Homogeneity

We should be clear about the implications of this result. The real world is not perfectly Robertson-Walker. If there are super-Hubble-radius perturbations (which are not sup- pressed, according to the analysis in the next section), in any one patch the measured value of the curvature parameter will deviate from unity. However, we draw the lesson that it is worthwhile doing a careful analysis of cosmological fine-tuning using a well-defined measure on the space of histories, as the results can differ substantially from a naive analysis.

6.5.1 Action for a perfect fluid background

We will first calculate the measure for the solutions of scalar perturbations to Einstein’s equations for a flat FRW background filled with a perfect fluid. The metric for this setting is

ds² =a²(η)

−(1 + 2φ)dη²+ 2B,idηdx²+ ((1−2ψ)δij+ 2E,ij)dxⁱdx^j

, (6.39)

whereφ,ψ,E_,ij, and B_,i are scalar perturbations to the metric. In this section, we will be using the conformal time η in addition to t. Derivatives with respect to η are denoted by the superscript ⁰.

Up to second order in the perturbations, the gravitational part of the action is

δ⁽²⁾Sgr = 1 2

Z

d⁴xa²(−6ψ⁰²−12 ¯H(φ+ψ)ψ⁰−9 ¯H²(φ+ψ)²

−2ψ_,i(2φ,i−ψ,i)−4 ¯H(φ+ψ)(B−E⁰),ii+ 4 ¯Hψ⁰E,ii

−4ψ⁰(B−E⁰),ii−4 ¯Hψ,iB,i+ 6 ¯H²(φ+ψ)E,ii

−4 ¯HE,ii(B−E⁰),jj + 4 ¯HE,iiB,jj+ 3 ¯H²E_,ii² + 3 ¯H²B,iB,i)

+total derivatives, (6.40)

where ¯H = a⁰/a (while H = ˙a/a). The dynamical quantity for hydrodynamical matter is ξ^α(x^β), the deviation of test particles from their trajectory in the unperturbed FRW universe. From this we can compute the matter part of the action,

δ⁽²⁾S_m = Z

d⁴x[1

2ρφ²+p(3

2ψ²−3φψ+φE_,ii−ψE_,ii+1

2E_,iiE_,jj

−E_,ijE,ij+ 1

2B,iB,i) + (ρ+p)(1

2ξⁱ⁰ξⁱ⁰+B,iξⁱ⁰+φξ_,iⁱ)

−1

2c²_s(ρ+p)(3ψ−E,ii−ξ_,iⁱ)²]a⁴+ total derivatives, (6.41)

where cs is the adiabatic speed of sound in the fluid; β ≡ H¯²−H¯⁰; and ρ and p are the unperturbed energy density and pressure of the fluid. Combining (6.41) with (6.40), we obtain the total action quadratic in the scalar perturbations,

δ⁽²⁾S = δ⁽²⁾Sgr+δ⁽²⁾Sm

= 1

2 Z

d⁴x(a²(−6[ψ⁰²+ 2 ¯Hφψ⁰+ ( ¯H²− β 3c²_s)φ²]

−4(ψ⁰+ ¯Hφ)(B−E⁰),ii−2ψ,i(2φ,i−ψ,i)

+2β(ξⁱ⁰+B_,i)(ξⁱ⁰+B_,i)−2βc²_s(3ψ−E_,ii−ξ_,iⁱ + 1 c²_sφ²)²)

+total derivatives. (6.42)

We now introduce the Mukhanov-Sasaki variablev:

v= 1

√2(φ_v−2zψ), (6.43)

wherez≡aβ^1/2/Hc¯ s andφv=−2a(ψ⁰+ ¯Hφ)/(csβ^1/2) is the velocity potential of the fluid.

Using constraints obtained by varying (6.42) with respect toφ,ψ, andE_,ii, the action takes on the simple form

δ⁽²⁾S = 1 2

Z d⁴x

v⁰²−c²_sv,iv,i+ z⁰⁰

z v²+ total derivatives

. (6.44)

This is the just the action for a scalar field with a time-varying mass. The fact that the we can express the action in terms of the Mukhanov-Sasaki variable v alone implies that there is only one dynamical degree of freedom present. The momentum p_v conjugate tov

is simply v⁰, and the Hamiltonian is given by

H= p²_v 2 −

c_sk²+z⁰⁰ z

v². (6.45)

