Nonlinear reconstruction Yu Yu (SJTU) Tianlai Workshop, 17 Sep 2018 - PowerPoint PPT Presentation

Nonlinear reconstruction Yu Yu (SJTU) Tianlai Workshop, 17 Sep 2018

OUTLINE ➤ Motivation ➤ Nonlinear reconstruction ➤ On BAO ➤ On RSD ➤ On velocity field ➤ Conclusion

Nonlinear evolution Correlation between different k-bins Information saturation Rimes & Hamilton 2005

BAO reconstruction Padmanabhan+ 2012 linear continuity equation

BAO reconstruction ➤ Sharper peak / weaker damping damped BAO wiggles = linear wiggle * damping z=49 recon. w/ 10 Mpc/h s recon. w/ 20 Mpc/h s z=0.3 Seo+ 2016 Eisenstein+ 2007

BAO reconstruction Padmanabhan+ 2012 heavy smooth to validate the Linear displacement linear continuity equation

OUTLINE ➤ Motivation ➤ Nonlinear reconstruction ➤ On BAO ➤ On RSD ➤ On velocity field ➤ Conclusion

NEW reconstruction method ➤ solve for a curvilinear coordinate, in which the mass per grid is constant.

NEW reconstruction method ➤ solve for a curvilinear coordinate, in which the mass per grid is constant.

Algorithm ➤ Moving mesh approach ➤ Originally used in moving mesh simulation (N-body and hydrodynamic; see Pen 1995 & 1998) ➤ solve for a mesh following the nonlinear density evolution ➤ to keep (approximately) constant mass/energy resolution ➤ In our case, we need solve for a mesh consistent with the nonlinear density field, perturbatively and iteratively. ➤ POTENTIAL ISOBARIC GAUGE/COORDINATE

NEW reconstruction method Provide us an estimate of the E-mode displacement. ➤

Displacement ➤ We could reconstruct the E-mode component, ψ ENL + ψ EL

OUTLINE ➤ Motivation ➤ Nonlinear reconstruction ➤ On BAO ➤ On RSD ➤ Other applications ➤ Conclusion

Comparison with linear field Zhu, YU and Pen (2016) point to point comparison Cross-correlation coefficient up-limit

Acoustic peak recovery linear up-limit This method standard nonlinear Wang et al. (2017)

Fractional error on the distance measurement nonlinear density Eisenstein’s a factor of 1.8 improvement New recon. a factor of 2.7 improvement linear limit Wang et al. (2017)

Halo fields Yu et al. (2017)

dark matter Cross- correlation coefficient n h =2.77 × 10 -2 (h/Mpc) 3 n h =2.77 × 10 -3 (h/Mpc) 3 n h =2.77 × 10 -4 (h/Mpc) 3

Summary for BAO reconstruction ➤ For DM field, the fractional error in BAO measurement is reduced by a factor of 2.7, close to the linear limit. ➤ For SDSS-like sample, the characteristic scale is improved to k>0.36 h/Mpc, a factor of 2.29 improvement in scale, or 12 in number of linear modes. ➤ We expect this to substantially improve the BAO accuracy of dense, low redshift survey, including the SDSS main sample, 6dFGS and 21cm intensity mapping initiatives. ➤ move on SDSS MGS, BOSS LOWZ, CMASS … ➤ forecast for 21cm initiatives (see Xin’s talk)

OUTLINE ➤ Motivation ➤ Nonlinear reconstruction ➤ On BAO ➤ On RSD ➤ On velocity ➤ Conclusion

Reconstruction with RSD Algorithm works with RSD

Lagrangian ➤ Thus, RSD resides in the reconstructed density field. ➤ Before reconstruction: nonlinear density + nonlinear RSD ➤ After reconstruction: more linear density + more linear RSD?

