Quasiperiodic oscillations, weak field lensing and shadow cast around
black holes in symmergent gravity

Black holes, whose horizons are defined as the shell of the points of no return, are one of the most fundamental predictions of general relativity (GR). The first direct detection of gravitational waves in 2015 by the LIGO-Virgo collaboration Abbott et al. (2016) and the first-ever observation of an image of the black hole in 2019 by the Event Horizon Telescope (EHT) Akiyama et al. (2019, 2022) signaled the start of a new era in gravitational physics. These experiments open up a new methodology to test theories of gravity in the strong field (high curvature) domains.

In the present work, we will test a recent emergent gravity model in appropriate black hole spacetimes. In general, the reconciliation of quantum matter and gravitation is a thorny fundamental problem, and one plausible approach is provided by emergent gravity models Sakharov (1967); Visser (2002); Verlinde (2017). Indeed, quantum field theories (QFT) by formulation are intrinsic to flat spacetime Macias and Camacho (2008); Wald (2018) because they rest on a unique Poincaré-invariant vacuum state. Flat spacetime means the sheer absence of gravity. The incorporation of gravity requires QFT to be taken to curved spacetime. This, however, is simply impossible because Poincaré symmetry gets broken in curved geometry and special states like vacuum get banned by general covariance Dyson (2013); Wald (2018). In view of this impasse and view also of the absence of a quantum theory of gravity ’t Hooft and Veltman (1974), one is left with essentially emergent gravity Sakharov (1967); Visser (2002) as the most plausible option. In recent years, Sakharov’s setup has been extended by the inclusion of gauge sector Demir (2021a), and it has been shown Demir (2021a), that gravity could emerge in a way erasing loop-induced gauge boson masses and stabilizing scalar masses, which leads to the naturalization of the effective field theories regarding their sensitivities to high scales Weinberg (1979); Burgess (2020). This mechanism leads to a new framework Demir (2021a, 2019) in which one obtains the usual Einstein gravity (Einstein-Hilbert term) plus quadratic curvature terms plus a ${\overline{MS}}$-renormalized QFT. It possesses sui generis features that can be searched at the collider, astrophysical and cosmological experiments and observations. This emergent gravity, termed as “symmergent gravity” for its gauge symmetry-restoring nature Demir (2019, 2016), has all its couplings induced by the flat spacetime quantum loops. Symmergent gravity is not an effective field theory constructed in curved spacetime Visser (2002); Birrell and Davies (1984). It is, in fact, flat spacetime effective field theory taken to curved spacetime in a way restoring gauge symmetries Demir (2021a, 2019). Its various couplings are related to the flat spacetime QFT so that its curvature sector enables one to extract information about the particle spectrum, mass spectrum, and other details of the QFT. Since the new particles do not have to couple to the already known particles, one ends up with a wide spectrum of masses and couplings Demir (2021a, b). As a matter of fact, quadratic curvature terms in symmergent gravity, different from those in the other approaches Visser (2002); Birrell and Davies (1984), have been shown to lead to successful Starobinsky inflation Çimdiker (2020). In addition, the same quadratic curvature terms have recently been shown to give distinctive signatures on black hole shadow Çimdiker et al. (2021). In the present work, we shall probe the symmergent gravity model in the quasi-periodic oscillations, weak field lensing, and shadow cast around the black holes. Our work can be viewed are “doing particle physics via black hole properties” as it will reveal salient features of the QFT sector through various black hole features.

