key: cord-0640248-wppciseh
authors: Vagnozzi, Sunny; Roy, Rittick; Tsai, Yu-Dai; Visinelli, Luca
title: Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A$^*$
date: 2022-05-16
journal: nan
DOI: nan
sha: 2c569b64f3a16343df9df5fed167deadd39ddd56
doc_id: 640248
cord_uid: wppciseh

Horizon-scale images of black holes (BHs) and their shadows have opened an unprecedented window onto tests of gravity and fundamental physics in the strong-field regime, allowing us to test whether the Kerr metric provides a good description of the space-time in the vicinity of the event horizons of supermassive BHs. We consider a wide range of well-motivated deviations from classical General Relativity solutions, and constrain them using the Event Horizon Telescope (EHT) observations of Sagittarius A$^*$ (SgrA$^*$), connecting the size of the bright ring of emission to that of the underlying BH shadow and exploiting high-precision measurements of SgrA$^*$'s mass-to-distance ratio. The scenarios we consider, and whose fundamental parameters we constrain, include various regular BH models, string- and non-linear electrodynamics-inspired space-times, scalar field-driven violations of the no-hair theorem, alternative theories of gravity, new ingredients such as the generalized uncertainty principle and Barrow entropy, and BH mimickers including examples of wormholes and naked singularities. We demonstrate that SgrA$^*$'s image places particularly stringent constraints on models predicting a shadow size which is larger than that of a Schwarzschild BH of a given mass: for instance, in the case of Barrow entropy we derive constraints which are significantly tighter than the cosmological ones. Our results are among the first tests of fundamental physics from the shadow of SgrA$^*$ and, while the latter appears to be in excellent agreement with the predictions of General Relativity, we have shown that various well-motivated alternative scenarios (including BH mimickers) are far from being ruled out at present.

Black holes (BHs) are among the most extreme regions of space-time, and are widely believed to hold the key towards unraveling various key aspects of fundamental physics, including the behavior of gravity in the strongfield regime, the possible existence of new fundamental degrees of freedom, the unification of quantum mechanics and gravity, and the nature of space-time itself [1] . We have now been ushered into an era where BHs and their observational effects are witnessed on a regular basis on a wide range of scales. Perhaps the most impressive example in this sense are horizon-scale images of supermassive BHs (SMBHs), delivered through Very Long Baseline Interferometry (VLBI), and containing information about the space-time around SMBHs. The first groundbreaking SMBH images were delivered by the Event Horizon Telescope (EHT), a millimeter VLBI array with Earthscale baseline coverage [2] , which in 2019 resolved the near-horizon region of the SMBH M87 * [3] [4] [5] [6] [7] [8] , before later revealing its magnetic field structure [9, 10] . This was recently followed by the EHT's first images of Sagittarius A * (SgrA * ), the SMBH located at the Milky Way center [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] .

The main features observed in VLBI horizon-scale images of BHs are a bright emission ring surrounding a central brightness depression, with the latter related to the BH shadow [21, 22] . On the plane of a distant observer, the boundary of the BH shadow marks the apparent image of the photon region (the boundary of the region of space-time which supports closed spherical photon orbits), 1 and separates capture orbits from scattering orbits: for detailed reviews on BH shadows, see e.g. Refs. [23] [24] [25] . Under certain conditions and after appropriate calibration, the radius of the bright ring can serve as a proxy for the BH shadow radius, with very little dependence on the details of the surrounding accretion flow [7, 8, 22, [26] [27] [28] [29] [30] [31] [32] . 2 This is possible if i) a bright source 1 In the case of spherical symmetry, as with Schwarzschild BHs, the photon region is simply referred to as photon sphere. 2 Some concerns on whether VLBI horizon-scale images are really seeing the image of the BH shadow surrounded by the photon ring were raised in Refs. [33] [34] [35] [36] [37] , but have been addressed in most of the above works. See also theoretical concerns on the possibility of testing gravity using BH images raised in Ref. [38] .

