Exploring degeneracies in modified gravity with weak lensing

0.35 MB·English

by  C. Danielle Leonard

Exploring degeneracies in modified gravity with weak lensing
C. Danielle Leonard,∗ Tessa Baker,† and Pedro G. Ferreira‡
Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK
By consideringlinear-orderdeparturesfrom general relativity,wecomputeanovelexpression for
theweak lensing convergencepower spectrum underalternativetheories of gravity. This comprises
anintegralovera‘kernel’ofgeneralrelativisticquantitiesmultipliedbyatheory-dependent‘source’
term. Theclearseparationbetweentheory-independentand-dependenttermsallowsforanexplicit
understandingofeachphysicaleffectintroducedbyalteringthetheoryofgravity. Wetakeadvantage
of this to explore thedegeneracies between gravitational parameters in weak lensing observations.
5
1
0
I. INTRODUCTION This paper is structured as follows: in Section II, we
2
detail the derivation of the expressionfor P (ℓ). Section
κ
pr Inrecentyears,weakgravitationallensinghasbeenput III discusses how weak lensing degeneracy directions be-
A forthasapromisingmethodoftestinggravitationoncos- tween gravitational parameters can be understood with
mologicalscales[1–8],withsomeexcitingfirstconstraints the help of our expression. Finally, Section IV provides

1 havingbeenfoundalready[9,10]. Moreover,advancesin forecastconstraintsondeviationsfromGR+ΛCDMfrom
2 relevantdataanalysis(forexample,[11])andthecoming futuresurveys,andinterpretstheseconstraintsusingour

next generation of lensing-optimised surveys mean that expression for Pκ(ℓ). We conclude in Section V.
]
O wewillsoonbeinapositiontotakefulladvantageofthe
C potential of weak lensing.
II. CONVERGENCE IN MODIFIED GRAVITY:
Stronger constraints on gravity are obtained by com-
h. biningweakgravitationallensingwithotherprobes. One THE LINEAR RESPONSE APPROACH
p
observable which is commonly touted as providing par-
o- ticularly complementary constraints to weak lensing is In what follows,we use the scalarperturbed Friedmann-
r fσ (a). Here, f(a) is the lineargrowthrate ofstructure, Robertson-Walker (FRW) metric in the conformal New-
st defi8ned as: tonian gauge, with the following form:
a
[ f(a)= dln∆M(a) (1) ds2 =a(τ)2 −(1+2Ψ)dτ2+(1−2Φ)dxidxi . (2)
dlna
2
Our parame(cid:2)terisation of alternative theories(cid:3) of gravity
v where ∆ (a) is the amplitude of the growing mode of
M makes use of the quasistatic approximation (see, for ex-
9 the matter density perturbation, and σ (a) is the am-
0 plitude of the matter power spectrum w8ithin spheres of ample, [14]). The quasistatic approximation states that
5 withintherangeofscalesrelevantforcurrentgalaxysur-
radius 8 Mpc/h. The combination fσ (a) can be con-
3 8 veys, the most significant effects of a sizeable class of
strained through measurements of redshift-space distor-
0 modified theories can be captured by introducing two
. tions in galaxy surveys.
1 functions of time and scaleinto the linearisedfield equa-
In [12], an expression for fσ (a) was derived in the
0 8 tions of GR. These functions play the role of a modified
case of linear deviations from the model of general rela-
5 gravitationalconstant, and a non-unity (late-time) ratio
tivity (GR) with ΛCDM. Here we build on this work by
1 of the two scalar gravitationalpotentials:
: constructing a similar expression for Pκ(ℓ), the angular
v
i power spectrum of the weak lensing observable conver- 2 2Φ(a,k)=8πGa2µ(a,k)ρ¯M∆M(a,k)
X gence (κ). The main advantage of our expressionis that ∇
Φ(a,k)
r itclearlydistinguishesthephysicalsourceofallmodified =γ(a,k). (3)
a gravity effects to P (ℓ), which allows for a more thor- Ψ(a,k)
κ
ough interpretation and understanding of these effects
In GR, both γ(a,k) and µ(a,k) are equal to 1.
than previously possible. While we focus on P (ℓ) in
κ
Clearlyequation3 canonly be aneffective description
this work, recall that the two main weak lensing observ-
of more complicated, exact sets of field equations [15–
ables, convergence and shear, can be trivially intercon-
24]. However,severalworkshavenumericallyverifiedthe
verted [13]. Therefore, we treat convergence as a proxy
validityofthequasistaticapproximationinmanygravity
forweaklensingmoregenerally,andallexpressionswhich
theories (notably those with one new degree of freedom)
we derive could be equivalently and easily formulated in
on the distance scales considered here [25–29].
terms of shear.
We first compute the power spectrum of the conver-
gence in general relativity, and then generalise to al-
ternative theories of gravity. We make the simplify-
ing assumption that radiation can be neglected for all
∗Electronicaddress: [e-mail]
†Electronicaddress: [e-mail] redshifts of interest in this paper. That is, we take
‡Electronicaddress: [e-mail] ΩGR(z)+ΩGR(z)=1.
M Λ
2
A. Calculating convergence: general relativity B. Calculating convergence: modified gravity
The convergence, κ, describes the magnification of an As indicated in equation 3, generally in non-GR theo-
image due to lensing. This effect is captured by the ries Φ=Ψ. So, in modified gravity equation 4 becomes:
6
geodesicequationforthedisplacementofaphotontrans-
versetothelineofsight. Inthecosmologicalweaklensing
d2
context of general relativity, this is given by: χθb = (Φ +Ψ ). (10)
dχ2 − ,b ,b
d2
χθb = 2Φ (4) (cid:0) (cid:1)
dχ2 − ,b The convergence then becomes:
(cid:0) (cid:1)
where,bindicatesapartialderivativewithrespecttoθb, 1 χ∞
χ is the radial comoving distance, and χ~θ = (χθ1,χθ2) κMG = dχ 2 Φ(~θ,χ)+Ψ(~θ,χ) g(χ)
4 ∇
is a two-component vector representing on-sky position. Z0 h i
1 0 c
This equation can be integrated to obtain the ‘true’ on- = dx 2 Φ(~θ,x)+Ψ(~θ,x) g(χ(x))
sky position of the light source as a function of the ob- 4Z−∞ H(x)∇ h i
servedon-skyposition. Theconvergenceisthengivenby (11)
takingthetwo-dimensionalon-skyLaplacian( 2)ofthis
∇ where hereafter we will use x = ln(a) instead of χ or
expression:
a, and we have converted the integration measure to x
1 χ∞
κ (~θ)= dχ 2Φ(~θ,χ)g(χ) (5) using dχ = c/ dx, where = aH is the conformal
GR 2Z0 ∇ Hubble facto−r. HNote that xHhere is distinct from the
where g(χ) is the lensing kernel: three-dimensional position variable ~x.