Since the verification of the GR as the relativistic theory of gravity by Eddington in 1919, gravitational lensing has remained one of the main tools for observing and understanding various phenomena in astronomy and cosmology (like galaxies, dark matter, dark energy and the evolution of the Universe Bozza (2003)). Over the decades, various methods have been devised to calculate the deflection angle by black holes Virbhadra and Ellis (2000, 2002); Bozza et al. (2001); Bozza (2002); Hasse and Perlick (2002); Perlick (2004); He et al. (2020); Bozza (2008). In 2008, Gibbons and Werner devised a different method to determine the deflection angle in weak fields using the Gauss-Bonnet theorem (GBT) on the optical geometries in asymptotically flat spacetimes Gibbons and Werner (2008). To achieve this, the GBT needs to be integrated into an infinite domain bounded by the light ray. This method has been applied to various related phenomena Övgün (2018, 2019a, 2019b); Javed et al. (2019a, b, c, 2020a, d); Övgün et al. (2019); Javed et al. (2020b). In Werner’s work Werner (2012), for instance, the GBT method for deflection angle was extended to stationary spacetimes in the Finsler-Randers type optical geometry on Nazim’s osculating Riemannian manifolds. Following this, Ishihara and others have found a way to apply the GBT method to finite-distances differently than the use of asymptotic receiver and source Ishihara et al. (2016, 2017). Next, Ono and others have extended Ishihara method to axisymmetric spacetimes Ono et al. (2017), with further applications to various non-asymptotic spacetimes, such as those with dark matter contributions in their metric function Pantig and Rodulfo (2020a); Pantig et al. (2022); Pantig and Övgün (2022a, b). Finally, the finite distance method has been studied by Li and others by using massive particles and Jacobi-Maupertuis Randers-Finsler metric within the GBT Li and Övgün (2020); Li et al. (2020a).

Recently, an eloquent step has been taken to probe black hole models in modified gravity theories with the EHT, which has showcased the first image (a 2D dark zone) of the supermassive black hole M87* Akiyama et al. (2019, 2022). Black holes have a dark region as seen from the observer in the celestial sky, which is known as the BH shadow. It can reveal a fine structure near the horizon of the black hole, and can therefore be used to test gravity theories in the strong gravity regime. Black hole shadow was first studied by Synge in 1966 for the specific case of the Schwarzschild black hole Synge (1966). After Synge, in 1979, Luminet gave the formula for the angular radius of the shadow Luminet (1979). It is well known that the shadow of a non-rotating black hole is just a standard circle. The shadow of a rotating black hole, on the other hand, is an elongated shape in the direction of rotation. To this end, theoretically-predicted shadow sizes of various black hole solutions have been contrasted with the 2017 measurements of M87* and various constraints have been put using the data. The shadows of modified gravity black holes are smaller and distorted compared to the shadow of the Kerr black hole. Black hole properties have been analysed by various recent works which focus on the black hole shadow cast Tsupko et al. (2020); Hioki and Maeda (2009); Dymnikova and Kraav (2019); Wei et al. (2019); Xu et al. (2018); Hou et al. (2018); Bambi et al. (2019); Konoplya (2019); Pantig (2021); Okyay and Övgün (2022); Uniyal et al. (2022); Kuang and Övgün (2022); Kumar et al. (2020); Belhaj et al. (2020); Li et al. (2020b); Övgün and Sakallı (2020); Övgün et al. (2020, 2018); Övgün and Sakallı (2020); Ling et al. (2021); Belhaj et al. (2021); Abdikamalov et al. (2019); Abdujabbarov et al. (2016); Atamurotov and Ahmedov (2015); Papnoi et al. (2014); Abdujabbarov et al. (2013); Atamurotov et al. (2013); Cunha and Herdeiro (2018); Gralla et al. (2019); Perlick et al. (2015); Nedkova et al. (2013); Li and Bambi (2014); Cunha et al. (2017); Johannsen et al. (2016); Johannsen (2016); Shaikh (2019); Allahyari et al. (2020); Yumoto et al. (2012); Cunha et al. (2016a); Moffat (2015); Cunha et al. (2016b); Zakharov (2014); Tsukamoto (2018); Hennigar et al. (2018); Chakhchi et al. (2022); Pantig and Övgün (2022c); Pantig and Rodulfo (2020b).

