Galaxy-galaxy lensing data: f ( T ) gravity challenges General Relativity

,


INTRODUCTION
The concordance model of cosmology, namely ΛCDM paradigm, incorporates General Relativity as the underlying gravitational theory, the standard model of particles, cold dark matter (CDM) and the cosmological constant Λ, and it has been well verified by various observational datasets, such as Cosmic Microwave Background (CMB) (Ade et al. 2016;Aghanim et al. 2020), Baryon Acoustic Oscillations (BAO) (Eisenstein et al. 2005;Alam et al. 2017), Type Ia supernovae (Riess et al. 2004;Astier et al. 2006), galaxy formation and evolution theory (Davis et al. 1985), as well as weak lensing observation (Heymans et al. 2012;Shi et al. 2018).However, recent observations have revealed possible tensions (Bullock & Boylan-Kolchin 2017;Perivolaropoulos & Skara 2022), such as the Hubble tension be-tween early-time measurements under ΛCDM and latetime local distance-ladder measurements (Wong et al. 2020;Abdalla et al. 2022), and the σ 8 tension between the relative clustering level found in CMB experiments and the late time large-scale structure observations (Di Valentino et al. 2021;Yan et al. 2020).While, the non-renormalizability of General Relativity and the difficulties in bringing it closer to a quantum description is a potential disadvantage for the theory (Addazi et al. 2022).Hence, a significant amount of research has been devoted in the construction of various gravitational modifications, aiming to alleviate or solve the above issues (Capozziello & De Laurentis 2011;Akrami et al. 2021).
Every theory of gravity, including General Relativity, ought to pass various tests, using a variety of observational data (Berti et al. 2015), from the expansion of the universe to the formation of large-scale structures.Tests of gravity on small scales usually study the consistency of its cosmological feasibility alongside Solar System experiments, and in order to achieve this it is necessary to first quantify possible deviations from General Relativity and then use the data to constrain them (Will 2014;Chan & Lee 2022;Chiba et al. 2007).As it is known, at the Solar System level, General Relativity is always inside the obtained parametric contours for the various modified gravity parameters (for the case of f (T ) gravity see (Iorio & Saridakis 2012;Bahamonde et al. 2020)).
On the other hand, with the development of largescale galactic surveys, weak gravitational lensing has become increasingly important in delineating matter distribution, leading it to become a powerful tool in constraining modified gravity (Bacon et al. 2000;Luo et al. 2018;Cai et al. 2023).In (Chen et al. 2020) the authors performed for the first time a novel test on possible deviations form General Relativity using galaxy-galaxy weak gravitational lensing.They used f (T ) framework to quantify these deviations and then used the deflection angle at non-cosmological scales (Ruggiero 2016) to approximately calculate the lensing potential and the effective surface mass density, and thus extract the upper bound on the deviation parameter with the weak lensing data from SDSS DR7.Additionally, the perturbative spherically symmetric solution of the covariant formula of f (T ) theory was extracted within T + αT 2 deviation from GR in (Ren et al. 2021b), where the deflection angle and the difference in position and magnification in the lensing frame were calculated.
In this work we desire to employ galaxy-galaxy weak gravitational lensing in order to extract more accurate constraints on possible deviations from General Relativity.Using f (T ) gravitational theories in order to quantify the deviation, interestingly enough we find that the quadratic correction on top of GR is favored.The plan of the manuscript is the following: In Section 2 we briefly review f (T ) gravity and we present the spherically symmetric solutions.Then, in Section 3 we calculate the corresponding gravitational potential and the weak lensing shear signal, and we derive a correction term related to the negative quadratic radius.In Section 4 we introduce the group catalog (Yang et al. 2008) and the shear catalog from the SDSS DR7 (Abazajian et al. 2009) in order to extract observational constraints.Hence, in Section 5 we fit the ESD and we provide the estimation results for the model parameters, alongside the application of AIC and BIC information criteria.Finally, Section 6 is dedicated to conclusions and outlook.

