Gravitational redshifting of galaxies in the SPIDERS cluster catalogue
C. T. Mpetha, C. A. Collins, N. Clerc, A. Finoguenov, J. A. Peacock, J. Comparat, D. Schneider, R. Capasso, S. Damsted, K. Furnell, A. Merloni, N. D. Padilla, A. Saro
MMNRAS , 1–10 (2020) Preprint 24 February 2021 Compiled using MNRAS L A TEX style file v3.0
Gravitational redshifting of galaxies in the SPIDERS cluster catalogue
C. T. Mpetha , , ★ C. A. Collins , N. Clerc , A. Finoguenov , J. A. Peacock , J. Comparat ,D. Schneider , R. Capasso , S. Damsted , K. Furnell , A. Merloni , N. D. Padilla , A. Saro , , , Astrophysics Research Institute, Liverpool John Moores University, IC2 Building, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, United Kingdom IRAP, Universite de Toulouse, CNRS, UPS, CNES, F-31028 Toulouse, France Department of Physics, University of Helsinki, PO Box 64, FI-00014 Helsinki, Finland Max-Planck-Institut für extraterrestrische Physik (MPE), Giessenbachstrasse 1, D-85748 Garching bei München, Germany Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, PA 16802, USA The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden Instituto de Astrofisica, Universidad Catolica de Chile, V. Mackenna 4860, Santiago, Chile Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy INAF - Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy INFN - National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy
24 February 2021
ABSTRACT
Data from the SPectroscopic IDentification of ERosita Sources (SPIDERS) are searched for a detection of the gravitationalredshifting of light from ∼
20 000 galaxies in ∼ Δ , the averageof the radial velocity difference between the cluster members and the cluster systemic velocity, reveals information on the size ofa combination of effects on the observed redshift, dominated by gravitational redshifting. The change of ˆ Δ with radial distanceis predicted for SPIDERS galaxies in General Relativity (GR), and 𝑓 ( 𝑅 ) gravity, and compared to the observations. The valuesof ˆ Δ = − . ± . − , ˆ Δ = − . ± . − , and ˆ Δ = − . ± . − for the BCG, X-ray and CG cases respectivelybroadly agree with the literature. There is no significant preference of one gravity theory over another, but all cases give a cleardetection ( > . 𝜎 ) of ˆ Δ . The BCG centroid is deemed to be the most robust method in this analysis, due to no well definedcentral redshift when using an X-ray centroid, and CGs identified by redMaPPer with no associated spectroscopic redshift. Forfuture gravitational redshift studies, an order of magnitude more galaxies, ∼
500 000, will be required—a possible feat with theforthcoming
Vera C. Rubin Observatory , Euclid and eROSITA . Key words: gravitation – galaxies: clusters: general – galaxies: kinematics and dynamics
Galaxy clusters are the largest gravitationally bound systems in theUniverse, making them an excellent test-bed for theories of gravity.They are composed of ∼ − galaxies and a large dark matterhalo.ln ( + 𝑧 obs ) = ln ( + 𝑧 cos ) + ln ( + 𝑧 pec ) + ln ( + 𝑧 grav ) . (1)There are various effects that contribute to the observed redshiftingof light from galaxies in clusters ( 𝑧 obs ), shown in equation (1). Thereis of course the cosmological redshift ( 𝑧 cos ) due to the expansion ofthe Universe, which will be the same for both the galaxy and the hostgalaxy cluster. After this, the most prominent is the peculiar redshift( 𝑧 pec )—random isotropic motions of galaxies within the cluster inthe line of sight. Galaxies are in motion around the minimum of the ★ E-mail: [email protected] cluster’s potential well, its dynamical centre, and so the average offsetbetween a galaxy’s peculiar redshift and that of the cluster centre willbe zero. To test this, a distribution of line-of-sight velocity offsets,found from observed redshifts, can be created. If the peculiar redshiftwere the only contribution along with the cosmological redshift, thisdistribution would be centred on zero, due to isotropy. But this isnot the case, and so the shift of the centre of this distribution isinformative of the size of other contributions, namely gravitationalredshifting ( 𝑧 grav ) whose possibility of detection was investigated byNottale (1983) and Cappi (1995). This shifting of the average is thequantity of interest in this study; the size of the shift and its evolutionwith distance from the cluster centre are both informative on thetheory of gravity governing the observed redshifts of these galaxies.To create a distribution of line-of-sight velocity offsets using galaxyredshifts, we define the quantity Δ = 𝑐 [ ln ( + 𝑧 obs ) − ln ( + 𝑧 cen )] , (2) © a r X i v : . [ a s t r o - ph . C O ] F e b C. T. Mpetha et al. where 𝑧 cen is the redshift of the galaxy cluster’s centre. Differencesin the logarithm of the redshifts have been used instead of simplyassuming 𝑧 = 𝑣 / 𝑐 as using the natural logarithm provides a betterapproximation to the line-of-sight velocity (Baldry 2018):ln ( + 𝑧 ) (cid:39) 𝑣 los 𝑐 + (cid:32) 𝑣 𝑐 − 𝑣 𝑐 (cid:33) + · · · . (3)Another advantage of this definition is that it removes some of thedependence on the cosmological redshift, as otherwise the expressionwould be Δ = 𝑐 ( 𝑧 obs − 𝑧 cen )/( + 𝑧 cen ) . This is advantageous if thecosmological redshift has a large uncertainty. Δ is a combination of the peculiar velocity, gravitational redshiftand other effects that will be detailed in section 3. Then, to find thelocation of a distribution of Δ values, we defineˆ Δ ≡ (cid:104) Δ (cid:105) . (4)Following Beers et al. (1990), Tukey’s biweight average is used asa minimum variance estimator for the location and scale of galaxyvelocity distributions, which are in general not Gaussian due to thepresence of interlopers, and dynamical instability of the cluster.This paper will use those SDSS Data Run 16 galaxies and clustersthat have been spectroscopically measured as part of the SPIDERSprogramme (Clerc et al. 2016), to explore the size of ˆ Δ at differentdistances from the centre of a cluster. Wojtak et al. (2011) made thefirst tentative detection of gravitational redshifting in galaxy clusters,with data from SDSS Data Run 7 (Abazajian et al. 2009). A similaranalysis has been performed on SDSS DR10 galaxies and clusters(Sadeh et al. 2015). This paper aims to repeat and build upon theseanalyses. The large amount of information available from the spec-troscopic follow up of X-ray selected galaxy clusters allows for novelmethods of calculating ˆ Δ as a function of distance from the clustercentre.The outline of this paper is as follows; in section 2 the SPIDERScatalogue is introduced and the properties of its clusters are discussed,including three different definitions of the centre of a cluster, thenthe data reduction process is presented. Next in section 3 the variouscontributions to ˆ Δ are described, before its variation with distancefrom the centre of the cluster is predicted for two theories of gravityin section 4. In section 5 ˆ Δ is found for each of the three centroid casesfrom observations of galaxies and clusters in the SPIDERS catalogue,and compared to the predictions. We conclude with closing remarksand future prospects in section 6.Planck Collaboration (2018) values of 𝐻 = . − Mpc − and Ω m , = .
