A Closure Theory for Non-linear Evolution of Power Spectrum

1. 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）

2. 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

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

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

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

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

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

8. 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

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

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

11. Summary Non-linear evolution of BAOs based on closure theory known as efficient treatment in subject of turbulence and non-equilibrium statistics “mode-coupling theory” “direct-interaction approximation” (e.g., Kraichnan 1959; Kawasaki 1970) Derivation of closure equations (Gaussian initial conditions) P(k) and x(r) Analytic treatment ※ 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