INCLUDING INTRINSIC ALIGNMENTS
SYSTEMATICS
arXiv:1501.03119
1
Mustapha Ishak
The University of Texas at Dallas
Jason Dossett
INAF – Osservatorio Astronomico di Brera, Italy
M. Ishak & J. Dossett - TG 2015
CONSTRAINTS AND TENSIONS IN MG
PARAMETERS FROM PLANCK,
CFHTLENS AND OTHER DATA SETS
2015 IS THE 100 ANNIVERSARY OF EINSTEIN’S
GENERAL RELATIVITY
TH
2
M. Ishak & J. Dossett - TG 2015
A.
2
2
k
Φ
=
−4πGa
ρi ∆i Q(k,ofa)ISiTGR
The Modified Growth
Formalism
Redshift bins
i
Scale bins 0.0 < z ≤
1 1<z≤2
!
2
2
− R(k,
−12πGa
ρiQ
(13 ,+Σw
0.0a)order
<Φ)
k ≤=0.01
Q1 , ΣEinstein’s
d versionskof(Ψthe
first
perturbed
from the p
1
3equations
i )σi Q(k, a).
ODIFIED
G0.01
ROWTH
QUATIONS
< ka<flat
∞Euniverse
Q2 , Σ2 i thisQ4metric
, Σ4
WalkerM(FLRW)
metric.
In
is written in the
Flat
Perturbed
FLRW
ing a particular
species,
ρi is the
∆i parameterizations.
is the rest-frame overd
TABLE I: matter
The subscript
numbering
indensity,
theMetric.
binned
) and R(k, a) are2the time2 and scale dependent
modified gravity parameters
ds = a(τ ) [−(1 + 2Ψ)dτ 2 + (1 − 2Φ)dxi dxi ],
ation to what is often referred to as the Poisson equation (though as noted in
[33] as:it relates the
nISiTGR
equation
overdensity
to Equations
the
potential,
which
Modified
Growth
time,
a(τ ) as
is the scale factor
normalized
to space-like
one today,
the xi ’sΦare
the on
c
represents describing
an inherentthe
inequality
between
the metric
two potentials
that may be c
!
potentials
of
the
perturbations.
2scalar modes 2
k Φ = −4πGa
ρi ∆i Q(k, a)
gravitational slip [9].
rameters have been introduced in various
interrelated notations that m
ibetween the parameters Q and R
nly avoid a strong parameter degeneracy !
ween the two2 potentials in the metric (for
example, the gravitational slip
2
irectly probed
by−observations,
does not
directly
use
Eq.a).
(3) in its co
k (Ψ
R(k,the
a) Φ)
=ISiTGR
−12πGa
ρi (1
+ w(or
Q(k,
i )σigravitational
at characterizes
how
spacetime
curvature
side
pote
(2) and (3)[32]:
i
a the perturbed Einstein equations (sometimes
this parameter can be re
!
!
).
Due
to
degeneracies
between
some
of
these
parameters,
some other
2
2
2
ting
matter species,
rest-frame
overd
i is the
k a(Ψparticular
+ Φ) = −8πGa
ρi ∆iρΣ(k,
a) −density,
12πGa ∆i isρthe
i (1 + wi )σi Q(k, a),
the
enddependent
of this sub-section.
We refer
the reade
)for
andexample
R(k, a) exemplified
are the time iat
and
scale
modified
gravity
parameters
(
i
tionships
between
thereferred
variousto
parameterization
can be found
in as
[32].
It isina
ation to what
is often
as the Poisson
(though
noted
ef f equation
Q(1 + R)
G
=
µ
=
QR
matter
rnthese
Σ
= Q(1
+
This
parameter
is directly
probedform.
by
observations
such
parameterizations
in
a functional
or
binned
We use
here
th
equation
asR)/2.
it relates
overdensity
to the
space-like
potential,
Φ which
ona
⌃
=the
1
3
2
nrepresents
our previous
works
Σ inequality
was referred
to as D,
but
an
havemay
a consist
⌘ two
=in potentials
= effort tothat
an inherent
between
the
be ca
or D
ng forward weslip
are[9].
now using the much more commonRand intuitive, Σ.
gravitational
We saw some review talks by Koyama, Bean, and Silvestri.
