The Fifth Force in the Local Cosmic Web
Harry Desmond, Pedro G Ferreira, Guilhem Lavaux, Jens Jasche
MMNRAS , 1–6 (2018) Preprint 26 November 2018 Compiled using MNRAS L A TEX style file v3.0
The Fifth Force in the Local Cosmic Web
Harry Desmond ∗ , Pedro G. Ferreira , Guilhem Lavaux , , andJens Jasche , Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France Sorbonne Universités, Institut Lagrange de Paris (ILP), 98 bis bd Arago, 75014 Paris, France The Oskar Klein Centre, Department of Physics, Stockholm University, Albanova University Center, SE 106 91 Stockholm, Sweden Excellence Cluster Universe, Technische Universität München, Boltzmannstrasse 2, D-85748 Garching, Germany
26 November 2018
ABSTRACT
Extensions of the standard models of particle physics and cosmology often lead tolong-range fifth forces with properties dependent on gravitational environment. Fifthforces on astrophysical scales are best studied in the cosmic web where perturbationtheory breaks down. We present constraints on chameleon- and symmetron-screenedfifth forces with Yukawa coupling and megaparsec range – as well as unscreenedfifth forces with differential coupling to galactic mass components – by searching forthe displacements they predict between galaxies’ stars and gas. Taking data fromthe
Alfalfa H i survey, identifying galaxies’ gravitational environments with the mapsof Desmond et al. (2018a) and forward-modelling with a Bayesian likelihood frame-work, we set upper bounds on fifth-force strength relative to Newtonian gravity from ∆ G/G N < few × − for range λ C = 50 Mpc, to ∆ G/G N (cid:46) . for λ C = 500 kpc. In f ( R ) gravity this requires f R < few × − . The analogous bounds with-out screening are ∆ G/G N < few × − and ∆ G/G N < few × − . These are thetightest and among the only fifth-force constraints on galaxy scales. We show how ourresults may be strengthened with future survey data and identify the key features ofan observational programme for furthering fifth-force tests beyond the Solar System. Key words: gravitation – galaxies: kinematics and dynamics – galaxies: statistics –cosmology: theory
Despite fundamental open questions, almost all attempts atextending the standard models of particle physics and cos-mology have proven unsatisfactory. Nevertheless, a genericfeature of such extensions is the introduction of extra degreesof freedom. These arise by replacing dimension-full param-eters with dynamical fields [e.g. lepton masses (Weinberg1967), dynamical dark energy (Ratra & Peebles 1988) orthe gravitational constant (Brans & Dicke 1961; Wetterich1988)], and embody higher derivatives and extra dimensions.As any generalisation of the Einstein-Hilbert action mustevolve new fields (Clifton et al. 2012), practically all at-tempts to extend the standard model add scalar, vector ortensor fields that influence the dynamics of the Universe andits contents.Extra fields couple naturally to the Ricci scalar R in thegravitational action. For example, a scalar φ may generate ∗ E-mail: [email protected] a non-minimal coupling of the form φ R , which complicatesdynamics: not only will it source energy and momentum(along with all other constituents of the Universe) but itwill also modify the gravitational force. Taking the simplestcase of standard kinetic energy and potential V ( φ ) , the New-tonian potential Φ of a point mass M is modified to Φ tot = GMr (cid:18)
GG e − mr (cid:19) (1)where G is the bare (Newtonian) gravitational constant, m ∼ d V /dφ and ∆ G/G depends on the magnitude of thenon-minimal coupling and the background field value rela-tive to the Planck mass M pl . m sets the range of the fifthforce and ∆ G its strength. The General Relativistic (GR)result is recovered for ∆ G → , and also for m → ∞ sothat the fifth force is confined to a narrow radius aroundthe source. The scalar Higgs field for example generates avery short-range fifth force (Herranen et al. 2015).There are extremely stringent constraints on fifth forcesover a wide range of scales (see Adelberger et al. 2003 for c (cid:13) a r X i v : . [ a s t r o - ph . C O ] N ov H. Desmond, P. G. Ferreira, G. Lavaux, J. Jasche
