Constraints on f(R) and nDGP Modified Gravity Model Parameters with Cluster Abundances and Galaxy Clustering
Rayne Liu, Georgios Valogiannis, Nicholas Battaglia, Rachel Bean
CConstraints on f ( R ) and nDGP Modified Gravity Model Parameters with ClusterAbundances and Galaxy Clustering Rayne Liu, ∗ Georgios Valogiannis, ∗ Nicholas Battaglia, and Rachel Bean Department of Astronomy, Cornell University, Ithaca, NY 14853, USA Department of Physics, Harvard University, Cambridge, MA 02138, USA (Dated: January 22, 2021)We present forecasted cosmological constraints from combined measurements of galaxy clusterabundances from the Simons Observatory and galaxy clustering from a DESI-like experiment ontwo well-studied modified gravity models, the chameleon-screened f ( R ) Hu-Sawicki model and thenDGP braneworld Vainshtein model.A Fisher analysis is conducted using σ constraints derived from thermal Sunyaev-Zel’dovich(tSZ) selected galaxy clusters, as well as linear and mildly non-linear redshift-space 2-point galaxycorrelation functions. We find that the cluster abundances drive the constraints on the nDGP modelwhile f ( R ) constraints are led by galaxy clustering. The two tracers of the cosmological gravitationalfield are found to be complementary, and their combination significantly improves constraints onthe f ( R ) in particular in comparison to each individual tracer alone.For a fiducial model of f ( R ) with log ( f R ) = − n = 1 we find combined constraintsof σ (log ( f R )) = 0 .
48 and σ ( n ) = 2 .
3, while for the nDGP model with n nDGP = 1 we find σ ( n nDGP ) = 0 . f ( R ) of f R ≤ . × − . Our results present the exciting potential to utilizeupcoming galaxy and CMB survey data available in the near future to discern and/or constraincosmic deviations from GR. I. INTRODUCTION
The ΛCDM model accredits the acceleration of cosmicexpansion [1–9] to the negative pressure exerted by anunknown dark energy, either as a cosmological constantΛ with a canonical equation of state w = −
1, or as avariable scalar field known as quintessence [10–12]. How-ever, more direct evidence for of the underlying nature ofdark energy remains absent. The cosmological constantexplanation suffers from stark incompatibility since thevalue inferred from astronomical observations is ∼
120 or-ders of magnitude greater than that predicted in particlephysics; the quintessence field theories that attempt toresolve the discrepancy face subsequent fine-tuning prob-lems [13].Modified gravity (MG) theories attempt to avoid thisextra energy component by explaining the accelerat-ing universe with altering the standard theory of grav-ity, namely Einstein’s General Relativity (GR), in largescales [14–16]. While GR has been meticulously testedwith astrophysics on smaller scales, such the solar sys-tem tests [17] and strong-gravity tests via gravitationalwaves [18, 19], MG can potentially be applicable tolarger cosmic scales with relatively weak gravitationalfields. Nevertheless, such remarkable tests of GR onsmall scales have already imposed stringent constraints[20–25], leaving a limited parameter space for mostMG models. Two particularly well-studied MG mod-els that survive are the Hu-Sawicki f ( R ) model se-ries [26], which feature a Chameleon mechanism, and ∗ Equal contribution. the normal-branch Dvali-Gabadadze-Porrati braneworldmodel (nDGP) [27], which introduces a fifth dimensionalforce (Vainshtein mechanism). They successfully evadethe above small-scale tests, while also reproducing anexpansion history indistinguishable from ΛCDM. Hence,constraints via other independent observational quanti-ties, especially the growth of the cosmic large-scale struc-ture (LSS), are of crucial importance [28]. Complemen-tary to constraints via geometric distance measurementsof the expansion history, LSS growth are very sensitiveto the phenomenology of the cosmological MG models ofinterest [28–30].Current and future LSS surveys will measure the abun-dance of galaxy clusters, as well as the 3-dimensional po-sitions and velocities of galaxy halos. Such measurementsare powerful probes of the LSS growth and clustering,and subsequently the nature of gravity and dark energy.In this work, we explore the constraining power of clusterabundances from upcoming observations of the thermalSunyaev-Zel’dovich (tSZ) effect by the Simons Observa-tory [31] and galaxy clustering from spectroscopic ob-servations by the Dark Energy Spectroscopic Instrument(DESI) [32] , both when considered independently andcombined with each other.Galaxy clusters have long been regarded a promisingset of observables to test MG, and their abundances rep-resented as number counts, as well as mass profiles, bothserve as powerful tools. Potential constraints on MG us-ing a wide variety of signal types have been considered,including X-rays [33–36], the tSZ effect [34, 37–39], andweak lensing [40, 41]. In this work, the constraints from a r X i v : . [ a s t r o - ph . C O ] J a n abundances of galaxy clusters over the large linear scalesis inferred through constraints on measurements of σ ,the mean amplitude of matter energy density fluctua-tions. We base these constraints on the forecasted weak-lensing and CMB-halo lensing calibrated tSZ galaxy clus-ter abundances in [39, 40].Meanwhile, mapping out the three-dimensional clus-tering of galaxies across the cosmic history offers anotherwindow into the underlying physical processes, includ-ing the gravity models, that shaped the LSS. Buildingupon the legacy of the recently completed analysis bythe Extended Baryon Oscillation Spectroscopic Survey(eBOSS) [42–44], DESI is expected to constrain the prop-erties of gravity and dark energy at unprecedented levelsof accuracy [45], in combination also with the next gen-eration of cosmological surveys, such as Euclid [46], theVera C. Rubin Observatory Legacy Survey of Space andTime (LSST) [47, 48], the Nancy Grace Roman SpaceTelescope [7] and SPHEREx [49]. A necessary require-ment for the optimal interpretation of this upcomingwealth of observational data is the ability to reliablymodel the clustering statistics in the variety of compet-ing scenarios, in our case the landscape of MG models.When galaxies are identified through spectroscopic mea-surements, in particular, one needs to take into accountnot only the non-linear growth of structure, but also thefact that galaxy peculiar velocities induce an observedanisotropy in the clustering pattern, the Redshift-SpaceDistortions (RSD) [50–52]. In this study we capture theseeffects closely following the recent work of [53], that em-ployed the Gaussian Streaming Model (GSM) approach[54–56] with Lagrangian Perturbation Theory [57–65] inthe context of MG [66–68], in order to successfully modelthe multipoles of the anisotropic correlation function ofhalos in theories beyond GR.We forecast individual and joint constraints on MGfrom these two probes using Fisher analysis. In addi-tion to obtaining model-dependent constraints througha state-of-the-art treatment of the galaxy bias and RSDeffects in MG, this work explores tests of gravity throughthe combination of these two complementary and promis-ing probes of the LSS. Our paper is structured as follows:first, in Section II, we outline our theoretical and observa-tional assumptions. We then present our analysis resultsin Section III, with a subsequent discussion including im-plications for future work in Section IV. The details ofthe particular model used to obtain the galaxy clusteringcovariance matrices are presented in Appendix A. II. FORMALISMA. Modified Gravity Models
We focus on two quintessential models in the literatureof MG, the Hu-Sawicki f ( R ) and the nDGP braneworldmodels, which respectively realize the Chameleon andVainshtein classes of screening. f ( R ) Hu-Sawicki model
In the Hu-Sawicki f ( R ) model, a nonlinear modifica-tion function, f ( R ), of the Ricci scalar is added to thestandard Einstein-Hilbert action: S = (cid:90) d x √− g (cid:20) R + f ( R )16 πG + L m (cid:21) , (1)where G is the Newtonian gravitational constant, L m thematter Lagrangian, and f ( R ) induces the acceleratinguniverse instead of a cosmological constant Λ. Througha conformal transformation, (1) as the Einstein frameexpression can be cast into the form of a scalar-tensortheory with the scalaron, f R ≡ df ( R ) dR , acting as the MG-induced degree of freedom [69]. Imposing an expansionhistory identical to ΛCDM in the high curvature limit, apresent day value of the scalaron can be obtained as [53] f R = − n c c (cid:18) Ω m m + Ω Λ0 ) (cid:19) n +1 , (2)where Ω m and Ω Λ0 are the normalized density parame-ters for nonrelativistic mass and dark energy today; theyas ΛCDM parameters appeared in (2) as a result of ourassumption on the expansion history. Instead of using c /c as the free parameter in (2), the pair of f R (typi-cally | f R | in the literature, and for the rest of this paperwe only consider f R >
0) and n are commonly used. Werecover the ΛCDM (GR) model when f R →
0, which isthe case in regions of high Newtonian potential, wherethe chameleon field becomes very massive due to the ef-fect of the screening mechanism [70, 71].Extensive studies of the Hu-Sawicki model in the pastdecade have tightened constraints on the available pa-rameter space of the model [72, 73], but have still left themodel observationally viable and theoretically attractive,as it is devoid of instabilities [74]. For these reasons, itserves as the ideal test bed for us to explore constraintson MG with upcoming surveys of the LSS and CMB.
2. nDGP model
The Dvali-Gabadadze-Porrati (DGP) model is a repre-sentative example of the Vainshtein screening mechanism[75, 76], and features a modification to gravity due to alarge extra fifth dimension of spacetime. The modifiedEinstein-Hilbert action is in this case S = (cid:90) d x √− g (cid:20) R πG + L m (cid:21) + (cid:90) d x √− g R πGr c , (3)where R and g denote respectively the correspond-ing Ricci scalar and metric determinant of the fifth di-mension, and r c the cross-over distance, a characteristicscale below which GR model becomes four-dimensional.A more appealing self-accelerating DGP model branch(sDGP), which requires no dark energy, has been shownto suffer from undesirable instabilities [77], hence westudy the “normal” branch (nDGP) coupled with a darkenergy component to match the desired ΛCDM expan-sion history, which still remains interesting due to priorsimulation investments. In this case, the only free pa-rameter to constrain is n = H r c ( H being the Hubbleconstant), of which the extensively studied values are 1and 5. GR is recovered when n → ∞ , corresponding tothe presence of large gradients of gravitational forces inVainshtein screening. B. Cluster Abundances and σ The constraints by cluster abundances are modeled af-ter results obtained from [40], in which a Fisher forecastbased on tSZ-selected galaxy clusters from a CMB-S4-like experiment is extended to model-independent con-straints on the time-evolution of σ ( z ). Specifically, weuse forecast errors on σ ( z ) from the predicted tSZ clus-ter abundances and include mass calibrations from opti-cal weak-lensing and CMB-halo lensing while marginal-izing over ΛCDM and cluster nuisance parameters [31]. σ is the amplitude of matter energy density fluctuationssmoothed out over a scale R = 8 M pc/h , and its evolutionover redshift z is a promising probe of structure growthin the linear density perturbation regime.To predict σ ( z ) from MG models, we calculate σ through the standard deviation of the probability den-sity function of the matter density fluctuations, convo-luted with a spherical top-hat window function W ( r , R )with radius R : W ( r , R ) = 14 πR / (cid:40) , | r | ≤ R, , | r | > R. (4)Fourier transforming, Parseval’s theorem gives σ R ( z ) = (cid:90) ∞ P ( k, z )2 π (cid:20) j ( kR ) kR (cid:21) k dk, (5)where P ( k, z ) is the matter power spectrum at wavenum-ber k and redshift z , j ( kR ) is the spherical Bessel func-tion of the first kind, and 3 j ( kR ) / ( kR ) is the Fouriertransform of the window function.In general, the power spectrum in MG can be obtainedfrom the ΛCDM one by considering the modifications tothe linear growth factor D ( k, z ): P MG ( k, z ) = P ΛCDM ( k, z = 0) · (cid:18) D MG ( k, z ) D ΛCDM ( z = 0) (cid:19) . (6)The growth factors, more commonly expressed as D ( k, a )( a = 1 / (1 + z ) being the scale-factor), are obtained bysolving the modified linear density evolution equations,extracted from the work of [78]:¨ D + 2 H ˙ D − πGρ m (1 + g eff ) D = 0 , (7) where H is the Hubble parameter, ρ m is the non-relativistic matter density, and dots are derivatives withrespect to time t . The effective gravitational factor g eff for f ( R ) [79] is g eff = k k + a m ( a )) (8)with the associated mass term m ( a ) = 1 c (cid:115) [Ω m, + 4(1 − Ω m, )] − ( n +1) ( n + 1) | f R | × (cid:115)(cid:20) Ω m, a + 4(1 − Ω m, ) (cid:21) n +2 , (9)and for nDGP, g eff = 13 β ( a ) (10)where β ( a ) = 1 + 2 Hr c (cid:32) H H (cid:33) = 1 + 2 HH n (cid:32) H H (cid:33) . (11)For more direct comparison with the standard ΛCDMmodel, we evaluate σ /σ . We obtain theΛCDM linear matter power spectrum at z = 0, P ΛCDM ( k, z = 0), from the Boltzmann code CAMB [80–82] as a starting point, and then utilize (5) and (6) todetermine σ /σ . Based on the assumptionsin MG, the structure growth at early times should be in-distinguishable from that in ΛCDM, hence we normalizethe ratio to 1 at redshift z = 10, high enough to set theinitial conditions of structure growth. We note that thegrowth factors between the GR limit solution of (7) andthe ΛCDM prediction from CAMB differ at the level farbelow the statistical uncertainities to affect the results ofthe Fisher analysis, and are corrected by normalizationusing the former.Our solution of D ( k, z ) (or D ( k, a )) for the f ( R ) modelis checked against the work of [83] in which the code forlinear perturbation in MG is slightly modified for ourpurpose. For both the ΛCDM and the nDGP models,the growth factor D ( z ) (or D ( a )) are scale-independent,and are checked against the empirical fitting function pro-posed by [29]: g ( a ) ≡ D ( a ) /a = exp (cid:20)(cid:90) a da (cid:48) a (cid:48) (cid:2) Ω m ( a (cid:48) ) γ − (cid:3)(cid:21) , (12)where Ω m ( a ) = Ω m a − / ( H/H ) , γ is taken as 0.55for ΛCDM, and 0.68 for both sDGP and nDGP with amodified expansion history [29]. This agreement remainsstable when Ω m is varied in a small range around ourfiducial value Ω m ∼ . σ ( z ) to summarizethe constraints from cluster abundances has the follow-ing advantages. There is not much information lost sincewe are examining the linear regime, despite the fact thatthe constraints are compressed into a root-mean-squaredquantity as σ . Performing a Fisher analysis using σ based on [40] is not only faster, but also more conser-vative in the sense that it does not introduce extra de-generacy breaking as is the case for full Fisher analyseswhere ΛCDM and nuisance parameters are not alreadymarginalized over. C. Galaxy Clustering Correlations
The LSS of the universe, as traced by the observedinhomogeneous clustering pattern of galaxies, has beenformed by the nonlinear gravitational collapse of the pri-mordial density distribution. We can model the observedclustering statistics of galaxies in MG, by taking into ac-count the crucial effects of non-linear clustering, large-scale galaxy bias and redshift space distortions (RSD).Our modeling procedure summarized below is tailored toDESI observations, and is heavily based upon the previ-ous works of [53, 68].In the intermediate, quasi-linear scales higher orderperturbation theory can substantially improve upon theaccuracy of the simple linear treatment, allowing for arobust modeling of the clustering statistics, without theneed to resort to computationally expensive N-body sim-ulations. In this work we focus on the Lagrangian Per-turbation Theory (LPT), in which the expansion param-eter is a vector field, Ψ , which displaces each fluid parti-cle from its initial position, q , to its final, Eulerian one, x ( q , t ), through the mapping: x ( q , t ) = q + Ψ ( q , t ) . (13)The first order LPT solution is the famous Zel’dovich ap-proximation [57]. In MG theories, an additional degree offreedom is present, altering the perturbed Einstein equa-tions and the nonlinear gravitational evolution of darkmatter overdensities, and subsequently the framework ofLPT, as detailed in [66–68, 84].The galaxies observed by surveys of the LSS do notperfectly trace the underlying dark matter density dis-tribution, but rather are biased tracers of it [85]. In thesimpler picture of linear perturbation theory, the large-scale overdensity of biased tracers (i.e. galaxies) is pro-portional to the underlying dark matter overdensity [86],while a wide range of more sophisticated treatments havebeen developed in the literature [87]. When working inLagrangian space as is in this work, biased tracers areidentified as regions of the primordial density field pre-selected by a biasing function, F , that depends on thelocal matter density [63, 64]. Given the statistical nature of cosmic density fields, the simplest meaningful observ-able statistic (in the configuration space) is the two-pointcorrelation function, ξ X ( r ), of tracers correlated over adistance r : ξ X ( r ) := (cid:104) δ X ( x ) δ X ( x + r ) (cid:105) , (14)where the angle brackets denote an ensemble average. InLagrangian space, the “Convolution Lagrangian Pertur-bation Theory” (CLPT) [64, 88, 89] was shown to workparticularly well at recovering the correlation function ofhalos from N-body simulations, in ΛCDM cosmologies.Building upon these works, [67, 68] then expanded CLPTin the case of MG theories and successfully recovered thereal-space two-point correlation function of dark matterhaloes across the parameter space of the f ( R ) and nDGPMG scenarios.In addition to imperfectly tracing the dark mat-ter distribution of the cosmic web, galaxies identifiedthrough spectroscopic means are observed in redshiftspace, rather than in real space, which further distortsthe observed clustering pattern, known as the RedshiftSpace Distortions (RSD) [50–52]. Due to its peculiar ve-locity about the Hubble flow, v ( x ), a galaxy with realspace position x will be instead observed at a redshiftspace position: s = x + ˆ z · v ( x ) aH ( a ) ˆ z, (15)with H ( a ) the Hubble factor at a given scale-factor a .As a consequence, the redshift-space 2-point correlationfunction for biased tracers ξ sX ( r ) = (cid:104) δ X ( s ) δ X ( s + r ) (cid:105) , (16)becomes directionally dependent, unlike the real-spaceexpression (14). In large linear scales, coherent infallleads to the “Kaiser boost”, an enhancement on the am-plitude of the two-point correlation function, whereas inthe non-linear scales, the random velocities within viri-alized structures lead to the ”Fingers-Of-God” (FOG)suppression effect.The Gaussian Streaming Model (GSM) [54–56] hasbeen shown to be very successful at modeling theanisotropic RSD correlation function of halos, througha convolution of the halo real space correlation functionwith the probability velocity distribution of tracers, thatis approximated as a Gaussian [90]. In particular, giventhe real-space mean pairwise velocity along the pair sep-aration vector of a pair of tracers, v ( r ), as well as itspairwise velocity dispersion, σ ( r ), the GSM gives theexpression for the anisotropic RSD correlation function:1 + ξ sX ( s ⊥ , s (cid:107) ) = (cid:90) ∞−∞ dy (cid:112) πσ ( r ) [1 + ξ rX ( r )] × exp (cid:34) − (cid:0) s (cid:107) − y − µv ( r ) (cid:1) σ ( r ) (cid:35) , (17)where s ⊥ , s (cid:107) are the perpendicular and parallel to theline-of-sight components of the redshift-space separations, with s = (cid:113) s ⊥ + s (cid:107) , r = (cid:112) s ⊥ + y and µ = ˆ r · ˆ z = yr .Using CLPT to model the 3 ingredients that enter theprescription (17), ξ rX ( r ) , v ( r ) , σ ( r ), and based on theMG implementations of [67, 68], [53] was able to suc-cessfully model the simulated RSD correlation functionof haloes in the f ( R ) and nDGP gravity scenarios up to1-loop order in PT. For the purposes of this work, weonly use the first order LPT solution (Zel’dovich approx-imation [57]) to evaluate the GSM ingredients, mainlybecause our model for the evaluation of the clustering co-variance matrix does not incorporate non-Gaussian cor-rections, as we explain in the Appendix A.Finally, we model the observed RSD two-point cor-relation function of galaxies in MG (and ΛCDM) for asurvey such as DESI, using the CLPT & GSM frame-work laid out in this section. Before this, we point outthat the elements entering the final GSM expression (17)also depend, in addition to the known LPT growth fac-tors and the linear power spectrum, on the Lagrangianbias parameters up to second order (for a 1-loop predic-tion), b L and b L , that we also need to account for. In ourDESI-type investigation, and as we will explain in moredetail in the next section, we consider constraints fromtwo types of objects, the Luminous Red Galaxies (LRGs)and the Emission Line Galaxies (ELGs), following [91].For these two types of objects, the linear Eulerian galaxybias can be modeled as [91, 92]: b E ( z ) D ( z ) = (cid:40) . , LRG , . , ELG , (18)where D ( z ) is the linear growth factor under ΛCDM,normalized to unity at z = 0. The second order biasparameter is not treated as an independent parameter,but is rather determined in terms of the b E predictionfrom Eqn. (18), through the fitting formula b E = 0 . − . b E + 0 . b E ) + 0 . b E ) , (19)that has been calibrated from N-body simulations [93].Finally, the Eulerian bias values b E and b E can be con-verted to the corresponding ones in the Lagrangian spacethrough the known conversion relationships [94, 95]: b L = = b E − ,b L = = b E − b L . (20)Combining eqs. (18-20), we can determine the necessarybias parameters for a given galaxy sample and redshift z in terms of the linear bias, that we treat as a nuissanceparameter and marginalize over, as we explain in the nextsection. We also note that throughout this analysis thebias is taken to be scale-independent, a good approxima-tion for our scales of interest, as was explained in [53]and also previously confirmed by simulations in [96], forboth GR and MG. The second nuisance parameter we include is a con-stant offset, α σ , that needs to be added to the modeledgalaxy pairwise velocity dispersion, σ → σ + α σ , (21)such that the latter matches the observed prediction fromsimulations (or observations) at the large scale limit, aswe found in [53]. This correction aims to capture un-known small-scale nonlinear effects and is essentially theequivalent of the “Fingers-of-God” free parameter, σ ,that is commonly employed in the simple phenomenolog-ical dispersion models [97, 98].Finally, all of the above ingredients are combined toproduce a prediction, by means of Eqn. (17), for the MGRSD galaxy correlation function for the ELG and theLRG DESI galaxy samples at a given redshift z . As com-monly performed in the literature, we further decomposethe correlation function through a multipole expansionin a basis of Legendre polynomials, L l ( µ s ): ξ ( s, µ s ) = (cid:88) l ξ l ( s ) L l ( µ s ) , (22)where the multipoles of order l will be given by ξ l ( s ) = 2 l + 12 (cid:90) − dµ s ξ ( s, µ s ) L l ( µ s ) , (23)with µ s = ˆ z · ˆ s = s (cid:107) /s . We restrict our analysis onvalues l = { , , } , which correspond to the monopole,the quadrupole and the hexadecapole, respectively (first3 non-vanishing multipoles). D. Fisher Analysis
For the forecasted constraints on the two MG modelswe use Fisher analyses on the tracers of the LSS, e.g.cluster abundances in the linear regime and nonlineargalaxy clustering. Our adopted fiducial background cos-mology and MG parameters are shown in Table I, follow-ing [40]. We consider three different f ( R ) scenarios with f R = 10 − (referred to as “F5”), 10 − (“F6”) and 0 (re-ferred to as “near-GR”). In the near-GR case at f R = 0,the parameter n is not well-defined and has no impact onthe growth, so we consider its value as fixed at n = 1 anddo not include it as a Fisher parameter. Furthermore,we consider two nDGP scenarios for n = { , } , that werefer to as N1 and N5, respectively.We utilize the Fisher formalism [99, 100], assuming aGaussian likelihood distribution: F ij = (cid:88) αβ ∂f α ∂p i Cov − [ f α , f β ] ∂f β ∂p j , (24)where f α,β are the observables in bins labeled by α, β ; Cov is the observable covariance matrix and p i and p j are a pair of the model parameters being constrained. Parameter FiducialValue(s)ΛCDM Ω c h b h H A s n s f ( R ) f R − , − , n n nDGP , b ( z ) Eqn. (18)parameters α σ . f ( R ) and nDGP MG scenarios. Weadd that the nuisance parameters refer only to the galaxyclustering evaluation. Constraints by cluster abundances, as discussed in II B,are represented by σ ( z ). In particular, the observ-ables f α are the set of { σ ( z ) /σ ( z ) } across30 linearly-spaced redshift bins centered on z = 0 . z = 2 .
95, which are predicted by the MG mod-els. The observable covariance matrix
Cov − [ f α , f β ] on σ /σ , obtained from [40], is diagonal, and is amodel-independent result where the errors from the back-ground ΛCDM parameters are marginalized over, re-ducing the parameters to constrain in (24) to the pair { f R , n } ( f ( R )) or n DGP (nDGP). Hence, the partialderivative stepsizes for the background cosmology param-eters are as specified in [40]. For F5 and F6, the partialderivatives are taken with respect to { log ( f R ) , n } , withthe respective stepsizes { . , . } . For the near-GRcase, log ( f R ) is not well-defined and we take a stepsizeof 10 − directly with respect to f R . In N1 and N5, thestepsize is 0 .
