Talk given at MIAPP workshop on Dynamics of Large-Scale Structure Formation in Munich, July 2019. Based on https://arxiv.org/abs/1811.10640.

1. Marcel Schmittfull (IAS) and Marko Simonović (CERN)
Based on arxiv:1811.10640 with
Valentin Assassi
Matias Zaldarriaga
MPIAA Workshop, Munich, July 2019
Modeling biased tracers at the field level
1

2. MPIAA Workshop, Munich, July 2019
Modeling biased tracers at the field level (Part I)
2
Biased tracers at the field level in PT

3. Overview
We calculate halo density field in PT and compare to simulations
1. How well does perturbative bias expansion work?
2. How well the halo density field correlates with the initial conditions?
3. What are the properties of the noise?

4. Overview
Most of the analyses use n-point functions. Disadvantages:
These questions have been extensively explored in the past
Desjacques, Jeong, Schmit: Large-Scale Galaxy Bias
— Cosmic variance, compromise on resolution/size of the box
— At high k hard to disentangle different sources of nonlinearities
— Overfitting (smooth curves, many parameters)
— Only a few lowest n-point functions explored in practice
— Difficult to isolate and study the noise

5. Overview
Advantages:
Use fields rather than summary statistics
— No cosmic variance, small boxes with high resolution are sufficient
— High S/N at low k, no need to go to the very nonlinear regime
— No overfitting, each Fourier mode (amplitude and phase) is fitted
— “All” n-point functions measured simultaneously
— Easier to isolate and study the noise
Baldauf, Schaan, Zaldarriaga (2015)
Lazeyras, Schmit (2017)
Abidi, Baldauf (2018)
McQuinn, D’Aloisio (2018)

6. Same initial conditions
Sim
ulation
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PT
Overview

7. Bias at the field level
5
s x of a halo at the initial position q are given by x = q + (q). The overdensity
1 + h(x) =
Z
d3
q (1 + h(q)) D(x q (q)) , (12)
Fourier transform of this field in Eulerian space is
d3
x (1 + h(x)) e ik·x
=
Z
d3
q (1 + h(q)) e ik·(q+ (q))
. (13)
d in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero
er our formulas. The nonlinear displacement from Lagrangian to Eulerian position
ve series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich
1(q) =
Z
k
eik·q ik
k2 1(k) . (14)
n be written as
3
Z
ik·q ik
linear displacement is large
5
rdinates x of a halo at the initial position q are given by x = q + (q). The overdensity
en by
1 + h(x) =
Z
d3
q (1 + h(q)) D(x q (q)) , (12)
. The Fourier transform of this field in Eulerian space is
k) ⌘
Z
d3
x (1 + h(x)) e ik·x
=
Z
d3
q (1 + h(q)) e ik·(q+ (q))
. (13)
ion and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero
not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian position
urbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich
1(q) =
Z
k
eik·q ik
k2 1(k) . (14)
ment can be written as
2(q) =
3
14
Z
k
eik·q ik
k2
G2(k) . (15)
ription of the nonlinear displacement field and expanding the exponent e ik· (q)
in Eq. (13)
5
f a halo at the initial position q are given by x = q + (q). The overdensity
h(x) =
Z
d3
q (1 + h(q)) D(x q (q)) , (12)
er transform of this field in Eulerian space is
x (1 + h(x)) e ik·x
=
Z
d3
q (1 + h(q)) e ik·(q+ (q))
. (13)
he rest of the paper we restrict the range of momenta to k 6= 0, so that the zero
ur formulas. The nonlinear displacement from Lagrangian to Eulerian position
ries = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich
1(q) =
Z
k
eik·q ik
k2 1(k) . (14)
written as
3
Z
ik·q ik
Lagrangian space
Eulerian space
halo
5
t the initial position q are given by x = q + (q). The overdensity
d3
q (1 + h(q)) D(x q (q)) , (12)
rm of this field in Eulerian space is
)) e ik·x
=
Z
d3
q (1 + h(q)) e ik·(q+ (q))
. (13)
the paper we restrict the range of momenta to k 6= 0, so that the zero
s. The nonlinear displacement from Lagrangian to Eulerian position
1 + 2 + · · · . At first order, we have the well-known Zel’dovich
such that the Eulerian coordinates x of a halo at the initial position q are given by x
generated in this way is given by
1 + h(x) =
Z
d3
q (1 + h(q)) D(x q (q)) ,
where D is the Dirac delta. The Fourier transform of this field in Eulerian space is
h(k) ⌘
Z
d3
x (1 + h(x)) e ik·x
=
Z
d3
q (1 + h(q)) e ik·(q+
For simplicity, in this equation and in the rest of the paper we restrict the range of mom
modes or mean density do not enter our formulas. The nonlinear displacement from L
such that the Eulerian coordinates x of a halo at the init
generated in this way is given by
1 + h(x) =
Z
d3
q (1 +
where D is the Dirac delta. The Fourier transform of this
h(k) ⌘
Z
d3
x (1 + h(x)) e ik·x
For simplicity, in this equation and in the rest of the paper
modes or mean density do not enter our formulas. The no
can be expanded in a perturbative series = 1 + 2
approximation [69]
1(q) =
Z
k
e
We need a hybrid scheme which takes into account large bulk flows

8. Bias at the field level
licity, in this equation and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the ze
r mean density do not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian positi
xpanded in a perturbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovi
mation [69]
1(q) =
Z
k
eik·q ik
k2 1(k) . (1
nd-order displacement can be written as
2(q) =
3
14
Z
k
eik·q ik
k2
G2(k) . (1
e perturbative description of the nonlinear displacement field and expanding the exponent e ik· (q)
in Eq. (1
sible to recover the usual Standard Eulerian bias expansion. This procedure also fixes the relation betwe
an bias parameters and their Standard Eulerian counterparts. Of course, this is not a surprise, as we expe
descriptions to agree order by order in perturbation theory.
e other hand we do not want to expand the full nonlinear displacement. We are going to keep the large
q) exponentiated and expand only the higher-order terms.3
In this way the largest part of the problema
acements is not expanded in perturbation theory. With this in mind, we can rewrite Eq. (13) in the followi
h(k) =
Z
d3
q
⇣
1 + bL
1 1(q) + bL
2 ( 2
1(q) 2
1) + bL
G2
G2(q) + · · ·
ik · 2(q) + · · ·
⌘
e ik·(q+ 1(q))
, (1
e new contributions come from expanding the second (and higher) order displacement field in the exponen
ortant to stress that at leading order this new term can be expressed through the second order operator
(15)). Therefore, at second order in perturbation theory, expanding the nonlinear terms in the displaceme
) only shifts some of the standard Lagrangian bias parameters by a calculable constant. We will give mo
The usual approximation in (C)LPT for example: Vlah, Castorina, White (2016)
art with the description of biased tracers in Lagrangian space. The displacement field is then
t linear contribution and smaller higher order corrections. The nonlinear corrections to are
while the linear piece is kept in the exponent. In this way, the dominant part of the large
treated exactly, and the resulting operators once written in Eulerian space are automatically
est of this section we give the details of this construction.
ity field at Lagrangian position q is modeled using a bias expansion in the linear Lagrangian-
L
h(q) = bL
1 1(q) + bL
2 ( 2
1(q) 2
1) + bL
G2
G2(q) + · · · , (8)
re Lagrangian bias parameters, 2
1 is the r.m.s. fluctuation of the linear density field
2
1 =
⌦ 2
1(q)
↵
=
Z 1
0
dk
2⇡2
k2
P11(k) , (9)
) is defined as2
G2(q) ⌘

