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No evidence for modifications of gravity from galaxy motions on cosmological scales

Abstract

Current tests of general relativity (GR) remain confined to the scale of stellar systems or the strong gravity regime. A departure from GR on cosmological scales has been advocated1 as an alternative to the cosmological constant Λ (ref. 2) to account for the observed cosmic expansion history3,4. However, such models yield distinct values for the linear growth rate of density perturbations and consequently for the associated galaxy peculiar velocity field. Measurements of the resulting anisotropy of galaxy clustering5,6 have thus been proposed as a powerful probe of the validity of GR on cosmological scales7, but despite substantial efforts8,9, they suffer from systematic errors comparable to statistical uncertainties10. Here, we present the results of a forward-modelling approach that fully exploits the sensitivity of the galaxy velocity field to modifications of GR. We use state-of-the-art high-resolution N-body simulations of a standard GR (Λ cold dark matter (CDM)) model11 and a compelling f(R) model12—one of GR’s simplest variants, in which the Ricci scalar curvature, R, in the Einstein–Hilbert action is replaced by an arbitrary function of R—to build simulated catalogues of stellar-mass-selected galaxies through a robust match to the Sloan Digital Sky Survey13. We find that f(R) fails to reproduce the observed redshift-space clustering on scales of ~1–10 Mpc h−1, where h is the dimensionless Hubble parameter. Instead, the standard ΛCDM GR model agrees impressively well with the data. This result provides strong confirmation, on cosmological scales, of the robustness of Einstein’s general theory of relativity.

Main

The simplest generalization of GR is represented by the so-called f(R) models (see ref. 12 for a review), in which f(R) gravity adheres to all of the pivotal principles that underpin GR, such as the equivalence principle. It also predicts gravitational waves travelling at the speed of light, another feature of GR recently confirmed in a spectacular way by the multi-messenger gravitational wave event GW170817 (refs 14,15). As such, f(R) gravity is arguably the ‘least unnatural’ and most compelling theory of gravity beyond Einstein’s GR. Moreover, the f(R) cosmological model includes a ‘screening mechanism’, which restores GR in dense environments, where strong constraints exist (see, for example, ref. 16). This means that deviations from GR, if any, can be detected by measurements on cosmological scales only, such as those probed by galaxy surveys, where such screening is, in general, ineffective.

However, a precise test of gravity from galaxy surveys is non-trivial. Usually, modifications of gravity are probed by measuring the linear growth of cosmic structure7. Structure growth induces large-scale peculiar velocities, namely, galaxy coherent motions superimposed on the pure cosmological expansion. This adds a Doppler contribution to the measured cosmological redshift, which induces so-called redshift-space distortions (RSD)5. A direct recovery of the linear growth rate from RSD is, however, impaired by large random velocities inside galaxy clusters and groups, resulting from the full nonlinear gravitational evolution. Their effects can extend to fairly large scales, resulting in a complicated scale-dependent clustering signal (see, for example, ref. 10 and references therein). The past decade saw substantial progress in modelling these nonlinear effects (see, for example, refs 8,9 and references therein), leading to a number of independent measurements of the growth rate of structure at different redshifts (for example, refs 17,18,19). Still, systematic errors in such modelling remain comparable to the achieved precisions (~3–5%)10, and it remains difficult to assess quantitatively how model-dependent assumptions globally affect the results.

Here, we take an alternative route that avoids most of the current difficulties. Rather than applying nonlinear corrections to linear models of the growth of structure to match the data, we take a forward-modelling approach that starts from the fully nonlinear description of matter clustering provided by numerical N-body simulations of structure formation in standard and non-standard cosmologies. The key problem becomes then to associate self-consistently and unambiguously observed galaxies with dark matter haloes in the simulations. This is achieved using the so-called subhalo abundance matching (SHAM) technique20 (see Methods). Following the standard paradigm of galaxy formation, galaxies reside in dark matter haloes (usually called subhaloes because they are often embedded in more massive parent haloes such as a group or a cluster). If a monotonic relationship exists between a property of the subhalo and a physical property of the galaxy, then there is a one-to-one match between simulated dark matter subhaloes and galaxies in a survey, selected according to the corresponding physical property of the galaxy. This implies, for example, that the clustering of subhaloes in a simulation can be directly and quantitatively compared with the measured clustering of a corresponding sample of galaxies. The relevant virtues of this approach are: (1) there is no ambiguity regarding galaxy bias, which links the clustering (and RSD) of galaxy tracers to the underlying dark matter field, and (2) nonlinear clustering and motions are fully reproduced up to the scales set by the mass resolution and volume of the simulation.

To implement SHAM in practice, the key point is to identify the specific physical property of observable galaxies that monotonically (or with very small scatter) depends on a property of the host dark matter halo. State-of-the-art hydrodynamic simulations such as EAGLE (Evolution and Assembly of GaLaxies and their Environments)21, in which dissipative processes and a comprehensive physical model of galaxy formation are included, show that a tight correlation (albeit with some scatter) exists between galaxy stellar mass (the total mass of a galaxy in stars) and the peak value of the maximum circular velocity over the subhalo’s merger history, vpeak, which is a robust proxy for the maximum gravitational potential attained over the existence of a simulated subhalo (see Methods).