05 for n DGP . A five-point central differencesscheme is applied to evaluating the partial derivativesover all the MG parameters, giving a third-order accu-racy, with the exception of the near-GR case where onlya one-sided derivative is feasible due to our assumptionthat f R ≥
0. For this, a corresponding four-point for-ward differences scheme is then applied to maintain thethird-order accuracy.For galaxy clustering the observables are thegalaxy correlation function multipoles, f α = { ξ ( s ) , ξ ( s ) , ξ ( s ) } , considered over 35 spatial sep-aration bins equally (logarithmically) spaced in therange 25 < s <
600 Mpc/h. The cosmological parame-ters, p i , are those given in Table I, while the covariancematrix for the monopole, quadrupole and hexadecapolemoments is described in the Appendix A. Our evaluationassumes a DESI-like survey with LRG and ELG galaxysamples in the redshift range 0 . < z < .
85, using18 linearly spaced z bins, as outlined in [91]. Our choices for the galaxy number density, survey volumeand linear bias as a function of redshift are informed by[91], in particular its Table V. The partial derivativesof the multipoles are evaluated with a 2-point centraldifferences scheme, with the derivative step-sizes withrespect to the background ΛCDM parameters and linearbias provided by [40]. With regards to the MG param-eters, the derivative steps for the { f R , n } pair in theF5 and F6 cases are { × − , . } and { × − , . } ,respectively, and for n DGP in the N1 and N5 modelsthey are 0 .
15 and 0 .
5. Again, in the near-GR casewith f R →
0, we employ a 3-point forward differencesscheme due to the restriction f R ≥
0, with a forwardstep of 10 − in f R while keeping n fixed.The step-size when differentiating with respect to thenuisance parameter α σ is 1 .
5, informed by the detailedstudy in [53]. We have checked that all of the choicesabove provide numerical stability in the derivatives.
III. RESULTS
In this section we present the forecasted constraintson the cosmological parameters of Table I following themethods outlined in the previous section.Figures 1 and 2 demonstrate the sensitivity of thegalaxy clustering constraints for the F5 and F6 cosmolo-gies. In Figure 1, we present 2-dimensional constraintsfrom each parameter pair in the Fisher analysis, as ob-tained by the first three non-vanishing multipoles of theredshift-space correlation function of the LRG and ELGgalaxy samples, both when considered separately andcombined. In addition to the constraints on the standardΛCDM parameters in line with previous works in the lit-erature (e.g. [91]), the complementarity of the LRG andthe ELG-derived contributions allows us to tightly con-strain the pair of the MG parameters { f R , n } , which arethe focus of our analysis. We see from the plot that thecombined constraints from the two samples on the MGparameters are much tighter than the individual ones. Itis also expected that using the ELGs produces tighterconstraints in all parameter planes, given that this sam-ple has a larger number density and redshift range, com-pared to the LRG counterpart (see Table V of [91]).Fig. 1 also shows, as we further quantify in Table II,that the forecasted constraints on the MG parameterlog ( f R ) are at least about an order of magnitudetighter compared to the parameter n . This finding isattributed to the fact that the 2-point function is moresensitive to variations of f R than of n , in particular forthe range of scales we consider in this work, as was foundby the sensitivity analysis of [101]. Due to this fact, mostprevious works in the literature (e.g. [83, 102]) have com-monly worked with a fixed value of n = 1, and onlyconsidered constraints on f R . Thanks to our flexibleanalytical model for the anisotropic correlation functionin any scalar-tensor theory, in this work we are able toprovide constraints on the fuller parameter space of the − . − . l og ( A s ) H . . . Ω c h . . Ω b h − . − . − . l og ( f R ) . . n s − . . . n − . − . log ( A s ) H . . Ω c h . . Ω b h − . − . − . log ( f R )LRG, r min ∼ . M pc/h
ELG, r min ∼ . M pc/h
LRG + ELG, r min ∼ . M pc/h
LRG + ELG, r min ∼ . M pc/h
FIG. 1. Galaxy clustering constraints on the parameters in the F5 case (fiducial values f R = 10 − , n = 1) using the DESILRG sample [red], the ELG sample [blue] and both combined [cyan] for a minimum survey scale of r min ∼ . Mpc/h , thesmallest scale we anticipate can be probed with the survey. Combined constraints for a more conservative minimum scale of r min ∼ . Mpc/h [yellow] are also presented to show the impact of scale on the constraints. f ( R ) Hu-Sawicki model.In addition, in Figure 1 we demonstrate how the choiceof the minimum survey scale impacts the constraints weobtain, finding that a more conservative value of r min ∼ . M pc/h dilutes the constraining power overall. Wefind that the constraints from LRG+ELG combined forthis more conservative value are comparable with theconstraints from LRGs alone for r min ∼ . M pc/h . Thisreduction in sensitivity is consistent with the predicteddeviations in the Hu-Sawicki model becoming progres-sively more pronounced, as one considers smaller scales.As a result, an analysis focusing on larger scales would re-strict the ability to probe MG signals, resulting in looserconstraints on the corresponding MG parameters.For the F6 model, we find that the 2D degeneracies be- tween the ΛCDM and MG parameters are qualitativelysimilar to those for F5. In Fig. 2 we show that the con-straints on the two MG parameters are comparativelylooser in F6 versus F5, consistent with the predicted de-viations from GR being smaller. We find that the F6 con-straints from LRG and ELG data are more comparable,with ELGs still being tighter, so that the combination ofthe two gives more notable relative improvements to theELG data alone than for the F5 scenario.We also performed the same analysis on the nDGPmodels, but will only present the final combined con-straints and omit showing the full corner plots for thesake of brevity. Our findings regarding nDGP are overallsimilar to the f ( R ) scenario: the direction of the degen-eracies for the background ΛCDM parameters are the − . − . − . − . − . log ( f R ) = 10 − , n = 1 − − − − log ( f R ) = 10 − , n = 1 − − LRGELGLRG+ELG
FIG. 2. Galaxy clustering constraints on { f R , n } in the F5(top) and F6 (bottom) cases (fiducial values f R = 10 − and10 − , n = 1) for the LRG sample [red], ELG sample [blue]and both combined [cyan] assuming r min ∼ . Mpc/h . same, while the degeneracies are stronger for the nDGPscenarios, presumably due to the scale-independence.The 1D projected uncertainties obtained from galaxyclustering for the F5, F6, near-GR f ( R ) model andnDGP are summarized in Table II. Across all the mod-els considered, the uncertainties are principally driven bythe ELG galaxy sample. For F5, the ELG constraints areroughly half the size of those from LRGs, and the combi-nation of the two only reduces the uncertainties by 7%.The impact of combining the two is more pronounced forF6, with ELG+LRG constraints about 30% tighter thanfor ELGs alone. For the near-GR case, the combinationof LRGs with ELGs tightens the constraint from ELGsalone by 25% (with the ELG constraint being about two-thirds that from LRGs). For nDGP, the ELG constrainton the n nDGP parameter is less than half of that fromLRGs alone, and only a 10% reduction is obtained bycombining the two samples.Our constraints for the f ( R ), near-GR and nDGPcases are of the same order as the ones presented in [102],which performed a Markov Chain Monte Carlo analysis. We now consider the constraints from cluster abun-dances, obtained from σ /σ . In Fig. 3, weprovide an overview comparison of the evolution of thepredicted σ ratio, over the 30 redshift bins from z = 0 . z = 2 .