315 have been used throughout.
SPectroscopic IDentification of eROSITA
Sources (SPIDERS: Clercet al. 2016, 2020; Chitham et al. 2020; Kirkpatrick et al. in prep.) isthe X-ray specific subprogramme of the extended Baryonic Oscilla-tion Spectroscopic Survey (eBoss: Dawson et al. 2016), which is apart of the Sloan Digital Sky Survey (SDSS: Gunn et al. 2006; Smeeet al. 2013). The SDSS is currently in its fourth generation, SDSS-IV(Blanton et al. 2017). SPIDERS is the spectroscopic follow up oflarge numbers of galaxies identified in the eBOSS survey. Galax-ies are assigned to clusters using the redMaPPer algorithm (Rykoffet al. 2014), which then uses identified members to estimate clusterproperties such as optical richness and redshift.SPIDERS will eventually include eROSITA
X-ray selected clusters. The most recent catalogue, Data Run 16, comprises a subset ofclusters that were identified in the CODEX program (Finoguenovet al. 2020), which searched ROSAT All Sky Survey data for extendedX-ray sources. Data Run 16 contains 2740 clusters with close to42 000 galaxies with spectroscopic redshifts. There are a number ofparameters measured for each cluster, including the virial mass 𝑀 estimated from its velocity dispersion, and both 𝑀 and the X-rayluminosity 𝐿 X are iteratively calibrated using an 𝑀 − 𝐿 X scalingrelation (Capasso et al. 2020). Also provided within SPIDERS arethree potential definitions of the centre of a cluster, each of whichhas been used for an independent measurement of ˆ Δ . There are numerous reasons why it is advantageous to have a largecluster sample. Firstly, even assuming that every galaxy in a clus-ter could be spectroscopically measured, there are simply too fewgalaxies to allow the statistical detection of a non-zero gravitationalredshift. Typically ˆ Δ ∼
10 km s − , and a cluster’s velocity disper-sion is 𝜎 v ∼ − . To have a standard error on the averagevalue of the distribution, 𝜎 v /√ 𝑁 , that is small enough to resolve thegravitational redshifting from no effect, around 10 000 galaxies arerequired. By stacking many galaxy measurements from many clustersinto a composite cluster, this requirement can be satisfied. Secondly,clusters do not in general exhibit spherical symmetry. There is oftenapshericity in the matter distribution leading to anisotropic velocitydistributions. By stacking a large number of clusters, these featureswill be smoothed out in the composite cluster.When stacking these clusters, simply using a distance in Mpc isnot ideal, as clusters can have a large range of sizes, and so havedifferent masses and densities at the same distance from the centre.Clusters show a high degree of similarity in their virialised region(e.g. Kaiser 1986). For this reason the ratio ˜ 𝑟 = 𝑟 / 𝑟 is used as adistance measure. The virial radius 𝑟 is the radius of the clusterwithin which the mean density is equal to the overdensity parameter 𝑣 =
200 multiplied by the critical density of the Universe 𝜌 c ; thedensity of a flat Friedmann-Lemaître-Robertson-Walker Universe,at the redshift of the cluster. Throughout this paper it is assumedthat clusters follow the Navarro-Frenk-White (NFW) density profile(Navarro et al. 1995); a similar density profile across all clusters isthus assumed at similar values of ˜ 𝑟 , and hence the effects in eachcluster can be stacked and compared. In SPIDERS, the size of acluster’s virial radius on the sky is measured in degrees. Hence tofind the distance of a galaxy from its parent cluster’s centre in unitsof the virial radius, the ratio Δ 𝜃 / 𝜃 is calculated:˜ 𝑅 = Δ 𝜃𝜃 . (5)˜ 𝑅 describes the projected distance of the galaxy from the clustercentre. By only knowing angular positions, information on the trueradial distance is lost. This is fine, so long as when calculating thesize of ˆ Δ as a function of distance in different theories of gravity, itis done using the projected distance from the cluster centre. Cluster miscentring is a leading cause of systematic error in clustervelocity dispersion analyses (Becker et al. 2007), warranting care-ful discussion of how the centre has been defined, and comparingpossible methods. Within the SPIDERS catalogue are three methodsof defining the cluster centre, and each one has been used for anindependent measurement of ˆ Δ in SPIDERS clusters. MNRAS , 1–10 (2020) ravitational redshifting of galaxies in SPIDERS The
Brightest Cluster Galaxy (BCG) is the most luminous galaxyin the cluster. This can be used as a proxy for the centre of a cluster,as it is formed through the merger of other large galaxies—and so islikely to trace the dynamical centre, which would lie at the minimumof the clusters substantial potential well (Oegerle & Hill 2001). Theredshift and central position of the cluster is then taken to be theredshift and position of the BCG. In previous ˆ Δ analyses this is themethod by which the cluster centre is defined.The Optical Centre definition is found from the red-sequenceMatched-filter Probabilistic Percolation (redMaPPer) algorithm,which uses a probabilistic approach to find the most likely Cen-tral Galaxy (CG) (Rykoff et al. 2014). It is assumed each clusterhas a single dominant galaxy at its centre, which is a red sequencegalaxy. Potential CGs are assigned a centre probability based onthree observables; their 𝑖 − band magnitude, red sequence photomet-ric redshift, and the cluster density around the candidate CG. Themost likely CG is identified, and thus used as the central position andredshift of the cluster. In many cases this coincides with the BCG,but there are enough differences for it to be an independent method.The X-ray centre is found using the peak of X-ray emission.This poses a problem of how the redshift of the centre is defined.One potential way of addressing this is by isolating the core of thecluster, centred on the X-ray peak, identifying all the galaxies lyingin this region, and using their average as a measure of the centralredshift. Typical core radii are in the range 𝑟 c (cid:39) ( . − . ) ℎ − Mpc(Bahcall 1996). As the typical cluster virial radius in the SPIDERScatalogue is 𝑟 (cid:39) . 