Common use of Bounce Clause in Contracts

Bounce. ‌ I will address the question of whether there are conditions under which there can be a big bounce as defined in section 2.8. I find in chapter 4 (and in ref. [56]) that a deformation function which depends on curvature terms can generate a bounce. Elsewhere in the literature on loop quantum cosmology the bounce happens in a regime when β < 0 because the terms depending on curvature or energy density overpower the zeroth order terms [40, 41]. However, I am not including derivatives in the deformation here so the effect would have to come from the non-minimal coupling of the scalar field or the zeroth order deformation. I take a˙ = 0 for finite a, include a deformation and I ignore the minimally coupled field for simplicity. From the ▇▇▇▇▇▇▇▇▇ equation (3.38) I find, 0 = ωψ ψ˙2 + σ √|β| U, (3.40) which implies that σβωψ < 0 for a bounce because otherwise the equation cannot balance for U > 0 and ψ ∈ R. Substituting (3.40) into the full equation of motion for the scale factor, and demanding that a¨ > 0 to make it a turning point, I find the following conditions, σβωψ < 0, (3.41a) ωRωψ + 3 ω′2 > 0, (3.41b) R σβ |β| ωψ + 2ωR √ ( ′′ ) σβωR′ 2ω ψ (√ ) aβ ∂ |β|ωψU + √|β| ( ωψ U ) > 0,