What WMAP taught us about inflation, and what to expect from Planck - PowerPoint PPT Presentation

What WMAP taught us about inflation, and what to expect from Planck Eiichiro Komatsu (Texas Cosmology Center, UT Austin) Aspects of Inflation, UT -TAMU Workshop, April 8, 2011 1

How Do We Test Inflation? • How can we answer a simple question like this: • “ How were primordial fluctuations generated? ” 2

Stretching Micro to Macro H –1 = Hubble Size δφ Quantum fluctuations on microscopic scales INFLATION! δφ 3 Quantum fluctuations cease to be quantum, and become observable

Power Spectrum • A very successful explanation (Mukhanov & Chibisov; Guth & Pi; Hawking; Starobinsky; Bardeen, Steinhardt & Turner) is: • Primordial fluctuations were generated by quantum fluctuations of the scalar field that drove inflation. • The prediction: a nearly scale-invariant power spectrum in the curvature perturbation, ζ =–(Hdt/d φ ) δφ • P ζ (k) = <| ζ k | 2 > = A/k 4–ns ~ A/k 3 • where n s ~1 and A is a normalization. 4

WMAP Power Spectrum Angular Power Spectrum Large Scale Small Scale about 1 degree on the sky 5

Getting rid of the Sound Waves Large Scale Small Scale Angular Power Spectrum Primordial Ripples 6

Inflation Predicts: Large Scale Small Scale Angular Power Spectrum l(l+1)C l ~ l ns–1 where n s ~1 7

Inflation may do this Large Scale Small Scale Angular Power Spectrum l(l+1)C l ~ l ns–1 “blue tilt” n s > 1 (more power on small scales) 8

...or this Large Scale Small Scale Angular Power Spectrum l(l+1)C l ~ l ns–1 “red tilt” n s < 1 (more power on large scales) 9

WMAP 7-year Measurement (Komatsu et al. 2011) Large Scale Small Scale Angular Power Spectrum l(l+1)C l ~ l ns–1 n s = 0.968 ± 0.012 (more power on large scales) 10

After 9 years of observations... WMAP taught us: • All of the basic predictions of single-field and slow-roll inflation models are consistent with the data • But, not all models are consistent (i.e., λφ 4 is out unless you introduce a non-minimal coupling) 11

Testing Single-field by Adiabaticity • Within the context of single-field inflation, all the matter and radiation originated from a single field, and thus there is a particular relation (adiabatic relation) between the perturbations in matter and photons: = 0 The data are consistent with S=0: | | < 0.09 (95% CL) 12

Inflation looks good • Joint constraint on the primordial tilt, n s , and the tensor-to-scalar ratio, r. • r < 0.24 (95%CL; WMAP7+BAO+H 0 ) 13

Gravitational waves are coming toward you... What do you do? • Gravitational waves stretch space, causing particles to move. 14

Two Polarization States of GW “+” Mode “X” Mode • This is great - this will automatically generate quadrupolar temperature anisotropy around electrons! 15

From GW to CMB Polarization Electron 16

From GW to CMB Polarization Redshift R e d s h i f t t f i h s Blueshift Blueshift e u l B t f i h s e u R l B e d s h i f t Redshift 17

From GW to CMB Polarization 18

“Tensor-to-scalar Ratio,” r ζ In terms of the slow-roll parameter: r=16 ε where ε = –(dH/dt)/H 2 = 4 π G(d φ /dt) 2 /H 2 ≈ (16 π G) –1 (dV/d φ ) 2 /V 2 19

Polarization Power Spectrum from ζ • No detection of polarization from gravitational waves (B-mode polarization) yet. 20

However • We cannot say, just yet, that we have definite evidence for inflation. • Can we ever prove, or disprove, inflation? 21

Planck may: • Prove inflation by detecting the effect of primordial gravitational waves on polarization of the cosmic microwave background (i.e., detection of r ) • Rule out single-field inflation by detecting a particular form of the 3-point function called the “local form” (i.e., detection of f NLlocal ) • Challenge the inflation paradigm by detecting a violation of inequality that should be satisfied between the local- form 3-point and 4-point functions 22

Planck might find gravitational waves (if r~0.1) If found, this would give us a pretty convincing proof that inflation did indeed happen. Planck? 23

And... • Typical “inflation data review” talks used to end here, but we now have exciting new tools: non-Gaussianity • To characterize a departure of primordial fluctuations from a Gaussian distribution, we use the 3-point function (bispectrum) and 4-point function (trispectrum) 24

Main Conclusions First: (Don’t worry if you don’t Eye-catchers understand what I am talking 4-point about here: I will explain it later.) amplitude ln( τ NL ) • The current limits 3.3x10 4 from WMAP 7-year (Smidt et are consistent with al. 2010) single-field or multi- field models. • So, let’s play around with the future. 3-point ln(f NL ) 74 amplitude 25 (Komatsu et al. 2011)

Case A: Single-field Happiness ln( τ NL ) • No detection of anything (f NL or τ NL ) after Planck. Single-field survived the test (for the moment: the future galaxy surveys can 600 improve the limits by a factor of ten). ln(f NL ) 10 26

