Biasing: Where are we going?

Transcription

1 Biasing: Where are we going? Daniel Green Stanford Based on: 1402:5916 with Assassi, Baumann and Zaldarriaga +work in progress with Assassi, Baumann, Pajer & Senatore

2 Biasing: the Lion in the Path Daniel Green Stanford Based on: 1402:5916 with Assassi, Baumann and Zaldarriaga +work in progress with Assassi, Baumann, Pajer & Senatore

3 Outline Eulerian Halo Biasing Galaxy Biasing Confirmation Biasing Outlook

4 Eulerian Halo Bias

5 What is the problem? In a 3d survey, we observe galaxies not dark matter How do we get from EFT to Galaxies?? Galaxy formation is complicated (can t even simulate)

6 Why should we care? SDSS is still not modeling bias very well Cosmology analysis stops at k =.2 h Mpc 1 because of modeling Plot from arxiv They can simulate the DM, so that isn t the problem

7 Why should we care? Nice argument from McDonald and Roy: Modes on the CMB ~ 10 6 Cost of CMB ~ $ 10 9 Modes in Euclid with k <.1 h Mpc -1 ~ 10 6 Cost of Euclid ~ $ 10 9 Cost per mode ~ $ 1000 Value of modes from k=.1-.6 h Mpc -1 ~ $200 billion

8 Strategy Initial Conditions Large-Scale Structure SPT, EFT-of-LSS N-body simulations halo biasing dark matter halos galaxy biasing RSD galaxies

9 EFT for Eulerian Halo Bias Key assumptions: DM halos should be determined by local physics local measurements n halos (L) n halo = F (δ, Φ,,...) L > halo size We want to predict correlation on much larger scales

10 EFT for Eulerian Halo Bias Key assumptions: Fluctuations are small so we can Taylor expand n g (x) =F [δ(y L),...] b 0 + b 1 δ(x)+c 1 L 2 2 δ(x)+... Move every point to the center by expanding in derivatives Any local quantity can appear on the RHS (e.g. density, baryon density, radiation fields,...)

11 EFT for Eulerian Halo Bias McDonald & Roy (2009): include everything consistent with the symmetries δ h = b 1 δ + b 2 δ b s2 i j Φ i j Φ b v2 i v j i v j b 2L 2 2 δ +... density tidal tensor velocity gradients We imposed: locality in space & time, rotations, translations, and galilean invariance

12 EFT for Eulerian Halo Bias The EFT of LSS relates many of these terms Newtonian gravity : 2 Φ = δ At linear order in SPT : δ = i v i θ At quadratic order in SPT: 5 7 δ(2) θ (2) = 2 7 i j Φ i j Φ The are 3 (& 7) terms at 2nd (& 3rd) order in δ

13 Comparison to EFT of LSS Biasing follows the same rules as τ ij (Φ) McDonald & Roy is local in time but easily changed: δ h = τ dτ b 1 (τ, τ )δ(τ, x fl )+... Power counting is identical Bias coefficients start at zeroth order in derivatives

14 Non-locality in time = Assembly Bias? Non-locality in time is also not new: Assembly bias Identify a halo by mass AND formation time δ h (M,τ f )=b 1 (M,τ f )δ +... Oldest Youngest Gao et al. (2005) Is this related to our non-locality?

15 Renormalization Halos are physical: i.e. counting them should be unambiguous The quantities in the bias expansion are ambiguous: e.g. δ 2 k = Λ d 3 q (2π) 3 δ qδ k q Λ δkδ 2 k 2P 11 (k) = P 11 (k)f(λ) depends on regulator d 3 p (2π) 3 F 2(k, p)p 11 (p) Cutoff is NOT a physical parameter

16 Renormalization McDonald (2006) showed we can remove this ambiguity with bias coefficients: δ h =[b 1 b 2 f(λ)]δ + b 2 δ Shift in linear bias will cancel the divergence: δ h,k δ k b 1 P 11 (k)+b 2 O(k 3 P 11 (k) 2 ) +contact terms Convergent and cutoff independent (up to contact terms)

17 Renormalization It is a bit easier to define renormalized operator [δ 2 ]=δ 2 counter-terms Compute counter-terms using [δ 2 ]δ (1) q δ(1) q n 0 = [δ2 ]δ (1) q δ(1) q n 0 tree [δ 2 ]=δ 2 σ 2 (Λ) [ δ δ ] 2205 ( i j Φ g ) 2 + σ 2 (Λ) = d 3 p (2π) 3 P 11(p) Assassi et al.

