Cosmological Tests of General Relativity
Rachel Bean (Cornell University)
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 The challenge: to connect theory to observation
•  While GR is a remarkably
beautiful, and comparatively
minimal metric theory, it is not
the only one. We should test it!
•  Rich and complex theory
space of cosmological
relevant modifications to GR
•  Motivated by cosmic
acceleration, but implications
from horizon to solar system
scales
Theory
Referenc
Scalar-Tensor theory
[21, 22]
(incl. Brans-Dicke)
f (R) gravity
[23, 24]
f (G) theories
[25–27]
Covariant Galileons
[28–30]
Horndeski Theories
The Fab Four
[31–34]
K-inflation and K-essence
[35, 36]
Generalized G-inflation
[37, 38]
Kinetic Gravity Braiding
[39, 40]
Quintessence (incl.
[41–44]
universally coupled models)
Effective dark fluid
[45]
Einstein-Aether theory
[46–49]
Lorentz-Violating theories
Hor̆ava-Lifschitz theory
[50, 51]
DGP (4D effective theory) [52, 53]
> 2 new degrees of freedom
EBI gravity
[54–58]
TeVeS
[59–61]
Category
Aim: look for and confidently
extract out signatures from
Baker et al 2011
TABLE
I:
A
non-exhaustive
list
of
theories
that
are
suitable
for
PPF
parameterization.
We will
cosmological observations?
2
µν
µνρσ
in the present paper. G = R − 4Rµν R
derlying
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 + Rµνρσ R
is the Gauss-Bonnet term.
our formalism are stated in Table II. PPN and
PPF are highly complementary in their coverage of different accessible gravitational regimes. PPN is restricted
formalism.
ii) A reader with a particu
ample theories listed in
m0 ⌦ 2k ⌘˜ k0#(h + 6˜
⌘ ) = (¯
⇢m + Pm )kv + m0 ⌦k ⇡˙ + [⇢Q + PQ ] k ⇡
(D.14)
2
ḧ
2k
2k
ynchronous) 2
0
mSpace-space
+ 2 (Synchronous)
⌘˜
⌘˜ = P + ṖQ ⇡ + (⇢Q + PQ ) ⇡˙
(D.15)
0⌦
h
i ḣHtrace
2
3
3a
a
2
2˙
2
2
2
"
#
˙ ⇡˙ + [⇢Q" + PQ ] k ⇡
˜ k0 (ḣ + 6⌘˜˙ ) = (¯
⇢ + P̄m )kv
+ m0 ⌦k
(D.14)
0 ⌦ 2k ⌘
✓
◆#
ḧ m
2k 2
2k0
2
2
¨
⌦
2
k
ḣ
k
m0 ⌦
ḣH + 2 ⌘˜
⌘
˜
=
P
+
Ṗ
⇡
+
(⇢
+
P
)
⇡
˙
(D.15)
0
2
2
Q
Q
Q
¨ + 3H ⇡˙ +
⇡ + + ⇡ 3H + 2
3
3a
a2 + m0 ⌦˙ ˙ ⇡˙ + ⇡
ace (Synchronous)
2
3
a
3
a
⌦ "
#
✓
◆#
2
¨
2k 2 Space-space
2k0
⌦
2
k
ḣ
k
0
2
2
˙
hH + 2 ⌘˜
⌘˜ = Ptraceless
+ ṖQ ⇡ +(Synchronous)
(⇢Q + PQ ) ⇡˙
⇡˙ + ⇡
¨ + 3H(D.15)
⇡˙ +
⇡ + + ⇡ 3H + 2
+ m0 ⌦
˙
 2
 2
3a
a2
3⇣a2
a
⌦
⌘3
2
m0 ⌦ 2k "
k
1
2
#
˙
˙
¨
˙
⌘˜ ¨3H(ḣ + 6⌘˜) (ḧ + 6⌘˜2) = P̄m ⇧ +✓m0 ⌦ 2 ⇡◆
+
ḣ + 6⌘˜
(D.16)
Space-space
(Synchronous)
⌦
2k
ḣ
2
a22traceless
2
2 a k0
˙
⇡˙ + ⇡
¨ + 3H ⇡˙ +
⇡ + + ⇡ 3H + 2 
+ m0⌦
2
˙
⌘
2⌦
2
3 agauge.
3The required gauge
a transformations
m20 ⌦ 2kthese
1⇣
We can translate
into Newtonian
are detailed
2˙ k
˙
¨
˙
⌘˜ 3H(ḣ + 6⌘˜) (ḧ + 6⌘˜) = P̄m ⇧ + m0 ⌦ 2 ⇡ +
ḣ + 6⌘˜
(D.16)
in Appendix B.2
a2
a
2
aceless (Synchronous)
 2
Time-time
(Newtonian)
⌘ required gauge transformations are detailed
We can
translate
these into
Newtonian
gauge. The
k2
1⇣
2˙ k

˙
¨
˙
⌘
˜
3H(
ḣ
+
6
⌘
˜
)
(
ḧ
+
6
⌘
˜
)
=
P̄
⇧
+
m
⌦
⇡
+
ḣ
+
6
⌘
˜
(D.16)
2
m
2k
6k0 N 0
in Appendix
B.
a2
˙aN2 + H 2 N )
m20 ⌦(t)
+
6H(
a2
Time-time (Newtonian)
late these into Newtonian
gauge.