𝑟 𝑐 (cid:39)( . − . ) 𝑟 . To give the best chance of observing sufficientgalaxies in the core region for a reasonable average, without dilutingthe ˆ Δ signal by using a core radius too large for the majority ofclusters, a core radius of 𝑟 c = . 𝑟 (6)is henceforth used. Galaxies within this region are used to find theredshift of the X-ray defined centre. There is an immediate problemwith this method: only information on the projected distance fromthe cluster centre is known, so it is likely that in some cases galaxiesnot in the core region are being used to estimate the central redshift.Furthermore, the ROSAT centroid for very faint sources is poorlydetermined, and so there is a strong possibility of miscentring inthese clusters.Each method for defining the centre of the cluster has advantagesand disadvantages. For the BCG and optical centroid cases, the effectscontributing to ˆ Δ depend on the motion of the CGs themselves, whichare not at rest relative to the clusters potential minimum, and eventhe increased internal dynamics of these large galaxies can have animpact. These effects cause various slight adjustments to the valuesshown in Fig. 7. However in Kaiser (2013) the net effect of thesevarious modifications due to CG properties is found to be small( (cid:46) − ), and only affecting the innermost region where theCGs lie, and so for brevity they have been neglected in this analysis.Regardless, numerous studies (e.g. Cui et al. 2015) have found theBCG correlates well with the minimum of the gravitational potential.Using an X-ray centroid can be preferable as it avoids miscentringon foreground/background galaxies, and represents a better tracerin highly dynamical clusters. The obvious drawback in this case isthe lack of a clear central redshift. In the optical centroid case, theredMaPPer algorithm is not perfect. It requires the central galaxy tobe a red-sequence member, and so fails when the CG is undergoingstrong star formation (Rykoff et al. 2014). Another problem is that forSPIDERS clusters there is a large discrepancy ( ∼ . 𝑟 − . 𝑟 )between the optical centre and the nearest spectroscopically observed Figure 1.
Cumulative distribution functions for all the galaxies used in eachcentroid analysis as a function of their projected separation from the clustercentre. A K-S test has been performed to check whether there is a significantdifference between the populations used in each case. An extremely small 𝑝 -value was found in each combination, so the null hypothesis that these aremembers of the same distribution can be rejected. galaxy in 188 clusters, with around 400 clusters showing a largestdifference between 0 . 𝑟 and 0 . 𝑟 . This would suggest that inthe more extreme of these cases redMaPPer has identified a CG thatis not spectroscopically measured, while the smaller offsets are likelyto be down to positional inaccuracies. To ensure there is no accidentalmiscentring, only galaxies within 0 . 𝑟 ∼
50 kpc, about the sizeof a large galaxy, of the optical centre have been identified as a CG.To test whether each choice of centroid creates statistically distinctgalaxy populations, a two sided Kolmogorov–Smirnov (K-S) test hasbeen performed on the cumulative distribution functions (CDFs) ofthe positions of the galaxies used in each independent analysis (afterthe data reduction described in section 2.4 has been performed)from the cluster centre, shown in Fig. 1. Each case is shown torepresent a statistically distinct population. Fig. 2 shows the CDFsof the difference of the central position for each centroid pair forclusters in SPIDERS, in units of the virial radius 𝑟 . As expectedfor the optical and BCG centres there are many cases of coincidence(61% of the cluster population).These centre offsets can be informative in their own right. Forexample, the BCG-X-ray centre offset could be a probe on clustersubstructure, and the dynamical state of a cluster (Lopes et al. 2018).The expectation is very small positional offsets for relaxed clusters,but non-negligible offsets for disturbed systems. The uncertaintiesof RASS X-ray positions translate to positional uncertainties rang-ing from 0 . − . 𝑟 . Hence from Fig. 2 it would appear that asignificant fraction of these clusters are likely to be disturbed sys-tems, consistent with De Propris et al. (2021), Seppi et al. (2020)and references therein. Furthermore, Seppi et al. (2020) demonstratethat the X-ray centroid uncertainty tends to increase the observedX-ray-BCG/optical offset. Table 1 demonstrates the different conditions imposed on the rawSPIDERS DR16 dataset in an attempt to obtain an uncontaminatedmeasurement of ˆ Δ . The effect of each condition, isolated from allthe others, is also shown. The choice of a redshift uncertainty limitof 𝜎 z < . .
5% of the population, and removes the
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Table 1.
Table of the different imposed conditions on all the SPIDERS DR16 galaxies and clusters, and the isolated effect of each. X = X-ray centroid, O =Optical centroid, B = BCG centroid.SPIDERS parameter Condition Explanation Isolated Effect NCOMPONENT = ALLZ_NOQSO
Use these redshifts Removes Quasar templates. Accurate spectroscopic redshifts.
ALLZWARNING_NOQSO = ALLZ_ERR_NOQSO < . R200C_DEG ˜ 𝑅 ≤ X / O / B . SCREEN_ISMEMBER_W = SCREEN_CLUVDISP_BEST | Δ | < . 𝜎 v Removes interlopers (Mamon et al. 2010, 2013). Removes 11 839 / 8 888 / 11 406 galaxies in X / O / B . Figure 2.
Cumulative distribution functions of the differences between theposition of the centre for each centroid combination, for all clusters used inthe analysis. upper tail of galaxies with large redshift uncertainties. There are alsoother, more subtle effects not included in the table. For the BCGand optical case 2512 galaxies are automatically removed so thecentral galaxy of a cluster is not compared with itself, furthermorein some cases the BCG or CG has a redshift uncertainty ≥ . 𝑅 ≤ . 𝑅 ≤ .