SPECIFICALLY SYMMETRIC SOLUTIONS IN F (T ) RAVITY
In this section we use f (T ) gravity in order to quantify possible deviations form General Relativity, and we extract the corresponding corrections on specifically symmetric solutions.
In the framework of teleparallel gravity one uses the tetrad field h A µ , related to the metric through g µν = h A µ h D ν η AD , where η AD = diag(1, −1, −1, −1).Concerning the connection, one uses the teleparallel one, namely (Cai et al. 2016) where ω A Dµ represents a flat metric-compatible spin connection, and therefore the torsion tensor is Furthermore, the torsion scalar is defined as where is the super-potential, with the contortion tensor being where h = det(h A µ ) is the tetrad determinant and G the gravitational constant.Finally, performing variation in terms of the tetrad, gives rise to the field equations of f (T ) gravity as T ρ µ . (5) In the following we will work in the Weitzenböck gauge, in which the spin connection vanishes (Cai et al. 2016).Solving the anti-symmetric part of the field equations with vanishing spin connection one can obtain the general spherically symmetric tetrad.There are two different branches of tetrad solution ansätze in this approach.The first branch, corresponding to real tetrad, is a familiar case and has been well studied (Tamanini & Boehmer 2012;DeBenedictis & Ilijic 2016;Bahamonde et al. 2019;Ruggiero & Radicella 2015;Ren et al. 2021b).The second branch is a complex tetrad, namely (Bahamonde et al. 2022) and the corresponding metric is where χ = ±1.We mention that although the tetrad is complex, all physical quantities, such as the metric, the torsion scalar and the boundary term are real and independent of the sign of χ.
One can see that the general field equations accept the exact solution for the metric function A(r): where P = Q 2 T 2 − 2QT + 4, in which case the torsion scalar is T = (4r 2 − 6Q)/(r 4 − 2Qr 2 ).Note that for Q ≪ 1 this model can be expanded as Q is the single parameter that quantifies deviations from General Relativity, and for Q = 0 the latter is recovered.
In summary, the physically meaningful metric solution of model ( 10) in complex tetrad ( 6) is written as (Bahamonde et al. 2022) which is an exact spherically symmetric solution in f (T ) gravity.We stress that Q is a parameter of the theory, completely independent of the electromagnetic charge, and the Schwarzschild solution is recovered for Q = 0.

WEAK LENSING
In this section we present the weak lensing machinery.The propagation of a photon in the universe is determined by the spacetime properties, and the local matter inhomogeneity leads to a deflection of its trajectory, resulting to the lensing effect (Bartelmann & Schneider 2001).Hence, the spacetime properties, and thus the underlying theory of gravity, leave imprints in the lensing signal contains, which can then be used as a test of the theory of gravity itself.
We consider that light from a distant source is affected by the gravitational field in the foreground during its propagation.Since the effective speed of light is reduced in a gravitational field, it will delay relatively to vacuum propagation.At the same time, photons are also deflected when they pass through a gravitational field.Thus, the difference between f (T ) gravity and General Relativity can be reflected in the deflection angle α in static spherically symmetric spacetime.
We assume that the whole lensing system lies in the asymptotically flat spacetime regime.The distance of the light ray of closest approach r 0 , as well as the impact parameter b, both lie outside the gravitational radius, and the deflection angle for metric (7) is given as (Keeton & Petters 2005) where the impact parameter b = is given by the vertical distance of the asymptotic tangent of the ray trajectory from the center of the lens to the observer with respect to the inertial observer.Imposing the identification m = GM c 2 , the bending angle can then be expressed as a series expansion in the single quantity m b as (Keeton & Petters 2005) For the spherically symmetric metric solution (11) the deflection angle can be calculated under the parametrized-post-Newtonian (PPN) formalism.Following (Keeton & Petters 2005), we calculate the coefficients of the first three orders as The effective lensing potential is defined as (Narayan & Bartelmann 1996): where D d , D ds , and D s denote the angular diameter distances between observer and lens, lens and source, and observer and source, respectively.The gravitational potential of the lens can be derived from the relationship between deflection angle and effective lensing potential under the second-order approximation like: and thus we acquire the three-dimensional gravitational potential: Here the first term is the Newtonian potential, the second one is the contribution from General Relativity, and the third term is the modification from f (T ) gravity.Note that the second term was neglected in (Chen et al. 2020), hence its incorporation in the present work will lead to significantly more accurate results.
