Observational test for f ( Q ) gravity with weak gravitational lensing

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INTRODUCTION
The Λ-Cold Dark Matter (ΛCDM) paradigm, based on GR, on the Standard Model of Particle Physics, on cold dark matter and on the cosmological constant, has received support from a series of experimental observations and stands as the current Standard Model of Cosmology.However, with the advancement of observational techniques, the ΛCDM scenario seems to be challenged by new observations, which may reveal possible tensions between its predictions and the data (Perivolaropoulos & Skara 2022;Abdalla et al. 2022;Wong et al. 2020;Di Valentino et al. 2021;Yan et al. 2020).These observational inconveniences, alongside the known theoretical problem of non-renormalizability of GR, led a significant amount of research to be devoted to the constructions of modified gravity (Akrami et al. 2021).
Although f (Q) gravity has been confronted in detail with data at cosmological scales (Anagnostopoulos et al. 2021;Lazkoz et al. 2019;Ayuso et al. 2021;Anagnostopoulos et al. 2023;Yang et al. 2024b,a;Capozziello & D'Agostino 2022), up to now it has not been tested using data from intermediate scales.In particular, it is known that in the framework of GR, light is bent in the presence of a gravitational field, distorting the image of a celestial object located behind intervening matter, a phenomenon known as gravitational lensing (Bacon et al. 2000;Hoekstra & Jain 2008;Luo et al. 2018;Cai et al. 2023;Jiang et al. 2024;Mo et al. 2024;Soares et al. 2023a,b).This phenomenon can be utilized to investigate the characteristics of dark matter and dark energy on the galactic and cosmological scale, which applications include cosmic shear (Asgari et al. 2021), galaxy-galaxy lensing (Alam et al. 2017) and CMB lensing (Lewis & Challinor 2006).Therefore, by examining the distortion of images from a multitude of background galaxies, weak lensing can directly map the gravitational field of the foreground objects (Salucci 2019), thus offering valuable insights into constraining modified gravity theories.
In (Ren et al. 2021b) the authors calculated the lensing effects, such as deflection angles and magnifications, under covariant f (T ) gravity.Moreover, in (Luo et al. 2021) emergent gravity was investigated through galaxygalaxy lensing where it was found that it is not capable to explain color dependence.Furthermore, (Chen et al. 2020) and (Wang et al. 2023) utilized weak lensing signals to constrain f (T ) gravity by modeling the effective ESD in real and complex tetrads, respectively.
In the present work we are interested in performing a confrontation of f (Q) gravity with galaxy-galaxy lensing data, in order to examine whether the theory can past the corresponding tests.The plan of the work is the following.In Section 2 we present f (Q) gravity and we extract the spherically symmetric solutions.In Section 3 we briefly review the weak lensing mechanism and we apply it for the case of f (Q) gravity, while we present the data that we use.Then, in Section 4 we perform the observational confrontation, extracting the corresponding constraints on the theory.Finally, in Section 5 we conclude.

SPHERICALLY SYMMETRIC SOLUTIONS IN F (Q) GRAVITY
In this section we introduce f (Q) gravity and we explore its application on spherically symmetric solutions.

