The basic idea of inflation is that very, very shortly (t≈10−35 seconds) after the Big Bang, the universe underwent a period of accelerated expansion (¨a >0). The universe increased in size by at least 60 e-folds in much less than a second. The effect of this expansion was to create an apparently flat, isotropic, and homogenous universe.
1.3.1 Basics
This section follows the arguments made in [15]. Another way to think about inflation is as a phase of decreasing Hubble radius (1/aH). The particle horizon ∆rmax(the maximum distance a particle can travel between an initial timeti = 0 and a later timet > ti) depends on evolution of the comoving Hubble radius:
∆rmax=c Z t
0
dt0 a(t0) =
Z
(aH)−1 d lna. (1.20)
A decreasing Hubble radius means that large scales that enter the universe at the present time were inside the horizon prior to inflation, and so had time to become homogenous. We describe a decreasing Hubble radius as
d
dt(aH)−1 <0. (1.21)
We will see that this statement implies accelerated expansion. We begin by evaluating the expression on the left-hand side:
d
dt(aH)−1 = d
dt( ˙a)−1 =−a¨
˙
a2. (1.22)
Then,
− a¨
˙
a2 <0⇒¨a >0, (1.23) which is our expression for accelerated expansion.
Another way to evaluate Eqn. 1.21 is d
dt(aH)−1 =−aH˙ +aH˙ (aH)2 =−1
a(1−), (1.24)
where
=−H˙
H2. (1.25)
Using the same inequality relation as above (Eqn. 1.21), we find that the necessary condition for inflation also corresponds to
− 1
a(1−)<0⇒ <1. (1.26) To solve the cosmological problems outlined in the previous section, we not only want the universe to exponentially expand, we also want it to expand for a sufficiently long period of time. That is, we wantto remain small through a minimum of 50-60 e-fold expansions of the universe. To achieve this condition, we define another parameter:
η= ˙
H. (1.27)
When|η|<1, the fractional change inper Hubble time is small, and inflation persists for
Figure 1.1: A slowly rolling inflaton potential. From [15].
many e-foldings.
As mentioned in §1.1.1, for cosmological calculations we typically assume the energy- momentum tensor is of the form that describes a perfect fluid. The Friedmann Equations can be recast to use the pressurep and densityρ of this fluid:
H2 = 1
2Mpl2ρ (1.28)
H˙ +H2=− 1
6Mpl2(ρ+ 3p), (1.29)
whereMpl is the Planck mass andH is the Hubble parameter.
If we use this form of the Hubble equations in the definition of (Eqn. 1.25), we find =−H˙
H2 = 3 2
1 +p
ρ
<1⇒ p ρ <−1
3, (1.30)
which indicates that inflation requires a negative pressure.
1.3.2 Slow-roll Inflation
So far we have not discussed the physics that lead to the conditions ( <1 and|η|<1) that ensure inflation. The most common inflationary theories are those involving slowly rolling scalar fields (see Fig. 1.1). In these theories, we consider a scalar field φ (the “inflaton”) with a potentialV(φ). The following discussion relies on [87].
The energy momentum tensor for φis Tµν = (∇µφ)(∇νφ)−gµν
1
2gαβ(∇αφ)(∇βφ) +V(φ)
. (1.31)
If we simplify to the homogenous case, all quantities depend only on the time, t. We can also set k = 0 by the following reasoning — even if the universe started out with a significant curvature, inflation will quickly drive it towards flatness and, hence, k = 0. A homogenous, real scalar field behaves as a perfect fluid with density and pressure given by
ρφ= 1
2φ˙2+V(φ) (1.32)
pφ= 1 2
φ˙2−V(φ). (1.33)
From these equations it is clear that if ˙φ2<< V(φ), then the potential of the scalar field will dominate both the pressure and density, with the result that ρφ ' −pφ. This is the same situation as described at the end of §1.3.1, which, as we have already noted, results in an accelerated expansion of the universe.