@i@j
@2 1(q)
2
2
1(q) . (10)
this operator in momentum space is given by
G2(k) =
Z
p

(p · (k p))2
p2|k p|2
1 1(p) 1(k p) . (11)
ng notation in which
R
p
⌘
R
d3
p/(2⇡)3
. In the bias expansion (8) we kept only terms up to
bation theory. We will continue to work at this order throughout this section, because it is
ng notation and motivating the bias model that we are going to use to make comparisons with
er order or higher derivative operators needed for the consistent one-loop calculation can be
ded. We will come back to this in Section VIII.
n Eq. (8) is in Lagrangian space. In order to go to Eulerian space, let us start from Eq. (8) and
split into the dominant linear contribution and smaller higher order corrections. The nonlinear corrections
treated perturbatively, while the linear piece is kept in the exponent. In this way, the dominant part of
displacements can be treated exactly, and the resulting operators once written in Eulerian space are auto
IR-resummed. In the rest of this section we give the details of this construction.
The proto-halo density field at Lagrangian position q is modeled using a bias expansion in the linear La
space density 1(q):
L
h(q) = bL
1 1(q) + bL
2 ( 2
1(q) 2
1) + bL
G2
G2(q) + · · · ,
where bL
1, bL
2, bL
G2
, . . . are Lagrangian bias parameters, 2
1 is the r.m.s. fluctuation of the linear density field
2
1 =
⌦ 2
1(q)
↵
=
Z 1
0
dk
2⇡2
k2
P11(k) ,
and the operator G2(q) is defined as2
G2(q) ⌘

@i@j
@2 1(q)
2
2
1(q) .
The representation of this operator in momentum space is given by
G2(k) =
Z
p

(p · (k p))2
p2|k p|2
1 1(p) 1(k p) .
Notice that we are using notation in which
R
p
⌘
R
d3
p/(2⇡)3
. In the bias expansion (8) we kept only te
second order in perturbation theory. We will continue to work at this order throughout this section, bec
su cient for introducing notation and motivating the bias model that we are going to use to make compar
simulations. The higher order or higher derivative operators needed for the consistent one-loop calculati
straightforwardly included. We will come back to this in Section VIII.
The bias expansion in Eq. (8) is in Lagrangian space. In order to go to Eulerian space, let us start from E
include the gravitational evolution. The gravitational evolution is encoded in the nonlinear displacement fi
5
Eulerian coordinates x of a halo at the initial position q are given by x = q + (q). The overdensity
is way is given by
1 + h(x) =
Z
d3
q (1 + h(q)) D(x q (q)) , (12)
Dirac delta. The Fourier transform of this field in Eulerian space is
h(k) ⌘
Z
d3
x (1 + h(x)) e ik·x
=
Z
d3
q (1 + h(q)) e ik·(q+ (q))
. (13)
n this equation and in the rest of the paper we restrict the range of momenta to k 6= 0, so that the zero
density do not enter our formulas. The nonlinear displacement from Lagrangian to Eulerian position
ed in a perturbative series = 1 + 2 + · · · . At first order, we have the well-known Zel’dovich
[69]
1(q) =
Z
k
eik·q ik
k2 1(k) . (14)
er displacement can be written as
2(q) =
3
14
Z
k
eik·q ik
k2
G2(k) . (15)
ik· (q)

9. Bias at the field level
Perturbative, Eulerian space, easy to compare to simulations
o expanded in the basis of shifted operators. We show in Appendix A that the Zel’dovich
n as
Z(k) = ˜1(k) +
1
2
˜G2(k)
1
3
˜G3(k) + · · · , (20)
or analogous to ˜G2 (see Appendix D). In other words, Z(k) can be absorbed in the bias
ging the bias parameters. Of course, this is just a choice, and there is nothing wrong in
he formulas. As we are going to see later, di↵erent choices may be more appropriate for
us point out that in the formula (20) the displacements 1(q) are treated exactly. In other
k· 1(q)
is never expanded in 1(q). The only expansion parameter is the derivative of the
1(q), which is a small quantity.5
This is consistent with the way the shifted operators
asis of shifted operators we can write the bias expansion of the halo density field in Eulerian
order in perturbation theory, in the following way
h(k) = b1
˜1(k) + b2
˜2(k) + bG2
˜G2(k) + · · · . (21)
this section. Notice that the new bias parameters bi di↵er from the original Lagrangian
his di↵erence comes from expanding the nonlinear part of the displacement (Eq. (16)) and
sity field in terms of shifted operators (Eq. (20)). We give the explicit relation of bi and bL
i
(21) has a similar structure as the usual Standard Eulerian bias expansion
h(k) = bE
1 (k) + bE
2 2(k) + bE
G2
G2(k) + · · · , (22)
ce that all fields in this equation are nonlinear. Apart from the IR resummation of the large
ant di↵erence compared to the expansion in terms of ˜O is that in Eq. (21) all operators are
inear field 1. As we are going to see, for the purposes of describing the biased tracers on
her important virtue of the expansion (21).
+ noise
IR resummation, correct positions of halos, spread of the BAO peak
Shifted operators easy to generate, analytical calculations straightforward
Only linear fields used in the construction
ik · 2(q) + · · · e , (16)
me from expanding the second (and higher) order displacement field in the exponent.
t leading order this new term can be expressed through the second order operator G2
cond order in perturbation theory, expanding the nonlinear terms in the displacement
he standard Lagrangian bias parameters by a calculable constant. We will give more
s in Section VIII.
tivates us to write down the bias expansion in Eulerian space in terms of shifted
he following way
˜O(k) ⌘
Z
d3
q O(q) e ik·(q+ 1(q))
, (17)
2
1), G2, . . .}.4
We would like to stress again a few important advantages that this
d operators are written in Eulerian space and therefore allow for easy comparisons
tion of their importance. (b) The large displacement terms 1(q) are kept resummed,
s with simulations on the level of realizations. Notice that this also implies that in
ass filter, compared to the wavelength of a Fourier mode 1(k). For a given wavenumber k, the
nto the long-wavelength and short-wavelength part: 1 = L
1 + S
1 , where L
1 = W(k) 1 and
L
1 on the short modes is fixed by the Equivalence Principle. Therefore, strictly speaking, only L
1
in any perturbative calculation S
1 has to be expanded order by order in perturbation theory. The
y keeping the full 1 in the exponent is always higher order in S
1 than terms we calculate. Also,
all scales. In order to keep the formulas simple, we decide not to do the long-short splitting in our
not just given by a translation of the position argument because they implicitly include the inverse
n @xi/@qj due to the coordinate transformation. This is similar to the Zel’dovich density, which is
gian space shifted by 1(q).
This motivates the bias expansion in terms of “shifted” operators
PT prediction