There appears a series of peaks in the radio to X-ray band of the electromagnetic spectrum from astrophysical objects. These peaks are signatures of the quasi-periodic oscillations (QPOs). One observes QPOs with more than two peaks in low-mass X-ray binaries (LMXBs) such that at least one compact object in the binary is a neutron star Ingram and Done (2010); Schaab and Weigel (1999); Török and Stuchlík (2005). In particular, most of the microquasars reveal twin-peak QPOs in their electromagnetic spectrum. The nature of these peaks is related to the process of matter accretion into the compact objects Ingram et al. (2016); Stuchlík et al. (2013); Stella and Vietri (1998); Rezzolla et al. (2003a). It turns out that the twin-peak QPOs and QPOs with more than two peaks are different in nature. In fact, various proposed models describe them as two types of QPOs Rezzolla et al. (2003b); Germanà (2017). The observation and analysis of the twin-peak QPOs in microquasars could be a useful tool to test different properties of neutron stars and black holes such as their magnetic fields, accretion disk properties, and the underlying gravity models Török et al. (2019); Zdunik et al. (2000); van der Klis (2000). On the other side, depending on the frequencies, one may distinguish two types of QPOs: (i) high-frequency (HF) QPOs corresponding to the frequency range from 0.1 to 1 kHz, and (ii) low-frequency (LF) QPOs corresponding to the frequency less than 0.1 kHz.

It is worth noting that there have been numerous proposals for describing the QPOs. Despite this, those proposals are not able to take into account all the astrophysical variables in the description of QPOs. Numerous observations and their detailed analyses indicate that QPOs are related to the dynamics of the test particles around the black holes, in particular, to their harmonic oscillations in the radial, vertical and azimuthal directions. Analyzing the dynamics of these test particles, especially their orbital motions and oscillations, one may test the gravity theories as well as spacetime properties around compact objects through observational data on QPOs  Bambi (2017); Stuchlík and Kološ (2015); Silbergleit et al. (2001); Wagoner et al. (2001); Rayimbaev et al. (2022); Stuchlík and Vrba (2021a, b, c); Rayimbaev et al. (2021a, b, c). Studies of models of QPOs in view of the orbital motions of the test particles enable the study of the innermost stable circular orbit (ISCO) of the particles Stuchlík et al. (2011); Török, G. et al. (2011); Rayimbaev et al. (2021d).

In this paper, as mentioned before, we will perform a detailed test study of the symmergent gravity in the strong field (high-curvature) domains using the EHT observations Akiyama et al. (2019, 2022). We will determine, in particular, the implications of the symmergent gravity setup for the quasi-periodic oscillations, weak field lensing, and shadow cast around the black holes. The paper is organized as follows: In Sec. II, a summary is given of the symmergent gravity model with emphasis on its gravity (curvature) sector and associated model parameters. Sec. III is devoted to study of the dynamics of test particles. The fundamental frequencies due to Keplerian orbits and harmonic oscillations are discussed in Sec. IV. In Sec. V, orbits of quasiperiodic oscillations are explored, and the results of this exploration are used in Sec. VI to determine the constraints on the parameters of the symmergent gravity. Further examination of the effects of these parameters is performed through the study of the weak deflection angle in Sec. VII, and the shadow cast in Sec. VIII. We close the paper with a Conclusion section in which we give a summary and discuss future prospects regarding the topics covered.

En passant, we note that in this paper we use geometrized (natural) units $G=1$, $c=1$ and ($-,+,+,+$) metric signature.

Symmergent gravity is a quadratic-curvature gravity. It is a subclass of $f(R)$ gravity theories. This special emergent gravity theory has been discussed in detail in Demir (2021a) (see also Demir (2019, 2016) for earlier analyses). The latest detailed discussion of the symmergent gravity was given in Çimdiker et al. (2021) while analyzing its implications for black hole shadow. This introductory black hole article gives all the basic aspects of symmergent gravity. Below, we summarize symmergent gravity in view of the required parameters and relevant features of the model (and kindly refer the reader to Demir (2021a); Çimdiker et al. (2021) for other details).

