## Testing General Relativity on galaxy scales

Einstein's theory of gravity, General Relativity (GR), has been tested precisely within the Solar System. However, it has been difficult to test GR on the scale of an individual galaxy. Collett *et al.* exploited a nearby gravitational lens system, in which light from a distant galaxy (the source) is bent by a foreground galaxy (the lens). Mass distribution in the lens was compared with the curvature of space-time around the lens, independently determined from the distorted image of the source. The result supports GR and eliminates some alternative theories of gravity.

*Science*, this issue p. 1342

## Abstract

Einstein’s theory of gravity, General Relativity, has been precisely tested on Solar System scales, but the long-range nature of gravity is still poorly constrained. The nearby strong gravitational lens ESO 325-G004 provides a laboratory to probe the weak-field regime of gravity and measure the spatial curvature generated per unit mass, γ. By reconstructing the observed light profile of the lensed arcs and the observed spatially resolved stellar kinematics with a single self-consistent model, we conclude that γ = 0.97 ± 0.09 at 68% confidence. Our result is consistent with the prediction of 1 from General Relativity and provides a strong extragalactic constraint on the weak-field metric of gravity.

General Relativity (GR) postulates that mass deforms space-time, so that light passing near to a massive object is deflected. If two galaxies are almost perfectly aligned, the deformation of space-time near the center of the foreground galaxy can be large enough that multiple images of the background galaxy are observed. Such alignments are called strong gravitational lenses. In the case of a spherical foreground lens and a perfect alignment of lens and source, the background galaxy is distorted into an Einstein ring. The radius of this ring, the Einstein radius, is a function of the mass of the lens, the amount of spatial curvature produced per unit mass, and a ratio of three angular diameter distances between the observer, lens, and source.

Angular diameter distances are calculated from the redshifts (inferred from the wavelength shift of spectral lines owing to the expansion of the Universe) of the lens, *z*_{l}, and source, *z*_{s}, and the cosmological parameters of our Universe (*1*). Therefore, the combination of a nonlensing measurement of the mass of a strong lensing galaxy and a measurement of the Einstein radius constrains the amount of spatial curvature produced per unit mass and tests whether GR is the correct theory of gravity.

In the limit of a weak gravitational field, the metric of space-time is characterized by two potentials (*2*)—the Newtonian potential, Φ, and the curvature potential, Ψ—so that the comoving distance element is(1)where τ is the conformal time, *x ^{i}* and

*x*are the spacelike coordinates,

^{j}*g*is the three-metric of constant-curvature spaces, and

_{ij}*a*(τ) is the scale factor of the Universe.

In GR, the two potentials are the same, but many alternative gravity models invoked to explain the accelerated expansion of the Universe [such as *f*(*R*) gravity (*3*)] predict the ratio of the two potentials (γ = Ψ/Φ) to be scale-dependent. These alternative models of gravity remove the need for a dark energy to accelerate the expansion of the Universe. Testing the scale dependence of γ therefore discriminates between GR and these alternative gravities. The motion of nonrelativistic objects is governed by the Newtonian potential, whereas the motion of relativistic particles is sensitive to both potentials (*2*). Measuring γ therefore requires observations of the motions of both relativistic and nonrelativistic particles around the same massive object.

On Solar System scales, the GR prediction for γ has been verified to high precision. By measuring the travel time of radio photons passing close to the Sun, the Cassini mission found γ = 1 + (2.1 ± 2.3) × 10^{–5} (*4*). However, the extragalactic constraints on GR are much less precise. On scales of 10 to 100 Mpc, γ has been constrained to just 20% precision (*5**–**7*) by combining measurements of weak gravitational lensing and redshift space distortions. On megaparsec scales, 30% constraints on γ have been achieved using the mass profiles of clusters (*8*, *9*).

On kiloparsec scales, strong gravitational lensing, combined with stellar kinematics in the lens, allows a test of the weak-field metric of gravity. The kinematics are sensitive only to the Newtonian potential, whereas the lensing is sensitive to the sum of the potentials. The deflection angle of light, , caused by a galaxy with surface mass density Σ(**x**), is given by(2)where *G* is the gravitational constant, and *c* is the speed of light. Here, γ is explicitly assumed to be constant over the length scales relevant for lensing. In this case, the lensing and dynamical masses are related by(3)where *M*_{dyn} is the mass derived from the dynamics, and is the mass derived from lensing and assuming GR is correct.

Previous studies have combined Einstein radius measurements with stellar velocity dispersions to infer constraints on γ (*10*, *11*). Recent work (*12*) used a sample of 80 lenses to infer γ = 0.995 ± 0.04 (statistical) ± 0.25 (systematic) at 68% confidence. The systematic uncertainties are much larger than the statistical uncertainties, because the result relies on assumptions about the density profile of the lenses and the orbits of the stars within the lenses. The large distances to the lenses in these samples (0.1 < *z*_{l} < 1) make a more detailed study of their kinematics impractical with current instruments.

