Squeezed-limit bispectrum, Non-Bunch-Davies vacuum, Scale-dependent - PowerPoint PPT Presentation

Squeezed-limit bispectrum, Non-Bunch-Davies vacuum, Scale-dependent bias, and Multi-field consistency relation Eiichiro Komatsu (Texas Cosmology Center, Univ. of Texas at Austin) Cook’s Branch, March 23, 2012

This talk is based on... • Squeezed-limit bispectrum • Ganc & Komatsu, JCAP, 12, 009 (2010) • Non-Bunch-Davies vacuum • Ganc, PRD 84, 063514 (2011) • Scale-dependent bias [and μ -distortion] • Ganc & Komatsu, in preparation • Multi-field consistency relation 2 • Sugiyama, Komatsu & Futamase, PRL, 106, 251301 (2011)

Motivation • Can we falsify inflation? 3

Falsifying “inflation” • We still need inflation to explain the flatness problem! • (Homogeneity problem can be explained by a bubble nucleation.) • However, the observed fluctuations may come from different sources. • So, what I ask is, “can we rule out inflation as a mechanism for generating the observed fluctuations?” 4

First Question: • Can we falsify single-field inflation? 5

An Easy One: Adiabaticity • Single-field inflation = One degree of freedom. • Matter and radiation fluctuations originate from a single source. = 0 Cold Photon Dark Matter * A factor of 3/4 comes from the fact that, in thermal equilibrium, ρ c ~(1+z) 3 and ρ γ ~(1+z) 4 . 6

Komatsu et al. (2011) Non-adiabatic Fluctuations • Detection of non-adiabatic fluctuations immediately rule out single-field inflation models. The data are consistent with adiabatic fluctuations: | | < 0.09 (95% CL) 7

Komatsu et al. (2011) Single-field inflation looks good (in 2-point function) • P scalar (k)~k 4– ns • n s =0.968 ±0.012 (68%CL; WMAP7+BAO+H 0 ) • r =4P tensor (k)/P scalar (k) • r < 0.24 (95%CL; WMAP7+BAO+H 0 ) 8

So, let’s use 3-point function k 3 k 1 • Three-point function! k 2 • B ζ ( k 1 , k 2 , k 3 ) = < ζ k 1 ζ k 2 ζ k 3 > = (amplitude) x (2 π ) 3 δ ( k 1 + k 2 + k 3 )b(k 1 ,k 2 ,k 3 ) model-dependent function 9

MOST IMPORTANT, for falsifying single-field inflation

Curvature Perturbation • In the gauge where the energy density is uniform, δρ =0, the metric on super-horizon scales (k<<aH) is written as ds 2 = – N 2 (x,t)dt 2 + a 2 (t)e 2 ζ (x,t) dx 2 • We shall call ζ the “curvature perturbation.” • This quantity is independent of time, ζ (x), on super- horizon scales for single-field models. • The lapse function, N (x,t), can be found from the Hamiltonian constraint. 11

Action • Einstein’s gravity + a canonical scalar field: • S=(1/2) ∫ d 4 x √ –g [R–( ∂Φ ) 2 –2V( Φ )] 12

Maldacena (2003) Quantum-mechanical Computation of the Bispectrum (3) 3 3 13

Initial Vacuum State ζ • Bunch-Davies vacuum, a k |0>=0: [ η : conformal time] 14

Maldacena (2003) Result k 3 k 1 k 2 • B ζ ( k 1 , k 2 , k 3 ) = < ζ k 1 ζ k 2 ζ k 3 > = (amplitude) x (2 π ) 3 δ ( k 1 + k 2 + k 3 )b(k 1 ,k 2 ,k 3 ) • b(k 1 ,k 2 ,k 3 )= } x { Complicated? But... 15

Maldacena (2003) Taking the squeezed limit k 3 k 1 (k 3 <<k 1 ≈ k 2 ) k 2 • B ζ ( k 1 , k 2 , k 3 ) = < ζ k 1 ζ k 2 ζ k 3 > = (amplitude) x (2 π ) 3 δ ( k 1 + k 2 + k 3 )b(k 1 ,k 2 ,k 3 ) • b(k 1 ,k 1 ,k 3 ->0)= } x { 2k 13 2k 13 k 13 k 13 16

Maldacena (2003) Taking the squeezed limit k 3 k 1 (k 3 <<k 1 ≈ k 2 ) k 2 • B ζ ( k 1 , k 2 , k 3 ) = < ζ k 1 ζ k 2 ζ k 3 > = (amplitude) x (2 π ) 3 δ ( k 1 + k 2 + k 3 )b(k 1 ,k 2 ,k 3 ) [ ] k 13 k 33 1 • b(k 1 ,k 1 ,k 3 ->0)= 2 =1–n s (1–n s )P ζ (k 1 )P ζ (k 3 ) = 17

Maldacena (2003); Seery & Lidsey (2005); Creminelli & Zaldarriaga (2004) Single-field Theorem (Consistency Relation) • For ANY single-field models * , the bispectrum in the squeezed squeezed limit (k 3 <<k 1 ≈ k 2 ) is given by • B ζ ( k 1 , k 1 , k 3 ->0) = (1–n s ) x (2 π ) 3 δ ( k 1 + k 2 + k 3 ) x P ζ (k 1 )P ζ (k 3 ) * for which the single field is solely responsible for driving inflation and generating observed fluctuations. 18

Maldacena (2003); Seery & Lidsey (2005); Creminelli & Zaldarriaga (2004) Single-field Theorem (Consistency Relation) • For ANY single-field models * , the bispectrum in the squeezed squeezed limit (k 3 <<k 1 ≈ k 2 ) is given by • B ζ ( k 1 , k 1 , k 3 ->0) = (1–n s ) x (2 π ) 3 δ ( k 1 + k 2 + k 3 ) x P ζ (k 1 )P ζ (k 3 ) * for which the single field is solely responsible for driving inflation and generating observed fluctuations. 19