033 of the optical centre also removessome clusters from the optical analysis, resulting in an extra 4 405galaxies removed. The net effect of these conditions and those inTable 1 is the removal of 22 539, 20 563 or 17 052 galaxies in theX-ray, optical or BCG centroid case respectively, from the originaldataset which contains 41 663 spectroscopic redshifts. The remainingnumber of galaxies is greater than 10 000 in all cases, and so shouldyield a small enough standard error on ˆ Δ to resolve it from zeroeffect.High resolution X-ray data are needed to determine whether thelarge centroid offsets in Fig. 2 are caused by clusters being in adisturbed state, something which could in principle bias the gravi-tational redshift signal. As this information is not available for SPI-DERS DR16 clusters, no selection cut has been made based on thesize of these centroid offsets. Tests were however performed to en-sure that the removal of those clusters with the largest offsets had nosignificant effect on the results.Fig. 3 shows density maps of the remaining galaxies used in thefinal analysis for each centroid case. On the 𝑥 -axis is their projected Figure 3.
Density maps of the velocity offsets of all the remaining galaxiesafter refining the original SPIDERS catalogue, as a function of their projecteddistance from the cluster centre in units of the virial radius. In the BCG centrecase (top) there are 24 611 galaxies, there are 21 100 for the optical (middle)and 19 124 for the X-ray (bottom). distance from the cluster centre in units of the virial radius, while onthe 𝑦 -axis is the size of each galaxy’s velocity offset from the centreof its parent cluster. MNRAS , 1–10 (2020) ravitational redshifting of galaxies in SPIDERS ˆ Δ A distribution of velocity offsets between galaxies and their host clus-ter’s centre is expected to have an average value that is blueshifted,as light experiences the largest redshifting at the minimum of theclusters potential well. For a single galaxy, the gravitational redshift,expressed as a velocity offset, is given by the difference between thegravitational potential at the galaxies distance from the cluster centre(here the dimensionless distance in units of 𝑟 is used), and that atthe centre, Δ gz ( ˜ 𝑟 ) = ( Φ ( ) − Φ ( ˜ 𝑟 ))/ 𝑐 , (7)where the gravitational potential is that which is associated with anNFW dark matter density profile (more detail is given in AppendixA). Only line-of-sight information can be measured, and thereforeonly the projected distance from the centre of the cluster, ˜ 𝑅 , is known;see (5). The density along the line of sight to that distance must beintegrated along with the potential difference. Hence, for a singlecluster (Lokas & Mamon 2001), Δ c , gz ( ˜ 𝑅 ) = 𝑟 𝑐 Σ ( ˜ 𝑅 ) ∫ ∞ ˜ 𝑅 ( Φ ( ) − Φ ( ˜ 𝑟 )) 𝜌 ( ˜ 𝑟 ) ˜ 𝑟𝑑 ˜ 𝑟 √ ˜ 𝑟 − ˜ 𝑅 , (8)where Σ ( ˜ 𝑅 ) is the surface mass density profile found from integratingthe NFW density profile along the line of sight, Σ ( ˜ 𝑅 ) = 𝑟 ∫ ∞ ˜ 𝑅 ˜ 𝑟 𝜌 ( ˜ 𝑟 )√ ˜ 𝑟 − ˜ 𝑅 𝑑 ˜ 𝑟 . (9)By integrating with respect to ˜ 𝑟 , and not the vector ˜r , sphericalsymmetry of the clusters is being assumed. Although often not thecase for a single cluster, a stacked set of many clusters is expected toexhibit spherical symmetry.Following Wojtak et al. (2011) the gravitational redshift signal fora stacked cluster sample can be calculated using Δ 𝑔𝑧 ( ˜ 𝑅 ) = ∫ 𝑀 max 𝑀 min Δ c , gz ( ˜ 𝑅 ) Σ ( ˜ 𝑅 ) ( 𝑑𝑁 / 𝑑𝑀 ) 𝑑𝑀 ∫ 𝑀 max 𝑀 min Σ ( ˜ 𝑅 ) ( 𝑑𝑁 / 𝑑𝑀 ) 𝑑𝑀 , (10)where the gravitational redshift profile for a single cluster has beenconvolved with the cluster mass distribution to accurately representthe stacked signal. The peculiar redshift of a galaxy can be decomposed as follows: 𝜷 = v 𝑐 , (11)1 + 𝑧 pec (cid:39) + 𝛽 los + 𝛽 / + · · · , (12)where 𝛽 los gives the component in the line of sight. And so thereis a second-order term due to transverse motion of the galaxy. Thisgives rise to the transverse doppler effect, which will contribute asmall positive shift in the location of a velocity distribution; this istypically ∼ few km s − , and is relatively constant with distance fromthe cluster centre.To find the size of this effect for a set of galaxy velocity offsetsfrom their host cluster’s centre, we calculate Δ TD = (cid:68) 𝑣 − 𝑣 (cid:69) / 𝑐 . (13)Calculating this effect involves a similar integral over the line-of-sight density profile and a convolution with the mass distribution (Zhao et al. 2013). For a single cluster, Δ c , TD ( ˜ 𝑅 ) = 𝑄𝑟 𝑐 Σ ( ˜ 𝑅 ) ∫ ∞ ˜ 𝑅 (cid:16) ˜ 𝑟 − ˜ 𝑅 (cid:17) 𝑑 Φ ( ˜ 𝑟 ) 𝑑 ˜ 𝑟 𝜌 ( ˜ 𝑟 ) 𝑑 ˜ 𝑟 √ ˜ 𝑟 − ˜ 𝑅 , (14)where 𝑄 = / Δ TD ( ˜ 𝑅 ) = ∫ 𝑀 max 𝑀 min Δ c , TD ( ˜ 𝑅 ) Σ ( ˜ 𝑅 ) ( 𝑑𝑁 / 𝑑𝑀 ) 𝑑𝑀 ∫ 𝑀 max 𝑀 min Σ ( ˜ 𝑅 ) ( 𝑑𝑁 / 𝑑𝑀 ) 𝑑𝑀 . (15) The Universe is not static. Observations of galaxies lie in our pastlight cone, and as such there is some discrepancy between the distanceobserved between two sources, and the true distance. In betweenlight being emitted from both sources, the second emitter will havemoved a distance depending on its line-of-sight velocity. The relationbetween the separation expressed in light-cone coordinates and rest-frame coordinates is (Kaiser 2013): 𝑑𝑥 LC = 𝑑𝑥 RF − 𝑣 x / 𝑐 . (16)This extra factor of 1 /( − 𝑣 x / 𝑐 ) in the distance leads to an extrafactor of ( − 𝑣 x / 𝑐 ) in the number density as 𝜌 ∝ / 𝑉 and forthe cylindrical volume observed 𝑉 ∝ 𝑑𝑥 . This bias on the observeddensity of objects, dependent on their line-of-sight velocity, createsa bias on ˆ Δ .Integrating over the line-of-sight coordinate 𝑥 gives a contributionproportional to 𝑣 : Δ LC = (cid:68) 𝑣 , gal − 𝑣 , (cid:69) / 𝑐 . (17)Once again this gives a small positive contribution to the shifting ofthe location, opposite in sign to the effect of gravitational redshifting.Assuming isotropic orbits of the galaxies, we obtain Δ LC = Δ TD . (18) Galaxies in spectroscopic samples are chosen according to their ap-parent luminosity/magnitude. Due to the special relativistic beamingeffect, this apparent luminosity can be changed by the peculiar motionof the galaxies. For galaxies lying just below the required apparentluminosity, motion towards the observer could shift them inside thecut, while those moving away could be shifted just outside the cut.Generally, this creates a small preferential bias in favour of galax-ies moving towards the observer, with the overall effect of a smallblueshifting on the centre of a distribution of velocity offsets.The size of this effect depends strongly on the galaxy survey, forexample in Wojtak et al. (2011) the flux limit is an 𝑟 -band magnitudeof 𝑟 = .
