My main messages The N formalism covers all scalar-field cases - PowerPoint PPT Presentation

Aspects of the δN formalism David H. Lyth Particle Theory and Cosmology Group Physics Department Lancaster University Cambridge2006 – p.1/18

My main messages • The δN formalism covers all scalar-field cases • Slow-roll inf., k -inf., ghost inf., ( R 2 gravity etc. ??) Cambridge2006 – p.2/18

My main messages • The δN formalism covers all scalar-field cases • Slow-roll inf., k -inf., ghost inf., ( R 2 gravity etc. ??) • User-friendly formulas for spectral index, non-gaussianity • Cf. spectral tilt: n − 1 = 2 η − 6 ǫ ( Liddle/DHL 1992 ) Cambridge2006 – p.2/18

My main messages • The δN formalism covers all scalar-field cases • Slow-roll inf., k -inf., ghost inf., ( R 2 gravity etc. ??) • User-friendly formulas for spectral index, non-gaussianity • Cf. spectral tilt: n − 1 = 2 η − 6 ǫ ( Liddle/DHL 1992 ) • Trispectrum, even higher correlators, could be as important as the bispectrum Cambridge2006 – p.2/18

My main messages • The δN formalism covers all scalar-field cases • Slow-roll inf., k -inf., ghost inf., ( R 2 gravity etc. ??) • User-friendly formulas for spectral index, non-gaussianity • Cf. spectral tilt: n − 1 = 2 η − 6 ǫ ( Liddle/DHL 1992 ) • Trispectrum, even higher correlators, could be as important as the bispectrum • Need to specify box size L (infrared cutoff) • But parameters run with L Cambridge2006 – p.2/18

The correlators Spectrum P , bispectrum † f NL , trispectrum †† τ NL : (2 π ) 3 δ ( k + k ′ ) K 1 P � ζ k ζ k ′ � = 5 (2 π ) 3 δ ( k + k ′ + k ′′ ) K 2 P 2 f NL 3 � ζ k ζ k ′ ζ k ′′ � = (2 π ) 3 δ ( k + k ′ + k ′′ + k ′′′ ) K 3 P 3 τ NL � ζ k ζ k ′ ζ k ′′ ζ k ′′′ � c = Cambridge2006 – p.3/18

The correlators Spectrum P , bispectrum † f NL , trispectrum †† τ NL : (2 π ) 3 δ ( k + k ′ ) K 1 P � ζ k ζ k ′ � = 5 (2 π ) 3 δ ( k + k ′ + k ′′ ) K 2 P 2 f NL 3 � ζ k ζ k ′ ζ k ′′ � = (2 π ) 3 δ ( k + k ′ + k ′′ + k ′′′ ) K 3 P 3 τ NL � ζ k ζ k ′ ζ k ′′ ζ k ′′′ � c = where the kinematic factors depend on the wave-vectors: 2 π 2 /k 3 K 1 ≡ K 1 ( k ) K 1 ( k ′ ) + 5perms K 2 ≡ K 2 K 1 ( | k + k ′′ | ) + 23perms K 3 ≡ † Komatsu/Spergel 2000; Maldacena 2003 †† Boubekeur/DHL 2005 Cambridge2006 – p.3/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) • − 54 < f NL < 114 ≪ P − 1 / 2 (WMAP + SDSS) Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) • − 54 < f NL < 114 ≪ P − 1 / 2 (WMAP + SDSS) ∼ 10 4 ≪ P − 1 (WMAP) • τ NL < Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) • − 54 < f NL < 114 ≪ P − 1 / 2 (WMAP + SDSS) ∼ 10 4 ≪ P − 1 (WMAP) • τ NL < • From last two, ζ is almost gaussian. Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) • − 54 < f NL < 114 ≪ P − 1 / 2 (WMAP + SDSS) ∼ 10 4 ≪ P − 1 (WMAP) • τ NL < • From last two, ζ is almost gaussian. • Observation eventually will give (absent detection) | f NL | < ∼ 1 and | τ NL | < ∼ 300 Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) • − 54 < f NL < 114 ≪ P − 1 / 2 (WMAP + SDSS) ∼ 10 4 ≪ P − 1 (WMAP) • τ NL < • From last two, ζ is almost gaussian. • Observation eventually will give (absent detection) | f NL | < ∼ 1 and | τ NL | < ∼ 300 • Or | f NL | < ∼ 0 . 01 (Coory 06) ?? Cambridge2006 – p.4/18

Observation • P = (5 × 10 − 5 ) 2 (WMAP + SDSS) • n − 1 = − 0 . 035 ± 0 . 012 (WMAP + · · · ) ( n − 1 ≡ d P /d ln k ) • − 54 < f NL < 114 ≪ P − 1 / 2 (WMAP + SDSS) ∼ 10 4 ≪ P − 1 (WMAP) • τ NL < • From last two, ζ is almost gaussian. • Observation eventually will give (absent detection) | f NL | < ∼ 1 and | τ NL | < ∼ 300 • Or | f NL | < ∼ 0 . 01 (Coory 06) ?? Cambridge2006 – p.4/18

