Gravitational Waves and the Scale of Inflation

Gravitational Waves and the Scale of Inflation Mehrdad Mirbabayi with L. Senatore, E. Silverstein, M. Zaldarriaga Institute for Advanced Study COSMO2014, Chicago Aug 29, 2014 Mehrdad Mirbabayi (IAS) GW and H inf 1 / 15

Does a measurement of r uniquely fix H inf? Mehrdad Mirbabayi (IAS) GW and H inf 2 / 15

Does a measurement of r uniquely fix H inf? Yes, if tensor modes are in vacuum during inflation: γk s γs k = vac (2π)3 δ 3 (k + k )δ 1 H ss inf 2 k 3 Mpl 2 What if γ is not in vacuum? Mehrdad Mirbabayi (IAS) GW and H inf 2 / 15

Gravitational Waves in Solar System Jupiter Mehrdad Mirbabayi (IAS) GW and H inf 3 / 15

Gravitational Waves in Solar System Jupiter Bremsstrahlung at center of Sun (Weinberg 65). Mehrdad Mirbabayi (IAS) GW and H inf 3 / 15

Examples of Tensor Emission During Inflation 1) Particle Production: M 2 X = M2 sin 2 φ f Scattering of X particles emits gravity waves γ X. Senatore et.al 11 2) Pseudo-scalar Inflaton: L φa = α f φf F Growing helical gauge field A excites the metric. Sorbo 11, Barnaby et.al. 12, Mukohyama et.al. 14 Mehrdad Mirbabayi (IAS) GW and H inf 4 / 15

Can γ X be Larger than γ vac? Mehrdad Mirbabayi (IAS) GW and H inf 5 / 15

γ X > γ vac? 1. Available energy density: 1 2 φ 2 = M 2 pl H2 ɛ 2. The energy in the auxiliary sector ρ X M 2 pl H2 ɛ 3. Estimate emission by 2 γ ρ X /M 2 pl at frequency ω H γ X ρ X M 2 pl H2 ɛ Mehrdad Mirbabayi (IAS) GW and H inf 6 / 15

γ X > γ vac? 1. Available energy density: 1 2 φ 2 = M 2 pl H2 ɛ 2. The energy in the auxiliary sector ρ X M 2 pl H2 ɛ 3. Estimate emission by 2 γ ρ X /M 2 pl at frequency ω H γ X ρ X M 2 pl H2 ɛ There is a lot of room to outperform vacuum γ vac H M pl γ X ɛ. Mehrdad Mirbabayi (IAS) GW and H inf 6 / 15

Punch Line Scalars are emitted during energy transfer: 1 2 φ 2 ρ X = δφ X Mehrdad Mirbabayi (IAS) GW and H inf 7 / 15

Punch Line Scalars are emitted during energy transfer: 1 2 φ 2 ρ X = δφ X Large tensor emission generically leads to Very Large scalar emission γ X > γ vac = δφ X δφ vac r max ɛ 2 Mehrdad Mirbabayi (IAS) GW and H inf 7 / 15

1. Exponential Expansion Suppose emission is at a physical frequency k/a ω phys. 1.a. Then each mode can be excited in a period t H 1 Mehrdad Mirbabayi (IAS) GW and H inf 8 / 15

1. Exponential Expansion Suppose emission is at a physical frequency k/a ω phys. 1.a. Then each mode can be excited in a period t H 1 1.b. The waves redshift before horizon crossing γ freeze out = H ω γ ω. It also takes more energy to excite γ ω at higher ω. With fixed energy per Hubble volume: N γ E γh 3 ω 4. Mehrdad Mirbabayi (IAS) GW and H inf 8 / 15

2. Weak Gravity Tensor emission is governed by linearized Einstein equation Gµν lin. 1 Mpl 2 T µν Mehrdad Mirbabayi (IAS) GW and H inf 9 / 15

2. Weak Gravity Tensor emission is governed by linearized Einstein equation Gµν lin. 1 Mpl 2 T µν 2.a. N X Incoherent Localized Events of mass M X per Hubble volume : γ 2 M X X 2 H 2 NX Mpl 2 Mpl 2 Comment1) This is an upper bound. Comment2) This can exceed γ 2 vac. Mehrdad Mirbabayi (IAS) GW and H inf 9 / 15

3. Scalar Emission from Energy Conservation Scalar fluctuations δφ X lead to: δρ φ = φδ φ X. Energy conservation: d 3 xδρ φ = M X = δφ X M X φ Mehrdad Mirbabayi (IAS) GW and H inf 10 / 15

3. Scalar Emission from Energy Conservation Scalar fluctuations δφ X lead to: δρ φ = φδ φ X. Energy conservation: d 3 xδρ φ = M X = δφ X M X φ Converting to ζ 2 X N X (Hδφ X / φ) 2 results in: ζ 2 X NX M 2 X ɛm 2 pl Comparison with γ X gives r max ɛ 2. H 2 ɛm 2 pl Mehrdad Mirbabayi (IAS) GW and H inf 10 / 15

These scenarios can dominate vacuum tensor fluctuations and break the relation between r and H inf, but then they dominate vacuum scalar fluctuations and ɛ must be relatively large for observable values of r. However, scalar and tensor tilts are less sensitive to ɛ: n s 1 = 1 2 ɛ 5 ɛ 4 Hɛ n t = 1 2 ɛ 3 ɛ 4 Hɛ Mehrdad Mirbabayi (IAS) GW and H inf 11 / 15

Non-Gaussianity If there are N X incoherent emission events per H 4 : ζ 3 X = f NL ζ X 1 ζ 3 X f NL can be made small by increasing N X. N 1/2 X Mehrdad Mirbabayi (IAS) GW and H inf 12 / 15

Non-Gaussianity If there are N X incoherent emission events per H 4 : ζ 3 X = f NL ζ X 1 ζ 3 X N 1/2 X f NL can be made small by increasing N X. But there is an upper bound ρ X = N X H 3 M X M 2 pl H2 ɛ Combined with ζ X gives f NL 1 Away from the squeezed limit: B(k 1, k 2, k 3 ) 1 k 2 1 k2 2 k2 3 Mehrdad Mirbabayi (IAS) GW and H inf 12 / 15

Conclusions 1. It is possible to have γ X γ vac. 2. Then ζ X ζ vac such that r ɛ 2. Hence detectable r requires relatively large ɛ. 3. Tensor consistency condition r = 2n t is violated. 4. There is large non-gaussianity f NL 1. 5. Generically, the same bound applies to multi-field models, but models in which scalar emission is suppressed can be built. Mehrdad Mirbabayi (IAS) GW and H inf 13 / 15

Thank you! Mehrdad Mirbabayi (IAS) GW and H inf 14 / 15

Exception A two-field scenario Consider a two filed inflationary model with both fields φ and ψ slow-rolling. Suppose the energy source for the auxiliary sector X is ψ: M 2 X = M2 sin 2 ψ f. Energy transfer from 1 2 ψ 2 to X sector leads to δψ emission. If ζ ψ H ψ φ 2 + ψ 2 then the contribution to scalar spectrum can be made small. This seems non-generic, but can be realized for instance if the re-heating surface is determined by φ: V (φ, ψ) = θ(φ φ 0 )U(φ, ψ). Mehrdad Mirbabayi (IAS) GW and H inf 15 / 15