gauge
transformations are detailed
N The required
N
=
⇢N ⇢˙ Q⇡2k
⇡˙ N
)
2 2c(t)(
6k
0 N ◆

✓
m20 ⌦(t)
6H( ˙ N + H N )
⇣
⌘
2
k+
k2
0
2
N
2
N
a
˙
ewtonian)
+ m0 ⌦ 3⇡
H
Ḣ + 2 + 3H ⇡˙ N
+ 2 ⇡N 3 ˙ N + H N
(D.17)
a N
a
N
N
N
 2
=
⇢
⇢
˙
⇡
2c(t)(
⇡
˙
)
Q
2k
6k0 N
◆
+ 6H( ˙ N(Newtonian)
+ H N ) ✓
Time-space
⇣
⌘ 2
2
k0
k2 N
a
2˙
N i 2
N
N
h
˙N + H N
+
m
⌦
3⇡
H
Ḣ
+
+
3H
⇡
˙
+
⇡
3
(D.17)
0
N
˙ 2 ⇡˙ N
m⇡˙20N⌦(t)2kN2 ) ˙ N + H N = (¯
⇢m + P̄am2)kv N + [⇢Q + PQ ] k 2 ⇡ N a+2 m20 ⌦k
(D.18)
⇢˙ Q ⇡ N 2c(t)(

✓
◆
⇣
⌘
Time-space
(Newtonian)
k0 trace
k2
(Newtonian)
N
2Space-space
N
3⇡
H
Ḣ + 2 + 3H h⇡˙ N
+i 2 ⇡ N 3 ˙ N + H N
(D.17)
N 2
2 N
N
a 2 ⌦(t)2k⇣2 ˙ N + H ⌘N a
˙ 2 ⇡˙ N
m
=
(¯
⇢
+
P̄
)kv
+
[⇢
+
P
]
k
⇡ + m20 ⌦k
(D.18)
2k
m
m
Q
Q
m20 ⌦ 02 ¨ + 2 3H 2 + 2Ḣ
+ 2H( ˙ + 3 ˙ ) + 2 (
)
Newtonian)
3a
trace (Newtonian)
h
i Space-space
N
N
N
N
P⇢ +
ṖQ
N (⇢Q + PQ ) (⇡
Ng
m⇡)kv+
˙ N + H N == (¯
˙ 2 ⇡˙ N
⇣ [⇢Q + PQ˙] ⌘
+
k 2 ⇡ N +) m20 ⌦k
(D.18)
m + P̄
#
2k 2
✓
◆
2 " ¨
2
˙
˙
2
¨2 + 2 3H + 2Ḣ
m2 0 ⌦ ⌦
+
2H(
+
3
)
+
(
)
k
2
k
0
2
N
˙ N 2 ˙ N + 3H ⇡˙ N 3a5H N + ⇡ N 3H 2 +
˙
+ m0 ⌦
(⇡˙ N
)+⇡
¨N
+
⇡N
ace (Newtonian)
˙
a2
3 a2matter
⌦
N
N
N
N
⇣
⌘ = P + ṖQ ⇡ +2k
(⇢2Q + PQ ) (⇡˙
)
2
#
2 3H + 2Ḣ
+ 2H( ˙ +"3 ˙ ) + 2 (
)
✓
◆ (D.19)
2
¨
k
2
k
0
N 3a N
N
N
N
N
N
N
2
N
2˙ ⌦
˙
Space-space
+ m0Ntraceless
⌦
(⇡˙ (Newtonian)
)2
+⇡
¨
2 ˙ + 3H ⇡˙
5H 2+ ⇡
3H + 2 2+
⇡
N
˙
a
3 a2
+ (⇢Q + PQ ) (⇡˙ N
) ⌦
2
2
matter
k
2˙ k
N#
✓N = P̄ ⇧ ◆
m20 ⌦(t) 2 N
(D.20) (D.19)
mk + m02⌦k 22 ⇡
0
N
a ⇡N
˙ N 2 ˙ N + 3H ⇡˙ N 5Ha N + ⇡ N 3H 2 +
(⇡˙ N
)+⇡
¨N
+
Space-space
traceless (Newtonian)
a2 gauge.
3 a2
The matter terms also transform in going to Newtonian
2
(D.19)
k 2 N etNal 2012
2
2 Bloomfield
2˙ k
a
N
m
⌦(t)
=
P̄
⇧
+
m
⌦
⇡N
0
⇢0 = a
⇢2+ ⇢¯˙ m (ḣ + 6⌘˜˙ ) 2 m
(D.21) (D.20)
2
a
aceless (Newtonian)
2k
particles, in terms of the Newtonian
gauge
+ developed
)/2, differs
from that
thaton
deter
Bean
and potentials
her group (have
software
based
the
of non-relativistic particles, . Creating
/ 2000;
6= 1 during
anBridle
accelerative
is extraordi
(Lewis et al.
Lewis &
2002) era
to perform
like
fluid models of dark energy (Hu tometric
1998). Measuring
it, therefore,
could(Bean
be a smoking
gun of
and spectroscopic
surveys
& Tangmatith
Frameworks
towetie
observations
toThis
theory
GR. In Zhang
et al. (2007b)
proposed
way to
constrain
/ , by contrasting
the mo
Mueller
&aBean
2013).
includes
peculiar velocity,
and the lensing distortions of light
distant objects that
and MS-DESI
willd
andfrom
cross-correlations
withLSST
the CMB.
