Large-scale structure tests of inflation Marilena LoVerde - PowerPoint PPT Presentation

Large-scale structure tests of inflation Marilena LoVerde (Institute for Advanced Study)

Inflation as the origin of structure stars, galaxies, clusters of galaxies quantum small matter & energy fluctuations fluctuations Sloan gravitational inflation WMAP collapse δ T CMB δφ inflaton Φ curvature δρ matter δ n galaxies

Inflation as the origin of structure stars, galaxies, clusters of galaxies quantum small matter & energy fluctuations fluctuations Sloan gravitational inflation WMAP collapse statistics of these probe of this era δ T CMB δφ inflaton Φ curvature δρ matter δ n galaxies

Inflation as the origin of structure stars, galaxies, clusters of galaxies quantum small matter & energy fluctuations fluctuations Sloan gravitational inflation WMAP collapse statistics of these probe of this era δ T CMB δφ inflaton Φ curvature δρ matter δ n galaxies powerful probe

Inflation as the origin of structure stars, galaxies, clusters of galaxies quantum small matter & energy fluctuations fluctuations Sloan gravitational inflation WMAP collapse statistics of these probe of this era δ T CMB δφ inflaton Φ curvature δρ matter δ n galaxies So long as we can accurately model this relationship

Community Goal: Determine ``initial’’ conditions for density perturbations (learn about inflation) Intermediate Goal: Identify & model signatures of inflation (e.g. non- Gaussianity) that may persist in large- scale structure This talk: Analytic and N-body comparison of large-scale structure with 3 simple examples of non-Gaussian initial conditions

Lots of data!

Planck Lots of data! South Pole Telescope Atacama Cosmology Telescope

Dark Energy Survey Hobby-Eberly Telescope Dark Planck Energy EXperiment Sloan Lots of data! Digital Sky Survey South Pole Telescope Subaru Hyper Suprime Cam Atacama Cosmology Telescope

What can we learn about the statistics of the initial (post-inflationary) conditions?

10 2 Power spectum: 10 1 10 0 < Φ (k) Φ (k’)> ≡ P Φ (k) δ (k+k’) Mass Variance ∆ M / M 10 − 1 variance of fluctuations on 10 − 2 ~ scale k 10 − 3 SDSS DR7 (Reid et al. 2010) 10 − 4 LyA (McDonald et al. 2006) ACT CMB Lensing (Das et al. 2011) ACT Clusters (Sehgal et al. 2011) 10 − 5 CCCP II (Vikhlinin et al. 2009) BCG Weak lensing (Tinker et al. 2011) ACT+WMAP spectrum (this work) 10 − 6 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 21 10 22 10 23 Mass scale M [Msolar] Hlozek et al 2011 k

10 2 Power spectum: 10 1 10 0 < Φ (k) Φ (k’)> ≡ P Φ (k) δ (k+k’) Mass Variance ∆ M / M 10 − 1 variance of fluctuations on 10 − 2 ~ scale k 10 − 3 SDSS DR7 (Reid et al. 2010) 10 − 4 LyA (McDonald et al. 2006) ACT CMB Lensing (Das et al. 2011) ACT Clusters (Sehgal et al. 2011) 10 − 5 CCCP II (Vikhlinin et al. 2009) BCG Weak lensing (Tinker et al. 2011) ACT+WMAP spectrum (this work) 10 − 6 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 21 10 22 10 23 Mass scale M [Msolar] Hlozek et al 2011 k If the initial conditions were Gaussian, we’ d be done

10 2 Power spectum: 10 1 10 0 < Φ (k) Φ (k’)> ≡ P Φ (k) δ (k+k’) Mass Variance ∆ M / M 10 − 1 variance of fluctuations on 10 − 2 ~ scale k 10 − 3 SDSS DR7 (Reid et al. 2010) 10 − 4 LyA (McDonald et al. 2006) ACT CMB Lensing (Das et al. 2011) ACT Clusters (Sehgal et al. 2011) 10 − 5 CCCP II (Vikhlinin et al. 2009) BCG Weak lensing (Tinker et al. 2011) ACT+WMAP spectrum (this work) 10 − 6 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 21 10 22 10 23 Mass scale M [Msolar] Hlozek et al 2011 k If the initial conditions were Gaussian, we’ d be done but they don’ t have to be perfectly Gaussian!

