RESPONSE FUNCTION OF THE COSMIC DENSITY POWER SPECTRUM AND RECONSTRUCTION

Transcription

1 RESPONSE FUNCTION OF THE COSMIC DENSITY POWER SPECTRUM AND RECONSTRUCTION Takahiro Nishimichi (Kavli IPMU) w/ Francis Bernardeau (IAP) Atsushi Taruya (YITP) Based on PLB 762 (2016) 247 and arxiv: See also RESPRESSO webpage:

2 OBJECTIVES To extract information from large-scale clustering signals, we need an accurate theory/model to meet the low statistical error level of big surveys quick evaluation to explore multi-d cosmological parameter space Different approaches available Perturbation theory (and its variant) Accuracy: large scale good (?), small scale no Speed: high loop order (2 or 3 loops) takes time (N-body) simulations Accuracy: good (w/ sufficient volume/realizations, only after a careful convergence study) Speed: takes much more time RESPRESSO! A possible integration of the 2 approaches

3 PERTURBATION THEORY IS IN CRISIS N-body linear theory Blas, Garny, Konstandin 14

4 PERTURBATION THEORY IS IN CRISIS N-body Next to leading linear theory Blas, Garny, Konstandin 14

5 PERTURBATION THEORY IS IN CRISIS N-body Next to leading Next-to-nextto-leading order linear theory Blas, Garny, Konstandin 14

6 PERTURBATION THEORY IS IN CRISIS N-body Next to leading Next-to-nextto-leading order linear theory Next-to-next-to-next-toleading order Blas, Garny, Konstandin 14

7 PERTURBATION THEORY IS IN CRISIS N-body Next to leading The success of the next-to-next-toleading (2 loop) order calculation just an illusion? Next-to-nextto-leading order linear theory Next-to-next-to-next-toleading order Blas, Garny, Konstandin 14

8 WHY? Perturbation theory (PT) is fine only when the quantity of interest is small and the series expansion is convergent Overdensity can reach >> 1 at present The fluid is assumed to follow an irrotational single-streaming flow Enters to multi-streaming phase after shell crossing (even if the initial condition is cold) These 2 things happen together (small scale, late time) The breakdown can propagate to large scales due to mode coupling The way how PT breaks down is totally non-trivial > simulations!

9 WHAT TO MEASURE? Basic things that determine the nonlinear mode-coupling structure: {F n (q 1,...,q n ),G n (q 1,...,q n )} n (n) (q 1,...,q n ), o (n) (q 1,...,q n ) K(k, q) =q P nl (k) P lin (q)

10 STANDARD PT KERNEL Very, very basic things (not observable from sims, though ) {F n (q 1,...,q n ),G n (q 1,...,q n )} continuity + Euler + Poisson eqs. (x) = (x)/ 1 (x) = v(x)

11 GAMMA EXPANSION (eg., RegPT) n (n) (q 1,...,q n ), o (n) (q 1,...,q n ) Bernardeau + 09 final state (density or velocity) multi-point propagator initial state Can calibrate against sims Good ansatz known Simpler expression for the spectra Clear physical interpretation (Crocce & Scoccimarro 06 for the 2pt propagator) Bernardeau, Taruya & TN 14 Standard PT Exp. asymptote

12 VIRTUE OF RESUMMATION Better convergence: everything is positive definite BAO almost done at the lowest order ð Taruya, TN + 12 Standard PT Improved PT (RegPT) [ (1) ] 2 P 0 P nl (k) 1-loop negative positive positive 2-loop negative positive

13 MORE INTUITIVE QUANTITY? RESPONSE! large scale structure gravitational evolution Input ( m,h,...; z) P nl (k) Output

14 MORE INTUITIVE QUANTITY? RESPONSE! large scale structure gravitational evolution Input ( m,h,...; z) P lin (k) P nl (k) To a very good approximation Output

15 MORE INTUITIVE QUANTITY? RESPONSE! large scale structure gravitational evolution Input P lin (k) P nl (k) K(k, q) =q P nl (k) P lin (q) Output

16 MORE INTUITIVE QUANTITY? RESPONSE! large scale structure gravitational evolution Input P lin (k) + P lin (k) P nl (k) K(k, q) =q P nl (k) P lin (q) Output

17 MORE INTUITIVE QUANTITY? RESPONSE! large scale structure gravitational evolution Input P lin (k) + P lin (k) P nl (k) K(k, q) =q P nl (k) P lin (q) Output + P nl (k)

18 RESPONSE FUNCTION K(k, q) =q P nl (k) P lin (q) k I want to study this mode at some late time t q 1 q 2 q 3 what is the impact from wave mode q at the initial time t0?

19 RESPONSE FUNCTION: THE FIRST TRIAL From order-by-order to the full order discussion possible Can estimate the derivative from a simulation ensemble TN, Bernardeau, Taruya 16 PLB K > 0 K < 0 ˆK i,j P lin j P i nl [P lin +,j ] P i nl ln P lin ln q [P lin,j ] linear P(k) initial condition wave mode q nonlinear P(k) final state wave mode k final wave mode k initial wave mode q [h/mpc]

20 RESPONSE FUNCTION: SIM VS PT Rescaled quantity: q 2k are shown SPT (2-loop) >> high q This is exactly where PT breaks down What N-body tells us is: Physics at strongly nonlinear regime does not propagate to large scales initial wave mode [h/mpc]

21 HIGH RES RESPONSE FUNCTION 1400 N=512^3 simulations to study fine structures of the response function TN, Bernardeau, Taruya 17 (arxiv: ) Vs 2-loop calculation based on different schemes (SPT and RegPT) New phenomenological model introduced

22 RESPONSE FUNCTION AT q << k Response function goes to zero from simulations Extended galilean invariance This is nicely explained by SPT Not the case for RegPT

23 RESPONSE FUNCTION AT q~k Peaky structure decay as time goes by SPT behaves weirdly at late time RegPT has its strength in this regime Efficiently captures mode transfer between nearby modes

24 RESPONSE FUNCTION AT k << q We need phenomenology here anyway!

25 OUR MODEL Regularize both in k and q Well-behaved over all q Eventually fails at high k

26 PRACTICAL USAGE? RECONSTRUCTION From the definition of a functional derivative Use this to predict P nl for cosmological model p 1 given P nl for another model p 0

27 STARTING POINT: SIMULATION DATABASE P nl database for the fiducial Planck 2015 cosmology from 10 x 2048^3 sims Cosmic variance suppressed with Angulo-Pontzen technique Fractional error < 0.1% Can smoothly interpolate over k and time

28 A SIMPLE IMPLEMENTATION Double the reliable k range from the pure RegPT prediction

29 Employ multi-steps MORE EXTREME MODELS

30 Employ multi-steps MORE EXTREME MODELS Ωm = Ωm = 0.15

31 RESPRESSO PYTHON PACKAGE AVAILABLE! (Rapid and Efficient SPectrum calculation based on RESponSe function)

32 SUMMARY P(k) to a 2D quantity K(k,q): more physical insight Difficulty in perturbative approaches Suppress small to large scale mode transfer! SPT and RegPT have good and bad behavior in different regimes RESPRESSO package available Response function is a natural interpolator over the cosmological parameter space Can go to k ~ 0.44 (0.35) h/mpc at z=1 (0.5) within 1% You can put your own simulation data if you do not like mine ;)