A Closure Theory for Non-linear Evolution of Power Spectrum 2007/10/26 ROE-JSPS workshop 2007 arXiv:0708.1367 [astro-ph] A Closure Theory for Non-linear Evolution of Power Spectrum Atsushi Taruya （RESCEU, Univ.Tokyo） In collaboration with Takashi Hiramatsu （RESCEU, Univ.Tokyo）

Introduction and motivation Key ingredient in cosmology with galaxy redshift surveys: Baryon Acoustic Oscillations (BAOs) as cosmic standard ruler Constraining dark energy EOS: Seo & Eisenstein (2005) Rule-of-thumb Needs accurate theoretical predictions at least ~1% accuracy on P(k) or x(r) Among several systematic effects on P(k), “non-linear gravitational growth” we here consider

Theoretical approach to non-linear gravitational growth Perturbation theory (PT) Perturbative treatment of (CDM+baryon) fluid system （analytic) （e.g., Suto & Sasaki 1991) Based on field-theoretical approach, Crocce & Scoccimarro (2006ab,2007) “Renormalized Perturbation Theory (RPT)” 「New approach」 infinite class of perturbative corrections at all order. Standard PT calculation can be improved by re-summing an Related works: McDonald, Matarrese & Pietroni, Valageas, Matsubara (‘07) Fitting formulae Parameterized function calibrated by N-body simulation （semi-analytic） （e.g., Peacock & Dodds 1996; Smith et al. 2003) N-body simulation Particle-based simulations treating (CDM + baryon) as self-gravitating N-body system （Numerical）

Alternative approximate treatment is proposed based on the idea of RPT RPT： demonstration Amongst various theoretical predictions, RPT reproduces the non-linear behaviors of BAOs in N-body simulations quite well. z=0 One-loop PT Fitting formula Linear Renormalized perturbation theory RPT Several approximations in practical use of RPT Reliability of N-body simulations (still few %) Alternative approximate treatment is proposed based on the idea of RPT In this talk, Crocce & Scoccimarro (2007)

Basic Quantities in RPT Non-linear Power spectrum Non-linear propagator Non-linear vertex function (a,b,c=1,2)

Renormalized Expressions Power spectrum linear P(k) Self-energy Linear propagator Propagator Self-energy These are non-perturbative expressions in a sense that we need fully nonlinear theory for propagator and vertex function as well as power spectrum to predict something Needs some approximations

Crocce & Scoccimarro (2007) Vertex function: Lowest-order evaluation (tree approx.) Self-energy: Born approximation replace with Propagator: Approximately including full-order non-linearity Power spectrum self-energy linear P(k) Corrections up to two-loop order are essential to reproduce the N-body results

Closure Approximation AT & Hiramatsu, arXiv:0708.1367 Alternative self-consistent treatment to compute both non-linear power spectrum and propagator Lowest-order evaluation of vertex Truncation of higher-loop corrections than two-loop

Closure Equations Time variable Evolution equations corresponding to the truncated diagrams: Operator: subscripts 1, 2 indicate = Fourier kernel: ※

Analytic results: P(k) and x(r) Results based on the Born approximation of self-energy up to one-loop order (i.e., replacing with ) z=1 z=1 Standard PT （1-loop） Closure RPT RPT Linear Closure Linear N-body data: Jeong & Komatsu (2006)

Non-linear evolution of BAOs based on closure theory Summary Non-linear evolution of BAOs based on closure theory known as efficient treatment in subject of turbulence and non-equilibrium statistics “direct-interaction approximation” “mode-coupling theory” (e.g., Kraichnan 1959; Kawasaki 1970) Derivation of closure equations (Gaussian initial conditions) Analytic treatment P(k) and x(r) ※ Extention to non-Gaussian initial conditions is also possible For quantitative predictions for P(k) at k>0.2h/Mpc (z<1), full numerical treatment of closure equations is necessary Hiramatsu & AT (2007), in progress