Constraints on Barrow Entropy from M87* and S2 Star Observations

Abstract

We use data from M87* central black hole shadow, as well as from the S2 star observations, in order to extract constraints on Barrow entropy. The latter is a modified entropy arising from quantum-gravitational effects on the black hole horizon, quantified by the new parameter $\Delta$. Such a change in entropy leads to a change in temperature, as well as to the properties of the black hole and its shadow. We investigate the photon sphere and the shadow of a black hole with Barrow entropy, and assuming a simple model for infalling and radiating gas we estimate the corresponding intensity. Furthermore, we use the radius in order to extract the real part of the quasinormal modes, and for completeness we investigate the spherical accretion of matter onto the black hole, focusing on isothermal and polytropic test fluids. We extract the allowed parameter region, and by applying a Monte-Carlo-Markov Chains analysis we find that $\Delta \simeq {0.0036}_{-0.0145}^{+0.0792}$. Hence, our results place the upper bound $\Delta \lesssim 0.0828$ at 1$\sigma$, a constraint that is less strong than the Big Bang Nucleosynthesis one, but significantly stronger than the late-time cosmological constraints.

1. Introduction

Black holes are currently the leading astrophysical laboratories for testing general relativity as well as theories of modified and quantum gravity. In particular, recent advances in optical, radio, X-ray and gravitational wave astronomy [1,2,3] have confirmed the presence of supermassive black holes in the galactic centers of giant elliptical and spiral galaxies, as well as small astrophysical black holes. Due to the observation of the first radio images of the supermassive black hole that exists at the center of the M87* galaxy, by Event Horizon Telescope (EHT), black-hole shadows have become a very useful tool to test general relativity and examine whether possible deviations due to gravitational modifications [4,5] could indeed be the case. In such researches, one first calculates the shadows of various black hole solutions [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] and then confronts them with the M87* data [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65].

On the other hand, one of the most intriguing discoveries is the theoretical connection between thermodynamics and gravity, which may play a significant role to understand more deeply the nature of black holes. In the classical relativistic picture, black holes can decrease the entropy of the universe by swallowing objects and therefore violating the second law of thermodynamics. To resolve this problem, Bekenstein [66] conjectured that black holes should have entropy. This idea was shown by Hawking using the semi-classical approach to be correct, and it was found that black holes radiate away energy and consequently the external observer would associate a temperature to the black hole horizon [67]. The laws of black hole thermodynamics relate the horizon temperature with the surface gravity. Hence, the black hole entropy, namely the Bekenstein-Hawking entropy, is given by $S_B=A/4$, where $S_B$ is the entropy and A the surface area of the black hole (in units where $\hslash =G=c=1$).

Recently, Barrow argued that quantum-gravitational effects induce a fractal structure on the black hole horizon, which then acquires spatial dimension more than two but less than three, quantified by the parameter $\Delta$ [68]. Hence, such a complex structure leads to a modification of the black hole entropy. This idea may have interesting consequences in cosmological and holographic applications [69,70,71,72,73,74,75,76,77,78,79,80]. Nevertheless, it also has interesting implications on the black hole properties itself, since it changes the black hole temperature too [81,82,83].

In this work we are interested in extracting constraints on the Barrow exponent $\Delta$, using data form the M87* central black hole shadow, as well as from the S2 star observations. The manuscript is organized as follows. In Section 2, we review Barrow entropy. In Section 3 we apply the involved expressions in order to find the black hole properties and the shadow images. Moreover, in Section 4 we use the M87* observations and we analyze the motion of the S2 star orbit to fit the data and improve the constraints on the Barrow parameter. Finally, in Section 5 we conclude. For completeness, in the Appendix A, we consider the spherical accretion of isothermal and polytropic fluids onto black holes with Barrow entropy.

