - 𝑉(𝜙)],
Performing the same variation of the action as before, we find
(4.45) 𝛿𝑆 = ∫ d𝑡 d3𝑥 𝑎3(𝑡) [𝜙̇𝛿𝜙̇ - 𝑑𝑉/𝑑𝜙 𝛿𝜙].
(4.46) = ∫ d𝑡 d3𝑥 [-d/d𝑡 (𝑎3𝜙̇) - 𝑎3 d𝑉/d𝜙]𝛿𝜙
and the principle of least action leads to the following Klein-Gordon equation
(4.47) d𝜙̇/d𝑡 + 3𝐻𝜙̇ = -d𝑉/d𝜙.
The expansion has introduced one new feature, the so-called Hubble frictionassociated with the term 3𝐻ᾡ. This friction will play a crucial role in the inflationary dynamics.
Let us now assume that this scalar field dominates the universe and determine its effect on the expansion.Given the form of the action in (4.44), it is natural to guess that the energy density is the sum of kinetic and potential energy densities
(4.48) 𝜌𝜙 = 1/2 𝜙̇2 + 𝑉(𝜙).
Taking the time derivative of the energy density, we find
(4.49) 𝜌̇𝜙 = (𝑑𝜙̇/𝑑𝑡 + 𝑑𝑉/𝑑𝜙)𝜙̇ = -3𝐻𝜙̇2,
where the Klein-Gordon equation (4.47) was used. Comparing this to the continuity equation, ῤ𝜙 = -3𝐻(𝜌𝜙 + 𝑃𝜙) we infer the pressure induced by the field is
(4.50) 𝑃𝜙 = 1/2 ᾡ2 - 𝑉(𝜙).
This pressure determines the acceleration of the expansion 𝑑𝑎̇/𝑑𝑡 ∝ -(𝜌𝜙 + 3𝑃𝜙). We see that the density and pressure are in general not related by a constant equation of state as for a perfect fluid. Notice that if the kinetic energy of the inflation is much smaller than its potential energy, then 𝑃𝜙 ≈ -𝜌𝜙. The inflationary potential then acts like a temporary cosmological constant, sourcing a period of exponential expansion.
Exercise 4.3 The action of scalar field in a general curved spacetime is
(4.51) 𝑆 = ∫ 𝑑4𝑥√(-𝑔) [-1/2 𝑔𝜇𝜈∂𝜇𝜙∂𝜈𝜙 - 𝑉(𝜙)],
where 𝑔 ≡ det 𝑔𝜇𝜈, It is easy to check that this reduces to (4.43) for a flat FRW metric. Under a variation of the (inverse) metric, 𝑔𝜇𝜈 → 𝑔𝜇𝜈 + 𝛿𝑔𝜇𝜈, the action changes
(4.52) 𝛿𝑆 = -1/2 ∫ 𝑑4𝑥√(-𝑔) 𝛵𝜇𝜈𝛿𝑔𝜇𝜈,
where 𝛵𝜇𝜈 is the energy-momentum tensor.
Show that the energy-momentum tensor for a scalar field is
(4.53) 𝛵𝜇𝜈 = ∂𝜇𝜙∂𝜈𝜙 - 𝑔𝜇𝜈(1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 + 𝑉(𝜙)).
and confirm that this leads to the expressions for 𝜌𝜙 and 𝑃𝜙 found above. Hints: We have to use that 𝛿√(-𝑔) = -1/2 √(-𝑔) (𝑔𝜇𝜈𝛿𝑔𝜇𝜈).
Show that the conservation of the energy-momentum tensor, ∇𝜇𝛵𝜇𝜈 = 0, implies the Klein-Gordon equation (4.47).
[Solution] Under a variation of the inverse metric, the given action changes as follow
(a) 𝛿𝑆 = ∫ d4𝑥 [𝛿√(-𝑔) (-1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 - 𝑉(𝜙)) - √(-𝑔) 1/2𝛿𝑔𝜇𝜈∂𝜇𝜙∂𝜈𝜙] = 1/2 ∫ d4𝑥 √(-𝑔) [𝑔𝜇𝜈(1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 + 𝑉(𝜙)) - ∂𝜇𝜙∂𝜈𝜙]𝛿𝑔𝜇𝜈,
where we used 𝛿√(-𝑔) = -1/2 √(-𝑔) (𝑔𝜇𝜈𝛿𝑔𝜇𝜈). Comparing this (4.52) we get
(b) 𝛵𝜇𝜈 = ∂𝜇𝜙∂𝜈𝜙 - 𝑔𝜇𝜈(1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 + 𝑉(𝜙).