We adopt the SHAM technique in this work to make a quantitative comparison of redshift-space clustering measured from the Sloan Digital Sky Survey (SDSS) main galaxy sample13 and the predictions of two simulations, built in the standard ΛCDM scenario and in an f(R) cosmology. To construct our reference data sample, we use the New York University Value-Added Galaxy Catalogue (NYU-VAGC)22, which is an enhanced version of the SDSS Data Release 7 (ref. 13). At a limiting SDSS Petrosian magnitude r ≤ 17.60 mag, this catalogue is highly complete and uniform over an area of 7,732 deg2, including 542,432 galaxies with measured spectroscopy and a median distance of ~300 Mpc h−1. From the observed redshift of spectral lines, redshift-space distances are computed (note that given the low redshift of the sample, distances are only very mildly dependent on the assumed cosmology). From this catalogue, we then build volume-limited samples that are complete in stellar mass (see Methods). Unlike luminosity, galaxy stellar mass is not directly obtained from the observed flux, but needs to be derived from the fit of a stellar population synthesis model to multi-band photometric measurements (see, for example, ref. 23). This procedure is prone to systematic errors (see, for example, ref. 24 for a review). To mitigate these, we define our volume-limited samples in terms of a galaxy number density threshold, rather than applying a limit in stellar mass. As demonstrated in Methods, if a sufficiently high-density sample is used, the impact of stellar-mass systematic errors on clustering measurements is negligible. For this reason, our reference sample is defined by imposing a mean galaxy density ng = 1 × 10−2 (Mpc h−1)−3 (see Supplementary Fig. 1).

This reference data catalogue is first matched to a simulation performed in the standard ΛCDM GR scenario. Specifically, we choose the Small MultiDark Planck (SMDPL) simulation11, which adopts a Planck cosmology with h = 0.6777, matter density parameter Ωm = 0.3071, mass density fluctuation amplitude in an 8 Mpc h−1 sphere σ8 = 0.8228 and scalar spectral index ns = 0.96 (ref. 25). The simulation uses 3,8403 dark matter particles in a box of 400 Mpc h−1 length per side, giving a mass resolution of 9.6 × 107 solar masses (M) per h. Dark matter haloes and subhaloes in our analyses are identified using the ROCKSTAR (Robust Overdensity Calculation using K-Space Topologically Adaptive Refinement) halo finder26. The high mass resolution of this simulation is crucial for the application of the SHAM technique. A satellite subhalo close to the centre of its parent halo may lose substantial mass due to tidal stripping. At an earlier time in the simulation, a massive halo can even be completely disrupted by tidal stripping such that, at a later time, it is no longer detectable. This leads to what, in simulations, have been called orphan galaxies27, which are galaxies that may exist as baryonic objects today, but are devoid of an identifiable dark matter subhalo. Overlooking these objects in the simulation can underestimate the predicted clustering on small scales27. In our case, based on ref. 28 and given the high mass resolution of the SMDPL simulation, our reference galaxy number density ng = 1 × 10−2 (Mpc h−1)−3 guarantees that the fraction of orphan galaxies is less than 2.6%, which is a negligible contribution. Another technical issue is that, in the SHAM approach, a scatter is usually added to the relationship between vpeak and galaxy stellar mass (see Methods). This quantity essentially affects the selection of subhaloes near the sample mass threshold only, which at the high number densities we have in our samples, has a limited impact. For this reason, in our SHAM implementation, we avoid adding any scatter, and as a consequence, our model has no free parameters.

Our SHAM prediction for f(R) gravity is based on a second simulation, presented in ref. 29, coupled to an effective halo technique30. This is necessary as the circular velocity vcir of a baryonic particle in a subhalo in f(R) gravity is not directly related to the CDM mass but to an effective mass defined through a modified version of the Poisson equation30

$$\nabla^2{\varphi } = 4{\pi}Ga^2\delta \rho _{{\mathrm{eff}}}$$

with 2 being the Laplace operator, φ Newton’s gravitational potential, G Newton’s gravitational constant, a the scale factor of the cosmological background expansion and δρeff the perturbation of the effective energy density field ρeff. The simulation has a mass resolution of 1.52 × 108Mh−1, comparable to that of the SMDPL simulation. This is crucial for this kind of test because the screening mechanism, which substantially affects the velocity field in f(R) cosmology, can be accurately explored only if the resolution is sufficiently high (see Methods). Although the computational cost of this kind of simulation limits the box size to 64 Mpc h−1, which misses the long-wavelength modes (large scales) of the density field, the simulation can still produce reliable predictions for the higher-order multipoles of clustering (see Supplementary Fig. 5 and Methods), which are those containing most of the velocity field information and, as such, the most important quantities to test gravity. In any case, we further develop a self-consistent correction for the missing large-scale modes by building a test ΛCDM simulation with the same box size (64 Mpc h−1) and initial conditions as the f(R) simulation. In this case, the differences between their respective multipoles arguably reflect only the different nature of gravity. The ratio of the multipoles for the two (big and small) ΛCDM simulations can then be used to renormalize the corresponding f(R) multipoles such that they can be compared on an equal footing with the final SMDPL measurements. Details and robustness tests of this method are described in Methods.

