Low Redshift Baryon Acoustic Oscillation Measurement from the Reconstructed 6-Degree Field Galaxy Survey

MNRAS 000,1–13 (2018) Preprint 6 March 2018 Compiled using MNRAS LATEX style ﬁle v3.0

Low Redshift Baryon Acoustic Oscillation Measurement from the Reconstructed 6-degree Field Galaxy Survey

Paul Carter1?, Florian Beutler1,2, Will J. Percival3,4,1, Chris Blake5,8, Jun Koda6,7,8, Ashley J. Ross9 1Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK 2Lawrence Berkeley National Lab, 1 Cyclotron Rd, Berkeley CA 94720, USA 3Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada 4Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada 5Centre for Astrophysics & Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia 6Dipartimento di Matematica e Fisica, Universit´adegli Studi Roma Tre, via della Vasca, Navale 84, I-00146 Roma, Italy 7INFN Sezione di Roma 3, Via della Vasca Navale 84, Rome 00146, Italy 8ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) 9Center for Cosmology and AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT Low redshift measurements of Baryon Acoustic Oscillations (BAO) test the late time evolution of the Universe and are a vital probe of Dark Energy. Over the past decade both the 6-degree Field Galaxy Survey (6dFGS) and Sloan Digital Sky Survey (SDSS) have provided important distance constraints at z < 0.3. In this paper we re-evaluate the cosmological information from the BAO detection in 6dFGS making use of HOD populated COLA mocks for a robust covariance matrix and taking advantage of the now commonly implemented technique of density field reconstruction. For the 6dFGS data, we find consistency with the previous analysis, and obtain an isotropic volume fid averaged distance measurement of DV (zeff = 0.097) = 372 ± 17(rs/rs ) Mpc, which has a non-Gaussian likelihood outside the 1σ region. We combine our measurement from both the post-reconstruction clustering of 6dFGS and SDSS MGS offering the most fid robust constraint to date in this redshift regime, DV (zeff = 0.122) = 539±17(rs/rs ) Mpc. These measurements are consistent with standard ΛCDM and after fixing the standard 2 −1 −1 ruler using a Planck prior on Ωmh , the joint analysis gives H0 = 64.0±3.5 kms Mpc . In the near future both the Taipan Galaxy Survey and the Dark Energy Spectroscopic Instrument (DESI) will improve this measurement to 1% at low redshift. Key words: keyword1 – keyword2 – keyword3

1 INTRODUCTION perature of the Universe becomes comparable to the ion- isation energy of electrons and recombination occurs. The Utilisation of the baryon acoustic peak feature measured

arXiv:1803.01746v1 [astro-ph.CO] 5 Mar 2018 mean free path of photons becomes greater than the Hubble in the two-point statistics of redshift surveys, has been in- distance and they therefore decouple. This process leaves tegral in the advancement to precision cosmology. Within the baryonic matter distributed in overdensities in a sur- the framework of the concordance ΛCDM model the initial rounding spherical shell. These shells have a co-moving ra- matter perturbations were seeded through quantum fluctu- dius which corresponds to the sound horizon, r ∼ 150Mpc, ations during an early inflationary epoch. Following this the s at the surface of last scattering. Through mutual gravita- Universe existed in a plasma state, in which radiation and tional interaction the dark matter component grows with baryonic matter are strongly coupled through the process of the baryonic component to emulate this feature (Eisenstein Thompson scattering. et al. 2007a). Galaxies form in regions of overdensity and are Interplay between gravitational attraction and radia- biased tracers of the underlying dark matter field on large tion pressure introduces acoustic oscillations in the primor- scales. dial photon-baryon fluid. At z ≈ 1100 the background tem- Making use of the two-point correlation function ξ it is possible to see the effect of BAO as a peak, centered about ? E-mail: [email protected] separation rs. Alternatively the power spectrum gives equiv-

