Perturbations in the inflationary epoch

INFLATION

1.4 Perturbations in the inflationary epoch

While inflation provides a convenient picture for explaining away the horizon and flatness problems, inflation really gains traction by providing a mechanism for generating the near scale-invariant fluctuations that are observed throughout the universe. Perturbations arrise from zero-point vacuum fluctuations in the metric, which are stretched to cosmological scales by the process of inflation.

The amplitude and scale invariance of these fluctuations are critical for the growth of large-scale structure in the universe, and understanding the precise details of the origin of these fluctuations remains a critical challenge for modern cosmology. We describe in broad terms the predicted origin of perturbations from inflation. This discussion follows closely the arguments presented in Dodelson 2003.

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1.4.1 Tensor perturbations

We begin by reviewing a few key principles of the quantum simple harmonic oscillator. In particular, we would like to remind ourselves how the ground state position variance is related to the frequency.

We start by writing down the wave equation:

¨

x+!²x= 0. (1.36)

When quantized, the oscillator (famously) has non-zero zero-point energy and a non-zero ground state variance, which is given by:

h|x|²i=h0|X^†X|0i. (1.37) The position operatorX can be rewritten in terms of the ladder operators:

X = r 1

2!(a^†+a), (1.38)

which allows us to rewrite the variance:

h|x|²i= 1

2!h0|(a+a^†)(a^†+a)|0i. (1.39) Asa|0i= 0, and the ladder operators obey the commutation relation[a, a^†] = 1, we find that:

h|x|²i= 1

2!. (1.40)

We will use this result when deriving the variance of the tensor perturbations.

Quantum metric perturbations are the result of zero-point vacuum fluctuations, analogous to the quantized simple harmonic oscillator. The dynamics of inflation provide a mechanism whereby metric perturbations at the quantum scale are stretched to cosmological scales.

We consider a generic perturbation to the space-space component of the metric. The perturbation can be represented as a symmetric trace-free matrix (symmetric to satisfy Einstein’s field equation, traceless because we are considering a perturbation). As a result, the matrix can be uniquely decomposed into scalar, vector, and tensor components. We will consider tensor perturbations first, as they are the simplest. This is because even in the presence of matter, each tensor modekevolves independently. It can be shown that the amplitudeh for some given mode kevolves according to the wave equation:

¨h+ 2Hh˙ +k²

a²h= 0. (1.41)

Here derivatives are taken with respect to physical timet. This wave equation comes from considering

the space-space component of the Einstein equation, the details of which can be found in Liddle and Lyth 2000. It is illustrative to write this wave equation in terms of derivatives with respect to conformal time⌧:

d²h d⌧² +2

a da d⌧

dh

d⌧ +k²h= 0. (1.42)

To calculate the variance of the wave equation given by 1.42 in various limiting cases, we will perform a change of variable to simplify the differential equation. Our aim is to remove the term proportional todh/d⌧, allowing us to make a connection with the simple harmonic oscillator. As in Dodelson, we find that this is made possible by the following change in variable (similar in nature to changing to a co-moving quantity):

˜h⌘Mpl

p2ah. (1.43)

This substitution leads to the much simplified wave equation:

d²˜h d⌧² +

✓ k² 1

a d²a d⌧²

◆

˜h= 0. (1.44)

Drawing from our experience with the simple harmonic oscillator, we can write down the wave equation in terms of ladder operators, just as we did in Equation 1.38 for the quantum oscillator:

H˜ =v(k,⌧)a+v^⇤(k,⌧)a^†, (1.45)

with the variance given by:

h|˜h|²i=|v(k,⌧)|². (1.46) Let’s consider the solution to this equation in the scenario of slow-roll inflation. In this case, da/d⌧=a²H' a/⌧. We arrive at the even further simplified wave equation:

d²v d⌧² +

✓ k² 2

⌧²

◆

v= 0, (1.47)

which has the solution:

v=e ^ik⌧

p2k

✓

1 i

k⌧

◆

. (1.48)

We can now find the variance for the two limiting cases described earlier. Fork⌧ 1, Equation 1.48 reduces to the familiar harmonic oscillator. The variance, in analogy with Equation 1.40 is simply:

|v(k,⌧)|²= 1

2k. (1.49)

18 Similarly, fork⌧⌧1:

|v(k,⌧)|²= 1 2k

✓ 1 k²⌧²

◆

. (1.50)

We now define the power spectrumPh(k)as:

Ph(k)⌘ 2

M_pl²a²|v(k,⌧)|². (1.51)

