Why did the Universe Inflate? Why did the scalar field start with a high value of the potential, and run down to a minimum? Why didn’t it just start at the minimum?
No boundary amplitude [h ij, ] = Dg e -S[g] Sum over all metrics that have a boundary and where the induced metric on is h ij.
Wheeler DeWitt Equation -G ijkl δ 2 δh ij δh kl 3 R(h) h ½ + 2Λh ½ Ψ [h ij ] = 0 Where G ijkl is the metric on superspace, G ijkl = ½ h -½ (h ik h jl + h il h jk – h ij h kl ) and 3 R is the scalar curvature of the intrinsic geometry of the three-surface.
1 ∂ 2 1 ∂ 2 ∂a 2 a 2 ∂φ 2 – a 2 – + a 4 V Ψ(a, φ) = 0 Wheeler DeWitt Equation
WKB Approximation Ψ = B e iC Δ C·C· Δ C + J = 0 Δ B·B· Δ C = 0
No Boundary Amplitude N Amplitude Amplitude = e 1/m 2 N
Volume Weighting Surface of present matter density Hubble Volume P( Observing a Hubble Volume ) = | Ψ | 2 × Number of Hubble Volumes MIIMM∞ | Ψ | 2 e 3N Time Space
Volume Weighted Probability Initial φ Probability Planck Density Quadratic Potential
Volume Weighted Probability Initial φ Probability φ = φ m Potential with a Maximum at φ = φ m
Why did the Universe Inflate? Volume weighting gives a high probability of a large amount of Inflation