© 2018 The Authors 2 P. Carter et al. alent information in Fourier space, with the signal translat- ing to oscillations in amplitude with wavenumber. Measure- ments of the apparent scale of the BAO features, offer a robust standard ruler of the distance to the measured galaxy population’s effective redshift. Measurements of this feature at different redshifts allow for a test of the general cosmological model, especially inference regarding the equation of state of Dark Energy and spatial curvature. The BAO peak is a well studied probe of cosmology, first detected in both the initial Sloan Digital Sky Survey (SDSS; York et al. 2000) and 2-degree Field (2dFGRS; Colless et al. 2001) galaxy redshift surveys (Percival et al. 2001; Eisen- stein et al. 2005; Cole et al. 2005). Subsequently the BAO Figure 1. The sky coverage of the 6dFGS K-band sample, the peak has been detected in later SDSS data releases (Percival colour of each cell corresponds to the completeness in that re- et al. 2010; Kazin et al. 2010), 6dFGS (Beutler et al. 2011), gion. This total completeness constitutes a combination of sky WiggleZ (Blake et al. 2011), BOSS (LOWZ and CMASS) and magnitude completeness of galaxies. (Alam et al. 2017), eBOSS luminous red galaxies (LRGs) (Bautista et al. 2017) and quasars (QSOs) (Ata et al. 2018) and a higher redshift detection using Ly-α forest measure- vide a basis on which to develop high-fidelity COLA-based ments in BOSS (Slosar et al. 2013; Font-Ribera et al. 2014; mock catalogues. Section4 gives a summary of the cluster- Delubac et al. 2015). These measurements have constructed ing statistic measurement made during this work and the a distance ladder that spans from z = 0 out to z ∼ 0.8 using formalism used for density field reconstruction. Section5 conventional galaxy redshift surveys, z ∼ 1.5 through eBOSS provides the model used and constraints extracted both be- QSO and to z ∼ 2.3 when including Ly-α. fore and after reconstruction. Following this a number of tests regarding the mock population and robustness of re- In recent analyses of the BAO peak, a method of den- sults are conducted in Section6. Section7 provides a joint sity field reconstruction has been employed. Eisenstein et al. analysis with SDSS DR7 MGS data and the cosmological (2007b) proposed that, as the bulk flows that smear the interpretation of theses results. Finally, our conclusions are acoustic peak are sourced from the density field potential given in Section8. itself, the galaxy map can itself be used to estimate the dis- Our analysis uses a fiducial flat ΛCDM model cosmol- placement field. Removal of these shifts has been shown to ogy with parameters Ωfid = 0.31, Ωfid = 0.69, hfid = 0.67 reduce the damping of the BAO and increase the S/N of this m Λ −1 −1 feature. This increased S/N results from higher order statis- where H0 = 100 hkm s Mpc and a fiducial sound horizon tics which have been moved back into linear fluctuations size at the drag epoch is rs(zd) = 147.5 Mpc. (Schmittfull et al. 2015). Density field reconstruction has been applied in recent work including SDSS (Padmanabhan et al. 2012; Ross et al. 2 THE 6DF GALAXY SURVEY 2015), WiggleZ (Kazin et al. 2014) and throughout BOSS The 6dF Galaxy Survey1 (6dFGS; Jones et al. 2009) com- (Alam et al. 2017). These studies use either a perturbation bines peculiar velocity and redshifts for galaxies covering theory based approach that relies on the finite difference almost the entire southern hemisphere. The survey was con- method (Padmanabhan et al. 2009; Noh et al. 2009), or ducted between 2001 and 2006 using the 6-degree Field an alternative FFT-based iterative algorithm (Burden et al. multi-fibre instrument on the UK Schmidt Telescope and 2014, 2015). covers 17, 000 deg2 of the sky with a median redshift 0.053. This paper explores the application of density field re- The sample selected for this work is the same as that used construction to 6dFGS following from the initial detection of in Beutler et al.(2011), selected with a magnitude cut of the BAO peak by Beutler et al.(2011). Aside from provid- K < 12.9 and imposing a cut for regions with < 60% coming post-reconstruction constraints, the paper also improves pleteness (Jones et al. 2006). This gives a catalogue of 75,117 on the analysis of errors, using high fidelity COLA-based galaxies. In Figure1 the sky coverage and total completeness mock catalogues (Koda et al. 2016) to provide more robust distribution of galaxies in this final catalogue are presented. covariance matrices than the lognormal approach adopted Further details regarding the structure and systematics in previously. We also combine with the Ross et al.(2015) low- the full 6dFGS can be found in Jones et al.(2009). est redshift measurement, using the SDSS-II Main Galaxy Throughout this work galaxies have S/N (FKP) weights Sample (MGS). This combined low-redshift measurement is applied (Feldman et al. 1994). These weights w (z) = useful in providing further constraining power on a direct FKP 1/(1 + n(z)P ), optimise between sample variance and shot measurement of H for cosmological tests of dark energy and 0 0 noise on the k-scale which is specified by P the amplitude curvature. Also this offers an independent test of the Hub- 0 of the power spectrum and n(z) is the number density. 6dFGS ble constant at the redshift of the supernovae surveys, for has a high density gradient with redshift, and therefore this which there is currently tension with other BAO and CMB weighting scheme has a strong impact on the analysis. We studies. This paper is organised as follows: Section2 offers an overview of the final 6dFGS dataset and describes the spec- 1 The data, randoms and mock catalogues ifications of the catalogue used. Section3 outlines the use used during this work are available at: of halo occupation distribution (HOD) modelling to pro- http://www.6dfgs.net/downloads/6dFGS Recon Files.tar.gz

MNRAS 000,1–13 (2018) Low-z BAO from the Reconstructed 6dFGS 3

time-stepping for the small scales, COLA outperforms clas- 1600 120 sical N-body simulations in terms of speed of computation zeﬀ 1400 by orders of magnitude. 100 To run the COLA simulations we used 432 cores and 1200 8GB of memory per core on the Raijin supercomputer at the 80 Australian National Computational Infrastructure (NCI),

1000 )

z with each run taking ∼ 45 minutes. The mocks produced ( )

z 3 −1 N ( consist of (1728) particles in boxes with 1.2h Gpc on each

800 60 ) z N ( side. The number of time steps of the COLA simulation is 600 w 40 20, down to z = 0. The mass resolution of the mock catalogues is . × 10 h−1 M and a friends-of-friends (FoF) 400 2 8 10 finder is used to locate haloes that consist of a minimum of 20 200 32 dark matter particles. The initial fiducial cosmology used in these mocks is Ωm = 0.3, Ωb = 0.0478, h = 0.68, σ8 = 0.82 0 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 and ns = 0.96. To ensure our simulation matches the data z catalogue, snapshots at z = 0.1 were used to build the mock surveys, close to the effective redshift of the data. The fundamental mass resolution of the mock catalogue Figure 2. The unweighted (blue) and FKP weighted (red) red- does mean that we are not able to replicate the low mass shift distribution of 6dFGS, with the best fitting model of Nfit(z) haloes observed at low redshift. This will lead to a discrep- to the unweighted distribution. The effective redshift of the ancy between data and mock at z < 0.05, in the HOD pop- 6dFGS in given as the black vertical line. ulation. This does not impact the BAO analysis on large scales; the reduction of number density at low redshift from −3 3 a lack of these low mass galaxies only contributes to a re- chose to use P0 ∼ 10, 000 h Mpc approximately the ampli- −1 duction in effective volume of 1.0%. tude where the BAO signal peaks, keff ∼ 0.15 hMpc (Seo & Eisenstein 2007). A fit to the unweighted redshift distribution is used to calculate n(z) for the weighting. The form γ γ of this model is Nfit(z) = Az exp[−(z/zp) ] with the best fit parameters A = 456500, zp = 0.03967 and γ = 1.5369 (Jones 3.2 Halo Occupation Distribution Model et al. 2009). The redshift distribution both unweighted and once the FKP weighting scheme has been applied are shown As is common in HOD modelling, we separate the clustering in Figure2. The effective redshift of the total weighted sam- contribution between massive central and satellite galaxies. The probability of a dark matter halo of mass M, hosting ple is zeff = 0.097 where the effective redshift is defined as, each type is respectively (Zheng et al. 2005),