From the ΛCDM simulation, we build a SHAM mock survey that fully reproduces the survey mask, geometry and wide-angle effects, volume-limited to have ng = 1 × 10−2 (Mpc h−1)−3 (see Methods). We then compute the redshift-space two-point correlation function ξ(rσ, rπ) (where rσ and rπ give the separation of galaxy pairs split into the directions perpendicular and parallel to the line of sight, respectively) for the f(R) and ΛCDM mock surveys and for the real data, using the standard Landy and Szalay estimator (see Methods). In Fig. 1, we compare the measurement from the SDSS data with that of the ΛCDM mock sample. It is remarkable how well the data and the model agree, particularly on small scales in the highly nonlinear regime. The corresponding two-dimensional ξ(rσ, rπ) for f(R) cosmology is much noisier due to the limited box size of the simulation; to reduce the noise and obtain a better statistical comparison with the data, we compress the information into spherical harmonics moments, namely, the monopole ξ0, quadrupole ξ2 and hexadecapole ξ4, as detailed in Methods. We also describe in Methods how we account for the well-known effect of the SDSS spectrograph fibre collisions on very small scales, using the so-called truncated multipoles. The results are shown in Fig. 2 for both ΛCDM and f(R), compared with the SDSS measurements. As expected from Fig. 1, the ΛCDM model is also an excellent description of the clustering multipoles, within 1σ statistical uncertainty of the observational measurements, given by the small error bars. The f(R) prediction, however, has a significantly smaller amplitude on small scales (see Methods); despite its relatively larger theoretical uncertainty, this corresponds to a discrepancy of more than 5σ with the SDSS data on small scales (s < 6 Mpc h−1), as shown in Fig. 2 by the shaded bars. The robustness of this result with respect to systematic errors in stellar-mass estimators is also indicated by different symbols: different stellar-mass models yield consistent clustering for the corresponding mass-selected samples, indicating that they do not substantially change the rank order of the galaxies in the SHAM implementation (see also Supplementary Fig. 3).

Fig. 1: The small-scale galaxy clustering measured from the SDSS in redshift space, compared with a realization of the standard ΛCDM model.
figure1

The reference galaxy sample here is defined by imposing a threshold in galaxy stellar mass so as to reach a mean galaxy density ng = 1 × 10−2 (Mpc h−1)−3. The colour-coded contours show the amplitude of the two-point correlation function ξ(rσ, rπ) of a stellar-mass-selected sample of galaxies from the NYU-VAGC, as a function of the transverse rσ and radial rπ separation. The actual measurement for positive rσ and rπ is replicated over four quadrants to highlight deviations from circular symmetry, produced by galaxy peculiar velocities that add to cosmological expansion when we use galaxy redshifts as a proxy for distance. Without peculiar motions, the contours would be perfect circles. The dashed lines give the corresponding predictions of ΛCDM, obtained from a mock galaxy survey fully mimicking the real data, based on the SMDPL N-body simulation. Dark matter haloes in the simulation and galaxies have been matched through an implementation of SHAM without free parameters (no scatter). The ΛCDM predictions agree impressively well with the observations in redshift space, especially for the so-called ‘fingers of God’ feature at small rσ, namely, the stretching of the contours along the line of sight produced by high-velocity galaxies in groups and clusters. This highlights the advantage of our forward-modelling approach using simulations to allow a comparison of galaxy clustering and motions for data and models in the fully nonlinear regime.

Fig. 2: Multipoles of the redshift-space two-point correlation ξ(rσ, rπ).
figure2

Galaxy clustering predicted by standard ΛCDM (black solid line) and the f(R) model (red solid line) are compared with the SDSS measurements (symbols with errors), in terms of the monopole ξ0, quadrupole ξ2 and hexadecapole ξ4 (multiplied by the redshift-space separation s for convenience). These were estimated using the ‘truncation’ technique (see Methods). The monopole and hexadecapole lines have been shifted vertically as indicated, to aid visualization. As suggested by Fig. 1, ΛCDM is in excellent agreement with observational measurements, while the f(R) model is not. To test the robustness of this result with respect to different estimates of galaxy stellar mass, we compare three methods: a template-fit method39 as adopted in the NYU-VAGC with the SDSS model magnitudes (star symbols), the same but using SDSS Petrosian magnitudes (circles) and a single-colour method40 labelled as Y07 (triangles). Clearly, changing the estimates of stellar mass does not change the results appreciably; this happens because different mass estimations approximately preserve the same rank order of galaxies. Small error bars show the 1σ statistical error estimated using jackknife resampling with 133 realizations, while the large shaded bars indicate a 5σ statistical error. The black and red shaded corridors indicate the 1σ uncertainty in the theoretical predictions of the two cosmological models.