of photons is present and strongly lensed near the horizon, and especially ii) the surrounding emission region is geometrically thick and optically thin at the wavelength at which the VLBI network operates. Most SMBHs we know of, including SgrA * and M87 * , are low-luminosity Active Galactic Nuclei which operate at sub-Eddington accretion rates and are powered by radiatively inefficient advection-dominated accretion flows (ADAFs), satisfying both the previous conditions. The possibility of connecting the ring and BH shadow angular diameters opens up the fascinating prospect of using BH shadows to test fundamental physics, once the BH mass-to-distance ratio is known [39, 40] . For a Schwarzschild BH of mass M located at distance D, the shadow is predicted to be a perfect circle of radius r sh = 3 √ 3GM/c 2 , therefore subtending an angular diameter θ sh = 6 √ 3GM/(c 2 D) = 6 √ 3θ g , where θ g = GM/(c 2 D) is the angular gravitational radius of the BH. 3 For Kerr BHs, the shadow is slightly asymmetric along the spin axis, with r sh being marginally smaller compared to the Schwarzschild case, but still depending predominantly on the BH mass M , and only marginally on the (dimensionless) spin a and observer's inclination angle θ. Crucially, these figures can change considerably for other metrics, including those describing BHs in alternative theories of gravity, the effects of corrections from new physics possibly leading to violations of the no-hair theorem [41] [42] [43] [44] [45] , or "BH mimickers", i.e. (possibly horizonless) compact objects other than BHs [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] . This paves the way towards tests of fundamental physics from the sizes of BH shadows of known mass-to-distance ratio.

With the EHT image of M87 * , the prospects of testing fundamental physics with BH shadows have become reality, as demonstrated by a large and growing body of literature devoted to such tests, mostly focusing on the size of M87 * 's shadow, but in some cases also considering additional observables such as the shadow circularity and axis ratio (see e.g. Refs. ). The new horizon-scale image of SgrA * delivered by the EHT offers yet another opportunity for performing tests of gravity and fundamental physics in the strong-field regime, which we shall here exploit. Even though they have already been performed with M87 * , there are very good motivations, or even advantages, for carrying out these tests from the shadow of SgrA * (see also Refs. [133] [134] [135] [136] [137] ):

1. SgrA * 's proximity to us makes it significantly easier to calibrate its mass and distance, and therefore its mass-to-distance ratio, whose value is crucial to connect the observed angular size of its ring to theoretical predictions for the size of its shadow within different fundamental physics scenarios. This is a distinct advantage with respect to M87 * , whose mass is the source of significant uncertainties, with measurements based on stellar dynamics [138] or gas dynamics [139] differing by up to a factor of 2 (see also Sec. IIIA of Ref. [140] ).

2. SgrA * 's mass in the O(10 6 )M range is several orders of magnitude below M87 * 's mass in the O(10 9 )M range, allowing us to probe fundamental physics in a completely different and complementary regime (see e.g. the potential-curvature parameter space of Ref. [141] ). For the same reason, SgrA * 's shadow can potentially set much tighter constraints on dimensional fundamental parameters, which scale as a positive power of mass in the units we adopt (as is the case for several BH charges or "hair parameters").

3. Finally, independent constraints from independent sources are always extremely valuable in tests of fundamental physics, and there is, therefore, significant value in performing such tests on SgrA * independently of the results obtained from M87 * . As we shall explicitly show, unlike M87 * , the EHT image of SgrA * sets particularly stringent constraints on theories and frameworks which predict a shadow radius larger than 3 √ 3M , including the Kazakov-Solodukhin space-time, the mutated Reissner-Nordström wormhole and the negative tidal charge regime of the Randall-Sundrum II model, the BH with scalar hair considered in Sec. III O, Einstein-aether gravity, the generalized uncertainty principle, and Barrow's modification to the Bekenstein-Hawking entropy.