Tocalculatethepowerspectrumoftheconvergenceun-
χ∞ χ
g(χ)=2χ dχ′ 1 W (χ′), (6) der modifications to GR, we follow [12] and perturb our
− χ′ field equations about those of the GR+ΛCDM model.
Zχ (cid:18) (cid:19)
Ourreasoninghereisthatcurrentobservationsonlyper-
W(χ)isthenormalisedredshiftdistributionofthesource
mittheorieswhichcanmatchGR+ΛCDMpredictionsto
galaxies, and χ is the comoving distance at a 0.
∞
→ leading order; we are interested in determining next-to-
We compute the power spectrum of the convergence
leading order corrections that are still permitted. Note
followingcloselythemethodlaidoutin[30]. Inthesmall
that we are building a theory of linear perturbations in
angle approximation, it is straightforwardto find:
model space, which is distinct from spacetime perturba-
Pi,j(l)= 1 d2θ e−i~l·θ~ χ∞dχg (χ) χ∞dχ′ g (χ′) tion theory. We define the perturbations of the qua-
κ 4 i j sistatic functions µ and γ about their GR values using:
Z Z0 Z0
d3k P (k)k4ei~k·[~x−~x′] (7) µ(x,k)=1+δµ(x,k)
× (2π)3 Φ
Z γ(x,k)=1+δγ(x,k). (12)
where ~x labels three-dimensional position such that ~x=
(χθ1,χθ2,χ) and ~x′ = (0,0,χ′). i and j label the source Inaddition,we introduce aperturbationaboutthe stan-
redshift bins to be considered. dard value of the effective equation of state of the non-
Performing the integrals over θ1 and θ2 and then over matter sector, w(x):
k and k , we have:
1 2
Pi,j(l)= 1 χ∞dχ gi(χ) χ∞dχ′ g (χ′) w(x)=−1+β(x), (13)
κ 4 χ2 j
Z0 Z0 and we define the useful related quantity:
× d2kπ3 PΦ sk32+ χl22!k4eik3·[χ−χ′]. (8) u(x)= xβ(x′)dx′. (14)
Z Z0
Finally,the Limber approximation[31,32], validhereon
We now consider how these linear perturbation vari-
l '10 [1], is employed, such that k l, and therefore
k ≈ χl. The small angle limit also m3 ≪eanχs that Pκi,j(l) ≃ farbolmesepqruoaptaigoante3,twhreoucagnhwtoriκte:and hence Pκi,j(ℓ). Firstly,
Pi,j(ℓ),whereℓlabelsanangularmultipole[33]. Wefind:
κ
1
ℓ4 χ∞ g (χ)g (χ) ℓ Φ(x,k)+Ψ(x,k)= 1+ Φ(x,k)
Pi,j(ℓ)= dχ i j P ,χ . (9) γ(x,k)
κ 4 χ6 Φ χ (cid:18) (cid:19)
Z0 (cid:18) (cid:19) (2 δγ(x,k))Φ(x,k). (15)
We have computed here the power spectrum of the con- ≈ −
vergence; that of the shear could be straightforwardly In order to express our results as corrections to
calculated by replacing equation 5 with the appropriate, GR+ΛCDM, we need to relate Φ(x,k) to Φ (x,k).
GR
similar definition. There are two effects to be accounted for. Firstly, the
3
relationship between Φ(x,k) and matter density pertur- where we have defined G (χ(x))= gi(χ(x)).
i χ(x)3
bations can be altered. Secondly, if the field equations There are still two non-GR effects to account for,
are modified, ∆M(x,k) will evolve at a different rate, both originating from the modified expansion history. If
and hence will be displaced from its GR value. To ac- β(x)=0 in equation 13, (x) and χ(x) will scale differ-
count for this we introduce the deviation δ∆(x,k) = ently6with the time variaHble x. Using the expression for
δ∆∆M(x(,xk,)k)i/s∆giGMvRen(xb,yk)th−e f1o.llowIning[12in]teitgrawlaesxpshreoswsinont:hat AδHpp(xe)nd=ixH, w(xe)fi−ndHtGhRa(tx) derived in equation A4 in the
3 x
δ (x,k)= ΩGR(x˜)I(x,x˜)δS (x˜,k)dx˜. (16) 1 1 δ
∆ 2 M f = 1 H
Z−∞ H(x) HGR(x)(cid:18) − HGR(x)(cid:19)
The integrandaboveseparatesinto twoparts: δS (x˜,k), 1 3
f = 1 u(x)(1 ΩGR(x)) , (23)
whichencapsulatesalldeviations fromGR+ΛCDM, and (x) − 2 − M
ΩGR(x˜)I(x, x˜), which is a weighting function containing HGR (cid:18) (cid:19)
M
GR+ΛCDM quantities only. It will be useful for us to and hence
presenttheexplicitformofδSf(x,k)here,derivedin[12]: 0 c δ (x′)
χ(x) 1 H dx′
δS (x,k)=δµ(x,k) δγ(x,k) ≈ (x′) − (x′)
f − Zx HGR (cid:18) HGR (cid:19)
(1 ΩGR) 3 0 c
+ − M 3ΩGR (1+f (x))u(x)+f (x)β(x) . δχ(x) u(x′) 1 ΩGR(x′) dx′ (24)
ΩGMR h M GR GR (17i) ⇒ ≈ 2Zx HGR(x′) (cid:0) − M (cid:1)
where δχ=χ χ .
