Primordial Non-Gaussianity

Transcription

1 Primordial Non-Gaussianity Sam Passaglia 1 1 University of Chicago KICP

2 In This Discussion Non-Gaussianity in Single-Field Slow-Roll Non-Gaussianity in the EFT of Inflation Observational Constraints

3 Non-Gaussianity in Single-Field Slow-Roll

4 Gaussian and Non-Gaussian Curvature Perturbations Inflation produces curvature perturbations ζ. Are they Gaussian or non-gaussian? Fnl = Local (Quadratic) Ansatz: 2 hζ 2 i) ζ = ζg + 3/5fNL (ζg G Fnl = 0 Fnl = 500 For canonical single field slow roll inflation fnl is of order ns 1, simply understood from separate universe perspective - Wayne Hu Fnl = 5000 (Dalal et al 2008)

5 Single Field Consistency Relation (following de Putter et al. 2015) ds 2 = a 2 (τ)[ dτ 2 + e 2ζ(τ,x) dx 2 ] For superhorizon modes in Unitary Gauge (Equifield Surfaces set a clock ) ζ = ζ short + ζ long Locally, can t know about long-wavelength mode. In appropriate [ coordinate system: ] d s 2 = a 2 (τ) dτ 2 + e 2 ζ s(τ, x) d x 2 ds 2 = d s 2

6 Single Field Consistency Relation Apply a spatial dilation with ζ l : x x(1 ζ l ) (We can move freely from ζ l / ζ l (x) / ζ l ( x)) d s 2 = a 2 (τ)e 2 ζ s(τ, x) d x 2 = a 2 (τ)e 2 ζ s(τ, x)+2ζ l dx 2 So spatial dilations generate large-scale curvature. ζ s (x) = ζ s ( x) = ζ s (x(1 + ζ l )) = (1 + ζ l (x )) ζ s In terms of long/short split, ansatz is ζ s = ( ζ lf NL )ζ G s = f NL = 5 6 x 5 6 d ln 2 ζ (k) d ln k = 5 12 (n s 1)

7 Shapes of Non-Gaussianity This was squeezed limit. ζ(k 1 )ζ(k 2 )ζ(k 3 ) characterizes non-gaussianity Function of 3*3-3 (momentum conservation) - 3 (rotational invariance) - 1 (scaling invariance) = 2 variables = leg ratios We just computed k 2 /k 3 << k 1 /k 2 : i.e long modes interacting with short mode f Equil. NL O(1)

8 Non-Gaussianity in the EFT of Inflation

9 EFT Idea (Following Cheung et al Wayne s Slides) Write down most general L consistent with symmetries of the problem Inflation: Want L with broken time diffs describing perturbations around flat FRW L(R µνσρ, g 00, K µν, µ, t) Lowest Order Operators should be most important

10 Unitary Gauge Action S = d 4 x [ 1 g 2 M Pl 2 R + 1 n! M n(t)(g ) n n=0 1 2 M 1 (t) 3 (g )δK µ µ 1 2 M 2 (t) 2 (δk µ) µ M ] 3 (t) 2 δk ν µ δkµ ν +... (From now on neglect the extrinsic curvature terms) M 0 and M 1 terms are linear in perturbations: Fixed by requiring the background to be FRW Not diff invariant: Time symmetry broken by design Theory has 3 d.o.f. : 2 graviton helicities + 1 scalar mode

11 Stueckelberg Trick: Make the Scalar d.o.f into a Scalar Field Perform a time-diffeomorphism t t = t + ξ(t, x i ) (i.e. a change in time-slicing) See how the action changes: This is the scalar d.o.f. Promote the parameter ξ(t, x i ) to a scalar field π(t, x i ) (with a slightly strange transformation rule π(x) π(x) ξ 0 (x)). We will have a diff invariant theory with the symmetries/background/perturbations we want

12 How does the Action change? t t + π ; g 00 δ(t+π) δ(t+π) δx µ δx g µν ; Mn(t) 4 Mn(t 4 + π) ν In general, many time-space mixing terms in action like: δ i πg 0i, πg 00,... If we gauge-fix by choosing spatially flat gauge, can neglect all the spatial metric perturbations Justifies neglecting extrinsic curvature terms: their contributions will be... fourth order? See Wayne s notes for a better explanation!

13 Goldstone Action in Spatially Flat Gauge to Cubic Order S π = d 4 x g [(MPl 2 ɛ HH 2 + 2M2 4 ) π 2 MPl 2 ɛ HH 2 (δ iπ) 2 a 2 + 2M2 4 ( π 3 π (δ iπ) 2 a 2 ) 4 3 M 3 4 π 3 ] + fourth order Recall ɛ H = Ḣ/H2 Removed the background-fixed terms

14 Effective Sound Speed of π c 2 s = 1 + 2M 4 2 M 2 P l ɛ HH 2 temporal/spacial coefficients at quadratic order Because M 2 is both in cubic and quadratic terms in Lagrangian expect non-gaussianity if c s << 1 π(δ i π) 2 a 2 π 2 kπrms c sa (π related to curvature fluctuations by ζ = Hπ in single-field slow-roll) f equil NL 1 c 2 s

15 Observational Constraints

16 From CMB Measure from 3-point functions of T and E fnl local = 0.8 ± 5.0; f equil NL = 4 ± 43; fnl ortho = 26 ± 21 But most interesting parameter range is f NL O(1) How can we reach this? S4? (S4 science book)

17 From Large Scale Structure? Multi-field models often give a scale dependent halo bias b(k) f squeeze NL k 2 (Dalal et al. 2007) From 800,000 photometric quasars (Leistedt 2014): 49 < f squeeze NL < 31 LSST forecasts: Systematics free σ(f NL ) O(1). Systematics-full σ(f NL ) O(30). Systematics-cleaned? σ(f NL ) O(5). Is it worth doing the analysis for DES - test bed for LSST?

18 Conclusion Non-gaussianity is a powerful probe of inflation EFT language makes it clear f NL related to effective sound speed of inflation On the verge of an interesting parameter space

19 Primordial Non-Gaussianity Sam Passaglia 1 1 University of Chicago KICP