Ng Í 1 log10(M) − log10(Mmin) ziwi(z) hNC (M)i = 1 + erf , (2) i=0 2 σlog M zeff = . (1) Ng Í wi(z) and, i=0 α M − Mmin hNS(M)i = , (3) M1 3 COLA-BASED MOCK CATALOGUES in which M is the minimum dark matter halo mass that Mock catalogues are produced based on numerical simula- min can host a central galaxy, M is the mass of a halo that on tions and populated using a Halo Occupation Distribution 1 average contains one additional satellite member and σ (HOD) model. We construct our covariance matrix from 600 log M (Zehavi et al. 2011) allows for a gradual transition from of these realisations which simulate the survey volume. hNC (M)i = 0 to hNC (M)i = 1. When assigning the satel- lites to the halo, peculiar velocities are randomly selected based on the virial velocity of the halo assuming a spherical 3.1 COLA Simulations Navarro et al.(1996) profile. As a dark matter halo can only In an ideal world the underlying dark matter haloes would host a satellite if it already has a central galaxy, the total be generated using N-body simulations, to encapsulate fully halo occupation is, the non-linear regime, however this is computationally ex- pensive. Instead we make use of the COmoving Lagrangian hNt (M)i = hNC (M)i (1 + hNS(M)i) . (4) Acceleration (COLA) method (Tassev et al. 2013, Howlett et al. 2015, Koda et al. 2016) to produce our 6dFGS mock The halo model accounts separately for clustering be- catalogues. tween galaxies in the same halo (1-halo term ξ1h(r)) and The COLA method allows for an increase in effi- those in different haloes (2-halo term ξ2h(r)), which con- ciency by combining N-body simulations with 2nd-order La- tribute to the correlation function on small scales and large grangian Perturbation Theory (2LPT). By allowing 2LPT scales respectively. The total correlation function is just the to solve the large scale distribution exactly and only using superposition of these terms ξ(r) = ξ1h(r) + ξ2h(r).

MNRAS 000,1–13 (2018) 4 P. Carter et al.

3.2.1 Fitting of the HOD model where DD(s), DR(s) and RR(s) are the pair counts of data- data, data-random and random-random pairs split in co- To replicate the data, both the redshift distribution n(z) and moving space. The random catalogues used during this anal- projected correlation function w (r ), p p ysis were generated using the selection function measured πmax from the data catalogue. This Monte-Carlo samples the sky ¹ wp(rp) = 2 dπξ(rp, π), (5) coverage (as given in Figure1) and redshift distribution using 30 times as many random points as galaxies, to ensure 0 that the Poisson noise contribution from the randoms is sig- are used to measure the HOD parameters. rp and π corre- nificantly smaller than that from the galaxies. The normal- spond to the separation between galaxies perpendicular and isation of pair counts is performed with nd and nr which parallel to the line-of-sight respectively. When integrating are the weighted number of objects for data and random −1 along the line-of-sight we use πmax = 50h Mpc, producing a catalogues respectively. statistic that has negligible contribution from the Fingers- The data-data pair counts have been assigned a fibre of-God (FoG) redshift space distortion (RSD) effect. collision weight for each galaxy pair. The design of the 6dF The redshift dependence of the HOD model is parame- instrument only allowed fibres to be placed > 5.7 arcmin terised through a polynomial form on Mmin and M1, apart resulting in lost pairs on small angle separation. By 2 3 4 allowing for multiple passes in the targeting plan, this effect log (Mmin(z)) = a + bx + cx + dx + ex (6) 10 of lost pairs is minimised for 70% of the survey area. The where x = z − 0.05. We generate mocks for a grid of HOD remaining 30% requires an up-weighting of pair counts to parameters (log10 (M1/Mmin), σlog M , α) and measure the ensure we do not bias the clustering statistics. This angular projected correlation function and number density to fit weight wf (θ12) is calculated as a ratio between expected and against the data. We find the best fitting HOD parameters observed angular correlation functions (Jones et al. 2004; of α = 1.50 ± 0.05, σlog M = 0.50 ± 0.12 and log10 (M1/Mmin) = Beutler et al. 2013) where θ12 is the angle between galaxies in 1.50 ± 0.08 where the polynomial terms are a = 12.0448, the pair. This approximation works when radial clustering is b = 22.8194, c = 110.4364, d = −1435.6529 and e = 3679.3770. matched between observed and unobserved pairs. On small The match between the best fit mock and the data are scales this is not true and we should instead use a scheme shown in Figure3 and Figure4. The mocks are a good fit at such as that of Bianchi & Percival(2017). As we are only z > 0.05, but at a lower redshift this breaks down due to the interested in the BAO scale, this is not necessary for this small scale mass resolution issue in the COLA simulation analysis. mentioned earlier. We assign a central galaxy to a halo according to the 4.2 Covariance Matrix probability < NC (M) >, and draw a random number of satel- lites from a Poisson distribution with mean < NS(M) > if the A robust covariance matrix was generated from 600 mock halo hosts the central galaxy. The position and the velocity realisations, built on COLA-based simulations and popu- of the central galaxies are those of the mean of the halo lated through a HOD parameterisation. Further details of particles (center of mass). For satellite galaxies, we assume the mock production are given in Section3, along with the an isotropic NFW profile (Navarro et al. 1996) with a con- implementation of the HOD model. centration parameter (Bullock et al. 2001). We randomly The covariance matrix is given by, assign the distance from the halo centre based on the mass N distribution and a Gaussian random velocity based on the Õ (ξn(si) − ξ(si))(ξn(sj ) − ξ(sj )) C = , (8) radius-dependent velocity-dispersion profile in addition to ij N − 1 the halo velocity (van den Bosch et al. 2004; de la Torre n=1 et al. 2013). in which the summation runs over N mock realisations. ξn(si) is the ith separation bin of the nth mock correlation function and ξ(si) is the average in this bin. The uncertainty on the correlation functions shown in Figures6,7 and8 are from 4 OVERVIEW OF CLUSTERING √ the standard deviation on the diagonal C , while during MEASUREMENT ii the fitting of models the full covariance matrix is used. We choose to measure the BAO signal using the correlation The covariance matrix, calculation without reconstruc- function to facilitate comparison with (Beutler et al. 2011). tion is compared to that previously used based on lognormal In this section we briefly explain how we measure the corre- catalogues (Beutler et al. 2011) in Figure5. This compar- lation function and covariance matrix. We will also introduce ison shows that the correlation between neighbouring bins the density field reconstruction technique which we apply to has decreased using the COLA mocks. The correlation func- the dataset. tions of the mock catalogues are compared to that from the data in Figure6 showing good agreement. These mock catalogues have been used in other recent work on the 6dFGS 4.1 Two-Point Correlation Function (Achitouv et al. 2017; Blake et al. 2018). We measure the two-point correlation statistic using the Landy & Szalay(1993) estimator, 4.3 FFT-based Density Field Reconstruction DD(s) n 2 DR(s) n The formalism used for density field reconstruction follows ξ(s) = 1 + r − 2 r , (7) RR(s) nd RR(s) nd from (Burden et al. 2014, 2015). In a Lagrangian framework