95, for the MG models versus the model-independent forecasted errors for future observations atthe Simons Observatory [31]. The ratios predicted fromMG are normalized at z = 10, varying { f R , n } for f ( R )and n for nDGP, respectively. The σ is normal-ized to be that calculated from (7) (with g eff = 0).This overview provides an insight into the sensitivity of σ with respect to the parameters of the two MG modelswe consider. For both the f ( R ) and nDGP MG models,the constraining power mainly lies at lower redshifts, at z <
2, increasing as one approaches z = 0, where thedeviation of the MG-predicted σ is the highest, and theforecasted errors by cluster abundances are the tightest,notably for 0 . < z < .
5. Furthermore, by comparingthe signal to errors for the f ( R ) case in sub-figures (a)and (b), we anticipate that the σ data will be moresensitive to variations in f R than in n .As is the case for galaxy clustering, we also summa-rize the model parameter constraints from galaxy clusterabundances in Table II. We find that cluster abundancesare not as competitive as galaxy clustering in constrain-ing the model that deviates most greatly from GR, with f R = 10 − . For this model, the constraint on f R is afactor of 2.6 larger, while for n it is 40% larger. Interest-ingly, though, we find that cluster abundances providecomparatively tighter constraints for the weaker f ( R )model, with fiducial f R = 10 − , where the constraintson f R and n are roughly 10% smaller than those pre-dicted from galaxy clustering. We also find that abun-dances provide constraints on the nDGP parameters thatare 50 ∼
60% tighter than those from galaxy clustering.In Fig. 4 we present the 2D { f R , n } constraints forthe F5 and F6 f ( R ) scenarios for the two datasets, withthe combined 1D constraints summarized in Table II. ForF5 with cluster abundances we find that there is a strongdegeneracy in the log ( f R ) − n plane but with a well-measured combination in the direction orthogonal to thedegeneracy. This phenomenon has been tested to be rela-tively stable across the higher and lower redshift ranges.In contrast, constraints from galaxy clustering do notshow significant degeneracies in this parameter space andprovide tighter constraints on f R and n separately, butwith an overall likelihood ellipse that is wider (the bestconstraint in the 2D plane is weaker than for the clus-ter abundances). In combination, the galaxy clusteringconstraints help break the degeneracy from the clusterabundance data, and drive the constraint to f R . Theconstraints on n are improved, relative to those from thegalaxy clustering, by a factor of ∼
2. For F6, the con-straints from both cluster abundances and galaxy clus-tering are weaker, but the galaxy clustering constraintson f R are comparable to those from the cluster abun-dances. The net effect of combining the two datasets isless significant than for F5, however we see improvements Model Fiducial Galaxy clustering Cluster CombinedParameters LRG ELG ELG+LRG Abundances f ( R ) F5 log ( f R ) = − ± . ± . ± . ± . ± . n = 1 ± . ± . ± . ± . ± . f ( R ) F6 log ( f R ) = − ± . ± . ± . ± . ± . n = 1 ± . ± . ± . ± . ± . f ( R ) near-GR (2- σ ) f R = 0 ≤ . × − ≤ . × − ≤ . × − ≤ . × − ≤ . × − DGP N1 n nDGP = 1 ± . ± . ± . ± . ± . n nDGP = 5 ± . ± . ± . ± . ± . r min ∼ . Mpc/h ) and cluster abundances alone respectively, and cross-combining the two observables. The numerical values withinthe same row of a fiducial parameter denotes the 1- σ (68%) errors on the same parameter around that fiducial value. Specificallyfor the near-GR case ( f R = 0), the 2- σ (95% confidence) upper limits are reported. of ∼
30% in constraints on both f R and n .In Fig. 5, we see that for the near-GR f ( R ) case the rel-ative impacts of the cluster abundance and galaxy clus-tering constraints are similar to that for F6, in that theconstraints on f R are comparable for each dataset, andthe combination provides a more modest ∼
8% improve-ment in the 1- σ constraint relative to that from clusterabundances alone. This result also implies that a 2- σ (95% confidence) upper limit of f R ≤ . × − canbe placed for f R when we take the fiducial scenario asGR ( f R = 0). Fig. 5 also shows the constraints on thenDGP model parameter. Here we find that the clusterabundances drive the constraints and the galaxy cluster-ing plays a minimal role in affecting improvements.Spanning the two most popular classes of screeningin the literature, through the representative f ( R ) andnDGP MG models, our detailed analysis overall servesto highlight the ways in which the upcoming precise ob-servations of redshift-space galaxy clustering and clus-ter abundances will enable us to probe the landscapeof dark energy and MG parametrizations in the next10 years. The nDGP model realizes the Vainshteinscreening mechanism, which is harder to constrain us-ing other astrophysical probes, in comparison to thechameleon screening of the f ( R ) scenario. Here we findthat the cluster abundances are better able to constrainthe scale-independent effects of the nDGP model, whilethe galaxy clustering provides tighter constraints on thescale-dependent f ( R ) scenario. This complementarity ofthe two techniques in constraining these models, and thepotential for cluster abundances to constrain nDGP, areimportant outcomes from our findings. IV. DISCUSSION
