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In this paper we use a modification of the Newtonian gravitational potential with a non-linear Yukawa-like correction, as it was proposed by C. Will earlier to obtain new bounds on graviton mass from the observed orbits of S-stars around the Galactic Center (GC). This phenomenological potential differs from the gravitational potential obtained in the weak field limit of Yukawa gravity, which we used in our previous studies. We also assumed that the orbital precession of S-stars is close to the prediction of General Relativity (GR) for Schwarzschild precession, but with a possible small discrepancy from it. This assumption is motivated by the fact that the GRAVITY Collaboration in 2020 and in 2022 detected Schwarzschild precession in the S2 star orbit around the Supermassive Black Hole (SMBH) at the GC. Using this approach, we were able to constrain parameter $\lambda$ of the potential and, assuming that it represents the graviton Compton wavelength, we also found the corresponding upper bound of graviton mass. The obtained results were then compared with our previous estimates, as well as with the estimates of other authors., Comment: 16 pages, 4 tables, 3 figures. Accepted for publication in Symmetry

Here we study possible improvements of the existing constraints on the upper bound of graviton mass by the analysis of the stellar orbits around the SMBH at the GC in the framework of Yukawa gravity. A motivation for this study is a recent detection of Schwarzschild precession in the orbit of S2 star around the SMBH at the GC by the GRAVITY Collaboration. The authors indicated that the orbital precession of the S2 star is close to the General Relativity (GR) prediction, but with possible small deviation from it, and parametrized this effect by introducing an ad hoc factor in the parametrized PPN equations of motion. Here we use the value of this factor presented by GRAVITY in order to perform two-body simulations of the stellar orbits in massive gravity using equations of motion in the modified PPN formalism, as well as to constrain the range of massive interaction $\Lambda$. From the obtained values of $\Lambda$, and assuming that it corresponds to the Compton wavelength of graviton, we then calculated new estimates for the upper bound of graviton mass which are found to be independent, but consistent with the LIGO's estimate of graviton mass from the first GW signal GW150914 (later this graviton mass estimation was significantly improved with consequent observations of GW events). We also performed calculations including numerical simulations in order to constrain the bounds on graviton mass in the case of a small deviation of the stellar orbits from the corresponding GR predictions and showed that our method could further improve previous estimates for upper bounds on the graviton mass. It is also demonstrated that such analysis of the observed orbits of S-stars around the GC in the frame of the Yukawa gravity represents a tool for constraining the upper bound for the graviton mass, as well as for probing the predictions of GR or other gravity theories., Comment: 16 pages, 3 tables, 8 figures. Accepted for publication in Phys. Rev. D

We show that fundamental plane of elliptical galaxies can be used to obtain observational constraints on metric theories of gravity. Being it connected to global properties of ellipticals, it can fix parameters of modified gravity. Specifically, we use fundamental plane to constrain modified theories of gravity with Yukawa-like corrections which commonly emerge in the post-Newtonian limit. After giving examples on how these corrections are derived, we first analyze the velocity distribution of elliptical galaxies comparing theoretical results of modified gravity with Yukawa-like corrections with astronomical data. According to these results, it is possible to constrain the parameters of the corrections discriminating among classes of models compatible with astronomical observations. We conclude that fundamental plane can be used as a standard tool to probe different theories of gravity in the weak field limit., Comment: 13 pages, 7 figures, to be published in Phys. Dark. Univ. (2020)

The aim of our investigation is to derive a particular theory among the class of scalar-tensor(ST) theories of gravity, and then to test it by studying kinematics and dynamics of S-stars around supermassive black hole (BH) at Galactic Center (GC). We also discuss the Newtonian limit of this class of ST theories of gravity, as well as its parameters. We compare the observed orbit of S2 star with our simulated orbit which we obtained theoretically with the derived ST potential and constrained the parameters. Using the obtained best fit parameters we calculated orbital precession of S2 star in ST gravity,and found that it has the same direction as in General Relativity (GR), but causes much larger pericenter shift., Comment: 13 pages, 1 table, 6 figures. Accepted for publication in Facta Universitatis: Series Phys. Chem. Tech. (Invited lecture at Balkan Workshop 2018)