The rest of this paper is then organized as follows. In Sec. II we discuss the methodology and assumptions entering into the computation of the sizes of BH shadows, and give a very brief overview of the fundamental physics scenarios we consider. Sec. III is divided into a large number of subsections, one for each of the fundamental physics scenarios considered, for which we report constraints obtained adopting the previously discussed methodology. Finally, in Sec. IV we draw concluding remarks and outline future directions.

The methodology we shall follow relies on comparing the observed angular radius of the ring-like feature in the EHT horizon-scale image of SgrA * with the theoretically computed radius of the shadows of BHs (or other alternative compact space-times) within each fundamental physics scenario we consider, with prior knowledge of SgrA * 's mass-to-distance ratio. Requiring consistency between the two quantities, within the uncertainty allowed by the EHT observation, allows us to set constraints on the parameters describing the space-time in question. This simple methodology has been discussed several times in the past [39, 40] and has been successfully applied to the EHT image of M87 * , though for reasons discussed earlier in Sec. I, the application to SgrA * is in principle more robust.

To apply this methodology, we require two ingredients. The first is SgrA * 's mass-to-distance ratio. The mass and distance to SgrA * , M , and D, have been studied in detail over the past decades exploiting stellar cluster dynamics, and in particular the motion of S-stars, individual stars resolved within 1 of the Galactic Center. A significant role has been played by S0-2 which, with a K-band magnitude of ∼ 14, period of ∼ 16 yrs, and semimajor axis of ∼ 125 mas, is the brightest star with a relatively close orbit and short period close to the Galactic Center. Its orbit has been exquisitely tracked by two sets of instruments/teams, 4 which we shall refer to as "Keck" and "VLTI" respectively. Following Ref. [16] , we adopt the following (correlated) mass and distance estimates as in Table I , where uncertainties are quoted at 68% confidence level (C.L.) and the measurements for D report both the statistical and systematic uncertainties. We refer the reader to Sec. 2.1 of Ref. [16] for more detailed discussions on these measurements.

The second ingredient we require is a calibration factor connecting the size of the bright ring of emission with the size of the corresponding shadow, which quantifies how safe it is to use the size of the bright ring of emission as a proxy for the shadow size. This calibration factor depends on the near-horizon physics of image formation and, as already anticipated earlier, is expected to be very close to unity for optically thin, geometrically thick radiatively inefficient advection-dominated accretion flows, such as the one surrounding SgrA * . In practice, the calibration factor accounts multiplicatively for various sources of uncertainty, ranging from formal measurement uncertainties, to fitting/model uncertainties, to theoretical uncertainties pertaining to the emissivity of the plasma (see Sec. 3 of Ref. [16] ).

Detailed studies of various sources of uncertainty have been conducted by the EHT in Ref. [16] and used to determine the above calibration factor. Folding in the calibration factor with uncertainties in SgrA * 's massto-distance ratio, and the angular diameter of SgrA * 's bright ring of emission, the EHT inferred δ, the fractional deviation between the inferred shadow radius and 4 The motion of the S0-2 star has been used to set constraints on various aspects of fundamental physics, see e.g. Refs. [142] [143] [144] [145] [146] .

the shadow radius of a Schwarzschild BH of angular size θ (θ sh = 3 √ 3θ). The inferred value of δ depends on the mass-to-distance ratio assumed, with the Keck and VLTI measurements resulting in the following estimates [16] :

• Keck: δ = −0.04 +0.09 −0.10 ;

• VLTI: δ = −0.08 +0.09 −0.09 , with good agreement between the two bounds.

In order to be as conservative as possible, and to simplify our later discussion, we take the average of the Keckand VLTI-based estimates of δ, treating them as uncorrelated. This leads to the following estimate of δ which we shall adopt throughout this work:

As is clear from the previous Keck-and VLTI-based estimates which we have averaged, there is overall a very slight preference for SgrA * 's shadow being slightly smaller than the prediction 3 √ 3M for a Schwarzschild BH of given mass, with δ < 0 preferred at 68% CL. At the level of shadow radius r sh and assuming Gaussian errors, the bound in Eq. (1) leads to the following 1σ constraints:

as well as the following 2σ constraints:

These constraints are in good agreement with those reported in Ref. [16] , though with slightly smaller uncertainties (by a factor of ≈ √ 2) as a result of having taken the average between the Keck-and VLTI-based estimates. In what follows, we shall use the bounds in Eqs. (2)-(3) to constraint the parameters governing space-times beyond the Schwarzschild BH.