By considering the two limiting cases above and substituting⌧ = (aH) ¹, we find:

Ph(k) = H²

k³M_pl², k⌧⌧1 (superhorizon) (1.52) Ph(k) = 1

ka²M_pl², k⌧ 1 (subhorizon). (1.53) These wonderfully simple solutions have much to tell us about the evolution of quantum metric per- turbations during inflation. During slow-roll inflation,✏⌧1andH is nearly constant. As a result, when a wavemode exits the horizon during inflation (at quantum scales), its amplitude is frozen until the mode re-enters the horizon (at cosmological scales). In the other extreme, perturbative wavemodes much smaller than the horizon decay away as the universe expands (analogous to the cosmological redshift of the CMB). The process of inflation thus predicts a stochastic gravitational wave background at scales comparable to the cosmological horizon at the time of recombination.

Without inflation, any gravitational waves present at the beginning of physical time will have long since redshifted away by recombination.

Because the tensor power spectrum is only defined up to some overall normalization relative to the scalar power spectrum, it is common to define the tensor-to-scalar ratior, where the ratio is taken at one particularkvalue, typicallyk= 0.002Mpc ¹. Additionally, as one might suspect from the form of Equation 1.52, departures of the power spectrum fromk ³proportionality are indicative of time evolution ofH during inflation. This is typically parameterized as:

Ph(k)/k^{nT 3}. (1.54)

The spectral indexnT is related to the slow-roll parameter✏by:

nT = 2✏. (1.55)

Heuristically, this dependence comes about from the time evolution of the inflaton potential. If the inflaton potential is constant, then every scale is equivalent to every other scale and the spectral index is precisely zero. However, in this scenario inflation never ends. If instead the potential evolves over time, then small scales are differentiated from large scales and a spectral tilt is introduced.

As experimentalists, our goal is to measure the presence of this gravitational wave background in the early universe’s history. As we will discuss in the next section, this is possible through observations of the polarization of the CMB.

1.4.2 Scalar perturbations

Much of our experience in calculating the tensor perturbation power spectra carries over when calculating equivalent expressions for scalar perturbations. Scalar perturbations are made more complicated by a coupling between the scalar field and gravity, as well as the matter content in the universe. To illustrate the general behavior of the perturbation dynamics, we will proceed assuming zero coupling and zero matter.

We begin by considering quantum fluctuations of the scalar field, given by the Klein-Gordon equation in an FRW metric:

¨ + 3H˙ r² +V⁰( ) = 0, (1.56)

where dots represent derivatives taken with respect to physical time and V⁰( ) =dV /d . We can similarly write the quantum fluctuations in a first-order perturbation quantity, :

¨ + 3H ˙ r² +V⁰( ) = 0. (1.57)

We can simplify this expression by noting that:

V⁰( ) =V⁰( + ) V⁰( ) =V⁰⁰( ) . (1.58) Settingm²⌘V⁰⁰( )in analogy with the Klein-Gordon equation, we find:

¨ + 3H ˙ r² +m² = 0. (1.59)

As in the case of tensor perturbations, we explore the time-evolution of one Fourier mode of wavenumberk:

¨_k+ 3H ˙_k+

✓k a

◆2

k+m²

2 ^k = 0. (1.60)

We can begin examining the behavior of the time evolution in the slow-roll limit. Taking into account the second slow-roll condition V⁰⁰( )/V( ) ⌧ M_pl² (Equation 1.35), we find that we can

20 safely neglect the last term in Equation 1.60 to find:

¨_k+ 3H ˙_k+

✓k a

◆²

k = 0. (1.61)

This has a very similar form to Equation 1.41. We again approximate H to be constant during inflation. The solution, up to some overall normalization, turns out to be (we refer the reader to Dodelson 2003 for details):

v _k(k,⌧) = 1

k^3/2(i+k⌧)e^ik⌧. (1.62)

In the two limiting cases considered earlier, we find the variance to be:

|v (k,⌧)|²= H

k³, k⌧⌧1 (superhorizon) (1.63)

|v (k,⌧)|²= H

ka, k⌧ 1 (subhorizon). (1.64)

These limiting cases reveal the same behavior as we encountered in the treatment of tensor perturbations. After perturbations exit the horizon during inflation, their time evolution stops, and their amplitude is frozen. When these modes re-enter the horizon at late times, their amplitude decays with the scale factor. This again, is in the case of zero matter and zero coupling. In reality, these scalar perturbations undergo a more complicated evolution, the result of which is observed in the acoustic peaks of the CMB.