The code data
models
"
•  Modify homogeneous and perturbed Einstein’s equations
of phenomenological
including
Bean and her group have developed
software based onparameterizations,
the publicly available
CAMBthe
andeq
H(z),
the logarithmic
growth factor,
fg (z),
and
its expon
Lewis et al. 2000; Lewis & Bridle
2002)
to perform likelihood
analyses
and
forecasting
to modifications
of the perturbed
Einstein
GmaK
ometric and spectroscopic surveys
(Bean & Tangmatitham
2010;
Laszloequations,
et al. 2011;
Mueller & Bean 2013). This includes peculiar velocity, weak lensing and2galaxy cluste
k
= 4⇡Gmatter a ⇢ , k 2 (
and cross-correlations with the CMB. The code models dark energy and modified gravit
of phenomenological parameterizations,
including
the equation
w(z), the
Hubba
where ⇢ is
background
density, kofisstate
comoving
spatial,
H(z), the logarithmic growth factor,
f (z),
and its
exponent, It(z),
and ageneral
parameterizatio
invariant,
matter
perturbation.
includes
paramet
o modifications of the perturbed(Hirata
Einstein
G Laszlo
(z, et
k) al.
and2011),
Glighta(z,simple,
k), Ga
&equations,
Seljak 2004;
nonlinear model based on the ⇤CDM-modeled 2Halofit alg
k
= 4⇡G
a ⇢ , k ( + ) = 8⇡Glight a ⇢ ,
Proposed work: This existing software will be modulari
where ⇢ is background density, DESC
k is comoving
spatial,analysis
a, the scale
factorThree
and specific
is the rp
and MS-DESI
pipelines.
nvariant, matter
perturbation.
It
general
parameterizations
forwill
galaxy
bias that
and int
aincludes
The
going
to Newtonian
gauge.
sections
C.1-C.3.
The
improvements
ensure
the
k matter terms also transform
kP in
˙)
P
=
+
Ṗ
(
ḣ
+
6
⌘
˜
(D.22)
˙
m ⌦(t)
= P̄ ⇧ + m ⌦ ⇡
(D.20)
2k
See Alessandra’s Talk
a
a
a
⇢ = ⇢+
⇢¯˙ al.
(ḣ + 62011),
⌘˜˙ )
(D.21)
a
Hirata
&
Seljak
2004;
Laszlo
et
a
simple,
Gaussian
model
for photometricsys
re
level
modeling
and survey-specific
⇢Newtonian
+
P̄ )kv gauge.
=
(¯
⇢ +2P̄015 )kv + P̄ of
+2k
⇢¯ both
(ḣ +theoretical
6⌘˜˙ )
(D.23)
erms alsoRachel transform
goingGravity, to(¯
Bean, Tines-ng Vancouver, January 2
a
= ⇤CDM-modeled
P + Ṗ
(ḣ + 6⌘˜˙ )
(D.22)
aon isP
The matter
anisotropic
shear
stress
invariant.
nonlinear
model
based
the
(Smith
Zaldarriaga
20
˙
˙
science
requirements.
When
Stage
III &
data
is made pu
2k
⇢ = ⇢ + ⇢¯ (ḣ + 6⌘˜)
(D.21) Halofit algorithm
2k
2
2
0
N
N
N
m
2
m
N
2
N
N
m
2
m
2
0
2
2
2
N m
2
2
m
m
m
N
2
m
35
m
2
m
2
2
a2
and
ic and
spectroscopic
spectroscopic
surveys
surveys
(Bean
(Bean
& Tangmatitham
& Tangmatitham
2010;
2010;
Laszlo
Laszlo
et al.
et al.
2011;
2011;
Kirk
Kirk
et ae
&
er Bean
& Bean
2013).
2013).
This
This
includes
includes
peculiar
peculiar
velocity,
velocity,
weak
weak
lensing
lensing
andand
galaxy
galaxy
clustering
clustering
corrc
oss-correlations
-correlations with
with
thethe
CMB.
CMB.
TheThe
code
code
models
models
dark
dark
energy
energy
andand
modified
modified
gravity
gravity
using
usina
Phenomenological
modeling
of ofgravity
menological
nomenological
parameterizations,
parameterizations,
including
including
thethe
equation
equation
state
of state
w(z),
w(z),
thethe
Hubble
Hubble
expans
exp
the
logarithmic
logarithmic
growth
growth
factor,
factor,
fg (z),
fg (z),
andand
its its
exponent,
exponent,(z),(z),
andand
a parameterization
a parameterization
directly
dire
•  Modify perturbed Einstein’s equations
difications
cations of the
of the
perturbed
perturbed
Einstein
Einstein
equations,
equations,
Gmatter
Gmatter
(z, (z,
k) and
k) and
Glight
Glight
(z, (z,
k), k),
k 2 k 2= =4⇡G
4⇡G
a2 ⇢a2 ⇢, , k 2 (k 2 (+ +) =) =8⇡G
8⇡G
a2 ⇢a2 ⇢, ,
matter
matter
lightlight
s⇢ background
is background
density,
density,
k isk comoving
is comoving
spatial,
spatial,
a, the
a, the
scale
scale
factor
factor
andand is the
is the
rest-frame
rest-fra
•  A modification to Poisson’s equation, Gmatter
nt,
matter
matter
perturbation.
perturbation.