The symmergent gravity is governed by the action

in which $R(g)$ is the scalar curvature and $c_{i}$’s, and $G_{N}$ are loop induced quantities, that is, quantities produced by flat spacetime matter loops. Here $S$ is a generic scalar field, including the SM Higgs field. Two relevant parameters are given by,

where $n_{\text{B}}$ ($n_{\text{F}}$) stands for the total number of bosons (fermions) in Nature. The $n_{\text{B}}$ bosons plus $n_{\text{F}}$ fermions involve the known particles (quarks, leptons, photon, gluon and weak bosons) as well as completely new particles (massive as well as massless) that do not have to couple to the known particles non-gravitationally. The graded trace ${\rm str}[{\mathcal{M}}^{2}]=\sum_{s}(-1)^{s}\,{\rm tr}[{\mathcal{M}}^{2}]_{s}$, $s$ being the particle spin, is a sum over the mass-suqared matrix ${\mathcal{M}}^{2}$ of fields. The constant $c_{S}$ is around 0.019 when $S$ is the usual Higgs field Demir (2021a, 2019), and can take on different values for different scalars beyond the SM.

The curvature sector of the symmergent gravity, as follows from the action (1), contains linear and quadratic curvature terms. Their coefficients are loop-induced quantities defined in (2). The theory is a special case of generic $f(R)$ gravity theories, where instead of the standard Einstein-Hilbert action with a linear curvature, $R(g)$ the theory contains a covariant function of the curvature $f(R)$ as

after leaving aside the irrelevant scalar field sector. This action leads to the equations of motion

where $\Box$ is the d’Alembertian operator, and

is the derivative of $f(R)$ with respect to its argument. Making use of its trace, equation (4) leads to a traceless motion equation

whose solution is directly tied to the loop factors $G$ and $c_{O}$, and thus, varies with the particle spectrum of the underlying QFT. Now, using the ansatz,

the geometry of the black hole spacetime in spherical coordinates $x^{\alpha}=\{t,r,\theta,\phi\}$ is found to have the form

with the lapse function

becoming the Schwarzschild black hole spacetime when $a\to 1$ or/and $c_{\text{O}}\to\infty$. From now on, we set $G=1$. We plot the lapse function in Fig. 1 as a function of $r/M$.

For the Schwarzschild case, $r_{h}=2M$ and hence $f(r_{h})=0$. Inclusion symmergent gravity parameters makes the horizon shift slightly toward or away from the Schwarzschild case. In Figure 1, full curves correspond to $c_{\rm O}<0$, and mimic an AdS ($c_{\rm O}<0\Rightarrow n_{\text{B}}<n_{\text{F}}$) behaviour. The dashed curves, on the other hand, correspond to $c_{\rm O}>0$, mimic a dS ($c_{\rm O}>0\Rightarrow n_{\text{B}}>n_{\text{F}}$) structure, and develop a second horizon at large $r$. We observe that as $r/M$ gets too large, deviation from the Schwarzschild case is pronounced.

In this section, we give a detailed analysis of the motions of test particles in the black hole spacetime (8) of the symmergent gravity.

In this section, we study the fundamental frequencies of test particles around a black hole in symmergent gravity. In particular, we will explore the effects of the symmergent gravity parameters on the frequencies of the particles in their Keplerian orbits. We will also explore the frequencies of the radial and vertical oscillations of the particles around their stable circular orbits. We will apply our results to the analysis of QPO frequencies.

In general, QPOs refer to juddering of the light rays from astronomical objects about certain frequencies. As an illustrative case, QPOs in (micro)quasars have come into view in relation to the fundamental frequencies of the particles orbiting the black hole at the center of the (micro)quasar. The main point is that the frequencies of QPOs depend on the spacetime geometry in the black hole environment.