The global properties of elliptical galaxies are connected through the so-called fundamental plane of ellipticals, which is an empirical relation between their parameters: effective radius, central velocity dispersion and mean surface brightness within the effective radius. We investigated the relation between the parameters of the fundamental plane equation and the parameters of modified gravity potential $f(R)$. With that aim, we compared theoretical predictions for circular velocity in $f(R)$ gravity with the corresponding values from a large sample of observed elliptical galaxies. Besides, we consistently reproduced the values of coefficients of the fundamental plane equation as deduced from observations, showing that the photometric quantities like mean surface brightness are related to gravitational parameters. We show that this type of modified gravity, especially its power-law version - $R^n$, is able to reproduce the stellar dynamics in elliptical galaxies. Also, it is shown that $R^n$ gravity fits the observations very well, without need for a dark matter., Comment: 9 pages, 2 figures. Accepted for publication in Atoms

The relativistically broadened Fe K$\alpha$ line, originating from the accretion disc in a vicinity of a super massive black hole, is observed in only less than 50\% of type 1 Active Galactic Nuclei (AGN). In this study we investigate could this lack of detections be explained by the effects of certain parameters of the accretion disc and black hole, such as the inclination, the inner and outer radius of disc and emissivity index. In order to determine how these parameters affect the Fe K $\alpha$ line shape, we simulated about 60,000 Fe K $\alpha$ line profiles emitted from the relativistic disc. Based on simulated line profiles, we conclude that the lack of the Fe K$\alpha$ line detection in type 1 AGN could, be caused by the specific emitting disc parameters, but also by the limits in the spectral resolution and sensitivity of the X-ray detectors., Comment: Based on the talk presented Balkan Workshop BW2018 (10-14 June 2018, Ni\v{s}, Serbia), accepted for publishing in International Journal of Modern Physics A, 8 figures, 1 table, 15 pages

An experimental detection of graviton is extremely hard problem, however, there are different ways to evaluate a graviton mass if it is non-vanishing. Theories of massive gravity or theories with non-vanishing graviton mass initially have a number of pathologies such as discontinuities, ghosts etc. In last years theorists found ways to overcome weaknesses of such theories meanwhile observational features are also discussed. In the first publication reporting about the discovery of gravitational waves from the binary black hole system the LIGO-Virgo collaboration obtained the graviton mass constraint around $1.2 \times 10^{-22}$ eV (later the estimate was improved with new data). A comparable and consistent graviton mass constraint around $2.9 \times 10^{-21}$ eV has been obtained from analysis of the bright star S2 trajectory near the Galactic Center., Comment: 6 pages, presented as an invited talk at the XXXI International Workshop on High Energy Physics (IHEP, Protvino, Russia)

Recently LIGO collaboration discovered gravitational waves \cite{Abbott_16} predicted 100 years ago by A. Einstein. Moreover, in the key paper reporting about the discovery, the joint LIGO \& VIRGO team presented an upper limit on graviton mass such as $m_g < 1.2 \times 10^{-22} eV$ (Abbott et al. (LIGO collaboration) PRL 116 (2016) 061102). Since the graviton mass limit is so small the authors concluded that their observational data do not show violations of classical general relativity. We consider another opportunity to evaluate a graviton mass from phenomenological consequences of massive gravity and show that an analysis of bright star trajectories could bound graviton mass with a comparable accuracy with accuracies reached with gravitational wave interferometers and expected with forthcoming pulsar timing observations for gravitational wave detection. It gives an opportunity to treat observations of bright stars near the Galactic Center as a wonderful tool not only for an evaluation specific parameters of the black hole but also to obtain constraints on the fundamental gravity law such as a modifications of Newton gravity law in a weak field approximation. In particular, we obtain bounds on a graviton mass based on a potential reconstruction at the Galactic Center., Comment: 8 pages, 1 figure