Then, the shear tensor can be calculated by the linear combination of the second partial derivatives of the potential, namely where ∆Σ(R) is the ESD under Newtonian approximation.Hence, in order to distinguish we define an effective ESD as: with Σ crit = c 2 4πG Ds D ds D d the critical surface mass density.As we can see, the shear components γ 1 and γ 2 introduce anisotropy (or astigmatism) into the lens mapping, and the quantity γ = (γ 2 1 + γ 2 2 ) 1/2 describes the magnitude of the shear.Since our metric is chosen from the black hole solution, the result is the point source approximation (see (Turyshev & Toth 2023) for cases under the extended source in Solar gravitational lens circumstance).
Since we are dealing with possible deviations form General Relativity, our predictions should be confronted with observational data form galaxies, large-scale structure, and dark matter halo density profiles.The matter distribution of the lens can be described by the modification of the NFW density profile (Mandelbaum et al. 2005).In particular, the NFW profile consists of two parameters, the characteristic mass scale M 200 and the halo concentration c h , which is defined as the ratio between the virial radius of a halo and its characteristic scale radius.
In the following section we will use the ESD calculated under the NFW model in order to describe the Newtonian term, and we will adopt the oncentrationmass relation to reduce the degrees of freedom.

DATA CONFRONTATION
We have now all the machinery required for a confrontation with observational data.In order to test the excess surface density caused by f (T ) gravity, we use a weak lensing catalog provided by SDSS DR7.Concerning the lens samples, we use the ones from the spectroscopic group catalog (Yang et al. 2008) based on SDSS DR7, which applies a halo-based group finding algorithm (Yang et al. 2006).The algorithm first assumes that each galaxy is a potential candidate group of galaxies, and then it calculates the total luminosity of each galaxy.The various galaxies were identified by selecting information such as distance and redshift that meet the criteria, and the sample was selected from individual galaxy systems to attenuate the bias caused by other surrounding structures, finally resulting to a sample size of 400,608.
The galaxy's stellar mass is calculated by the stellar mass-to-light ratio from (Bell et al. 2003).Specifically, we use the data given by (Luo et al. 2021) which are divided into five stellar mass bins.In order to test the influence produced by f (T ) gravity on galaxy formation time, we further subdivide our sample into blue galaxies and red galaxies, based on a cut of a color-magnitude plane from (Yang et al. 2008), given by: 0.1 (g − r) = 1.022 − 0.0652x − 0.0031x 2 . (21) In this expression g and r are the different band magnitudes, x = 0.1 M r − 5logh + 23.0, and 0.1 M r − 5logh is the absolute magnitude of the galaxy after K correction, while the superscript describes evolution correction at redshift z = 0.1 (Blanton et al. 2005).We use the shape catalog created by (Luo et al. 2017), based on SDSS DR7 imaging data for the source.SDSS imaging data of DR7 contains about 230 million different luminosity objects, covers about 8,423 square degrees of LEGACY sky, including the u, g, r, i, and z five-band photometry.Thus, in (Luo et al. 2017) the authors built an image processing pipeline to correct systematic errors introduced mainly by the point spread function (PSF).This pipeline processes the SDSS DR7 r band imaging data, which can be used to generate the background galaxy catalog containing the shape information of each galaxy.The final shape catalog that we use contains the position, shape, shape error, and photometric redshift information of about 40 million galaxies, based on (Csabai et al. 2007).