In this section, we explore the upper and lower frequencies of twin-peaked QPOs in the framework of symmergent gravity. Here, the twin-peak QPOs are picked up on the basis of probing the symmergent gravity parameters. In general, the upper and lower frequencies are described in different ways by the frequencies of the radial, vertical and orbital oscillations:

• •

  The relativistic precession (RP) model has been first proposed in Ref. Stella and Vietri (1998) to explain the physical mechanism behind kHz twin-peak QPOs in the frequency range 0.2 to 1.25 kHz from neutron stars in LMXRBs. It has been shown that the RP model applies also to black hole candidates in the binary systems of black holes and neutron stars Stella (2001). Furthermore, an RP model has been developed in Ingram and Motta (2014) to obtain mass and spin measurements of black holes at the center of microquasars using the data from the accretion disk power-density spectrum. The upper and lower frequencies in the RP model are given by the fundamental oscillation frequencies $\nu_{U}=\nu_{\phi}$ and $\nu_{L}=\nu_{\phi}-\nu_{r}$, respectively.

• •

  The epicyclic resonance (ER) model is based on the resonances in the axisymmetric oscillation modes in the accretion disc of black holes Abramowicz and Kluźniak (2001). It has been shown that the disc oscillation modes are related to the frequencies of (quasi) harmonic oscillations of the circular geodesics of the test particles. Here, we consider two special cases of the ER model: ER2 and ER3. They differ from each other by their oscillation modes. ER2 has the classes of the upper and lower frequencies $\nu_{U}=2\nu_{\theta}-\nu_{r}\ \&\ \nu_{L}=\nu_{r}$ as well as $\nu_{U}=\nu_{\theta}+\nu_{r}\ \&\ \nu_{L}=\nu_{\theta}$. ER3, on the other hand, has only one class of the upper and lower frequencies $\nu_{U}=\nu_{\theta}+\nu_{r}\ \&\ \nu_{L}=\nu_{\theta}-\nu_{r}$ Abramowicz and Kluźniak (2001).

• •

  The warped disc (WD) model is structured upon the non-axisymmetric modes of the oscillations of the accretion disc around the black holes and neutron stars Kato (2004, 2008). According to the WD model assumptions, the upper and lower frequencies are $\nu_{U}=2\nu_{\phi}-\nu_{r}$, $\nu_{L}=2(\nu_{\phi}-\nu_{r})$, and vertical oscillations gives cause to a thin accretion disc to be warped  Kato (2004, 2008).

Now, using the three models mentioned above, we analyze possible upper and lower frequencies of the twin-peaked QPOs. These frequencies are generated by the flickering of the test particles in their stable circular orbits around symmergent black holes.

Possible values of the upper and lower frequencies of twin-peak QPOs around a symmergent gravity black hole are plotted in Figure 10 for the RP, WD, ER2, and ER3 models. The inclined blue-solid $\nu_{U}=\nu_{L}$ line is the deadline for twin-peak QPOs since their two peaks coincide for $\nu_{U}=\nu_{L}$. The light-blue shaded area is the graveyard for twin-peak QPOs in that no such QPOs can be observed in this domain of $\nu_{U}$ and $\nu_{L}$. It is observed from the figure that no low-frequency twin peak QPOs occur. This is due to the presence of a non-zero $a$ parameter because for $a<1$ ratios of the frequencies slightly increase but, for $a>1$, the same ratios decrease compared to the Schwarzschild case ($a=1$). It is also observed from the ER2 and ER3 models that two different QPOs could be observed with similar frequency ratios for upper (lower) frequencies above 450 Hz (250 Hz). Frequency degeneracies as such are observed also at low-frequency twin-peak QPOs around 100 Hz in the RP and WD models (similar low-frequency behavior occurs between 100-200 Hz in the ER3 model).