Overall, these results show that when data and simulations are self-consistently matched with physically motivated and robustly tested forward modelling, the ΛCDM model in the framework of GR provides an impressively accurate description of both galaxy clustering and motions on small scales. Conversely, a representative of the family of f(R) models, such as the one considered here31, is clearly ruled out. This incarnation of f(R) is characterized by a free parameter fR0 = −10−6 and an index n = 1. Such a small value of \(\left\lfloor {{f}_{R0}} \right\rfloor\) makes this f(R) model barely distinguishable from ΛCDM when using other cosmological probes such as cluster counts or weak lensing32. Yet, the impact on the observed redshift-space clustering is dramatic, as we can see from Fig. 2. The sensitivity of the nonlinear velocity field to modifications to the laws of gravity is remarkable33. Our result provides a strong confirmation on cosmological scales of Einstein’s general theory of relativity, substantially reducing the appeal of a modification to gravity as a solution to the conundrum of the cosmic acceleration.

Methods

Observational data

Our analysis uses the NYU-VAGC22, which is an enhanced version of the SDSS main galaxy sample Data Release 7 (ref. 13). Specifically, we use the bbright NYU-VAGC, which has a fairly homogeneous r-band (one of the five colour bands in the SDSS photometric system) Petrosian magnitude limit of r ≤ 17.60 mag over the whole survey footprint. The catalogue covers an area of 7,732 deg2; of these, 144 deg2 are masked due to bright stars. Galaxies in this catalogue are mainly located in a contiguous region in the north galactic cap. We also include the three strips in the south galactic cap, which account for about 10% of the total number of galaxies, for a total of Nt = 594,307 photometrically selected galaxies. Of these, Ns = 542,432 have a reliable spectroscopic redshift. All samples analysed here are limited to redshift z > 0.02.

Volume-limited samples complete in stellar mass

We follow the method proposed in ref. 34 to construct volume-limited samples that are complete in stellar mass. Since blue galaxies have a lower ratio of stellar mass to light, M/L, for a given stellar mass, a blue galaxy can be detected to a higher redshift than a red one (see Supplementary Fig. 1). This implies that the flux limit of a flux-limited survey translates into a minimum stellar-mass limit as a function of redshift, Mmin(z), that is higher for red galaxies than for blue ones. As shown in the same figure, below that limit, red galaxies disappear, and a sample naively selected in stellar mass would be biased towards blue objects. Clearly, if we choose a minimum mass limit defined as that corresponding to the detected object in the sample that has the most extreme red colour, all other (bluer) galaxies of the same stellar mass will be more luminous and so will also be in the sample. As such, the resulting sample will be statistically complete in stellar mass. In practice, the maximum M/L in the sample (corresponding, for the case just mentioned, to the reddest objects) can be estimated from the data34. Supplementary Fig. 2 shows the r-band M/L of galaxies in the NYU-VAGC for different stellar-mass models (see next section). The solid lines are our estimates of the maximum M/L as a function of the absolute r-band magnitude Mr, namely, Mmin(Mr). The stellar-mass limit above which the sample is complete in stellar mass is then given by Mmin(Mrmax(z)), where Mrmax(z) is the maximum (faintest) r-band absolute magnitude Mr that can be seen at z given an apparent flux limit mr. The solid curves in Supplementary Fig. 1 show the Mmin(z) obtained in this way for different stellar-mass models. The corresponding horizontal and vertical lines show stellar mass and maximum redshift limits (zmax) for six volume-limited subsamples that are complete in stellar mass.

Stellar-mass estimate systematic uncertainties

Unlike luminosity, a galaxy’s stellar mass cannot be directly measured, but has to be derived from a fit to its spectral energy distribution, using a stellar population synthesis model (for example, ref. 23), a modelling process that is prone to systematic errors. There are two main sources of errors. One lies in the theoretical uncertainty, in particular in the choice of the stellar initial mass function (IMF) (see ref. 24 for a review). Choosing a Chabrier35 or Kroupa IMF36 has a significant impact on the amplitude of the stellar-mass function. The second major source of systematic uncertainty lies in the way the total flux of a galaxy is estimated from the imaging data. In the SDSS, the aperture used to estimate the flux in all five photometric bands (u, g, r, i and z) is set by the galaxy surface brightness profile as measured in the r-band alone. This defines an r-band Petrosian radius rp. The total flux in all bands is then obtained by integrating out to twice this value, 2rp. This aperture is large enough to contain virtually all flux for objects with an exponential surface brightness profile. For objects with a de Vaucouleurs profile, however, this typically includes only 80% of the flux. As such, a significant fraction of light can be lost, and SDSS Petrosian magnitudes tend to underestimate the total flux of galaxies with such a profile, a problem that is particularly acute for massive elliptical galaxies (see, for example, ref. 37).

To mitigate these systematics, we define our galaxy samples in terms of their number densities, rather than through thresholds in stellar mass. The idea is to keep the rank order of galaxies stable. Changes in the IMF shift the absolute value of a galaxy stellar mass while not significantly changing its relative rank order. Therefore, by selecting galaxies in terms of number densities, the choice of IMF has little impact on the selected samples and, most importantly, does not perturb the rank order in the SHAM.