GR
−
The explicit form of I(x,x˜) can be found in [12]. Thedeviationofχ(x)fromitsGRvaluewillalsoaffect
With these modifications in hand, the parameterised quantities which depend on χ(x), such as G(χ(x)) and
Poisson equation becomes: PGR(ℓ/χ(x)) [47]. We allow for this by expanding these
Φ
in a Taylor series around χ , to first order:
2k2Φ(x,k)=8πGe2xρGR(x)∆ (x,k)(1+δµ(x,k)) GR
− M M
=3 2 (x)ΩGR(x)∆GR(x,k) ℓ ℓ ∂lnP δχ
HGR M M PGR PGR 1+ Φ
×(1+δ∆(x,k))(1+δµ(x,k)) (18) Φ (cid:18)χMG(cid:19)≈ Φ (cid:18)χGR(cid:19) ∂lnχ (cid:12)(cid:12)χGRχGR
where in going from the first to the second line, we have  (cid:12)(cid:12) (25)
usedthefactthatthecombination 2 (x)ΩGR(x)isun- (cid:12)
HGR M
changed by our modifications to the background expan-
G (χ)G (χ) G (χ )G (χ )
sion rate, as shown in Appendix A. Hence, Φ(x,k) is i j ≈ i GR j GR
given in terms of Φ (x,k) by:
GR ∂lnG (χ) ∂lnG (χ) δχ
i j
1+ +
Φ(x,k)≃ΦGR(x,k)(1+δ∆(x,k)+δµ(x,k)). (19) × (cid:18) ∂lnχ ∂lnχ (cid:19)(cid:12)(cid:12)χGRχGR
Combining equations 15 and 19, we now have an ex-  (cid:12)(cid:12) (26)
(cid:12)
pressionforΦ+ΨinmodifiedgravityintermsoftheGR
where δχ is given by equation 24 above. We have now
potential plus perturbative correction factors:
accounted for all modified gravity effects, and these are
Φ(x,k)+Ψ(x,k) Φ (x,k) 2 δγ(x,k) summarised in Table I.
GR
≃ − Finally, it will be more convenient for us to work in
(cid:16)
+2δ∆(x,k)+2δµ(x,k) . (20) terms of Pδ, the matter power spectrum, instead of PΦ.
We do so via the following expression, where for clarity
(cid:17)
So, referring to equation 11, κ becomes: we temporarily omit the label ‘GR’ on all quantities.:
1 0 cg(χ(x)) 4
κMG(~θ)= dx 2 ΦGR(x,k) P (k,x)= 1 9 H(x) Ω2 (x)D(x)2P (x=0,k).
4Z−∞ H(x) ∇ h Φ k44(cid:18) c (cid:19) M δ
(27)
(2+2δµ(x,k) δγ(x,k)+2δ (x,k)) . (21)
∆
× −
i Here D(x) is the usual growth factor of matter pertur-
At this stage, it becomes more convenient to work di-
bations. Inserting equation 27 into equation 25, we find:
rectlywiththepowerspectrumPi,j(ℓ). Thiscanbecom-
κ
puted to linear order in deviations from GR+ΛCDM, in ℓ ℓ
directanalogytothemethodoutlinedfortheGRcasein PΦGR χ ≈PΦGR χ ×
Section IIA. We find: (cid:18) (cid:19) (cid:18) GR(cid:19)
Pi,j(ℓ)= ℓ4 0 dx c G (χ(x))G (χ(x))PGR ℓ ,χ(x) 1 ∂ln(k−4PδGR(x=0,k)) δχ .
κ 4 (x) i j Φ χ(x)  − ∂lnk (cid:12) χGR
Z−∞ H (cid:18) (cid:19) (cid:12)k=ℓ/χGR
(1+2δµ(x,k) δγ(x,k)+2δ∆(x,k)) (22)  (cid:12)(cid:12) (28)
× − (cid:12)
4
Correction Description Equation
Φ(x,k)+Ψ(x,k)≃(1+ 1 )Φ(x,k) Non-unityratio of scalar potentials 15
γ(x,k)
Φ(x,k)≃Φ (x,k)(1+δ (x,k)+δµ(x,k)) Altered Poisson equation 19
GR ∆
1 ≃ 1 1− 3u(x)(1−ΩGR(x)) Altered H(x) 23
H(x) HGR(x)(cid:2) 2 M (cid:3)
χ(x)≃χ (x)+ 3 c u(x) 1−ΩGR(x) dx Altered χ 24
GR 2R HGR(x) (cid:0) M (cid:1)
G (χ)G (χ)≃G (χ )G (χ )1+ ∂lnGi(χ) + ∂lnGj(χ) (cid:12) δχ  Altered G(χ) 26
i j i GR j GR (cid:16) ∂lnχ ∂lnχ (cid:17)(cid:12)(cid:12) χGR
 (cid:12)χGR 
(cid:12)
PGR ℓ ≃PGR ℓ ×1− ∂ln(k−4PδGR(x=0,k))(cid:12) δχ  Altered PGR 28
Φ (cid:16)χ(cid:17) Φ (cid:16)χGR(cid:17) ∂lnk (cid:12)(cid:12) χGR Φ
 (cid:12)k=ℓ/χGR 
(cid:12)
TABLE I: Here we summarise the various corrections to the GR expression for Pi,j(ℓ), including a brief description and the
κ
numberof theequation in which they are introduced.
