Gravitational Waves and the Scale of Inflation

Transcription

1 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

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

3 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

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

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

6 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

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

8 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

9 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

10 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

11 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

12 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

13 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

14 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. Most efficient emission is at ω H Mehrdad Mirbabayi (IAS) GW and H inf 8 / 15

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

16 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

17 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. 2.b. Coherent emission by extended configurations (e.g. gauge field A): N X Number of Species M X Energy / Hubble volume Mehrdad Mirbabayi (IAS) GW and H inf 9 / 15

18 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

19 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

20 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 In the concrete example of localized incoherent events r max 0.3ɛ 2 Mehrdad Mirbabayi (IAS) GW and H inf 10 / 15

21 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

22 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

23 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

24 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 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

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

26 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