In addition, we use galaxy samples with a high mean number density, which has a valuable effect of reducing the impact of uncertainty (scatter) in the estimated stellar mass between different models. As shown in ref. 38, if the same stellar IMF is used, the overall distribution of the estimated stellar masses in the sample remains stable when changing the model, but on a galaxy-by-galaxy basis, there is still significant scatter. This scatter, however, affects the sample definition only near the mass threshold chosen. If the sample is very sparse, the fraction of objects going in and out of the sample due to scatter can be significant. However, if the mean number density of the selected samples is sufficiently high, this fraction will be small compared with the bulk of the sample.

We test directly the effectiveness of such a strategy by applying three different stellar-mass models to our galaxy samples. The first one is the default model used in the NYU-VAGC, which is based on the five-band SDSS Petrosian photometry and a fit to templates from a stellar population synthesis model39. The template fit yields the galaxy M/L, and the stellar mass can then be obtained by multiplying it by its luminosity. To address the impact of the fixed r-band aperture on the estimated stellar mass, we adopt the same template-fitting method, but use the SDSS model magnitudes instead of the Petrosian ones. The former are obtained by fitting the (point spread function convolved) exponential and de Vaucouleurs profiles to a galaxy and then adopting the one giving the best χ2. In contrast to the SDSS Petrosian magnitudes, SDSS model magnitudes better account for the loss of flux due to the fixed aperture. Finally, we also consider a third model based on a single-colour method, following ref. 40. In this case, the stellar mass is given by

$${\mathrm{log}}_{10}\left[ {M_ \ast \left( {h^{ - 2}{\kern 1pt} M_ \odot } \right)} \right] = - 0.406 + 1.097\left[ {{\,}^{0.0}\left( {g-r} \right)} \right] - 0.4\left( {{\,}^{0.0}M_{\mathrm{r}} - 5{\kern 1pt} {\mathrm{log}}_{10}{\kern 1pt} h - 4.64} \right)$$

where 0.0Mr − 5 log10h is the absolute magnitude that is K-corrected and evolution corrected to redshift zero, as

$${\,}^{0.0}M_{\mathrm{r}} - 5{\kern 1pt} {\mathrm{log}}_{10}{\kern 1pt} h = m_{\mathrm{r}} - {\mathrm{DM}}\left( z\right) - k_{0.0}\left( z\right) + 1.62z$$

Here, k0.0(z) is the K-correction to redshift zero, and DM(z) is the distance modulus

$${\mathrm{DM}}(z) \equiv 5{\kern 1pt} {\mathrm{log}}_{10}\left[ {D_{\mathrm{L}}} \right] + 25$$

with DL being the luminosity distance in Mpc h−1. Note that in this estimator, the stellar mass is explicitly dependent on only the rest frame 0.0(g − r) colour (g and r are mangnitudes in two different SDSS photometric bands as mentioned the above) and magnitude. It is also important to note that this single-colour estimator implicitly assumes the Kroupa IMF36, while the default NYU-VAGC uses the Chabrier IMF35. Supplementary Fig. 2 compares the M/L obtained for the three different models. Panel b shows the notable differences between the single-colour estimator and the photometric template-fit method: estimated stellar masses differ not only on a galaxy-by-galaxy basis, but also in the resulting global M/L relation.

Two-point correlation function estimators

To estimate the redshift-space two-point correlation function ξ(rσ, rπ) (ref. 41), we use the well-known Landy and Szalay estimator42. To reveal deviations from isotropy, ξ(rσ, rπ) can be conveniently expanded in terms of Legendre polynomials as

$${\mathrm{\xi }}_l\left( s \right) = \frac{{2l + 1}}{2}\mathop {\smallint }\limits_{ - 1}^1 {\kern 1pt} {\mathrm{d}}\mu \xi \left( {s,\mu } \right)P_l\left( \mu \right)$$

where Pl(μ) is the Legendre polynomial of order l, s = \(\sqrt {r_{\mathrm{\sigma }}^2 + r_{\mathrm{\pi }}^2}\) and μ = rπ/s. In Supplementary Fig. 3, we plot the monopole ξ0, quadrupole ξ2 and hexadecapole ξ4 of ξ(rσ, rπ) measured from different galaxy samples with varying mean densities (different colours) and based on different stellar-mass models (different line styles). For samples with very low mean densities, such as ng = 5 × 10−4 (Mpc h−1)−3 and ng = 1 × 10−3 (Mpc h−1)−3 (cyan and magenta groups of lines, respectively), the systematic errors in stellar-mass estimates do affect the resulting galaxy clustering. This is precisely the effect due to a significant fraction of objects in the sample, with stellar mass close to the mass selection threshold, that can move in or out of the sample depending on the method used to estimate their masses. However, this effect is minimized for much denser samples, corresponding to ng = 5 × 10−3 (Mpc h−1)−3 and ng = 1 × 10−2 (Mpc h−1)−3 (yellow and olive, respectively, in the figure), with the clustering properties remaining substantially unchanged with respect to the mass estimator used.