Drawingtogether,then,equations22,23,26and28,and convergencepowerspectrumundermodificationstogen-
using equation 27, we obtain our final expressionfor the eral relativity:
9 0 g (χ (x))g (χ (x)) ℓ 3 (x) 3
Pi,j(ℓ)= dx i GR j GR PGR D2 (x)HGR ΩGR(x)2 1+ u(x) 1 ΩGR(x)
κ 16Z−∞ χGR(x)2 δ (cid:18)χGR(x)(cid:19) GR c3 M ×" 2 − M
(cid:0) (cid:1)
∂lnG (χ) ∂lnG (χ) ∂ln(PGR(x=0,k)/k4) δχ(x)
+2δµ(x,k) δγ(x,k)+2δ (x,k)+ i + j δ . (29)
∆
− (cid:18) ∂lnχ ∂lnχ − ∂lnk (cid:19)(cid:12)(cid:12)χGRχGR(x)#
(cid:12)
(cid:12)
(cid:12)
Themajoradvantageofequation29isthatitneatlysep- Here we have defined the ‘kernel’ term:
arates the convergence power spectrum into the familiar
GR expression (the non-bracketed quantity) and a cor-
9 g (χ (x))g (χ (x) ℓ
rection factor (the bracketed terms). It is then easy to (x,ℓ)= i GR j GR PGR
pick out contributions from: K 16 χGR(x)2 δ (cid:18)χGR(x)(cid:19)
3 (x)
• themodifiedclusteringproperties(describedbyδµ ×DG2R(x)HGcR3 ΩGMR(x)2, (31)
and δγ),
the modified expansion history (described by β, u and the ‘source’ term:
•
and δχ), and
3
the modified growthrate of matter density pertur- δS (x,ℓ)= u(x) 1 ΩGR(x)
• bations (encapsulated in δ , see equation 16). WL 2 − M
∆
(cid:0) (cid:1) ∂lnG (χ)
i
+2δµ(x,k) δγ(x,k)+2δ (x,k)+
It will be useful for us to write equation 29 in a form ∆
− ∂lnχ
which explicitly highlights the GR expression and the
correction factor: ∂lnG (χ) ∂ln[PGR(x=0,k)/k4] δχ(x)
+ j δ .
0 ∂lnχ − ∂lnk !(cid:12) χGR(x)
Pκi,j(ℓ)= dxK(x,ℓ) 1+δSWL(x,ℓ) . (30) (cid:12)(cid:12)(cid:12)χGR (32)
Z−∞ (cid:12)
(cid:0) (cid:1)
5
III. UNDERSTANDING DEGENERACIES straightforwardalgebraallowsustofindanexpressionof
WITH THE LINEAR RESPONSE APPROACH the form
We have at hand an expression (equation 29) for Pκi,j(ℓ) a=D(ℓ)b (34)
under modifications to general relativity. Let us now in-
where (ℓ) may be a complicated expression, but de-
vestigate what this can teach us about the degeneracies D
pends only on GR+ΛCDM quantities. The degeneracy
between gravitational parameters in weak lensing obser-
direction, we see, depends on ℓ in the weak lensing case.
vations. Note that we restrictourselvesto discussing de-
In order to calculate δPi,j(ℓ), we need to specify W ,
generacies between parameters describing modifications κ i
thenormalisedredshiftdistributionofthesourcegalaxies
to gravity. We do not examine degeneracies between
in the redshift bin i. We select a source number density
gravitationalandcosmologicalparameters,nordowein-
with the following form:
vestigate degeneracies with the galaxy bias. We leave
theInsetqhuisessteicotnisonfo,rwfeutcuornesiwdoerrkt.he case in which δµ and n(z) zαe−(cid:16)zz0(cid:17)β, (35)
∝
δγ are independent of scale, due to the fact that the
and we select α = 2, β = 1.5, and z = z /1.412 where
scale-dependence of these functions is expected to be 0 m
z =0.9 is the median redshift of the survey, mimicking
sub-dominant to their time-dependence [14, 24, 34]. We m
the number density of a Dark Energy Task Force 4 type
will briefly investigate scale-dependence later, in Section
survey [2, 3]. In this section, we will simply consider all
IVC. Additionally, as we are working in the quasistatic
galaxies between z = 0.5 and z = 2.0 to be in a single
approximation,ouranalysisisrestrictedtotheregimeof
redshift bin, with W(χ) given by normalising equation
validity of linear cosmological perturbation theory. Var-
35.
ious values of ℓ which ensure this to be true are sug-
max To break parameter degeneracy in a two parameter
gestedintheliterature(seeforexample[1,2]). Adopting
case,asecondobservablewithadifferent(ideallyorthog-
a conservative approach, we select ℓ = 100 here and
max onal)degeneracydirectionisintroduced. Here,wechoose
for the remainder of this work.
thissecondobservabletoberedshift-spacedistortions,as
We first remind the reader of how degeneracy direc-
it is known to provide nearly orthogonal constraints to
tions may be calculated. Then, using equations 17 and
weaklensing. Wewillthereforeoftenemployresultsfrom
29, we explore how the degeneracy directions of weak
[12]. Particularly, we reproduce here their equation for
lensing and redshift-space distortions in the space of the
the deviation of fσ (x) from its GR value, analogous to
parameters of δµ(x) and δγ(x) are affected by the cho- 8
our equation 33:
sen ansatzes for these functions. Note that here and for
the remainderofthis work,wecompute theGR+ΛCDM fσ (x) fσGR(x) x
matter powerspectrumusing the publicly availablecode δfσ8(x)= 8 fσ−GR(x8) = Gf(x,x˜)δSf(x˜)dx˜
CAMB [35] and using the best-fit ΛCDM parameters of 8 Z−∞
(36)
the 2013 Planck release (including Planck lensing data)
[36].
whereδS (x)isgivenasinourequation17,andG (x,x˜)
f f
isageneralrelativistickernelgiveninequation34of[12].
Note that fσ (x) above is independent of k, because we
8
A. Calculating degeneracy directions
are considering a case where µ and γ are functions of
time only.
Degeneracies exist when an observation can probe only The degeneracy direction of a measurement of fσ (x)
8
some combination of the parameters we wish to con- canthenbecomputedinadirectlyanalogouswaytothat
strain. The degeneracy direction is the relationship described above for weak lensing. The sole difference is
between parameters in the fiducial scenario (here, in thatinsteadofdepending onmultipoleℓ,thedegeneracy
GR+ΛCDM). For example, if this relationship is a = b, direction is dependent on the time of observation, x.
then the relevant observation can probe only a b, not With this information in hand, we now explore de-
−
a or b individually. generacy directions of weak lensing and redshift-space
In the case of weak lensing, degeneracy directions can distortions in the space of the parameters of δµ(x) and
be understood in the following schematic way. First, de- δγ(x).