Figure 1. σ bound on fifth-force strength ∆ G/G and range λ C obtained from offsets between the stellar and gas mass centroidsof Alfalfa galaxies, both with and without screening. a review); on astrophysical scales the tightest constraintsfor low m come from Shapiro time delay measurementsfrom the Cassini satellite (Bertotti et al. 2003), which re-quire ∆ G/G (cid:46) − . Although this is sufficiently strongto make a universally-coupled fifth force cosmologically in-significant, a number of theories (for example generalisedscalar-tensor theories and massive gravity) evade Solar Sys-tem bounds by means of a screening mechanism wherebythe fifth-force strength or range becomes a function of envi-ronment. Chameleon screening (Khoury & Weltman 2004)arises when the effective mass, m eff , becomes dependent onlocal density (and thus on ∇ Φ , where Φ is the Newtonianpotential): in denser regions, m eff become large and the fifthforce has short range, while in empty regions (or on cos-mological scales) m eff → and the fifth force effectivelyemerges. By virtue of the ‘thin-shell effect’, and corroboratedin simulations (Zhao et al. 2011a,b; Cabré et al. 2012), anobject’s degree of screening is set by Φ = Φ in + Φ ex , where Φ in is the potential at the object’s surface due to its ownmass and Φ ex is the contribution from surrounding mass.The object is unscreened if | Φ | is less than a critical value | Φ c | . Conversely, in the Vainshtein (Vainshtein 1972) and symmetron (Hinterbichler & Khoury 2010) mechanisms thefifth-force strength depends on environment: near massivebodies ∆ G/G → , while away from them ∆ G/G (cid:54) = 0 .In the presence of screening, the laboratory, the SolarSystem and clusters will generally probe the screened regimeand hence be expected to yield the GR result. However, thisis not the case for a range of galaxy environments in thecosmic web, which probe very low density regions and shouldtherefore manifest a fifth force. In this Letter we use a map ofscreening proxies to identify these environments and henceforward-model a key signal of chameleon and symmetronscreening: a displacement between galaxies’ stellar and gasmass centroids. Comparing to optical and H i data, we set σ limits from ∆ G/G < few × − at range /m eff (cid:39) λ C = 50 Mpc to ∼ . for λ C = 500 kpc. In f ( R ) gravity, where ∆ G/G = 1 / , this corresponds to f R (cid:46) few × − . The detailed procedure for charting the gravitational envi-ronments of the local Universe is given in Desmond et al.(2018a) (building on earlier work in Cabré et al. 2012); weprovide a summary here. Our map encompasses a region outto approximately h − Mpc and is based on the 2M++galaxy catalogue (Lavaux & Hudson 2011), a synthesis of2MASS, 6dF and SDSS data. We connect the K -band lu-minosity function with the halo mass function from a highresolution Λ CDM N-body simulation ( darksky-400 ; Skill-man et al. 2014) by using abundance matching (AM) to as-sociate a dark matter halo to each galaxy, according to thespecific prescription of Lehmann et al. 2017. (We validatethis model in the K -band using a counts-in-cells clusteringstatistic in Desmond et al. (2018a).) The magnitude limitof the 2M++ survey (12.5 in K ) means that it misses faintgalaxies and their associated halos. To correct for this, weuse the abundance-matched simulation to estimate the dis-tribution and density of halos hosting galaxies above themagnitude limit, and fill these in through their probabilis-tic correlation with observables. Finally, we account for thematter not associated with resolved halos by means of aBayesian reconstruction of the density field with resolution . h − Mpc using the BORG algorithm (Jasche et al. 2010;Jasche & Wandelt 2012; Jasche et al. 2015; Jasche & Lavaux2018), which propagates information from the number den-sities and peculiar velocities of 2M++ galaxies assumingconcordance cosmology and a bias model. We call this the“smooth density field”. As each step in this chain is proba-bilistic, we generate many Monte Carlo realisations of thefields to sample the statistical uncertainties in the inputs.We focus here on a particular fifth-force signal: the dis-placement between galaxies’ optical (tracing stellar mass)and H i (tracing cold gas mass) centroids. Such a displace-ment may come about either from a difference in the cou-pling of the fifth force to stars and gas, or, more likely,from chameleon or symmetron screening (Jain & VanderPlas2011; Brax et al. 2012). In the latter, gas and dark matterin unscreened galaxies feel a fifth force due to neighbouringunscreened mass, leading to an effective increase in New-ton’s constant ∆ G = 2 β G for coupling coefficient β if thescalar field is light. Stars on the other hand self-screen andfeel only G . The result of this effective equivalence principleviolation (Hui et al. 2009) is an offset between the stellarand gas mass in the direction of the external fifth-force (cid:126)a .We search for such a displacement, and its correlationwith (cid:126)a , using the complete catalogue of Alfalfa (Giovanelliet al. 2005; Kent et al. 2008; Haynes et al. 2011), a blindH i survey out to z (cid:39) . conducted with the Arecibo ob-servatory. Optical counterparts (OCs) for the majority ofdetections were derived from cross-correlation with opticalsurveys and included in the catalogue. The uncertainty inthe H i centroid position is best estimated directly from itsdisplacement from the OC: we create 50 logarithmically uni-form bins in the signal to noise ratio of the detection (SNR)between the minimum and maximum values 4.6 and 1000respectively, calculate in each bin the standard deviation ofthe RA and DEC components of the H i -optical offset, andset the corresponding components of the H i centroid un-certainties to be twice these to ensure our constraints areconservative. This gives the uncertainty a median and stan- MNRAS000