f (Q) gravity
Let us briefly review the fundamentals and structure of gravity theories based on non-metricity.The general affine connection Γ α µν is expressible as where Γα µν denotes the Levi-Civita connection, represents the contortion tensor with T α µν being the torsion tensor, and is the disformation tensor, which arises from the nonmetricity tensor defined as where g µν is the metric tensor, and Greek indices cover the range of the coordinate space.
Employing the general affine connection yields the torsion and curvature tensors as and the non-metricity tensor as Setting non-metricity to zero recovers Riemann-Cartan geometry, with torsion nullified we obtain a torsion-free geometry, and with curvature nullified we acquire teleparallel geometry.On the other hand, making both non-metricity and torsion equal to zero (in this case the general connection is the Levi-Civita one) results in Riemann geometry, nullifying both nonmetricity and curvature (using Weitzenböck connection) leads to Weitzenböck geometry, and setting both curvature and torsion to zero (employing symmetric teleparallel connection) yields symmetric teleparallel geometry (Beltrán Jiménez et al. 2018;Järv et al. 2018).
Gravity in Riemann geometry is described by a Lagrangian based on the Ricci scalar derived from curvature tensor contractions, leading to GR.Moreover, in Weitzenböck geometry, gravity is described by a Lagrangian based on the torsion scalar derived from torsion tensor contractions, forming the TEGR.Similarly, therefore, in symmetric teleparallel geometry, gravity is described via a Lagrangian formed by contractions of the non-metricity tensor, specifically the non-metricity scalar Drawing inspiration from the modifications in f (R) and f (T ) gravity, the Lagrangian for STEGR, namely Q, can be generalized to a function of choice, leading to the formulation of f (Q) gravity.The action for this theory is given by (Beltrán Jiménez et al. 2018) where g is the determinant of the metric and κ is the gravitational constant.In this framework, the classical form of STEGR is recovered when the function f (Q) is set to Q. Variation of the total action S + S m , where S m is the action of the matter sector, gives rise to the field equations of f (Q) gravity as (Beltrán Jiménez et al. 2020;Dialektopoulos et al. 2019 where f Q = ∂f /∂Q and T µ ν is the matter energy momentum tensor.