In this work we performed a detailed study of our abil-ity to constrain the large-scale properties of gravity witha combination of two promising probes of the LSS: galaxy clustering from spectroscopic observations by DESI, aswell as cluster abundances from tSZ observations by theSimons Observatory.For galaxy clustering, we employ the Gaussian Stream-ing Model with Lagrangian Perturbation Theory (LPT)to predict the anisotropic redshift-space 2-point correla-tion function of biased tracers, which was recently gener-alized to support predictions for MG parametrizations.We apply the model to predict the multipoles of theRSD correlation function for the ELG and the LRG DESIspectroscopic galaxy samples, as well as their correspond-ing covariance matrices. Regarding cluster abundances,we use the amplitude of density fluctuations, σ ( z ), ob-tained by tSZ-selected galaxy clusters, as a window intothe nature of the underlying gravity model, expandingupon recent detailed studies in the context of standardcosmologies.We employ the Fisher forecasting formalism to obtaina set of joint constraints on two widely-studied MG mod-els, the f ( R ) Hu-Sawicki and the nDGP gravity models.We demonstrate that the two independent probes com-plement each other in constraining the f ( R ) Hu-Sawickimodel parameters, for varying degrees of deviation awayfrom a ΛCDM background, as well as in a near-GR fidu-cial scenario. We find that the tightest constraints areobtained in the large-deviation F5 scenario, at the level ofa ∼
2% forecasted joint constraint on the log ( f R ) pa-rameter, with the ELGs serving as the primary source ofdiscriminating power on the galaxy clustering side. Theconstraining power of both probes is primarily derivedfrom their corresponding lower redshift snapshots, whenthe MG deviations are more pronounced overall. We alsoconsider the full 2D parameter space, { f R , n } , for theHu-Sawicki model, and place corresponding constraints.In a similar manner, we find that the interplay betweenthe cluster abundance and galaxy clustering observablescan be utilized to constrain the parameter space of thenDGP gravity scenario. We forecast a combined relativeconstraint of 2% in the n nDGP = 1 case and that the0 . . . . . . . z . . . . . σ ( z ) MG /σ ( z ) ΛCDM f R = 10 − , n = 1 f R = 10 − , n = 1 f R = 10 − , n = 1 f R = 10 − , n = 1ΛCDM with forecasted errors (a) Varying f R , n = 1 . . . . . . . z . . . . . . . σ ( z ) MG /σ ( z ) ΛCDM f R = 10 − , n = 1 f R = 10 − , n = 2 f R = 10 − , n = 3 f R = 10 − , n = 4 f R = 10 − , n = 25ΛCDM with forecasted errors (b) Varying n , f R = 10 − . . . . . . . z . . . . . . . σ ( z ) MG /σ ( z ) ΛCDM nDGP , n = 1nDGP , n = 2nDGP , n = 3nDGP , n = 4nDGP , n = 5ΛCDM with forecasted errors (c) Varying n , for nDGP FIG. 3. The MG predicted σ ( z ) MG /σ ( z ) ΛCDM normalized at z = 10, plotted against the error of this ratio forecasted bycluster abundances. [Left, (a)] shows scenarios with different f R values while fixing n in f ( R ) as 1. [Center, (b)] showsscenarios fixing f R = 10 − while changing n . [Right,(c)] shows the case in nDGP, where n nDGP is varied. − . − . − . log ( f R ) n n . . . . . Galaxy clusteringCluster abundancesCombined (a) f ( R ): F5 (fiducial f R = 10 − ) . . . . . − − − log ( f R ) − n − n . . . . Galaxy clusteringCluster abundancesCombined (b) f ( R ): F6 (fiducial f R = 10 − ) FIG. 4. The constraints on the [Left] F5 and [Right] F6 f ( R ) scenarios are shown from [red] galaxy clustering, [yellow] clusterabundances and [cyan] the two combined. The covariance ellipses in the { f R , n } parameter space indicating the 1- σ (68%)confidence levels and their respective 1D Gaussian likelihoods are shown for each scenario, with the darker fill-in shades denotingthe corresponding 1- σ (68%) confidence levels, lighter shades the 2- σ (95%) confidence levels. cluster abundance observations would principally drivethese constraints. This, and the opposite phenomenonthat galaxy clustering drives the constraints for the F5scenario, are potentially due to the fact that the f ( R )model is scale-dependent in contrast to nDGP in linearand quasi-linear scales, which might also explain the rel- atively balanced constraining powers of the two observ-ables for the F6 and near-GR scenarios, in which thescale-dependence is weaker than in F5.There are many possible ways in which one can expandupon this line of enquiry. For galaxy clustering, the accu-racy of our model can be further improved by including1 . . . f R × − . . . . P × GalaxyclusteringClusterabundancesCombined (a) f R : near-GR (fiducial f R = 0) . . . n nDGP P (b) nDGP: fiducial n nDGP = 1 n nDGP . . . . P (c) nDGP: fiducial n nDGP = 5 FIG. 5. The 1D likelihood distribution for [Left, (a)] the near-GR f ( R ) model, with fiducial value f R = 0 (the value of n becomes redundant) and for the nDGP model with fiducial values [Center, (b)] n nDGP = 1 and [Right, (c)] n nDGP = 5. Theconstraints from [red] galaxy clustering (ELG+LRG), [yellow] cluster abundances and [cyan] the two combined are shown, withthe darker fill-in shades denoting the corresponding 1- σ (68%) confidence levels, lighter shades the 2- σ (95%) confidence levels. the 1-loop corrections of LPT [53] into the GSM pre-diction, as well as by introducing effective field theorycorrections to account for non-perturbative small-scalephysics. Such an approach would also need to be com-bined with a suitably improved treatment of the cluster-ing covariance matrix, that we assumed to be Gaussianin the present work. Furthermore, it would be very in-teresting to also explore the constraining power of theFourier space counterpart of the two-point function, theredshift space power spectrum, obtained either by an-alytical approaches (see e.g. [103]) or through emula-tors [101]. For galaxy cluster abundances, extending thiswork to a full Fisher analysis with MG requires halomass functions [e.g., 104] with fitting formulas in their ansatz for the f ( R ) Hu-Sawicki and the nDGP gravitymodels. We expect these constraints to further improvewith CMB-S4 cluster abundances in combination withphotometric and weak gravitational lensing observationsby Stage-IV surveys such as the V. Rubin ObservatoryLSST [47, 48]. Finally, it would be interesting to use aMarkov Chain Monte Carlo forecasting approach to seehow non-Gaussianities in the posterior likelihood impactthe constraints and degeneracies we present.In the near future, synergies between new cosmologicalsurveys will allow us to explore the vast landscape ofdark energy and MG scenarios, and shed new light onthe nature of cosmic acceleration. Our work serves tohighlight the great promise held in such considerations,as well as the optimal ways in which the vast amounts ofupcoming observations could be utilized. ACKNOWLEDGMENTS