fine the fractional difference between Pi,j(ℓ) in an alter-
κ
native gravity theory and in GR+ΛCDM:
B. Degeneracy directions in the µ¯ −Σ plane
0 0
Pi,j(ℓ) Pi,j (ℓ)
δPi,j(ℓ)= κ − κ,GR . (33)
κ Pi,j (ℓ) As mentioned above, redshift-space distortions are the
κ,GR
preferred choice of an additional observation to break
Tofindthedegeneracydirection,wefindtherelationship weaklensingdegeneracyinthisscenario. Uponcloserex-
which exists between parameters when δPi,j(ℓ) = 0. If amination, this statement hinges upon the chosen time-
κ
we consider a two parameter case (call them a and b), dependent ansatz for the functions which parameterise
6
deviations from GR. As there is no clear front-runner Wenowconsidertwoansatzesforµ¯(x)andΣ(x). First,
amongst alternative theories of gravity, typically a phe- consider a phenomenological ansatz, for which we know
nomenologicalansatzischosen,inwhichdeviationsfrom weaklensingandredshift-spacedistortionstobeaneffec-
GR become manifest at late times in order to mimic ac- tive combination in constraining gravity theories. This
celeratedexpansion. Itisforthistypeofphenomenologi- choice is a specific case of the form proposed in [37] and
calansatzthatredshift-spacedistortionandweaklensing has been used in, for example, [9]. It is given by:
observations are known to provide complimentary con-
ΩGR(x)
straints [9]. µ¯(x)=µ¯ Λ
However,it may also be desirable to constrain the pa- 0ΩGR(x=0)
Λ
rameters of a specific theory of gravity. The functions ΩGR(x)
which parameterise the deviation of an alternative grav- Σ(x)=Σ0ΩGRΛ(x=0) (39)
itytheoryfromGRcan,inprinciple,takeonawiderange Λ
of time-dependencies. Is the combination of weak lens- where ΩGR(x) is the time-dependent energy density of
Λ
ingandredshift-spacedistortionsstillaneffectivewayto dark energy in the fiducial ΛCDM cosmology.
break degeneracies and constrain the parameters of the WeinsertδS (x)(equation38)intoequations33and
WL
theory we consider? A priori, this is unknown. 36 with our chosen µ¯(x) and Σ(x). We then follow the
To explore this issue, we consider now the degeneracy proceduresketchedinSectionIIIAtofindthedegeneracy
directions of weak lensing and redshift-space distortions directionsofweaklensingandredshift-spacedistortionin
under two different ansatzes for the functions which pa- the µ¯ Σ plane. Inthis particularcasethe degeneracy
0 0
−
rameterise deviations from GR. For this section only, we direction of redshift-space distortions does not depend
make the simplifying assumption that β(x)=0 (i.e. the on time. This is because δS (x) is dependent on only
f
expansion history is ΛCDM-like). We expect that the one parameter, µ¯ , and therefore the only degeneracy
0
effect of this assumption on our qualitative findings will direction is µ¯ =0.
0
be small. The degeneracy directions for this ansatz can be seen
First, we perform a simple operation on δµ(x) and in Figure 1 (left). In the case of weak lensing, we have
δγ(x) to obtain a more observationally-motivated set of plottedthedegeneracydirectionforℓ=50;directionsfor
functions. Let us call these µ¯(x) and Σ(x), in keeping other multipoles ℓ=10 100 differ only within 5%. We
−
insofar as possible with the notation used in [9]. The seethat, indeed, the degeneracydirectionsarenearlyor-
choice of this set of functions allows nearly orthogonal thogonal,withonlyaslightcorrectionoftheweaklensing
constraints in the µ¯0−Σ0 plane for the phenomenologi- degeneracy direction away from Σ0 =0.
calchoiceoftime-dependence. Themappingbetweenthe Now,considerselectinganansatzwithaverydifferent
two sets of functions, as shown in Appendix B, is given time-dependence. To guide our selection, recall that we
by: expect choices of µ¯(x) which persist over longer times to
1 result in a greater value of the integral term in equation
Σ(x)=δµ(x) δγ(x) 38, and hence a greater deviation of the weak lensing
− 2
degeneracy direction from Σ = 0. Therefore, with no
µ¯(x)=δµ(x) δγ(x). (37) 0
− attempt to correspond to any particular gravity theory,
Wecanrewritethelinearresponse‘source’termsforboth we selectthe simplest possible choicewhichpersists over
weaklensing(equation32)andredshift-spacedistortions long times: constant µ¯(x) and Σ(x).
(equation36)intermsofµ¯(x) andΣ(x)(inthe β(x)=0
Σ(x)=Σ
case): 0
µ¯(x)=µ¯ . (40)
x 0
δS (x)=2Σ(x)+3 ΩGR(x˜)I(x,x˜)µ¯(x˜)dx˜
WL M In reality, we use step functions beginning at z = 15
Z−∞ rather than true constants to allow for the numerical
δS (x)=µ¯(x). (38)
f computation of the degeneracy directions. The degener-
We see that δS (x) depends solely on µ¯(x). acydirectionsarecalculatedasbefore,andareplottedin
f
The expression for δS (x) requires slightly more Figure1(right). Clearly,theyarelessorthogonalthanin
WL
pause. It depends on Σ(x), but it also contains another thepreviouscase,asexpectedfromthecommentsabove.
term, which comprises an integral over µ¯(x) and some What does this example tell us about the effective-
general relativistic quantities. By comparing with equa- ness of combining weak lensing and redshift-space dis-
tion16,wecaneasilyrecognisethistermas2δ (x). This tortions? The ansatz for µ¯(x) and Σ(x) given by equa-
∆
term quantifies a correction to the degeneracy direction tion 40 deviates from GR+ΛCDM at all times after
of weak lensing away from Σ = 0. It is clearly depen- z = 15. As mentioned above, most cosmologically-
0
dentupontheansatzoftime-dependencechosenforµ¯(x). motivated alternative theories of gravity present devi-
Particularly, we note that due to the integral nature of ations from GR+ΛCDM at late times only, mimicking
the correction term, choices of µ¯(x) which persist signif- accelerated expansion. Therefore, we treat the case of
icantly overlongertimes will result in greaterdeviations equation 40 as a heuristic ‘upper bound’ on the cumula-
to the degeneracy direction. tive effect produced by the integral term of equation 38.