10. Bias at the field level
Example of DM
re the large bulk flows are also treated nonperturbatively. This question has been explored
nce [70]) and in this section we review the main arguments and give some further details. We
lest case of dark matter only and then move to biased tracers.
A. Dark Matter
atter field is given by the same expression as h where all Lagrangian bias parameters are set
˜ = ˜1 +
2
7
˜G2
3
14
[ ˜G2 ]
2
9
˜G3 +
1
6
˜3
˜S3 . (101)
this field up to one-loop order is given by
h˜1
˜1i +
4
7
h˜1
˜G2i +
4
49
h ˜G2
˜G2i
3
7
h˜1[ ˜G2 ]i
4
9
h˜1
˜G3i +
1
3
h˜1
˜3i 2h˜1
˜S3i . (102)
ments about some of the terms in this expression. The kernel of the G3 operator is such that
plies that the cross spectrum of shifted operators h˜1
˜G3i is non-vanishing only at the two-loop
t this contribution. The cross spectrum h 1[G2 ]i is proportional to P11(k)
h 1[G2 ]i =
8
3
P11(k)
Z 1
0
p2
dp
4⇡2
P11(p) . (103)
expression for the shifted fields are of the two-loop order and we will ignore them. In the
the one-loop power spectrum for biased tracers this term renormalizes the linear bias b1.
his case we are calculating the power spectrum of the dark matter field, this contribution has
ancellation is ensured by the contribution from ˜S3. The symmetrized kernel of this operator
✓ 2
◆
the contribution from h˜1[ ˜G2 ]i in the power spectrum. Therefore, the nontrivial terms that
rder are
˜P(k) = h˜1
˜1i +
4
7
h˜1
˜G2i +
4
49
h ˜G2
˜G2i +
1
3
h˜1
˜3i 2h˜1
˜Snew
3 i . (106)
d from the ˜S3 operator by subtracting the constant 4/21 contribution from the kernel. This is
one-loop IR-resummed power spectrum from a realization of the shifted fields.
di↵erent contributions to the power spectrum. The thin blue line is the power spectrum of the
The thick brown line is the sum of all four terms in the previous equation which represent the
s.15
One interesting point to notice is that the total one-loop contribution is at least an order of
an the leading term in the power spectrum on all scales. This result is not surprising, since the
near density field in terms of shifted operators is closely related to the expansion of the nonlinear
Lagrangian perturbation theory, and it is well known that the one-loop power spectrum of the
maller than the linear prediction on all scales.
ontributions to the one-loop dark matter power spectrum evaluated using Eq. (106). The thin blue
spectrum of the shifted linear density field. Di↵erent dotted and dashed lines are di↵erent one-loop
d brown thick line is the sum of all one-loop terms.
˜
where
⌃2
⇤ =
1
6⇡2
Z ⇤
0
dp P11(p) (1 j0(p
The parameter in ⌃2
k is usually chosen to be smaller than
given wavenumber a↵ect only the fluctuations on shorter scales.
condition is not imposed, and for the purposes of the compariso
cosmology the di↵erence between the two definitions is small.
Figure 21. Comparison of the IR resummation and shifted fields, for
Figure 21 shows the comparison of the one-loop dark matter p
and the standard formula for the IR-resummation. The agreeme
The same results as in the standard PT approach with IR resummation

11. Bias at the field level
What operators are needed for the one-loop prediction?
7
B. Promoting Bias Parameters to Transfer Functions
as expansion in terms of shifted operators keeping only terms up to second order in perturbation
escribe the density field of biased tracers deeper in the nonlinear regime, we have to include
instance, even for the evaluation of the one-loop power spectrum one has to keep all cubic
closer look at this example
h(k) = b1
˜1(k) + b2
˜2(k) + bG2
˜G2(k) +
X
i
bi
3
˜Oi
3 , (23)
bic operators and bi
3 are the corresponding bias parameters. At lowest order in perturbation
ors correlate only with ˜1. We can split the cubic operators into parts parallel and orthogonal
˜Oi
3 =
h˜1
˜Oi
3i
h˜1
˜1i
˜1 + ˜Oi
3
h˜1
˜Oi
3i
h˜1
˜1i
˜1
!
⌘
h˜1
˜Oi
3i
h˜1
˜1i
˜1 + ˜Oi?
3 . (24)
a scale-dependent bias parameter b1(k), we can write
h(k) = b1(k) ˜1(k) + b2
˜2(k) + bG2
˜G2(k) +
X
i
bi
3
˜Oi?
3 . (25)
ew cubic operators are orthogonal to all other fields. This implies that even the bias expansion
he fields, with the appropriate b1(k), is su cient to describe the density field with the correct
m. Allowing for scale-dependent bias parameters e↵ectively allows us to reduce the order in
t we need to describe the density field of biased tracers at a given order in perturbation theory.
s motivation to promote all bias parameters to k-dependent functions
B. Promoting Bias Parameters to Transfer Functions
So far we wrote the bias expansion in terms of shifted operators keeping only terms up to second order
theory. If we want to describe the density field of biased tracers deeper in the nonlinear regime, we
higher order terms. For instance, even for the evaluation of the one-loop power spectrum one has t
operators. Let us take a closer look at this example
h(k) = b1
˜1(k) + b2
˜2(k) + bG2
˜G2(k) +
X
i
bi
3
˜Oi
3 ,
where ˜Oi
3 is a set of cubic operators and bi
3 are the corresponding bias parameters. At lowest order
theory the cubic operators correlate only with ˜1. We can split the cubic operators into parts parallel
to ˜1,
˜Oi
3 =
h˜1
˜Oi
3i
h˜1
˜1i
˜1 + ˜Oi
3
h˜1
˜Oi
3i
h˜1
˜1i
˜1
!
⌘
h˜1
˜Oi
3i
h˜1
˜1i
˜1 + ˜Oi?
3 .
In this way, allowing for a scale-dependent bias parameter b1(k), we can write
h(k) = b1(k) ˜1(k) + b2
˜2(k) + bG2
˜G2(k) +
X
i
bi
3
˜Oi?
3 .
At one-loop order, the new cubic operators are orthogonal to all other fields. This implies that even th
up to second order in the fields, with the appropriate b1(k), is su cient to describe the density field
one-loop power spectrum. Allowing for scale-dependent bias parameters e↵ectively allows us to red
perturbation theory that we need to describe the density field of biased tracers at a given order in pert
This example provides motivation to promote all bias parameters to k-dependent functions
h(k) = b1(k) ˜1(k) + b2(k) ˜2(k) + bG2
(k) ˜G2(k) + · · · ,
No contribution
at 1-loop
Keep the second order fields, promote biases to k-dependent functions
Make different operators “orthogonal” to each other

