Black hole formation by the waterfall field of hybrid inflation

Porto 2011 p.1/12 Black hole formation by the waterfall field of hybrid inflation David H. Lyth Particle Theory and Cosmology Group Physics Department Lancaster University

What happens Porto 2011 p.2/12 Classical waterfall field χ(x, t) is generated from the vacuum fluctuation.

What happens Porto 2011 p.2/12 Classical waterfall field χ(x, t) is generated from the vacuum fluctuation. It generates a contribution P ζχ (k) to the spectrum of the curvature perturbation ζ(x, t).

Porto 2011 p.2/12 What happens Classical waterfall field χ(x, t) is generated from the vacuum fluctuation. It generates a contribution P ζχ (k) to the spectrum of the curvature perturbation ζ(x, t). Cosmological constraints on abundance of black holes in early universe require P ζχ < 1 if P ζχ peaks at a superhorizon scale, otherwise P ζχ < 10 2. This constrains the hybrid inflation potential.

Porto 2011 p.2/12 What happens Classical waterfall field χ(x, t) is generated from the vacuum fluctuation. It generates a contribution P ζχ (k) to the spectrum of the curvature perturbation ζ(x, t). Cosmological constraints on abundance of black holes in early universe require P ζχ < 1 if P ζχ peaks at a superhorizon scale, otherwise P ζχ < 10 2. This constrains the hybrid inflation potential. If P ζχ is much smaller, practically no black hole formation. So to get significant black hole formation we need a borderline P ζχ seems unlikely!

Porto 2011 p.2/12 What happens Classical waterfall field χ(x, t) is generated from the vacuum fluctuation. It generates a contribution P ζχ (k) to the spectrum of the curvature perturbation ζ(x, t). Cosmological constraints on abundance of black holes in early universe require P ζχ < 1 if P ζχ peaks at a superhorizon scale, otherwise P ζχ < 10 2. This constrains the hybrid inflation potential. If P ζχ is much smaller, practically no black hole formation. So to get significant black hole formation we need a borderline P ζχ seems unlikely! Conclusion: we need to understand black hole formation but don t really expect it to have happened.

Porto 2011 p.3/12 Papers calculating P ζχ I ll describe my own work, JCAP 1107:035,2011 and arxiv:1107.1681v2. These papers explain what s wrong with the other 19 papers.

Porto 2011 p.4/12 Quantum field theory of hybrid inflation L = 1 2 M PR 2 + 1 2 ( φ)2 + 1 2 ( χ)2 V (φ, χ) ds 2 dt 2 a 2 (t)δ ij dx i dx j H(t) ȧ/a constant Even with linearized gravity we have NO PRECISE QUANTUM FIELD THEORY. Just ad hoc approximations.

Hybrid inflation potential (standard version) Porto 2011 p.5/12 V (φ, χ) = V 0 + V (φ) + 1 2 m2 (φ)χ 2 + 1 4 λχ4 m 2 (φ) g 2 φ 2 m 2 g 2 ( φ 2 φ 2 c )

Hybrid inflation potential (standard version) Porto 2011 p.5/12 V (φ, χ) = V 0 + V (φ) + 1 2 m2 (φ)χ 2 + 1 4 λχ4 m 2 (φ) g 2 φ 2 m 2 g 2 ( φ 2 φ 2 c At φ > φ c, χ = quantum fluctuation. Set it to zero. Then we have inflation with potential V 0 + V (φ) V 0. )

Hybrid inflation potential (standard version) V (φ, χ) = V 0 + V (φ) + 1 2 m2 (φ)χ 2 + 1 4 λχ4 m 2 (φ) g 2 φ 2 m 2 g 2 ( φ 2 φ 2 c At φ > φ c, χ = quantum fluctuation. Set it to zero. Then we have inflation with potential V 0 + V (φ) V 0. Inflaton field φ has φ vev = 0 and waterfall field χ has χ 2 vev = 4 V 0 m 2 12M 2 P ( ) 2 H m Sensible models have EITHER m H (eg. F-term GUT inflation) OR m H making χ a modulus ( supernatural / running mass inflation). ) Porto 2011 p.5/12

Porto 2011 p.6/12 Generation of χ(x, t) Waterfall begins when φ = φ c, ends when χ χ vev. Ignore back-reaction so φ + 3H φ 2 φ = V (φ) g 2 χ 2 φ χ + 3H χ 2 χ = m 2 (φ)χ λχ 3.