2. Black Holes with Barrow Entropy

Barrow proposed a modification of Bekenstein-Hawking black hole entropy induced by quantum gravity effects on its horizon [68]. The corresponding corrections change the exponent of the entropy-area law, leading to

$$S_B={\left(\frac{A}{4}\right)}^{1+\frac{\Delta }{2}},$$

(1)

where $\Delta$ is the new parameter, and with A the usual area of the black hole’s event horizon. $\Delta$ is restricted to the interval $0\le \Delta \le 1$, with $\Delta =0$ giving the standard Bekenstein–Hawking entropy, while $\Delta =1$ corresponding to the maximal deformation of the horizon structure.

In this work we will focus on Schwarzschild black hole solutions, with metric

$$d{s}^2=-f\left(r\right)d{t}^2+\frac{d{r}^2}{f\left(r\right)}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right).$$

(2)

If the mass parameter is M, the corresponding horizon is $r_H=2M$, and as usual we can express its area as $A=4\pi r_H^2=16\pi M^2$. In this case (1) can be re-written as $S_B\left(M\right)={\left(4\pi M^2\right)}^{1+\frac{\Delta }{2}}$. Hence, using that $\frac{1}{T}=\frac{\partial S_B}{\partial M}$, one can find the modified black hole temperature [84] arising from the modified Barrow entropy as [81].

$$T_B=\frac{1}{(\Delta +2){\left(4\pi \right)}^{1+\frac{\Delta }{2}}{M}^{1+\Delta }}.$$

(3)

In summary, the effect of Barrow entropy is to change the black hole temperature too, while in the case $\Delta =0$ we re-obtain the standard Hawking temperature $T=1/\left(8\pi M\right)$.

Let us proceed by considering a standard Schwarzschild black hole solution that would have the same temperature with the above Barrow temperature. Using the well-known expression for the black hole temperature $T=\frac{f^{\prime }\left(r\right)}{4\pi }{|}_{r={\tilde{r}}_H}$, with ${\tilde{r}}_H$ the horizon, we can easily see that in this case the corresponding metric function should be

$$f\left(r\right)=1-\frac{(\Delta +2)M^{\Delta +1}{\left(4\pi \right)}^{\frac{\Delta }{2}}}{r},$$

(4)

and thus the horizon should be

$$\begin{array}{c}\hfill {\tilde{r}}_H={\left(4\pi \right)}^{\frac{\Delta }{2}}(\Delta +2)M^{\Delta +1},\end{array}$$

(5)

and the mass

$$\tilde{M}={\left(4\pi \right)}^{\frac{\Delta }{2}}\left(\frac{\Delta }{2}+1\right)M^{\Delta +1}.$$

(6)

In the limiting case where Barrow entropy becomes standard Bekenstein-Hawking entropy, i.e., for $\Delta =0$, the above solution becomes the standard one.

Now, it is well known that the Hawking temperature can be also understood geometrically by Wick-rotating the time coordinate $t\to i\tau$ and $r\to {\tilde{r}}_H+\delta r$. Thus,

$$\begin{array}{c}\hfill f\left(r\right)=\frac{r-{\tilde{r}}_H}{r},\end{array}$$

(7)

and then near the horizon we have $f\left(r\right)\simeq f^{\prime }{\left(r\right)|}_{{\tilde{r}}_H}(r-{\tilde{r}}_H)$, and the metric (2) reads [84].

$$\begin{array}{c}\hfill d{s}^2=\frac{\delta r^2}{{\tilde{r}}_H}d{\tau }^2+\frac{{\tilde{r}}_H}{\delta r}d{\left(\delta r\right)}^2+{\tilde{r}}_H^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right).\end{array}$$

(8)

Defining a new radial coordinate $\rho$ as $\rho =2\sqrt{{\tilde{r}}_H\delta r},$ the line element acquires the form

$$d{s}^2\simeq \frac{{\rho }^2}{4{\tilde{r}}_H^2}d{\tau }^2+d{\rho }^2+{\tilde{r}}_H^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right).$$

(9)

