Baryon impact on the halo mass funcVon

Baryon impact on the halo mass func=on Fi?ng formulae and implica=ons for cluster cosmology Sebas=an Bocquet LMU Munich Alex Saro, Klaus Dolag, Joe Mohr arXiv: 1502.07357 WWW. see e.g., Hirschmann et al. 2014, McDonald et al. 2014, Saro et al. 2014 .ORG Dolag in prep. 2 dN/dM [h3 Mpc
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a priori clear that one
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3.1 The halo mass function
approach
r
function that is valid
for
M
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3.1 The halo mass function
(2013)
provid
tion
correc
ETHOD
andmass
cosmology
dependent
behm
pends
on
The
comoving
number
density
of
haloes
of
M
is
3.1
The
halo
mass
function
(2013) pro
The comoving number density of haloes of mass M is
very
different
redshift
evolution
For on
now
oretical background on the halo mass function
1
pends
1
dn
⇢mass
¯m d ln
comoving
haloes
M is Tinker et al. (2008)
provid
dn density
⇢of
¯mpresent
d=lnf ( of
proach:
Assu
ttingThe
form
we willnumber
adopt.
We
also
the
)
,
(1)
For
no
= f( )
,
(1)
dM
M
dM
1
different
,
and
one
uses
dMdn fits when
M⇢¯m
dM
theproach:
mass func
mean
orm the multi-dimensional
analysing
d ln
As
=
f
(
)
,
(1)
vertzfrom
critical
to the
mean
dens
Baryons
and
the
halo
mass
f
eswith
extracted
from
our
simulations.
the mean
matter
density
⇢¯m (at
redshift
= redshift
0),
and
with
the
mean
matter
density
⇢
¯
=
0),
and
mz(at
dM
M
dM
mass
fu
Comoving number dZensity of hZalos of mass M
d
epends o
n: approach relies on the implici
dM
with the mean
matter
density
⇢
¯
(at
redshift
z
=
0),
and
1
m
2
2
2
1
2
2
2
•  variance of ⌘the m
aWer density fi(k,
eld tion(kR)k
correctly
the
beha
P⌘(k,
z)Ŵ In
(M,
(kR)k
(2)dk,captures
this z)dk,
work
we
allow
departures
from
P
Ŵ
(M,
z)
(2)
Z
2
imulations used in
thisz)
work.
The
num2⇡ 1
2
sDMonly
function
2⇡
2
2
a dependence
correction to
parametrizing
a2(2013)
possibleprovide
redshift
as
runs, and M
200m
P
(k,
z)
Ŵ
(M,
z). ⌘
(kR)k
dk,
(2)
2
which is the variance of the2⇡
matter
density
field P (k, z)
smoothed
1 matter
+ z:
pends
onP (k,
(z).
mean
which
is
the
variance
of
the
density
field
z)
smoothed
ber
density
of
haloes
of
mass
M
is
M
N
(z
=
0)
lewith • 
Halo,
min
This mass
fun
mean m
aWer d
ensity the is
Fourier
transform
Ŵmatter
of thedensity
real-space
top-hat
window
A
For
now,
we
focus
on
z
50
which
the
variance
of
the
field
P
(k,
z)
smoothed
A(z)top-hat
= A0 (1window
+
z)
(M with
)
the Fourier
transform
1/3 Ŵ of the real-space
1
function
function
of radius
= (3M/4⇡
⇢¯mof
) the
. The
function
fproach:
( ) iswindow
comdnwith the
⇢¯m dR
lntransform
This
mass
Assuming mass
that
the
mas
1/3 top-hat
Fourier
Ŵ
real-space
a
z
=
f
(
)
,
(1)
function
of
radius
R
=
(3M/4⇡
⇢
¯
)
.