77, while in SPIDERS the limit is an 𝑖 -band fibre magnitudeof 𝑖 = . (cid:48)(cid:48) aperture (Clerc et al. 2016). To calculate the size ofthis effect, consider the fractional change in the apparent luminosityas a function of the spectral index at the cosmological redshift of thesource, as well as the peculiar velocity of the source galaxy (Kaiser2013), given by Δ 𝐿 / 𝐿 = ( + 𝛼 ( 𝑧 cos )) 𝑣 x 𝑐 . (19)The modulation of the number density of detectable objects is givenby ( Δ 𝐿 / 𝐿 ) 𝛿 ( 𝑧 ) = − ( + 𝛼 ( 𝑧 )) 𝑣 x 𝑐 𝑑 ln 𝑛 ( > 𝐿 lim ( 𝑧 )) 𝑑 ln 𝐿 . (20)
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Where 𝛿 ( 𝑧 ) is the redshift dependent logarithmic derivative of thenumber distribution of galaxies. The redshift dependence comes fromtranslating the apparent luminosity limit to an absolute luminositylimit that varies with redshift. Following Kaiser (2013), 𝛼 ( 𝑧 ) (cid:39) Δ SB = − (cid:104) 𝛿 ( 𝑧 )(cid:105) (cid:68) 𝑣 − 𝑣 , (cid:69) / 𝑐 , (21) Δ SB = − (cid:104) 𝛿 ( 𝑧 )(cid:105) Δ TD . (22)As (cid:104) 𝛿 ( 𝑧 )(cid:105) is O( ) , the shift due to the surface brightness modulationis the largest correction besides gravitational redshifting, and againis fairly constant with distance from the cluster centre.An exact expression involves an investigation into the variation of 𝛿 ( 𝑧 ) over the redshift range of the SPIDERS cluster catalogue, whichwe develop in section 4. These effects are not the only ones present. Cai et al. (2017) givea comprehensive summary of the different contributions to ˆ Δ , in-cluding cross-terms. These are shown to reduce the ˆ Δ signal by only (cid:46) − , and so for the purposes of this analysis will not beconsidered further.Importantly, the Transverse Doppler effect discussed in section 3.2assumes that differences in redshift are being used to approximate ve-locity differences. The choice of using logarithmic differences altersthe size of the Transverse Doppler effect. Assuming isotropy:ln ( + 𝑧 pec ) (cid:39) 𝛽 x + 𝛽 + · · · , (23) Δ TD ∗ = (cid:68) 𝑣 , gal − 𝑣 , (cid:69) / 𝑐 , (24) Δ TD ∗ = Δ TD . (25)The combination of the effects considered in this analysis givesˆ Δ = Δ gz + Δ LC + Δ SB + Δ TD ∗ , (26)ˆ Δ = Δ gz + ( − (cid:104) 𝛿 ( 𝑧 )(cid:105)) Δ TD . (27)And so from assuming isotropy only the gravitational and transversedoppler effects need to be calculated, as well as the logarithmicderivative 𝛿 ( 𝑧 ) , to account for the four largest contributions. ˆ Δ ( ˜ 𝑅 ) To predict the size of ˆ Δ for the stacked SPIDERS clusters, first anexpression for the mass distribution is needed for the convolution in(10). An advantage of such a well studied cluster sample is there ispre-existing knowledge on the mass distribution, found from X-rayproperties of the clusters. In Wojtak et al. (2011) the mass distributionneeded to be estimated directly from the velocity distribution. Themass distribution of SPIDERS clusters used in the final analysis isshown in Fig. 4. It is approximated as a Gaussian with a mean oflog ( 𝑀 / 𝑀 (cid:12) ) = . .
3. A skewednormal would provide a better fit, but the effect on the final resultwould be marginal.The next step towards a prediction of ˆ Δ requires the calculationof (cid:104) 𝛿 ( 𝑧 )(cid:105) in equation (27). To find the surface brightness modulationin SPIDERS clusters the process demonstrated in Kaiser (2013) hasbeen closely followed, using values from Montero-Dorta & Prada(2009); the result is shown in Fig. 5. Two functions have been cal-culated: the logarithmic derivative 𝛿 ( 𝑧 ) described in section 3.4, and Figure 4.
Distribution of the logarithm of virial masses of the SPIDERSclusters used in this analysis. The distribution can be approximately fittedwith a Gaussian.
Figure 5.