In this section, we determine constraints on the symmergent gravity parameters $a$ and $c_{\rm O}$ by using the data from QPOs in the following astrophysical objects:

• •

  GRS 1915-105 is observed in upper and lower frequencies $168\pm 5$ & $113\pm 3$, respectively, with central black hole mass $12.4^{+2.0}_{-1.8}M_{\odot}$ Reid et al. (2014)

• •

  GRO J1655-40, powered by central black hole with mass $(5.4\pm 0.4)M_{\odot}$, observed in the high frequencies $441\pm 2$ & $298\pm 4$ and low frequency $17.3\pm 0.1$ Hz Stuchlík and Kološ (2015, 2016).

• •

  XTE J1550-564, mass of the black hole at the centre of the microquasar is $(9.1\pm 0.61)M_{\odot}$ and it is detected in $276\pm 3$& $184\pm 5$ Hz  Stuchlík et al. (2013); Miller et al. (2001),

• •

  H1743+322 microquasar has been found in the frequency band of electromagnetic spectrum $242\pm 3$ & $166\pm 5$ Hz, mass of the central black hole lies in the range of between 8 and 14.07 solar mass  Stuchlík and Kološ (2015); Remillard et al. (2006),

• •

  a QPO have been detected around SgrA* located at the center of Milky Way galaxy, with the low frequency $1.445\pm 0.16$ and $0.866\pm 0.04$ mHz  Stuchlík and Vrba (2021c); Abramowicz et al. (2004).

Once the radii of the QPO orbits are determined, it becomes possible to reveal the constraints on the symmergent gravity parameters $a$ and $c_{\rm O}$ by solving the equations

for each of the RP, WD, ER2 and ER3 models. In these equations, $\nu_{\rm U}^{ob}$ and $\nu_{\rm L}^{ob}$ are observational values of the upper and lower frequencies of a given QPO. In order to get relations between the allowed values of the symmergent parameters, we use observational results on the above-mentioned astrophysical objects.

Figure 12 shows relationships between $a$ and $c_{\rm O}$ parameters in symmergent gravity for the twin-peak QPO objects GRS 1915-105, GRO J1655-40, XTE J1550-564, H1743+322, and Sgr A* in the RP (top-left), WD (top-right), ER2 (bottom-left), and ER3 (bottom-right) models. It is clear that $a$ and $c_{\rm O}$ exhibit a linear relationship. It is observed that the data from GRS 1915-105, GRO J1655-40, XTE J1550-564, and H1743+322 exhibit a nearly model-independent behavior. These microquasars have a frequency ratio of 3:2 and stellar-mass black holes at their centers. The data from Sgr A* exhibits strong model dependence and deviates significantly from the microquasar data. It may be that the existing QPO models (RP, WD, ER2, and ER3) are not sufficiently accurate to cover the Sgr A* data.

In this section, we determine the weak deflection angle by a symmergent black hole using the Gauss-Bonnet theorem (GBT) approach developed by Ishihara and others Ishihara et al. (2016, 2017). To this aim, we explore weak deflection angle as a sensitive probe of the loop factor $c_{\rm O}$. The GBT is expressed as Gibbons and Werner (2008)

in which $T$ is a two-dimensional orientable surface with boundaries $\partial T_{a}$ ($a=1,2,\cdots,N$) that are differentiable curves, $\theta_{a}$ ($a=1,2,\cdots,N$) denote jump angles, $K$ denotes the Gaussian curvature of the surface $T$, $dS$ is the area element of the surface, $\kappa_{g}$ means the geodesic curvature of $\partial T_{a}$, and $\ell$ is the line element along the boundary. By applying this fundamental formula to the quadrilateral ${}^{\infty}_{R}\Box^{\infty}_{S}$, which consists of the spatial curve for the light ray, two outgoing radial lines from $R$ and $S$ and a circular arc segment $C_{r}$ of coordinate radius $r_{C}$ ($r_{C}\to\infty$) centered at the lens $L$ which intersects the radial lines through the receiver or the source, Ishihara et al. Ishihara et al. (2016) obtained the formula

from which it follows that $\alpha$ is an invariant and remains well-defined even if $L$ is a singularity point since the domain ${}^{\infty}_{R}\Box^{\infty}_{S}$ does not contain the point $L$. It follows that $\alpha=0$ in Euclidean space. This is because $K$ vanishes in Euclidean space, and the area integral of $K$ thus vanishes. Furthermore, (40) proves useful for asymptotically flat spacetimes, and cannot be applied to non-asymptotically flat ones that have the metric (lapse) function in Eq. (9).