12. Bias at the field level
How much of the true halo density field correlates with this model?
rithm:
˜?
1 (k) = ˜1(k) , (27)
˜?
2 (k) = ˜2(k) + M10(k)˜1(k) , (28)
˜G?
2 (k) = ˜G2(k) + M20(k)˜1(k) + M21(k)˜2(k) . (29)
rotation matrix Mij(k) is M10(k) = P˜2
˜1
(k)/P˜1
˜1
(k) etc., and can be computed using a
ition of the 3 ⇥ 3 correlation matrix between the three shifted fields {˜1, ˜2, ˜G2} in every k-bin as
dix C. The bias expansion in this orthogonal basis is then
h(k) = 1(k) ˜1(k) + 2(k) ˜?
2 (k) + G2
(k) ˜G?
2 (k) + · · · . (30)
rameters, or transfer functions, i(k) are independent from each other. We can therefore add
ors using the same procedure without changing any of the lower-order bias parameters, which is
n our framework, where transfer functions are determined by minimizing the mean-square model
vel, the change of basis, i.e., going from bi to i, does not change the predicted halo density; it
more convenient way to interpret the numerical values of bias parameters. Also notice that the
mains unchanged, 1(k) = b1(k). In Section VIII we will present one-loop perturbation theory
) and compare against measurements of i(k) from N-body simulations.
s
2(k) = b2 + bG2
h˜2
˜G2i
h˜2
˜2i
. (87)
! 0 the numerator of the second term scales like O(k2
) while the denominator approaches a constant.
second term vanishes on very large scales. Notice that this contribution is not suppressed by loop
both numerator and denominator are of the same order in perturbation theory. For this reason, when
nctions are measured at not-so-large scales where the scaling O(k2
) is not valid, the second term is
negligible. However, because of the large constant contribution to h˜2
˜2i, the second term turns out
mall enough, given the size of the higher loop corrections that we are neglecting and final error bars
determine the bias parameters.
ze, we use the following minimal model to fit the k-dependent transfer functions
1(k) = b1 + c2
sk2
+ b2
h˜1
˜2i
h˜1
˜1i
+ bG2
h˜1
˜G2i
h˜1
˜1i
+ b 3
h˜1
˜3i
h˜1
˜1i
b1
h˜1
˜S3i
h˜1
˜1i
, (88)
2(k) = b2 , and G2
(k) = bG2
. (89)
s 5 free parameters, the same as the one-loop power spectrum. When we use the cubic bias model, we
parameter, b3, which is fitted from the low-k limit of 3(k).
C. Power Spectra of Shifted Fields from Theory and on a Grid
ansfer functions with Eq. (88) we need to calculate the power spectra h ˜Oa
˜Obi of shifted operators that
As we already mentioned, this calculation is the same as in [57, 70], and more details can be found
summarize only the main steps. Let us start from the definition
h ˜Oa
˜Obi(k) =
Z
d3
q e ik·q
⌦
Oa(q) Ob(0) e ik·( 1(q) 1(0))
↵
. (90)
1
operator ˜S3. Even though this may not be obvious from just a few leading
hoice is imposed by the fact that ˜S3 comes from the shift of the halo density field
This term is fixed and has no extra free parameters, even when renormalization
ave to add a k2
term to the transfer function 1(k) with a free coe cient. In
or the one-loop matter power spectrum we label this parameter c2
s even though
all UV contributions from correlation functions of the form h˜1
˜O3i and the bias
e bias operators such as r2
.
sfer function. This expression can be further simplified. The first step is to write
h˜?
2
˜?
2 i = h˜2
˜2i
h˜2
˜1i2
h˜1
˜1i
, (86)
?
i = h˜2
˜2i because the second term is higher order in perturbation theory. For
can replace h˜?
2
˜G?
2 i with h˜2
˜G2i. As a result, we can write the transfer function
2(k) = b2 + bG2
h˜2
˜G2i
h˜2
˜2i
. (87)
the second term scales like O(k2
) while the denominator approaches a constant.
on very large scales. Notice that this contribution is not suppressed by loop
denominator are of the same order in perturbation theory. For this reason, when
at not-so-large scales where the scaling O(k2
) is not valid, the second term is
because of the large constant contribution to h˜2
˜2i, the second term turns out
e size of the higher loop corrections that we are neglecting and final error bars
rameters.
g minimal model to fit the k-dependent transfer functions
h˜2
˜2i because the second term is higher order in perturbation theory. For
place h˜?
2
˜G?
2 i with h˜2
˜G2i. As a result, we can write the transfer function
2(k) = b2 + bG2
h˜2
˜G2i
h˜2
˜2i
. (87)
econd term scales like O(k2
) while the denominator approaches a constant.
very large scales. Notice that this contribution is not suppressed by loop
minator are of the same order in perturbation theory. For this reason, when
ot-so-large scales where the scaling O(k2
) is not valid, the second term is
use of the large constant contribution to h˜2
˜2i, the second term turns out
of the higher loop corrections that we are neglecting and final error bars
ters.
nimal model to fit the k-dependent transfer functions
2
+ b2
h˜1
˜2i
h˜1
˜1i
+ bG2
h˜1
˜G2i
h˜1
˜1i
+ b 3
h˜1
˜3i
h˜1
˜1i
b1
h˜1
˜S3i
h˜1
˜1i
, (88)
and G2
(k) = bG2
. (89)
me as the one-loop power spectrum. When we use the cubic bias model, we
ed from the low-k limit of 3(k).
ra of Shifted Fields from Theory and on a Grid
8) we need to calculate the power spectra h ˜Oa
˜Obi of shifted operators that
The number of parameters the same as in the 1-loop power spectrum

13. Bias at the field level
Mass and momentum conservation — noise suppressed on large scales
Deterministic part of the shot “noise”
Two different kinds of long modes on large scales

14. MPIAA Workshop, Munich, July 2019
Modeling biased tracers at the field level (Part II)
14
Results and lessons learned

15. Numerical setup
Same initial conditions
Sim
ulation
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truth<latexit sha1_base64="IB7CHTpqOeRx7pTZPfheU1Oiklg=">AAACCnicbVC7TgJBFJ3FF+JrhdJmIjExFmQXC+kk0cISE3kksCGzwwATZnY3M3eNZMMf+A22amtnbP0AW0v/xFmgEPAkNzk5596cm+NHgmtwnG8rs7a+sbmV3c7t7O7tH9iH+YYOY0VZnYYiVC2faCZ4wOrAQbBWpBiRvmBNf3SV+s17pjQPgzsYR8yTZBDwPqcEjNS1850eE0C6SUdJDCqG4aRrF52SMwVeJe6cFKuFyvVX4fWs1rV/Or2QxpIFQAXRuu06EXgJUcCpYJNcJ9YsInREBqxtaEAk014y/X2CT4zSw/1QmQkAT9W/FwmRWo+lbzYlgaFe9lLxP68dQ7/iJTyIYmABnQX1Y4EhxGkRuMcVoyDGhhCquPkV0yFRhIKpayHFl2kn7nIDq6RRLrnnpfKtKecSzZBFR+gYnSIXXaAqukE1VEcUPaAn9IxerEfrzXq3PmarGWt+U0ALsD5/AQRancY=</latexit>
PT