Raising one index, this yield
(c) 𝛵𝜇𝜈 = 𝑔𝜇𝛼∂𝛼𝜙∂𝛽𝜙 - 𝛿𝜇𝜈(1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 + 𝑉(𝜙).
For a homogeneous field configuration 𝜙(𝑡), this takes the form of a perfect fluid
(d) 𝛵𝜇𝜈 = diag (-𝜌𝜙, 𝑃𝜙, 𝑃𝜙, 𝑃𝜙).
(e) 𝜌𝜙 = -𝛵00 = -𝑔00∂0𝜙∂0𝜙 + 𝛿00(1/2 𝑔00∂0𝜙∂0𝜙 + 𝑉(𝜙) = 1/2 ᾡ2 + 𝑉(𝜙). [with 𝑔00 = -1, ∂0 ≡ ∂/∂(-𝑡)]
(f) 𝑃𝜙 = 1/3 𝛵𝑖𝑖 = -1/3 𝛿𝑖𝑖 (1/2 𝑔00∂0𝜙∂0𝜙 + 𝑉(𝜙)) = 1/2 ᾡ2 - 𝑉(𝜙).
This confirms the expressions derived in the text. We can now compute the covariant derivative of (4.53 or c)
(g) ∇𝜇𝛵𝜇𝜈 = 𝑔𝜇𝛼𝑔𝜈𝛽∇𝜇(∂𝛼𝜙∂𝛽𝜙) - 𝑔𝜇𝜈∇𝜇(1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 + 𝑉(𝜙) = 𝑔𝜇𝛼𝑔𝜈𝛽[∂𝜇(∂𝛼𝜙∂𝛽𝜙) - (𝛤𝜌𝜇𝛼∂𝛽𝜙 + 𝛤𝜌𝜇𝛽∂𝛼𝜙)∂𝜌𝜙] - 𝑔𝜇𝜈(1/2 𝑔𝛼𝛽∂𝛼𝜙∂𝛽𝜙 + 𝑉(𝜙)).
Using that 𝑔0𝑖 = 0 and ∂𝑖𝜙 = 0, this expression simplifies to
(h) ∇𝜇𝛵𝜇𝜈 = 𝛿𝜈0∂0ᾡ2 + 𝛤0𝜇𝛼𝑔𝜇𝛼𝛿𝜈0ᾡ2 + 𝛤00𝛽𝑔𝜈𝛽ᾡ2 + 𝛿0𝜈∂0(-1/2 ᾡ2 + 𝑉(𝜙)).
Recall that the Christoffel symbols with two times indies vanish. This implies that 𝛤00𝛽 = 0 and 𝛤0𝜇𝛼 = 𝛤0𝑖𝑗𝑔𝑖𝑗 = 𝐻𝑔𝑖𝑗𝑔𝑖𝑗 = 3𝐻, so that
(i) ∇𝜇𝛵𝜇𝜈 = 𝛿𝜈0[2𝜙̇ d𝜙̇/d𝑡 + 3𝐻𝜙̇2 + (-𝜙̇ d𝜙̇/d𝑡 + 𝜙̇ d𝑉/d𝜙)] = 𝛿𝜈0𝜙̇(d𝜙̇/d𝑡 + 3𝐻𝜙̇ + d𝑉/d𝜙).
The conservation of the energy-momentum tensor, ∇𝜇𝛵𝜇𝜈 = 0, then implies the Klein-Gordon equation
(j) d𝜙̇/d𝑡 + 3𝐻𝜙̇ + d𝑉/d𝜙 = 0. ▮
4.3.2 Slow-Roll Inflation
The dynamics during inflation is then determined by a combination of the Friedmann and Klein Gordon equations
(4.54) 𝐻2 = 1/3𝑀𝑃𝑙2 [1/2 𝜙̇2 + 𝑉], [RE (2.136) and (4.48)]
(4.55) d𝜙̇/𝑑𝑡 + 3𝐻𝜙̇ = -d𝑉/d𝜙
These equations are coupled. The energy stored in the field determines the Hubble rate, [𝐻], which in turn induces friction and hence affects the evolution of the field. We would like to determine the precise conditions under which this feedback leads to inflation.
A slowly rolling field
First, (4.54) and (4.55) can be combined into an expression for the evolution of the Hubble parameter
(4.56) Ḣ = -1/2 𝜙̇2/𝑀𝑃𝑙2. [from (4.54), 2𝐻Ḣ = 1/3𝑀𝑃𝑙2 [𝜙̇ d𝜙̇/d𝑡 + d𝑉/d𝜙 𝜙̇] → Ḣ = 1/6𝐻Ḣ𝑀𝑃𝑙2 [𝜙̇ d𝜙̇/d𝑡 - (d𝜙̇/d𝑡 + 3𝐻𝜙̇)𝜙̇] = -1/2 𝜙̇2/𝑀𝑃𝑙2.]