It includes
It includes
general
general
parameterizations
parameterizations
for for
galaxy
galaxy
biasbias
andand
intrinsic
intrinsic
alig
–  Relevant at sub-horizon scales
a Seljak
& Seljak
2004;
2004;
Laszlo
et
et
al.
2011),
2011),
a simple,
a simple,
Gaussian
Gaussian
model
model
for for
photometric
photometric
redshift
redshift
er
– Laszlo
Gmatter
≠1:al.
can
be
mimicked
by additional,
dark
sector
clustering
ear
model
model
based
based
on on
thethe
⇤CDM-modeled
⇤CDM-modeled
Halofit
Halofit
algorithm
algorithm
(Smith
(Smith
& Zaldarriaga
& Zaldarriaga
2011).
2011).
•  An inequality between Newton’s potentials, Glight/Gmatter
sed
work:
work:
This
This
existing
software
software
will
will
beatbe
modularized
modularized
documented
documented
to integrate
to integrate
intointo
th
–  existing
Remains
relevant
even
horizon
scalesandand
dand
MS-DESI
MS-DESI
analysis
analysis
pipelines.
pipelines.
Three
specific
specific
projects
projects
to enhance
to enhance
thethe
software
software
areare
desc
d
–  G
≠1: notThree
easily
mimicked.
light/G
matter
•  potential smoking
gun
for
modified
gravity?
C.1-C.3.
ns C.1-C.3.
TheThe
improvements
improvements
willwill
ensure
ensure
that
that
thethe
analysis
analysis
pipelines
pipelines
areare
ableable
to meet
to meet
theth
•  Significant stresses exceptionally hard to create in non-relativistic
oth
f both
theoretical
theoretical
modeling
modeling
andand
survey-specific
survey-specific
systematic
systematic
error
error
characterization
characterization
necessary
necessart
fluids e.g. DM and dark energy.
equirements.
e requirements.When
When
Stage
Stage
III III
datadata
is made
is made
public,
public,
expected
expected
on on
thisthis
proposal’s
proposal’s
timefra
time
yze
nalyze
the
the
datadata
using
using
thisthis
software
software
pipeline,
pipeline,
andand
integrate
integrate
improvements
improvements
in the
in the
intrinsic
intrinsic
ali
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 metric
ric error
error
andand
nonlinear
nonlinear
modeling
modeling
intointo
thethe
code.
code.
papers (e.g. Daniel and Linder ones, or others?) will
[CHECK THI
Alikely
minimal
approach
to
modifying
the
growth
history
is
discuss this, perhaps Amendola et al also].
to parameterize
a deviation
from
itscombined
impact on
On subhorizon
scales, (8)
andGR
(10)by
are
to the
give
linear
CDM growth
rate directly.
In the
epochs in growth
which
a second
order equation
for CDM
perturbation
Alternative
descriptor:
the
rate,
fg & γ
2growth
˙ and
the [CHECK
modification
isfor
considered,
the
fiducial
form
for the
THIS
XH
±k
]
history is
GM therefore
to Einstein’s
matter on su
scales time de
G
therefore
parameterizes
the
e↵ect
of
a
modification
M
S
rbations,
and GM , a↵ec
d
ln
c the clustering properties of
to
Einstein’s
equations
on
fg ⌘
= ⌦m (a) ,
ly fit by
For GR(15)
γ~0.55 on Equation (2
d lnscales.
a
matter on subhorizon
Note on super
horizon
subhorizon scales
fg and GM on
time even
derivatives
of both
hence
withscales
= 0.55
to horizon
scalesand
[ADD, and
CITES]
. GL
and GM , a↵ect the growth of matter perturbations.
(15) [DISCUSS alternative models of gravity and the values
Equation
provides
an
explicitGconnection
between
•  offgfrelated
to
G(21)
but does
describe
Get
matter
light/
matter
or
they
produce,
e.g.
Amendola
al
p88-89
GM ⇡
fgg and GM on sub horizon scales
ES] .
mention some].
"
# that (8)
he values To calculate the transfer
functions we assume
2fg
d ln fg
Ḣ
l p88-89 and (13) hold
and
use
Photon geo
GM ⇡
1 + fg +
+ 2 .
(22)
3⌦m
d ln a
H
potentials,
that (8)
to how an ov
˙ S = fg H S = ḣ
(16)
c
c
Photon geodesics are determined
2 by the sum of the propagation o
potentials, + . Hence GL describes the alteration
In comparis
to define
ḣ,an
and
assume
the method
to calculate
⌘˙ is the
to
how
over
or
under
dense
region
will
modify
the
(16)
scribe the equ
same
as in GR. of light.
