Prepared for submission to JCAP Constraining models of f(R) gravity with Planck and WiggleZ power 4 spectrum data 1 0 2 r p A 7 Jason Dossetta Bin Hub David Parkinsona ] O aSchool of Mathematics and Physics, University of Queensland, C Brisbane, QLD 4072, Australia . bInstitute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands h p E-mail: [email protected], [email protected], [email protected] - o r t Abstract. In order to explain cosmic acceleration without invoking “dark” physics, we s a consider f(R) modified gravity models, which replace the standard Einstein-Hilbert action [ in General Relativity with a higher derivative theory. We use data from the WiggleZ Dark 2 Energy survey to probe the formation of structure on large scales which can place tight con- v straintsonthesemodels. Wecombinethelarge-scalestructuredatawithmeasurementsofthe 0 8 cosmic microwave background from the Planck surveyor. After parameterizing the modifica- 9 tion of the action using the Compton wavelength parameter B0, we constrain this parameter 3 using ISiTGR, assuming an initial non-informative log prior probability distribution of this . 1 cross-over scale. We find that the addition of the WiggleZ power spectrum provides the 0 4 tightest constraints to date on B0 by an order of magnitude, giving log10(B0) < 4.07 at − 1 95% confidence limit. Finally, we test whether the effect of adding the lensing amplitude : v ALens and the sum of the neutrino mass mν is able to reconcile current tensions present i X in these parameters, but find f(R) gravity an inadequate explanation. P r a Keywords: gravitation: f(R) gravity, perturbation: linear, dark energy, modified gravity, large-scale structure Contents 1 Introduction 1 2 Theory 2 2.1 The growth of perturbations 2 2.2 f(R) gravity theories 4 3 Datasets 6 4 Results & Discussion 7 4.1 Model I - f(R) gravity 8 4.2 Model II - f(R) gravity + ALens 9 4.3 Model III - f(R) gravity + ALens + mν 12 5 Conclusions P 14 A Marginalised parameter constraints 19 1 Introduction Thediscovery of the late-time acceleration of the Universe through measurements of Type-Ia supernovae [1–7] and confirmed by both Cosmic Microwave Background (CMB) [8, 9] and large-scale structure experiments [10–15] is one of the most important discoveries in modern cosmology. However, the question of what mysterious force is actually responsible for the acceleration remains an open question. Many suggestions have been made, yet at the most fundamental level we are unsure whether the acceleration arises from some extra dark fluid componentpresentintheuniverseorsomemodificationofEinstein’stheoryofgravity. There isafundamentaldegeneracybetweentheoriesthatcannotbebrokenusingonlydistancedata, as the dynamics of most general Modified Gravity (MG) theories can easily be replicated by some fluid dark energy with an equation of state that varies with scale factor, w(a). There is therefore a ‘theory degeneracy’ in using only distance data, that will require some new form of information to break it. A large number of different MG theories exist in the literature (for a review, see [16]). For example, higherderivative theories suchas f(R)[17,18]and Galileon [19]models modify the Einstein field equations to be higher than second order. Another type, higher dimension models such as DGP [20], change the propagation of the gravity theory by changing the dimensionality of space-time. All of these theories have accelerating Friedmann-Lemaˆıtre- Robertson-Walker (FLRW)-like solutions, which can be tuned to match the distance data without needing to introduce a cosmological constant (Λ). However, by changing the the- ory of gravity, they affect the motion of particles on all scales, beyond the expansion of the homogenousuniverse. Theclusteringofmatterandgrowthoflarge-scale structureintheuni- verse is also changed. As such MG theories make very different predictions for the clustering of matter. In contrast, most theoretically-motivated dark fluidmodels (such as quintessence) haveaverylargeclusteringscale, andsohaveonlyasmalleffectontheformationofstructure beyond their contributions to the Hubble expansion. In this way, measurements of the large – 1 – scale structure of matter can be used to break this ‘theory degeneracy’ between the MG and dark fluid hypotheses, and also distinguish between the different MG theories. Cosmological observations have already been used to test and constrain MG theories. The cosmic microwave background (CMB) provides a very clean probe of