Talking the ratio of (4.56) and (4.54), we then find
(4.57) 𝜀 ≡ -Ḣ/𝐻2 = (1/2 𝜙̇2)/𝑀𝑃𝑙2𝐻2 = (3/2 𝜙̇2)/(1/2 𝜙̇2 + 𝑉).
Inflation (𝜀 ≪ 1) therefore occurs if the kinetic energy density, 1/2 𝜙̇2, only makes a small contribution to the total energy density, 𝜌𝜙 = 1/2 𝜙̇2 + 𝑉. For obvious reasons, this situation is called slow-roll inflation.
In order for slow-roll behavior to persist, the acceleration of the scalar field also has to be small. To assess this, it is useful to define the dimensionless acceleration per Hubble time
(4.58) 𝛿 ≡ -(d𝜙̇/d𝑡)/𝐻𝜙̇.
When 𝛿 is small, the friction term in (4.55) dominates and the inflation speed is determined by the slope of the potential. As long as the inflation kinetic energy stays subdominant and the inflationary expansion continues. To see this more explicitly, we take the time derivative of (4.57),
(4.59) 𝜀̇ = 𝜙̇ (d𝜙̇/d𝑡)/𝑀𝑃𝑙2𝐻2 - 𝜙̇2Ḣ/𝑀𝑃𝑙2𝐻3,
and substitute it into (4.36):
(4.60) 𝜅 = 𝜀̇/𝐻𝜀 = 2 (d𝜙̇/d𝑡)/𝐻𝜙̇ - 2 Ḣ/𝐻2 = 2(𝜀 - 𝛿).
This shows that {𝜀,∣𝛿∣} ≪ 1 implies {𝜀,∣𝜅∣} ≪ 1. If both the speed and the acceleration of the inflation field are small, then the inflationary expansion will last for a long time.
Slow-roll approximation
Now, we will use these condition to simplify the equations of motion. This is called the slow-roll approximation.
First, we note that 𝜀 ≪ 1 implies 𝜙̇2 ≪ 𝑉, which leads to the following simplification of Friedmann equation (4.54)
(4.61) 𝐻2 ≈ 𝑉/3𝑀𝑃𝑙2.
Next, we see that the condition ∣𝛿∣ ≪ 𝑉 simplifies the Klein-Gordon equation (4.55) to
(4.62) 3𝐻𝜙̇ ≈ -𝑉,𝜙,
where 𝑉,𝜙 ≡ d𝑉/d𝜙. This provides a simple relationship between the slope of the potential and the speed of the inflation. Finally, substituting (4.61) and (4.62) into (4.57) gives
(4.63) 𝜀 = (1/2 𝜙̇2)/3𝑀𝑃𝑙2𝐻2 ≈ 𝑀𝑃𝑙2/2 (-𝑉,𝜙/𝑉),
which express the parameter 𝜀 purely in terms of the potential.
To evaluate the parameter 𝛿 in (4.58), in the slow-roll approximation, we take the time derivative of (4.62), 3Ḣ𝜙̇ + 3𝐻d𝜙̇/d𝑡 = -𝑉,𝜙𝜙ᾡ. This leads to
(4.64) 𝜂 ≡ 𝛿 + 𝜀 = -(d𝜙̇/d𝑡
)/𝐻𝜙̇ -Ḣ/𝐻2 ≈ 𝑀𝑃𝑙2 𝑉,𝜙𝜙/𝑉.
Hence, a convenient way to judge whether a given potential 𝑉(𝜙) can leads to slow-roll inflation is to compute the potential slow-roll parameters
(4.65) 𝜀𝑉 ≡ 𝑀𝑃𝑙2/2 (𝑉,𝜙/𝑉)2, 𝜂𝑉 ≡ 𝑀𝑃𝑙2 𝑉,𝜙𝜙/𝑉.
Successful inflation occurs when these parameters are much smaller than unity.
Exercise 4.4 Show that in the slow-roll regime, the potential slow-roll parameters and the Hubble slow-roll parameters are related as follows:
(4.66) 𝜀𝑉 ≈ 𝜀 and 𝜂𝑉 ≈ 2𝜀 - 1/𝜅.