In order to avoid the conical singularity we can impose the periodicity of the Euclidean time coordinate $\tau$ as

$$\frac{\tau }{2{\tilde{r}}_H}\sim \frac{\tau }{2{\tilde{r}}_H}+2\pi ,$$

(10)

and then we can identify the inverse of the period of the Euclidean time coordinate to correspond to the temperature [84]. In particular, in the Euclidean path integral formulation we can make the identification for the finite temperature field theory using the relation.

$$\int \left[D\varphi \right]{\mathrm{e}}^{{\int }_0^{t_0}dtL\left(\varphi \right)}=\mathrm{Tr}\left({\mathrm{e}}^{-t_{0}H}\right)=\mathrm{Tr}\left({\mathrm{e}}^{-\frac{H}{T}}\right),$$

(11)

which holds for any field $\varphi$, with which one finds that the Schwarzschild black hole (2) has temperature $T=1/\left(4\pi {\tilde{r}}_H\right)$, which using (5) gives exactly a Hawking temperature that coincides with (3).

3. Barrow Entropy Effect on Black Hole Shadows

In this section we use the Schwarzschild-like metric which we found by using the Barrow corrected black hole temperature in order to study the shadow of a black hole possessing Barrow entropy. As it is known, there are two constants of motion for particle motion in spherically symmetric geometry, due to the existence of the timelike and spacelike Killing vectors, namely the energy E and the angular momentum L of the particle, in our case photon, respectively. Following the standard procedure it is straightforward to obtain the equations of motion for the photon [12].

$$\begin{array}{ccc}\hfill \frac{dt}{d\lambda }& =& \frac{E}{f\left(r\right)},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill \frac{dr}{d\lambda }& =& \frac{\sqrt{R\left(r\right)}}{r^2},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \frac{d\theta }{d\lambda }& =& \frac{\sqrt{\mathsf{\Theta }\left(\theta \right)}}{r^2},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \frac{d\varphi }{d\lambda }& =& \frac{L{csc}^2\theta }{r^2},\hfill \end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill R\left(r\right)& \equiv & E^2{r}^4-(\mathcal{K}+L^2)r^{2}f\left(r\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \mathsf{\Theta }\left(\theta \right)& \equiv & \mathcal{K}-L^2{csc}^2\theta {cos}^2\theta ,\hfill \end{array}$$

(17)

where $\mathcal{K}$ is a constant of integration known as the Cartan constant. It simply follows from the separation of the Hamilton-Jacobi equations into a radial part and a polar part setting each part equal to $\mathcal{K}$ [85]. Using the above equations we can further study the radial geodesics by introducing the effective potential $V_{\mathrm{eff}}\left(r\right)$ as follows

$${\left(\frac{dr}{d\lambda }\right)}^2+V_{\mathrm{eff}}\left(r\right)=0,$$

(18)

where

$$V_{\mathrm{eff}}\left(r\right)=-1+\frac{f\left(r\right)}{r^2}({\xi }^2+\eta ),$$

(19)

and

$$\xi =\frac{L}{E},\eta =\frac{\mathcal{K}}{E^2}.$$

(20)

We can use the two parameters $\xi$ and $\eta$ in order to analyze the motion of photons around the black hole. Since we are interested to explore the effect of the Barrow parameter on the shadow of the black hole, we need to use the conditions for unstable orbit. As we know, in the observer’s sky, we can observe the black hole shadow due to the fact that some of the scattered photons escape from the black hole and some of the photons are captured by the black hole geometry. In other words, the black hole shadow is obtained as a union of the dark spots in the observer’s sky. While it is straightforward to see that the critical orbits are characterized by certain critical values in terms of the impact parameters $\xi$ and $\eta$. To determine the critical orbits or the unstable circular photon orbits, we simply need to study the effective potential, that is we need to find the maximum of the effective potential $V_{\mathrm{eff}}$ yielding the unstable orbits. Thees unstable circular photon orbits can be obtained by applying the following conditions:

$$V_{\mathrm{eff}}=0,V_{\mathrm{eff}}^{\prime }\left(r\right)=0,V_{\mathrm{eff}}^{″}\left(r\right)\le 0.$$