The blue-green solid curve gives the galaxy number distribution 𝑑𝑁 / 𝑑𝑧 = 𝑧 𝑛 ( 𝑧 ) , truncated at the host cluster maximum and minimum red-shifts in SPIDERS. The purple dot dash cure shows the logarithmic derivativeof the number density with respect to magnitude, limited by the absolute mag-nitude sensitivity, which is a function of redshift. Its average value over therelevant redshift range, restricted by 𝑑𝑁 / 𝑑𝑧 = 𝑧 𝑛 ( 𝑧 ) , is (cid:104) 𝛿 ( 𝑧 ) (cid:105) (cid:39) . the redshift dependent number distribution 𝑑𝑁 / 𝑑𝑧 = 𝑧 𝑛 ( 𝑧 ) . Thefibre magnitude limit of 21 . ∼ 𝛿 ( 𝑧 ) at low redshiftand hence have a minor impact on the result. The 𝑦 -axis in Fig. 5shows the value of 𝛿 ( 𝑧 ) , and the number distribution has been scaledup for comparison. To find the average value of 𝛿 ( 𝑧 ) over the redshiftrange of the cluster catalogue, restricted by the number distribution, (cid:104) 𝛿 ( 𝑧 )(cid:105) = 𝑧 u ∫ 𝑧 l 𝑧 𝑛 ( 𝑧 ) 𝛿 ( 𝑧 ) 𝑑𝑧 𝑧 u ∫ 𝑧 l 𝑧 𝑛 ( 𝑧 ) 𝑑𝑧 , (28)where 𝑧 l = .
016 and 𝑧 u = .
667 corresponding to the lower andupper redshift limit in the SPIDERS cluster catalogue. The result ofthis integration is (cid:104) 𝛿 ( 𝑧 )(cid:105) (cid:39) . . (29) MNRAS , 1–10 (2020) ravitational redshifting of galaxies in SPIDERS Figure 6.
A redshift-mass relation for the clusters in the SPIDERS catalogue.The line of best fit is found using the BCES bisector method.
Finally, reasonable values for the concentration parameter 𝑐 = 𝑟 / 𝑟 s , which gives the ratio between the virial radius and the so-called ‘scale radius’ 𝑟 s are needed. This relates to the form of the NFWdensity profile, and the explicit dependence can be seen in AppendixA. In Dutton & Macciò (2014) a redshift dependent 𝑀 − 𝑐 relation is found:log ( 𝑐 ) = . + ( . − . ) 𝑒 − . 𝑧 . − ( . − . 𝑧 ) log (cid:18) 𝑀 ℎ − 𝑀 (cid:12) (cid:19) . (30)This replaces 𝑐 in the integration’s in (10) and (15). A 𝑧 − 𝑀 relation for the SPIDERS clusters, seen in Fig. 6, is used to replace 𝑧 with 𝑀 in (30). There is of course uncertainty in this relation—andthe errors from the BCES bisector fit will be propagated through tothe predictions for General Relativity and 𝑓 ( 𝑅 ) gravity. Using (29), the combination of the effects in section 3.5 givesˆ Δ GR = Δ gz − Δ TD . (31)The results using fiducial values are demonstrated in Fig. 7. The neteffect of the other contributions is a small added blueshifting to thegravitational redshift. 𝑓 ( 𝑅 ) gravity For General Relativity with a cosmological constant—the standardmodel of cosmology Λ CDM—the Einstein-Hilbert action, which isintegrated over all coordinates of spacetime, describes the interactionbetween matter and gravity: 𝑆 = ∫ 𝑑 𝑥 √− 𝑔 𝑀 ( 𝑅 − Λ ) + L m . (32) Figure 7.
Different contributions to a General Relativity predicted shift of theaverage of galaxy velocities in the stacked SPIDERS clusters as a functionof the projected distance from the centre of the cluster, in units of the virialradius 𝑟 . Here, 𝑀 = / 𝜋𝐺 is the reduced Planck mass; 𝑅 is the Ricci scalar,which gives information on the curvature of spacetime; Λ is thecosmological constant; L m is the matter Lagrangian and 𝑔 is thedeterminant of the Friedmann-Lemaître-Robertson-Walker (FLRW)metric describing a homogeneous and isotropic expanding Universe.A simple modification can be made to this action, representing asimple modification to gravity (Starobinsky 1980): 𝑆 = ∫ 𝑑 𝑥 √− 𝑔 𝑀 ( 𝑅 + 𝑓 ( 𝑅 )) + L m , (33)where the cosmological constant Λ has been replaced by some un-known function of the Ricci scalar. In Schmidt (2010) it is shownthat, in the limit where the background value of 𝑓 ( 𝑅 ) is much largerthan a cluster’s potential well, the effect on the gravitational forceexperienced by a test mass in the cluster due to this modification togravity is 𝐺 → / 𝐺 , and in the reverse scenario there is no mod-ification to 𝐺 . A strong field model with | 𝑓 R0 | = − is shown tocause the 4 / Δ in General Relativityand 𝑓 ( 𝑅 ) gravity. It should be noted that constraints on 𝑓 ( 𝑅 ) gravity(e.g. Cataneo et al. 2015) rule out this universal 4 / Δ to variations onthe theory of gravity. In Fig. 8 the galaxy velocity offsets for each centroid, shown inFig. 3, have been split into three bins with equal numbers. Thebiweight average of the distribution of Δ ’s in each bin is found,giving the 𝑦 -value of each data point, while its value on the 𝑥 -axis isthe average projected distance of the binned galaxies from the clustercentre in units of the virial radius 𝑟 . It is the projected distanceas our observations only measure angles on the sky, so there is someignorance as to how far away from the cluster centre each galaxytruly lies. The 𝑦 -error gives the standard error of the average value,while the 𝑥 -error gives the dispersion of galaxy positions withineach bin. All errors give regions of 68% probability. Three bins have MNRAS000
Different contributions to a General Relativity predicted shift of theaverage of galaxy velocities in the stacked SPIDERS clusters as a functionof the projected distance from the centre of the cluster, in units of the virialradius 𝑟 . Here, 𝑀 = / 𝜋𝐺 is the reduced Planck mass; 𝑅 is the Ricci scalar,which gives information on the curvature of spacetime; Λ is thecosmological constant; L m is the matter Lagrangian and 𝑔 is thedeterminant of the Friedmann-Lemaître-Robertson-Walker (FLRW)metric describing a homogeneous and isotropic expanding Universe.A simple modification can be made to this action, representing asimple modification to gravity (Starobinsky 1980): 𝑆 = ∫ 𝑑 𝑥 √− 𝑔 𝑀 ( 𝑅 + 𝑓 ( 𝑅 )) + L m , (33)where the cosmological constant Λ has been replaced by some un-known function of the Ricci scalar. In Schmidt (2010) it is shownthat, in the limit where the background value of 𝑓 ( 𝑅 ) is much largerthan a cluster’s potential well, the effect on the gravitational forceexperienced by a test mass in the cluster due to this modification togravity is 𝐺 → / 𝐺 , and in the reverse scenario there is no mod-ification to 𝐺 . A strong field model with | 𝑓 R0 | = − is shown tocause the 4 / Δ in General Relativityand 𝑓 ( 𝑅 ) gravity. It should be noted that constraints on 𝑓 ( 𝑅 ) gravity(e.g. Cataneo et al. 2015) rule out this universal 4 / Δ to variations onthe theory of gravity. In Fig. 8 the galaxy velocity offsets for each centroid, shown inFig. 3, have been split into three bins with equal numbers. Thebiweight average of the distribution of Δ ’s in each bin is found,giving the 𝑦 -value of each data point, while its value on the 𝑥 -axis isthe average projected distance of the binned galaxies from the clustercentre in units of the virial radius 𝑟 . It is the projected distanceas our observations only measure angles on the sky, so there is someignorance as to how far away from the cluster centre each galaxytruly lies. The 𝑦 -error gives the standard error of the average value,while the 𝑥 -error gives the dispersion of galaxy positions withineach bin. All errors give regions of 68% probability. Three bins have MNRAS000 , 1–10 (2020)