propagation
there have be
Rachel Bean, TNote
es-ng In
Gravity, Vancouver, 2015 that
theJanuary fiducial
fit universal
in (15) should
not be
applied
comparison
to
the
adoption
of
w
to
dee ⌘˙ is the
alent notation
to conformal
on horizon
scribe the Newtonian
equations ofCDM
state perturbations
parameter for dark
energy,
growth rate of the rest frame CDM density perturbations,
ct on the
2 rate
¨ccS+in
˙CAMB,
•  which
Alternative
parameterization:
the
are
equal
to
is
reasonably
fit (21)
by
H
+
k
⇡ of
0 CDM growth
c
in which
m for the modeling the logarithmic growth factor,
Four distinct cosmological tracers
1.  Standard candles and rulers
2.  Clustering of matter
3.  Motion of non-relativistic tracers
4.  Lensing distortion of light
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Different relations to gravitational map
II: Growth, up to some
normalization
I: Background expansion
Probes of homogeneous
geometry:
•
•
•
CMB angular diameter
distance
Supernovae luminosity
distance
BAO angular/radial scale
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Biased tracers of perturbed
gravitational field
III: Growth directly
Direct tracers of perturbed
gravitational field
•
Galaxy distribution
•
CMB ISW correlation
•
Xray and SZ selected
galaxy cluster distribution
•
Weak galaxy and cluster
lensing correlation
•
Peculiar velocities/ bulk
flows of galaxies
Complementary tracers for testing gravity
Masamune et al 2010
•  Non-relativistic: Galaxy
positions & motions
–  Sensitive to ψ ∼ Gmat
–  Biased tracer
–  Can be measured at specific z
•  Relativistic: Weak lensing &
CMB lensing & ISW
–  Sensitive to (φ+ψ): Glight
–  Direct tracer of potential,
–  but still plenty of systematics
(more on this…)
–  Integrated line of sight info
•  Contrasting both can get at Glight/Gmatter
Zhang et al 2007
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Complementary tracers for testing gravity
Are these the unrealis-c dreams of a theorist? Masamune et al 2010
•  Non-relativistic: Galaxy
positions ‘g’& motions ‘θ’
–  Sensitive to ψ ∼ Gmat
–  Biased tracer
–  Can be measured at specific z
•  Relativistic: Weak lensing &
CMB lensing & ISW
–  Sensitive to (φ+ψ): Glight
–  Direct tracer of potential, but
still plenty of systematics (more
on this…)
–  Integrated line of sight info
•  Contrasting both can get at Glight/Gmatter
Zhang et al 2007
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 CANDELS photo-z investigation
9
Complications: Photometric redshifts
Quicker (more galaxies) and concurrent with
imaging, but less accurate than spectral z
•
Challenge to refine photo-z estimates given
disparate and incomplete spectroscopic
samples
•
Redshift probability distribution dispersed and
biased relative to the true spectroscopic z.
z

z
= rms
1 + zspec
z = (zphot
biasz = mean
10
1 + zspec
zspec )
2
Fig. 14.— An example of the photometric redshift probability distributions for one galaxy with spectroscopic redshift z=0.734. Blue lines
show five individual codes (code 3B, 6E ,7C ,11H, and 13C) without correcting distributions so that they match the 68.3% confidence interval
criterion. Red lines show the distributions after corrections. Finally, the black line shows the sum of the individual distributions.
Dahlen et al.
In this example of the hierarchical Bayesian method,
we have used a simple assumption for U(z), i.e., that we
have no information if the measured Pm (z)i is wrong.
Furthermore, we have allowed fbad in the whole range
fbad =[0.0,1.0]. Alternatively, we can assume that there
is at least some minimum probability that the actual measurement are correct and let the bad fraction vary in the
range fbad =[0.0,x]. Repeating our analysis after varying
x does not change results significantly, however, there is
a slight decrease in the outlier fraction and full rms when
setting 0.3 < x < 0.5, i.e., assuming that the measured
P(z)i are correct at least 50-70% of the times. Setting
x=0.0, equivalent to assuming that all measured P(z)i are
always correct, does, however, result in a significant increase in the outlier fraction (from 3.4% to 4.9%) and full
rms (σF =0.10 to σF =0.36).
The example above illustrates that the hierarchical
Bayesian approach does indeed provide means for improving results. It is possible to assume a more advanced guess
for the shape of U(z). For example, if the measurement
is bad, one could use a redshift probability following the
volume catalog.
element Results
redshiftaredependence.
thiscodes,
assumpFig. 4.— The mean biasz in the photometric redshift determinations for the H-selected
shown for all Using
individual
tion,scatter.
we find
that
the
outlier the
fraction
decreases
as well as the median of all codes and the median of the five codes with the smallest
Error
bars
represent
error inslightly
the mean.
(from 3.4% to 3.1%), while the full rms show a marginal
increase ( σF =0.10 to σF =0.11) and (after excluding outtheeffect.
rms, σO , remains unchanged. Since we do not
icanceliers)
of this
with the ACS z-band selected catalog (Table 3), we find
expect the spectroscopic control sample to follow the disTo quantify
the magnitude dependence of the photothat the scatter is similar for each code. This is not unextribution of the volume element, we do not expect this
metricexample
redshifts,
we divide the spectroscopic sample from
pected since most of the photometry in the two cases are
necessarily reflects the true expected effect of the
zspec
z )
z = (zphot
CANDELS photo-z investigation
z/(1 + zspec )
•
combined with a magnitude limit appropriate for this pa
ticular survey. In addition, it should be possible to le
the expected distribution be dependent on, e.g., apparen
magnitude or color.
Instead of using a generic form for U(z), another po
sibility is to dilate the given P(z) and use this for U(z
In this case we assume that the errors are underestimate
if the measurement is bad, rather than having no info
mation. There are many possibilities when applying th
hierarchical Bayesian method as discussed in Lang & Hog
(2012).