Figure 1. σ bound on fifth-force strength ∆ G/G and range λ C obtained from offsets between the stellar and gas mass centroidsof Alfalfa galaxies, both with and without screening. a review); on astrophysical scales the tightest constraintsfor low m come from Shapiro time delay measurementsfrom the Cassini satellite (Bertotti et al. 2003), which re-quire ∆ G/G (cid:46) − . Although this is sufficiently strongto make a universally-coupled fifth force cosmologically in-significant, a number of theories (for example generalisedscalar-tensor theories and massive gravity) evade Solar Sys-tem bounds by means of a screening mechanism wherebythe fifth-force strength or range becomes a function of envi-ronment. Chameleon screening (Khoury & Weltman 2004)arises when the effective mass, m eff , becomes dependent onlocal density (and thus on ∇ Φ , where Φ is the Newtonianpotential): in denser regions, m eff become large and the fifthforce has short range, while in empty regions (or on cos-mological scales) m eff → and the fifth force effectivelyemerges. By virtue of the ‘thin-shell effect’, and corroboratedin simulations (Zhao et al. 2011a,b; Cabré et al. 2012), anobject’s degree of screening is set by Φ = Φ in + Φ ex , where Φ in is the potential at the object’s surface due to its ownmass and Φ ex is the contribution from surrounding mass.The object is unscreened if | Φ | is less than a critical value | Φ c | . Conversely, in the Vainshtein (Vainshtein 1972) and symmetron (Hinterbichler & Khoury 2010) mechanisms thefifth-force strength depends on environment: near massivebodies ∆ G/G → , while away from them ∆ G/G (cid:54) = 0 .In the presence of screening, the laboratory, the SolarSystem and clusters will generally probe the screened regimeand hence be expected to yield the GR result. However, thisis not the case for a range of galaxy environments in thecosmic web, which probe very low density regions and shouldtherefore manifest a fifth force. In this Letter we use a map ofscreening proxies to identify these environments and henceforward-model a key signal of chameleon and symmetronscreening: a displacement between galaxies’ stellar and gasmass centroids. Comparing to optical and H i data, we set σ limits from ∆ G/G < few × − at range /m eff (cid:39) λ C = 50 Mpc to ∼ . for λ C = 500 kpc. In f ( R ) gravity, where ∆ G/G = 1 / , this corresponds to f R (cid:46) few × − . The detailed procedure for charting the gravitational envi-ronments of the local Universe is given in Desmond et al.(2018a) (building on earlier work in Cabré et al. 2012); weprovide a summary here. Our map encompasses a region outto approximately h − Mpc and is based on the 2M++galaxy catalogue (Lavaux & Hudson 2011), a synthesis of2MASS, 6dF and SDSS data. We connect the K -band lu-minosity function with the halo mass function from a highresolution Λ CDM N-body simulation ( darksky-400 ; Skill-man et al. 2014) by using abundance matching (AM) to as-sociate a dark matter halo to each galaxy, according to thespecific prescription of Lehmann et al. 2017. (We validatethis model in the K -band using a counts-in-cells clusteringstatistic in Desmond et al. (2018a).) The magnitude limitof the 2M++ survey (12.5 in K ) means that it misses faintgalaxies and their associated halos. To correct for this, weuse the abundance-matched simulation to estimate the dis-tribution and density of halos hosting galaxies above themagnitude limit, and fill these in through their probabilis-tic correlation with observables. Finally, we account for thematter not associated with resolved halos by means of aBayesian reconstruction of the density field with resolution . h − Mpc using the BORG algorithm (Jasche et al. 2010;Jasche & Wandelt 2012; Jasche et al. 2015; Jasche & Lavaux2018), which propagates information from the number den-sities and peculiar velocities of 2M++ galaxies assumingconcordance cosmology and a bias model. We call this the“smooth density field”. As each step in this chain is proba-bilistic, we generate many Monte Carlo realisations of thefields to sample the statistical uncertainties in the inputs.We focus here on a particular fifth-force signal: the dis-placement between galaxies’ optical (tracing stellar mass)and H i (tracing cold gas mass) centroids. Such a displace-ment may come about either from a difference in the cou-pling of the fifth force to stars and gas, or, more likely,from chameleon or symmetron screening (Jain & VanderPlas2011; Brax et al. 2012). In