M. Ishak
& J. Dossett - TG Q
2015and R
only avoid a strong parameter degeneracy between the
parameters
are the
The
to the Poisson equatio
1and
+ XR (k)
X MG
(k) parameters.
− X (k)
z −parameter
z
1 −Q
X represents
(k)
za−modification
z
z1
z2
z1
div
z2
T GR
+ techniques,
tanh
+
tanh of our ,code. One (12)
emeter
of
all
these
we
have
developed
two
versions
version of the
R quantifies
slip
(at have
late developed
times,
stress
is negligible,
ψ/φ
To 2
take
advantage the
of 2allgravitational
these techniques,
twoanisotropic
versions
code.
One versionRof=the
co
ztwwe
2 when
ztw of our
orm
(11).
It
provides
the
option
to
apply
the
functional
form
(11)
to
either
Q
and
R
(as
don
VOLVING
THE
RAVITY
ses the
functional
form
(11). It
provides
theODIFIED
option
to applywith
the functional
form (11)
either
Q and R D
(as=done
her
than
using the
parameter
R which
is
degenerate
Q, we instead
usetothe
parameter
Q(1
he
other
version
of
the
code
is
based
on
binning
methods.
It
provides
the
option
of
the
se
where
the
transition
between
the
two
redshift
bins
occurs
and
z
is
the
redshift
below
50])
or
Q
and
D.
The
other
version
of
the
code
is
based
on
binning
methods.
It
provides
the
option
of
the
seco
sted in [50] (this parameter is equivalent to the parameter Σ TinGR[45, 53] or G of [52]). Combining
pproach
with
traditional
(described
or bins,
alternatively
the
hybrid
approach
with two
We
hard
code
zT GR
= 2zbinning
toabove),
give
usINNING
equally
sized
but
this
of third,
course
is optional
andwith
ARAMETERS
tional
binning
(described
or above),
alternatively
the
third,
hybrid
approach
tworedsh
red
div
we
arrive
at the
second
modified
growth
equation
usedETHODS
in this
paper:
ins, represents
but the evolution
in scalemethod
evolves for
monotonically.
(k)
the binning
k in the ith z bin. For the suggested hybrid method
on
in
scale evolves
monotonically.
!
!only Q and D. Transitions
In the binning version
of
the
code,
we
evolve
between the redshift bin are evolv
2
2
2
rsion
of
the
code,
we
evolve
only
Q
and
D.
Transitions
between
redshift
bin
arebinni
evo
ρ
(1
+
k
(ψ
+
φ)
=
−8πGa
ρ
∆
D
−
12πGa
i i with a transition width
i
i )σ
i Q.
ollowing¢ [44,
and use a hyperbolic tangent
function
ztwwthe
=
0.05.
In this
way the
2 53]
scale-dependent
parameterizations
that
binned
in
i ztw = 0.05. In this way the bin
irepresenting
danuse
a hyperbolic
tangent
function
with
a ctransition
width
−k/k
actually
be
functionally
Q or D)
c as (with X
Xzwritten
+ X2 (1 −
e−k/k
)
(13)
1 (k) = X1 e
redshift:
ten functionally
as (with−k/k
X representing
Q
or
D)
−k/kc
c
E
P
M
:B
M
G
modified growth
equations
are
(8)
and
(10),
1 +eX
Xz(1(k)
− zare
1 −the
Xz2 MG
(k) parameters.