The ESD ∆Σ(R) that we use as the shear implies that it is measured by the weighted mean of source galaxy shapes, namely where w is the weight for each source galaxy, calculated by the shape noise σ shape and the noise from sky σ sky w = 1 (σ 2 shape + σ 2 sky ) . (23)

RESULTS
In order to detect the distribution mechanism of excess surface density calculated under modified gravity, we use the signal of the weak gravitational lensing to constrain the model parameter Q.Firstly, we consider a modified NFW model with General Relativity and f (T ) corrections.Secondly, since the lens sample is selected from an individual galaxy system, and the contribution of the modified model parameters only played a significant role on small scales, we perform the analysis without taking into account the contribution of the two halo terms.The distribution of galaxies determines the location of the center of the lens gravitational potential, and the selected central galaxy is not necessarily at the center of the gravitational potential, which is called the off-center effect.The off-center effect can influence the estimation of the mass distribution because it adds significant system uncertainty.Therefore, we consider a slightly off-center effect of about R sig = 0.01h −1 M pc for a conservative estimation (Luo et al. 2018).
We proceed by using ten catalogs of the five mass intervals of the red and blue galaxies in order to test ESD calculated under f (T ) gravity, and then we obtain the best-fit results of the model by using the Chi-square test.The χ 2 is calculated as where C −1 is the inverse covariance matrix.
We mention that the measurements of the small stellar mass bin have very high signal-to-noise ratios.Moreover, the correction term of f (T ) gravity contributes significantly to the small mass end, hence it plays an important role in reducing χ 2 .Furthermore, we take into account the dependence of the weak lensing data and the f (T ) correction on the color characteristics of the galaxies in the fitting process.As can be seen from Fig. 1, since there are fewer blue galaxies in the large mass bin, the ESD profile carries less information.This results in a decrease in the limit level of the model parameters in the last two bins of the blue galaxy, while the red galaxy has a better overall signal quality.
Since the signal-to-noise ratio of red galaxies is significantly better than that of blue galaxies, we mix them up to extract information from five different stellar mass bins to constrain the model parameters.Finally, as a very conservative estimation, we adopt the concentration-mass relationship proposed by (Neto et al. 2007) as a Gaussian before eliminating degeneracies with other parameters, and we use the MCMC method to obtain the final parameter space of Q.
In summary, after performing the above steps, we finally obtain the estimation for the Q interval within 1σ confidence level as Additionally, in Fig. 2 we present the two-dimensional posterior of the fitting, as well as the constraints for various quantities.
As we observe, the f (T )-gravity parameter Q that quantifies the deviation from General Relativity, and interestingly enough the value zero is excluded at 1σ confidence level.This suggests that f (T ) corrections on top of General Relativity are favoured, at least at the galaxy scales, where galaxy-galaxy lensing data are sensitive to the gravitational potential.This is one of the main result of the this work.In order to compare the fitting quality and the model performance with that of standard ΛCDM concordance scenario, we apply the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), (Liddle 2007).The AIC criterion provides an estimator of the Kullback-Leibler information, it exhibits the property of asymptotic unbiasedness, and is defined as (Akaike 1974): where L max represents the maximum likelihood of the model, and p tot represents the total number of free parameters.Additionally, the BIC criterion provides an estimator of the Bayesian evidence, and is defined as (Kass & Raftery 1995) where N tot is the number of samples, while the other parameters are the same as in AIC.According to Jeffreys classification (Kass & Raftery 1995), if ∆IC ≤ 2 then the scenario is statistically compatible with the best-fit model, if 2 < ∆IC < 6 then we have a moderate tension between the two scenarios, while for ∆IC ≥ 10 we obtain significant tension.Hence, we apply these criteria to draw a comparison between the NFW model in ΛCDM and the modified NFW model in f (T ) gravity.