linear structure formation at high redshift, but either by natureor by design it is the case that most MG the- ories make the same predictions at high-redshift as the standard concordance model ΛCDM. Even so, measurements of the CMB power spectrum and secondary bispectrum still have some sensitivity to MG growth of structure through the large-scale integrated Sachs-Wolfe effect and weak lensing [21–28]. At low redshifts, data that have been used included the power spectrum of luminous red galaxies [28–33], cluster abundances [33–35], Coma cluster [36], weak lensing [32, 33, 37–42], redshift-space distortions [42–44], 21 cm line [45, 46] and matter bispectrum [47, 48]. In this paper, we choose to test the f(R) gravity theory, as it is one of the simplest theoriesavailable, andsincethefunctionoftheRicciscalarRcanbechosen,itcanbe“tuned” to reproduce any background expansion history needed [49–51]. We make predictions of the growth of structure under this theory, and compare them to observation to constrain the associated parameters. To do this, we use measurements of the galaxy power spectrum made by the WiggleZ Dark Energy Survey [52] and combine it with recent measurements of the CMB power spectrum from the Planck surveyor [9]. The galaxy power spectrum from the WiggleZ Dark Energy Survey has advantages over some other large-scale survey data, as the effect of non-linearites (redshift-space distortions and non-linear structure formation) seem to be small [53]. The outline of our paper is as follows. In section 2 we describe the theoretical basis of the f(R) theory we are considering and the predictions that it makes. In section 3 we describe the different data sets we used in the analysis. In section 4 we give the results from our likelihood analysis, and discuss them. Finally, we conclude in section 5. 2 Theory In this section, we will describe the theoretical framework in testing gravity by using cosmo- logical data. First, we will introduce the generic formalism in the subsection 2.1. Then, in subsection 2.2 we will study one of the explicit parameterizations of f(R) gravity under the quasi-static (QS) approximation. 2.1 The growth of perturbations Let us consider perturbations of the flat FLRW metric in the conformal Newtonian Gauge. In this gauge the metric is written ds2 = a(τ)2[ (1+2ψ)dτ2 +(1 2φ)dxidx ], (2.1) i − − where τ is conformal time, a(τ) is the scale factor normalized to one today, ψ and φ are the potentials describing the scalar modes of the metric perturbations, and the x ’s are the i comoving coordinates. Applying Einstein’s field equations to this metric, one can obtain the Poisson and anisotropy equations. The first comes from combining the time-time and time-space equa- tions and the second from the traceless space-space equation. In the context of testing general relativity, these equations are modified to include terms which will mimic the effects – 2 – of a modified gravity model on the growth in the linear regime. The modified Poisson and anisotropy equations as written in the formalism of [54] are given, respectively, by 2 2 k φ = 4πGa ρ ∆ Q(k,a) (2.2) i i − i X 2 2 k (ψ R(k,a)φ) = 12πGa ρ (1+w )σ Q(k,a), (2.3) i i i − − i X whereσ istheshearstress,andρ isthedensity(with denotingaparticularmatterspecies). i i i Themodifiedgrowthfunctions,Q(k,a)andR(k,a),canhavebothtime(describedintermsof scalefactor, a)andscale dependence. AmodificationtothePoisson equation isquantifiedby the parameter Q(k,a), while R(k,a) quantifies the so-called gravitational slip (a term coined by [55] to refer to the ratio between the two metric potentials) as at late times, assuming negligible anisotropic stress from normal matter components, ψ = Rφ. It is worth noting that eq. (2.2), above, cannot truly be called the Poisson equation, as it often is. It relates the space-like potential, φ, (the one only affecting relativistic particles) to the overdensity, while the Poisson equation should relate the overdensity to a potential which influences the dynamics of all particles. The potential that does this is the time-like, Newtonian potential, ψ. Asimilar modifiedgrowth formalism hadbeenintroducedby[56]priorto theformalism described above. This formalism however was introduced under the assumption of negligible anisotropic stress and thereforecould only beapplied at late times. Underthese assumptions the modified Poisson and anisotropy equations are recast to include two observation-related variables, µ(a,k), which defines a time and scale-dependent