Inflation as an easy target

6.7 What is Inflation Good For?

6.7.2 Inflation as an easy target

Given that we need some theory of initial conditions to explain why our universe was not chosen at random, the question becomes whether inflation provides any help to this unknown

3For a more familiar example, consider a glass of water with an ice cube that melts over the course of an hour. At the end of the melting process, if we reverse the momentum of every molecule in the glass, we will describe an initial condition that evolves into an ice cube. But there’s no way of knowing that, just from the macroscopically available information; it looks just like a regular glass of water.

theory. We would like to suggest that it does, in two familiar ways: the required initial state does not need to be as big, or as smooth, as in conventional Big Bang cosmology.

First, inflation allows the initial patch of spacetime with a Planck-scale Hubble param- eter to be physically small, while conventional cosmology does not. If we extrapoloate a matter- and radiation-dominated universe from today backwards in time, a comoving patch of sizeH₀⁻¹ today corresponds to a physical size∼10⁻²⁶H₀ ∼10³⁴m⁻¹_{P l} when H=m_{P l}. In contrast, with inflation, the same patch needs to be no larger than the Planck length when H=m_{P l}, as emphasized by Kofman, Linde, and Mukhanov [77, 81]. If our purported the- ory of initial conditions, whether quantum cosmology or baby-universe nucleation or some other scheme, has an easier time making small patches of space than large ones, inflation would be an enormous help.

The other advantage is in the degree of smoothness required. At the end of the previous section we calculated that a perfect-fluid universe with Planckian Hubble parameter would have to be extremely homogeneous to be compatible with the current universe, while an analogous inflationary patch could accommodate any amount of sub-Planckian perturba- tions. While the actual number of trajectories may be smaller in the case of inflation, there is a sense in which the requirements seem more natural. Within the set of initial condi- tions that experience sufficient inflation, all such states give us reasonable universes at late times; in a more conventional Big Bang cosmology, the perturbations require an additional substantial fine tuning. Again, we have a relatively plausible target for a future theory of initial conditions: as long as inflation occurs, and the perturbations are not initially super-Planckian, we will get a reasonable universe.

These features of inflation are certainly not novel; it is well-known that inflation allows for the creation of a universe such as our own out of a small and relatively small bubble

of false vacuum energy. We are nevertheless presenting the point in such detail because we believe that the usual sales pitch for inflation is misleading; inflation does offer important advantages over conventional Friedmann cosmologies, but not necessarily the ones that are often advertised. In particular, inflation does not by itself make our current universe more likely; the number of trajectories that end up looking like our present universe is unaffected by the possibility of inflation, and even when it is allowed only a tiny minority of solutions feature it. Rather, inflation provides a specific kind of “nice” set-up for a true theory of initial conditions — one that is yet to be definitively developed.

Appendix A

Solutions to the Linearized Equations of Motion

We start by finding the solution to the equations of motion, linearized about a timelike, fixed-norm background, Aµ. Then, showing less details, we find the solutions to the equa- tions of motion linearized about a spacelike background. Finally, we put the solutions in both cases into the compact form of (A.26)–(A.28). Our results agree with the solutions for Goldstone modes found in [44].

The equations of motion for a timelike (+) or spacelike (−) vector field are (3.16),

Qµ≡

ηµν±A_µA_ν m²

(β1∂ρ∂^ρA^ν+ (β∗−β1)∂^ν∂ρA^ρ+β4G^ν) = 0, (A.1)

whereG^ν is defined in (3.14) andA^µQ_µ= 0 identically.

Timelike background. Consider perturbations about an arbitrary, constant (in space and time) timelike backgroundA_µ= ¯A_µthat satisfies the constraint: ¯A_µA¯^µ=−m². Define perturbations by A_µ = ¯A_µ+δA_µ. Then, to first order in these perturbations, ¯A^µQ_µ = 0 identically, andη^µνA¯µδAν = 0 by the constraint. We can define a basis set of four Lorentz

4-vectorsn^α, with components

n⁰_µ= ¯A_µ/m , nⁱ_µ ; i∈ {1,2,3}, (A.2)

such that

η^µνn^α_µn^β_ν =η^αβ. (A.3)

The independent perturbations are δa^α ≡ η^µνn^α_µδAν for α = 1,2,3. (δa⁰ is zero at first order in perturbations due to the constraint.) It is then clear that there are three independent equations of motion at first order in pertubations (assuming the constraint) for the three independent perturbations,

δQⁱ ≡nⁱ_ν β₁∂_ρ∂^ρδA^ν + (β∗−β₁)∂^ν∂_ρδA^ρ+β₄n⁰_µn⁰_ρ∂^µ∂^ρδA^ν

= 0, (A.4)

wherei∈ {1,2,3}. We look for plane wave solutions for the δA:

δA_µ= Z

d⁴k q_µ(k)e^ikνxν. (A.5)