6. comparison to earlier work
Over the years, there has been a number of investiga
tions comparing results from different codes in order t
assess the accuracy of and the consistency between di
ferent photometric redshift codes. This includes Hogg e
al. (1998), Abdalla et al. (2008), and Hildebrandt et a
(2008, 2010). The most comprehensive previous investiga
tion of photometric redshift methods conducted in a sim
lar way to what presented here is described in Hildebrand
et al (2010). In that investigation, the result of twelv
different runs, representing eleven codes, are presented
Of these codes, three are common to this investigatio
(EAZY, LePhare, and HyperZ). Photometric redshifts ar
calculated using an R-filter selected 18-band photometr
catalog covering the GOODS-North field. The wavelengt
range covered is the same as here, i.e., U-band to the IRA
Comparison
photometric
and January spectroscopic
redshifts for a sample of 589 WFC3 H-band Dahlen
selected et
galaxies
with highest
al 1308.5353
Rachel between
Bean, Tes-ng Gravity, Vancouver, 2015 ra. Figure shows codes as listed in Table 1. Bottom right panel shows the result after taking the median of the five codes with
atter.
nformation about The observed ellipticity ϵ has contributions from the gravitameasurement of tional shear γG and an intrinsic shear γI , which is caused by
elations between the alignment of a galaxy in its surrounding gravitational field.
ubstantial exten- Moreover, ϵ is assumed to have an uncorrelated component ϵrnd ,
smological infor- which accounts for the purely random part of the intrinsic orienComplications:
Lensed
galaxies
nables one
to in- tations and shapes of
galaxies. Note
that (1) is only valid if the
2004; Bernstein gravitational shear is weak, see e.g. Seitz & Schneider (1997);
mation one adds Bartelmann & Schneider (2001) and for certain definitions of el•  ofLensing
distorts observed volume and
nctional form
lipticity.
another systemmagnifies
faintthe
galaxies
& be used to construct
Likewise,
positions(Moessner
of galaxies can
ck of knowledge
estimate of the number density contrast
Jainan1997)
matter, follow the
(i)
(i)
n(i) (θ) = n(i)
(2)
m (θ) + ng (θ) + nrnd (θ) ,
date the perfornd number den- which is determined by the intrinsic number density contrast
•  Number depends on slope of
onstrain cosmo- of galaxies ng and the alteration of galaxy counts due to lensluminosity
function
d flexible moding magnification
nm . An uncorrelated shot noise contribution is
In doing so we added via nrnd . In contrast to ϵ (i) (θ) the number density contrast
hich have been n(i) (θ) can obviously not be estimated from individual galaxies.
•  Galaxy correlations must factor this in
osmology them- One can understand n(i) (θ) as the ensemble average over a hypohotometric red- thetical, Poisson-distributed random field of which the observed
t al. 2006; Zhan
particular
representation.
The formal
+ Cm
+ Cgi mjis+one
Cm
Cni nj galaxy
= Cgi gdistribution
i gj
i mj
j
2007) galaxy- relation between the projected number density contrast as used
Guzik & Seljak in (2) and the three-dimensional galaxy number density fluctuaYoo et al. 2006; tions will be provided below, see (12).
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Wolf et al 2003 s of the transformed data set
by the GI signal. While this Cosmological imaging surveys observe the angular positions and
-known redshift dependence the projected shapes of huge numbers of galaxies over increas05) projects out the GI term ingly large areas on the sky. In addition, by means of multiComplications: Intrinsic alignments
ions. Furthermore the work colour photometry, it is possible to perform a tomographic analHirata et al. (2007) suggests ysis, i.e. obtain coarse information about the line-of-sight dimension in terms of photometric redshifts (photo-z). From the galaxy
e dominated
by
luminous
red
•  Galaxy Intrinsic shapes
are
aligned
in of
their
host
halo
shapes
in
a
given
region
space,
one
can infer the ellipticity
d from the cosmic shear catues require excellent redshift
(i)
ϵ (i) (θ) = γG(i) (θ) + γI(i) (θ) + ϵrnd
(θ) ,
(1)
use a significant reduction in
where the superscript in parentheses assigns a photo-z bin i.
y provide • information
about
Observed
shapeThe
correlations
include both
and intrinsic
terms
observed ellipticity
ϵ haslensed
contributions
from the
gravitaddition for a measurement of tional shear γG and an intrinsic shear γI , which is caused by
s cross-correlations between the alignment of a galaxy in its surrounding gravitational field.
ation. This substantial exten- Moreover, ϵ is assumed to have an uncorrelated component ϵrnd ,
eases the cosmological infor- which accounts for the purely random part of the intrinsic orienmportantly, enables one to in- tations and shapes of galaxies. Note that (1) is only valid if the
s (Hu & Jain 2004; Bernstein gravitational shear is weak, see e.g. Seitz & Schneider (1997);
density information one adds Bartelmann & Schneider (2001) and for certain definitions of elC i ✏j = C G i G j + C G i I j + C I i G j + C I i Ij
down the functional form ✏of
lipticity.
ntroduces as another systemLikewise,
theg positions
ofg galaxies
can be used to construct
+
C
C
=
C
I
n
✏
G
i
j
i
j
i
j
tifies the lack of knowledge an estimate of the number density contrast
e baryonic • matter,
follow the of intrinsic alignments is a function galaxy type,
The amplitude
(i)
(i)
.