The detection of gravitationally redshifted optical emission lines has been reported just for a few active galaxies. In this paper we give a short overview of studies that analyzed or exploited the detection of the gravitational redshift in optical AGN spectra. In addition, we tested the consistency of gravitational redshift as the physical origin of the redward shifts observed in their spectra using a sample of $\approx$ 50 Hamburg-ESO intermediate to high redshift quasars that are among the most luminous quasars known ($10^{47} \lesssim L \lesssim 10^{48}$ erg/s), and are expected to host very massive black holes. To this aim we modeled the line profile with accretion disk models around a black hole., Comment: 10 pages, 2 figures, 3 tables

Here we investigate possible applications of observed stellar orbits around Galactic Center for constraining the R$^n$ gravity at Galactic scales. For that purpose, we simulated orbits of S2-like stars around the massive black hole at Galactic Center, and study the constraints on the R$^n$ gravity which could be obtained by the present and next generations of large telescopes. Our results show that R$^n$ gravity affects the simulated orbits in the qualitatively similar way as a bulk distribution of matter (including a stellar cluster and dark matter distributions) in Newton's gravity. In the cases where the density of extended mass is higher, the maximum allowed value of parameter $\beta$ in R$^n$ gravity is noticeably smaller, due to the fact that the both extended mass and $R^n$ gravity cause the retrograde orbital precession., Comment: 10 pages, 4 figures, accepted for publication in Advances in Space Research

We investigated the gravitational microlensing of active galactic nucleus dusty tori in the case of lensed quasars in the infrared domain. The dusty torus is modeled as a clumpy two-phase medium. To obtain spectral energy distributions and images of tori at different wavelengths, we used the 3D Monte Carlo radiative transfer code SKIRT. A ray-shooting technique has been used to calculate microlensing magnification maps. We simulated microlensing by the stars in the lens galaxy for different configurations of the lensed system and different values of the torus parameters, in order to estimate (a) amplitudes and timescales of high magnification events, and (b) the influence of geometrical and physical properties of dusty tori on light curves in the infrared domain. We found that, despite their large size, dusty tori could be significantly affected by microlensing in some cases, especially in the near-infrared domain (rest-frame). The very long time-scales of such events, in the range from several decades to hundreds of years, are limiting the practical use of this method to study the properties of dusty tori. However, our results indicate that, when studying flux ratios between the images in different wavebands of lensed quasars, one should not disregard the possibility that the near and mid-infrared flux ratios could be under the influence of microlensing., Comment: 10 pages, 6 figures, 1 table, MNRAS accepted. V3: corrected values in Table 1

Here we consider the influence of the microlensing on the spectrum of a lensed object taking into account that composite emission is coming from different regions arranged subsequently around the central source. We assumed that the lensed object has three regions with the black body emission; first the innermost with the highest temperature of $10^4K$, second and third (located around the central) with slightly lower temperatures $7.5\cdot10^3$ and $5\cdot10^3$K, respectively. Than we explore the flux anomaly in lensed object due to microlensing. We compare U,V and B spectra of a such source. This results show that, due to microlensing, in a spectroscopically stratified object a flux anomaly is present., Comment: 4 pages, 2 figures, 1 table

Here we present an overview of some of the most significant observational and theoretical studies of the broad Fe K{\alpha} spectral line, which is believed to originate from the innermost regions of relativistic accretion disks around central supermassive black holes of galaxies. The most important results of our investigations in this field are also listed. All these investigations indicate that the broad Fe K{\alpha} line is a powerful tool for studying the properties of the supermassive black holes (such as their masses and spins), space-time geometry (metric) in their vicinity, their accretion physics, probing the effects of their strong gravitational fields, and for testing the certain predictions of General Relativity., Comment: 13 pages, 15 figures, accepted for publication in New Astronomy Reviews