Spherically symmetric solutions
Let us now proceed to the investigation of spherically symmetric solutions in the framework of f (Q) gravity.Although the coincident gauge is widely used in the cosmological Friedmann-Robertson-Walker Cartesian coordinates, the spherical symmetry is not compatible with it.Therefore, we need to choose the affine connection corresponding to the spherical symmetric coordinate system.For the spherically symmetric we choose the diagonal form of the metric is: here g tt and g rr are functions only of r.
Additionally, concerning the connection, one can perform the symmetry reduction imposing the additional assumptions that it is torsionless and stationary.We refer to (Hohmann 2020;D'Ambrosio et al. 2022) for details of this reduction.After choosing the formula of Γ r θθ , all the non-zero connection components can be driven by where Γ t θθ is an independent function.Then the nonmetricity scalar Q is solvable and can be expressed as (D'Ambrosio et al. 2022) In order to proceed we must choose a specific f (Q) form.In principle, we expect that every realistic modified theory of gravity should be a small deviation from GR.Hence, we deduce that every realistic f (Q) theory ought to have the form where α the model parameter.Thus, the spherically symmetric solutions can be approximated as tt , g rr = g (0) rr + α g (1) rr , where g (0) tt and g (0) rr correspond to the standard Schwarzschild solution, namely The general solution for the corrections can be extracted as (D'Ambrosio et al. 2022) with c 1 , c 2 constants.For large r, the above solution can be further approximated as (D'Ambrosio et al. 2022) with the renormalized mass In summary, the approximate spherically symmetric solution for realistic f (Q) theories is 3. WEAK LENSING THEORY AND DATA In this section, we introduce the weak lensing mechanism.Weak lensing refers to the shearing of distant galaxy images due to the differential deflection of neighboring light rays.This effect results in a small signal, usually causing an ellipticity at a level of around 1%.Although this distortion is minor compared to the inherent shape of individual galaxies, it can be detected statistically by analyzing the coherence of the lensing shear across the sky (Bartelmann & Schneider 2001).Therefore, the characteristics of spacetime, affected by the underlying theory of gravity, manifest in the lensing signal, providing a way to test the theory of gravity itself.
We are interested in exploring the effects on the light traveling from a distant celestial body, as it propagates through a foreground gravitational field on the way, within the framework of f (Q) gravity.Based on the fact that photons undergo deflection upon encountering foreground gravitational fields, to identify and understand the differences between f (Q) gravity and GR, a comparative study can be conducted by meticulously calculating the deflection angle, symbolized as α, within a static spherically symmetric spacetime.The deflection angle can be calculated from the geodesic equation under a certain spherically symmetric spacetime.The geodesic equation can be written using the corresponding affine connection and disformation tensor forms in non-metric gravity.According to the relation between affine connection and Levi-Civitia connection, the geodesic equation can be converted to the form under curvature gravity, which is essentially the same.Under modified gravity theories, the spherically symmetric metric solution is different, which results to the modification of the weak lensing formula.
The effective lensing potential is defined as (Narayan & Bartelmann 1996): where D d , D ds , and D s denote the angular diameter distances between the observer and the lens, the corresponding distance between the lens and source, and the one between the observer and the source.Moreover, the gravitational potential of the lens can be derived from the deflection angle as With the assumptions that the whole lensing system lies in the asymptotically flat spacetime regime, and that the distance of the light ray of closest approach r 0 , as well as the impact parameter b, both lie outside the gravitational radius, the deflection angle for metric (19) can be expressed as (Keeton & Petters 2005) where A = g tt ,B = g rr , and the impact parameter defined by b = r 2 0 A(r0) shows the perpendicular distance from the central axis of the lensing object to the asymptotic tangent of the light ray's trajectory when viewed with respect to an inertial observer.
Hence, for the spherically symmetric metric solution ( 19), the bending angle can be calculated under the parametrized-post-Newtonian (PPN) formalism, and can be expressed as a series expansion in the single quantity m b as (Keeton & Petters 2005) where we only wrote the coefficients of the first three orders.Therefore, we can acquire the three-dimensional gravitational potential as Here, the first term represents the Newtonian potential, the second term constitutes the contribution from GR, and the third term embodies the modification arising from f (Q) gravity.Subsequently, the shear tensor can be computed as a linear combination of the second partial derivatives of the potential, namely where ∆Σ(R) is the ESD under Newtonian approximation.Therefore, to account for differences, we define an effective ESD as follows, where Σ crit = c 2 4πG Ds D ds D d is the critical surface mass density.The quantity γ = (γ 2 1 + γ 2 2 ) 1/2 in the above expression describes the magnitude of the shear, since the shear components γ 1 and γ 2 introduce anisotropy (or astigmatism) shape distortion of the background galaxies, i.e. it quantifies how much the apparent shape of a galaxy has been altered compared to its intrinsic shape.
Having assembled all the requisite theoretical apparatus for the weak lensing phenomenon, let us now present the corresponding data.We choose the data available within the Seventh Data Release of the SDSS DR7 (Abazajian et al. 2009).Positioned at Apache Point Observatory near Sacramento Peak in southern New Mexico, the SDSS leveraged a specialized 2.5-meterwide-field telescope (Eisenstein et al. 2005) to capture CCD imagery across five broad bands photometry including the u, g, r, i, and z.The telescope employed twin instruments: a wide-field imager operating in drift scan mode to compile photometric catalogs, and a dual multi-object spectrograph system feeding 640 fibers, delivering spectra spanning wavelengths from 3800 to 9200 angstroms.These spectroscopic capabilities allowed for targeted analysis of celestial objects identified from the photometric data.
The lenses sample for our study are selected from a galaxy group catalog built upon the spectroscopic data from the SDSS DR7 (Yang et al. 2008), utilizing a sophisticated halo-based group detection algorithm originally developed in (Yang et al. 2006).The robustness of this algorithm lies in its iterative process and the utilization of an adaptive filter tailored to the inherent characteristics of dark matter halos, while the halo mass in the catalog are approximated via abundance matching techniques.Following the estimation of the halo mass, further properties such as the velocity dispersion, virial radius, and other relevant metrics are inferred accordingly.This algorithm has undergone an enhancement (Yang et al. 2021) to effectively handle galaxies with both photometric and spectroscopic redshifts, proving its efficacy in data analysis for the Dark Energy Spectroscopic Instrument (DESI) legacy imaging surveys DR8.Based on this sample, we have selected lens galaxies suitable for testing gravitational theories.The various galaxies were identified by selecting information such as distance and redshift.Since we consider the gravitational potential generated by a spherically symmetric solution, the sample was selected from individual galaxy systems, leading to a 400,608 sample size.
We employ the shape catalog constructed by (Luo et al. 2017), leveraging the SDSS DR7 imaging data as our source material, which comprises precise positional, morphological, shape uncertainty, and photometric redshift data for approximately 40 million galaxies, drawing upon the foundational work of (Csabai et al. 2007).Furthermore, the ESD ∆Σ(R) that we use is measured through the weighted mean of source galaxy shapes as where w = 1 (σ 2 shape +σ 2 sky ) is the corresponding weight, with σ shape =< e 2 > the shape noise and σ sky = σpix RF

√
4πn the sky noise (with σ pix the galaxy size in pixels, R the resolution factor, F the galaxy flux, and n the sky and dark current in Analog-to-Digital Units.