To solve this problem, Ishihara and others Ishihara et al. (2016) used the main definition $\Psi_{R}-\Psi_{S}+\phi_{RS}$ in Eq. (40), and focused on the calculation of the positional angle $\Psi_{S}$ ($\Psi_{R}$) between the radial direction and the light ray at the source (receiver) position, where the relative angle between the sources and the receiver is given by $\phi_{RS}$. Rewriting the metric in Eq. (8) as

with the solid angle $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\phi^{2}$ and azimuthal angle $\phi$. Considering the photon orbital plane at $\theta=\pi/2$, the angle $\Psi$ is found to be Ishihara et al. (2016)

where $b$ is the impact parameter of light. Using the lapse function (9), one finds

as the explicit expression of $\Psi$, with the variable $u=1/r$. Taking into account the finite distance between the lensing object and the source and the receiver, one gets

where $u_{R}$ ($u_{S}$) is the inverse-position of the receiver (source). The methodology used by Ishihara and others Ishihara et al. (2016) determines the source-receiver angle $\phi_{RS}$ as,

where the orbit equation $F(u)$ is given by

after setting $ds^{2}=0$ in (41). The inverse-position $u_{o}$ in (45) is the inverse of the closest approach, which can be determined by an iterative solution of $F(u)$. In fact, $u_{o}=\sin\phi/b$ turns out to be sufficient so that the azimuthal angle is found to be

with which one finds

such that $\phi_{\text{R}}=\pi-\phi|_{u\rightarrow u_{\text{R}}}$ and $\phi_{\text{S}}=\phi|_{u\rightarrow u_{\text{S}}}$ are the azimuthal angles of the receiver and the source, respectively Takizawa et al. (2020); Ono and Asada (2019). Having determined all three angles in Eq. (40), one arrives at the angle formula

whose second line is seen to diverge as $r\to\infty$. This apparent divergence implies that the symmergent black hole does not allow $r\to\infty$ ($u\to 0$) limit. In spite of this, it is possible to regularize $\hat{\alpha}$ by choosing $u$ small yet finite (in accordance with the realistic situations) so that the formula (49) takes its small-$u$ apporximate form

The deflection angle ${\hat{\alpha}}$ is depicted in Figure 13. The left-panel shows the exact formula (49) as a function of the impact parameter $b$ for nearby observers ($u_{\rm R}=u_{\rm S}=0.5/b$), with the parameter values $a=0.9$ and $c_{\rm O}/M^{2}=\pm 1.99$ and $\pm 3.98$. The right-panel of Figure 13, on the other hand, shows the small-$u$ limit in (50) as a function of the impact parameter $b$ for faraway observers ($u_{R}=u_{S}=0.02/b$), with the parameter values $a=0.9$ and $c_{\rm O}/M^{2}=\pm 1.99$ and $\pm 3.98$. In both panels, the Schwarzschild case (black curve) decreases monotonically as $b/M\to\infty$. It is observed that $c_{\rm O}>0$ ($c_{\rm O}<0$) mimics the dS (AdS) behaviour. This comes to mean that $n_{\rm B}>n_{\rm F}$ ($n_{\rm B}<n_{\rm F}$) leads to dS (AdS) behaviours. Interestingly, for an observer near the black hole, when $b/M\to b_{\rm crit}/M$, $\hat{\alpha}$ remains below the Schwarzschild deflection angle such that the AdS case ($c_{\rm O}<0\Rightarrow n_{\rm B}<n_{\rm F}$) stays slightly higher. At high values of $b/M$, the AdS (dS) case leads to an increasing (decreasing and vanishing) $\hat{\alpha}$. Meanwhile, an observer far from the black hole perceives a higher value for $\hat{\alpha}$ around $b_{\text{crit}}/M$ for the AdS case. Clearly, $\hat{\alpha}$ does not exist at all in the dS case, even for low values of $b/M$. These two panels reveal the effects of $a$ and $c_{\text{O}}$ on the deflection angle. Here, we remark that negative deflection angle in the dS ($c_{\rm O}>0\Rightarrow n_{\rm B}>n_{\rm F}$) case comes to mean that at such values of $b/M$ the photons are repelled. There are certain cases where repulsive deflection can also happen Nakashi et al. (2019). Nonetheless, one notes that negative deflection angle does not produce the Einstein ring, as we shall see in the next discussion.