MNRAS 000,1–13 (2018) Low-z BAO from the Reconstructed 6dFGS 5

103 103 103 0 < z < 0.2 0.05 < z < 0.1 0.1 < z < 0.2

102 102 102 ) p r

( 1 1 1 p 10 10 10 w

100 100 100

10 1 10 1 10 1 − 100 101 − 100 101 − 100 101 2.0 2.0 2.0 p w 1.5 1.5 1.5 /σ ) 1.0 1.0 1.0 Mock ) p 0.5 0.5 0.5 r ( p

w 0.0 0.0 0.0

− 0.5 0.5 0.5 − − − Data

) 1.0 1.0 1.0 p

r − − − ( p 1.5 1.5 1.5 w

( − − − 2.0 2.0 2.0 − 100 101 − 100 101 − 100 101 1 1 1 rp (h− Mpc) rp (h− Mpc) rp (h− Mpc)

Figure 3. Comparison between the projected correlation function of the data realisation and the mean of the mock catalogues where the blue line shows the COLA mock and the black points the data. There is good agreement at z > 0.05, the mocks show a larger amplitude in the clustering in the 1-halo component because of the low mass resolution. The shaded grey regions correspond to the standard deviation.

0 10 200 Mocks 1 mock 10− 180 1.0 0.9 2 data 160 10− 0.8 3 140 0.7 10−

Mpc) 0.6 ) 120 1

z 4 − ( 10− 0.5 h n ( 100

5 s 0.4 10− 80 0.3 6 10− 60 0.2 7 0.1 10− 40 Lognormal 0.0 8 40 60 80 100 120 140 160 180 200 10− 1 0.05 0.10 0.15 0.20 s (h− Mpc) z

Figure 5. Comparison of the COLA mock and lognormal based Figure 4. The redshift distribution of the data realisation and the correlation matrices. The correlation in the mock covariance ma- mean of the mock catalogues. Consistency is seen in the redshift trix has been reduced in comparison to the lognormal based ma- range z > 0.05, while at z < 0.05 the mock shows a lower density trix. to the resolution limit of the COLA mocks.

the Eulerian position of a particle is given by, where the q is the Lagrangian position and Ψ is the displacement vector ﬁeld. Implementing ﬁrst order Lagrangian x(q, t) = q + Ψ(q, t), (9) Perturbation Theory (LPT) the standard Zel’dovich approx-

MNRAS 000,1–13 (2018) 6 P. Carter et al. imation (Zel’dovich 1970) can be obtained, covariance matrix used was built from the set of 600 mocks ik correcting for statistical bias in the standard way (Hartlap Ψ( )(k) = − δ( )(k), (10) et al. 2007). 1 k2 1 which relates the Fourier transform of the overdensity field to the displacement field in k-space. To linear order galaxies 5.1 Template Formalism trace the matter density field as δg = bδm where b is the bias. Because of the redshift space distortions (RSDs), to The fitting of the correlation function relies primarily on obtain the displacement field Ψ we actually have to solve a template model with damped BAO given in Eisenstein the differential equation, et al.(2007a). The model power spectrum that is used in this template contains a linear model P (k) from CAMB f δg lin ∇ · Ψ + ∇ · (Ψ · rˆ)rˆ = − , (11) (Lewis et al. 2000) and a no-wiggle power spectrum P (k) b b nw (Eisenstein & Hu 1998), On linear scales RSD enhances the clustering along the line- of-sight, dependent on the amplitude of f = d ln(D(a))/d ln(a) mod Plin(k) − 1 k2Σ2 P (k) = Pnw(k) 1 + − 1 e 2 nl . (14) the growth rate, D(a) the growth function, a the scale factor Pnw(k) and σ describes the amplitude of the linear power spectrum 8 The no-wiggle model has had the BAO feature removed on scales of 8 h−1Mpc. whilst retaining the overall shape. The template mod( ) is Equation 11 can be solved as in Padmanabhan et al. ξ s then obtained from this through a Hankel transform and (2012) using a finite difference approximation to compute used with 5 nuisance parameters ( , , , , Σ ) which the gradients. This sets up a grid in configuration space Bξ a1 a2 a3 nl allow marginalisation of the BAO damping (the scale of through which the potential can be described as a linear which is set by Σ ) and broadband shape of the correlation system of equations. This methodology was chosen because nl function, although Ψ is irrotational, the term (Ψ · rˆ)rˆ is not, hence you cannot locate the solution directly with Fourier meth- fit 2 mod a1 a2 ξ (s) = B ξ (αs) + + + a3. (15) ods. However Burden et al.(2015) showed that by making s2 s the approximation that (Ψ · ) can be irrotational and iter- rˆ rˆ α is defined as, ating after correcting, one can efficiently obtain the correct solution using FFTs (with IFFT referring to the inverse fast D (z )rfid α = V eff s , (16) fourier transform) with β = f /b, fid( ) DV zeff rs ikδ(k) β ikδ(k) Ψ = IFFT − − IFFT − · rˆ rˆ, (12) and allows one to freely scale the BAO feature in our model. 2 1 + β 2 k b k b A best fit value of α < 1 corresponds to a larger BAO scale The displacement field calculated from this form of the al- compared to the fiducial cosmology used to construct our gorithm has been shown to agree with the finite difference template and α > 1 indicates a smaller BAO scale. DV is approach and causes negligible differences between post- the volume averaged distance, reconstruction 2-point statistics Burden et al.(2014). 1/3 h 2 2 −1 i To also remove RSD we modify the displacement vector DV = cz(1 + z) D (z)H (z) . (17) final A as Ψ = Ψ+ΨRSD (Kaiser 1987; Padmanabhan et al. 2012) where,