the latter, gas and dark matterin unscreened galaxies feel a fifth force due to neighbouringunscreened mass, leading to an effective increase in New-ton’s constant ∆ G = 2 β G for coupling coefficient β if thescalar field is light. Stars on the other hand self-screen andfeel only G . The result of this effective equivalence principleviolation (Hui et al. 2009) is an offset between the stellarand gas mass in the direction of the external fifth-force (cid:126)a .We search for such a displacement, and its correlationwith (cid:126)a , using the complete catalogue of Alfalfa (Giovanelliet al. 2005; Kent et al. 2008; Haynes et al. 2011), a blindH i survey out to z (cid:39) . conducted with the Arecibo ob-servatory. Optical counterparts (OCs) for the majority ofdetections were derived from cross-correlation with opticalsurveys and included in the catalogue. The uncertainty inthe H i centroid position is best estimated directly from itsdisplacement from the OC: we create 50 logarithmically uni-form bins in the signal to noise ratio of the detection (SNR)between the minimum and maximum values 4.6 and 1000respectively, calculate in each bin the standard deviation ofthe RA and DEC components of the H i -optical offset, andset the corresponding components of the H i centroid un-certainties to be twice these to ensure our constraints areconservative. This gives the uncertainty a median and stan- MNRAS000 , 1–6 (2018) he Fifth Force in the Local Cosmic Web L3 dard deviation across the sample of (cid:48)(cid:48) and (cid:48)(cid:48) respectively.(We briefly mention the results of a less conservative choicebelow, and note that similar results are obtained by fittingfor the uncertainty as a zeroth, first or second order polyno-mial in SNR.) We cut the catalogue at 100 Mpc where thefixed angular uncertainty leads to an unacceptably large spa-tial uncertainty, yielding a sample of size , . We thencut a further 1,355 galaxies with poor SNR ( Alfalfa qual-ity flag 2 or 9) and 262 galaxies where the optical and H i images are likely misidentified ( > (cid:48) H i -OC offset), whichcorresponds roughly to a σ outlier clip. We have checkedthat our analysis is not especially sensitive to this: even cut-ting at (cid:48) (a < σ clip), removing . of our sample, doesnot appreciably alter our results. Our final sample has size N Alf = 10 , . We supplement the Alfalfa information for of our galaxies with structural galaxy properties fromthe
Nasa Sloan Atlas (NSA; stellar mass M ∗ , half-light ra-dius R eff , apparent axis ratio b/a , aperture velocity disper-sion σ d and Sérsic index n ), which will improve the precisionof the predicted H i -OC offset as calculated below.To constrain the fifth-force strength ∆ G and range λ C we proceed as follows. First, assuming a Compton wave-length for the scalar field in the range . < λ C / Mpc < we set the screening threshold | Φ c | /c = 32 × − (cid:18) λ C Mpc (cid:19) . (2)This is exact for the case of Hu-Sawicki f ( R ) (Hu & Saw-icki 2007) (where | Φ c | is . times the background scalarfield value φ = f R ) and also applicable more generallywith λ C interpreted in terms of the self-screening parame-ter φ / (2 βM pl ) . We use our gravitational maps to determinewhich halos, and portions of the smooth density field, are un-screened given these parameters by calculating Φ ex as a sumover all mass within λ C of the test point. We take Φ in = − σ d for galaxies with NSA information and Φ in = − V max forthose without, where V max is the maximum rotational ve-locity estimated by correcting the full-width half-max of theradio detection for turbulence and projection effects (Tully& Fouque 1985). These contributions to the total potentialderive primarily from the test galaxy’s dark matter. Notethat in the case of cluster galaxies, the potential of the clus-ter itself is part of the external contribution. We calculate (cid:126)a by summing the contributions of all unscreened mass within λ C . We then calculate the equilibrium H i -OC offset (cid:126)r ∗ pre-dicted for a given galaxy: M ( < r ∗ ) r ∗ ˆ (cid:126)r ∗ = ∆ GG (cid:126)a (3)if it is unscreened and otherwise, where M ( < r ∗ ) is thedark matter plus gas mass between the H i and optical cen-troids. This follows from the requirement that the extra forceon the stellar disk due to its offset from the halo centre com-pensate for its not feeling the fifth force, so that the stars, gasand dark matter continue to move together (Jain & Vander-Plas 2011). We calculate M ( < r ∗ ) by assuming a constantdensity ρ within r ∗ (justified post-hoc: r ∗ for the fifth-forcemodels we are sensitive to is − − − kpc, much less thanthe halo scale radius r s ), and estimate it separately for eachgalaxy using the empirical relation between central baryonicand dynamical surface mass densities (Lelli 2014; Lelli et al. Figure 2.