C. T. Mpetha et al.
Figure 8.
The biweight location of the distribution of SPIDERS galaxy ve-locity offsets using three definitions of the cluster centre, as a function oftheir projected distance from the centre in units of the virial radius 𝑟 .Equal numbers of galaxies have been used for each bin. The observations arecompared with predictions for ˆ Δ from General Relativity (GR) and 𝑓 ( 𝑅 ) gravity. The highlighted regions around these predictions demonstrate errorbounds propagated through from the redshift-mass relation. been chosen to maximise the number of galaxies per bin, while stillallowing easy visual comparison with the predictions of two theoriesof gravity: General Relativity (solid black line) and 𝑓 ( 𝑅 ) gravity(blue dashed line). The highlighted region around each predictionshows error bounds caused by the uncertainty in the redshift-massrelation in Fig. 6.In the BCG and the X-ray case there is good agreement with theGeneral Relativity predicted variation of ˆ Δ with ˜ 𝑅 . The apparent ten-sion between the GR prediction and the first optical data point is notsignificant, and is lessened when the data are rebinned. Dependenceof the optical result on the allowed difference between a cluster’soptical centroid and the nearest spectroscopically observed galaxywas also tested. All results agreed to within 1 𝜎 .Another informative way of presenting these results is through thetotal integrated ˆ Δ over a defined range of ˜ 𝑅 . These results are shownin Fig. 9, alongside the distribution of galaxies used for each centroidto find the integrated effect. This further highlights the difference innumbers for each case. Only galaxies in the range 0 < ˜ 𝑅 ≤ 𝑓 ( 𝑅 ) gravity.The integrated General Relativity and 𝑓 ( 𝑅 ) signals in the range0 < ˜ 𝑅 ≤ Δ GR = − . + . − . km s − , (34)ˆ Δ f ( R ) = − . + . − . km s − . (35)Hence all centroid cases are consistent with both GR and 𝑓 ( 𝑅 ) to1 𝜎 , while the BCG and X-ray centre cases show more similarity tothe GR prediction. The optical integrated effect appears to be moreconsistent with the 𝑓 ( 𝑅 ) prediction, but in Fig. 8 the evolution of ˆ Δ in the optical case does not particularly follow that of 𝑓 ( 𝑅 ) , whilethe BCG and X-ray cases do have similar evolution to GR. Further,the BCG, X-ray and optical cases show a ∼ . 𝜎 , ∼ . 𝜎 and ∼ . 𝜎 clear detection of ˆ Δ respectively in the range 0 < ˜ 𝑅 ≤ Figure 9.
The velocity distributions and the integrated ˆ Δ shift over the range0 < ˜ 𝑅 ≤ − . clusters in Wojtak et al. (2011), using the combination of ef-fects described in section 3.5 (and other small contributions) givesˆ Δ GR = − . − . Also the observational result in Wojtak et al.(2011) is ˆ Δ = − . ± . − . Considering they quote a meanmass of ∼ × 𝑀 (cid:12) , and the distribution in Fig. 4 peaks around3 . × 𝑀 (cid:12) , and is skewed towards higher masses, the GR pre-diction for the SPIDERS clusters and the size of the observed ˆ Δ seem to follow the expected behaviour of a higher mass sample lead-ing a larger predicted ˆ Δ . Furthermore, Sadeh et al. (2015) foundˆ Δ = − + − km s − using SDSS Run 10 galaxies and clusters. Thisresult is also in good agreement, albeit with large uncertainties. A positive detection of the gravitational redshift effect, along withother small contributions to a shift of the average of a distributionof galaxy velocity offsets, denoted ˆ Δ , is reported using SPIDERSDR16 galaxies and clusters. This work considered three definitionsof the centre of a cluster: using the Brightest Cluster Galaxy; thougha probabilistic determination of a red-sequence Central Galaxy; orfrom using the peak of X-ray emission. Each definition provides a dis-tinct galaxy population, and produces results for ˆ Δ largely consistentwith one another. Most notably the X-ray and BCG centroid casespredict a very similar change of ˆ Δ with projected distance from thecluster centre, ˜ 𝑅 . This is despite the need for a slightly cumbersomedefinition of the central redshift in the X-ray case.Galaxy redshift errors have not been used when finding ˆ Δ as theyare likely to be correlated with the apparent magnitude and galaxytype, which could introduce a bias on ˆ Δ . However it is important tonote that the uncertainty in observed redshifts could still introduce abias, yet it is hoped the large numbers of galaxies used beats downthis systematic.The result with the smallest error (largest sample of galaxies andclusters) and most robust methods comes from using a BCG to tracethe centre of a cluster. Using the centre of X-ray emission to tracethe cluster centre is a promising method: it removes the issue ofaccidental miscentring on foreground or background galaxies, andin dynamic clusters where the BCG is unlikely to trace the centre ofmass, X-ray centres may be a more accurate measure. The downsidein this analysis was the large X-ray centroid uncertainty in faint MNRAS , 1–10 (2020) ravitational redshifting of galaxies in SPIDERS ROSAT sources and a cumbersome central redshift definition—fromfinding the average redshift of galaxies in the core region. In general,a combination of these two methods—using the BCG closest to theX-ray centre, could provide a powerful hybrid, combining X-ray’slack of contamination and the ease of observing a BCG.For the optical case, although redMaPPer assigns an optical cen-tre based on a most likely Central Galaxy, in many cases there wasno spectroscopically observed galaxy near to the optical centre. Forsome clusters this could simply be due to positional errors, but inothers it is likely that the CG identified by redMaPPer has not beenspectroscopically observed by SPIDERS. Despite this shortcoming,finding the CG using a probabilistic approach has potential bene-fits over simply using a BCG. In cases where the cluster is highlydynamic, the filters used by redMaPPer may identify a CG moreappropriately.The integrated results for ˆ Δ in the range 0 < ˜ 𝑅 ≤ 𝑓 ( 𝑅 ) gravity—to within 1 𝜎 ; however, ˆ Δ ( ˜ 𝑅 ) slightlyfavours General Relativity in the BCG and X-ray cases. Each cen-troid case demonstrates a significant ( > . 