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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 convergence in general relativity, and then generalise to alternative theories of gravity. We make the simplifying assumption that radiation can be neglected for all ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] redshifts of interest in this paper. That is, we take ‡Electronicaddress: [email protected] 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 scenarios. 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 information 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 single 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 expansion 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 component(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