Using Equations (13) and (18) it is easy to combine $V_{\mathrm{eff}}$ and $R\left(r\right)$. If we express the above conditions in terms of $R\left(r\right)$ we obtain:

$$R\left(r\right)=0,\frac{dR\left(r\right)}{dr}=0,\frac{d^{2}R\left(r\right)}{d{r}^2}>0.$$

(21)

In terms of the above conditions one can easily show that the photon radius is determined by the following algebraic condition

$$2f\left(r\right)-r{f}^{\prime }\left(r\right)=0.$$

(22)

By solving (22) under (4), we obtain a simple relation for the radius of the photon sphere $r_{\mathrm{ph}}$ given by

$$\begin{array}{c}\hfill r_{\mathrm{ph}}=\frac{3}{2}(2+\Delta )M^{\Delta +1}{\left(4\pi \right)}^{\frac{\Delta }{2}}=\frac{3}{2}{\tilde{r}}_H.\end{array}$$

(23)

The radius of the photon sphere can be used to find the size of black hole shadow. In order to describe the shadow as seen by large distances, one introduces the two celestial coordinates X and Y [85], namely $X={lim}_{r_{*}\to \infty }\left(-r_{*}^{2}sin{\theta }_0\frac{d\varphi }{dr}\right)$ and $Y={lim}_{r_{*}\to \infty }{r}_{*}^2\frac{d\theta }{dr}$, with $r_{*}$ the distance between the black hole and the observer, and ${\theta }_0$ the inclination angle between the observer’s line of sight and the black hole rotational axis. Using the geodesics equations we finally obtain [86].

$$\begin{array}{ccc}& & X=-\xi \left(r_{\mathrm{ph}}\right)csc{\theta }_0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & Y=\sqrt{\eta \left(r_{\mathrm{ph}}\right)-{\xi }^2\left(r_{\mathrm{ph}}\right){cot}^2{\theta }_0},\hfill \end{array}$$

(25)

and thus we have $X^2+Y^2={\xi }^2\left(r_{\mathrm{ph}}\right)+\eta \left(r_{\mathrm{ph}}\right)$. Hence, the event horizon (i.e., shadow) radius $R_{\mathrm{sh}}$ can finally be found as [12].

$$R_{\mathrm{sh}}\left(r_{\mathrm{ph}}\right)=\sqrt{{\xi }^2\left(r_{\mathrm{ph}}\right)+\eta \left(r_{\mathrm{ph}}\right)}=\frac{r_{\mathrm{ph}}}{\sqrt{f\left(r_{\mathrm{ph}}\right)}},$$

(26)

which explicitly yields

$$\begin{array}{c}\hfill R_{\mathrm{sh}}=3\sqrt{3}\left(\Delta +2\right)2^{\Delta -1}{M}^{\Delta +1}{\left(4\pi \right)}^{\frac{\Delta }{2}}.\end{array}$$

(27)

We can see that the event horizon radius is expected to increase due to the effect of quantum gravity corrections, since $M>0$ and $\Delta \ge 0$.

We continue by using the inverse relationship between $R_{\mathrm{sh}}$ and the real part of quasinormal modes given by [87,88].

$${\omega }_{\Re }=\underset{l\gg 1}{lim}\frac{l+\frac{1}{2}}{R_{\mathrm{sh}}},$$

(28)

with l the multipole numbers, which in our case gives

$$\begin{array}{c}\hfill {\omega }_{\Re }=\underset{l\gg 1}{lim}\frac{l+\frac{1}{2}}{3\sqrt{3}\left(\Delta +2\right)2^{\Delta -1}{M}^{\Delta +1}{\left(4\pi \right)}^{\frac{\Delta }{2}}}.\end{array}$$

(29)

In

Table 1. The photon sphere radius $r_{\mathrm{ph}}$, the event horizon radius $R_{\mathrm{sh}}$ and the real part of quasinormal modes ${\omega }_{\Re }$, for different values of $\Delta$, with $M=1$ and $l=1$.