Shift velocity v s ψ v sEz v sEx ψ E-mode B-mode ➤ The E-mode part of shift velocity could be reconstruction (v sEz ). ➤ Unwanted byproducts, v sEx, v sEy

Shift velocity total want miss don’t want

1D Cross-correlation coefficient ➤ 1D cross-correlation coe ffi cient with linear field + linear RSD, i.e., < δ NL (1+f μ 2 ) δ L > and < δ recon (1+f μ 2 ) δ L >

2D Cross-correlation coefficient ➤ 2D cross-correlation coe ffi cient with linear field + linear RSD, i.e., < δ NL (1+f μ 2 ) δ L > and < δ recon (1+f μ 2 ) δ L >

Reconstruction with RSD ➤ Usually we use upto k=0.1, since nonlinear RSD is not well understood. ➤ RSD is more linear in the reconstructed ANISOTROPIC density field. ➤ More linear RSD means more robust constrain on modified gravity

0.8 GR F6 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.8 0.8 F5 F4 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Multipole moments ➤ l=0, 2, 4 (monopole, quadrupole, hexadecapole) P 2 (k)/P 0 (k) —> β

Multipole moments simulated reconstructed P 0 (k) 0.2 0.2 P 2 (k)

Multipole moments ratio P 2 (k)/P 0 (k) 0.2 0.2 linear FR linear GR simulated reconstructed

[ P 2 ( k ) / P 0 ( k ) ] f(R) [ P 2 ( k ) / P 0 ( k ) ] GR Multipole moments ratio ratio

OUTLINE ➤ Motivation ➤ Nonlinear reconstruction ➤ On BAO ➤ On RSD ➤ On velocity field ➤ Conclusion

Velocity reconstruction ➤ Standard method: linear continuity equation ➤ Our attempts: relation with displacement v(q) v.s. Ψ (q) v(q) v.s. Ψ recon (q)

Velocity reconstruction ➤ Use the relation (transfer function) to convert the reconstructed displacement to the reconstructed velocity field.

Velocity reconstruction Eulerian Lagrangian

Velocity reconstruction n a i g n a r g a L Nonlinear Recon. Standard Recon. n a i r e l u E

CONCLUSION

Conclusion ➤ Reconstruction is useful in ➤ extracting BAO signal in low-redshift high-density survey ➤ using RSD to di ff er gravity models ➤ velocity reconstruction ➤ reconstruction of the initial condition ➤ mock generation ➤ Quite new area to explore

Constrained simulations ➤ Good cross-correlation coe ffi cient with initial density (DM). ➤ RAW reconstructed field is nonGaussian (uplimit of 3) ➤ Need de-noise (Wiener filter)

Constrained simulations original new simulation simulation

Fast mock generation ➤ DESI, PFS ➤ Synergetic surveys, spectroscopic galaxy survey X weak lensing, need correct density field and velocity field simultaneously. ➤ LPTs have long-standing problems, unable to produce correct density field, while PM unable to produce correct velocity. ➤ Improved understanding on the displacement will be helpful.

THANK YOU

Improvement P(k) = P s (k) + P n (k) linear signal term k*=0.157 h/Mpc P s (k) = C(k) P lin (k) k*=0.360 h/Mpc noise term a factor of 2.29 P n (k) = P(k) - P s (k) improvement in scale, look at k* or P s (k*) = P n (k*) 12 in number of linear modes scale both P(k) to P NGP (k) dashed line shot noise 1/n h Yu et al. (2017)

halo v.s. downgraded DM ➤ n m =b 2 n h ➤ This downgraded DM sample shares the same e ff ective shot noise as the halo sample ➤ This result tells us that the main limitation comes from the shot- noise, but the bias. Yu et al. (2017)

v.s. Eisenstein’s ➤ indeed outperform over the standard BAO reconstruction method for dense sample (n g >10 -3 (Mpc/h) -3 ). Yu et al. (2017)

modeling of the power spectrum shape by transfer functions TF L = < δ sr δ sL >/< δ sr δ sr > TF E = < δ sr δ sE >/< δ sr δ sr > δ sL =TF L * δ sr δ sE =TF E * δ sr P( δ sL )/P( δ sL ) P( δ sE )/P( δ sE )

Reconstructed V.S. real displacement ➤ Reconstructed displacement Φ v.s. real displacement Ψ ➤ Downgraded correlation in z-direction

Multipole moments simulated reconstructed 0.2 0.2 P 4 (k)

Multipole moments ratio Reference: 2012MNRAS.425.2128J arXiv:1205.2698 linear prediction linear prediction for f(R) for GR linear measurement measurement P 2 (k)/P 0 (k) for f(R) for GR (nonlinear) (nonlinear)

Multipole moments ratio ratio ] f(R) ] GR linear prediction ) ) k k ( ( P 0 P 0 / / ) ) k k ( ( P 2 P 2 [ [ Reference: 2012MNRAS.425.2128J arXiv:1205.2698

Elucid analog ➤ To fully use the correlation, add Gaussian to fill the power gap.