ΨRSD = − f (Ψ · rˆ)rˆ, (13) 5.2 Pre-Reconstruction using the already calculated displacement field along the line-of-sight. This retrieval of the real-space post- To provide constraints on the pre-reconstruction BAO peak reconstruction density field results in the reduction of am- the above template was utilised through a Likelihood based plitude in the correlation function (Figure6). routine similar to that outlined in Ross et al.(2015). The model has 6 free parameters (B, α, a1, a2, a3, Σnl) and we assume a Gaussian prior on the non-linear damping scale Σ = 10.3 ± 2.0 h−1Mpc as measured from the mean of the 5 CORRELATION FUNCTION MODELLING nl mocks (discussed in Section6). The 1σ and 2σ uncertainties In Beutler et al.(2011) a model for the pre-reconstruction are defined at the ∆χ2 = 1 and 4 levels respectively after correlation function was constructed from the formalism marginalisation over the nuisance parameters. of Crocce & Scoccimarro(2008). Here we use a model The best fit correlation function is compared to the data in that marginalises over the shape of the correlation func- Figure7. We also show the χ2 distribution. The marginalised tion through a number of nuisance parameters (Anderson constraint on the shift parameter α pre-reconstruction et al. 2014). This freedom to marginalise over the broadband is α = 1.018 ± 0.067 with χ2/ν = 8.86/11 = 0.81, shape is important when modelling the post-reconstruction which is consistent with Beutler et al.(2011) who used correlation function because we do not have a physically a WMAP7 fiducial cosmology and found α = 1.036 ± motivated model. 0.062 (the α fit from our work translates to αWMAP7 = ( ) ( ) [( Planck/ Planck)/( WMAP7/ WMAP7)] ± The fit of the model m s to the data d s is determined αPlanck DV rs DV rs = 1.047 0.069 through a minimum-χ2 fitting, χ2 = DT C−1D, where D = for comparison in the WMAP7 fiducial model). The differ- d(s) − m(s). In our standard analysis we use a fitting range ence in uncertainty is due to the revised covariance matrix of 30 h−1Mpc < s < 200 h−1Mpc over 17 bins. The inverse (see Section 4.2 for a comparison).

MNRAS 000,1–13 (2018) Low-z BAO from the Reconstructed 6dFGS 7

6 TESTS ON THE MOCK CATALOGUES 200

150 6.1 Fitting the mean of the mocks We ﬁt both the pre/post reconstruction average correla- 100 tion functions from the mock population. This ﬁt makes use of the covariance matrices, which have been rescaled 50 by the number of realisations N, Cmean = Cone/N. The best ) s

( ﬁt model is shown in Figure8 with αpre = 0.999 ± 0.0065

ξ 0 2 and α = 0.997 ± 0.0035 giving an improvement factor of s post 50 I = σα,pre/σα,post ∼ 1.86. − 100 − Pre Recon Mock − Post Recon Mock − 150 Pre Recon Data − − Post Recon Data 200 − − 0 50 100 150 200 1 6.2 Comparison between data and fits to s(h− Mpc) individual mocks To make comparisons between the mock population and the Figure 6. The distribution of all 600 mock correlation functions, data, each of the 600 realisations used to produce the co- both pre- (blue) and post-reconstruction (red). The data points variance matrices were individually fit. ∼ 30% of the mock show the 6dFGS correlation function with errors from the diago- catalogues in the population did not have a well constrained nal of the constructed COLA mock covariance matrix. The post- measurement of α, a similar fraction to that found for the −1 reconstruction data points have been displaced by +1 h Mpc for SDSS MGS analysis (Ross et al. 2015). To ensure that this clarity. comparison was performed only on the mocks that have a relevant detection of the BAO feature, we select a subsample having 1σ contours (∆χ2 = 1) within the prior region 5.3 Post-Reconstruction 0.7 < α < 1.3 (both pre- and post-reconstruction). This cut reduces the population to 70% of its original number. Com- We apply density field reconstruction to both the data and parisons of the distributions of best-fitting χ2, the value of α COLA mock catalogues. In doing so a galaxy bias of b = 1.82 pre/post-reconstruction and σα the 1σ error bound (Fig.9) (Beutler et al. 2011) and a growth rate of f (zeff = 0.097) = show that the data realisation is within the locus of mea- 0.579 were assumed. Our results are independent of this surements from this subsample of the mocks. choice as shown later in Section 6.3. When calculating the The distribution of errors on α, shows that for our displacement field, the overdensity field was smoothed us- mocks we see 80% of these mocks have σ /σ > 1, 2 α,pre α,post ing a Gaussian smoothing kernel, S(k) = e−(kR) /2, with the as would be expected after applying density field recon- smoothing scale of R = 15 h−1Mpc. Using the calculated struction. Our detection in the data of the BAO peak us- displacement field as a proxy for the non-linear evolution, ing our model, which marginalises over broadband shape, the catalogues were shifted to move the field back into the is at the ∼ 1.9σ level pre-reconstruction and ∼ 1.75σ psuedo-linear regime. The reconstruction algorithm gave a post-reconstruction. In the subsample of the mocks pre- mean galaxy shift of s¯ = 5.87 h−1Mpc. reconstruction 18% have a detection higher than this and The correlation function post-reconstruction has an es- post-reconstruction this increases to 52%. The mean detec- timator that includes both a shifted random S and indepen- tion level pre/post-reconstruction for the mocks is 1.5σ and dent unshifted random catalogues R (Padmanabhan et al. 1.8σ respectively. This increase in the number of high signifi- 2012), cance detections in the mocks shows the expected trend from density field reconstruction to, on average, enhance the sig- ! nificance of detection. The lower left plot in Figure9 shows SS(s) n 2 DD(s) n 2 DS(s) n2 ξ(s) = r + r − 2 r . (18) however that in 28% of cases, in the mock sample, recon- RR(s) ns RR(s) nd RR(s) ndns struction lowers the significance of detection. The reduction of significance seen in the data realisation is therefore likely As in the pre-reconstruction case, we use the smoothing a case of the data being one of these “unlucky” samples. parameter fit from the mean of the mocks with a prior Σnl = When comparing the pre-reconstruction significance detec- −1 4.8 ± 2.0h Mpc. The marginalised constraint on the shift tion to Beutler et al.(2011) we find a lower value, this is due ± (±0.235) parameter α post-reconstruction is α = 0.895 0.042 0.079 to a number of conservative alterations. These include (1) with χ2/ν = 9.49/11 = 0.86 (the error in the bracket show- a change in the fitting model to include polynomial terms ing the non-Gaussianity at the 98% confidence level). The that marginalise over the shape giving more freedom, (2) a model, data and χ2 distribution are displayed in Figure7. change of fitting range from 10 h−1Mpc < s < 200 h−1Mpc The α post-reconstruction found has a non-Gaussian likeli- to 30 h−1Mpc < s < 200 h−1Mpc and (3) the new robust co- hood beyond the 1σ region which means that although the variance matrix made from the COLA-based mocks rather value is displaced from the average, it is still consistent with than log-normal realisations (but which has slightly larger the mock catalogues (discussed further in Section6). covariance amplitude).