What do non-Gaussian initial conditions look like?

“f NL ” probability skewness ~ f NL Φ (x) ∼ δσ (x)+ f NL δσ (x) 2 kurtosis ~ f NL2 . . . Salopek and Bond 1990; Gangui, Lucchin, Matarrese, Mollerach 1994; Komatsu and Spergel 2001 Φ value “g NL ” skewness ~ 0 probability kurtosis ~ g NL Φ (x) ∼ δσ (x) + g NL δσ (x) 3 + . . . . . . (Okamoto and Hu 2002; Enqvist and Nurmi 2005) Φ value “ τ NL ” probability ~ Φ (x) ∼ δφ (x)+ δσ (x) + f NL δσ (x) 2 + . . . ~ f NL = f NL (1 + P φφ /P σσ ) 2 skewness ~ f NL τ NL = f NL2 (1 + P φφ /P σσ ) kurtosis ~ τ NL and P φσ = 0 Φ value ( Φ =primordial gravitational potential) (Lyth and Wands 2002; Ichikawa, Suyama, Takahishi, Yamaguchi (2008); Tseliakhovich, Hirata, Slosar 2010)

More generally, non-trivial multi-point correlation functions (or polyspectra) are introduced

Bispectrum: 〈 Φ ( k ) Φ ( k ’) Φ ( k ’’) 〉 = 2 f NL (P Φ (k) P Φ (k’) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’) } } k’’ k function of largest in the k k’’ triangle “squeezed” limit k’ k’ ( Φ =primordial gravitational potential)

Bispectrum: 〈 Φ ( k ) Φ ( k ’) Φ ( k ’’) 〉 = 2 f NL (P Φ (k) P Φ (k’) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’) } } k’’ k function of largest in the k k’’ triangle “squeezed” limit k’ k’ Trispectrum: 〈 Φ ( k ) Φ ( k ’) Φ ( k ’’) Φ ( k ’’’) 〉 = g NL (P Φ (k) P Φ (k’) P Φ (k’’) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’+ k ’’’) c } + 2 τ NL (P Φ (k) P Φ (k’) P Φ (| k + k ’’|) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’+ k ’’’) function of a quadrilateral peaks in the peaks in the limit k’ squeezed limits k’ k’’ k’ k k k’’ k’’’ k’’ k k’’’ k’’’ ( Φ =primordial gravitational potential)

Bispectrum: 〈 Φ ( k ) Φ ( k ’) Φ ( k ’’) 〉 = 2 f NL (P Φ (k) P Φ (k’) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’) } } k’’ k function of largest in the k k’’ triangle “squeezed” limit k’ k’ Trispectrum: 〈 Φ ( k ) Φ ( k ’) Φ ( k ’’) Φ ( k ’’’) 〉 = g NL (P Φ (k) P Φ (k’) P Φ (k’’) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’+ k ’’’) c } + 2 τ NL (P Φ (k) P Φ (k’) P Φ (| k + k ’’|) + . . .) (2 π ) 3 δ ( k + k ’+ k ’’+ k ’’’) function of a quadrilateral peaks in the peaks in the limit k’ squeezed limits k’ k’’ k’ k k k’’ k’’’ k’’ k k’’’ k’’’ so g NL and τ NL different ``shape” trispectra ( Φ =primordial gravitational potential)

Helpful to consider how polyspectra couple different physical scales scales

“f NL ” Φ ∼ δσ + f NL δσ 2 〈 Φ short2 〉 = 〈 σ G,short2 〉 (1 + 4 f NL σ G,long (x)) small-scale power depends on large-scale fluctuations! ( Φ =primordial gravitational potential) Slosar, Hirata, Seljak, Ho, Padmanabhan 2008