$\mathbf{\Delta }$	${\mathit{r}}_{\mathbf{ph}}$	${\mathit{R}}_{\mathbf{sh}}$	${\mathit{\omega }}_{\mathit{\Re }}$
0	3	5.196152424	0.5000000000
0.001	3.005300838	5.205333745	0.4991180853
0.005	3.026590476	5.242208479	0.4956071896
0.008	3.042648663	5.270022075	0.4929915236
0.010	3.053397641	5.288639851	0.4912560290
0.030	3.162827055	5.478177156	0.4742592542
0.050	3.275860255	5.673956398	0.4578949904
0.080	3.452414952	5.979758107	0.4344784798
0.100	3.574958849	6.192010363	0.4195852494
0.120	3.701516690	6.411214974	0.4052392912
0.150	3.899154632	6.753533931	0.3846987723
0.170	4.036303659	6.991083014	0.3716271436
0.200	4.250450140	7.361995599	0.3529037985

we present the numerical values for the photon radius, the values for the shadow radius, and the real part of quasinormal modes by varying the Barrow parameter. One can see that while the shadow radius increases by increasing $\Delta$, the value of ${\omega }_{\Re }$ decreases. As we already mentioned, relation (29) is precise in the eikonal limit, namely $l\to \infty$, however it has been shown that in many cases it gives satisfactory results even for small l, which are most important for observations [87,88]. Finally, the decrease in ${\omega }_{\Re }$ is therefore simply explained from the inverse relation between the real part of the quasinormal modes and the shadow radius, according to (28).

We close this section by considering the scenario where the black hole is surrounded by an infalling/radiating accretion flow. Via this simple model, we can extract valuable information about the intensity of the radiation which can be detected by a distant observer. In order to achieve this we need to estimate the specific intensity at the observed photon frequency ${\nu }_{\mathrm{obs}}$ at the point $(X,Y)$ of the observer’s image [37,47,89,90,91,92].

$$\begin{array}{c}\hfill I_{obs}({\nu }_{obs},X,Y)={\int }_{\gamma }{\mathrm{g}}^{3}j\left({\nu }_e\right)d{l}_{\mathrm{prop}}.\end{array}$$

(30)

The freely falling gas has the four-velocity components written as

$$\begin{array}{c}\hfill u_e^{\mu }=\left(\frac{1}{f\left(r\right)},-\sqrt{1-f\left(r\right)},0,0\right),\end{array}$$

(31)

with $f\left(r\right)$ given in (4). In addition we need to use the condition $p_{\mu }{p}^{\mu }=0$, from which one can easily show that

$$\begin{array}{c}\hfill \frac{p^r}{p^t}=\pm f\left(r\right)\sqrt{f\left(r\right)\left(\frac{1}{f\left(r\right)}-\frac{b^2}{r^2}\right)},\end{array}$$

(32)

with b the impact parameter. It is important to mention here that sign $+(-)$ describes the case when the photon approaches (or draws away) from the black hole. The redshift function $\mathrm{g}$ can be calculated using [37,47,89,90,91,92].

$$\begin{array}{c}\hfill \mathrm{g}=\frac{p_{\mu }{u}_{obs}^{\mu }}{p_{\nu }{u}_e^{\nu }},\end{array}$$

(33)

with $u_{obs}^{\mu }$ the 4-velocity of the observer. For the specific emissivity we assume a simple model in which the emission is monochromatic, with emitter’s-rest frame frequency ${\nu }_{☆}$, and the emission has a $1/r^2$ radial profile:

$$\begin{array}{c}\hfill j\left({\nu }_e\right)\propto \frac{\delta ({\nu }_e-{\nu }_{\star })}{r^2},\end{array}$$

(34)

where $\delta$ denotes the Dirac delta function. Expressing the proper length in terms of radial coordinate for observed flux, we find

$$\begin{array}{c}\hfill F_{obs}(X,Y)\propto -{\int }_{\gamma }\frac{{\mathrm{g}}^3{p}_t}{r^2{p}^r}dr.\end{array}$$

(35)

In order to show all the above in a more transparent way, in Figure 1 we present the black hole shadow for fixed M and various values of Barrow exponent $\Delta$, according to (27). Additionally, we have numerically calculated and depicted the intensity from (35). As we observe, with increasing Barrow parameter the size of the shadow increases, while the intensity decreases.