MNRAS 000,1–13 (2018) 8 P. Carter et al.

0.05 0.05

0.04 0.04

0.03 0.03

0.02 0.02 ) ) s s ( ( ξ 0.01 ξ 0.01

0.00 0.00

0.01 0.01 − − 0.02 0.02 − 40 60 80 100 120 140 160 180 200 − 40 60 80 100 120 140 160 180 200 1 1 s(h− Mpc) s(h− Mpc) 18 17 16 16 15

14 14 2 2

χ χ 13 12 12 11 10 10 8 9 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 α α

Figure 7. Left: In the upper plot the pre-reconstruction correlation function is given with the best fit model, corresponding to α = 1.018 ± 0.066. This shows consistency with the results from Beutler et al.(2011) in which they used a lognormal covariance matrix. The plot below provides the marginalised χ2 distribution for α. Right: In the upper plot the post-reconstruction correlation function is given with the best fit model, corresponding to α = 0.895 ± 0.041. The plot below provides the marginalised χ2 distribution for α. There is a strong bi-modality in this likelihood surface, which translates to strongly non-gaussian errors at the 2σ (∆χ2 = 4) level. This means that although the best fit for the 6dFGS post-reconstruction constraint was found at α < 0.9, the likelihood also gives weight to α ∼ 1.04 when combining with MGS measurements (Section 7.2).

6.3 Robustness Tests binning, the post-reconstruction analysis was conducted with correlation function binning ∆s = 8 h−1Mpc, 12 h−1Mpc To investigate the robustness of the post-reconstruction α and 10 h−1Mpc displaced by 5 h−1Mpc in comparison to the detection and the uncertainty on this parameter, tests are standard pipeline. run where parameters are systematically varied in both the When testing the robustness of the post-reconstruction fitting procedure and density field reconstruction. result the input survey linear bias is varied from b = 1.82 by ± During the post-reconstruction analysis, our default 0.15 and also the smoothing kernel used on the density field is varied from −1 by ± −1 . It should be noted procedure uses 17 bins between 30 h−1Mpc < s < 200 h−1Mpc 15h Mpc 5h Mpc and takes the non-linear damping as that from the mean of that the covariance matrix used during these reconstruction the mocks with a small prior. To test the robustness of the input parameter tests is still derived from the default recon- procedure, the scale over which the fitting occurs and also struction. Hence this mostly tests the robustness of α and the damping scale are varied. The damping scale was fixed does not test σα. Test results are collated in Table.2 and both at the value for mean of the mocks and also held at described below: −1 Σnl = 0 h Mpc (the value that given complete freedom this parameter tends towards). In testing the robustness against (i) Having a best-fit α ∼ 0.9 is robust for all test cases.

MNRAS 000,1–13 (2018) Low-z BAO from the Reconstructed 6dFGS 9

120 Table 1. The results of a number of robustness tests applied to Pre Recon the 6dFGS post-reconstruction data. The first set of tests vary − 100 Post Recon the scale over which our template is fit, the second set uses vari- − 80 ations of the damping scale fixed at both the exact value located from the mean of the mocks and 0 h−1Mpc (which is the value 60 preferred if left completely free). Thirdly we vary the binning used in the correlation function when fitting against the model

) 40 −1 −1 −1 s from 10 h Mpc to 8 h Mpc, 12 h Mpc and also displacing the ( − ξ bin centres by h 1 . During reconstruction parameter tests 2 5 Mpc

s 20 we vary the linear galaxy bias b input ±0.15 and ﬁnally the scale −1 0 of smoothing R in the Gaussian kernel by ±5 h Mpc.

20 2/ − Test α χ ν 40 Normal Pipeline 0.895 ± 0.042 9.49/11 − 60 Fitting Procedure − 40 60 80 100 120 140 160 180 200 1 s(h− Mpc) (i) 20 h−1Mpc < s < 200 h−1Mpc 0.902 ± 0.041 10.78/12 40 h−1Mpc < s < 190 h−1Mpc 0.895 ± 0.032 8.33/9 −1 −1 Figure 8. The best fit to the pre/post-reconstruction mean of the 50 h Mpc < s < 180 h Mpc 0.895 ± 0.035 8.20/7 mock correlation functions. The improvement factor is I ∼ 1.86 −1 (ii) and the best fit non-linear damping scales of Σnl = 10.3 h Mpc Σ −1 ± / −1 nl = 4.5 h Mpc fixed 0.902 0.041 9.50 12 and Σnl = 4.8 h Mpc, respectively. −1 Σnl = 0 h Mpc 0.887 ± 0.031 9.42/12 (iii) ∆ −1 ± / However the uncertainty of α depends on the scales fitted. s = 8 h Mpc 0.913 0.112 11.93 15 ∆s = 12 h−1Mpc 0.929 ± 0.036 5.91/7 Changing the scale used in these three cases varied the un- ∆s = 10 h−1Mpc (+5 h−1Mpc) 0.865 ± 0.050 9.44/11 certainty by ∆(σα) = 0.01. (ii) Given complete freedom to the non-linear damping Reconstruction Parameters the scale which minimises χ2 is 0h−1Mpc, although this is (iv) only weakly preferred in comparison to the value found from b = 1.67 0.895 ± 0.052 9.89/11 2 the mean of the mocks, ∆χ = 0.08. However, although it b = 1.97 0.894 ± 0.044 9.893/11 2 does not make a large difference in χ it does impact σα −1 at the ∆(σ ) = 0.01 level. As the use of Σ = 0h−1Mpc R = 10 h Mpc 0.908 ± 0.044 8.95/11 α nl −1 would likely underestimate the uncertainty present, our de- R = 20 h Mpc 1.053 ± 0.045 9.88/11 fault procedure is to place a small prior on the value cen- tred on the mean recovered from the mocks. The value of cal constraints. Combining the joint clustering measurement 2 α though remains robust against the chosen value of the with Ωmh from the CMB calibrates the standard ruler. smoothing scale. (iii) We have also tested changing the binning of the cor- 7.1 6dFGS post-reconstruction only relation function and covariance matrix from our default of 10 h−1Mpc to 8 h−1Mpc, 12 h−1Mpc and also displacing the The best fit constraint post-reconstruction translates to −1 ( ) fid( )( / fid) ± bin centres by 5 h Mpc whilst retaining the default width. DV zeff = 0.097 = αDV zeff = 0.097 rs rs = 372 In all cases the value of α post-reconstruction varies by up to ( / fid) fid( ) 17 rs rs Mpc with DV zeff = 0.097 = 416 Mpc. This result ∼ 4% about the default case and σα is within ∆(σα) = 0.01, is consistent with our fiducial cosmology at the 2σ level. Al- −1 except for the 8 h Mpc. In this case the non-Gaussian like- though the measured value of α is reasonably far from 1, the lihood has given more weighting to the sub-dominant peak likelihood shows a strong bimodal shape (see Figure7). widening the overall distribution at the 1σ level. (iv) When varying the linear bias used during density field reconstruction by ±0.15 and also changing the smoothing 7.2 Combined Analysis with SDSS DR7 MGS kernel scale, α is found to only change by 1.5% except for in −1 The SDSS DR7 Main Galaxy Sample was analysed using the the case of using R = 20h Mpc. In this case there is a large density field reconstruction technique in Ross et al.(2015), shift in α which is driven by the sub-dominant peak in the finding a 4% distance measurement at zeff = 0.15. By com- non-Gaussian likelihood of α becoming dominant. bining the likelihoods from SDSS MGS and 6dFGS post- reconstruction we consider the relative contributions of both surveys. As there is less than 3% angular overlap between SDSS MGS and 6dFGS and the redshift distribution is suf- ficiently different, the covariance between surveys has a neg- 7 COSMOLOGICAL INTERPRETATION ligible contribution to the result. In this section we convert our measurements of α to a vol- A joint constraint between the post-reconstruction ume averaged distance measurements and combine with a 6dFGS and SDSS MGS is obtained by multiplicatively com- 2 Ωmh prior from Planck 2015 to offer H0 and Ωm cosmologi- bining the likelihoods, taking the consensus ξ(s)+ P(k) result