“f NL ” Φ ∼ δσ + f NL δσ 2 〈 Φ short2 〉 = 〈 σ G,short2 〉 (1 + 4 f NL σ G,long (x)) small-scale power depends on large-scale fluctuations! “g NL ” Φ ∼ δσ + g NL δσ 3 + . . . 2 2 〈 Φ short3 〉 = 18 g NL 〈 σ G,short2 〉 σ G,long (x) ≡ f NLeff (x) 〈 σ G,short2 〉 small-scale skewness depends on large-scale fluctuations! ( Φ =primordial gravitational potential)

“f NL ” Φ ∼ δσ + f NL δσ 2 〈 Φ short2 〉 = 〈 σ G,short2 〉 (1 + 4 f NL σ G,long (x)) “g NL ” Φ ∼ δσ + g NL δσ 3 + . . . 2 2 〈 Φ short3 〉 = 18 g NL 〈 σ G,short2 〉 σ G,long (x) ≡ f NLeff (x) 〈 σ G,short2 〉 “ τ NL ” ~ Φ ∼ δφ + δσ + f NL δσ 2 + . . . ~ 〈 Φ s2 〉 = 〈 Φ G,short2 〉 (1 + 4 f NL σ G,long (x)) Φ = φ + σ ( Φ =primordial gravitational potential)

“f NL ” “g NL ” “ τ NL ” definition Φ (x)= φ G (x)+ σ G (x) Φ (x)= σ G (x)+f NL σ G (x) 2 Φ (x)= σ G (x)+g NL σ G (x) 3 +f NL (1+ ξ 2 ) σ G (x) 2 ξ 2 =P φφ /P σσ τ NL = f NL2 (1+ ξ 2 ) skewness 2 〈 Φ 3 〉 = 0 2 〈 Φ 3 〉 ≈ 〈 Φ 2 〉 6f NL 6f NL 〈 Φ 3 〉 ≈ 〈 Φ 2 〉 3 kurtosis 3 3 〈 Φ 4 〉 ≈ 24 g NL 〈 Φ 2 〉 〈 Φ 4 〉 ≈ 48 f NL2 〈 Φ 2 〉 〈 Φ 4 〉 ≈ 48 τ NL 〈 Φ 2 〉 c c c short-long scale 〈 Φ s2 〉 = 〈 Φ s2 〉 (1+4f NL Φ l ) 〈 Φ s2 〉 = 〈 Φ s2 〉 (1+6g NL Φ l2 ) 〈 Φ s2 〉 = coupling 〈 Φ s2 〉 (1+4(1+ ξ 2 )f NL σ l ) 2 〈 Φ s3 〉 = 18 g NL 〈 Φ s2 〉 Φ l -10 < f NL < 74 -12.34 × 10 5 < g NL < 15.58 × 10 5 -6000 < τ NL < 33000 Komatsu et al 2010 Fergusson et al 2010 Smidt et al 2010

These are meant to be cartoon examples, but these types of initial condition can arise from real models

For instance total energy dominated by inflaton: H 2 = 8 π G/3 V( φ , σ ) potential ∼ V( φ , σ ) δφ perturbations from inflaton Gaussian δσ σ curvaton φ curvature perturbations from curvaton can be non-Gaussian inflaton “f NL ” Φ ∼ δσ + δσ 2 “g NL ” Φ ∼ δσ + δσ 3 + . . . “ τ NL ” Φ ∼ δφ + δσ + δσ 2 + . . . Linde and Mukhanov 1997; Lyth and Wands 2002

But “local” models (i.e. Φ NG (x)=F( σ G (x))) of non-Gaussianity are just some examples Can also get non-Gaussianity from self interactions of inflaton (e.g. k-inflation, DBI, . . . ) that has a very different ``shape’’ But single-field models do not generate such extreme couplings of perturbations on short and long length scales