16. Simulations
log M[h 1
M ] ¯n [(h 1
Mpc) 3
] ¯n is comparable to
10.8 11.8 4.3 ⇥ 10 2
LSST [80, 81], Billion Object Apparatus [82]
11.8 12.8 5.7 ⇥ 10 3
SPHEREx [83, 84]
12.8 13.8 5.6 ⇥ 10 4
BOSS CMASS [85], DESI [86, 87], Euclid [88–90]
13.8 15.2 2.6 ⇥ 10 5
Cluster catalogs
Table I. Simulated halo populations at z = 0.6.
. The shifted squared density ˜2 and shifted tidal field ˜G2 are computed similarly, using 2
1(
e mass.
Ran 5 MP-Gadget1 DM-only N-body sims with 15363 DM particles,
30723 mesh for PM forces, L=500 Mpc/h,
~4000 time steps to evolve z=99 to z=0.6
4 FoF halo mass bins
1Feng et al. https://github.com/bluetides-project/MP-Gadget
[derived from P-Gadget]
mptcle = 2.9 ⇥ 109
M /h<latexit sha1_base64="5gOaYc51Lo066Ov3uoAnov8D0Bc=">AAACH3icbVDLSgMxFM3UV62vqks3wSK6kDpTBS0oFNy4ESrYB3TqkEnTNnQyGZI7Qhn6CX6E3+BW1+7EbZf+ieljYVsPBA7n3JuTHD8KuAbbHlqppeWV1bX0emZjc2t7J7u7V9UyVpRVqAykqvtEs4CHrAIcAlaPFCPCD1jN792O/NozU5rL8BH6EWsK0gl5m1MCRvKyx8JLXCVwBDRgg5tCvugCF0xjx34quqf3nitbEs66XjZn5+0x8CJxpiSHpih72R+3JWksWGhuJlo3HDuCZkIU8FFSxo01iwjtkQ5rGBoSE9pMxh8a4COjtHBbKnNCwGP170ZChNZ94ZtJQaCr572R+J/XiKF91Ux4GMXAQjoJascBBolH7eAWV4xC0DeEUMXNWzHtEkUomA5nUnwxMJ048w0skmoh75znCw8XudL1tJ00OkCH6AQ56BKV0B0qowqi6AW9oXf0Yb1an9aX9T0ZTVnTnX00A2v4CxZtokU=</latexit>

17. Model on the grid
Distribute 15363 particles on regular grid
Assign artificial particle masses
Displace by linear displacement
Interpolate to Eulerian grid using CIC weighted by particle masses
[Very similar to generating N-body initial conds./Zeldovich density]
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1(q)<latexit sha1_base64="TJaAHFoYRyspIjBVZgsQyBThdy0=">AAACKXicbVDLSsNAFJ34rPVVdekmWIS6KUkVdGfBjcsK9gFNKJPppB06j3RmUikh3+FH+A1ude1O3Yk/4rSNYFsPDBzOuZdz5wQRJUo7zoe1srq2vrGZ28pv7+zu7RcODhtKxBLhOhJUyFYAFaaE47ommuJWJDFkAcXNYHAz8ZsjLBUR/F6PI+wz2OMkJAhqI3UKrsfxAxKMQd5NvNEwTTwGdT8Ik2Ga5n+5V1Mk7bglM3DWKRSdsjOFvUzcjBRBhlqn8OV1BYoZ5hpRqFTbdSLtJ1Bqgig2IbHCEUQD2MNtQzlkWPnJ9GupfWqUrh0KaR7X9lT9u5FAptSYBWZycqta9Cbif1471uGVnxAexRpzNAsKY2prYU96srtEYqTp2BCIJDG32qgPJUTatDmXErDUdOIuNrBMGpWye16u3F0Uq9dZOzlwDE5ACbjgElTBLaiBOkDgETyDF/BqPVlv1rv1ORtdsbKdIzAH6/sHurmolw==</latexit>
O(q)<latexit sha1_base64="fNNlinCcTxszFIVZ7rBkeZk8/5Y=">AAACJXicbVDLSgMxFM3UV62vqks3g0WoC8tMFXRnwY07K9gHtEPJZDJtaB7TJFMpw3yFH+E3uNW1OxFciX9i+ljY1gOBwzn3cm6OH1GitON8WZmV1bX1jexmbmt7Z3cvv39QVyKWCNeQoEI2fagwJRzXNNEUNyOJIfMpbvj9m7HfGGKpiOAPehRhj8EuJyFBUBupkz9rc/yIBGOQB0l7OEiTNoO654fJIE1zE44gTe7SojFPO/mCU3ImsJeJOyMFMEO1k/9pBwLFDHONKFSq5TqR9hIoNUEUm4BY4QiiPuzilqEcMqy8ZPKt1D4xSmCHQprHtT1R/24kkCk1Yr6ZHN+pFr2x+J/XinV45SWER7HGHE2DwpjaWtjjjuyASIw0HRkCkSTmVhv1oIRImybnUnyWmk7cxQaWSb1ccs9L5fuLQuV61k4WHIFjUAQuuAQVcAuqoAYQeAIv4BW8Wc/Wu/VhfU5HM9Zs5xDMwfr+Ba4ppw4=</latexit>
˜O(x)<latexit sha1_base64="jp0htYvlhK/ioIFAQhH9fbs5T6c=">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</latexit>

18. Overview
1. What works
2. What doesn’t work / lessons learned

19. Overview
1. What works
2. What doesn’t work / lessons learned

20. Cubic bias with shifted operators
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model(x)<latexit sha1_base64="xXWfnBvGKftBpWdQBwua7QKJibc=">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</latexit>
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21. Heavier halos
truth(x)<latexit sha1_base64="UsUOUE0HIJAIppSt49vlRs+X3zI=">AAACOnicbVBNSyNBEO1x3TVGd4169NIYBL2EGRX0puDFYwSjQiaEnp4a06S7Z+iuUcMwv8uLv8Kbh714UMSrP8CeGPDzQcPjvaquqhdlUlj0/Ttv6tf07z8ztdn63PzffwuNxaUTm+aGQ4enMjVnEbMghYYOCpRwlhlgKpJwGg0PKv/0AowVqT7GUQY9xc61SARn6KR+4yjUcMlTpZiOi/DiqixCxXAQJcVVWdY/m8N3c1iZMUhk/SI0iqLJcVCuuw82+o2m3/LHoN9JMCFNMkG737gN45TnCjRyyaztBn6GvYIZFFyCm5NbyBgfsnPoOqqZAtsrxqeXdM0pMU1S455GOlY/dhRMWTtSkausVrdfvUr8yevmmOz2CqGzHEHzt0FJLimmtMqRxsIARzlyhHEj3K6UD5hhHF3adRdC8PXk7+RksxVstTaPtpv7e5M4amSFrJJ1EpAdsk8OSZt0CCfX5D95II/ejXfvPXnPb6VT3qRnmXyC9/IKYSewtA==</latexit>
model(x)<latexit sha1_base64="xXWfnBvGKftBpWdQBwua7QKJibc=">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</latexit>
model(x)<latexit sha1_base64="xXWfnBvGKftBpWdQBwua7QKJibc=">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</latexit>

