Features in Inflation and Generalized Slow Roll

Transcription

1 Features in Inflation and Generalized Slow Roll Power l [multipole] Wayne Hu CosKASI, April 2014

2 BICEP Exercise

3 Year of the B-Mode Gravitational lensing B-modes (SPTPol, Polarbear...) detected Gravitational wave B-modes (BICEP2) measured 0.2 BB l(l+1)c l /2π [µκ 2 ] BICEP2 Polarbear SPTpol (lensing xcorr) l [multipole] BB l(l+1)c l /2π [µκ 2 ] 0.06 BICEP l [multipole]

4 Tensor Tension In ΛCDM with power law scalar power spectra, Planck temperature power spectrum in tension with BICEP2 polarization detection Planck SPT/ACT BICEP2

5 Tensor Temperature Excess r=0.2 and fixed acoustic peaks produces an excess in temperature power spectrum that is not observed (limits r< % CL) Exacerbates a prexisting 2-3σ tension in ΛCDM at r=0 C l /C l l

6 Running of the Tilt Introducing scale by running tilt changes inferences from temperature spectrum, weakening upper limit on r r=0.2 requires a large running of order the tilt, not compatible with scale-free potentials, more indicative of transient feature scale free n s =0.96 scalar power dn s /dlnk=-0.02 suppresses scalar spectrum above recombination horizon k [Mpc -1 ]

7 Inflationary Features and Generalized Slow Roll

8 Ordinary Slow-Roll Approximation Curvature power spectrum given by 2 H 2 R(k) = 8π 2 Mpl 2 ɛ Hc s k=1/s where s = dnc s /(ah) is the inflationary sound horizon and ɛ H = d ln H dn 1 The tensor-scalar ratio r = 16ɛ H c s Scalars can be suppressed by making r evolve strongly with scale Evolution of slow-roll parameters violates the ordinary slow roll approximation but does not interrupt inflation if ɛ H 1 Not sufficient to introduce features directly into 2 R

9 Generalized Slow-Roll Approximation Transient evolution in ɛ H c s preserves approximate de Sitter background Solve the Mukhanov-Sasaki equation iteratively with Green function technique using deviations from de Sitter as external source (Stewart 2002; Choe, Gong, Stewart 2004) Single source function G captures power spectrum deviations up to order unity with percent level accuracy (Dvorkin & Hu 2010) Valid for any P (X, φ) or inflation EFT described by (g ) n operators (Hu 2011) Bispectrum for all terms including leading order modefunction correction terms for c 2 s enhanced operators (Adshead, Hu, Miranda 2013)

10 GSR and the Potential GSR source function G vs potential combination 3(V /V)2-2V /V Potential Dvorkin & Hu (2009) s

11 Featuring the BICEP

12 Tensor Temperature Excess Prefers sharper change than running, suppression over 1efold (excess exists even without tensors) Steps in power from steps in tensor scalar ratio ε H c s l(l+1)cl/2π (µk 2 ) χ 2 =17 Power law r=0 Power law r=0.2 Step r=0.2 frac. C l /C l l [multipole]

13 Freezeout of Curvature Source function G deviations from de Sitter G =1-ns is tilt in slow roll 0.2 source function G slow roll de Sitter inflaton sound horizon s [Mpc]

14 Freezeout of Curvature De Sittter mode functions give W, linear transfer to curvature power For sharp features, curvature power oscillates or rings transfer function W(ks) smooth changes freeze out at sound horizon crossing rapid changes cause oscillation in k ks

15 Freezeout of Curvature De Sittter mode functions give W, linear transfer to curvature power For sharp features, curvature power oscillates or rings 30 2 curvature power R [10-10 ] k [Mpc]

16 Transfer to Anisotropy Radiation transfer projects to temperature, E-polarization anisotropy Projection is sharp in the acoustic temperature regime, everywhere in E-polarization Acoustic Acoustic Feature range Feature range ISW SW Reionization (b)

17 Transfer to Anisotropy Radiation transfer projects to temperature, E-polarization anisotropy Projection is sharp in the acoustic temperature regime, everywhere in E-polarization l(l+1)cl/2π (µk 2 ) frac. C l /C l Power law r=0 Step r=0.2 beyond slow roll l [multipole] Miranda, Hu, Adshead (2014)

18 Polarization Predictions Matching tensor excess in E-mode polarization If scalars are suppressed by feature, E-modes compensated as well (as opposed to TT statistical fluke) EE l(l+1)c l /2π (µk 2 ) EE EE frac. C l /C l Power law r=0.2 Power law r=0 Step r=0.2 tensor excess scalar suppression l [multipole] Miranda, Hu, Adshead (2014)

19 Reconstructing the Source

20 Inverse Problem Source function completely describes observable scalar properties Invert directly from observables to inflationary source Use transfer functions of a precomputed basis for rapid MCMC 0 de Sitter l(l+1)cl/2π (µk2) slow roll inflaton sound horizon s [Mpc] transfer function W(ks) source function G 0.1 curvature power 2R [10-10] k [Mpc] Acoustic 0.5 Feature range ISW 0.0 SW sound horizon crossing ks l [multipole] 1000

21 WMAP Basis Complete principal component basis for any observable feature with N>1/4 Cosmic variance and WMAP beam limit number to 20 components Maximum likelihood 2 lnl=17 maximum likelihood model Dvorkin & Hu (2011)

22 Functional Constraints on Source 20 PC filter on source function from WMAP data Suppression at s>1000mpc consistent with and Planck (in progress) BICEP2 changes preference for oscillation as tensors absorb and modify TT features G 20 [Mpc] Dvorkin & Hu (2011) s [Mpc]

23 Predicted Polarization If features are due to single field inflation (GSR) there must be corresponding ones in polarization Dvorkin & Hu (2011)

24 Bispectrum Features Predicts features in the bispectrum Efficiently calculated through generalized slow-roll Bispectrum features related to the l~20-40 glitch are large but confined to too small a range to be observed Adshead, Hu, Dvorkin, Peiris (2011) equilateral configuration Chen et al (2007)

25 Sharp Steps

26 Oscillations in Planck Data Sharp step preferred in fits at χ2=11-15 Chance noise realizations can produce spurious fits, matching E-polarization is key test

27 Sharp Real Space Feature Oscillatory high k power represents the Fourier transform of sharp correlation function feature with mild log divergence r 1 k 1 k 2 r θ 13 k θ 23 2s s k 2 r 3 k 1 θ 23 θ 13 k 3 Adshead, Hu, Huterer Dvorkin, & Hu, Smith Lim (2006) (2011)

28 Sharp Step Oscillatory high k power damped by finite width of feature For theoretical maximum k 2 to k d s s ~100; S/N ~ power spectrum Numerical GSR1 k d s s 3 G/k eq ~f nl eq Adshead Hu, & Huterer (2012;2014) & Smith (2006) 10 k eq s s 100

29 Summary Planck-BICEP2 tension between TT and BB may indicate features in inflationary spectrum Step in tensor scalar ratio r ɛ H c s significantly favored in ΛCDM Relatively sharp features preferred, beyond scope of ordinary slow roll approximation Ringing in spectrum highly constrained in acoustic regime, matching E-polarization predicted, testable Generalized slow roll technique accurately computes power spectrum and bispectrum Extremely sharp step separately preferred but can be mimicked by noise Conclusive tests in polarization and bispectrum