n(i) (θ) = n(i)
(2)
m (θ) + ng (θ) + nrnd (θ) ,
luminosity and redshift
k to elucidate the perforaxy shape
and
number
den-January which
is determined by the intrinsic number density contrast
Rachel Bean, Tes-ng Gravity, Vancouver, 2015 ability to constrain cosmo- of galaxies ng and the alteration of galaxy counts due to lens-
Complications: Shear contamination
•  Incomplete correction of the atmospheric and instrumental PSF
can induce additive and multiplicative shear errors
✏i = (1 + mi )✏i + a
Spurious Shear in Weak Lensing with LSST
15
Figure 9. Absolute spurious shear correlation function after combining 10 years of r- and i-band LSST data when a standard
KSB algorithm is implemented and the PSF is modeled at different levels: non-stochastic PSF knowledge only (dashed) and
partial stochastic PSF knowledge from polynomial interpolation
of stars
(solid).
RedVlines
indicate
the pessimistic
case assuming
Rachel Bean, Tes-ng Gravity, ancouver, January 2015 Ne↵ = 184 while the green lines show the optimistic case when
Ne↵ = 368 is assumed. All curves are the medians of 20 different
realisations under the fiducial observing condition, with the error
Chang et al 2012
LSST Project
σ
0.1
0.98
0.97
0.0
0.96
0.95
−2
10
−0.1
Complications: Non linear scales
0.005
0.010
0.020
0.050
0.100
0.200
0.500
1.000
−1
10
0
10
k [1/Mpc]
F IG . 10.— M004, M008, M013, M016, M020, and M026. For each simulation, we show six residuals corresponding to the different values of the
scale factor a. The errors are on the order of 1% for the bulk of the domain
of interest. Considering that the tested emulators are built on an incomplete
design, this result is remarkably good.
2.000
k [1/Mpc]
•  F IGCosmological
extraction
requires modeling to
. 8.— Median MCMC draw for the
bandwidth function σ in
(blackNL
line). scales
of the codes, to structure the directories where different runs
Small values on this plot correspond to places where the spectra are comparare performed in, and to submit the simulations to the computcommensurate
accuracy
of:
atively
less smooth because of the baryon wiggles.
In addition, all the
power
ing queue system. For future efforts of this kind the adoption
and development
of dedicated
workflow capabilities
for these
–  the underlying CDM nonlinear power
spectrum
(for LCDM
and other
models)
tasks must be considered. The number of tasks to be carout uncertainties,
will become too large to
keep
track of
without such
–  Modeling baryonic effects, signal ried
and
on
halo
concentration
tools. In addition, since large projects will require extensive
collaborations, software tools will make it easier to work in 14
a
team environment since each collaborator will have informa1.05
tion about previous tasks and results. An example of such a
a=0.5
a=0.6
tool for cosmological simulation analysis and visualization is
1.04
a=0.7
a=0.8
given in Anderson et al. (2008).
a=0.9
1.03
a=1.0
We carried out the data analysis after the runs were finished.
1.02
For very large simulations this is not very practical, and onthe-fly analysis tools are required to minimize read and write
1.01
times and failures. This in turn requires that the code infras1
tructure be tailored to the problem under consideration.
(3) Serving the Data: Clearly, large simulation efforts can0.99
not be carried out by a few individuals, and require possi0.98
bly community wide coordination. The simulation data will
0.97
be valuable for many different projects. It is therefore necessary to make the data from such simulation efforts pub0.96
licly available and serve them in a way that new science can
0.95
be extracted from different groups of simulations. Transfer10
10
10
k [1/Mpc]
ring large amounts of data is difficult because of limitations
in communication bandwidth and also because of the large
storage requirements. It would therefore be desirable to have
F IG . 9.— Ratio of the emulator prediction to the smooth simulated power
spectra for the M000 cosmology at six values of the scale factor a. The error
computational resources dedicated to the database. In such
exceeds 1% very slightly in only one part of the domain for the scale factors
a situation, researchers would be able to run their analysis
a = 0.7, 0.8, and 0.9.
etonal
1212.1177
codes
on machines
to the
thebias
database
anda perLawrence
al 0912.4490
FIG. 8: Biaseset
induced
in the dark energy equation
of state
FIG. 9: with
Samedirect
as Fig.access
8,Zentner
but for
wa from
Stage
parameter
w
from
the
analysis
of
a
STAGE
IV
dark
energy
IV
experiment.
Notice
that
the
vertical
axis
is
asymmetric
0
form queries on the data easily. We are planning to make
the
in terms of storage. From the Coyote Universe runs we stored
experiment as a function of the maximum multipole used to
about zero.
Coyote
Universe
database
available
in
the
future
and
use
it
as
11 time snapshots plus theinfer
initial
conditions
(particle
posicosmological parameters. For STAGE IV experiments,
manageable testbed for such services.
tions and velocities and halofour
information)
to 250
of the OWLSleading
simulations
lead GB
to biases a
significantly
Communication
withcomputational
other Communities:
Theanother
complexlarger
than the
others run
and itthis
is useful
to emphasize(4)
this.