22. Measures of success
Small and scale-independent error power spectrum
High cross-correlation between model and truth
rcc(k) =
h truth(k) ⇤
model(k)i
p
Ptruth(k)Pmodel(k)<latexit sha1_base64="cRIq0DTnUaRL8LfD6bOtGkmKbM8=">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</latexit>
For best-fit model, , so focus on herePerr = Ptruth(1 r2
cc)<latexit sha1_base64="tRkpT8H4AD9a4kRuQlWo+CYRCgo=">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</latexit>
Perr(k) ⌘ h| truth(k) model(k)|2
i<latexit sha1_base64="okwU7UK76wFFi51s3hFUJl6za4k=">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</latexit>
Perr<latexit sha1_base64="HbsFOTY+FC07lz5Qau+blG/LzoY=">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</latexit>

23. Error power spectrum
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
isson noise 1/¯n
h model
h |2
i
s (model)
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
s (model)
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
Quite flat, similar to Poisson shot noise

24. Heavier halos
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
Poisson noise 1/¯n
| truth
h
model
h |2
i
(truth)
Quadr. bias (model)
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
(truth)
Quadr. bias (model)
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
Casas-Miranda et al. (2002); Baldauf et al. (2013, 2016)
Ginzburg et al. (2017)

25. Heavier halos
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
Poisson noise 1/¯n
Perr = h| truth
h
model
h |2
i
Simulated halos (truth)
Quadr. bias (model)
Casas-Miranda et al. (2002); Baldauf et al. (2013, 2016)
Ginzburg et al. (2017)

26. Modeling transfer functions 35
igure 16. Transfer functions i(k) of the cubic bias model, h = 1
˜1 + 2
˜?
2 + G2
˜G?
2 + 3
˜?
3 , for the four mass bins. Treating
l k bins as independent and minimizing the power of the model error in each k bin gives the black lines, with uncertainty
Using the same number of parameters (5 or 6) as usual bias
expansion, can fit transfer functions s.t. Perr unchanged
i(k)
35
k) of the cubic bias model, h = 1
˜1 + 2
˜?
2 + G2
˜G?
2 + 3
˜?
3 , for the four mass bins. Treating
inimizing the power of the model error in each k bin gives the black lines, with uncertainty
he scatter between the five independent simulations). When fitting these transfer functions

27. Modeling transfer functions 35
igure 16. Transfer functions i(k) of the cubic bias model, h = 1
˜1 + 2
˜?
2 + G2
˜G?
2 + 3
˜?
3 , for the four mass bins. Treating
l k bins as independent and minimizing the power of the model error in each k bin gives the black lines, with uncertainty
Using the same number of parameters (5 or 6) as usual bias
expansion, can fit transfer functions s.t. Perr unchanged
i(k)
35
k) of the cubic bias model, h = 1
˜1 + 2
˜?
2 + G2
˜G?
2 + 3
˜?
3 , for the four mass bins. Treating
inimizing the power of the model error in each k bin gives the black lines, with uncertainty
he scatter between the five independent simulations). When fitting these transfer functions

28. Scale dependence of the error
±1% of Phh
Expanding Z
Quadratic bias
Cubic bias
Linear Std.
Eul. bias
Scale dependence important around the nonlinear scale
Potentially dangerous because can bias cosmological parameters

29. Overview
1. What works
2. What doesn’t work / lessons learned

30. Dropping nonlinear terms
Linear Std. Eul. bias
Linear bias
Poisson prediction
Quadr. bias
Error power spectrum 2-6x larger, even on large scales, and not flat.
Quadratic operators improve model even on very large scales.
Linear Std. Eul. bias
Linear bias
Cubic bias
Poisson prediction
Quadr. bias
d. Eul. bias
Linear bias
bias
Poisson prediction
bias

31. Do we really need shifted operators?
So far used shifted operators
What if we instead use Standard Eulerian bias expansion,
expanding in the Eulerian density?
h(x) = b1
˜1(x) + b2
˜2(x) + · · ·
˜1(k) =
Z
d3
q 1(q) e ik·(q+ 1(q))
˜2(k) =
Z
d3
q 2
1(q) e ik·(q+ 1(q))
Std.Eul.
h (x) = b1 (x) + b2
2
(x) + · · ·

32. Standard Eulerian bias with
Figure 20. Left panel: Model error power spectrum for Standard Eulerian bias
nonlinear dark matter NL from simulations as the input for the Standard Eul
Large displacements are treated perturbatively, leading to
decorrelation at the field level (bump in Perr)
PT<latexit sha1_base64="BNxYJZLtgsSvC69UM/3qqOUbA2c=">AAACB3icdVDLSsNAFJ3UV62PRl26GSyCq5CmrY27gi5cVugL2hImk2k7dCYJMxOhhH6A3+BW1+7ErZ/h0j9x0lawogcuHM65l3vv8WNGpbLtDyO3sbm1vZPfLeztHxwWzaPjjowSgUkbRywSPR9JwmhI2ooqRnqxIIj7jHT96XXmd++JkDQKW2oWkyFH45COKEZKS55ZHASEKeSlA8FhszX3zJJt2fWqU7uEtlVxK86Vq0m96tYcF5Yte4ESWKHpmZ+DIMIJJ6HCDEnZL9uxGqZIKIoZmRcGiSQxwlM0Jn1NQ8SJHKaLw+fwXCsBHEVCV6jgQv05kSIu5Yz7upMjNZG/vUz8y+snauQOUxrGiSIhXi4aJQyqCGYpwIAKghWbaYKwoPpWiCdIIKx0VmtbfJ5l8v04/J90HKtcsZy7aqlxs0onD07BGbgAZVAHDXALmqANMEjAI3gCz8aD8WK8Gm/L1pyxmjkBazDevwAJO5n0</latexit>