We examine the applicability of the models in the following cases: no off-center effect, off-center distance dispersion given by R sig = 0.01, and by R sig = 0.05.The results are presented in Table 1.The ΛCDM scenario in the first case has AIC/BIC values of 54.411 and 61.417, respectively.When R sig = 0.01 the values for AIC/BIC are 60.918 and 67.924, while for R sig = 0.05 ditionally incorporate the off-center effects.This is one of the main result of the this work.

CONCLUSIONS
In this work we used weak gravitational lensing data to test General Relativity and f (T ) gravity at galaxies scales.We considered an exact spherically symmetric solution of f (T ) theory under the complex tetrad.which is obtained from an approximate quadratic correction to teleparallel equivalent of General Relativity, and thus it is expected to hold for every realistic deviation from General Relativity.Firstly, following the post-Newtonian approximation, we calculated the deflection angle and the shear signal of the weak lensing under f (T ) gravity.In particular, quantifying the deviation from General Relativity by a single parameter Q, we obtained the corresponding deviation in the gravitational potential, in the shear component and in the effective ESD profile, which is mainly affected at small scales.
We divided each stellar mass sample into blue and red to test the model's dependence on galaxy color.We modeled the ESD profiles using the NFW profiles, and we found that except that the ESD profiles differ significantly in red and blue galaxies, the modified gravitational model does not depend significantly on the color of the galaxies.In addition, based on the group catalog of SDSS DR7, we used the weak lens data to give the tight constraints on the parameter Q.In the end, we extracted the estimation for the deviation parameter from General Relativity with the latest measurement as Q = −2.138+0.952  −0.516 × 10 −5 M pc −2 at 1σ confidence.Such a constraint suggest that the deviation parameter is constrained to negative values, and the value zero is excluded at 1σ confidence level.Therefore, f (T ) corrections on top of General Relativity are favoured, at least at galaxy scales.
In order to compare the fitting accuracy of f (T ) gravity with that of ΛCDM cosmology, and examine the overall efficiency of the model, we applied information criteria, and we calculated the AIC and BIC values in three different cases.Although the dependence of f (T ) gravity on the off-center effect implies that its optimality needs to be carefully studied, our analysis showed that f (T ) gravity is more consistent with observational data when the off-center effect is considered.In summary, the application of information criteria verifies our aforementioned result, that f (T ) gravity is more efficient than General Relativity in fitting the weak lensing data, and thus it offers a challenge to the latter.
We would like to comment here that since our results are extracted under the black hole metric solution with the point source approximation, one could study the influence of the extended source.Additionally, since our analysis applies at galactic scales, and on large scales current observations of filament lensing suggest distributions of matter that are hard to be explained under the General Relativity framework, one could try to construct f (T ) models that could describe them more efficiently.Finally, one could try to perform the same analysis, quantifying the deviations from General Relativity using other frameworks, such as f (R) and f (Q) gravity.These subjects will be investigated in future works.
We close this manuscript by mentioning that the scaledependent matter distribution and galaxy clustering is a powerful probe of cosmological models and gravitational theories.For instance, the Bullet Cluster (Wik et al. 2014), as the first clear example of a merging galaxy cluster, provides strong evidence for the existence of dark matter by featuring at cluster scales (Brownstein & Moffat 2007).The data samples used in this work mainly reflect the features of galaxy scales, which cannot effectively reflect the cluster scales.Nevertheless, the investigation of galaxy mergers in clusters themselves is both necessary and interesting and could shed light on the galaxy evolution and dynamics within the corresponding gravity theories, and will be performed in a future project.

Figure 1 .
Figure1.Best fits to the ESD profile.The horizontal axis is the projected distance away from the lens galaxy.The blue and orange points with error bars represent the weak lensing data for blue and red galaxies, while the purple and orange lines are the fit curves.Here we choose Rsig = 0.01.