Offsets r ∗ between optical and H i centroids predictedfor Alfalfa galaxies within 100 Mpc by a model with λ C = 5 Mpc and ∆ G/G = 1 , as a function of total Newtonian potential Φ . The green points are for the full model with screening; theblue points show the case where screening is switched off. Thebars in the legend show the average size of the uncertainties in Φ ,defined as the minimal widths enclosing 68% of the Monte Carlorealisations of the model. The y-uncertainties are too small tobe visible on this plot, and are subdominant to the measurementuncertainties. The vertical dashed line shows the threshold | Φ c | above which galaxies in the model with screening are screened. (cid:126)r ∗ = 34 π ρ ∆ GG (cid:126)a . (4)As (cid:126)r ∗ spans a very small angle on the plane of the sky wecompare separately its orthogonal RA ( r ∗ ,α ) and DEC ( r ∗ ,δ )components with those of the measured displacement foreach galaxy.We feed these calculations into a Bayesian likelihoodformalism. First, we generate N MC = 1000 Monte Carlo re-alisations of the predicted signal (cid:126)r ∗ for each Alfalfa galaxy,sampling independently for each one the galaxy–halo con-nection (from 200 independent AM realisations), the distri-bution of mass in the smooth density field (from 10 particle-mesh BORG realisations), the contribution to Φ ex and (cid:126)a from halos too faint to be recorded in 2M++ (calibratedwith the darksky-400 N-body box), and the Gaussian ob-servational uncertainties on the structural galaxy proper-ties used to derive M ( < r ∗ ) and Φ in . The full probabil-ity distributions that we marginalise over are given in ta-ble 1 of Desmond et al. (2018b). We estimate the proba-bility that a given galaxy is unscreened as f ≡ N ( | Φ ex | + | Φ in | < | Φ c | ) /N MC . The likelihood function then has sepa-rate screened ( r ∗ = 0 ) and unscreened (Eq. 4) components,with relative weights − f and f respectively. We modelthe unscreened component using a normalised histogram ofthe distributions of r ∗ ,α and r ∗ ,δ over all N MC realisations,obviating the need for assumptions on the form of the like-lihood function such as Gaussianity. We convolve this like-lihood with the Gaussian H i measurement uncertainty foreach galaxy, θ i , and treat galaxies as uncorrelated and RAand DEC components as independent. This gives the total MNRAS , 1–6 (2018) H. Desmond, P. G. Ferreira, G. Lavaux, J. Jasche likelihood of the
Alfalfa data under the fifth-force modelspecified by { λ C , ∆ G } . Finally, we take 20 logarithmicallyuniformly spaced values of λ C between 400 kpc and 50 Mpcand constrain λ C and ∆ G/G by MCMC.Our study greatly extends previous work testingchameleon screening by means of this signal (Vikram et al.2013), in which M ( < r ∗ ) and (cid:126)a were not modelled. We de-scribe our method exhaustively in Desmond et al. (2018b). In Fig. 1 we show our σ constraint in the λ C − ∆ G/G plane, with and without screening. The dependence of the ∆ G/G limit on λ C may be understood as a combination oftwo effects. First, when λ C is smaller less mass contributesto (cid:126)a , leading to a smaller predicted signal at fixed ∆ G/G (Eq. 4). This allows ∆ G/G , which simply scales the pre-dicted (cid:126)r ∗ , to be larger while keeping the prediction consis-tent with the observations. Second, a smaller λ C correspondsto a smaller | Φ c | (Eq. 2), making both the test galaxy it-self and the surrounding mass less likely to be unscreened,and hence to contribute to (cid:126)a . In the case without screen-ing, (cid:126)a is calculated from all mass within λ C (rather thanonly unscreened mass), and each test galaxy is consideredfully unscreened ( f = 1 ). Removing screening strengthensthe ∆ G/G constraints at low λ C but does not change themsignificantly for λ C (cid:38) Mpc, because at higher λ C mostmasses are unscreened anyway. Instead, the factor limitingthe constraint is the volume around a galaxy within whichmatter contributes to (cid:126)a , which is set by λ C and is the samebetween the screening and no-screening runs. Similar resultsare obtained from resamples of the Alfalfa data with repeats(bootstraps) and from parts of the full dataset (jackknifes).In Fig. 2 we show the correlation with Φ of the signal r ∗ predicted for the Alfalfa galaxies by a fiducial model with λ C = 5 Mpc, ∆ G/G = 1 . Green points are for the casewith screening included (so that r ∗ → for | Φ | > | Φ c | ) andblue for the case without. For this relatively high value of ∆ G/G the predicted signal is typically O (kpc). The trendwith Φ derives from r ∗ ∝ a (Eq. 4) combined with thepositive correlation of a with | Φ | ; in the case with screening,however, the signal vanishes for | Φ | /c > | Φ c | /c = 3 . × − .Many chameleon constraints have focused on f ( R ) grav-ity where ∆ G/G = 1 / ; in this case we require λ C (cid:46) . Mpc( σ ), or equivalently