7
The effect of this term can be quantified by considering where n¯ is the number density of galaxies per steradian
i
the angle of the weak lensing degeneracy direction with inbiniand γ2 isthermsintrinsicshear,equalto0.22
h inti
respect to the vertical. We find that for the range of ℓ for a DETF4-type survey.
which we consider and for the ansatz given by equation Computing the appropriate value of n¯ requires the
i
40, the maximum possible value of this angle is θ 50◦. selectionofsourceredshiftbins. Inpractice,onceinpos-
≈
Although the degeneracy directions in this case are cer- sessionofdata,theselectedbinsarethosewhicharemax-
tainlynolongerorthogonal(θ =0◦),theyaresufficiently imal in number while maintaining shot noise sufficiently
distinct that we expect the resulting constraints to be below the signal. For our forecasting purposes, we in-
reasonable (if not ideal). We have therefore shown that stead follow, for example, [2] and [3]. We select redshift
theeffectivenessofcombiningweaklensingwithredshift- bins by subdividing n(z) of equation 35 into 5 sectors,
spacedistortionsinthe β(x)=0caseisrelativelyrobust such that the number of galaxies in each bin is equal.
to the chosen form of µ¯(x) and Σ(x). The value of n¯ for the total redshift range for a DETF4-
type survey is given by n¯ = 3.55 108, so the value in
×
each tomographic bin is simply n¯ =n¯/5.
i
IV. FORECAST CONSTRAINTS FROM
FUTURE SURVEYS Inthe followingsubsections,we use the Fisher formal-
ism to compute forecast constraints in a number of sce-
narios. We first consider constraints on the parameters
In addition to providing an understanding of degenera-
ofδµ(x)andδγ(x)inthecasewherewefixtheexpansion
cies, our expression for Pi,j(ℓ) enables the forecasting of
κ history to mimic ΛCDM. We then incorporate expected
constraints. Thestraightforwardformofequation30ren-
measurementsofw andw fromBaryonAcousticOscil-
ders the calculation of Fisher matrices very simple, and 0 a
lationstoforecastconstraintsontheparametersofδµ(x)
clarifies the interpretation of the resulting forecasts. We
and δγ(x) in the case where we marginalise over the pa-
takeadvantageofthesefeaturestoforecastconstraintson
rameters of β(x). We finish by discussing the directions
gravitational parameters for a Dark Energy Task Force
ofbestconstraintinthe parameterspaceofthe scalede-
4 (DETF4) type survey, as defined in the classification
pendent ansatz for µ(x,k) and γ(x,k) put forth in [14].
of [38]. We focus on combined constraints from weak
lensing and redshift-space distortions,with some consid-
eration given as well to baryon acoustic oscillations.
As mentioned above, the forecasts presented here em-
ploythetechniqueofFisherforecasting(see,forexample, A. ΛCDM-like expansion history: β(x)=0
[39]). The key quantity of this method is the Fisher in-
formation matrix:
We first consider constraints on the parameters of δµ(x)
∂2ln andδγ(x)inthecasewheretheexpansionhistoryisfixed
= L (41)
ab tobeΛCDM-like. Asinequation37,wetransformδµ(x)
F − ∂p ∂p
a b
D E and δγ(x) to µ¯(x) and Σ(x), and we choose the time-
where p are the relevant parameters, and is the like-
i dependence given by equation 39.
L
lihood. For redshift-space distortions, we straightfor-
To compute these constraints, we calculate the 2 2
wardlybuildontheresultsof[12]toconstructtheappro- ×
Fisher matrix for lensing, for redshift-space distortions,
priate Fisher matrix. However, for weak lensing we re-
andforbothobservationscombined. Forthis,werequire
quire a slightly different expression. Although our equa-
expressions for the derivatives ∂fσ8(x), ∂fσ8(x) and
tion 29 allows for the cross-correlation of source galaxy ∂µ¯0 ∂Σ¯0
redshift bins, we have until now considered only a sin-
gle wide redshift bin. In practice, weak lensing data are ∂ γ2 δ ∂Pi,j(ℓ)
normallyconsideredinanumberoftomographicredshift Pi,j(ℓ)+ h inti ij = κ (44)
∂µ¯ κ n¯ ∂µ¯
bins. In [40], the Fisher matrix for such a situation is 0 (cid:18) i (cid:19) 0
shown to be given by: ∂ Pi,j(ℓ)+ hγi2ntiδij = ∂Pκi,j(ℓ) (45)
∂Σ κ n¯ ∂Σ
ℓmax 1 0 (cid:18) i (cid:19) 0
= ℓ+ f Tr C−1C C−1C (42)
Fab 2 sky GR ,a GR ,b These are found in a straightforwardmanner from equa-
ℓ=Xℓmin(cid:18) (cid:19) (cid:2) (cid:3) tions 29 and 36; we present them in Appendix C.
where ,a is a derivative with respect to p , f is re-
a sky TheresultingforecastconstraintsareillustratedinFig-
lated to the fraction of the sky observed (f = 0.375
sky ure 2. As discussed in Section III, the degeneracy direc-
for a DETF4-type survey), and C is an N N matrix
b× b tionsofthe twoobservablesarenearlyorthogonalinthis
where N is the number of tomographic redshift bins. C
b case. Combining them results in promising forecast con-
represents the observed power spectrum of the conver-
straints on µ¯ and Σ . We see that we can expect a
gence, and is given by the following expression [40]: 0 0
DETF4-type survey to provide constraints at a level of
γ2 δ approximately 4% in this plane, in the case where β(x)
Ci,j(ℓ)=Pi,j(ℓ)+ h inti ij (43)
κ n¯ is assumed to be fixed at 0.
i
8
0.15 0.15
0.10 0.10
0.05 0.05
0 0.00 0 0.00
¯ ¯
(cid:2) (cid:5)
0.05 0.05
(cid:0) (cid:3)
0.10 0.10
(cid:0) (cid:3)
0.15 0.15
(cid:0) (cid:3)
0.15 0.10 0.050.00 0.05 0.10 0.15 0.15 0.10 0.050.00 0.05 0.10 0.15
(cid:0) (cid:0) (cid:0) (cid:3) (cid:3) (cid:3)
0 0
(cid:1) (cid:4)
FIG. 1: Degeneracy directions of weak lensing (ℓ=50, dashed red) and redshift-space distortions (solid green) in theµ¯ −Σ
0 0
plane, where µ¯(x) and Σ(x) scale as ΩGR(x) (left), and as constants (right).
