Measure and probability in cosmology

General relativity has a Hamiltonian formulation, which formally provides a canonical (Liouville) measure on the space of solutions. In ordinary statistical physics, the Liouville measure is used to compute probabilities of macrostates, and it would seem natural to use the similar measure arising in general relativity to compute probabilities in cosmology, such as the probability that the Universe underwent an era of inflation. Indeed, a number of authors have used the restriction of this measure to the space of homogeneous and isotropic universes with scalar field matter (minisuperspace)—namely, the Gibbons-Hawking-Stewart measure—to make arguments about the likelihood of inflation. We argue here that there are at least four major difficulties with using the measure of general relativity to make probability arguments in cosmology: (1) Equilibration does not occur on cosmological length scales. (2) Even in the minisuperspace case, the measure of phase space is infinite and the computation of probabilities depends very strongly on how the infinity is regulated. (3) The inhomogeneous degrees of freedom must be taken into account (we illustrate how) even if one is interested only in universes that are very nearly homogeneous. The measure depends upon how the infinite number of degrees of freedom are truncated, and how one defines “nearly homogeneous.” (4) In a Universe where the second law of thermodynamics holds, one cannot make use of our knowledge of the present state of the Universe to retrodict the likelihood of past conditions.

Figure 1

The black curves in the left plot are some numerically calculated (flat FLRW-$\phi$) solutions, chosen so as to be evenly spaced in $\phi$ at the early time $H=100m$, and with $\dot{\phi }\ge 0$ at this early time (the solutions with $\dot{\phi }\le 0$ can be obtained by reflection). Only the behavior of $\phi$ (horizontal axis) as a function of $H$ (vertical axis) is shown; the behavior of $a$ is not shown, as the $\phi$ dynamics is independent of the value of $a$. A solution will undergo at least 60 $e$-folds of slow-roll inflation if and only if it originates in one of the shaded regions. The right plot is a blowup of the left plot near the origin. The solid curves are the solutions that have exited slow roll (i.e., they contain all the solutions drawn in the left plot, which are too close together at late times to distinguish on the plot) and the dashed curves are solutions that have never inflated. For the universes that undergo inflation, we see that $\phi$ remains practically constant until $\dot{\phi }$ reaches to the slow-roll value. Then the solution undergoes slow-roll inflation along one of the diagonals (which are close to, but slightly offset from, the turning points $\phi =\pm \sqrt{6}H/m$) until $H$ becomes small enough ($H_f\sim m$) and inflation ends. If we take the measure surface at an early time (large $H/m$), most of the solutions start out in the shaded region, but if we take the measure surface at a later time, a smaller fraction start out in the shaded region. As we discuss, this does not violate Liouville’s theorem, since solutions are “spreading out” in the other phase space dimension, $a$.

Figure 2

A region of the plane ${\mathbf{R}}^2$. The black line $y=0$ lies more in the yellow region than in the red region, but the red region is larger. In the “real” measure, $\Omega =dxdy$, a randomly chosen point is more likely to lie in the red region. But in the restricted measure, $\Omega {|}_{y=0}=dx$, a randomly chosen point is more likely to lie in the yellow region.