The
function
f
(
)
is
com7
11
a(z) = a0 (1 + z) The cru
6.2parametrized
⇥ 10 M
835
monly
as = (3M/4⇡ ⇢¯ )1/3 . Themfunction
mass
dM
dM
the
mass
function
in M
can
500cfunct
function
of
radius
R
f
(
)
is
comm
8
13
•  ⇥Fi?ng fparametrized
unc=on b
monly
asparameters) z
1.1
10
1049 ⇣(8 ⌘fit M500c /M
200c
b(z)
=
b
⇣
⌘
The
0 (1 + z)
a
monly
parametrized
as
10
14
r density
¯10
z = 0), and
c
m (at redshift
dn
dn otherczassumi
2.2 ⇥⇢
8824
⇣
⇣
⌘
⌘
f
(
)
=
A
+
1
exp
(3)
Z
M500c /M2
a⇣ 2
b⇣ ⌘
c(z)
= c0 (1 +=
z)
c
⌘
a
dM500cWhite
dM
1997)
200m
1
( )=
(3)
2f
2 A + 1 exp + 1 c exp
other
assu
2
f
(
)
=
A
(3)
P (k, z)Ŵ (kR)k
) ⌘ •  2 Simultaneously dk,all pbarameters (2) subscript
fit for ata from 72008).
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atThere
reds
⇢¯
with2⇡
four parameters A, a, b, c bthat needwhere
to be the
calibrated
(e.g.
Jenk- the values
White
199
=
f ( ) is
, az , bz ,aczand
arebadditional
fit
parameters
where
Aazpproach redshi[s i
n B
ayesian l
ikelihood ⌦
(which
m
ins
et
al.
2001).
Here,
A
sets
the
overall
normalization,
are
M5
with
four
parameters
a,smoothed
b, c to
that
need
to be(e.g.
calibrated
(e.g. 2008).
Jenk- The
inthe
thisfour
analysis
is highlighted
incA,
parameters
A, a,Pb,(k,
that
need
be
calibrated
Jenkesed
ofwith
matter
density
field
z)
authorspower
assume
theand
cutoff
scale c to ing
be constant
un
prescripti
(Cash s
ta=s=cs, e
mcee c
ode) theins
slope
and
normalization
of
the
low-mass
law,
c
sets
⌦
(which
ntified
through
a
parallel
FoF
algoins2001).
etthe
al.Here,
2001).
sets
the
normalization,
bmarehave
etŴ
al.
A Here,
setstop-hat
theAoverall
normalization,
and bTinker
are aetand
Thisamass
function
should
nsform
of
real-space
window
tion
of overall
self-similarity
(e.g.
al.
2008;
Wat
0
<
z prescri
<
thethe
scale
of
a
high-mass
exponential
The
function
f
(
)
has
SnowCluster 2links
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ing
0.16.
The
FoF
over
dark
mat1/3
slope
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of the
power mass
law, power
and
c sets
the⇢¯and
slope
the low-mass
law,in and
c sets
.
function
M200m
= (3M/4⇡
. and
The normalization
function
f ( low-mass
) isofcomm)
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that lensing
is valid
for M
might miss some redshift
200m , as
mass function
Mof
cangravitational
be expressed
as shear
500c
dispersionsinand
weak
profiles.
It one
is not
andsimply
cosmology
behavior.
Remember, for example, the
a priori clear that one can
use thedependent
same form
of the fitting
dnforvery
dMdifferent
dn that is valid
200m redshift evolutions of ⇢
¯m (z) and ⇢crit (z).
function
M
200m , as one might miss some redshift
=
f mass M is
Tinker
al. (2008) provide
the mass
dM
dM
500c
200m dM
500c etRemember,
and
cosmology
dependent
behavior.
for example,
the function for a range of
different of mean
, and
one
uses mean (z) = crit /⌦m (z) to con1
very different redshift evolutions
⇢¯m1(z)
and
⇢500c
crit (z).