33. doesn’t change, so remains good
But field-level model error explodes b/c fields incoherent
—> Modeling the field is harder than modeling
Shifting fields
Good model of
x ! x +<latexit sha1_base64="+3MORj4q/XkrwqAejYnO59kK720=">AAACAHicbVBNS8NAEJ34WetX1IMHL4tFEISSVEGhBwtePFawH9CEstlu2qWbTdjdaEvpxb/ixYMiXv0Z3vw3btsctPXBwOO9GWbmBQlnSjvOt7W0vLK6tp7byG9ube/s2nv7dRWnktAaiXksmwFWlDNBa5ppTpuJpDgKOG0E/ZuJ33igUrFY3OthQv0IdwULGcHaSG37cOCVPcm6PY2ljB+9MhqceVXF2nbBKTpToEXiZqQAGapt+8vrxCSNqNCEY6VarpNof4SlZoTTcd5LFU0w6eMubRkqcESVP5o+MEYnRumgMJamhEZT9ffECEdKDaPAdEZY99S8NxH/81qpDq/8ERNJqqkgs0VhypGO0SQN1GGSEs2HhmAimbkVkR6WmGiTWd6E4M6/vEjqpaJ7XizdXRQq11kcOTiCYzgFFy6hArdQhRoQGMMzvMKb9WS9WO/Wx6x1ycpmDuAPrM8fPGuWKA==</latexit>
Ptruth(k)<latexit sha1_base64="FOMz+2fqzsab3fgcaMHNsJQA77o=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahXspuFfRmwYvHCvYD2mXJpmkbmmSXZLZQlv4TLx4U8eo/8ea/MW33oK0PBh7vzTAzL0oEN+B5305hY3Nre6e4W9rbPzg8co9PWiZONWVNGotYdyJimOCKNYGDYJ1EMyIjwdrR+H7utydMGx6rJ5gmLJBkqPiAUwJWCl23EWY9LTHoFEazyvgydMte1VsArxM/J2WUoxG6X71+TFPJFFBBjOn6XgJBRjRwKtis1EsNSwgdkyHrWqqIZCbIFpfP8IVV+ngQa1sK8EL9PZERacxURrZTEhiZVW8u/ud1UxjcBhlXSQpM0eWiQSowxHgeA+5zzSiIqSWEam5vxXRENKFgwyrZEPzVl9dJq1b1r6q1x+ty/S6Po4jO0DmqIB/doDp6QA3URBRN0DN6RW9O5rw4787HsrXg5DOn6A+czx8JFZNB</latexit>
Pmodel(k)<latexit sha1_base64="Wkkhl1QO/+EDzUB82BmflyI5OVc=">AAAB+XicbVDLSsNAFJ3UV62vqEs3g0Wom5JUQXcW3LisYB/QhjCZTNqh8wgzk0IJ/RM3LhRx65+482+ctllo64ELh3Pu5d57opRRbTzv2yltbG5t75R3K3v7B4dH7vFJR8tMYdLGkknVi5AmjArSNtQw0ksVQTxipBuN7+d+d0KUplI8mWlKAo6GgiYUI2Ol0HVbYT5QHHIZEzarjS9Dt+rVvQXgOvELUgUFWqH7NYglzjgRBjOkdd/3UhPkSBmKGZlVBpkmKcJjNCR9SwXiRAf54vIZvLBKDBOpbAkDF+rviRxxrac8sp0cmZFe9ebif14/M8ltkFORZoYIvFyUZAwaCecxwJgqgg2bWoKwovZWiEdIIWxsWBUbgr/68jrpNOr+Vb3xeF1t3hVxlMEZOAc14IMb0AQPoAXaAIMJeAav4M3JnRfn3flYtpacYuYU/IHz+QPOgJMb</latexit>
✏(x)<latexit sha1_base64="h9H07/Hh2aJKk976iJ2sT2SQv+I=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BItQL2W3Cnqz4MVjBfsB26Vk09k2NJssSVYsS3+GFw+KePXXePPfmLZ70NYHA4/3ZpiZFyacaeO6305hbX1jc6u4XdrZ3ds/KB8etbVMFYUWlVyqbkg0cCagZZjh0E0UkDjk0AnHtzO/8whKMykezCSBICZDwSJGibGS34NEMy5F9em8X664NXcOvEq8nFRQjma//NUbSJrGIAzlRGvfcxMTZEQZRjlMS71UQ0LomAzBt1SQGHSQzU+e4jOrDHAklS1h8Fz9PZGRWOtJHNrOmJiRXvZm4n+en5roOsiYSFIDgi4WRSnHRuLZ/3jAFFDDJ5YQqpi9FdMRUYQam1LJhuAtv7xK2vWad1Gr319WGjd5HEV0gk5RFXnoCjXQHWqiFqJIomf0it4c47w4787HorXg5DPH6A+czx/3kZEI</latexit>
Translate by a shifttruth<latexit sha1_base64="dorPueCGyAIyn+LUjl5+vXhBTBQ=">AAAB+3icbVBNS8NAEN34WetXrEcvi0XwVJIq6M2CF48V7Ac0IWw2m3bpZhN2J2IJ/StePCji1T/izX/jts1BWx8MPN6bYWZemAmuwXG+rbX1jc2t7cpOdXdv/+DQPqp1dZoryjo0Fanqh0QzwSXrAAfB+pliJAkF64Xj25nfe2RK81Q+wCRjfkKGksecEjBSYNe8iAkgQeGpBIPKYTQN7LrTcObAq8QtSR2VaAf2lxelNE+YBCqI1gPXycAviAJOBZtWvVyzjNAxGbKBoZIkTPvF/PYpPjNKhONUmZKA5+rviYIkWk+S0HQmBEZ62ZuJ/3mDHOJrv+Ayy4FJulgU5wJDimdB4IgrRkFMDCFUcXMrpiOiCAUTV9WE4C6/vEq6zYZ70WjeX9ZbN2UcFXSCTtE5ctEVaqE71EYdRNETekav6M2aWi/Wu/WxaF2zyplj9AfW5w9uWJSv</latexit>
truth<latexit sha1_base64="dorPueCGyAIyn+LUjl5+vXhBTBQ=">AAAB+3icbVBNS8NAEN34WetXrEcvi0XwVJIq6M2CF48V7Ac0IWw2m3bpZhN2J2IJ/StePCji1T/izX/jts1BWx8MPN6bYWZemAmuwXG+rbX1jc2t7cpOdXdv/+DQPqp1dZoryjo0Fanqh0QzwSXrAAfB+pliJAkF64Xj25nfe2RK81Q+wCRjfkKGksecEjBSYNe8iAkgQeGpBIPKYTQN7LrTcObAq8QtSR2VaAf2lxelNE+YBCqI1gPXycAviAJOBZtWvVyzjNAxGbKBoZIkTPvF/PYpPjNKhONUmZKA5+rviYIkWk+S0HQmBEZ62ZuJ/3mDHOJrv+Ayy4FJulgU5wJDimdB4IgrRkFMDCFUcXMrpiOiCAUTV9WE4C6/vEq6zYZ70WjeX9ZbN2UcFXSCTtE5ctEVaqE71EYdRNETekav6M2aWi/Wu/WxaF2zyplj9AfW5w9uWJSv</latexit>
<latexit sha1_base64="0kQQ4bQnE6qcjjlOoHqoSjdsqzk=">AAAB63icbVBNSwMxEJ34WetX1aOXYBE8ld0q6M2CF48V7Ae0S8mm2TY0yS5JVihL/4IXD4p49Q9589+YbfegrQ8GHu/NMDMvTAQ31vO+0dr6xubWdmmnvLu3f3BYOTpumzjVlLVoLGLdDYlhgivWstwK1k00IzIUrBNO7nK/88S04bF6tNOEBZKMFI84JTaX+k3DB5WqV/PmwKvEL0gVCjQHla/+MKapZMpSQYzp+V5ig4xoy6lgs3I/NSwhdEJGrOeoIpKZIJvfOsPnThniKNaulMVz9fdERqQxUxm6Tkns2Cx7ufif10ttdBNkXCWpZYouFkWpwDbG+eN4yDWjVkwdIVRzdyumY6IJtS6esgvBX355lbTrNf+yVn+4qjZuizhKcApncAE+XEMD7qEJLaAwhmd4hTck0Qt6Rx+L1jVUzJzAH6DPH/PXjis=</latexit>
P(k)<latexit sha1_base64="IPjeipLCYvYNMcy5WVDRIvD5NRs=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXspuK+jNghePFewHtEvJptk2NMkuSVYoS/+CFw+KePUPefPfmG33oK0PBh7vzTAzL4g508Z1v53CxubW9k5xt7S3f3B4VD4+6egoUYS2ScQj1QuwppxJ2jbMcNqLFcUi4LQbTO8yv/tElWaRfDSzmPoCjyULGcEmk1rV6eWwXHFr7gJonXg5qUCO1rD8NRhFJBFUGsKx1n3PjY2fYmUY4XReGiSaxphM8Zj2LZVYUO2ni1vn6MIqIxRGypY0aKH+nkix0HomAtspsJnoVS8T//P6iQlv/JTJODFUkuWiMOHIRCh7HI2YosTwmSWYKGZvRWSCFSbGxlOyIXirL6+TTr3mNWr1h6tK8zaPowhncA5V8OAamnAPLWgDgQk8wyu8OcJ5cd6dj2VrwclnTuEPnM8fN2uNrw==</latexit>