The nearby (lens redshift *z*_{l} = 0.035) galaxy ESO 325-G004 (hereafter, E325) is located at right ascension , declination –38°10′34′′ (J2000 equinox). Hubble Space Telescope (HST) imaging of E325 serendipitously revealed the presence of an Einstein ring with a 2.95′′ radius around the center of the galaxy (*13*), shown in Fig. 1. Follow-up observations have revealed that the source is an extended star-forming galaxy at source redshift *z*_{s} = 2.1 (*14*). E325 provides a laboratory for testing GR on kiloparsec scales because the small distance to the lens enables spatially resolved measurements of the stellar kinematics; this places tight constraints on the three-dimensional mass structure of the lens. E325 also has an extended arc system, which provides tight constraints on the two-dimensional surface mass profile of the lens. The small distance also means that the Einstein ring appears in the baryon-dominated central region of the lens galaxy: The uncertain distribution of dark matter is much less important than for more distant lenses.

E325 has previously been observed with the Advanced Camera for Surveys on the HST (*13*). We used the HST image observed in the red F814W filter to describe the light profile of the galaxy for our mass modeling, and we produced an image of the arc light by subtracting the F814W image from that in the blue F475W filter, with the F814W image radially rescaled to account for a slight color gradient in the lens galaxy (*15*). Previous strong lensing constraints on γ have relied only on Einstein radius measurements (*10*–*12*), yielding a constraint only on the total lensing mass within the Einstein ring. However, the extended arcs in E325 also allow for a detailed reconstruction of the unlensed source from the HST observations. This reconstruction places additional constraints on the radial magnification across the image plane wherever lensed images are present (*16*). The radial magnification is a weighted integral of the mass within the Einstein ring; hence, reconstructing the arcs excludes many density profiles that would have the same Einstein radius but produce arcs of different shapes and widths. This approach is now mature in studies of strong lenses where high-precision constraints are required (*17*–*20*).

To constrain the dynamical mass, we used observations acquired with the Multi Unit Spectroscopic Explorer (MUSE) (*21*) on the European Southern Observatory (ESO) Very Large Telescope to obtain spatially resolved spectroscopy across the lens (*15*). From the spectra, we extracted the velocity dispersion of the stars in each pixel. We use Jeans axisymmetric modeling (*22*) to infer the dynamical mass of the lens from these data (*15*).

We simultaneously fit a 20-parameter model to both the MUSE and the HST data (*15*). The parameters describe the mass-to-light ratio of the stellar component observed with the HST, including a mass-to-light gradient, a dark matter halo, and a central supermassive black hole. We also include parameters that describe external lensing shear from masses close to the line of sight, the radial profile of the stellar orbital anisotropy, and the unknown inclination of the lens galaxy with respect to the line of sight. The final parameter of our model is γ, the ratio of the Newtonian and curvature potentials. We assume that this ratio is constant across the relevant length scales of this lens. The smallest angular scale that the MUSE data resolve corresponds to a physical scale of 100 pc in the lens plane, and the Einstein radius corresponds to 2000 pc.

By sampling the posterior probability of our model fit to the data, using a Markov chain Monte Carlo method (*23*), we infer the uncertainties on the model parameters. Our best-fitting reconstruction of the HST image is shown in Fig. 2. Our model simultaneously reproduces the observed lensing and dynamical data. Our reconstruction of the MUSE data is shown in Figs. 3 and 4.

Within the constraints of our fiducial model, we infer that it is only possible to simultaneously reconstruct the lensing and kinematics if the stellar mass-to-light ratio of the lens increases within the central kiloparsec, with an observed F814W-band stellar mass-to-light ratio at large radius of 2.8 ± 0.1*M*_{☉}/*L*_{☉} (in units of solar mass over solar luminosity; consistent with a Milky Way–like stellar initial mass function), rising to 6.6 ± 0.1*M*_{☉}/*L*_{☉} in the center (consistent with an excess of mass in low-luminosity stars relative to the Milky Way). We infer that the central black hole has a mass of 3.8 ± 0.1 × 10^{9 }*M*_{☉}, consistent with expectations (*24*) given the mass of E325. We find that the lens must be inclined close to the plane of the sky, with an inclination angle of 90 ± 15°. The typical orbits of the stars are mildly radial at most radii, although they are poorly constrained in the central 0.5 kpc. The dark matter in our fiducial model only accounts for of the mass within the Einstein radius, consistent with expectations extrapolating from higher-redshift lenses (*25*), given that the 2.95′′ Einstein radius is a quarter of the 13′′ radius that contains half of the light of the lens.

Our fiducial model assumes that the dark matter follows the Navarro-Frenk-White (NFW) profile (*26*) that is found in cosmological dark matter–only simulations; however, the baryonic processes of galaxy formation are capable of resculpting the dark matter profile (*27*–*29*). An alternative model without a stellar mass-to-light gradient is also able to reconstruct the data, although the central dark matter density would have to fall with radius to the power –2.6 (ρ_{DM} ∝ *r*^{–2.6})—much steeper than the fiducial (ρ_{DM} ∝ *r*^{–1}) profile. In this model, the baryons are consistent with a Milky Way–like stellar initial mass function, and the dark matter accounts for 31 ± 4% of the mass within the Einstein radius (*15*).