⇢
¯
M
d
ln
m
,
(1)
from the
critical
mean density
as a(5)
function
of redshift. Their
f ((2008)
) vert
⇥tofunction
. a range
Tinker =
et al.
provide
mass
for
of
M
Mapproach
dM
M
• 
HMF a
pproximately u
niversal f
or F
oF (
b ≈
0
.2) or Δfunc- ≈ 200 500c
500c on the200m
relies
implicit
different mean , and one uses mean
(z) = crit
/⌦m (z)assumption
to con- that the fitting mean
ft z = 0), and
correctly
for
every mean . Watson et al.
from
critical
density
acaptures
functionthe
ofbehavior
redshift.as
Their
s massvert
should
the
sameasuniversal
properties
the
• function
Tinker et tohave
amean
l. tion
2
008: (2013) provide
a correction
their func178m mass function that deapproach
relies
on
the implicit
assumption
that thetofitting
2
.
s2 (kR)k
function
in
M
200m
or f(σ,z) pends
for don
ifferent values of Δmean (et200-­‐3200) dk,–  Fit f(2)
mean
tion correctly captures the
behavior
for (z).
every mean . Watson
al.
The crucial, evolving part is
now
captured
in 500c
the, and
factor
we
For
now,
we
focus
on
Interpolate to Δ
(2013)– provide
a correction
to500
theircrit 178m mass function that de-choose the following ap/MPpends
.onThese
masses proach:
can beAssuming
converted
onefunction
to thedn/dM200m is universal,
00c
200m
field
(k, z)
smoothed
that from
the mass
mean (z). assump=on that f(σ,z) is universal for every Δ
– 
Implicit mean l-space
top-hat
window
the mass
function
M500c can Frenk
be expressed
r assuming
cluster
density
profile
(e.g.in Navarro,
& as
Foranow,
we focus
on
500c , and we choose the following aphe
( Assuming
) aisa
comtefunction
1997)
mass-concentration
relation
(e.g.
et al.
•  fand
Our pproach: proach:
that the mass function
isdM
universal,
200m
dnDuffy
dn dn/dM
200m
=as
the mass
in M500c
can
be expressed
8). Therefore,
the conversion
depends
on mass,
redshift,
and
–  function
Propagate universal proper=es Δ200
dM
dMin dMmean
500c
200m
500c to Δ500crit ⇣ c is
⌘ involved in the overdensity conversion). The follow- 1
(which
⇢¯m d ln
M500c
dn
p
(3) = dn dM200m
=
f
(
)
⇥
.
(5)
2
prescription is adM
good
fit dM
at the
few
percent level in
the dM
range
M
M
dM
500c
500c
200m
500c
200m
500c
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500c
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uffy ass-­‐concentra=on rela=on as the
be calibrated– 
(e.g.
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This
mass
should
have
universal properties
f( )
⇥
. the same(5)
ormalization, a and M
b are
500c dMin
500c
. 200m
massMfunction
M200mM
500c
⌘ ↵ have
+The
ln same
M500c
.evolving
(6)
ss power
law,
andfunction
c sets should
part is as
now
This
mass
thecrucial,
universal
properties
the captured in the factor
M
200m
. The function
f ( ) has
mass function
in M200m .M500c /M200m . These masses can be converted from one to the
are evolving
func=ons of isΩnow
al (Jenkins The
et– al.α,β 2001),
otherpart
assuming
a captured
cluster density
m, z
crucial,
in theprofile
factor (e.g. Navarro, Frenk &
on redshift
White
1997)
a mass-concentration
relation (e.g. Duffy et al.
– 
Characteris=c change oconverted
f few percent M500cand
/Mcosmolcan beand
from one to the
200m . These masses
2008).
Therefore,
the conversion
depends
other assuming a cluster
density
profile (e.g.
Navarro, Frenk
& on mass, redshift, and
fromWhite
universality
by a mass-concentration
⌦m (which is involved
the overdensity
1997) and
relationin(e.g.
Duffy et al.conversion). The followence as2008).
a power
SnowCluster 2law
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Munich physics 5 ing prescription
isBaocquet the few
percent level in the range
Therefore,
the conversion
depends
ongood
mass,
redshift,
and
Universality 13
16
Mass definition: M200, mean
2
Mass definition: M500, crit
z=2
Hydro
DMonly
Watson et al.