(Suyama & Yamaguchi 2008; Komatsu 2010; Sugiyama, Komatsu & Futamase 2011) Case B: Multi-field Happiness(?) ln( τ NL ) • f NL is detected. Single-field is gone. • But, τ NL is also detected, in accordance with τ NL >0.5(6f NL /5) 2 expected from most 600 multi-field models. ln(f NL ) 30 27

(Suyama & Yamaguchi 2008; Komatsu 2010; Sugiyama, Komatsu & Futamase 2011) Case C: Madness • f NL is detected. Single- ln( τ NL ) field is gone. • But, τ NL is not detected, or found to be negative, inconsistent with τ NL >0.5(6f NL /5) 2 . • Single-field AND 600 most of multi-field models are gone. ln(f NL ) 30 28

Bispectrum 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 )F(k 1 ,k 2 ,k 3 ) model-dependent function Primordial fluctuation ”f NL ” 29

MOST IMPORTANT

Probing Inflation (3-point Function) • Inflation models predict that primordial fluctuations are very close to Gaussian. • In fact, ALL SINGLE-FIELD models predict the squeezed- limit 3-point function to have the amplitude of f NL =0.02. • Detection of f NL >1 would rule out ALL single-field models! • No detection of this form of 3-point function of primordial curvature perturbations. The 95% CL limit is: • –10 < f NL < 74 • The WMAP data are consistent with the prediction of simple single-field inflation models: 1–n s ≈ r ≈ f NL 31

A Non-linear Correction to Temperature Anisotropy • The CMB temperature anisotropy, Δ T/T, is given by the curvature perturbation in the matter-dominated era, Φ . • One large scales (the Sachs-Wolfe limit), Δ T/T=– Φ /3. For the Schwarzschild • Add a non-linear correction to Φ : metric, Φ =+GM/R. • Φ ( x ) = Φ g ( x ) + f NL [ Φ g ( x )] 2 (Komatsu & Spergel 2001) • f NL was predicted to be small (~0.01) for slow-roll models (Salopek & Bond 1990; Gangui et al. 1994) 32

“Local Form” B ζ • Φ is related to the primordial curvature perturbation, ζ , as Φ =(3/5) ζ . • ζ ( x ) = ζ g ( x ) + (3/5)f NL [ ζ g ( x )] 2 • B ζ ( k 1 , k 2 , k 3 )=(6/5)f NL x (2 π ) 3 δ ( k 1 + k 2 + k 3 ) x [P ζ (k 1 )P ζ (k 2 ) + P ζ (k 2 )P ζ (k 3 ) + P ζ (k 3 )P ζ (k 1 )] 33

f NL : Shape of Triangle • For a scale-invariant spectrum, P ζ (k)=A/k 3 , • B ζ ( k 1 , k 2 , k 3 )=(6A 2 /5)f NL x (2 π ) 3 δ ( k 1 + k 2 + k 3 ) x [1/(k 1 k 2 ) 3 + 1/(k 2 k 3 ) 3 + 1/(k 3 k 1 ) 3 ] • Let’s order k i such that k 3 ≤ k 2 ≤ k 1 . For a given k 1 , one finds the largest bispectrum when the smallest k, i.e., k 3 , is very small. • B ζ ( k 1 , k 2 , k 3 ) peaks when k 3 << k 2 ~k 1 • Therefore, the shape of f NL bispectrum is the squeezed triangle! (Babich et al. 2004) 34

B ζ in the Squeezed Limit • In the squeezed limit, the f NL bispectrum becomes: B ζ ( k 1 , k 2 , k 3 ) ≈ (12/5)f NL x (2 π ) 3 δ ( k 1 + k 2 + k 3 ) x P ζ (k 1 )P ζ (k 3 ) 35

Maldacena (2003); Seery & Lidsey (2005); Creminelli & Zaldarriaga (2004) Single-field Theorem (Consistency Relation) • For ANY single-field models * , the bispectrum in the squeezed limit is given by • B ζ ( k 1 , k 2 , k 3 ) ≈ (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 ~0.02. * for which the single field is solely responsible for driving inflation and generating observed fluctuations. 36

Suppose that single-field models are ruled out. Now what? • We just don’t want to be thrown into multi-field landscape without any clues... • What else can we use? • Four-point function! 37

Trispectrum: Next Frontier • The local form bispectrum, Β ζ ( k 1 , k 2 , k 3 ) =(2 π ) 3 δ ( k 1 + k 2 + k 3 )f NL [(6/5)P ζ (k 1 )P ζ (k 2 )+cyc.] • is equivalent to having the curvature perturbation in position space, in the form of: • ζ ( x )= ζ g ( x ) + (3/5)f NL [ ζ g ( x )] 2 • This can be extended to higher-order: • ζ ( x )= ζ g ( x ) + (3/5)f NL [ ζ g ( x )] 2 + (9/25)g NL [ ζ g ( x )] 3 This term is probably too small to see, so I don’t talk much about it. 38