In
of data
perGravity, run. For
the 300
particle
would
tions in larger
volumes. On
front,
Rachel Bean, Tes-ng Vancouver, Jbillion
anuary 2015 main
panel,
the biases that
result be
from analyzing
ity of those
the analysis
task
makes ittonecessary
to efficiently
collabit may
be possible
develop more
sophisticated
models
increase to 75 TB. Only verythe
few
places
worldwide
would
four simulations without any model to account for baryonic
for the influence
baryons
on lensing power
that
orate and communicate
withofother
communities,
forspectra
example
able to manage the resultingeffects.
large databases.
These biases are clearly very large. Each line is lacan minimize
biases in and
inferred
cosmological
parameters
statisticians,
computer
scientists,
applied
mathematicians.
(2) Simulation Infrastructure:
Running
a
very
large
numbeled by the name of the corresponding OWLS simulation in
∆2emu / ∆2sim
spectra are shown (37 models, 6 redshifts each) in light gray. The power
spectra a scaled and shifted to fit the plot. σ is low on the baryon acoustic
oscillation scale to accommodate local wiggles, as expected. The vertical
line shows the matching point between perturbation theory and the simulation outputs. On the left of this, each replicate is identical (compared to the
realizations from the different simulation boxes) and smooth, therefore the
behavior of σ has not much information.
−2
−1
0
without a significant cost in statistical errors. As the cos-
A single survey can’t give the full picture
•  Trade offs in
–
–
–
–
–
Techniques (SN1a, BAO,RSD, WL, Clusters + lensing, motions, positions)
Photometric speed vs. spectroscopic precision
Angular and spectral resolution
Astrophysical tracers used (LRGs, ELGs, Lya/QSOs, clusters)
Epochs, scales and environs being studied (cluster vs dwarf galaxies)
•  Much more than a DETF FoM:
–  Astrophysical & instrumental systematic mitigation as so easily summarized.
–  Readiness vs technological innovation
–  Survey area vs depth - repeat imaging, dithering, cadence and survey area
overlap/config.
•  Future surveys distinct and complementary, both with each other and
•  ground based LSS surveys (LSST, DESI and others)
•  Planck and ground-based CMB gravitational lensing measurements
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 15
Upcoming Surveys:
Different strengths & systematics
(based on publicly available data) Stage IV DESI LSST Euclid WFIRST-­‐AFTA Starts, dura-on ~2018, 5 yr 2020, 10 yr 2020 Q2, 7 yr ~2023, 5-­‐6 yr Area (deg2) 14,000 (N) 20,000 (S) 15,000 (N + S) 2,400 (S) FoV (deg2) 7.9 10 0.54 0.281 Diameter (m)
4 (less 1.8+) 6.7 1.3 2.4 Spec. res. Δλ/λ
3-­‐4000 (Nfib=5000) 250 (slitless) 550-­‐800 (slitless) Spec. range
360-­‐980 nm 1.1-­‐2 µm 1.35-1.95 µm BAO/RSD 20-­‐30m LRGs/[OII] ELGs 0.6 < z < 1.7, 1m QSOs/Lya 1.9<z<4 ~20-­‐50m Hα ELGs Z~0.7-­‐2.1 20m Hα ELGs z = 1–2, 2m [OIII] ELGS z = 2–3 pixel (arcsec) 0.7 0.13 0.12 Imaging/ weak lensing (0<z<2.) ~30 gal/arcmin2 5 bands 320-­‐1080 nm 30-­‐35 gal/arcmin2 1 broad vis. band 550– 900 nm 68 gal/arcmin2 3 bands 927-­‐2000nm SN1a 104-­‐105 SN1a/yr z = 0.–0.7 photometric Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 2700 SN1a z = 0.1–1.7 IFU spectroscopy Example constraints on Gmatter, Glight
WFIRST photometric survey + CMB. GMatter,Glight constant 0<z<2
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Bean and Mueller, in prep
Spec & Photo survey: constraints on γ0-γa
Beware a strong theoretical prior significantly biases redshift range of
interest for growth studies (as with equation of state)
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Bean and Mueller, in prep
Clusters as a probe of gravity
•  Cluster dynamics provide an
alternative direct probe of the
gravitational potential.
CMB γ
Hot cluster gas Energe-c e-­‐ •  Measurable via the kinetic Sunyaev
Zeldovich (kSZ) effect
Blue-­‐shiied γ
Observatory
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Clusters as a probe of gravity
•  Solution: isolate kSZ through
cross-correlation
mean pairwise momentum •  Problem: kSZ has weak
frequency dependence and
small amplitude (compared to
thermal SZ)
ned.ipac.caltech.edu Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Hand et. al 2012 kSZ pairwise dark energy constraints
Current constraints:
BOSS:
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 Mueller, De Bernardis, Bean, Niemack 2014
To conclude
•  Phenomenology, a powerful tool to connect theoretical modeling of
gravity to observations on cosmological scales
•  Variety is critical to achieve required sensitivities over range of
redshifts and environments:
–
–
–
–
Non-relativistic (peculiar velocities) and relativistic (lensing)
Different systematics (e.g. CMB and galaxy lensing)
Different epochs and scales
Different tracers: halo and field galaxies, clusters and CMB
•  Understanding and incorporating instrumental and astrophysical
uncertainties key to realizing cosmological aims
–  Photometric redshift uncertainties
–  Effect of baryons processes on dark matter clustering
–  Disentangling instrumental (e.g. PSF) and astrophysical (e.g IAs) signals
from cosmological
Rachel Bean, Tes-ng Gravity, Vancouver, January 2015