z − zT GR As discusse
Xz2 (k)
= X
+X
−−
e Xand
),Q andz D
z1 (k)
z1 (k)
div now
X(k, z) = 3
+ 42
tanh
+
tanh
,
(1
work [56], this approach of2 using the parameter
D instead
of R is 2also useful because
observation
2
z
z
tw
tw
1 + Xz1 (k)
Xasz2 (k) − Xz1 (k)
z − zdiv
1 − Xz2 (k)
z − zT GR
ing
inand
principle
evolves
sing
ISW are
sensitive
to the sum of
the metric
potentials
φ + ψtanh
and its time
derivative
resp
a) =
+
tanh
+
,
P1
is
binned
in
scale
as:
where zdiv is— the
redshift
where
the
transition
between
the
two
redshift
bins
occurs
and
z
is
the
redshift
bel
TzGR
2 able to give us!direct
2
ztw
2
tw
ervations are
measurements
of this parameter.
which GR is to be tested. We hardX
code
zT GR
this of course
is optional a
< k=c 2zdiv to give us equally sized bins, but Redshift
bins
1 if k
(k)
=represents
(14)
z1X
an easily
be changed.
method
for kbins
in theoccurs
ith z bin.
Forzthe
suggested
hybrid
meth
zi (k)
dshift
where
theXtransition
between
the
redshift
and
is
the
redshift
b
X2 if the
k ≥binning
kc , two
Tz
GR
Scale
bins
0.0
<
≤
1
1
<
z
≤
2
t has the form
sted.
hard code
zT GR!=X2z
us equally
sized
but this
optional
divk to
0.0
< kbins,
≤ 0.01
Q1in
, of
ΣTime
Q
, ΣScale
C. We
Different
Approaches
to
Modified
Growth
Parameters
and
1course
3is
3
< kgive
3 ifEvolving
c
−k/k
−k/k
X
(k)
=
c
z2
d. Xzi (k) represents
the binning
ith
For the
me(1
0.01
< bin.
k c<
Q2 ,suggested
Σ2
Q4hybrid
, Σ4
X2 (1
− ez
) ∞
1 e k in+the
X4 Xifzmethod
≥ k=c . Xfor
1k(k)
−k/kc
c
Xapproaches
+ X4 (1 the
− e−k/k
),parameters in time and scale; o
e, there have
primarily
been
two
MG
z2 (k) = X3 eto evolving
— implement
P2 evolves
monotonically
inI:scale:
TABLE
The
subscript
numbering
in thethe
binned
paramet
er
chosen
to
the
traditional
binning
method
with
some
control
transition
uous functional form and the other −k/k
basedc on binning. −k/k
We cimplementoninthe
ISiTGR
two approac
while with traditional binning
Xz1 (k)in principle
= we
X1explain
e evolvesbelow.
+asX2 (1 − e
)
ally, a new hybrid approach,
as
−k/kc form!for each
−k/kc
rst approach involves
defining
a
functional
that allows it to evolve mono
X
+
X
X
−
kX−
kX
if k <parameter
kc),
X
+
(1
2z2 (k)1 = X
2 3 e X1
c−
1 e
4
X
(k)
=
(1
by
[29]
and
used
in
ISiTGR
[33]
as:
X
(k)
=
+
tanh
z
z1
time and scale.
This allows
one to2make1 no assumptions
from general rela
X2 if k ≥as
kc ,to when a deviation (15)
2
ktw
!
!
an
was
taken
inXfor
example and
[50].
that workin
theredshift
functional form,
¢  approach
P3
scale
independent
evolves
only as:
alSuch
binning
in is
principle
evolves
as
X
X4 +
X
k − kIn
3
4 − X3
c 3 if k < kkc2 Φ = −4πGa2
ρi ∆i Q(k, a)
X
(k)
=
z
Xz2 (k) =
+
tanh
2
X,4 if ks ≥ kc .
4
2
2 !
k
X(a) = (Xtw − 1) a + 1
i
X1 if k 0< kc
!
