New Probes of Initial State of Quantum Fluctuations during - PowerPoint PPT Presentation

New Probes of Initial State of Quantum Fluctuations during Inflation Eiichiro Komatsu (Texas Cosmology Center, Univ. of Texas at Austin; Max-Planck-Institut für Astrophysik) C-lab, Nagoya University, July 23, 2012

This talk is based on... • Squeezed-limit bispectrum • Ganc & Komatsu, JCAP, 12, 009 (2010) • Non-Bunch-Davies vacuum and CMB • Ganc, PRD 84, 063514 (2011) • Scale-dependent bias and μ -distortion • Ganc & Komatsu, PRD 86, 023518 (2012) 2

Question • Did inflation really occur? 3

Question • Did inflation* really occur? * By “inflation,” I mean a period of the early universe during which the expansion of the universe accelerates. (Quasi-exponential expansion.) 4

Does this plot prove inflation? (Temperature Fluctuation) 2 5 =180 deg/ θ

Komatsu et al. (2011) Inflation looks good (in 2-point function) • P scalar (k)~k ns –4 • 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 ) 6

Motivation • Can we falsify inflation? 7

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?” 8

First Question: • Can we falsify single-field inflation? *I will not be talking about multi-field inflation today: for potentially ruling out multi-field inflation, see Sugiyama, Komatsu & Futamase, PRL, 106, 251301 (2011) 9

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 ~ ρ γ 3/4 10

Komatsu et al. (2011) Non-adiabatic Fluctuations • Detection of non-adiabatic fluctuations immediately rule out single-field inflation models. The current CMB data are consistent with adiabatic fluctuations: | | < 0.09 (95% CL) 11

Let’s use 3-point function k 3 k 1 • Three-point function (bispectrum) 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 12

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<< a H) 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. 14

Action • Einstein’s gravity + a canonical scalar field: • S=(1/2) ∫ d 4 x √ –g [R–( ∂Φ ) 2 –2V( Φ )] 15

Maldacena (2003) Quantum-mechanical Computation of the Bispectrum (3) 3 3 16

Initial Vacuum State ζ • Bunch-Davies vacuum, a k |0>=0 with [ η : conformal time] 17

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... 18

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 19

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 ) = 20

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. 21

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. 22

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. 23

Limits on f NL When f NL is independent of wavenumbers, it is called the “ local type .”

Komatsu&Spergel (2001)

Komatsu et al. (2011) Limits on f NL • f NL = 32 ± 21 (68%C.L.) from WMAP 7-year data • Planck’s CMB data is expected to yield Δ f NL =5. • f NL = 27 ± 16 (68%C.L.) from WMAP 7-year data combined with the limit from the large-scale structure (by Slosar et al. 2008) • Future large-scale structure data are expected to yield Δ f NL =1.

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.” 28

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 29 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 30 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 31 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 > 32

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) 33

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 34

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) 35

Ganc & Komatsu, JCAP, 12, 009 (2010) Result • where 36