[Solution] In the slow-roll approximation, the Friedmann and Klein-Gordon equations are
(a) 𝐻2 ≈ 𝑉/3𝑀𝑃𝑙2, 3𝐻𝜙̇ ≈ -𝑉,𝜙
(b) 𝜀 = (1/2 𝜙̇2)/3𝑀𝑃𝑙2𝐻2 ≈ 𝑀𝑃𝑙2/2 (-𝑉,𝜙/𝑉), 𝜅 = 𝜀̇/𝐻𝜀 = 2 (d𝜙̇/d𝑡)/𝐻𝜙̇ - 2 Ḣ/𝐻2 = 2(𝜀 - 𝛿), 𝛿 ≡ -(d𝜙̇/d𝑡)/𝐻𝜙̇.
In the slow-roll approximation we have
(c) 𝛿 + 𝜀 = --(𝑑𝜙̇/𝑑𝑡)/𝐻𝜙̇ - -Ḣ/𝐻2 ≈ 𝑀𝑃𝑙2 𝑉,𝜙𝜙/𝑉 ≡ 𝜂𝑉
and hence
(d) 𝜀𝑉 ≈ 𝜀, 𝜂𝑉 ≈ 𝛿 + 𝜀 = -(𝜅/2 - 𝜀) + 𝜀 = 2𝜀 - 1/2 𝜅. ▮
The total number of 𝑒-foldings of accelerated expansion is
(4.67) 𝑁tot ≡ ∫𝑎𝑖𝑎𝑒 𝑑 ln 𝑎 = ∫𝑡𝑖𝑡𝑒 𝐻(𝑡) d𝑡 = ∫𝜙𝑖𝜙𝑒 𝐻/𝜙̇ 𝑑𝜙,
where 𝑡𝑖 and 𝑡𝑒 are defined as the time when 𝜀(𝑡𝑖) = 𝜀(𝑡𝑒) ≡ 1. In the slow-roll regime, we can use (4.63) to write the integral over the field space as
(4.68) 𝑁tot ≈ ∫𝜙𝑖𝜙𝑒 1/√(2𝜀𝑉) ∣𝑑𝜙∣/𝑀𝑃𝑙,
where 𝜙𝑖 and 𝜙𝑒 are the field values at the boundaries of the interval where 𝜀𝑉 < 1. As we have seen above, a solution to the horizon problem requires 𝑁tot ≳ 60, which provide an important constraint on successful inflation models.
Case study: quadratic inflation
As an example, let us give the slow-roll analysis of arguably the simplest model of inflation: single-field inflation driven by a mass term
(4.69) 𝑉(𝜙) = 1/2 𝑚2𝜙2.
As we will see in Chapter 8, this models is ruled out by CMB observation, but it still provides a useful example to illustrate the mechanism of slow-roll inflation. The slow-roll parameter for the potential are
(4.70) 𝜀𝑉(𝜙) = 𝜂𝑉(𝜙) = 2 (𝑀𝑃𝑙/𝜙)2.
To satisfy the slow-roll conditions {𝜀𝑉, ∣𝜂𝑉∣} < 1, we need to consider super-Planckian value for the inflation
(4.71) 𝜙 > √2𝑀𝑃𝑙 ≡ 𝜙𝑒.
Let the initial field value be 𝜙𝑖, The 𝑒-foldings of inflationary expansion are
(4.72) 𝑁tot = ∫𝜙𝑖𝜙𝑒 d𝜙/𝑀𝑃𝑙 1/√(2𝜀𝑉) = 𝜙2/4𝑀𝑃𝑙2∣𝜙𝑒𝜙𝑖 = 𝜙𝑖2/4𝑀𝑃𝑙2 - 1/2.
To obtain 𝑁tot > 60, the initial field value must satisfy
(4.73) 𝜙𝑖 > 2√60 𝑀𝑃𝑙 ~ 15𝑀𝑃𝑙.
We note that the total field excursion is super-Planckian, ∆𝜙 = 𝜙𝑖 - 𝜙𝑒 ≫ 𝑀𝑃𝑙. In Section 4.4.1, we will discuss whether this large field variation should be a cause for concern.
4.3.3 Creating the Hot Universe
Most of the energy density during inflation is the form of the inflaton potential 𝑉(𝜙). Inflation ends when the potential sleeps and the field picks up kinetic energy. The energy in the inflation sector then has to transferred to the particles of the Standard Model. This process is called reheating and starts the hot BB.
