The integral of B projected into a hypersurface inC is independent of deformations of the hypersurface, and is equal to the integral of Ω. Hence, given some transverse hypersurfacee Σ inC representingM, the measure can be written in either of two forms,
µ= Z
Σ
Ω =e Z
Σ
Bini, (6.19)
where ni is a unit vector in C orthogonal to Σ. With this formalism established, we can apply the measure to cosmological spacetimes.
where we have chosen units where 8πG = 1. The canonical coordinates can be taken to be the lapse function N, the scale factora, and the scalar field φ. We can do a Legendre transformation to get the conjugate momenta,
pN = 0, pa=−6N−1aa ,˙ pφ=N−1a3φ .˙ (6.23)
The Hamiltonian is then given by
H=N − p2a 12a+ p2φ
2a3 +a3V(φ)−3ak
!
. (6.24)
Varying with respect to N gives the Hamiltonian constraint, H = 0, which is just the Friedmann equation,
H2 = 1 3
ρφ˙+ρV +ρk
, (6.25)
where we have defined
ρφ˙= 1
2φ˙2 , ρV =V(φ) , ρk=−3k
a2. (6.26)
Henceforth we will set N = 1, and we are left with a four-dimensional phase space,
Γ ={φ, pφ, a, pa}, (6.27)
with the canonical measure
ω = (dpa∧da+dpφ∧dφ)|H=0, (6.28)
which is just the Liouville measure subject to the constraint that H = 0. To enforce the constraint, we can use the Friedmann equation to eliminate a from the measure, which yields
a=
s 3k
V + ˙φ2/2−3H2 . (6.29)
Upon substitution into (6.28), the measure simplifies to
ω= 1
|k|
3k
V + ˙φ2/2−3H2 5/2
1
3(V −φ˙2−3H2)dφ˙∧dφ+ (V0+ 3Hφ)dH˙ ∧dφ+ ˙φdH∧dφ˙
. (6.30) The corresponding magnetic field is
Bi≡
Bφ, Bφ˙, BH
= 1
|k|
−3k 3H2−V −φ˙2/2
5/2
−φ, V˙ 0+ 3Hφ,˙ −1
3(V −φ˙2−3H2)
. (6.31) We are now left with a three-dimensional reduced phase space, and a two-dimensional space of trajectories. The measure is defined by choosing some transverse surface Σ, and integrating theB-field dotted into an orthogonal one-formni.
µ= Z
Σ
Bini. (6.32)
One possible choice of the transverse surface Σ is to fix the Hubble parameter,
Σ :{H =H∗}. (6.33)
The measure evaluated on a surface of constant H is then
µ = Z
H=H∗
BHdφ dφ˙ (6.34)
= Z
H=H∗
1
|k|
−3k 3H∗2−V −φ˙2/2
5/2
(V −φ˙2−3H∗2)dφdφ .˙ (6.35)
It is convenient to rewrite this by changing variables from (φ, ˙φ) to (ρ{φ˙ ,ρk), and using the Friedmann equation (6.25). For simplicity, we will look at the potential V(φ) = m2φ2/2, although our results don’t depend on this choice. The measure then becomes
µ= 33/2 2m|k|
Z
H=H∗
−k ρk
5/2 3ρφ˙+ρk
ρφ˙1/2(3H∗2−ρφ˙−ρk)1/2dρφ˙dρk (6.36)
It is clear that the integrals over both ρφ˙ and ρk diverge. The divergence with respect to ρφ˙ occurs at large values, and is easily regulated by limiting our attention to densities smaller than some fixed number. With respect to curvature, however, there is a divergence as
ρk→0, (6.37)
where the integrand goes asρ5/2k . We might imagine regularizing this divergence by removing a region of size around ρk = 0, and letting →0. We would find that all of the measure is dominated by nearly flat universes, in the following sense: Let µ(a, b) be the measure obtained by integrating over all values of ρφ˙ less than the cutoff, and values of ρk with a < ρk< b. Then we have
→0lim µ(a, b)
µ(, a) = 0 (6.38)
for anyb > a >0. (An analogous conclusion holds for negative curvatures.) In other words,
solutions with ρk6= 0 are a set of measure zero.
There is a straightforward interpretation of this result: the flatness problem does not exist. If we were to somehow imagine randomly choosing a Robertson-Walker universe, it would be spatially flat with probability one. We feel that this interpretation is the most sensible one, even though it runs counter to the conventional presentation of the flatness problem.2 The usual statement of the flatness problem notes that even a very small deviation from flatness at early times grows into an appreciable amount of curvature at late times. While this is true, it only becomes a “problem” when we presume a measure — in particular, some approximately-flat measure over values of the curvature parameter on some initial-condition surface in the early universe. The lesson of the GHS measure is that this reasonable-seeming intuition is wrong; the correct measure is very far from flat, and is strongly concentrated on precisely flat universes.
Notice that the hypersurface H = H∗ intersects all trajectories exactly once if k ≤0.
However, our conclusion remains valid even for closed universes, since the divergenceρ−5/2k is present in all three components of (6.31). In principle, one can imagine deforming the H = H∗ surface to one that intersects all trajectories exactly once, and the divergence still remains. Alternatively, we could have chosen to eliminatepa orpφ instead of ain the measure. In this case, since dtd( ˙φa3) =V0a3, the transverse surface ˙φa3 = constant would intersect all trajectories once, as long as the potentialV for φis monotonic. However, the physical meaning of this transverse surface is less intuitive, so we use instead theH =H∗
surface in our analysis.
2This divergence was noted in the original GHS paper [73], where it was attributed to “universes with very large scale factors” due to a different choice of variables. This seems to be beside the point, as any open universe will eventually have a large scale factor. It is also discussed by Gibbons and Turok [80], who correctly attribute it to nearly-flat universes. However, they advocate discarding all such universes as physically indistinguishable, and concentrating on the non-flat universes. To us, this seems to be throwing away almost all the solutions, and keeping a set of measure zero.
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.