𝜎 ) detection of the grav-itational redshifting of galaxies in SPIDERS clusters.Possible improvements to this work include a more robust predic-tion for the size of ˆ Δ in the theories of gravity used, involving bettertreatment of the redshift dependence, and a skewed normal fit to themass distribution. Furthermore, comparison with the predicted valueof ˆ Δ in other theories of gravity than the two considered could revealthe usefulness of this approach. If most other alternative theorieshave very similar predictions of ˆ Δ ( ˜ 𝑅 ) to GR, then because it is such asmall effect with often large uncertainties, the efficacy of the methodmay be limited.SDSS-V using eROSITA X-ray data (Kollmeier et al. 2017)promises more galaxy clusters with lower masses up to largerredshifts. More clusters means a better constrained ˆ Δ , and betterprospects for using this signal to distinguish between theories ofgravity. eROSITA will also have much better X-ray resolution, givingmore localised X-ray central positions. The 4MOST eROSITA GalaxyCluster Redshift Survey (Finoguenov et al. 2019) aims to providespectroscopic redshifts for ∼
40 000 eROSITA galaxy groups/clusters,including their BCG and >15 cluster members for 𝑧 < .
7. By com-bining an X-ray central position found from eROSITA data and thenearest 4MOST BCG or redMaPPer identified CG to this X-rayposition, there is the potential for accurate identification of clustercentres, even in dynamic systems where simply using a BCG causesmiscentring.To obtain a very strong positive detection of ˆ Δ , say > 𝜎 , con-sider the BCG case with an uncertainty of ± − . Assuming asimilar velocity dispersion, there needs to be ∼
530 000 galaxies inthe whole sample. While this is around an order of magnitude largerthan what is currently possible with SPIDERS, with forthcomingdeep optical telescopes such as
The Vera C. Rubin Observatory forgalaxy identification and, for example, the
Euclid satellite for spec-troscopic follow-up, this is certainly an achievable goal.
The Vera C.Rubin Observatory will overall observe billions of galaxies (Ivezićet al. 2019), and
Euclid ’s Near Infrared Spectrometer plans to mea-sure ∼
50 million spectroscopic redshifts of galaxies (Laureijs et al.2011). These numbers, coupled with well measured X-ray selectedclusters from eROSITA , promise tightly constrained measurementsof ˆ Δ in the near future.The same error of ∼ − for the BCG result in this analysis,compared with the 𝑓 ( 𝑅 ) prediction, would indicate a ∼ 𝜎 devia-tion between observations and the prediction of this example of analternative theory of gravity. Although these considerations demonstrate the sensitivity of grav-ity theories to gravitational redshifting using galaxy clusters, an im-portant caveat is that kinematic data alone is insufficient to provideadequate discrimination between theories of gravity. There is a de-generacy between the size of 𝐺 affecting the velocity distributionand the mass of the cluster—both GR and 𝑓 ( 𝑅 ) can give rise to thesame gravitational redshift signal but with different dark matter halofunctions (Zhao et al. 2013). Knowledge on how the X-ray inferredcluster mass changes for a given 𝑓 ( 𝑅 ) is needed (Li et al. 2016;Mitchell et al. 2020). Furthermore, as weak lensing based clustermass estimates are unaffected by an extension to 𝑓 ( 𝑅 ) (Barreiraet al. 2015; Lubini et al. 2011), this is a potential method by whichthis degeneracy can be broken. ACKNOWLEDGEMENTS
We thank the referee for helpful suggestions leading to tests thatimproved the robustness of our results. CTM and CAC acknowledgesupport from Liverpool John Moores University. JAP was supportedby the European Research Council under grant no. 670193. AS is sup-ported by the ERC-StG ‘ClustersXCosmo’ grant agreement 716762,and by the FARE-MIUR grant ’ClustersXEuclid’ R165SBKTMA.
DATA AVAILABILITY
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The NFW density profile gives the mass density as a function of thedistance from the centre of a cluster in units of its virial radius (Lokas& Mamon 2001): 𝜌 ( ˜ 𝑟 ) = 𝑀 𝑐 𝑔 ( 𝑐 ) 𝜋𝑟 ˜ 𝑟 ( + 𝑐 ˜ 𝑟 ) , (A1) Φ ( ˜ 𝑟 ) = − 𝑔 ( 𝑐 ) 𝐺 𝑀 𝑟 ln ( + 𝑐 ˜ 𝑟 ) ˜ 𝑟 . (A2)Important parameters here are the concentration parameter 𝑐 andthe function 𝑔 ( 𝑐 ) : 𝑐 = 𝑟 𝑟 𝑠 , (A3) 𝑔 ( 𝑐 ) = ( ln ( + 𝑐 ) − 𝑐 /( + 𝑐 )) − . (A4)The concentration parameter gives the ratio between the virial radiusof an astronomical body and its so called scale radius, and givesan indication of the mass concentration of the object. Typically, forclusters 𝑐 ∼ 𝑐 ∼
10 .
APPENDIX B: BOOTSTRAPPING TEST
Sampling with replacement was used to obtain 10 values for ˆ Δ forthe galaxy samples in each of the centroid cases to test the consistencyof the quoted uncertainties in Fig. 8; histograms of the results areshown in Fig. B1. The average values and standard deviations of theresulting Gaussians are consistent with the quoted results. This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS , 1–10 (2020) ravitational redshifting of galaxies in SPIDERS Figure B1.
Bootstrapping with 10 samples on the galaxy velocity offsetdistributions for each centroid case. MNRAS000