Xz1 (k)
=
Here though, we have rather chosen
to implement
the 2traditional binning method with some
2 control on the transiti
(Ψ
−kR(k,
Φ) a=total
−12πGa
(1 + wi )σi are
Q(
X2 kif(8)
k≥
, a)
c(9).
med,
where X denotes either Q or R in Eqs.
and
Thus
of six modelρiparameters
s:
!
i - TG 2015
M. Ishak & J. Dossett
Q , R , Q , R , k , and s. The parameters s and k parametrize time and scale dependence resp
5
http://isit.gr M. Ishak & J. Dossett - TG 2015
GALAXY INTRINSIC ALIGNMENTS (IA) AS A
CONTAMINANT TO WEAK LENSING (WL) SIGNAL
¢
¢
¢
¢
Contaminates WL signal by up
to 15-20%.
2 pt. IA biases cosmological
parameters from WL at the
10%-50% level
Measured correlation function
is sum of GG, GI and II signals.
Used a model for IA with
amplitude parameter ACFHTLenS
6
M. Ishak & J. Dossett - TG 2015
DATA SETS USED
¢  CMB
temperature anisotropy power-spectrum
from Planck 2013
¢  Low-l WMAP Polarization data
¢  Weak lensing tomography shear-shear cross
correlations from the CFHTLenS
¢  Galaxy power spectrum from the WiggleZ survey
¢  ISW-galaxy cross correlations
¢  BAO data from 6dF, SDSS DR7, and BOSS DR9
7
M. Ishak & J. Dossett - TG 2015
RESULTS OVERVIEW
¢
¢
¢
We find a 40-50% improvement on figure of merit (FoM) for
the MG parameters over previous results.
GR is consistent with the data at the 95% CL when
considering 2D contours.
With current data, only 1-5% change in the FoM for the
MG parameters when ACFHTLenS fixed to zero.
8
P1
P2
P3
M. Ishak & J. Dossett - TG 2015
RESULTS: IA AND DIFFERENT LENSING
DATASETS.
10
P1
GR
P2
P1
P3
¢
FIG. 6: 68% and 95% 2-D confidence contours for the intrinsic alignment amplitude parameter ACF HT LenS
9 of [73] thou
ROW: the theory is fixed to GR and the constraints obtained are in good agreement with those
to more precise recent data. To the left are the results for the IA-optimized red galaxy sample of [73]. T
of non-zero ACF HT LenS is also in agreement with [73]. SECOND and THIRD ROWS: Similar constraints
for the scale-dependent parameterizations P2 and P3 that model any deviation from GR. The bounds are
ACF HT LenS parameter is practically on the the 95% CL boundary line for the optimized red galaxy sample
results for the scale independent MG parameterization.M.
The
constraints
are very similar
to the GR case with
Ishak
& J. Dossett
- TG 2015
The contours in the GR case are in agreement with
Heymans et al. 2013
6
RESULTS CONT’D
¢  P1
6
FIG. 1: 68% and 95% 2-D confidence contours for the parameters Qi and Σi from parameterization P1 for redshift and scale
dependence of the MG parameters. All of the constraints for this evolution method are fully consistent with GR at the 68%
level.
IV.
RESULTS AND ANALYSIS
For all results presented, in addition to the MG parameters and relevant nuisance parameters for the various data
sets, we vary the six core cosmological parameters: Ωb h2 and Ωc h2 , the baryon and cold dark matter physical density
FIG. 1: 68%respectively;
and 95% 2-D θ,
confidence
contours
the parameters
Σi fromdiameter
parameterization
i and
parameters,
the ratio
of the for
sound
horizon toQthe
angular
distanceP1
of for
theredshift
surfaceand
of scale
last
10
dependence
of
the
MG
parameters.
All
of
the
constraints
for
this
evolution
method
are
fully
consistent
with
GR
at
the
68%
scattering; τrei , the reionization optical depth; ns , the spectral index; and ln 10 As , the amplitude of the primordial
level.
power spectrum.