We study the photocenter position variability due to variations in the quasar inner structure. We consider variability in the accretion disk emissivity and torus structure variability due to different illumination by the central source. We discuss possible detection of these effects by Gaia. Observations of the photocenter variability in two AGNs, SDSS J121855+020002 and SDSS J162011+1724327 have been reported and discussed. With investigation of the variations in the quasar inner structure we explore how much this effect can affect the position determination and whether it can be (or not) detected with Gaia mission. We used (a) a model of a relativistic disk, including the perturbation that can increase brightness of a part of the disk, and consequently offset the photocenter position, and (b) a model of a dusty torus which absorbs and re-emits the incoming radiation from accretion disk. We estimated the value of the photocenter offset due to these two effects. We found that perturbations in the inner structure can significantly offset the photocenter. It depends on the characteristics of perturbation and accretion disk and structure of the torus. In the case of two considered QSOs the observed photocenter offsets cannot be explained by variations in the accretion disk and other effects should be considered. We discussed possibility of exploding stars very close to the AGN source, and also possibility that there are two variable sources in the center of these two AGNs that may indicate a binary super-massive black hole system on a kpc (pc) scale. The Gaia mission seems to be very perspective, not only for astrometry, but also for exploring the inner structure of AGNs. We conclude that variations in the quasar inner structure can affect the observed photocenter (up to several mas). There is a chance to observe such effect in the case of bright and low-redshifted QSOs., Comment: 12 pages, 8 figures, 3 tables. Accepted for publication in Astronomy and Astrophysics. Language improved, typos corrected

Here we give a brief overview of some investigations of the gravitational microlensing influence on broad absorption spectral lines of lensed QSOs. Especially, we consider the microlensing influence on the Fe K$\alpha$ broad absorption lines using a model of an accretion disk covered by an absorption region. Gravitational microlensing is modeled by ray shooting method which enables us to obtain realistic microlensing patterns. We obtain that microlensing can affect both emission and absorption component of line that depends on dimensions on emission and absorption line regions. Here we give detailed analysis of emission and absorption line shape variations due to gravitational microlensing., Comment: 7 pages, 5 figures. Accepted for publication in New Astron. Rev. (Procceedings of the 7th Serbian Conference on Spectral Line Shapes in Astrophysics)

In this chapter we discuss the X-ray radiation from relativistic accretion disks around supermassive black holes, supposed to exist in the centers of Active Galactic Nuclei (AGN). Our focus is on the X-ray radiation, especially in the Fe K$\alpha$ line which originates in the innermost parts of an accretion disk. Moreover, here we discuss some effects which can disturb the Fe K$\alpha$ profile and cause its rapid and irregular variability, observed in the X-ray spectra of some AGN. We will pay attention to three such effects: perturbations in the disk emissivity, absorbtion by warm absorbers and gravitational microlensing. The X-ray emission from accretion disks around non-rotating (Schwarzschild metric), as well as rotating (Kerr metric) supermassive black holes, is discussed. The X-ray radiation of AGN is probably produced in a compact region near their central supermassive black holes, and can provide us some essential information about the plasma conditions and the space-time geometry in these regions. The goal of this chapter is mainly to present a short overview of some important and recent investigations in this field., Comment: 62 pages, 24 figures, to appear as a chapter of the book "Black Holes: Properties, Formation and Features" (to be published by Nova Science Publishers, Inc)

Gravitational microlensing is a very useful tool for investigating the innermost part of lensed quasars, especially for studying a relativistic accretion disk around a massive black hole (BH) supposed to exist in a quasar's center. Here we present a short overview of our recent investigations in this field., Comment: Ph.D. dissertation summary, 6 pages, 2 figures, accepted for publishing in Publications of the Astronomical Society of the Pacific

We consider a contribution of microlensing to the X-ray variability of high-redshifted QSOs. Cosmologically distributed gravitational microlenses could be localized in galaxies (or even in bulge or halo of gravitational macrolenses) or could be distributed in a uniform way. We have analyzed both cases of such distributions. We found that the optical depth for gravitational microlensing caused by cosmologically distributed deflectors could be significant and could reach $10^{-2} - 0.1$ at $z\sim 2$. This means that cosmologically distributed deflectors may contribute significantlly to the X-ray variability of high-redshifted QSOs ($z>2$). Considering that the upper limit of the optical depth ($\tau\sim 0.1$) corresponds to the case where dark matter forms cosmologically distributed deflectors, observations of the X-ray variations of unlensed QSOs can be used for the estimation of the dark matter fraction of microlenses., Comment: 6 pages, to appear in "Impact of Gravitational Lensing on Cosmology", IAU Symposium 225, eds. Y. Mellier & G. Meylan