Newton’s constant through the product µ(a,k)G, and the gravitational slip γ(a,k). These equations are written 2 2 k ψ = 4πGµ(a,k)a ρ∆ , (2.4) − φ = γ(a,k) . (2.5) ψ As pointed out in [57], functions Q and R are simply related to µ and γ in the limit of negligible matter anisotropic stress, σ via the following relations: −1 Q = µγ , R = γ . (2.6) In this paper, we primarily use the extended formalism eq. (2.2) and (2.3). Above, ∆ = δ +3 (1+w )θ /k2 is the comoving overdensity, with δ the overdensity, i i i i i H θ the divergence of the peculiar velocity, and the Hubble parameter in conformal time. i H Enforcingconservation of energy momentum on aperturbedfluidgives theevolution of these quantities as [58]: δP ′ ′ δ = (1+w)(θ 3φ)+3 (w )δ (2.7) − − H − δρ ′ w δP/δρ ′ 2 2 θ = (1 3w)θ θ+ k δ+k (ψ σ). (2.8) −H − − 1+w 1+w − where w is the equation of state which relates the pressure of a fluid, P to its density, ρ via the usual relation P = wρ and primes denote derivatives with respect to conformal time, τ. On linear scales, the growth of cosmological structures can be described almost entirely by the growth of the overdensity for cold dark matter (CDM), δ . Taking into account that m – 3 – CDM is pressureless and shear free, we can use the above two equations and (after switching to proper time) obtain the usual equation for the growth: k2ψ δ¨ +2Hδ˙ + = 0. (2.9) m m a2 where dots denote derivatives with respect to proper time, t. IntheGRcase, subbinginfork2ψ usingeqs.(2.2)and(2.3)gives theevenmorefamiliar equation δ¨ +2Hδ˙ 4πGρ δ = 0, (2.10) m m m m − which is independent of scale, k. This scale independence does not necessarily hold for modified gravity models. Such is the case for the f(R) modified gravity models that we discuss below. 2.2 f(R) gravity theories Due to the simplicity of its Lagrangian, f(R) gravity obtained a lot of attention, (see the recent review [59]and references therein) especially as an illustration of the chameleon mech- anism. Besides the simplicity of the structure of this theory, there exist some other reasons for the interest it attracted. First of all, the form of the function f(R) can be engineered to exactly mimic any background history via a one-parameter family of solutions [49–51]. Second, f(R) gravity can provide a slightly better fit to the CMB data than flat ΛCDM, which can be attributed to the lowering of the temperature anisotropy power spectrum at the low-multipole regime [33]. Its Lagrangian in the Jordan frame could be written as R+f(R) 4 S = d x√ g + , (2.11) m − 16πG L Z (cid:20) (cid:21) where is the minimally coupled matter sector. m L Becauseofthehigherorderderivativenatureoff(R)gravity, thereexistsascalardegree of freedom, named the scalaron f df/dR with mass R ≡ 2 ∂2Veff 1 1+fR m = R . (2.12) fR ≡ ∂f2 3 f − R (cid:18) RR (cid:19) The corresponding Compton wavelength reads −1 λ m . (2.13) fR ≡ fR Usually, it is convenient to use the dimensionless Compton wavelength in Hubble units f H RR ′ B R , (2.14) ′ ≡ 1+f H R with f = d2f/dR2 and ′ = d/dlna. In the GR limit, the scalar field disappears due to RR the infinite mass, i.e. zero Compton wavelength (B(a) = 0). Thanks to this extra scalar degree of freedom as well as the background symmetries (homogeneity and isotropy), on the backgroundlevel thefunctionformoff(R)could beengineered tomimicthegiven expansion history [49–51]. Hence, we could not distinguish dark energy models via the background kinematic tests alone. Fortunately, at the perturbation level breaking of homogeneity and – 4 – isotropy will also break this theoretical degeneracy, i.e. we could recognize different dark energy models via their growth structure dynamics. Armed with these theoretical observations, let us now turn to the Q(k,a) and R(k,a) parameterization in f(R) gravity. In general, a reasonable parameterization is usually based on some approximations. Here, it is the quasi-static (QS) assumption, which means that we only keep the spatial derivative terms in equations while neglecting the temporal derivatives. For a small deviation from GR (B(a) 1), it has been proven that the QS description for ≪ f(R) gravity is satisfactory for several ongoing projects [60]. This is because in f(R) gravity, asidefromthehorizonscale,inthelinearregimethereexistsanewscale—theComptonscale — which characterizes the deviation from GR. Above this scale, its deviation is tiny; while belowit, themodifieddynamicscouldbeobvious. WhentheComptonscalesitsdeeplyinside the horizon, i.e. small B(a) case, the information of modified gravity is mainly characterized by the sub-horizon dynamics, the temporal evolution of which, is insignificant. Hence, for this case theQS approximation holds. However, whenthe Compton scale is comparable with horizon scale (B(a) 1), the temporal evolution of gravitational potentials are no longer ∼ negligible and the QS assumption breaks down. From a practical observational point of view, the current cosmological probes already rule out the large Compton wavelength case. For example, the recently released Planck CMB temperature and lensing power spectra data gives thepresentComptonwavelength boundB0 < 0.1 at95%C.L.