Once the inflation field reaches the bottom of the potential, it begins to oscillate. Near the minimum, the potential can be approximated as 𝑉(𝜙) ≈ 1/2 𝑚2𝜙2 and the equation of motion of the inflation is
(4.74) d𝜙̇/d𝑡 + 3𝐻𝜙̇ = -𝑚2𝜙. [RE (4.47) ]
The energy density evolves according to the continuity equation [RE (4.50) (4.69)]
(4.75) 𝜌̇𝜙 + 3𝐻𝜌𝜙 = -3𝐻𝑃𝜙 = 3/2 𝐻 (𝑚2𝜙2 - ᾡ2),
where the right-hand side averages to zero over one oscillation period. This averaging has ignored the Hubble friction in (4.74), which is justified on timescales shorter than the expansion time. The oscillating field behaves like pressureless matter, with 𝜌𝜙 ∝ 𝑎-3 (See problem 4.1 for the derivation). As the energy density drops, the amplitude of the oscillation decreases.
To avoid that the universe ends up completely empty, the inflaton has to couple to Standard Model fields. The energy stored in the inflaton filed will then be transferred to ordinary particles. If the decay is slow, then the inflaton's energy density follows the equation
(4.76) 𝜌̇𝜙 + 3𝐻𝜌𝜙 = -𝛤𝜙𝜌𝜙,
where 𝛤𝜙 parameterizes the inflaton decay rate. A slow decay of the inflaton typically occurs if the coupling is only to fermions. If the inflaton can also decay into bosons then the decay rate may be enhanced by Bose condensation and parametric resonance effects. This kind of rapid decay is called preheating, since the bosons are created far from thermal equilibrium.
the new particles will interact with each other and eventually reach the thermal state that characterizes the hot Big Bang. The energy density at the end of reheating epoch is 𝜌𝑅 < 𝜌𝜙,𝑒 is the energy density at the end of inflation, and reheating temperature 𝛵𝑅 is determined by
(4.77) 𝜌𝑅 = π2/30 𝑔*(𝛵𝑅) 𝛵𝑅4.
If reheating takes a long time, then 𝜌𝑅 ≪ 𝜌𝜙,𝑒 and the reheating temperature gets smaller. At a minimum, the reheating temperature has to be larger than 1 MeV to allow for successful BBN, and most likely it is much larger than this to also allow for baryongenesis after inflation.
This completes our highly oversimplified sketch of the reheating phenomenon. In reality, the dynamics during reheating can be very rich, often involving non-perturbative effects captured by numerical simulation (for review see [2-5]). Describing this phase in the history is essential for understanding how the hot Big Bang began.With accelerated expansion having ended, there is no "cosmic amplifier" to make the microscopic physics of reheating easily accessible on cosmological length scales. Moreover, the high energy scale associated with the end of inflation, as well as the subsequent thermalization, can further hide details of this era from our low-energy probes. Nevertheless, in some cases, the dynamics during this period can generate relics such as isocurvature perturbations, stochastic gravitational waves, non-Gaussianities, dark matter/radiation, primordial black holes, topological and non-topological solitons, matter/antimatter asymmetry, and primordial magnetic fields. Detecting any of these relics would give us an interesting window into the reheating era.
4.4 Open Problems*
Despite its phenomenological success, inflation is not yet a complete theory. To achieve inflation we had to postulate new physics at energies far above those probed by particle colliders and its success is sensitive to assumption about the physics at even higher energies.
4.4.1 Ultraviolet Sensitivity
The most conservative way to describe the inflationary era is in terms of an effective field theory (EFT). In the EFT approach, we admit that we don't know the details of the high-energy theory and instead parameterize our ignorance. We begin by defining the degrees of freedom and symmetries of the inflationary theory. This theory is valid below a cutoff scale 𝛬, and we should ask how the unknown physics above the scale 𝛬 can affect the low-energy dynamics during inflation. This means writing down all possible corrections to the low-energy theory that are consistent with the assume symmetries. The effective Lagrangian of the theory can then be written as
(4.78) 𝓛eff[𝜙] = 𝓛0[𝜙] + ∑𝑛 𝑐𝑛 𝑂𝑛[𝜙]/𝛬𝛿𝑛 - 4,
where 𝓛0 is the Lagrangian of the inflationary model and 𝑂𝑛 are a set of "operators" that parameterize the corrections coming from the couplings to additional high-energy degrees of freedom. The dimensionless parameters 𝑐𝑛 are usually taken to be of order one.8 If the cutoff scale 𝛬 is much larger than the typical energy 𝐸 at which the low-energy theory is being probed, then the corrections are small, suppressed by powers of 𝐸/𝛬. This is why quantum gravity doesn't affect our everyday lives, or even those of our friends working at particle colliders.9 A remarkable feature of inflation, however, is that it is extremely sensitive even to effects suppressed by the Planck scale.