The authors would like to thank Mathew Mad-havacheril for helpful discussions regarding the clus- ter abundances analysis, as well as code resources thathelped plotting the covariance ellipses. Georgios Valo-giannis would like to thank Sukhdeep Singh and Yin Lifor useful discussions on analytical models of covariancematrices. This work is not an official Simons Observatorypaper.Georgios Valogiannis’ work is supported by NSF grantAST-1813694. Georgios Valogiannis and Rachel Beanacknowledge support from DoE grant DE-SC0011838,NASA ATP grant 80NSSC18K0695, NASA ROSES grant12-EUCLID12-0004 and funding related to the Ro-man High Latitude Survey Science Investigation Team.Nicholas Battaglia acknowledges support from NSF grantAST-1910021.2
APPENDIXAppendix A: Covariance matrix calculation
This Appendix provides the details of the analyticalmodel we use to evaluate the covariance matrix of themultipoles of the anisotropic correlation function. Webegin with the known expression for the Poisson errormatrix of the power spectrum, assuming Gaussian den-sity perturbations [105, 106]:
Cov [ P ( k ) , P ( k (cid:48) )] =(2 π ) V s (cid:18) P ( k + 1 n (cid:19) ( δ D ( k − k (cid:48) ) + δ D ( k + k (cid:48) ))+ 1 n V s [ P ( | k − k (cid:48) | ) + P ( | k + k (cid:48) | + 2 P ( k ) + 2 P ( k (cid:48) )]+ 1 n V s , (A1)with n the number density of the galaxies in a givensample, V s the survey of the volume and δ D the Diracdelta-function. The second and third lines on the r.h.sof Eqn. (A1) encode the Poisson shot noise contribu-tions to the covariance matrix [107]. Eqn. (A1) has ne-glected contributions from nonlinear gravitational evo-lution [106, 108–110], super sample covariance [111–115]and effects of the survey nontrivial window function [116].Our goal is to Fourier transform the result (A1), so thatwe obtain the configuration space equivalent expressionfor the covariance matrix of the anisotropic correlationfunction. In the simpler case of real space considera-tions, with the correlation function being isotropic, [107]demonstrated that, by angle-averaging the Fourier trans-form of (A1), the oscillatory Bessel function dependenciescan be eliminated (unlike in the RSD case, as shown be-low), and a more compact expression is possible. [107]also found the Poisson shot-noise contributions to be di-agonal for the correlation function. In redshift space,which is what we are interested in in this work, the equiv-alent configuration space expression for (A1) has beenderived in [117, 118], assuming only Gaussian shot-noisecontributions (i.e. neglecting the second and third linesin the r.h.s of (A1)), and is the following: Cov [ ξ l ( s i ) , ξ l ( s j )] = i l + l π (cid:90) ∞ k σ l l ( k ) j l ( ks i ) j l ( ks j ) dk, (A2)where we defined the multipole per-mode covariance: σ l l ( k ) = (2 l + 1)(2 l + 1) V s × (cid:90) − (cid:20) P ( k, µ k ) + 1 n (cid:21) L l ( µ k ) L l ( µ k ) dµ k , (A3)where j l ( ks i ) , j l ( ks j ) are the spherical Bessel functionsof the first kind. Poisson shot-noise contributions can potentially become significant, as pointed out by [107].To that end, we proceed to expand the expression (A2)to also include the Poisson terms to the shot-noise con-tributions, just as in the real-space version of [107]. Todo so, we first adopt our convention for the Fourier trans-formation, applied on the correlation function: ξ ( s ) = (cid:90) d k (2 π ) e i k · s P ( k ) , (A4)and label the terms reflecting the Poisson shot-noisecontributions in (A1) (second and third lines of r.h.s)as Cov [ P ( k ) , P ( k (cid:48) )] (cid:12)(cid:12)(cid:12)(cid:12) Poisson . Fourier transforming bothsides then gives Cov [ ξ ( s i ) , ξ ( s j )] (cid:12)(cid:12)(cid:12)(cid:12) Poisson =2 n V s ξ ( s i ) δ D ( s i − s j ) . (A5)Finally, we aim to project out the correlation functionmultipoles, for which we integrate the ξ terms on the l.h.sabove (after multiplying both sides with the appropriateLegendre polynomials), as in (23), which gives Cov [ ξ l ( s i ) , ξ l ( s j )] (cid:12)(cid:12)(cid:12)(cid:12) Poisson =(2 l + 1)(2 l + 1) n V s πs i δ D ( s i − s j ) (cid:90) − ξ ( s i , µ s ) L l ( µ s ) L l ( µ s ) dµ s , (A6)where we have made use of the delta-function property: δ D ( s i − s j ) = δ D ( s i − s j ) s i δ D (Ω i − Ω j ) , (A7)with Ω denoting the corresponding solid angles. Com-bining (A6) with (A2), we arrive at the desired result: Cov [ ξ l ( s i ) , ξ l ( s j )] = i l + l π (cid:90) ∞ k σ l l ( k ) j l ( ks i ) j l ( ks j ) dk +(2 l + 1)(2 l + 1) n V s πs i δ D ( s i − s j ) (cid:90) − ξ ( s i , µ s ) L l ( µ s ) L l ( µ s ) dµ s , (A8)which is the equation we use to evaluate the covariancematrix of the multipoles of ξ in this work. The last termin Eqn. (A8) expands the Gaussian expression (A2) of[117, 118] in order to also capture the Poisson shot-noisecontributions in the anisotropic case, and exhibits thesame diagonal nature as the corresponding real space ex-pression of Eqn. (32) in [107], which it recovers in the The Fourier transformation of the r.h.s gives rise to additionalterms involving delta-functions, as in [107], that only contributeat separations r = 0, and are thus dropped. s , in order to avoidoverestimating the error predictions, as in [107, 119].To summarize our procedure, after getting an analyt-ical prediction for the RSD correlation function for ourdesired cosmology from Eqn. (17), we use it to predictthe covariance matrix from Eqn. (A8) (combined with the input from (A3)). An intermediate step is to FourierTransform to also obtain P ( k, µ k ) from ξ ( s, µ s ), which isrequired in Eqn. (A3), and can be easily performed withthe publicly available package mcfit . 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