MNRAS 000,1–13 (2018) 10 P. Carter et al.

1.3 0.30 1.2 0.25 22% 78% 1.1 0.20 post

post 1.0 0.15 α, α σ 0.9 0.10 0.8 0.05 0.7 0.00 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 αpre σα,pre 100 80 70 80 60 60 50

N N 40 40 30 20 20 10 0 0 0 5 10 15 20 25 30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 χ2 ∆χ2 10 8 27% p 73% 6

2 pre 4 χ

∆ 2 0 2 − 2 0 2 4 6 8 10 − 2 ∆χpost

2 p 2 Figure 9. The distribution of (upper left) α, (upper right) σα, (middle left) χ , (middle right) ∆χ (significance of detetction), (lower left) ∆χ2 comparison pre/post-reconstruction for the mock sample that have been cut for outliers and to realisations that have 1σ contours fully within the prior range (0.7 < α < 1.3). The data (red crosses and dashed lines (blue/red:pre/post) respectively) is within the range of the mock catalogues, that show improvement following density field reconstruction. for SDSS MGS available as a supplement from Ross et al. from SDSS MGS, 6dFGS adds enough information to pro- (2015), which alone gives α = 1.040 ± 0.037. When these vide an improvement of ∼ 16% on the SDSS MGS result. likelihoods are combined a constraint of α = 1.040 ± 0.032 This combined result offers the most robust BAO constraint is obtained. The individual ∆χ2 distributions are shown in at low redshift to date. Figure 10. Although the majority of the information comes Taking this joint constraint on α we can provide a dis-

MNRAS 000,1–13 (2018) Low-z BAO from the Reconstructed 6dFGS 11

over periodic boxes of size 1.2 h−1Gpc, populated with galax- 10 ies to describe the 6dFGS data sample through the application of a HOD modelling on top of a FoF halo finder. These robust mocks offer a more accurate covariance ma- 8 trix with which to preform the analysis in comparison to the log-normal realisations generated in previous work. 6 • A FFT version of density field reconstruction has been

2 applied to the 6dFGS data and mock realisations. χ

∆ • We make use of a correlation function model that is now 4 commonly used in SDSS (MGS and BOSS) BAO analyses. This model unlike the previous work, allows for freedom in the broadband shape by introducing polynomial terms to 2 marginalise over. This is imperative when dealing with the post-reconstruction 2-point statistics as a well deﬁned phys- ical model for the broadband shape is still not known and it 0 ensures that the information being used in the constraint is 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 coming solely from the peak position. α