f R ≡ df/dR | R < few × − , where R is the current cosmological value of the Ricci scalar. Thisis stronger than cluster and cosmology constraints by twoorders of magnitude (Song et al. 2007; Schmidt et al. 2009;Yamamoto et al. 2010; Ferraro et al. 2011; Lombriser et al.2012a,b; Lombriser 2014; Terukina et al. 2014; Dossett et al.2014; Wilcox H. et al. 2015) and by distance indicators (Jainet al. 2013) and rotation curves (Vikram et al. 2018) by one,and operates in a fully complementary regime to laboratoryfifth-force searches (Adelberger et al. 2003; Burrage & Sak-stein 2016; Burrage & Sakstein 2017; Brax et al. 2018). For λ C → ∞ , which holds for a light scalar field, we expect a ∆ G/G constraint better than − . These results extend di-rect constraints on fifth forces from Solar System to galacticscales, helping to fill the gap in the parameter space of testsof gravity (Baker et al. 2015). The strength of our bounds owes to the large sample size, great range of gravitationalenvironments probed (including with very low | Φ | ), and avector rather than scalar observable, which effectively af-fords two orthogonal signals in the plane of the sky.We have checked that our analysis is converged withnumber of Monte Carlo realisations, that the AM galaxy–halo connection and smooth density field from BORG arethoroughly sampled, that our MCMC is converged with thenumber of steps, and that our principal results are insensi-tive to reasonable variations in M ( < r ∗ ) and the assumeduncertainties in galaxy and halo properties. We have marginalised over the statistical uncertainties inmost of the model inputs, including the galaxy–halo connec-tion, the smooth density field and the observed properties ofgalaxies. Nevertheless, we make three key assumptions thatmay lead to systematic error in our results:1) We assume that H i -optical offsets generated by non-fifth-force effects follow the Gaussian likelihood model wecreated for the noise. While baryonic processes such as hy-drodynamical drag, ram pressure and stellar feedback mayinduce a stronger signal than fifth forces, their environment-dependence is unlikely to mimic the effect of screening: ourconstraints derive primarily from the correlation betweenthe direction of the H i -OC offset and (cid:126)a , as well as boththe relative magnitude of these vectors over all galaxies andthe precise dependence of the prediction on gravitationalpotential. Indeed, our model for the uncertainty θ in theH i centroid implies that on average the entire signal canbe accounted for by non-fifth-force effects; that strong con-straints are nonetheless attainable attests to the specificityof the features of the signal that fifth forces should induce.2) To calculate Φ and (cid:126)a we assume Λ CDM structureformation. Although the fifth-force scenarios we investigatewould alter cosmology, this is a small effect for { λ C , ∆ G } as low as is in question here (Lombriser 2014); this system-atic error is almost certainly subdominant to the statisticalerrors in the Λ CDM galaxy–halo connection and smoothdensity field. Our method should not therefore be consid-ered a means of probing modified gravity in cosmology, butrather of unearthing any galaxy-scale fifth force in the low- z Universe, of gravitational or non-gravitational origin.3) Our fiducial noise model sets the positional uncer-tainty of the H i centroid to be twice as large on a galaxy-by-galaxy basis as the H i -OC displacement itself. If we removethe factor of two in our θ assignment – as would roughlybe derived by fitting θ to the data as a zeroth, first or sec-ond order polynomial in SNR – we find . σ evidence for ∆ G/G > . This reflects a positive correlation between (cid:126)a and the observed (cid:126)r ∗ over the unscreened part of thesample across the lower portion of our λ C range ( λ C (cid:46) Mpc), with a maximum log-likelihood at λ C (cid:39) . Mpcand ∆ G/G (cid:39) . that is larger than that obtainedby ∆ G = 0 . We describe and validate this possible detec-tion fully in Desmond et al. (2018b), and note that a similarsignal is found in Desmond et al. (2018c). MNRAS000
Alfalfa data under the fifth-force modelspecified by { λ C , ∆ G } . Finally, we take 20 logarithmicallyuniformly spaced values of λ C between 400 kpc and 50 Mpcand constrain λ C and ∆ G/G by MCMC.Our study greatly extends previous work testingchameleon screening by means of this signal (Vikram et al.2013), in which M ( < r ∗ ) and (cid:126)a were not modelled. We de-scribe our method exhaustively in Desmond et al. (2018b). In Fig. 1 we show our σ constraint in the λ C − ∆ G/G plane, with and without screening. The dependence of the ∆ G/G limit on λ C may be understood as a combination oftwo effects. First, when λ C is smaller less mass contributesto (cid:126)a , leading to a smaller predicted signal at fixed ∆ G/G (Eq. 4). This allows ∆ G/G , which