The weak deflection angle has direct application in a phenomenon called Einstein ring. Here, we will determine the effects of the symmergent gravity parameters on the Einstein ring. To begin with, let us define the distance of the source and the receiver with respect to the lensing object as $d_{\text{S}}$ and, $d_{\text{R}}$ respectively. Through the thin lens equation, we then have $d_{\text{RS}}=d_{\text{S}}+d_{\text{R}}$ so that the position of the weak field images is given by Bozza (2008)

It is well known that an Einstein ring is formed when $\beta=0$, and the above equation leads then to the angular radius Virbhadra and Ellis (2000, 2002); Bozza et al. (2001); Bozza (2002); Hasse and Perlick (2002); Perlick (2004)

In addition, since the Einstein ring is assumed to be small it is safe to take relation $b=d_{\text{R}}\sin\theta\sim d_{\text{R}}\theta$. Then, the weak deflection angle (52) takes the explicit form

after introducing $\epsilon=b^{2}/c_{\text{O}}$. This angular radius is depicted in Figure 14 as a function of the impact parameter for $a=0.9$ and $c_{\rm O}/M^{2}=\pm 1.99$ and $\pm 3.98$. This plot corresponds to the right-panel in Figure 13. The sensitivity of the Einstein ring to the model parameters ensures that it can be used to probe the symmergent gravity framework.

Let us now consider Sgr A*. Our location is $d_{\text{R}}\approx 8.3$ kpc from the galactic center. In the Schwarzschild limit, the angular radius is $\theta_{\text{E}}^{\text{Schw}}=1.186$ arc sec. Using equation (53), one finds that small deviations from $\theta_{\text{E}}^{\text{Schw}}$ at $a=0.90$ requires $c_{\text{O}}\sim 4\times 10^{31}$, which means that Nature contains much more bosonic degrees of freedom than the fermionic ones.

The perihelion shift of planetary motion around a symmergent black hole can qualify as a sensitive probe of the model parameters. To calculate it, we start with the effective potential for radial motion (in geometrical units $G=1$ and $c=1$)

in which ${\cal L}$ is the specific orbital angular momentum, $R$ is the semimajor axis of the elliptical orbit, and $e$ is the orbit’s eccentricity Arakida (2013). To determine the orbit, we use the variable $u=1/r$ so that $\frac{dr}{d\lambda}=-{\cal L}\frac{du}{d\phi}$. Then, for non-circular orbits and massive particle’s the equation of radial motion becomes

whose first term at the right-hand side suffices for determining the contribution of the quasi-Newtonian orbit:

in which $\phi_{0}$ is a constant of integration. The most direct way to determine the eccentricity $e$ is to apply successive approximations about the quasi-Newtonian orbit in (56). This is because the last two terms in equation (55) are small in comparison to the Newtonian contribution and, using the standard perturbation techniques, one can easily obtain the first order correction to be $u\cong u_{0}+u_{1}$ ($u_{1}<<u_{0}$), where Islam (1983); Freire et al. (2001). The perturbation $u_{1}(\phi)$ (orbit of small eccentricity) obeys the equation

with the solution

Having this correction at hand, one obtains

with

so that the radial coordinates takes the form

after introducing the angular shift