We also investigated another model that does not partition the mass into baryonic and dark matter; this model also required a steeply rising total mass-to-light ratio in the central regions. Our current data cannot distinguish between highly concentrated dark matter, a steep stellar mass-to-light gradient, and an intermediate solution, but E325 is definitely not consistent with a NFW dark matter halo and a constant stellar mass-to-light ratio.

In our fiducial model, we find at 68% confidence, with the uncertainties estimated from the Markov chain Monte Carlo analysis. Owing to the small statistical uncertainties, we also carried out a thorough analysis of our systematic uncertainty relating to the parameterization of the lens model.

In principle, the parameterization of the mass profile should not affect the inferred value of γ, because both the lensing and the kinematics are sensitive to the total mass distribution, rather than the partition between dark matter, black hole, and baryons. The alternative models investigated (*15*) show an excess scatter on γ of 0.01, in addition to the statistical uncertainties.

Our model assumes the fiducial cosmology from the Planck satellite’s cosmic microwave background observations (*1*). Our inference of the lensing mass is inversely proportional to the assumed value of the current expansion rate of the Universe, *H*_{0}. The low redshift of the lens causes our inference on γ to be almost independent of the other cosmological parameters. The 1.3% uncertainty on the Planck determination of *H*_{0} is equivalent to a 2.6% uncertainty on γ.

Our dynamical model assumes a particular stellar spectral library to extract the kinematics. For our data, there is a 2.2% systematic shift between velocity dispersions measured with theoretically motivated stellar libraries (*30*) and those empirically derived from observations of Milky Way stars (*31*). This translates to a 4.4% uncertainty on the dynamical mass and hence 8.8% uncertainty on γ. This systematic uncertainty dominates the final uncertainty on γ.

By simultaneously reconstructing the extended light profile of the arcs and spatially resolved kinematics of E325, we precisely constrained γ with a single system. The high-resolution data allow us to fit for both the density profile and the anisotropy profile of the lens, removing these as sources of systematic uncertainty that had limited previous strong lensing constraints on γ (*10*–*12*). We conclude that γ = 0.97 ± 0.09 (68% confidence limits), including our identified systematic uncertainties. This constraint on the ratio of the two potentials outside the Solar System confirms the prediction of GR in the local Universe for galactic masses and kiloparsec scales. Our result implies that large deviations from γ = 1 can only occur on scales greater than ~2 kpc, thereby excluding alternative gravity models that produce the observed accelerated expansion of the Universe but predict γ ≠ 1 on galactic scales.

## Supplementary Materials

www.sciencemag.org/content/360/6395/1342/suppl/DC1

Materials and Methods

Figs. S1 to S6

Tables S1 to S4

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We are grateful to S. Carniari for help with reducing the MUSE data and to D. Newman for discussions. We thank the three anonymous referees for their constructive feedback.

**Funding:**T.E.C. is funded by a Dennis Sciama Fellowship from the University of Portsmouth. R.J.S. acknowledges support from the STFC through grant ST/P000541/1. D.B., R.C.N., K.L.M., and K.K. are supported by STFC through grant ST/N000688/1. K.K. is supported by the European Research Council (ERC) (grant agreement 646702 “CosTesGrav”). Numerical computations were performed on the SCIAMA High Performance Compute (HPC) cluster, which is supported by the ICG, SEPNet, and the University of Portsmouth.

**Author contributions:**T.E.C. performed the lensing and dynamical modeling, led the proposal for the MUSE data, and wrote the paper. L.J.O. implemented the Jeans modeling of E325, extracted the MUSE velocity dispersions, and cowrote the paper. R.J.S. performed the initial kinematic fits and produced the lens-subtracted HST data. M.W.A. reduced the HST and MUSE data. K.B.W. helped design and implement the MUSE observations. All authors provided ideas throughout the project and commented on the manuscript and observing proposal.

**Competing interests:**The authors have no competing interests.

**Data and materials availability:**This work is based on observations made with ESO telescopes at the La Silla Paranal Observatory; those data are available at http://archive.eso.org under program ID 097.A-0987(A). This work is also based on observations made with the NASA/ESA Hubble Space Telescope and obtained from the Hubble Legacy Archive, http://hla.stsci.edu [under programs GO 10429 (PI: Blakeslee) and GO 10710 (PI: Noll)], which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope–European Coordinating Facility (ST-ECF/ESA), and the Canadian Astronomy Data Centre (CADC/NRC/CSA). Our lens modeling code, PYLENS, is available at http://github.com/tcollett/pylens. The MGE_FIT_SECTORS software and JAM dynamical modeling code are available at www-astro.physics.ox.ac.uk/~mxc/software/. The ensemble sampler EMCEE is available at http://dfm.io/emcee/. The output parameters of our models are given in tables S2 to S4.