We have seen that slow-roll inflation requires a flat potential (in Planck scale), but his is hard to control and sensitive to very small corrections. In the absence of any special symmetries, the EFT of slow-roll inflation takes the following form
(4.79) 𝓛eff[𝜙] = -1/2 (𝜕𝜇𝜙)2 - 𝑉(𝜙) - ∑𝑛 𝑐𝑛 𝑉(𝜙) 𝜙2𝑛/𝛬4𝑛 + ∙ ∙ ∙ ,
where a representative set of higher-dimension operators is written. If the inflation field value is smaller than the cutoff scale we can truncate the 𝐸𝐹𝛵 expansion in (4.79) and the leading effect comes from the dimension-six operator
(4.80) ∆𝑉 = 𝑐1𝑉(𝜙) 𝜙2/𝛬2.
Since 𝜙 ≪ 𝛬, this is a small correction to the inflation potential, ∆𝑉 ≪ 𝑉. Nevertheless, it affects the delicate flatness of the potential and hence is relevant for the dynamics during inflation. In particular, the slow-roll parameter 𝜂𝑉 receives the correction
(4.81) ∆𝜂𝑉 = 𝑀𝑃𝑙2/𝑉 (∆𝑉),𝜙𝜙 ≈ 2𝑐1 (𝑀𝑃𝑙/𝛬)2 > 1,
where the final inequality comes from 𝑐1 = 𝑂(1) and 𝛬 < 𝑀𝑃𝑙. Notice that this problem is independent of the energy scale of inflation. All inflationary models have to address this so-called eta problem.
One promising way to solve the eta problem is to postulate a shift symmetry for he inflation, 𝜙 → 𝜙 + 𝑐, where 𝑐 is constant. If this symmetry is respected by the couplings to massive fields, then having 𝑐𝑛 ≪ 1 is technically natural [6]. Another way to ameliorate the eta problem is supersymmetry. Above the scale 𝐻, the theory is supersymmetric and the contribution from bosons and fermions precisely cancel as a minor miracle. However, supersymmetry is spontaneously broken during inflation, leading yo an inflaton mass of order the Hubble scale, 𝑚𝜙 ~ 𝐻, and an eta parameter of order one, ∆𝜂𝑉 ~ 1. Although the eta problem is still there, it is much less severe and requires just a percent-level fine-tuning of inflaton mass parameter.
We have seen that in quadratic inflation the inflaton moves over a super-Planckian distance in field space, ∆𝜙 > 𝑀𝑃𝑙. This is a general feature of all inflationary models with observable gravitational wave signals. The sper-Planckian excursion of the field should make us nervous. When the field value isn't smaller than the cutoff, we cannot trust the EFT expansion in (4.79). Models of large-field inflation therefore usually invoke symmetries-such as an approximate shift symmetry 𝜙 → 𝜙 + 𝑐-to suppress the couplings of the inflaton to massive degree off freedom and reduce the size of the corrections in (4.79). An important question is whether the assumed symmetries are respected by the UV-completion.
The UV sensitivity of inflation is a challenge, because we either need to work in a UV-complete theory of quantum gravity or make assumptions about the form that such a UV-completion might take. It is also an opportunity,, because it suggests the exciting possibility of using cosmological observations to learn about fundamental aspects of quantum gravity [7].
4.4.2 Initial Conditions
First, Imagine that the inflation field initially takes different values in different regions of space, some at top of the potential, some at the bottom. The regions of space with large vacuum energy will inflate and grow. Weighted by volume, these regions will dominate, explaining why most f space experience inflation. Alternatively, it is sometimes assumed that slow-roll inflation was proceeded by an earlier phase of false vacuum domination (see Fig. 4.9. In quantum mechanics, there is a small probability that the field will tunnel through the potential energy barrier. so the question becomes why the quantum mechanical tunneling is more likely to put the inflaton field at the top of the potential than at its minimum. We should still ask why the universe started in the high-energy false vacuum. Most ambitiously, the no-boundary proposal by Hartle and Hawking gives a prescription for evaluating the provability that a universe is spontaneously created from nothing [8].
Second, if the initial inflaton velocity is non-negligible, then it is possible that the field will overshoot the region of the potential where inflation is supposed to occur, without actually sourcing accelerated expansion. In small-field models the Hubble friction is often not efficient enough to slow down the field before it reaches the region of interest. But in large-field models, on the other hand, Hubble friction is usually very efficient and the slow-roll solution becomes an attractor.