95% confidence limits on MG parameters
IV. RESULTS
AND
evolved using
formANALYSIS
P1
Q1
[0.49,2.56]
Σ1
[0.97,1.14]
Σ2and relevant
[0.84,1.22]
For all results presented, in additionQto2 the [0.05,3.08]
MG parameters
nuisance parameters for the various data
2
2
[0.30,1.78]
[0.97,1.06]
3
sets, we vary the six core cosmological Q
parameters:
Ωb h andΣΩ3 c h , the
baryon and cold dark matter physical density
10
[0.28,2.88]
parameters, respectively; θ, the ratio Q
of4 the sound
horizon Σto4 the [0.90,1.12]
angular diameter distance of the surface of last
scattering; τrei , the reionization optical depth; ns , the spectral index; and ln 1010 As , the amplitude of the primordial
TABLE
II: We list the 95% confidence limits for the MG parameters from using form P1 to define their time and scale
power spectrum.
dependence. In this traditional binning approach we find that all of the MG parameters are fully consistent with their GR
values of 1 at the 95% level.
M. Ishak & J. Dossett - TG 2015
7
95% confidence limits on MG parameters
evolved using form P2
Q1
[0.38,3.43]
Σ1
[1.03,1.37]
Q2
[0.00,2.86]
Σ2
[0.75,1.07]
Q3
[0.28,2.46]
Σ3
[0.93,1.14]
Q4
[0.05,1.99]
Σ4
[0.86,1.14]
RESULTS CONT’D
P2
TABLE
¢  III: We list the 95% confidence limits for the MG parameters from using form P2 to define their time and scale
dependence. While most of the MG parameters for this hybrid evolution method are consistent with their GR values of 1 at
the 95% level, we find a significant tension for the MG parameter Σ in the large scale (small k) low redshift bin (Σ1 ).
FIG. 2: 68% and 95% 2-D confidence contours for the parameters Qi and Σi from parameterization P2 for redshift and scale
dependence of the MG parameters. As you can see in the first bin, there a tension with the GR value of 1. However, contrary
7
to the marginalized 1-D constraints given in Table III the GR point is still within the 95% confidence region.
Correlation
95% confidence
limits ontable
MG parameters
Binning
parameterization
evolved
using form P2(P1)
Q1
Q3
Q
Σ1
Σ2
Σ3
Σ4
QQ
[0.38,3.43]
Σ41
[1.03,1.37]
12
ACF HT LenS -0.021162 -0.29209
0.015916 0.0056355
-0.0014863
Q2
[0.00,2.86]
Σ2
[0.75,1.07] 0.083586 0.015755 0.066954
σ8
-0.012168 -0.53048
0.044293 -0.43088
0.045781
Q3
[0.28,2.46]
Σ3
[0.93,1.14] -0.61952 0.048845 -0.29894
11
Ωm
-0.0012586 -0.072645
-0.051569 0.11762
-0.08057
Q4
[0.05,1.99]
Σ4
[0.86,1.14] -0.085185 -0.033916 -0.17292
Hybrid parameterization (P2)
TABLE III: We
the
95%
confidence
limits
for
the MG 0.095984
parameters -0.14858
from using0.20636
form P2-0.086038
to define 0.10421
their time and scale
ACFlist
0.058535
-0.29535
-0.052588
HT LenS
dependence. While σmost
of the0.2655
MG parameters
this hybrid
evolution 0.32713
method are
consistent0.1362
with their
GR values of 1 at
-0.70809for 0.12172
-0.33026
-0.59009
-0.20504
8
M. Ishak
J. Dossett
TG
the 95% level, we find
tension
for the MG
parameter
Σ in the0.01565
large scale
(small
k)0.14932
low&redshift
bin -(Σ
1 ).2015
Ω a significant
0.027229
-0.065934
-0.028016
0.0803
-0.15645
-0.26513
RESULTS CONT’D
8
FIG. 2: 68% and 95% 2-D confidence contours for the parameters Qi and Σi from parameterization P2 for redshift and scale
dependence of the MG parameters. As you can see in the first bin, there a tension with the GR value of 1. However, contrary
to¢
the marginalized 1-D constraints given in Table III the GR point is still within the 95% confidence region.
