Abstract
Cosmological models in 1+1 dimensions are an ideal setting for investigating the quantum structure of inflationary dynamics -- gravity is renormalizable, while there is room for spatial structure not present in the minisuperspace approximation. We use this fortuitous convergence to investigate the mechanism of slow-roll eternal inflation. A variant of 1+1 Liouville gravity coupled to matter is shown to model precisely the scalar sector of cosmological perturbations in 3+1 dimensions. A particular example of quintessence in 1+1d is argued on the one hand to exhibit slow-roll eternal inflation according to standard criteria; on the other hand, a field redefinition relates the model to pure de Sitter gravity coupled to a free scalar matter field with no potential. This and other examples show that the standard logic leading to slow-roll eternal inflation is not invariant under field redefinitions, thus raising concerns regarding its validity. Aspects of the quantization of Liouville gravity as a model of quantum de Sitter space are also discussed.
Highlights
Background solutionsWorking in conformal gauge for the background, Nt = 1, Nx = 0, the background equations of motion are 0 = φ′′ + χ′′ χ′′Λe2φ γ2 4 α e2αφV (X ) X ′′ e2αφV,X (3.3)(here prime denotes derivative with respect to conformal time); one has the Hamiltonian constraint (χ′)2 2χ′φ′(X ′)2 + e2αφV(X ) (3.4)For instance, when V = 0 one has deSitter solutions
A particular example of quintessence in 1+1d is argued on the one hand to exhibit slow-roll eternal inflation according to standard criteria; on the other hand, a field redefinition relates the model to pure de Sitter gravity coupled to a free scalar matter field with no potential
We develop a variant of Liouville theory for which there is a one-to-one correspondence between the scalar geometric perturbations in Liouville gravity and those of four-dimensional Einstein gravity
Summary
A situation in which quantum effects play a key role is that of inflation. The solutions to the classical equations of motion in an inflation model involve a scalar field, the inflaton X, slowly descending its smooth potential V(X). In the quantum theory of a scalar field rolling down its potential in curved spacetime, there are fluctuations about the classical field value. The idea of slow-roll eternal inflation is that large coherent fluctuations δX over Hubble-size volumes, seemingly far out on the tails of the scalar field probability distribution, have an extraordinary effect on the structure of the wavefunction [5,6,7,8,9]. To investigate the mechanism of slow-roll eternal inflation in the full quantum theory, in section 4 we focus on the particular choice of potential V(X) = exp[−βX] for a scalar matter field X in a modified two-dimensional Liouville gravity.
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