Porto 2011 p.6/12 Generation of χ(x, t) Waterfall begins when φ = φ c, ends when χ χ vev. Ignore back-reaction so φ + 3H φ 2 φ = V (φ) g 2 χ 2 φ χ + 3H χ 2 χ = m 2 (φ)χ λχ 3. Assume waterfall begins with a linear era when last terms negligible. Choose slicing of uniform φ. Use dη dt/a. Mode function χ k satisfies d 2 (aχ k ) dη 2 + ω 2 k (η)(aχ k) = 0, ω 2 k k2 + a 2 (t) ˆm 2 (φ(t)) 2H 2

Porto 2011 p.6/12 Generation of χ(x, t) Waterfall begins when φ = φ c, ends when χ χ vev. Ignore back-reaction so φ + 3H φ 2 φ = V (φ) g 2 χ 2 φ χ + 3H χ 2 χ = m 2 (φ)χ λχ 3. Assume waterfall begins with a linear era when last terms negligible. Choose slicing of uniform φ. Use dη dt/a. Mode function χ k satisfies d 2 (aχ k ) dη 2 + ω 2 k (η)(aχ k) = 0, ω 2 k k2 + a 2 (t) ˆm 2 (φ(t)) 2H 2 A mode of χ becomes classical if/when phase of χ k becomes constant. Classical field χ(x, t) is GAUSSIAN with spectrum P χ (k, t) = (k 3 /2π 2 ) χ k (t) 2

Porto 2011 p.7/12 Curvature perturbation ζ(x, t) ζ(x, t) δ[ln a(x, t)] a(x, t) is local scale factor defined on slicing of uniform energy density ρ. Comoving volume element a(x, t) 3. Equivalent definition: ζ(x, t) = δn, N(x, t) ln[a(x, t)/a(t 0 )] N is the number of e-folds from ANY slice of uniform a ( flat slice)

Porto 2011 p.8/12 Evolution of curvature perturbation ζ(x, t) Now assume ρ has negligible gradient. This usually requires smoothing of ρ on a super-horizon scale.

Porto 2011 p.8/12 Evolution of curvature perturbation ζ(x, t) Now assume ρ has negligible gradient. This usually requires smoothing of ρ on a super-horizon scale. Then there s negligible energy flow, d(ρa 3 ) = pd(a 3 ). Evolution of ζ given by ζ(x, t) ζ(x, t 1 ) = δ [N(x, t) N(x, t 1 )] N(x, t) N(x, t 1 ) = t ȧ(x, t) t 1 a(x, t) dt = 1 3 ρ(t) ρ(t 1 ) dρ ρ + p(ρ, x)

Porto 2011 p.8/12 Evolution of curvature perturbation ζ(x, t) Now assume ρ has negligible gradient. This usually requires smoothing of ρ on a super-horizon scale. Then there s negligible energy flow, d(ρa 3 ) = pd(a 3 ). Evolution of ζ given by ζ(x, t) ζ(x, t 1 ) = δ [N(x, t) N(x, t 1 )] N(x, t) N(x, t 1 ) = t ȧ(x, t) t 1 a(x, t) dt = 1 3 ρ(t) ρ(t 1 ) dρ ρ + p(ρ, x) Use this to calculate ζ χ, the contribution to ζ generated during the linear era of the waterfall.

Porto 2011 p.9/12 Case m H: result for ζ χ Assume adiabatic condition d ω k /dη ωk 2, requires m(t) H. Then χ(x, t) has negligible spatial gradient.

Porto 2011 p.9/12 Case m H: result for ζ χ Assume adiabatic condition d ω k /dη ωk 2, requires m(t) H. Then χ(x, t) has negligible spatial gradient. Find NON-GAUSSIAN ζ χ : ζ χ (x) ( H ) ( χ 2 ) ( ) χ 2 (x) χ 2 2 m(t) χ 2 + φ 2 χ 2 where is spatial average and EVERYTHING IS EVALUATED AT END OF NON-LINEAR ERA.

Porto 2011 p.9/12 Case m H: result for ζ χ Assume adiabatic condition d ω k /dη ωk 2, requires m(t) H. Then χ(x, t) has negligible spatial gradient. Find NON-GAUSSIAN ζ χ : ζ χ (x) ( H ) ( χ 2 ) ( ) χ 2 (x) χ 2 2 m(t) χ 2 + φ 2 χ 2 where is spatial average and EVERYTHING IS EVALUATED AT END OF NON-LINEAR ERA. Spectrum P ζχ peaks at some k = k. Have to assume scale k is within observable universe (typically satisfied). At k k P ζχ (k) = 1 ( ) 2 ( ) H χ 2 2 ( ) 3 k π 2 m(t) χ 2 + φ 2 k

Porto 2011 p.9/12 Case m H: result for ζ χ Assume adiabatic condition d ω k /dη ωk 2, requires m(t) H. Then χ(x, t) has negligible spatial gradient. Find NON-GAUSSIAN ζ χ : ζ χ (x) ( H ) ( χ 2 ) ( ) χ 2 (x) χ 2 2 m(t) χ 2 + φ 2 χ 2 where is spatial average and EVERYTHING IS EVALUATED AT END OF NON-LINEAR ERA. Spectrum P ζχ peaks at some k = k. Have to assume scale k is within observable universe (typically satisfied). At k k P ζχ (k) = 1 ( ) 2 ( ) H χ 2 2 ( ) 3 k π 2 m(t) χ 2 + φ 2 k On cosmological scales typically k k, making ζ χ negligible.