6.5.2 Action during inflation

We now repeat the calculation for the case where the background is filled with a canonical scalar field S instead of a perfect fluid. The gravitational part of the action remains the same. The scalar-field contribution to the action is

SS =d⁴x√

−g 1

2S_;αS^;α−V(S)

. (6.46)

Expanding all quantities to second order in the perturbations, we have

δ⁽²⁾S = δ⁽²⁾S_gr+δ⁽²⁾SS

= 1

2 Z

a²[−6ψ⁰²−12 ¯Hφψ⁰−2ψ_,i(2φ_,i−ψ_,i)−2( ¯H⁰+ 2 ¯H²)φ² +(δS⁰²−δS_,iδS_,i−V,SSa²δS²) + 2( ¯S⁰(φ+ 3ψ)⁰δS −2V,Sa²φδS)

+4(B−E⁰),ii( ¯SδS/2−ψ⁰−Hφ)] + total derivatives.¯ (6.47)

As before, we introduce a gauge-invariant quantity analogous to the Mukhanov-Sasaki variable,

v=a(δS+ ( ¯S⁰/H)ψ).¯ (6.48)

In terms of v, the action (6.47) simplifies to

δ⁽²⁾S = 1 2

Z

v⁰²−v,iv,i+z⁰⁰

zv²+ total derivatives

, (6.49)

wherez=aS¯⁰/H. Similar to the perfect fluid case, the action is just that for a scalar field¯ with a time-varying mass, only that we now havec²_s = 1.

6.5.3 Computation of the measure

Given the actions (6.42) and (6.47), we can straightforwardly compute the invariant mea- sure on phase space. One caveat is that now the Hamiltonian is time-dependent, so the carrier manifold of the Hamiltonian has an odd number of dimensions. We can retain the symplecticity of a time-dependent Hamiltonian system (which requires an even num- ber of dimensions) by promoting time to be an addition canonical coordinate qⁿ⁺¹ = t.

The conjugate momentum is then the negative value of the Hamiltonian, pn+1 = −H.

We can then construct an extended Hamiltonian H₊ =H(p, q, t) +p_n+1, which is explic- itly time-independent, and from which we can derive the original Hamiltonian’s equations ( ˙qⁱ = ∂H₊/∂pi and ˙pi = −∂H₊/∂qⁱ), plus two additional trivial equations ˙t = 1 and H˙ =∂H/∂t.

With t promoted to a coordinate, the time-dependent Hamiltonian system also comes equipped naturally with a closed symplectic two-form, now with an additional term:

ω=

n

X

a=1

dp_a∧dq^a−dH ∧dt. (6.50)

The invariance of the form of Hamilton’s equations ensures that the Lie derivative of ω with respect to the vector field generated by H₊ vanishes. The top exterior power of ω is then guaranteed to be conserved under the extended Hamiltonian flow, and can thus play the role of the Liouville measure for the augmented system. The GHS measure can then be obtained by pulling-back the Liouville measure onto a hypersurface intersecting the

trajectories and satisfying the constraintH₊= 0.

In our case, the original system, with coordinate v and conjugate momentum p_v, is augmented to one with two coordinates v and η, and their conjugate momenta p_v and

−H. The extended Hamiltonian,H₊=p²_v/2−(c²_sk²−z⁰⁰(t)/z(t))v²− H, is explicitly time- independent (identically zero), and its conservation is analogous to the Friedmann equation constraint in the analysis of the flatness problem. Using (6.50), the GHS measureω_GHS for the perturbation is

ω_GHS = dp_v∧dv−(dH ∧dη)|_H=p2

v/2−(c²_sk²−z⁰⁰(t)/z(t))v²

= dp_v∧dv−d p²_v

2 −

c²_sk²+z⁰⁰ z

v²

∧dη

= dp_v∧dv−p_v(dp_v∧dη) + 2v

c²_sk²+z⁰⁰ z

dv∧dη . (6.51)

One convenient hypersurface in which we can evalute the flux of trajectories is η = constant. As dη = dt/a is always positive, this surface intersects all trajectories exactly once. The flux of trajectories crossing this surface is unity, the coefficient of the first term in (6.51). This implies that all values for v and pv are equally likely. There is nothing in the measure that would explain the small observed values of perturbations at early times.

Hence, the observed homogeneity of our universe does imply considerable fine-tuning; unlike the flatness problem, the horizon problem is real.