Since η^µνn⁰_µδAν = 0, at first order,

q_µ=c_jn^j_µ where j∈ {1,2,3}. (A.6)

The equations of motion become the algebraic equations:

0 = β₁k_ρk^ρnⁱ_νn^jν+ (β∗−β₁)nⁱ_νk^νn^j_µk^µ+β₄n⁰_µn⁰_ρk^µk^ρnⁱ_νn^jν

c_j (A.7)

= β₁k_ρk^ρδ^ij + (β∗−β₁)nⁱ_νk^νn^j_µk^µ+β₄n⁰_µn⁰_ρk^µk^ρδ^ij

c_j (A.8)

≡M^ijcj. (A.9)

The three independent solutions to these equations are given by setting an eigenvalue of the matrixM to zero and settingci to the corresponding eigenvector. Setting an eigenvalue of M equal to zero gives a dispersion relation,

β1kρk^ρ+β4(n⁰_µk^µ)²= 0, (A.10)

with two linearly independent eigenvectors,

(e₂)_i =_2ijn^j_µk^µ ; (e₃)_i=_3ijn^j_µk^µ. (A.11)

The second eigenvalue ofM gives the dispersion relation,

β∗k_ρk^ρ+ (β∗−β₁+β₄)(n⁰_µk^µ)² = 0, (A.12)

with corresponding eigenvector,

c_i =nⁱ_µk^µ. (A.13)

Spacelike background. The first-order linearized equations of motion about a spacelike background are:

δQ^a≡n^a_ν β1∂ρ∂^ρδA^ν+ (β∗−β1)∂^ν∂ρδA^ρ+β4n³_µn³_ρ∂^µ∂^ρδA^ν

= 0 (A.14)

wherea∈ {0,1,2}and where, similarly to the timelike case, we have defined the set of four Lorentz 4-vectors, n^α_µ, to be

n³_µ= ¯Aµ/m and n^a_µ; a∈ {0,1,2} (A.15)

such that

η^µνn^α_µn^β_ν =η^αβ. (A.16)

The independent perturbations are δa^α ≡ η^µνn^α_µδAν for α = 0,1,2. (δa³ is zero at first order in perturbations due to the constraint.)

Again we look for plane wave solutions of the form in (A.5). But now, sinceη^µνn³_µδA_ν = 0, at first order,

qµ=can^a_µ where a∈ {0,1,2}. (A.17)

The equations of motion become the algebraic equations:

=

β₁k_ρk^ρn^a_νn^bν+ (β∗−β₁)n^a_νk^νn^b_µk^µ+β₄n³_µn³_ρk^µk^ρn^a_νn^bν

c_b (A.18)

=

β₁k_ρk^ρη^ab+ (β∗−β₁)n^a_νk^νn^b_µk^µ+β₄n³_µn³_ρk^µk^ρη^ab

c_b (A.19)

≡M^abc_b. a, b∈ {0,1,2} (A.20)

Two independent solutions correspond to the dispersion relation (a∈ {0,1,2})

β₁k_ρk^ρ+β₄(n³_µk^µ)² = 0, (A.21)

with corresponding eigenmodes

(e₁)_a=_a1b3n^b_µk^µ ; (e₂)_a=_ab23n^b_µk^µ. (A.22)

The third solution corresponds to the dispersion relation

β∗k_ρk^ρ−(β∗−β₁−β₄)(n³_µk^µ)² = 0, (A.23)

with corresponding eigenmode

ca=ηabn^b_µk^µ. (A.24)

General expression. We can express the solutions in the timelike and spacelike cases in a compact form by using the orthonormality of then^α_µ, (A.3), along with (A.2), (A.15), and the fact that,¹

_αβρσn^α_µn^β_ν =_µναβn^α_ρn^β_σ. (A.25)

Then plugging (A.6) and (A.17) into (A.5) yields the solutions,

δAµ= Z

d⁴k qµ(k)e^ikνxν (A.26)

1This follows from the invariance of the Levi-Civita tensor, αβγδn^αµn^βνn^γρn^δσ =µνρσ

plus orthonormality, (A.3).

where either,

q_µ(k) =iα^νk^ρA¯^σ

m_µνρσ and β₁k_ρk^ρ+β₄

A¯_µk^µ m

2

= 0 and α^νA¯_ν = 0, (A.27)

whereα^ν are real-valued constants or,

q_µ=iα

η_µν±A¯_µA¯_ν m²

k^ν and β∗k_ρk^ρ±(β∗−β₁±β₄)

A¯_µk^µ m

2

= 0, (A.28)

whereαis a real-valued constant. The reality of theα’s follows from the condition,qµ(k) = q_µ^∗(−k), that holds if and only if δAµ in (A.5) is real. In (A.28), the “+” sign corresponds to the timelike background and the “−” sign to a spacelike background.

Appendix B

Additional Properties of the

Goldstone Modes