OBSERVATIONAL CONSTRAINTS
Let us now apply the steps described in the previous section, for the f (Q) gravity.In particular, we desire to extract constraints on the model parameter α that quantifies the deviation from GR.Our approach consists of three main stages: i) Model formulation: We adopt a modified NFW profile that incorporates both GR and additional f (Q) corrections.This choice allows us to assess the impact of deviations from standard gravity on the inferred mass distribution.ii) Sample consideration: Recognizing that the modifications to the model parameters primarily manifest on smaller scales, we select our lens sample drawn from a single galaxy system, and in this case we can exclude the two-halo term from our analysis.This decision acknowledges the dominant role of host halo effects in the context of the chosen galaxy system and allows us to excavate the specific nature of the modified gravity under the galactic scale.iii) Off center reconstruction: The distribution of galaxies within the system serves as a proxy for the center of the gravitational potential.However, it is crucial to acknowledge the off-center effect, where the designated central galaxy may not precisely coincide with the true center of the gravitational potential.This misalignment introduces considerable uncertainty in the estimation of halo mass and the model parameter α.In that case, we consider the off-center radius a free parameter and to be constrained during the MCMC process.
To evaluate ESD predictions our analysis employs ten catalogs of red and blue galaxies, divided into five mass intervals.We firstly employ the chi-square method to perform the quality-of-fit test of our model.Notably, the small stellar mass bin exhibits high signal-to-noise ratios and the f (Q) correction significantly impacts the low mass regime, contributing to reduced χ 2 .Fig. 1 reveals that the poor signal quality of blue galaxies in the large mass bin leads to decreased information content in their ESD profiles, while the red galaxies generally display superior signal quality.Therefore, we combine the data from red and blue galaxies, and we employ the concentration-mass relation from (Neto et al. 2007 50.447 40.639 -10.970 -13.772Table 1.The information criteria BIC and AIC for the standard NFW model in ΛCDM scenario, and the effective one in the framework of f (Q) gravity, alongside their corresponding differences.We have performed our analysis without and with the off-center effect.
model parameters.Then we use the MCMC method to derive the final parameter space for both halo properties and model parameter α.
Performing the above steps, we eventually extract the constraint for the α parameter, which is at 1σ confidence level.Moreover, for the off-center radius we obtain R sig = 0.02Mpc.Finally, in Fig. 2 we present the two-dimensional fitting posterior, and the constraints for various quantities.Interestingly enough, we deduce that the parameter α is constrained to small, but non-zero values, which implies that f (Q) corrections on top of GR are favoured by the weak lensing data.
In order to assess the efficiency of the fittings in comparison with ΛCDM paradigm, we employ the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).These statistical measures allow us to quantify the trade-off between model complexity and its ability to explain the observed data.The AIC (Akaike 1974) is derived from the Kullback-Leibler information, serving as an asymptotically unbiased estimator of model fitness.It is given by: where L max denotes the maximum likelihood achieved by the model when fitted to the data, and p tot represents the total number of free parameters in the model.The BIC (Kass & Raftery 1995) is related to the Bayesian evidence and is defined as: where N tot is the number of samples.Finally, based on their information criterion differences ∆IC, and using Jeffreys' classification (Anagnostopoulos et al. 2019), we compare f (Q) gravity and ΛCDM scenario.We consider two cases.Firstly, without considering off-center effect, and secondly considering the offcenter distance dispersion given by the best value R sig = The ESD profile and its best fits, as a function of the projected distance from the lens galaxy.The blue and orange alongside the corresponding errors, mark the weak lensing data for blue and red galaxies.For completeness, we have added the fit curves, shown with purple and orange lines.We have chosen Rsig = 0.02, which is the best fit value from MCMC constrain.0.02Mpc.The obtained results are summarized in Table 1.As we can see, in the absence of off-center effects, the fitting procedure indicates a reduced fit quality or increased complexity, which makes ΛCDM scenario more favourable.Nevertheless, in the presence of off-center effects, the information criteria analysis reveals a substantial improvement for f (Q) gravity, thus it is favored in this case compared with the ΛCDM paradigm.Hence, the combination of f (Q) gravity alongside off-center effects yields the best balance between model fit and complexity, making it the most favored model according to both information criteria.