Lastly, since we have extracted the black hole profile and properties we can straightforwardly investigate the accretion of matter onto it. For completeness, we provide this analysis in the Appendix A.

4. Observational Constraints on the Barrow Parameter

In this section we proceed to the use of the Event Horizon Telescope observations for the shadow of the M87${}^{*}$ central black hole in order to impose constraints on the Barrow parameter $\Delta$. As we will see, this will not be edequate and thus we need to incorporate additional data from the S2 star orbit observations [93,94].

The M87${}^{*}$ central black hole has angular diameter ${\theta }_{\mathrm{sh}}=(42\pm 3)\mu as$, is at distance $D=16.8$ Mpc, and its mass is $(6.5\pm 0.9)\times {10}^9{M}_{\odot }$. We equate this to $\tilde{M}$ given in (6) in terms of the parameters $\Delta$ and M. Thus, we treat M as a parameter and not the true mass of the system. From a theoretical point of view this can be advantageous, since spherical solutions may be modeled differently, where each theoretical model introduces a set of parameters that have to be constrained to fit observational data. In this work we model M87${}^{*}$ as a Barrow quantum-corrected two-parameter black hole, while one could model it using alternative theories of gravity too (see e.g., [95,96]).

Combining the observational parameters allows us to introduce the single quantity $d_{M87*}$, which accounts for the size of the M87*’s shadow in unit mass, as [48].

$$\begin{array}{c}\hfill d_{M87}=\frac{D{\theta }_{\mathrm{sh}}}{M_{87}}=11.0\pm 1.5.\end{array}$$

(36)

In particular within $1\sigma$ confidence level one has the range $9.5\le d_{M87}\le 12.5$.

Let us now use the theoretically predicted shadows of the previous section, in order to calculate the predicted diameter per unit mass $d_{\mathrm{sh}}$ for black holes with Barrow entropy. In Figure 2 we depict $d_{\mathrm{sh}}$ as a function of $\Delta$, for fixed $M=1$, alongside the observational bounds according to (36).

Nevertheless, as one can see, in general the results depend on both $\Delta$ and M. Indeed, in Figure 3 we present the parameter region which is consistent with M87* data.

Additionally, in Figure 4 we present the predicted combined diameter $d_{\mathrm{sh}}$ as a function of M and $\Delta$.

In order to break the degeneracy, and constrain $\Delta$ more efficiently, we have to use the S2 star orbit data [93,94]. In particular, using solution (4), we can study the motion of the S2 star restricted in the equatorial plane $(\theta =\pi /2,\dot{\theta }=0)$. From the Lagrangian it follows that

$$\begin{array}{ccc}\hfill 2\mathcal{L}& =& -f\left(r\right){\dot{t}}^2+\frac{{\dot{r}}^2}{f\left(r\right)}+r^2{\dot{\varphi }}^2.\hfill \end{array}$$

For the two constants of motion, namely total energy E and total angular momentum L of the star, we have $\frac{\partial \mathcal{L}}{\partial \dot{t}}=-E$ and $\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}=L$. Using the above we find that

$$\dot{t}=\frac{E}{1-\frac{\left(\Delta +2\right)M^{\Delta +1}{\left(4\pi \right)}^{\frac{\Delta }{2}}}{r}},$$

(37)

along with $\dot{\varphi }=\frac{L}{r^2}$. Finally, we have the following equation of motion for a massive particle (S2 star in our case) [97,98,99].

$$\ddot{r}={\displaystyle \frac{1}{2g_{11}\left(r\right)}}\left[g_{00,r}\left(r\right){\dot{t}}^2+g_{11,r}\left(r\right){\dot{r}}^2+g_{33,r}\left(r\right){\dot{\varphi }}^2\right].$$