Λ
are expected to provide the best constraints on the ex-
pansion history of the universe.
0.2
In this section, we use a CPL-type ansatz for β(x) as
proposed in [41, 42]: β(x) = w +1+w (1 ex). We
0 a
−
incorporate forecast BAO constraints on w and w and
0 a
0.1 use these to obtain expected constraints in the µ¯ Σ
0 0
−
plane. We first marginalise over only w , while holding
0
w to its fiducial value of 0; then we examine the effect
a
¯0 0.0 of allowing wa to vary as well.
(cid:8)
1. Marginalising over w ; w =0
0.1 0 a
(cid:6)
Wefirstdemonstratehowconstraintsintheµ¯ Σ plane
0 0
−
are affected by marginalising over w when w is held
0 a
0.2
fixed to its fiducial value of 0.
(cid:6)
Because we are now incorporating information about
0.2 0.1 0.0 0.1 0.2 three parameters (µ¯ , Σ and w ), our Fisher matrices
0 0 0
(cid:6) (cid:6)
are 3 3 in dimension. In order to compute these, we
(cid:7)0 now re×quire additional derivatives of Pi,j(ℓ) and fσ (x)
κ 8
withrespecttow ;allarelistedinAppendix C.Because
0
FIG. 2: Forecast constraints for weak lensing (orange),
transversemeasurementsofBAOareindependentofnon-
redshift-space distortions (green) and both observables com-
background gravitational effects [43], the Fisher matrix
bined (blue) for a DETF4-type survey, in the µ¯ −Σ plane
0 0 ofBAOisnon-zeroonlyinthe(w ,w )component. The
withβ(x)fixedto0. Contoursrepresentthe68.3%and95.4% 0 0
valueofthismatrixcomponentisequalto 1 ,where
confidenceregions. σ2
w0,BAO
σ isthe1-σerroronw fromBAOmeasurements.
w0,BAO 0
To explorethe effectof marginalisingoverw , we con-
0
sider three levels of constraint from BAO:
B. The effect of marginalising over {w ,w }
0 a
1. For comparison: the case where w is fixed to its
0
fiducial value. This is identical to the case consid-
In reality, β(x) is not fixed to zero, but rather the as- ered in Section IVA.
sociated parameters will also be constrained with some
non-zero error. While weak lensing and redshift-space 2. Thecasewhereσ =1%. Thisscenariomim-
w0,BAO
distortions will provide some constraints on these, it is ics best-case constraints from a DETF4-type sur-
baryon acoustic oscillation (BAO) measurements which vey.
9
3. The case where σ = 5%. This lies between Finally, we note that there is clearly a directionin the
w0,BAO
current best constraints and scenario 2 above. µ¯ Σ planewhichisentirelyinsensitivetothechangein
0 0
−
w . Thisisinfactexpectedduetothenatureofthecon-
The resulting constraints from the combination of weak 0
tours displayed. Given the hypothetical 3D confidence
lensing, BAO, and redshift-space distortions are shown
region in the space of µ¯ , Σ and w , the marginalised
inFigure3andFigure4. Figure3showsintheleft-hand 0 0 0
constraintofscenario3isequivalenttoprojectingthisel-
panelthe forecastconstraintsonµ¯ forcases1 3above
whenmarginalisingoverw andΣ0;theright-h−andpanel lipsoidintotheµ¯0−Σ0 plane. Whenwereducetheerror
0 0 in only the w direction as in scenario 2 – that is, re-
displaysthesameforΣ whenmarginalisingoverw and 0
0 0 ducing the errorin the direction orthogonalto the plane
µ¯ . Figure 4 shows the 68.3% forecast joint constraints
0 of projection – the resulting projection will, by simple
onµ¯ Σ inscenarios1 3while marginalisingoverw
0− 0 − 0 geometrical considerations, coincide with the first pro-
only.
jection in two locations. The same argument can then
We note from Figure 4 that the degeneracy direction
be extended to the case of fixed w , which involves sim-
of the combined constraint in the µ¯ Σ plane changes 0
0− 0 ply taking a slice of the 3D ellipsoid at the location of
considerably between the three scenarios. µ¯ and Σ are
0 0 the µ¯ Σ plane.
mildlynegativelycorrelatedinscenario1,whereasinsce- 0− 0
nario 2 they are positively correlated, and in scenario 3
even more so. This can be understood by considering
2. Marginalising over {w ,w }
0 a
the jointforecastconstraintsinthe µ¯ w and Σ w
0 0 0 0
− −
planes, marginalised in each case over the other non-w
a We now consider the case where we do not fix w to
parameter. These are displayed at a 68.3% level in Fig- a
zero. Inthis scenario,there is informationpresentabout
ure 5 for scenario 3. Both µ¯ and Σ are shown therein
0 0 4 parameters (µ ,Σ ,w ,w ), so all Fisher matrices are
to exhibit a positive correlation with w . This implies 0 0 0 a
0 4 4. In addition to the previous derivative expressions,
that µ¯0 and Σ0 are also positively correlated with each w×enowneedderivativeswithrespecttow ofPi,j(ℓ)and
other,exceptinthecasewherew isfixedorconstrained a κ
0 fσ (x). Once again, these are computed from equations
sotightlythatthiseffectisnegated. Astheconstrainton 8
29and36,andlistedinAppendixC.Inthisscenario,the
w is loosened, moving from scenario 1 through scenario
0 BAO Fisher matrix is slightly more complicated, as the
2 to scenario 3, this positive correlation becomes more
entire 2 2 block related to w and w is non-zero.
pronounced. × 0 a
In analogy to the above, we consider three scenarios:
WenoticealsofromFigure4thattheconstraintonΣ
0
is relativelyinsensitiveto the levelofBAO constrainton 1. The scenario where w and w are fixed to their
0 a
w0, whereas the constraint on µ¯0 changes considerably fiducial values. Again, this for comparison, and is
between scenarios 1 3. This is consistent with Figure identical to the case considered in Section IVA.