The time evolution of the scalar field is described by the Klein-Gordon equation, which can be thought of as the equation of motion for a scalar field in Minkowski space, but with a friction term due to the expansion of the universe:
φ¨+ 3a˙ a
φ˙+dV
dφ = 0. (1.34)
To simplify the equation of motion, we make the slow-roll approximation. One of the assumptions made in the slow-roll approximation is that ¨φ ≈ 0. So this equation can be rearranged as
φ˙' −dV /dφ
3H . (1.35)
The Friedmann equation with such a field as the sole energy source is H2 = 8πG
3 1
2
φ˙2+V(φ)
. (1.36)
The other assumption made by using the slow-roll approximation is to neglect the kinetic
energy of φ compared to the potential energy (drop the ˙φ2 in the equation above). The Friedmann equation then becomes
H2' 8πG
3 V(φ). (1.37)
Plugging Eqn. 1.37 and Eqn. 1.35 into the definitions of and η gives = Mp2
2 V0
V 2
(1.38) and
η≡Mp2V00
V , (1.39)
whereMp is the Planck mass and the prime mark indicates a derivative with respect to φ.
The slow roll conditions are satisfied if||<<1 and|η|<<1.
1.3.3 Perturbations
During inflation, quantum fluctuations in the inflaton field are expanded to cosmological scales. The decay of the inflaton field results in a spectrum of remnant density and grav- itational wave perturbations. Most inflationary models predict that this spectrum will be scale invariant (the same at all wavelengths).
Scale invariance implies that the inflaton field experiences fluctuations that are the same for every wavenumber, δφk = constant. Those fluctuations can be related to the ones in density by
δρ= dV
dφδφ. (1.40)
From this we expect nearly scale-invariant density perturbations (the scale factor dVdφ evolves with time, which is why we do not get exactly scale-invariant perturbations). The density perturbation (or “scalar”) spectrum is related to the inflationary potential as follows:
A2S≈ V3 Mp6(V0)2
k=aH
, (1.41)
where k = aH indicates that the values of V0 and V are to be evaluated at the moment when the physical scale of the perturbation λ= a/k is equal to the Hubble radius H−1. These density fluctuations are also known as scalar fluctuations, since they are scalar fluc-
tuations of the metric. The density fluctuations produced by inflation are adiabatic (by which we mean that perturbations in the density of all components of the universe are correlated), Gaussian, and uncorrelated (i.e., the phases of the Fourier modes describing the fluctuations at different scales are uncorrelated). It should be noted that inflation does predict some small amount of nongaussianity, but the fluctuations should be nearly Gaus- sian. These predictions of inflation — an adiabatic, nearly scale-free spectrum of density perturbations with a Gaussian distribution, have been confirmed to new precision by the Planck instrument [3, 2]. The graviton is also excited during inflation, which creates tensor perturbations in the metric, or gravitational waves. Their spectrum is described by
A2T ≈ V Mp4
k=aH
. (1.42)
The existence of tensor perturbations is one of the most crucial predictions of the theory of inflation, since it can be observationally verified via measurements of the polarization of the cosmic microwave background (to be explained in the next section). Note that the tensor perturbation spectrum depends only on the potential V, and not its derivatives. So observations of the tensor modes are directly related to the energy scale of inflation:
Vinflation1/4 ∼1/41016 GeV, (1.43)
where the calibration factor comes from the measurements of AS from the COBE experi- ment.
The ratio of the tensor and scalar perturbation spectra, known as the tensor-to-scalar ratior can be related back to the slow roll parameters:
r= A2T
A2S = constant×, (1.44)
where the constant depends on the exact approximations used to calculate the tensor and scalar spectra and typically takes a value between 12 and 16. By combining Eqn. 1.43 and Eqn. 1.44, we can get an equation that directly relates the tensor-to-scalar ratio, r to the inflationary potential:
Vinflation1/4 ∼ r 10
1/4
1016 GeV. (1.45)
Figure 1.2: The anisotropies of the CMB as observed by Planck. Image courtesy of ESA and Planck Collaboration.