P3
Correlation table
Binning parameterization (P1)
Q1
Q2
Q3
Q4
Σ1
Σ2
Σ3
Σ4
ACF HT LenS -0.021162 -0.29209 0.015916 0.0056355 -0.0014863 0.083586 0.015755 0.066954
σ8
-0.012168 -0.53048 0.044293 -0.43088 0.045781 -0.61952 0.048845 -0.29894
Ωm
-0.0012586 -0.072645 -0.051569 0.11762
-0.08057 -0.085185 -0.033916 -0.17292
Hybrid parameterization (P2)
FIG. 3: 68% and
2-D confidence
for the
parameters
Q0 , Σ0 , -0.14858
and R0 from
the scale-0.086038
independent
parameterization,
ACF95%
0.058535contours
-0.29535
-0.052588
0.095984
0.20636
0.10421
HT LenS
P3, for the MG parameters.
These
constraints
are
consistent
with
GR
a
the
95%
level,
but
a
tension
is
evident.
The tension is
σ8
0.2655
-0.70809 0.12172 -0.33026
0.32713 -0.59009 0.1362 -0.20504
evident when viewing
these
plots
is
not
easily
seen
using
the
1-D
constraints
given
in
Table
V.
This
is
due
to
the
non-Gaussianity
Ωm
0.027229 -0.065934 -0.028016 0.0803
0.01565 -0.15645 0.14932 -0.26513
9
of the probability distribution for these parameters as further Fig. 4.
TABLE IV: Correlations between ACF HT LenS , σ8 , Ωm versus the MG parameters of P1 and P2.
Correlation table
Functional parameterization (P3)
Q0
Σ0
C. P3 Scale Independent
Evolution
ACF HT LenS -0.023164 0.10624
σ8
-0.66775 -0.75738
The results from the scale independent evolution
method,
P3, are also much more insightful than those from P1.
Ωm
-0.072171 0.052317
They offer some insight as to the origin of the strong tensions exhibited by parameters from P3. The marginalized
95% confidence limits
forVI:
theCorrelations
parametersbetween
Q0 , Σ0 Aand
R0 are presented in Table V. Looking only at these constraints,
TABLE
CF HT LenS , σ8 , Ωm versus the MG parameters of P3
one might conclude that nothing interesting is happening in this parameterization of the MG parameters as all of the
parameters are completely consistent with one at the 95% level.
12
constraints is illustrated even better in Fig. 4 where we plot the 1-D probability distributions for the MG parameters
confidence limits
MG parameters
FIG.
1-Devolution
probability
distributions
for 95%
the parameters
Q0parameters
, Σon
the scale independent parameterization for
0 , and R0
used 4:
in this
method.
The distributions
for the
Qfrom
0 and R0 are both quite skewed, with a large
evolved
using
form
P3
the
parameters,
P3. of
These
constraints are while
consistent
with GR for
a the
level, but As
a tension
is evident
in
tail MG
in one
end of each
theQdistributions,
distribution
Σ0 95%
isRbimodal.
discussed
brieflyparticularly
in §IV B the
[0.77,1.99]
Σ0 display
[0.79,1.16]
0level [-0.23,1.18]
the
parameters
Q
a
significant
of
non-Gaussianity,
especially
Σ
0 and R0 . The0probability distributions
0 which
cause of these tensions in the MG parameter space seems to be caused by a tension between the CMB and weak
is somewhat bimodal. This non-Gaussianity explains why the tension seen in these plotsM.is Ishak
not seen
the -marginalized
& J.using
Dossett
TG 2015
TENSIONS BETWEEN THE DATA SETS
¢  We
have seen indications of tensions in the MG
parameter space for P2 and P3.