Finally, since inflation is supposed to explain the homogeneous initial conditions of the hot BB, we must ask what happens when allow for large inhomogeneities in the inflaton field, These inhomogeneities carry gradient energy, which might hinder the accelerated expansion. The sensitivity of an inflationary model to initial perturbations of the inflaton field is best addressed by numerical simulations. In 1989 the earliest such simulations were performed in 1 + 1 dimensions [9, 10]. Recently the problem was revisited using simulation in 3 + 1 dimensions [11, 12]. It was again found that large-field models are more robust than small-field models.
4.4.3 Eternal Inflation
A rather dramatic consequence of the quantum dynamics of inflation may be that globally that it never ends, but is eternal. There are two types of eternal inflation, both are illustrated in Fig. 4.9. First, inflation can occur while the field is stuck in false vacuum. Any region of space has a finite probability to decay and reach the true vacuum via quantum tunneling. If the decay rate is 𝛤, then the probability that the region will survive in the inflationary state is 𝑒-𝛤𝑡. The volume of inflating region increases as 𝑒3𝑡. If the rate of exponential expansion is larger than the rate of decay, then inflation does not end globally. There are many pockets of space where inflation does end-one of these pockets is so-called our "universe"-but they are embedded in a much larger "multiverse." The space between the pocket universes is rapidly expanding, creating new space, where new pocket universes can form. This never ends.
We will study the effects of quantum mechanics on the dynamics of inflation and see that the field experiences random fluctuations up and down the potential. Eternal inflation occurs if these quantum fluctuation dominate over the classical rolling. Consider the evolution of the field in one Hubble volume 𝐻-3 during one Hubble time ∆𝑡 = 𝐻-1. Classically, the field will change by ∆𝜙𝑐𝑙 ~ ᾡ𝐻-1. At the same time, quantum mechanics induces random jitters ∆𝜙𝑄, so that the total change of the field is
(4.82) ∆𝜙 = ∆𝜙𝑐𝑙 + ∆𝜙𝑄.
How big does this probability have to be in order for inflation to be eternal? During a Hubble time, the volume increases by a factor of 𝑒3 ≈ 20, meaning that the inflationary region breaks up into 20 Hubble-sized regions. As we will see in Chapter 8, averaged over one of these 20 regions, inflationary quantum fluctuations have a Gaussian probability distribution with width 𝐻/2π. If the probability to have ∆𝜙 < 0 is greater than 1 in 20, then the volume of space that is inflating will increase. When
(4.83) ∆𝜙𝑄 = 𝐻/2π > 0.6 ∆𝜙𝑐𝑙 = 0.6 ∣𝜙̇∣𝐻-1.
If the condition is satisfied, inflation will continue forever, This arguments for slow-roll eternal inflation was admittedly somewhat heuristic and it is sill being debated the reliability.
Exercise 4.5 Show that, in quadratic inflation, slow-roll eternal inflation occurs for
(4.84) 𝜙2/𝑀𝑃𝑙2 ≳ 13 𝑀𝑃𝑙/𝑚.
In Chapter 8, we will find that in order for the predicted CMB fluctuations to have the right amplitude, we must have 𝑚 ~ 10-6 𝑀𝑃𝑙. This implies 𝜙 > 3600 𝑀𝑃𝑙. In that case the regime of eternal inflation would not exist.
[Solution] The condition for eternal inflation is and assuming a quadratic potential 𝑉(𝜙) = 𝑚2𝜙2/2
(a) ∆𝜙𝑄 = 𝐻/2π > 0.6 ∆𝜙𝑐𝑙 = 0.6 ∣𝜙̇∣𝐻-1 ⇒ 𝐻2 > 0.6 × 2π∣𝜙̇∣ ≈ 3.8∣𝜙̇∣
(b) 𝐻2 ≈ 𝑉/3𝑀𝑃𝑙2 = 𝑚2𝜙2/3𝑀𝑃𝑙2
(c) ∣𝜙̇∣ ≈ ∣𝑉,𝜙∣/3𝐻 ≈ √3𝑀𝑃𝑙∣𝑉,𝜙∣/3√𝑉 = 𝑀𝑃𝑙 𝑑(𝑚2𝜙2/2)/d𝜙 × 1/[√(3/2)𝑚𝜙] = √(2/3) 𝑚𝑀𝑃𝑙.
The condition 𝐻2 > 3.8∣𝜙̇∣ implies
(d) 𝑚2𝜙2/3𝑀𝑃𝑙2 > 3.8√(2/3)𝑚𝑀𝑃𝑙 ⇒ 𝜙2/𝑀𝑃𝑙2 > 6/𝑚2 × 3.8√(2/3)𝑚𝑀𝑃𝑙 = 3.8√24 𝑀𝑃𝑙/𝑚 ≈ 18.6 𝑀𝑃𝑙/𝑚.