−
5, in which we see that the degeneracy direction in the
µ¯ w plane has a far greater positive slope than that 2. The scenario where the BAO Fisher matrix rep-
0 0
in t−he Σ w plane. These degeneracy directions, and resents the best-case expected constraints from a
0 0
hence the−relative sensitivity of µ¯ and Σ constraints DETF4-type survey. In this scenario, the compo-
0 0
to w constraints, can be understood by considering the nents of the BAO-only covariance matrix (the in-
0
expressions for Pκi,j(ℓ) (equation 29) and δfσ8(x) (equa- verse of the Fisher matrix) are given by: Cw0,w0 =
tions 17 and 36). Both Pκi,j(ℓ) and δfσ8(x) are given by 0.0010, Cwa,w0 = −0.0038, and Cwa,wa = 0.016
integrals in time over a kernel and a source term. In the [44].
case of δfσ (x), the general relativistic kernel G (x,x˜)
8 f 3. The scenario where the BAO-only covariance ma-
issignificantbacktoz 15,whereasinthe weaklensing
≃ trix is obtained by multiplying the covariance ma-
case,thekernelisnon-zeroonlyasfarbackinredshiftas
trixlistedaboveinscenario2byanoverallfactorof
the furthest source galaxies (z = 2 in this case). In the
(8.2)2. Thiscorrespondstothecasewherethepro-
current model of β(x), deviations from a ΛCDM expan-
jected 68.3% error on w from BAO is 5% and all
0
sion history are more significant at early times, whereas
other elements of the covariance matrix are scaled
µ¯(x) and Σ(x) are both chosen to be significant only at
up accordingly.
latetimes(belowz 5). Therefore,theδfσ (x)integra-
8
≃
tion from z 15 favours sensitivity to the background The left-handpanelofFigure 6 presentsthe combined
≃
expansionvariable w over µ¯ , whereas the weak lensing weaklensing,redshift-spacedistortionandBAOforecast
0 0
integral, significant only from z 2, results in relatively constraints on µ¯ while marginalising over w , w and
0 0 a
≃
greatersensitivitytoΣ . This resultsinthe relativesen- Σ ; the right-hand panel does the same for constraints
0 0
sitivity of the µ¯ constraint to the w constraint level, onΣ while marginalisingoverw , w and µ¯ . Figure 7,
0 0 0 0 a 0
as seen in Figure 4. Note that we have not accounted meanwhile, presents the 68.3% confidence regions in the
here for any uncertainty in galaxy bias models at high µ¯ Σ plane while marginalising over w and w .
0 0 0 a
−
redshifts, which may have significant effects on the sen- We see that the forecast constraint on Σ is now
0
sitivity of fσ (x) to the background expansion at early slightly more sensitive to the level of BAO constraint
8
times. on w and w than in the above case where w is fixed.
0 a a
10
50
40
30
20
10
0
0.4 0.3 0.2 0.10.0 0.1 0.2 0.3 0.4 0.10 0.05 0.00 0.05 0.10
(cid:9) (cid:9) (cid:9) (cid:9) ¯ (cid:9) (cid:9)
(cid:10)0 (cid:11)0
FIG.3: Forecastconstraintsfromweaklensing,redshift-spacedistortions,andBAOinthecasewherew hasbeenmarginalised
0
over and w has been fixed to 0. The left-hand panel shows the confidence region for µ¯ when Σ is marginalised over, while
a 0 0
the right-hand panel shows the confidence region for Σ with µ¯ marginalised over. Black, solid: w fixed; red, dashed: BAO
0 0 0
error on w =1% (DETF4); green, dotted: BAO error on w =5%.
0 0
than for scenario 1, which is the same in both figures by
0.15
design).
We surmise that allowing for a time-dependence in
0.10
the equation of state of the effective dark energy com-
ponent(via β(x)) loosens the expected constraints on µ¯
0
0.05
and Σ , but not catastrophically so. In fact, the level
0
ofconstraintprovidedby BAO measurementson the ex-
¯0 0.00 pansionhistoryofthe universeappearstohaveagreater
(cid:14)
effect on forecast constraints in the µ¯ Σ plane than
0 0
0.05 −
(cid:12) does our assumption regarding the time-dependence of
that expansion history.
0.10
(cid:12)
0.15
(cid:12) C. Scale-dependent µ(x,k) and γ(x,k)
0.15 0.10 0.050.00 0.05 0.10 0.15
(cid:12) (cid:12) (cid:12)
(cid:13)0 Untilthispoint,wehaveneglectedanyscale-dependence
ofµ(x,k)andγ(x,k),focusingonlyontime-dependence.
FIG. 4: Forecast 68.3% confidence regions in the µ¯0 −Σ0 We now consider a scale-dependent ansatz.
plane, marginalising over w , for the case where w = 0.
0 a It has been shown that in the quasistatic regime and
Black, solid: w fixed; red, dashed: BAO error on w = 1%
0 0 for local theories of gravity, µ(x,k) and γ(x,k) can be
(DETF4); green, dotted: BAO error on w =5%.
0 expressed as a ratio of polynomials in k with a specific
form [14]:
p (x)+p (x)k2
Thisisparticularlynoticeableinscenario3,inwhichthe γ(x,k) 1 2
≃ 1+p (x)k2
expansion history is the least well-constrained. Turning 3
to µ¯ , we see from Figure 6 that the forecast constraint 1+p (x)k2
0 µ(x,k) 3 . (46)
remains sensitive to our knowledge of the expansion his- ≃ p (x)+p (x)k2
4 5
toryinmuchthe same wayasin the w fixedcase. That
a
is, the constraint in scenario 3 is broadened consider- This form has recently been considered in [24], in which
ably relative to that in scenario 2, and both are slightly a Principle Component Analysis was undertaken for a
broader than in the above case where w fixed. Finally, combined future data set including weak lensing and
a
examiningthecombinedplotinFigure7,weseethatthe galaxycountmeasurementsfromtheLargeSynopticSur-
confidence regions therein are slightly larger than those vey Telescope (LSST), as well as Planck measurements
in the corresponding Figure 4, where w is fixed (other andupcomingsupernovadata. Therein,theprimarygoal
a