[25]andthejointanalysis of several LSS tracers combined with WMAP data gives a even more stringent constraint B0 < 1.1 10−3 at 95%C.L. [33]. Given the above theoretical explanations and current data × analysis status, we will adopt this QS approximation in our following study. For f(R) modified gravity models, by assuming quasi-staticity and ΛCDM background, the functional form of the modified growth parameters, Q(k,a) and R(k,a), has been given by [56, 61, 62] and can be written as: 1 1+ 2λ2k2as 3 1 Q(k,a) = , (2.15) 1 1.4 10−8λ2a3 1+λ2k2as 1 1 − × 1+ 4λ2k2as 3 1 R(k,a) = , (2.16) 1+ 2λ2k2as 3 1 where the empirical prefactor in eq. (2.15) corresponds to corrections to more accurately modeltheISWcontributionsinf(R)gravity[62]. Andλ1isnothingbutthepresentCompton wavelength λ21 = B0c2/(2H02). As demonstrated in [51], this parameterization assumes a power-law growth of the Compton wavelength fRR = B0 as+2. (2.17) 1+fR 6H02 We shouldemphasizethataconstant valueforswillnotingeneralbecapable ofreproducing the fullΛCDM expansion history. However, it works as a good approximation for each epoch ˜ alone [63], as can be inferred from eq.(2.17). Indeed a reasonable value of s is given by s 5 ≈ during radiation domination, s 4 during matter domination and s < 4 during the late ≥ time phase of accelerated expansion. For small values of B0, it is customary to fix s = 4 as discussed in [60]. In this paper we adopt this choice and then play with a single parameter B0. Combining the above eqs. (2.15) and (2.16) with eqs. (2.2) and (2.3) and subbing into eq. (2.9) gives the following equation for the growth of matter perturbations in linear regime – 5 – in f(R) gravity: 4πG 1+ 4λ2k2a4 δ¨ +2Hδ˙ 3 1 ρ δ = 0. (2.18) m m− 1 1.4 10−8λ2a3 1+λ2k2a4 m m 1 1 − × In contrast to eq. (2.10) which is scale independent, growth in f(R) modified gravity models is dependent upon scale as seen above. More useful is the fact that the growth of matter perturbations are dependent upon the the value of λ21 or equivalently B0. A larger B0 will boostgrowth,albeitinascale-dependentway. Assuchcosmologicalobservationswhichprobe the growth of matter perturbations, such as the matter power spectrum (MPK) should be useful in placing constraints on the f(R) parameter B0. 3 Datasets We use the measurements of CMB temperature anisotropy1 [64] from the first data release of the Planck surveyor. Its temperature power-spectrum likelihood is divided into low-l (l < 50) and high-l (l 50) parts. This is because the central limit theorem ensures that the ≥ distributionofCMBangularpowerspectrumC inthehigh-lregimecanbewellapproximated l by a Gaussian statistics. However, for thelow-l part theC distribution is non-Gaussian. For l these reasons the Planck team adopts two different methodologies to build the likelihood. In detail, for the low-l part, the likelihood exploits all Planck frequency channels from 30 to 353 GHz, separating the cosmological CMB signal from diffuse Galactic foregrounds through a physically motivated Bayesian component separation technique. For the high-l part, a correlated Gaussian likelihood approximation is employed. This is based on a fine-grained set of angular cross-spectra derived from multiple detector power-spectrum combinations between the 100, 143, and 217 GHz frequency channels, marginalizing over power-spectrum foreground templates. In order to break the well-known parameter degeneracy between the reionization optical depthτ andthe scalar index n , thelow-l WMAP polarization likelihood s (WP) is used[64]. Finally, the unresolved foregrounds are marginalized over, assuming wide priors on the relevant nuisance parameters as described in [65]. In order to take advantage of the constraining power of data from large-scale structure, we use measurements of the galaxy power spectrum as made by the WiggleZ Dark Energy