34. Standard Eulerian bias with
Figure 20. Left panel: Model error power spectrum for Standard Eulerian bias
nonlinear dark matter NL from simulations as the input for the Standard Eul
Squaring nonlinear field is UV sensitive, leading to large error on
large scales
NL<latexit sha1_base64="y6rSth5MBCQqOXYXSK6nidTcons=">AAACB3icdVDLSsNAFJ3UV62PRl26GSyCq5CmrY27gi5ciFSwtdCWMJlM26EzSZiZCCX0A/wGt7p2J279DJf+iZO2ghU9cOFwzr3ce48fMyqVbX8YuZXVtfWN/GZha3tnt2ju7bdllAhMWjhikej4SBJGQ9JSVDHSiQVB3Gfkzh+fZ/7dPRGSRuGtmsSkz9EwpAOKkdKSZxZ7AWEKeWlPcHh9NfXMkm3Z9apTO4W2VXErzpmrSb3q1hwXli17hhJYoOmZn70gwgknocIMSdkt27Hqp0goihmZFnqJJDHCYzQkXU1DxInsp7PDp/BYKwEcREJXqOBM/TmRIi7lhPu6kyM1kr+9TPzL6yZq4PZTGsaJIiGeLxokDKoIZinAgAqCFZtogrCg+laIR0ggrHRWS1t8nmXy/Tj8n7Qdq1yxnJtqqXGxSCcPDsEROAFlUAcNcAmaoAUwSMAjeALPxoPxYrwab/PWnLGYOQBLMN6/APlimeo=</latexit>

35. Smoothing does not rescue Std. Eul. bias41
for Standard Eulerian bias models, for the lowest halo mass bin. Using the
nput for the Standard Eulerian bias model (purple) creates a large error on
Smoothing before squaring can reduce UV junk, but also kills signal
—> Always get larger model error than shifted operators

36. Weighting halos by their mass
Halo number density
(how many halos per cell)
Halo mass density
(how much halo mass per cell)
used so far more similar to dark matter
smaller shot noise
Seljak, Hamaus & Desjacques (2009)
Hamaus, Seljak & Desjacques (2010, 2011, 2012)
Cai, Bernstein & Sheth (2011)

37. Weighting halos by their mass
Shot noise (squared
model error) 17x
lower for light halos,
2-7x lower for heavy
halos
With 60% halo mass
scatter (green), still
get factor few
How well can we do
observationally?
49
mass weighting on the mean-square model error divided by the Poisson expectation, h| obs
h (k)
obs obs truth ?

38. Mass weighting questions
How well can halo masses be measured (e.g. BOSS, DESI)?
What observable properties of galaxies can we use? What sims?
New ideas to get halo masses?
For shot noise limited applications, gain may be large
What if mass estimates are biased?
Use for BAO reconstruction? (Suffers from high shot noise)

39. Conclusions
Bias model at the field level requires IR bulk flows
Avoid squaring nonlinear density
Model error is quite scale-independent, roughly
But scale-dependence important at nonlinear scale
Nonlinear bias terms reduce model error substantially, at expense of
more coefficients
Halo mass weighting reduces noise
Maybe useful for field-level likelihood, BAO reconstruction
1/¯n<latexit sha1_base64="iG9SZ2d2+sWzkpKj3qRksqRBXfc=">AAAB/nicbVC7SgNBFL3rM8ZX1EawcDAIVnE3FtoZsLGMYB6QLGF2MpsMmZldZmaFsAT8BLFVLO3E1sYPsbTyN5w8CpN44MLhnHu5954g5kwb1/1yFhaXlldWM2vZ9Y3Nre3czm5VR4kitEIiHql6gDXlTNKKYYbTeqwoFgGntaB3NfRrd1RpFslb04+pL3BHspARbKxU906bAVZItnJ5t+COgOaJNyH50uHL58P+T73cyn032xFJBJWGcKx1w3Nj46dYGUY4HWSbiaYxJj3coQ1LJRZU++no3gE6tkobhZGyJQ0aqX8nUiy07ovAdgpsunrWG4r/eY3EhBd+ymScGCrJeFGYcGQiNHwetZmixPC+JZgoZm9FpIsVJsZGNLUlEAObiTebwDypFgveWaF4Y8O5hDEycABHcAIenEMJrqEMFSDA4RGe4Nm5d16dN+d93LrgTGb2YArOxy/d5pl9</latexit>

40. Discussion
Bias parameters that minimize Perr differ from the usual ones
measured from N-point functions or responses
Especially b2 (constant at low k but still part of model, not noise;
usually only get from k-dependence of P22 at high k)
Which bias parameters give best cosmology constraints?
Is field-level likelihood better than P(k) analysis?
Extensions:
- Add stochastic k2 term to the noise?
- Galaxies instead of halos
- RSD
Schmidt, Elsner et al. 2018
Elsner, Schmidt et al. 2019