In addition, in SHAM (see next section), with a higher mean density, we move further down the galaxy stellar-mass function, thus increasing the fraction of satellite galaxies, that is, the population dominating nonlinear RSD effects. This population of high-speed satellites is also expected to be free of the so-called velocity bias, which is a potential issue for central galaxies43. For this reason, for our reference sample, we choose ng = 1 × 10−2 (Mpc h−1)−3, which is dense and has a large enough volume to allow a robust clustering measurement.

SHAM

The SHAM method is based on the simple assumption that there is a monotonic relationship between a property of dark matter subhaloes and a property of the corresponding galaxies. Usually, these are identified, respectively, with the subhalo circular velocity (as a proxy for the self-gravity of the halo) and the galaxy total stellar mass. In its original form44, the actual maximum circular velocity of subhaloes measured in the simulation (vmax) was used. In this case, however, it has been shown that the corresponding SHAM catalogue of galaxies underpredicts the observed clustering on small scales44. The reason is that, in contrast to its more tightly bound stellar component, the gas component and dark matter halo of a galaxy can easily be disrupted by the tidal field of a nearby or parent massive halo. A better proxy is shown to be provided by the subhalo’s maximum circular velocity at the epoch of accretion (vacc), before this disruption happens45. This allows us to recover, for example, galaxies associated with subhaloes in the central region of a host halo. Even better results are obtained if vpeak is used46. The reason for this is that, at the epoch of vpeak, the subhalo has the strongest binding force and, hence, is most stable against tidal stripping. Its properties are thus expected to be more tightly correlated with the galaxy stellar mass at this epoch. However, as shown in ref. 47, during the lifetime of a subhalo, vpeak could sometimes show spikes, which might not reflect its typical status during most of its existence. This analysis, based on the state-of-the-art hydrodynamical simulation EAGLE21, shows that the strongest correlation with galaxy stellar mass is obtained by using the highest value of the circular velocity satisfying a relaxation criterion (vrelax). At the same time, however, vrelax is shown to be only marginally better than vpeak in reproducing the simulated galaxy clustering. Furthermore, baryonic effects are shown to introduce only a small perturbation in a vpeak-ranked halo catalogue and have a limited impact on the positions of dark matter haloes. This results in an overall accuracy in the predicted redshift-space galaxy clustering that is better than 10% above 1 Mpc h−1 when using a vpeak-ranked halo catalogue in building the SHAM47.

Numerical stability of SHAM predictions

The SHAM method is based on subhaloes and their merger histories, which are in turn derived from high-resolution N-body simulations. It is, therefore, important to test the robustness of SHAM predictions against different N-body codes, halo finders and methods of constructing halo merger trees. To do this, we perform a test simulation with 1,0243 dark matter particles in a box of 150 Mpc h−1 length per side. In our test simulation, rather than using the GADGET code48 as adopted in the SMDPL simulation, we instead use the RAMSES code49. RAMSES uses a multigrid relaxation method for solving the Poisson equation, which is different from the Tree-Particle-Mesh method used in GADGET. Moreover, we use the AMIGA (Adaptive Mesh Investigations of Galaxy Assembly) Halo Finder50 to identify haloes and construct the halo merger tree using the MERGERTREE code, which is part of the AMIGA Halo Finder package. Supplementary Fig. 4 compares the multipoles of the two-point correlation function of our test simulation (dashed lines) with those obtained from the SMDPL simulation (solid lines). In both cases, the multipoles are estimated using the distant observer approximation (see ‘Survey geometry and wide-angle effects’). Despite the significant differences in the numerical methods used, the two simulations yield very similar predictions for the clustering of derived SHAM galaxies when a large enough number density, ng = 1 × 10−2 (Mpc h−1)−3, is adopted, as in the main paper.

SHAM predictions in f(R) gravity

Unlike in the ΛCDM case, the circular velocity of a baryonic particle in a subhalo in f(R) gravity is not directly related to the true CDM mass of the subhalo but to an effective mass that is defined through a modified version of the Poisson equation30

$$\nabla ^2{\varphi } = 4\pi Ga^2\delta \rho _{{\mathrm{eff}}}$$

By definition, ρeff incorporates all the effects of modified gravity. The circular velocity is then given by \(v_{{\mathrm{cir}}}^2\left( x \right)\) = \({\textstyle{{GM_{{\mathrm{eff}}}\left( { < x} \right)} \over x}}\), where Meff(<x) is the effective mass enclosed in radius x for a dark matter halo. In practice, it is more convenient to calculate the circular velocity for each dark matter halo in an f(R) simulation using the effective halo catalogue technique described in refs 30,51.