Survey2. As described in [53], we use the power spectrum measured from spectroscopic redshifts of 170,352 blue emission line galaxies over a volume of 1 Gpc3 [52]. The covariance matrices as given in [53] are computed using the method described by [66]. The best model proposedfornon-linearcorrectionstothematterpowerspectrumwasonethatwascalibrated against simulations (model G in [53]), and so this model may not be appropriate for this situation. Instead, it has already been demonstrated that linear theory predictions are as good a fit to the data as the calibrated model to k 0.2h/Mpc [53, 67]. Furthermore, ∼ recent work [68] has shown that the linear power spectrum produced by codes using the modified growth formalism detailed above reproduce quite accurately the power-spectrum obtained using N-Body simulations for f(R) gravity models out to k 0.2h/Mpc. The ∼ linear predictions, in fact, are more accurate than those made using the non-linear matter power spectrum fitting formula module, Halofit. For these reasons we restrict ourselves to scales lessthatkmax = 0.2h/Mpc andusethelineartheorypredictiononly. Wedoinvestigate 1http://pla.esac.esa.int/pla/aio/planckProducts.html 2http://smp.uq.edu.au/wigglez-data – 6 – the effect of a different cut-off scale of kmax = 0.1h/Mpc on our results, to test for possible non-linear systematic error. We also marginalise over a linear galaxy bias for each of the four redshift bins, as in [53]. Finally, in order to break other parameter degeneracies relating to late-time observables suchasΩ wealsousebaryonacousticoscillation(BAO)measurementsfromthe6dFGalaxy m Survey measurement at z = 0.1 [69], the re-analyzed SDSS DR7 [70, 71] at effective redshift zeff = 0.35, and the BOSS DR9 [13] surveys at zeff = 0.2 and zeff = 0.35. 4 Results & Discussion We used a Markov-Chain Monte Carlo method to obtain posterior constraints on the param- eters, using two separate modifications of the cosmological analysis code CosmoMC [72, 73], namely ISiTGR3 [74, 75] and MGCAMB [76, 77]. Since the constraints from the two codes were quite consistent, in what follows we primarily show the results from ISiTGR. In table 1 we list the priors on the various parameters used in our analysis. Cosmological Parameter symbol prior min max ΛCDM parameters Physical baryon density 100Ωbh2 0.5 10. Physical CDM density ΩCDMh2 0.001 0.99 Angularsize of thesound horizon at decoupling 100θ 0.5 10 Opticaldepth of reionization τ 0.01 0.8 Scalar spectral index ns 0.9 1.1 Amplitudeof thescalar perturbations log(As) 2.7 4 Parameters to extend the model Compton wavelength of thef(R) theory log(B0) -10 1 CMB lensing parameter ALens 0 10 Combined mass of theneutrino species (eV) Pmν 0 5 Derived parameters Matter density Ωm – – Hubbleparameter (kms−1Mpc−1) H0 – – Matter power dispersion at 8h−1Mpc σ8 – – Table 1. Cosmologicalparameters and prior ranges. The logarithmic prior on B0 is a non-informative prior, which gives equal weighting to the largest values allowed for the Compton wavelength (log10(B0) = 1 is equivalent to the gigaparsec scale) and the smallest values (log10(B0) = 10 tens of kiloparsecs). In − ∼ contrast a uniform prior on B0 would give additional probability weight to the largest scales, and would possibly bias the result. We do not expect to be able to detect the Compton wavelength on the kiloparsec scale, but assume this as a reasonable lower limit given we observe no modified gravity signal on galactic scales. Throughout this work we will use three different combinations of data sets. The first combination uses Planck + WP + BAO and is denoted PLC (standing for Plank likelihood code) hereafter. Next, in addition to the PLC data we add data from the WiggleZ galaxy power spectrum with data points out to kmax = 0.1h/Mpc We denote this data set as PLC + WiggleZ0.1. Finally, we include data from the WiggleZ galaxy power spectrum out to kmax = 0.2h/Mpc and denote this combination of data sets PLC + WiggleZ0.2. We analyze constraints on the parameters for three different models as well: 3http://isit.gr – 7 – Model I - f(R) gravity only, adding the parameter log10(B0); • Model II - f(R) + ALens, where we vary the additional CMB lensing amplitude • parameter ALens; and Model III - f(R) + ALens + mν, where we also vary the sum of the mass of active • neutrino species. P Note that we assume three massive, degenerate active neutrinos, and in Models I & II where the mass remains fixed we assume a value of m = 0.06 eV. ν 4.1 Model I - f(R) gravity P 1.0 