(38)

In general, one cannot find an analytical expression for $r\left(\varphi \right)$ and, therefore, one must elaborate numerically the equations of motion. In the present work we apply the Bayesian theorem with the likelihood function as given in [99,100], with the observational data for $(X_{obs},Y_{obs})$ given in [94,97], considering $\Delta$ and M as free parameters. In order to find the best-fit values we use the Monte-Carlo-Markov Chains analysis. For the central mass object we take $4.1\times {10}^6{M}_{\odot }$ along with the uniform priors $0<\Delta <1$ and $0<M<2$. In Figure 5 we present the region of the parameter space in agreement with S2 star data. Concerning Barrow parameter, in which we are interested in this manuscript, the best fit value and 1$\sigma$ errors are

$$\Delta \simeq {0.0036}_{-0.0145}^{+0.0792},$$

(39)

which is the main result of the present work.

We can now combine the above result with the black hole shadow. In particular, applying the best-fit parameters we easily find the shadow radius $R_{\mathrm{sh}}=5.3127$, measured in units of black hole mass. Hence, in Figure 6 we depict the shadow image and intensity for a black hole with Barrow entropy, for the best-fit values of (39) and Figure 5.

In summary, as we observe, although the standard value $\Delta =0$, in which Barrow entropy becomes Bekenstein-Hawking entropy, lies inside the obtained 1$\sigma$ region, the best-fit value is $\Delta =0.0036$, while the 1$\sigma$ upper bound is $\Delta \lesssim 0.0828$. Such constraint is stronger than the late-time cosmological ones from Supernovae (SNIa) Pantheon sample and cosmic chronometers (CC) datasets, namely $\Delta \lesssim 0.188$ [101,102], but less strong than the Big Bang Nucleosynthesis (BBN) one, namely $\Delta \lesssim 1.4\times {10}^{-4}$ [103], since the latter is known to lead to very strong constraints. Hence, it reveals the capabilities of black hole shadow and S2 star observations, since they can lead to significantly improved constraints although the data points are for the currently relatively few.

5. Conclusions

In this work we used data from M87* central black hole shadow, as well as from the S2 star observations, in order to extract constraints on Barrow entropy. The latter is a modified entropy relation arising from quantum-gravitational effects that induce a intricate, fractal structure on the black hole horizon, quantified by the new Barrow parameter $\Delta$. Such a change in entropy leads to a change in temperature, as well as to the properties of the black hole and its shadow.

We investigated the photon sphere and the shadow of a black hole with Barrow entropy, and assuming a simple model for infalling and radiating gas we estimated the corresponding intensity. Furthermore, we used the radius in order to extract the real part of the quasinormal modes, and for completeness we investigated the spherical accretion of matter onto the black hole, focusing on isothermal and polytropic test fluids.

We used the EHT data from the M87* black hole extracting the allowed parameter region, and then we additionally incorporated data from the motion of S2 star around the Sgr A* black hole, through a Monte-Carlo-Markov Chains analysis, in order to break the degeneracies and extract the final constraints on the Barrow exponent. We found that $\Delta \simeq {0.0036}_{-0.0145}^{+0.0792}$ at $1\sigma$ confidence level. Hence, our analysis places the upper bound $\Delta \lesssim 0.0828$, a constraint that is less strong than the Big Bang Nucleosynthesis (BBN) one, but significantly stronger than the late-time cosmological ones.

In summary, black-hole related data can serve as a new tool in order to test general relativity and examine if modifications of various kinds are allowed. Although the data points are currently few, they can be very efficient in constraining the theoretical parameters. The significant improvement of the datasets expected in the near future makes the corresponding analyses both interesting and necessary.

Although the M87* is shown and expected to rotate, the Sgr A* black hole might rotate very slowly compared to M87*. In most applications pertaining to Sgr A* rotation is dropped from consideration as in Refs. [98,99,104]. In that sense, the constraint we found for the S2 star is justified by assuming a nonrotating black hole in our galaxy. Rotation will be considered in a subsequent work.