Figure 10. The ∆χ2 distributions of α post-reconstruction 6dFGS (red), SDSS MGS (green) and joint (purple). Overall we find that, in the subsample of the mock catalogues that show a detection of the BAO peak both pre/post- reconstruction, 80% show an improvement after the appli- tance measurement at the effective volume weighted red- cation of density field reconstruction to remove non-linear shift of joint 6dFGS and SDSS MGS, zjoint = 0.122, of shifts of the galaxies, and the significance of detection im- eff proves on average. In the mean of the mock population an D (z = 0.122) = 539 ± 17(r /rfid) Mpc, with Dfid(z = V eff s s V eff improvement factor of I = σ /σ ∼ 1.86 is seen. In the 0.122) = 519 Mpc. The joint post-reconstruction measure- α,pre α,post data catalogue, pre-reconstruction we find consistent results ment of the distance is compared with the other survey re- to the previous analysis of Beutler et al.(2011). In perform- sults in Figure 11, and also to the fiducial flat ΛCDM cos- ing density field reconstruction to the data realisation we mology in the redshift range z < 0.8. introduce a bi-modality into the likelihood that we argue is likely a statistical fluctuation against the mock population. At the 1σ confidence interval the error drops from ∼ 6.6% to 2 7.3 Combining with the Planck 2015 Ωmh prior ∼ 4.1%, however due to the non-Gaussian nature of the post- reconstruction likelihood the significance of detection drops The ratio of DV /rs can be constructed by using a numerically calibrated analytical approximation (Aubourg et al. 2015) from ∼ 1.9σ to ∼ 1.8σ. This decrease in statistical signifi- to CAMB of the sound horizon at the drag epoch (fiducial cance post-reconstruction is displayed in 28% of the mock fid sample suggesting it is consistent with statistical fluctua- rs (zd) = 147.5 Mpc), tions within the population. The measurements agree with the flat ΛCDM Planck based cosmology at the 2σ level. exp(−72.3(Ω h2 + 6 × 10−4)2) r = 55.154 ν , (19) By combining the post-reconstruction result with that s 2 0.12807 2 2 0.2535 (Ω h ) (Ωmh − Ων h ) b from SDSS MGS at zeff = 0.15, we obtain a joint constraint which can be propagated through to give a degenerate con- of α(zeff = 0.122) = 1.040 ± 0.032. The likelihood of 6dFGS ∼ tour in H0 − Ωm parameter space. In the above analytical offers a 16% improvement on the SDSS MGS result at the Ω 2 expression Ων and Ωb are the density parameters for neu- 1σ level. Using the mh prior from Planck 2015 results, trinos and baryons with values 1.4 × 10−3 and 0.0484 respec- the degeneracy of the joint 6dFGS and SDSS MGS DV /rs 2 tively. Taking the Ωmh prior from the Planck 2015 results constraint in the Ωm − H0 plane is broken, as shown in Fig- 2 (Ωmh = 0.1417 ± 0.024)(Planck Collaboration et al. 2015) ure. 12. This provides cosmological parameter constraints of −1 the degeneracy can be broken to provide a measurement of H0 = 64.0 ± 3.5 kms Mpc and Ωm = 0.346 ± 0.045. The fi- −1 H0 = 64.0 ± 3.5 kms Mpc and Ωm = 0.346 ± 0.045. A plot nal measurements of α are collated in Table2. While these 2 showing the joint fit H0 − Ωm contour with the Planck Ωmh results are consistent with the currently accepted Planck prior is given in Figure 12. ΛCDM model, they are in 2.4σ tension with the latest Su- −1 pernova Ia results (H0 = 73.24 ± 1.74 kms Mpc; Riess et al. (2016)). This work constitutes the current best constraint avail- 8 CONCLUSION able from BAO in the low redshift (z < 0.3) regime. In the We have updated the BAO analysis of 6dFGS presented in near future both the Taipan Galaxy Survey (da Cunha et al. Beutler et al.(2011). Changes made to the analysis pipeline 2017) and DESI (DESI Collaboration et al. 2016) will tar- in comparison to that work include: get this low redshift regime providing measurement of BAO at the 1% level. These will further study any potential late • Production of 600 COLA based fast mock catalogues time deviations from the currently accepted ΛCDM model.

MNRAS 000,1–13 (2018) 12 P. Carter et al.

1.20

1.15

1.10 Planck ) s /r

) 1.05 z ( V

D 1.00 ( / ) s 0.95 /r ) z ( 6dFGS only (P ost) V 0.90 − 6dFGS only (P re)

D −

( Joint 6dFGS + MGS MGS only − 0.85 SDSS BOSS DR12 WiggleZ eBOSS LRGs 0.80 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z

Figure 11. Distance constraints of low redshift measurements BOSS DR12, eBOSS LRGs, WiggleZ and the joint 6dFGS/SDSS MGS constraint of α = 1.040±0.032. For comparison purposes we focus on measurements at z < 0.8, higher redshift measurements exist for both eBOSS QSO and BOSS Lyα. The dashed line corresponds to a ﬂat ΛCDM model matching the Planck best-ﬁt. The pre-reconstruction 6dFGS result is from Beutler et al.(2011) and the post-reconstruction is from this work. Error-bars are solid for 1σ and dashed for 2σ (focussing on the datasets used) it should be remembered that the post-reconstruction result from 6dFGS alone has a non-Gaussian likelihood such that the ΛCDM cosmology is consistent at the 2σ level.

74 Table 2. Summary table of results from the analysis. 72 Result α χ2/ν ± (±0.179) 6dF only - Pre-recon 1.018 0.066 0.116 0.81 70 ± (±0.235) 6dF only - Post-recon 0.895 0.041 0.079 0.86

) Joint Result - Post-recon ± (±0.068) 1 1.040 0.032 1.14 68 0.076 − Cosmological Parameters (combined with Planck prior)

Mpc 66 1 ± − Ωm 0.346 0.045 −1 64 H0 64.0 ± 3.5 kms Mpc (kms

0 62 H also acknowledges support from the UK Science and Tech- 60 nology Facilities Council grant ST/N000668/1 and the UK 58 Space Agency grant ST/N00180X/1. FB is a Royal Soci- ety University Research Fellow. JK is supported by MUIR 56 PRIN 2015 ”Cosmology and Fundamental Physics: Illumi- 0.25 0.30 0.35 0.40 0.45 nating the Dark Universe with Euclid” and Agenzia Spaziale Ωm Italiana agreement ASI/INAF/I/023/12/0. Parts of this research were conducted by the Australian Research Coun- cil Centre of Excellence for All-sky Astrophysics (CAAS- Figure 12. H0 − Ωm parameter contours from the Planck 2015 TRO), through project number CE110001020. The 6dF Ω h2 . ± . prior (red) and the D /r . ± . m = 0 1417 0 024 v s = 3 66 0 12 Galaxy Survey had many contributions to the instrument, (blue) measurements from the combined constraint of 6dFGS and SDSS MGS. Combining these individual constraints breaks the survey and science, in particular we thank Matthew Colless, degeneracy to give a combined contour (purple) that allows for Heath Jones, Will Saunders, Fred Watson, Quentin Parker, a measure of H0 and Ωm. The corresponding results are given in Mike Read, Lachlan Campbell, Chris Springob, Christina Section 7.3 and Table2 Magoulas, John Lucey, Jeremy Mould, and Tom Jarrett, as well as the staﬀ of the Australian Astronomical Observa- tory and other members of the 6dFGS team. This project ACKNOWLEDGEMENTS has made use of the SCIAMA High Performance Comput- PC and WJP acknowledge support from the European Re- ing (HPC) cluster at the ICG and also the GREEN-II Su- search Council through the Darksurvey grant 614030. WJP percomputer in Swinburne. This work was supported by the

MNRAS 000,1–13 (2018) Low-z BAO from the Reconstructed 6dFGS 13

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