The δN formula • Choose comoving x but generic t Cambridge2006 – p.5/18

The δN formula • Choose comoving x but generic t • Write g ij = a 2 ( x , t ) γ ij ( x , t ) with || γ || = 1 • So a ( x , t ) is local scale factor. Cambridge2006 – p.5/18

The δN formula • Choose comoving x but generic t • Write g ij = a 2 ( x , t ) γ ij ( x , t ) with || γ || = 1 • So a ( x , t ) is local scale factor. • At t 1 choose a ( x , t 1 ) = a ( t 1 ) (‘flat’ slice) Cambridge2006 – p.5/18

The δN formula • Choose comoving x but generic t • Write g ij = a 2 ( x , t ) γ ij ( x , t ) with || γ || = 1 • So a ( x , t ) is local scale factor. • At t 1 choose a ( x , t 1 ) = a ( t 1 ) (‘flat’ slice) • At t choose δρ = 0 (uniform density slice) • And write a ( x , t ) = a ( t ) e ζ ( x ,t ) Cambridge2006 – p.5/18

The δN formula • Choose comoving x but generic t • Write g ij = a 2 ( x , t ) γ ij ( x , t ) with || γ || = 1 • So a ( x , t ) is local scale factor. • At t 1 choose a ( x , t 1 ) = a ( t 1 ) (‘flat’ slice) • At t choose δρ = 0 (uniform density slice) • And write a ( x , t ) = a ( t ) e ζ ( x ,t ) • Then ζ ( x , t ) = δN where Cambridge2006 – p.5/18

The δN formula • Choose comoving x but generic t • Write g ij = a 2 ( x , t ) γ ij ( x , t ) with || γ || = 1 • So a ( x , t ) is local scale factor. • At t 1 choose a ( x , t 1 ) = a ( t 1 ) (‘flat’ slice) • At t choose δρ = 0 (uniform density slice) • And write a ( x , t ) = a ( t ) e ζ ( x ,t ) • Then ζ ( x , t ) = δN where � t d ln a ( x , t ) N = dt dt t 1 Salopek & Bond 1990; DHL, Malik & Sasaki 2005 (non-perturbative refs.) Cambridge2006 – p.5/18

The family of unperturbed universes • Use (inverse) smoothing scale k ≪ aH Cambridge2006 – p.6/18

The family of unperturbed universes • Use (inverse) smoothing scale k ≪ aH • Invoke separate universe assumption • Local evolution is that of an unperturbed universe • Zeroth order gradient expansion plus local isotropy Cambridge2006 – p.6/18

The family of unperturbed universes • Use (inverse) smoothing scale k ≪ aH • Invoke separate universe assumption • Local evolution is that of an unperturbed universe • Zeroth order gradient expansion plus local isotropy • Assume some light fields φ i ( x , t 1 ) define subsequent expansion N ( x , t ) • Choose c s a 1 H 1 /k ∼ a few, so that that δφ i is classical Cambridge2006 – p.6/18

The family of unperturbed universes • Use (inverse) smoothing scale k ≪ aH • Invoke separate universe assumption • Local evolution is that of an unperturbed universe • Zeroth order gradient expansion plus local isotropy • Assume some light fields φ i ( x , t 1 ) define subsequent expansion N ( x , t ) • Choose c s a 1 H 1 /k ∼ a few, so that that δφ i is classical • Then N ( x , t ) = N ( φ i ( x ) , ρ ( t )) the expansion of a family of unperturbed universes DHL, Malik & Sasaki 2005 (non-perturbative) Cambridge2006 – p.6/18

The standard scenario • Light fields φ i = { φ, σ i } • φ is the inflaton • σ i (if they exist) are Goldstone Bosons, no potential Cambridge2006 – p.7/18

The standard scenario • Light fields φ i = { φ, σ i } • φ is the inflaton • σ i (if they exist) are Goldstone Bosons, no potential • Everything determined by φ • identical separate universes • constant ζ Cambridge2006 – p.7/18

The standard scenario • Light fields φ i = { φ, σ i } • φ is the inflaton • σ i (if they exist) are Goldstone Bosons, no potential • Everything determined by φ • identical separate universes • constant ζ ∂ 2 N ζ = ∂N ∂φ δφ + 1 ∂φ 2 ( δφ ) 2 + · · · 2 Cambridge2006 – p.7/18

The standard scenario • Light fields φ i = { φ, σ i } • φ is the inflaton • σ i (if they exist) are Goldstone Bosons, no potential • Everything determined by φ • identical separate universes • constant ζ ∂ 2 N ζ = ∂N ∂φ δφ + 1 ∂φ 2 ( δφ ) 2 + · · · 2 • Slow-roll, GR ⇒ P δφ = ( H/ 2 π ) 2 and ∂N/∂φ = V/V ′ • First term of ζ dominates � 2 1 � H ∗ P ( k ) = 2 ǫ ∗ 2 π n − 1 = 2 η ∗ − 6 ǫ ∗ Cambridge2006 – p.7/18

Non-gaussianity in the standard scenario • In the δN approach, non-gaussianity from • non-linearity of ζ in terms of δφ • non-gaussianity of δφ Cambridge2006 – p.8/18