simply scales the pre-dicted (cid:126)r ∗ , to be larger while keeping the prediction consis-tent with the observations. Second, a smaller λ C correspondsto a smaller | Φ c | (Eq. 2), making both the test galaxy it-self and the surrounding mass less likely to be unscreened,and hence to contribute to (cid:126)a . In the case without screen-ing, (cid:126)a is calculated from all mass within λ C (rather thanonly unscreened mass), and each test galaxy is consideredfully unscreened ( f = 1 ). Removing screening strengthensthe ∆ G/G constraints at low λ C but does not change themsignificantly for λ C (cid:38) Mpc, because at higher λ C mostmasses are unscreened anyway. Instead, the factor limitingthe constraint is the volume around a galaxy within whichmatter contributes to (cid:126)a , which is set by λ C and is the samebetween the screening and no-screening runs. Similar resultsare obtained from resamples of the Alfalfa data with repeats(bootstraps) and from parts of the full dataset (jackknifes).In Fig. 2 we show the correlation with Φ of the signal r ∗ predicted for the Alfalfa galaxies by a fiducial model with λ C = 5 Mpc, ∆ G/G = 1 . Green points are for the casewith screening included (so that r ∗ → for | Φ | > | Φ c | ) andblue for the case without. For this relatively high value of ∆ G/G the predicted signal is typically O (kpc). The trendwith Φ derives from r ∗ ∝ a (Eq. 4) combined with thepositive correlation of a with | Φ | ; in the case with screening,however, the signal vanishes for | Φ | /c > | Φ c | /c = 3 . × − .Many chameleon constraints have focused on f ( R ) grav-ity where ∆ G/G = 1 / ; in this case we require λ C (cid:46) . Mpc( σ ), or equivalently f R ≡ df/dR | R < few × − , where R is the current cosmological value of the Ricci scalar. Thisis stronger than cluster and cosmology constraints by twoorders of magnitude (Song et al. 2007; Schmidt et al. 2009;Yamamoto et al. 2010; Ferraro et al. 2011; Lombriser et al.2012a,b; Lombriser 2014; Terukina et al. 2014; Dossett et al.2014; Wilcox H. et al. 2015) and by distance indicators (Jainet al. 2013) and rotation curves (Vikram et al. 2018) by one,and operates in a fully complementary regime to laboratoryfifth-force searches (Adelberger et al. 2003; Burrage & Sak-stein 2016; Burrage & Sakstein 2017; Brax et al. 2018). For λ C → ∞ , which holds for a light scalar field, we expect a ∆ G/G constraint better than − . These results extend di-rect constraints on fifth forces from Solar System to galacticscales, helping to fill the gap in the parameter space of testsof gravity (Baker et al. 2015). The strength of our bounds owes to the large sample size, great range of gravitationalenvironments probed (including with very low | Φ | ), and avector rather than scalar observable, which effectively af-fords two orthogonal signals in the plane of the sky.We have checked that our analysis is converged withnumber of Monte Carlo realisations, that the AM galaxy–halo connection and smooth density field from BORG arethoroughly sampled, that our MCMC is converged with thenumber of steps, and that our principal results are insensi-tive to reasonable variations in M ( < r ∗ ) and the assumeduncertainties in galaxy and halo properties. We have marginalised over the statistical uncertainties inmost of the model inputs, including the galaxy–halo connec-tion, the smooth density field and the observed properties ofgalaxies. Nevertheless, we make three key assumptions thatmay lead to systematic error in our results:1) We assume that H i -optical offsets generated by non-fifth-force effects follow the Gaussian likelihood model wecreated for the noise. While baryonic processes such as hy-drodynamical drag, ram pressure and stellar feedback mayinduce a stronger signal than fifth forces, their environment-dependence is unlikely to mimic the effect of screening: ourconstraints derive primarily from the correlation betweenthe direction of the H i -OC offset and (cid:126)a , as well as boththe relative magnitude of these vectors over all galaxies andthe precise dependence of the prediction on gravitationalpotential. Indeed, our model for the uncertainty θ in theH i centroid implies that on average the entire signal canbe accounted for by non-fifth-force effects; that strong con-straints are nonetheless attainable attests to the specificityof the features of the signal that fifth forces should induce.2) To calculate Φ and (cid:126)a we assume Λ CDM structureformation. Although the fifth-force scenarios we investigatewould alter cosmology, this is a small effect for { λ C , ∆ G } as low as is in question here (Lombriser 2014); this system-atic error is almost certainly subdominant to the statisticalerrors in the Λ CDM galaxy–halo connection and smoothdensity field. Our method should not therefore be consid-ered a means of probing modified gravity in cosmology, butrather