Scalar field Clause Samples

Scalar field. ‌ I now investigate the effects that the inverse-volume and holonomy corrections can have when I couple gravity to an undeformed scalar field. In this case, the energy and pressure densities are given by ρ = φ˙ 2 2 + U (φ), P = φ˙ 2 − U (φ), (4.48) and the continuity equation gives us the equation of motion for the scalar field, φ¨ + 3Hφ˙ + U′ = 0, (4.49) where U′ = ∂U . Let us investigate the era of slow-roll inflation. Using the assumptions |φ¨/U ′| ≪ 1 and 1 φ˙2 ≪ U , I have the slow-roll equations, φ˙ = −U′ , (4.50a) 3H 2 H = σ∅γ∅ U 3ω (1 − BI σ∅γ2 ♢˜ 3ω2 a2δ ∅ γ2 U ). (4.50b) If I substitute (4.50b) into (4.50a), take the derivative with respect to time and substitute in (4.50b) and (4.50a) again, I find where the slow-roll parameters are ▇▇ = ▇ ▇, (▇.▇▇) ▇ := 1 ( ω U′′ − (1 − 2ς)ϵ + χ − δς) , (4.52a) 1 − ς γ∅ U 1 − ς 2γ∅ ( U′ )2 χ := 1 − 3ς ∂ log γ∅ BI ∅ 2 ∂ log a γ2 ♢˜ 3ω2 a2δ γ2 U, (4.52d) and the conditions for slow-roll inflation are