Using 𝑚 ~ 10-6𝑀𝑃𝑙, this implies 𝜙 > 4300 𝑀𝑃𝑙. ▮
The concept of eternal inflation is almost as old as inflation itself, it recently revived in the context of string theory. the string theory seems to give rise to a large number of metastable solutions.The space vacua is called the string landscape. The physics may be different in each of these vacua. Eternal inflation provides the mechanism by which this landscape populated. The fact that eternal inflation combined with the string landscape allows for a solution the the cosmological constant problem (see Section 2.3.3) is the single most important reason to take either of these ideas seriously.
Let us finally address the elephant in the room: the measure problem of the inflationary multiverse. We have seen that eternal inflation produces an infinite number of universe. The fraction of universes with a certain observable property is equal to infinity divided by infinity. Unfortunately, the answer tends to depend sensitively on type of regulator that is being used.
Although we have remarkable observational evidence that something like inflation occurred in the early universe (see Chapter 8), inflation cannot yet be considered a part of the standard model of cosmology, Inflation is still a work in progress.
4.5 Summary
In this chapter, we have studied inflation. We stared with a discussion of the causality problems of the hot BB, highlighting the crucial role played by the evolution of the comoving Hubble radius, (𝑎𝐻)-1. We showed that this evolution determines the particle horizon-the maximal distance that light can propagate between an initial time 𝑡𝑖 and a later time 𝑡:
(4.85) d𝘩(𝜂) = ∫𝑡𝑖𝑡 d𝑡/𝑎(𝑡) = ∫𝑁𝑖𝑁 (𝑎𝐻)-1 d𝑁,
where 𝑁 = ln 𝑎.Since the comoving Hubble radius is monotonically increasing in the conventional BB theory, the particle horizon is dominated by the contributions from late times. Most parts of the CMB would then never have been causal contact, yet they have nearly the same temperature. This is the horizon problem.
Inflation is a period of accelerated expansion during which the comoving Hubble radius is shrinking. The particle horizon receives most of its contribution from early times and can be much larger than naively assumed. Causal contact is established in the time before the hot BB. The shrinking of the comoving Hubble radius also solves the closely related flatness problem and allows a causal mechanism to generate superhorizon correlations.
Inflation occurs if all physical quantities vary slowly with time. For example, the fractional change of the Hubble rate during inflation must be small,
(4.86) 𝜀 ≡ -Ḣ/𝐻2 ≪ 1.
To solve the problems of the ht BB, this condition must be maintained for about 60 𝑒-foldings of expansion. A popular way to achieve this is through a slowly rolling scalar field (the inflaton). The evolution of the inflaton is governed by the Klein-Gordon equation
(4.87) d𝜙̇/d𝑡 + 3𝐻𝜙̇ = -𝑉,𝜙.
The field rolls slowly if the dynamics is dominated by the friction term, ∣d𝜙̇/d𝑡∣ ≪ ∣3𝐻𝜙̇∣ ≈ ∣𝑉,𝜙∣. The size of the friction is determined by the energy density associated with the inflation itself, which by the Friedmann equation determines the expansion rate
(4.88) 𝐻2 = 1/3𝑀𝑃𝑙2 [1/2 𝜙̇2 + 𝑉].
Inflation requires that the kinetic energy is much smaller than the potential energy, 1/2 ᾡ2 ≪ 𝑉. The conditions for successful slow-roll inflation can be characterized by the slow-roll parameters
(4.89) 𝜀𝑉 ≡ 𝑀𝑃𝑙2/2 (𝑉,𝜙/𝑉)2, 𝜂𝑉 ≡ 𝑀𝑃𝑙2 𝑉,𝜙𝜙/𝑉.
Inflation will occur in regions of potential where {𝜀𝑉, ∣𝜂𝑉∣} ≪ 1. Achieving such flat potentials in a theory of quantum gravity is challenging.
7 This parameter is often denoted 𝜂. the parameter 𝜀 and 𝜅 are often called Hubble slow-roll parameters to distinguish potential slow-roll parameters defined in Section 4.3.1.
8 Some new degrees of freedom must appear below the Planck scale, 𝛬 ≲ 𝑀𝑃𝑙, because gravity is not renormalizable and needs a high-energy (or "ultraviolet") completion. If 𝜙 has order-one couplings to any heavy fields, with masses of order 𝛬, then integrating out these fields yields the effective action (4.78), with order-one couplings 𝑐𝑛.
9 Note that the largest energies proved at the Large Hadron Collier (LHC) are still tiny relative to the Planck scale, 𝐸/𝑀𝑃𝑙 < 10-16.
