Unitarity, clock dependence and quantum recollapse in quantum cosmology

Abstract

We continue our analysis of a quantum cosmology model describing a flat Friedmann–Lemaˆıtre–Robertson–Walker universe filled with a (free) massless scalar field and an arbitrary perfect fluid. For positive energy density in the scalar and fluid, each classical solution has a singularity and expands to infinite volume. When quantising we view the cosmological dynamics in relational terms, using one degree of freedom as a clock for the others. Three natural candidates for this clock are the volume, a time variable conjugate to the perfect fluid, and the scalar field. We have previously shown that requiring unitary evolution in the “fluid” time leads to a boundary condition at the singularity and generic singularity resolution, while in the volume time semiclassical states follow the classical singular trajectories. Here we analyse the third option of using the scalar field as a clock, finding further dramatic differences to the previous cases: the boundary condition arising from unitarity is now at infinity. Rather than singularity resolution, this theory features a quantum recollapse of the universe at large volume, as was shown in a similar context by Pawlowski and Ashtekar. We illustrate the properties of the theory analytically and numerically, showing that the ways in which the different quantum theories do or do not depart from classical behaviour directly arise from demanding unitarity with respect to different clocks. We argue that using a Dirac quantisation would not resolve the issue. Our results further illustrate the problem of time in quantum gravity.

Quantum analysis of the recent cosmological bounce in the comoving Hubble length

“…For a fluid with equation of state parameter w, ρ(n) = ρ 0 n 1+w for some ρ 0 where n = U/(a 3 V c ) is the particle number density. Now introducing a new variable m = 8πGρ0 3Vc ( U Vc ) 1+w , conservation of U is equivalent to conservation of m, and we can define an equivalent fluid action (see also [17,18])…”

“…Importantly, this Hamiltonian is linear in m and Λ; for a suitable choice of lapse given by the appropriate power of a, the equations of motion for χ 1 and χ 2 can be brought into the form χi = −1; if one allows for a negative lapse χi = 1 would also be possible. Hence, in such a gauge either χ 1 or χ 2 are identified with (minus) the time coordinate [5,18]. We could apply any canonical transformation upon these variables, in particular point transformations from constants to functions of themselves (inducing time conjugates proportional to the original one, the proportionality factor being a function of the constants).…”

“…For this argument to be valid without the introduction of boundary conditions as in, e.g., [18], here and in the following we must assume that X(b) takes values over the whole real line and is monotonic. This is true for many cases including the ones studied here, namely radiation and Λ with k = 0 (and also in the case of dust with k = 0, studied in [15]).…”

Lower bound on the cosmological constant from the classicality of the Early Universe

“…Upon quantization they become duals satisfying commutation relations. If q A represents the other degrees of freedom of matter and geometry (metric or connection), the Hamiltonian constraint can either be written in terms of Λ (resulting in the standard Wheeler-DeWitt equation for timeless ψ s (q A , Λ)) or in terms of its conjugate time T (leading to a Schrödinger-like equation for ψ(q A , T )) [13][14][15]. The general solution takes the form:…”

“…In standard unimodular theory this is the only quantum clock. In other theories one could consider multiple clocks at different epoch of the Universe[13][14][15][16][17], or even at the same epoch[18]. The constraints on each of these different theories are specific to each of them.…”