The f(R) simulation used in our analyses is the one described in ref. 29. The simulation has 5123 dark matter particles in a box of 64 Mpc h−1 length per side; this allows us to reach a mass resolution of 1.52 × 108Mh−1. However, due to its limited box size, the predicted galaxy clustering cannot be directly compared with observations, since missing long-wavelength modes on scales larger than the box size also have an effect on small-scale clustering. To overcome this, we ran a further ΛCDM simulation with the same box size and—most importantly—the same initial conditions as the f(R) simulation. This allows a comparison of the small-scale behaviour of the two models on equal footing: the missing long-wavelength Fourier components on scales larger than the box size will be, by construction, the same for the two simulations. As such, we expect the ratio of the two-point correlation function multipoles (monopole ξ0, quadrupole ξ2 and hexadecapole ξ4) of the SHAM galaxy catalogues built from these two simulations to be virtually independent of the box size and reflect only the differences between the intrinsic gravity models. Thus, in practice, the full f(R) predictions to be compared with the full SMDPL (Fig. 2) are obtained as

$$\left( {\xi _{0,2,4}^{f\left( R \right)}} \right)_{{\mathrm{True}}} = \left( {\frac{{\xi _{0,2,4}^{f\left( R \right)}}}{{\xi _{0,2,4}^{{\mit{\Lambda}} {\mathrm{CDM}}}}}} \right)_{64{\kern 1pt} {\mathrm{Mpc}}\,h^{-1}}\xi _{0,2,4}^{{\mathrm{SMDPL}}} = \left( {\xi _{0,2,4}^{f\left( R \right)}} \right)_{64\,{\mathrm{Mpc}}\,h^{-1}}\frac{{\xi _{0,2,4}^{{\mathrm{SMDPL}}}}}{{\left( {\xi _{0,2,4}^{{\mit{\Lambda}} {\mathrm{CDM}}}} \right)_{64\,{\mathrm{Mpc}}\,h^{-1}}}}$$

where \(\xi _{0,2,4}^{f\left( R \right)}\) and \(\xi _{0,2,4}^{{\mit{\Lambda}} {\mathrm{CDM}}}\) are obtained from the 64 Mpc h−1 box simulations as shown in Supplementary Fig. 5.

From this figure, we can see that the box size affects mainly the monopole ξ0, while the quadrupole ξ2 and hexadecapole ξ4 are essentially preserved. This is very important, beyond any correction we may apply, as the last two are the quantities that specifically measure the deviation from an isotropic distribution produced in redshift space by the peculiar velocities of galaxies (if there were no peculiar velocities, there would be no RSD effects and only the monopole would be non-zero). As such, higher-order multipoles are, in the first place, less sensitive to the underlying real-space positions of galaxies. This is particularly true on small scales, where the RSD effects are dominated by random motions of high-speed galaxies, in contrast to the coherent motion of galaxies on large scales. They thus contain most of the velocity field information and are the most robust quantities to test gravity. In view of this, we further remark that in this kind of comparison, the leading consideration for an f(R) simulation is its mass resolution, rather than the box size: the strength of gravity and the velocity field are substantially affected by the f(R) screening mechanism, which can be accurately explored only if the mass resolution is sufficiently high, as in our simulation.

The screening mechanism plays an important role in SHAM predictions for f(R) gravity. Very massive main (distinct) haloes in f(R) gravity are usually screened. Subhaloes in these main haloes would feel the same strength of gravity as in standard gravity. However, less massive main (distinct) haloes are usually unscreened, so that subhaloes with similar (or even slightly smaller) true dark matter mass can have higher circular velocities due to enhanced gravity. As in SHAM, we select subhaloes using circular velocity (which is related to the effective density field and is tightly correlated to a galaxy’s stellar mass in the processes of galaxy formation), so that the subhaloes in the less massive unscreened main (distinct) haloes will be selected. For a fixed number density of haloes, this leads to the relatively weaker clustering of SHAM predictions in f(R) gravity, which is in contrast to the ΛCDM case (see Fig. 2 and ref. 51).

Survey geometry and wide-angle effects

Given the significantly different box sizes between the SMDPL and our f(R) simulations, implementing redshift-space effects on their corresponding SHAM catalogues of artificial galaxies requires two different approaches. Specifically, in the case of the SMDPL, given its large volume, we can build realistic SHAM mock surveys that fully reproduce the geometry and angular mask of the real SDSS data. However, for the f(R) simulation, due to its limited box size, a more idealized approximation necessarily has to be used. In this section, we test the robustness of the approach we used to implement redshift-space effects in the f(R) simulation and clearly identify the range of scales where the f(R) measurements can be compared on an equal footing with the ΛCDM simulation predictions and the real data.

Our ‘standard cosmology’ ΛCDM simulated galaxy sample is built from the very large box of the SMDPL simulation11. In this case, we can construct a full-sky SHAM mock SDSS survey that reproduces the SDSS data selection function without introducing simplifying assumptions. We collate eight replicas of the box and place the observer at the centre of this super-box. Redshift distortion effects are then obtained for each SHAM galaxy by projecting its velocity along the line of sight to the observer. Note that the large box size of the SMDPL simulation would by itself be sufficient for our purposes, given that we are interested in only small scales (<20 Mpc h−1). However, by combining eight replicas, we can more easily accommodate the irregular geometry of the real SDSS data. This is an important check as our SDSS volume-limited samples are, in general, fairly shallow and characterized by an irregular geometry. Their median redshift is only around z ≈ 0.1, that is, ~250 Mpc h−1 in a standard ΛCDM cosmology with Ωm = 0.3, with a large fraction of the galaxies closer than this distance.