¢  Known tension between CMB and weak lensing,
notably in constraints on σ8.
¢  For
P3 we get a bimodal σ8, hinting the tension in
MG parameter space is likely related to known
tension between the data sets.
13
M. Ishak & J. Dossett - TG 2015
TABLE III: We list the 95% confidence limits for the MG parameters from using form P2 to define their time and scale
dependence. While most of the MG parameters for this hybrid evolution method are consistent with their GR values of 1 at
the 95% level, we find a significant tension for the MG parameter Σ in the large scale (small k) low redshift bin (Σ1 ).
RESULTS: CORRELATIONS WITH MG
PARAMETERS
¢
We find only weak to moderate correlations between MG
parameters and the IA parameter.
8
With current data, only 1-5% change in the FoM for the MG
parameters when ACFHTLenS fixed to zero.
—  Both
dependent
parameterizations
correlation
FIG. 2: 68%
and 95%scale
2-D confidence
contours
for the parameters Qi andshow
Σi frommost
parameterization
P2 for redshift and scale
dependence ofin
thelow-z,
MG parameters.
you can
see probed
in the first most
bin, there
tension with
the GR
high-kAsbins
(bin
bya lensing
data)
. value of 1. However, contrary
to the marginalized 1-D constraints given in Table III the GR point is still within the 95% confidence region.
—
Correlation table
Binning parameterization (P1)
Q1
Q2
Q3
Q4
Σ1
Σ2
Σ3
Σ4
ACF HT LenS -0.021162 -0.29209 0.015916 0.0056355 -0.0014863 0.083586 0.015755 0.066954
σ8
-0.012168 -0.53048 0.044293 -0.43088 0.045781 -0.61952 0.048845 -0.29894
Ωm 2-D confidence
-0.0012586
-0.072645
-0.051569
0.11762
-0.08057
-0.085185
-0.17292
FIG. 3: 68% and 95%
contours
for the
parameters
Q0 , Σ0 , and
R0 from
the scale-0.033916
independent
parameterization,
Hybrid
parameterization
(P2)
P3, for the MG parameters. These constraints are consistent with GR a the 95% level, but a tension is evident. The tension is
ACF HTthese
0.058535
-0.29535
-0.052588
0.20636
-0.086038
0.10421
evident when viewing
is not easily
seen using
the 1-D0.095984
constraints-0.14858
given in Table
V. This
is due to the
non-Gaussianity
LenS plots
σ8
0.2655
-0.70809 as
0.12172
-0.33026
0.32713 -0.59009 0.1362 -0.20504
of the probability distribution
for
these parameters
further Fig.
4.
Ωm
0.027229 -0.065934 -0.028016 0.0803
0.01565 -0.15645 0.14932 -0.26513
Correlation table
TABLE IV: Correlations between
ACF HT LenS
, σ8 , Ωm versus
the MG parameters of P1 and P2.
Functional
parameterization
(P3)
14
Q0
Σ0
ACF HT LenS -0.023164 0.10624
C. P3 σScale
Independent
Evolution
-0.66775 -0.75738
8
Ωm
-0.072171 0.052317
M. Ishak & J. Dossett - TG 2015
SUMMARY
¢  We
find a 40-50% improvement on figure of merit
for the MG parameters over previous results.
¢  The intrinsic alignment amplitude shows weak to
moderate correlation with the MG parameters
(Q2 & Σ2 most correlated).
¢  GR & P3 show a clear IA signal for the optimized
early-type galaxy sample
¢  GR is consistent with the data at the 95% CL
when considering 2D contours.
¢  A clear tension is present in the parameter Σ
apparently related to the known tension between
CMB and weak lensing.
15
M. Ishak & J. Dossett - TG 2015