18 Renormalization Depends on regulator and renormalization scheme E.g. In a scaling universe P 11 (k) k n σ 2 d d p (2π) d pn =0 No counter terms are needed for n> 3 Results also depend on the renormalization scale and choice of renormalization condition (scheme)

19 Renormalization: Summary After renormalization, write: δ h (x) = O b (R) O [O](x) Sum over all possible operators Each term is independent of the cutoff Counter-terms can include all possible operators: Consistency requires tidal tensor and velocity field

20 Relation to Time Dependence? The tidal tensor arises from time dependence Assume halos are conserved δ h + θ = i (δ h v i ) Chan et al. ; Baldauf et al. Plugging in the ansatz assuming no velocity bias b s2 2 7 (b 1 1) This is a different result from renormalization

21 Contact Terms Halo correlation functions are predicted up to δ h (x)o(0) (a 0 + a a )δ(x) These give rise to stochastic bias In momentum space, these are given by δ h (k)o(k ) a 0 + a 2 k 2 + a 4 k 4... Included via stochastic parameters or added by hand Can forbid a 0 if conserved: δ h = i j i

22 How many parameters Want 1-loop halo-halo power spectrum (z=0) Using n=-3/2, there are 5 bias coefficients (cubic bias parameters don t matter in 2 point function) δ h = b 1 δ + b 2 [δ 2 ]+b 2 2 δ + b s2 i j Φ i j Φ +b s3 [ i j Φ i j Φ i v j i v j ] Six more parameters for non-locality in time

23 Velocity Bias Redshifts are sensitive to the velocity of the galaxy Velocity of the galaxy need not match DM A general biasing formula: v i h = v i + c 1 i δ + c 2 i δ The leading term is fixed by Galilean Invariance θ h = θ + O( 2 δ) Biasing is quadratic in derivatives

24 Velocity Bias Redshifts are sensitive to the velocity of the galaxy Velocity of the galaxy need not match DM An alternative biasing formula: θ h = b 1 θ + b 2 θ Now, biasing at leading order in derivatives This bias is non-local in terms of the velocity

25 Galaxy Bias

26 Halos to Galaxies There are many different approaches to galaxy bias: Halo occupation distribution (HOD) Conditional Luminosity Function (CLF) Abundance Matching Machine learning (SVR, k-nn, etc.) Galaxy bias

27 HOD and CLF HOD is defined as : P (N M) e.g. Berlind & Weinberg, Zheng et al. Probability of finding N galaxies in a mass M halo CLF is defined as: Φ(L M) e.g. Yang et al. Mean galaxies of luminosity L in a mass M halo Both assume halo mass determines everything Calibrated to data

28 Abundance Matching A number of variants e.g. Vale and Ostriker, Conroy et al.,... Galaxies depend halo properties along its history e.g. depend on halo formation time, for time when it attained maximum mass Depends on non-observable quantities Results compared to late time statistics

29 Machine Learning Given a training set with galaxies and halos Xu et al. Use standard algorithms to make a model E.g. use SVR or k-nn given 6 halo properties Requires a reliable training set (including properties)

30 Galaxy Biasing We can also just bias the galaxies directly δ g = b 1 δ + b 2 δ +... Bias coefficients may depend on many parameters b 1 = f(m,τ f,m halo,...) We may end up averaging over parameters Compute just like for halos

31 Confirmation Bias

32 There are a lot of bias parameters Want to compute halo power spectrum at 1-loop: 5 bias kernels 11 bias parameters (5 extra moments) 4 bias parameters for tree-level bispectrum Need 11 parameters for 7 correlation functions Serious danger of over-fitting

33 Confirmation Bias Every bias model has assumptions: Halo Bias: locality, equivalence principle, neglect baryons Galaxy Bias : neglect baryons, details of halo Are we really going to test these assumptions? OR do we have enough parameters to fit anything?

34 Strategy 1: reduce number of parameters Maybe we can relate the parameters e.g. assuming halo conservation we had: b s2 = 2 7 (b 1 1) b s3 = (b 1 1) fix the 2nd and 3rd order tidal tensor terms This is what BOSS does in DR11 Beutler et al. Is conservation reliable (especially at large z)?

35 Strategy 2: calibrate halos in N-body Assume LRGs follow halos Completely determine halo biasing from N-body Measure bias by running different densities or different initial conditions? Allow for small LRG - halo bias (?)

36 What is the worry? Wrong models may work fine on some data (false confidence in procedure / bias future results) Every method seems to have a lot of assumptions Still enough free parameter to fit most data e.g. In some analyses, one use the model δ h = b 1 (1 + a 1 k + a 2 k 2 )δ Cosmological parameters were still consistent

37 What is the worry? WiggleZ - arxiv:

38 Outlook

39 What can EFT bring to bias? The EFT philosophy gives us a tools Check that every term has been included How many terms to include at a given order Method for calculating perturbatively Renormalization

40 What can EFT bring to bias? It would be too easy to fit for 11 parameters at agree with data There are already models that work to k = 1 h Mpc -1 Are there non-trivial tests of the approach? Are there ways to reduce the number of parameters, without adding unnecessary assumptions?