Case m H: black hole bound Porto 2011 p.10/12 Two cases. 1. If k ah, horizon-sized black holes may form when k enters horizon after inflation. Taking non-gaussianity into account, bounds on their abundance give P ζχ (k ) < 1: SATISFIED.

Case m H: black hole bound Porto 2011 p.10/12 Two cases. 1. If k ah, horizon-sized black holes may form when k enters horizon after inflation. Taking non-gaussianity into account, bounds on their abundance give P ζχ (k ) < 1: SATISFIED. 2. If k ah black holes may form as soon as inflation ends on scales horizon and maybe also on scales horizon.

Case m H: black hole bound Porto 2011 p.10/12 Two cases. 1. If k ah, horizon-sized black holes may form when k enters horizon after inflation. Taking non-gaussianity into account, bounds on their abundance give P ζχ (k ) < 1: SATISFIED. 2. If k ah black holes may form as soon as inflation ends on scales horizon and maybe also on scales horizon. Can t discuss formation on scales horizon because it would required a detailed model of end of inflation.

Case m H: black hole bound Porto 2011 p.10/12 Two cases. 1. If k ah, horizon-sized black holes may form when k enters horizon after inflation. Taking non-gaussianity into account, bounds on their abundance give P ζχ (k ) < 1: SATISFIED. 2. If k ah black holes may form as soon as inflation ends on scales horizon and maybe also on scales horizon. Can t discuss formation on scales horizon because it would required a detailed model of end of inflation. So I smooth ζ on horizon scale. Then it s practically gaussian and usual black hole bound applies, P ζχ (ah) < 10 2 : SATISFIED

Porto 2011 p.11/12 Case m H ASSUME χ has negligible gradient. ( ) ( ) 1 χ 2 (x) χ 2 ζ χ (x) 2s(t) χ 2 s(t) 3 ( ) 1/2 9 2 + 4 + m2 (t) < H 2 1

Porto 2011 p.11/12 Case m H ASSUME χ has negligible gradient. ( ) ( ) 1 χ 2 (x) χ 2 ζ χ (x) 2s(t) χ 2 s(t) 3 ( ) 1/2 9 2 + 4 + m2 (t) < H 2 1 Spectrum P ζχ (k) peaks at a scale k, assumed to be well inside observable universe. P ζχ (k) 1 4s 2 (t) ( k k ) 3 (k < k ) Scale k IS super-horizon. Black hole bound P ζχ (k ) < 1 may be violated.

Closing remarks Porto 2011 p.12/12 According to folklore, excessive black hole production WILL occur for waterfall mass m H but WILL NOT occur for m H. No valid proof in literature (except Bugaev and Klimai 1107.3754 for V (φ) φ 2 and particular parameters: see remarks in DHL arxiv:1107.1681v2).

Porto 2011 p.12/12 Closing remarks According to folklore, excessive black hole production WILL occur for waterfall mass m H but WILL NOT occur for m H. No valid proof in literature (except Bugaev and Klimai 1107.3754 for V (φ) φ 2 and particular parameters: see remarks in DHL arxiv:1107.1681v2). I show that excessive black hole production MAY occur for waterfall mass m H but WILL NOT occur for m H. (Proof should hold for any V (φ) in at least part of parameter space.)

Porto 2011 p.12/12 Closing remarks According to folklore, excessive black hole production WILL occur for waterfall mass m H but WILL NOT occur for m H. No valid proof in literature (except Bugaev and Klimai 1107.3754 for V (φ) φ 2 and particular parameters: see remarks in DHL arxiv:1107.1681v2). I show that excessive black hole production MAY occur for waterfall mass m H but WILL NOT occur for m H. (Proof should hold for any V (φ) in at least part of parameter space.) The case m H requires further investigation. It s in progress: Abolhasani, Firouzjahi, DHL, Sasaki and Wands.

Porto 2011 p.12/12 Closing remarks According to folklore, excessive black hole production WILL occur for waterfall mass m H but WILL NOT occur for m H. No valid proof in literature (except Bugaev and Klimai 1107.3754 for V (φ) φ 2 and particular parameters: see remarks in DHL arxiv:1107.1681v2). I show that excessive black hole production MAY occur for waterfall mass m H but WILL NOT occur for m H. (Proof should hold for any V (φ) in at least part of parameter space.) The case m H requires further investigation. It s in progress: Abolhasani, Firouzjahi, DHL, Sasaki and Wands. THANK YOU FOR YOUR ATTENTION!