Maldacena (2003); Seery & Lidsey (2005); Creminelli & Zaldarriaga (2004) Single-field Theorem (Consistency Relation) • For ANY single-field models * , the bispectrum in the squeezed squeezed limit (k 3 <<k 1 ≈ k 2 ) is given by • B ζ ( k 1 , k 1 , k 3 ->0) = (1–n s ) x (2 π ) 3 δ ( k 1 + k 2 + k 3 ) x P ζ (k 1 )P ζ (k 3 ) • Therefore, all single-field models predict f NL ≈ (5/12)(1–n s ). • With the current limit n s =0.96, f NL is predicted to be 0.017. * for which the single field is solely responsible for driving inflation and generating observed fluctuations. 20

Understanding the Theorem • First, the squeezed triangle correlates one very long- wavelength mode, k L (=k 3 ), to two shorter wavelength modes, k S (=k 1 ≈ k 2 ): • < ζ k 1 ζ k 2 ζ k 3 > ≈ <( ζ k S ) 2 ζ k L > • Then, the question is: “why should ( ζ k S ) 2 ever care about ζ k L ?” • The theorem says, “it doesn’t care, if ζ k is exactly scale invariant.” 21

ζ k L rescales coordinates Separated by more than H -1 • The long-wavelength curvature perturbation rescales the spatial coordinates (or changes the expansion factor) within a given Hubble patch: • ds 2 =–dt 2 +[ a (t)] 2 e 2 ζ (d x ) 2 x 1 = x 0 e ζ 1 x 2 = x 0 e ζ 2 ζ k L 22 left the horizon already

ζ k L rescales coordinates Separated by more than H -1 • Now, let’s put small-scale perturbations in. • Q. How would the ( ζ k S1 ) 2 ( ζ k S2 ) 2 conformal rescaling of coordinates change the amplitude of the small-scale perturbation? x 1 = x 0 e ζ 1 x 2 = x 0 e ζ 2 ζ k L 23 left the horizon already

ζ k L rescales coordinates Separated by more than H -1 • Q. How would the conformal rescaling of coordinates change the amplitude of the small-scale ( ζ k S1 ) 2 ( ζ k S2 ) 2 perturbation? • A. No change, if ζ k is scale- invariant . In this case, no correlation between ζ k L and x 1 = x 0 e ζ 1 x 2 = x 0 e ζ 2 ( ζ k S ) 2 would arise. ζ k L 24 left the horizon already

Creminelli & Zaldarriaga (2004); Cheung et al. (2008) Real-space Proof • The 2-point correlation function of short-wavelength modes, ξ =< ζ S ( x ) ζ S ( y )>, within a given Hubble patch can be written in terms of its vacuum expectation value (in the absence of ζ L ), ξ 0 , as: • ξ ζ L ≈ ξ 0 (| x – y |) + ζ L [d ξ 0 (| x – y |)/d ζ L ] • ξ ζ L ≈ ξ 0 (| x – y |) + ζ L [d ξ 0 (| x – y |)/dln| x – y |] • ζ S ( y ) • ξ ζ L ≈ ξ 0 (| x – y |) + ζ L (1–n s ) ξ 0 (| x – y |) • ζ S ( x ) 3-pt func. = <( ζ S ) 2 ζ L > = < ξ ζ L ζ L > = (1–n s ) ξ 0 (| x – y |)< ζ L2 > 25

This is great, but... • The proof relies on the following Taylor expansion: • < ζ S ( x ) ζ S ( y )> ζ L = < ζ S ( x ) ζ S ( y )> 0 + ζ L [d< ζ S ( x ) ζ S ( y )> 0 /d ζ L ] • Perhaps it is interesting to show this explicitly using the in-in formalism. • Such a calculation would shed light on the limitation of the above Taylor expansion. • Indeed it did - we found a non-trivial “counter- example” (more later) 26

Ganc & Komatsu, JCAP, 12, 009 (2010) An Idea • How can we use the in-in formalism to compute the two-point function of short modes, given that there is a long mode, < ζ S ( x ) ζ S ( y )> ζ L ? • Here it is! (3) S S ζ L 27

Ganc & Komatsu, JCAP, 12, 009 (2010) Long-short Split of H I (3) S S ζ L • Inserting ζ = ζ L + ζ S into the cubic action of a scalar field, and retain terms that have one ζ L and two ζ S ’s. (3) 28

Ganc & Komatsu, JCAP, 12, 009 (2010) Result • where 29

Result • Although this expression looks nothing like (1–n S )P(k 1 ) ζ kL , we have verified that it leads to the known consistency relation for (i) slow-roll inflation, and (ii) power-law inflation. • But, there was a curious case – Alexei Starobinsky’s exact n S =1 model. • If the theorem holds, we should get a vanishing bispectrum in the squeezed limit. 30

Starobinsky (2005) Starobinsky’s Model • The famous Mukhanov-Sasaki equation for the mode function is where • The scale-invariance results when So, let’s write z=B/ η 31

Starobinsky’s Potential • This potential is a one-parameter family; this particular example shows the case where inflation lasts very long: φ end -> ∞ 32

Ganc & Komatsu, JCAP, 12, 009 (2010) Result • It does not vanish! • But, it approaches zero when Φ end is large, meaning the duration of inflation is very long. • In other words, this is a condition that the longest wavelength that we observe, k 3 , is far outside the horizon. • In this limit, the bispectrum approaches zero. 33