1.0 f(R):Planck+WP+BAO f(R):Planck+WP+BAO 0.8 f(R):Planck+WP+BAO+WiggleZkmax=0.1 0.8 f(R)+AL:Planck+WP+BAO f(R):Planck+WP+BAO+WiggleZkmax=0.2 max0.6 max0.6 P P / / P P 0.4 0.4 0.2 0.2 0.0 0.0 −10 −8 −6 −4 −2 0 −10 −8 −6 −4 −2 0 Log10(B0) Log10(B0) Figure 1. Left panel: 1D posterior distribution of log10(B0) for our three different combinations of datasets. ThePLC+WiggleZ0.1 datasetyeildsa3σ detectionofanon-zeroB0. Extendingthedata points of the WiggleZ data set up to kmax =0.2h/Mpc, however,removesthis detection and places a very stringent constraint on B0. Right panel: The 1D posterior distribution of log10(B0) for Model I and II for the PLC data set. The high-probability peak at larger values of B0 for the PLC data set vanishes the addition of the ALens parameter, as discussed in section 4.2. Wefirstconsiderthecosmological constraints onf(R)gravity alone, withnoadditional, beyond the standard model, parameters. Our constraints on the f(R) parameter log10B0 are given in figure 1. We can quickly see that the that the addition of galaxy power spectrum form WiggleZ can improve the 95% upper bound of B0 by over three orders of magnitude log10(B0) < 0.44 ,(95%, Model I:PLC), (4.1) − log10(B0) < 4.07 ,(95%, Model I:PLC+WiggleZ0.2). (4.2) − This represents one of the tightest constraints B0 and thus f(R) models to date as previous constraints from CMB data sets alone gave B0 < 0.1 at 95%C.L. [25], and joint analysis of several large-scale structure tracers combined with CMB data from WMAP had given constraints of B0 < 1.1 10−3 at 95%C.L [33]. × The most noticeable results, as seen in the left panel of figure 1, are the two high probability peaks for non-zero values for B0 for the PLC and PLC + WiggleZ0.1 data set combinations. For the PLC data set this peak does not represent a significant detection of a non-zero B0 and is removed by adding ALens to the parameter analysis as discussed more in the next section. However, for PLC + WiggleZ0.1 this represents a more than 2σ detection of a non-zero B0 as log10(B0) = −1.42+−11..5656 (95%, Model I:PLC+WiggleZ0.1). (4.3) – 8 – f(R):Planck+WP+BAO+WiggleZkmax=0.1 f(R):Planck+WP+BAO+WiggleZkmax=0.2 1.20 σ81.05 0.90 0.0 B)0−2.5 g(10−5.0 o L −7.5 0.28 0.30 0.32 0.34 0.90 1.05 1.20 −7.5 −5.0 −2.5 0.0 Ωm σ8 Log10(B0) Figure 2. Plot showing joint constraints on cosmological parameters from combining Planck CMB andWiggleZgalaxypowerspectradata. The WiggleZ datais fit upto kmax =0.1h/Mpc(black)and up to kmax =0.2h/Mpc (red). We argue that this is due to the fact that for kmax = 0.1h/Mpc the power spectra data has a slight preference for a lower value Ωm, and hence a higher value of σ8 and B0 are needed to accommodate a fit to the CMB. This is illustrated well in figure 2 where we plot the two-dimensional joint posterior parameter distributions between the matter density, Ω , the m amplitude of clustering, σ8, and the f(R) parameter, log10(B0). This difference in best fit between WiggleZ0.1 and WiggleZ0.2 comes from the scale-dependent growth in f(R). In the f(R) model power on small scales is boosted, but when this combined with our free linear bias parameter, it results in a better fit at larger scales by predicting a slightly lower power for the small k-values. However, the scale-dependent growth of f(R) provides a worse fit for the smaller scale data, and so when we include the data out to kmax = 0.2h/Mpc, the data disfavors larger values of the Compton wavelength. Finally, in figure 3, we plot the 1D probability distributions for the core cosmological parameters using the WiggleZ power spectrum out to kmax = 0.2. With the exception of the low-probability tail for σ8, the best fit values for the standard cosmological parameters do not change much compared to the ΛCDM case. This is because the constraints on most of the core cosmological parameters come from the CMB, and are very well-constrained by Planck. We found this to be generally the case throughout our analysis. For the purposes of completeness, the means and standard deviations of all the cosmological parameters used in our fits are given in table 2 in appendix A. 4.2 Model II - f(R) gravity + ALens Wenowconsidermodelswhereinadditiontothef(R)modelparameterlog10(B0)wevarythe parameter ALens, the normalized the CMB lensing amplitude. This parameter has garnered a lot of attention since the Planck 2013 data release wherea 2σ deviation from unity ALens = 1.23 0.11 was found. This is significant because a deviation in the value of this parameter ± – 9 –