As anticipated, such an approach is not possible for an f(R) cosmology. Compared with ΛCDM ones, f(R) simulations are expensive, typically requiring 20 times more CPU time than a ΛCDM simulation with the same box size and mass resolution. This severely limits the maximum box achievable for an f(R) simulation with sufficiently high mass resolution, given the current state of the art in supercomputers. As such, it is currently not possible to build an f(R) SHAM mock sample that is large enough to accommodate the volume of a survey such as the SDSS we are using here. Given this situation, to analyse RSD effects in an f(R) cosmology, we adopt the commonly used approach known as the distant observer approximation. This is the assumption originally adopted to derive the classic linear model of RSD5, which assumes that the box is at such a large distance compared with its size that both sides of the box along one direction can be considered to be parallel to the line of sight. This approach was used to produce the results of Supplementary Figs. 4 and 5, as explicitly indicated by the label. These figures are fully self-consistent as we are comparing quantities from simulated samples treated in the same way (for example, placed at the same large distance). We test the robustness of this approximation against the geometry of our specific SDSS samples by applying both approaches to the SMDPL simulation. The comparison of the two outcomes is shown in Supplementary Fig. 6. Again, there are some noticeable effects on the monopole ξ0; however, over the range ~2–10 Mpc h−1, both the quadrupole and the hexadecapole (the two most important statistics in our gravity test) show little difference between the two ways of implementing the RSD effects, indicating that our comparisons of RSD effects in the f(R) and ΛCDM SHAM catalogues with the corresponding SDSS data are robust.

Fibre collisions mitigation

In the SDSS twin multi-object fibre spectrographs on the 2.5 m aperture Sloan Telescope at Apache Point Observatory, two fibres cannot be placed closer than 55″ on the spectroscopic plate in the same observation, due to the physical size of the fibre plugger52. Thus, any two galaxies separated by 55″ or less cannot be observed simultaneously. In the SDSS observing strategy, this effect is alleviated by increasing the overlaps of adjacent tiles such that close pairs can be targeted from different observations in the overlaps. However, there are still about 7% of the galaxies in close angular pairs that do not have a measured redshift in the survey. These missing fibre-collided pairs of galaxies introduce a systematic effect on two-point statistics, which cannot be simply accounted for through a homogeneous weighting, since the missing objects are not randomly distributed. One way to correct for this is to use an angular weight based on the ratio, as a function of separation, of the numbers of observed pairs to the total number of targets in the original survey parent catalogue53. This method works for a flux-limited sample, but cannot be directly generalized to subsamples such as volume-limited ones, at least without any assumptions43. A better way, which we adopt here, is—rather than using the conventional multipole expansion—to adopt the so-called truncated multipoles as proposed in refs 43,54. These are defined as

$$\xi _l\left( s \right) = \frac{{2l+1}}{2}\mathop {\smallint }\limits_{ - 1}^1 {\kern 1pt} {\mathrm{d}}\mu W\left( {s,\mu } \right)\xi \left( {s,\mu } \right)P_l\left(\mu \right)$$

where W(s, μ) is a mask used to exclude the unreliable small-scale measurements from the integration

$${\mathrm{W}}( {{\mathrm{s}},{\mathrm{\mu }}}) = \left\{ {\begin{array}{c} {1,{\mathrm{r}}_{\mathrm{\sigma }} \ge 0.2\,{\mathrm{Mpc}}\,h^{-1}}\\{0,{\mathrm{r}}_{\mathrm{\sigma }} < 0.2\,{\mathrm{Mpc}}\,h^{-1}} \end{array}} \right.$$

The choice of the truncation scale of 0.2 Mpc h−1 is very conservative, since at the maximum redshift of our galaxy data (z ≈ 0.1 for the ng = 1 × 10−2 (Mpc h−1)−3 sample), the fibre angular scale of 55″ corresponds to rσ ≈ 0.08 Mpc h−1. Note also that W(s, μ) is used to obtain the multipoles of ξ(s, μ) only. The two-dimensional correlation function ξ(s, μ), in the first place, is calculated in the usual way, using galaxy pairs from all scales. This is important because the missing galaxies in close pairs do impact clustering on large scales as well. However, the missing power on large scales can be corrected for by properly down-weighting randoms, as implemented in our estimate of ξ(s, μ).

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

J.H. is supported by a Durham co-fund Junior Research Fellowship; J.H. and B.L. acknowledge support by the European Research Council (ERC-StG-716532-PUNCA); L.G. acknowledges support by the European Research Council (ERC-AdG-291521-Darklight) and by the Italian Space Agency (ASI Grant I/023/12/0); B.L. and C.M.B. are also supported by UK STFC Consolidated Grants ST/P000541/1 and ST/L00075X/1.

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All authors contributed to the development and writing of this paper. J.H. led the data analysis. B.L. conducted and provided the simulations. L.G. led the writing of the paper. C.M.B. and J.H. conceived the idea of data analysis.

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Correspondence to Jian-hua He.

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He, Jh., Guzzo, L., Li, B. et al. No evidence for modifications of gravity from galaxy motions on cosmological scales. Nat Astron 2, 967–972 (2018). https://doi.org/10.1038/s41550-018-0573-2

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