of unearthing any galaxy-scale fifth force in the low- z Universe, of gravitational or non-gravitational origin.3) Our fiducial noise model sets the positional uncer-tainty of the H i centroid to be twice as large on a galaxy-by-galaxy basis as the H i -OC displacement itself. If we removethe factor of two in our θ assignment – as would roughlybe derived by fitting θ to the data as a zeroth, first or sec-ond order polynomial in SNR – we find . σ evidence for ∆ G/G > . This reflects a positive correlation between (cid:126)a and the observed (cid:126)r ∗ over the unscreened part of thesample across the lower portion of our λ C range ( λ C (cid:46) Mpc), with a maximum log-likelihood at λ C (cid:39) . Mpcand ∆ G/G (cid:39) . that is larger than that obtainedby ∆ G = 0 . We describe and validate this possible detec-tion fully in Desmond et al. (2018b), and note that a similarsignal is found in Desmond et al. (2018c). MNRAS000 , 1–6 (2018) he Fifth Force in the Local Cosmic Web L5 We use the observed displacements between galaxies’ stel-lar and gas mass centroids in the
Alfalfa catalogue to con-strain fifth forces that couple differentially to stars, gasand dark matter. As a case study we consider chameleonand symmetron screening, in which stars in otherwise un-screened galaxies self-screen. We deploy the gravitationalmaps of Desmond et al. (2018a) to determine screened andunscreened regions of the d <
Mpc Universe, and calcu-late the acceleration that would be induced at the positionof each
Alfalfa galaxy by a fifth force with strength ∆ G andrange λ C . Comparing to the data with a Monte Carlo like-lihood formalism, we require ∆ G/G (cid:46) . for λ C = 500 kpcand ∆ G/G (cid:46) few × − for λ C = 50 Mpc. In f ( R ) gravitythis is f R (cid:46) few × − . The corresponding bounds withoutscreening are ∆ G/G (cid:46) few × − and ∆ G/G (cid:46) few × − .These are the strongest and among the only fifth-force con-straints at astrophysical scales.While our results reveal the gravitational informationthat can currently be extracted with this signal, they may bestrengthened as data from future galaxy surveys is broughtto bear. The principal factors limiting the inference in Fig. 1are the large uncertainty θ that we use for the angular po-sition of the H i centroid (with average ¯ θ = 36 (cid:48)(cid:48) ), and thenumber of galaxies in the sample. To forecast the improve-ment afforded by future surveys, we generate mock datasetswith N gal = f × N Alf galaxies ( − < f < ), and H i an-gular uncertainty Θ × θ i ( − < Θ < ) for galaxy i . Wegenerate a mock signal for each galaxy by randomly scat-tering around 0 by this uncertainty, and select the galaxiesrandomly from the full Alfalfa sample. We rederive posteri-ors on ∆ G/G (at λ C = 5 Mpc) for each mock dataset, andfit to this data a power-law of the form σ (cid:18) ∆ GG (cid:19) (cid:39) . × − (cid:18) N gal (cid:19) . (cid:18) ¯ θ (cid:19) . , (5)where the left hand side is the σ constraint on ∆ G/G .To project constraints for N gal > N Alf we extrapolate thisrelation: for N gal ∼ , ¯ θ ∼ . (cid:48)(cid:48) – achievable by next-generation radio surveys such as SKA (Santos et al. 2015;Yahya et al. 2015) – the constraints on ∆ G/G should be O (10 − ) . This would be competitive with proposed SolarSystem tests involving laser ranging to Phobos and opticalnetworks around the Sun (Sakstein 2018). We caution how-ever that further modelling will be required to extend thegravitational maps to the higher redshift ( z ∼ . ) that this N gal requires, and also that the time-dependence of param-eters such as f R may impact the inference.Our analysis is the first to employ “big data” fromgalaxy surveys to constrain gravitational physics with anintra-galaxy signal. We have shown this to afford tighterconstraints on fifth forces than other methods involving ei-ther cosmological information or cherry-picked astrophysicalobjects. Nevertheless, the power of tests of this type remainslargely unexplored: many more galactic signals – includingdisk warps, mass discrepancies, dynamical asymmetries andoffsets between kinematics at different wavelengths – willbring further and independent constraining power. Our workpaves the way for fundamental physics to be incorporatedas a key science driver in upcoming survey programmes. ACKNOWLEDGEMENTS
HD is supported by St John’s College, Oxford. PGF ac-knowledges support from Leverhulme, STFC, BIPAC andthe ERC. GL acknowledges support by the ANR grant num-ber ANR-16-CE23-0002 and from the Labex ILP (refer-ence ANR-10-LABX-63) part of the Idex SUPER (ANR-11-IDEX-0004-02). We thank Martha Haynes for sharingthe complete
Alfalfa catalogue before public release, PhilBull for information on SKA and Jeremy Sakstein and TessaBaker for comments on the draft. Computations were per-formed at Oxford and SLAC.
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