1
0.5
2
z = 1.2
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SnowCluster 2015 1012
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1013
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Sebas=an Bocquet – LMU Munich physics M500, crit /M
M200, mean /M
1015
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6 Cosmological impact 1015
Hydro
DMonly
Tinker+08
input
⌦m
SPT-SZ
Planck
eROSITA
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Simulated Planck-­‐like survey 1014
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1.2
1.4
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•  High-­‐mass sample •  Baryonic impact is negligible •  Systema=c difference with Tinker et al. (2008) •  Our mass func=on resolves a large por=on of the tension between clusters and CMB anisotropies SnowCluster 2015 8 (⌦m /0.27)
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0.30 0.32 0.34 0.36
0.750 0.775 0.800 0.825 0.850
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Sebas=an Bocquet – LMU Munich physics 8
0.84
0.86
0.88
8 (⌦m /0.27)
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7 Cosmological impact Planck Collaboration: Cosmology from SZ cluster counts
⌦m
Simulated Planck-­‐like survey Hydro
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Tinker+08
input
0.850
8
0.825
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•  High-­‐mass sample Fig. 7: Comparison of constraints from the CMB to those from
•  Baryonic mpact negligible the cluster
counts in ithe
(⌦m , i8s )-plane.
The green, blue and
violet contours give the cluster constraints (two-dimensional
•  Systema=c ifference with and CMB lenslikelihood)
at 1 and 2 d
for
the WtG, CCCP,
ing massTinker calibrations,
et arespectively,
l. (2008) as listed in Table 2. These
constraints are obtained from the MMF3 catalogue with the
•  Our mdata
ass setfunc=on resolves from
a large SZ+BAO+BBN
and ↵ free. Constraints
the Planck
TT, TE, EE+lowP
likelihood
(hereafter,
Planck primary
por=on CMB
of the tension between CMB) are shown as the dashed contours enclosing 1 and 2 conclusters and CMB anisotropies fidence regions
(Planck
Collaboration
XIII 2015), while the grey
0.3
Planck 2015 XXIV 0.88
0.86
Fig. 8:0.84Comparison of cluster and primary CMB constraints in
the base ⇤CDM model expressed in terms of the mass bias,
0.82
1 b. The solid black curve shows the distribution of values re0.30 0.32 0.34 0.36
0.750 0.775 0.800 0.825 0.850
0.84 0.86 0.88
quired to reconcile
the counts
and primary0.82CMB
in ⇤CDM; it
0.3
⌦
m
8
is found as the posterior on the 1 b from8 (⌦
am
joint
analysis
of
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the Planck cluster counts and primary CMB when leaving the
mass bias free. The coloured dashed curves show the three prior
distributions on the mass bias listed in Tab. 2.
shaded region also include BAO. The red contours give results sion with the primary CMB, and then consider one-parameter
from a joint
analysis of the cluster counts, primary
CMB and extensions
the base ⇤CDM model, varying the curvature,
the
SnowCluster 2015 Sebas=an Bocquet – LMU Munich pto
hysics 8 the Planck lensing power spectrum (Planck Collaboration XV Thomson optical depth to reionization, the dark energy equation-
Cosmological impact 1015
Hydro
DMonly
Tinker+08
input
⌦m
SPT-SZ
Planck
eROSITA
0.81
8
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Simulated eROSITA-­‐like survey 1014
0.80
0.79
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•  Low-­‐mass sample •  Baryonic impact is visible •  Neglec=ng baryonic effects leads to underes=mate ΔΩm = -­‐ 0.01 •  This is the expected level of uncertain=es from eROSITA (Pillepich et al. 2012) SnowCluster 2015 0.78
1.4
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Sebas=an Bocquet – LMU Munich physics 0.79
0.80
8
0.81 0.765 0.780 0.795 0.810
8 (⌦m /0.27)
0.3
9 Summary •  Magne*cum hydro sims: up to (896 Mpc/h)3 •  Baryonic effects on the mass func=on will be important for surveys like eROSITA: expect ΔΩm = -­‐ 0.01 •  There are systema=c differences between different mass func=on fits that shi[ cosmological results •  The corresponding level of systema=c uncertainty is roughly comparable to current constraints •  Using our mass func=on instead of Tinker et al. (2008) would resolve a large por=on of the difference between Planck clusters and CMB (for our simplified analysis) SnowCluster 2015 Sebas=an Bocquet – LMU Munich physics 10