CONCLUSIONS
In this work we performed a confrontation of f (Q) gravity with observations at galactic scales, namely us-ing galaxy-galaxy lensing data.We considered a small quadratic f (Q) correction on GR, quantified by the new model parameter α, an ansatz which is expected to hold in every realistic modification of gravity, and then we extracted the resulting spherically symmetric solutions.As we saw, these correspond to a deviation from Schwarzschild solution that depends on α in a two-fold way, namely a renormalized mass and a new term proportional to r −2 .
As a next step we calculated the effective lensing potential as well as the deflection angle, and thus we extracted the correction on the shear component, and effective ESD profile.As we saw, the r −2 correction amplifies the effect of weak lensing, predominantly manifesting at small scales.We employed the group catalog and shape catalog from the SDSS DR7 for the lens and source samples respectively.Lens galaxy samples were segregated into blue star-forming galaxies and red passive galaxies and then divided into five stellar mass bins to investigate the model dependence on galaxy color.Modeling the ESD with the NFW profile containing f (Q) correction terms, we found a minimal dependence of the modified gravitational model on galaxy color.
Furthermore, in order to investigate the influence of the off center effect on the parameter estimation, we set the off center radius to be an free parameter and try to constrain it together with other model and halo properties using the MCMC program.Finally, we extracted the most recent estimation for the deviation parameter as α = 1.202 +0.277  −0.179 × 10 −6 Mpc −2 at 1 σ confidence level, with the off-center radius estimation as R sig = 0.02 +0.01 −0.01 Mpc.To our knowledge, this is the first constraint for f (Q) gravity at galactic scales.The fact that the zero value is excluded at 1 σ confidence interval, indicates that small corrections on top of GR are favored.
In order to rigorously evaluate the fitting efficiency of f (Q) gravity comparing to that of ΛCDM scenario, we calculated the AIC and BIC information criteria values with and without considering the off-center effect.Considering the importance of the intrinsic off-center effect of the galaxy sample to constrain f (Q) gravity, we chose the best fit value of the off-center radius for the calculation.Our analysis revealed that f (Q) gravity demonstrates a heightened congruence with the galaxy-galaxy lensing data when the off-center effect is appropriately taken into account.Interestingly enough, f (Q) correc-tions alongside off-center effects yield a scenario that is slightly favoured over ΛCDM paradigm, and hence can challenge the latter.
Since the contribution of modified gravity is more pronounced at small scales, the effects of baryonic feedback effects (Semboloni et al. 2011;Duffy et al. 2010) at that scale may show up in different samples of galaxies.In that case, we acknowledged that a comprehensive joint analysis incorporating all relevant astrophysical influences would likely enhance the precision of our findings, constituting a valuable pursuit for future investigations.Additionally, our analysis tested f (Q) gravity at galaxy scales, yet validation at smaller solar-system scales and larger cluster/filament scales is imperative, especially since future research is expected to leverage strong or filamentary gravitational lensing.Finally, we mention that our methodology is applicable to other extended gravity theories, at low-redshifts and weak-field contexts, and thus exhibits considerable potential for broad implementation in the wealth of data from upcoming galaxy surveys.

ACKNOWLEDGMENTS
Figure1.The ESD profile and its best fits, as a function of the projected distance from the lens galaxy.The blue and orange alongside the corresponding errors, mark the weak lensing data for blue and red galaxies.For completeness, we have added the fit curves, shown with purple and orange lines.We have chosen Rsig = 0.02, which is the best fit value from MCMC constrain.

Figure 2 .
Figure2.The 1σ, 2σ and 3σ, constraints for the halo mass deviation parameter and the off-center radius.
) as a Gaussian prior to mitigate degeneracies with other