For simplicity, throughout this work we shall restrict ourselves to spherically symmetric metrics, i.e. neglecting the effect of spin. The reason behind this choice is two-fold. First, the effect of spin on the shadow radius is small: we illustrate this in Fig. 1 for the case of a Kerr BH, where we plot the predicted shadow radius as a function of the dimensionless spin a , for different values of the observer's inclination angle i. Clearly, the effect is more pronounced for high inclination angles (i.e. for almost edge-on viewing), but remains small ( 12%).

Second, and perhaps most importantly, there is no clear consensus on the value of SgrA * 's spin and inclination angle. The EHT images are in principle consistent with large spin and low inclination angle, but are far from being inconsistent with low spin and large inclination angle [11, 15] . Independent works based on radio, infrared, and X-ray emission, as well as millimeter VLBI, exclude extremal spin (1 − a 1), but have been unable to place strong constraints otherwise [149] [150] [151] . Estimates based on semi-analytical models, magnetohydrodynamics simulations, or flare emissions, have reported constraints across a wide range of spin values [152, 153] . One of the most recent estimates of SgrA * 's spin was reported in Ref. [154] , based on the impact on the orbits of the S-stars of frame-dragging precession, which would tend to erase the orbital planes in which the Sstars formed and are found today: these considerations require SgrA * 's spin to be very low, a 0.1. Constraints on the inclination angle are even more uncertain. The inconsistency across different estimates of SgrA * 's spin prompts us to a conservative approach where we neglect the effect of spin, while taking the very recent estimate of Ref. [154] as indication that the spin may be low: for a 0.1, as Fig. 1 clearly shows, the effect of spin on the shadow radius is negligible at all inclination angles.

Finally, shadow-based observables which are most sensitive to the spin are those related to the shadow's circularity, such as the deviation from circularity ∆C discussed in Ref. [50] and adopted by several works which used the shadow of M87 * to constrain fundamental physics, which for a Kerr BH should be 4%, increasing with spin and inclination angle. However, the sparse interferometric coverage of the EHT's 2017 observations of SgrA * , and the associated significant uncertainties in circularity measurements, prevented the collaboration from quantifying the circularity of the shadow based on observational features. In closing, we note that most of the space-times considered in Ref. [16] are also spherically symmetric, for reasons similar to the ones we outlined, providing an independent validation of our approach.

A. Black hole shadow radius in spherically symmetric space-times

Here we briefly review the calculation of the shadow radius in spherically symmetric space-times. If such a space-time possesses a photon sphere, the gravitationally lensed image thereof as viewed by an observed located at infinity will constitute the BH shadow. 5 Let us consider a generic static, spherically symmetric, asymptotically flat space-time, i.e. one admitting a global, nonvanishing, time-like, hypersurface-orthogonal Killing vector field, with no off-diagonal components in the matrix representation of the metric tensor. Without loss of generality, the space-time line element in Boyer-Lindquist coordinates can be expressed as:

where dΩ is the differential unit of solid angle. In the following, we shall henceforth refer to the function A(r) = −g tt (r) as the "metric function". If the metric in Eq. (4) possesses a photon sphere, the radial coordinate thereof, r ph , is given by the solution to the following implicit equation:

The shadow radius r sh corresponds to the gravitationally lensed image of the surface defined by r ph , and is therefore given by (see e.g. Refs. [23-25, 103, 155] ):

In the case of the Schwarzschild metric, where the metric function is A(r) = 1−2M/r [156] , combining Eqs. (5)- (6) leads to the well-known result r sh = 3 √ 3M . For obvious symmetry reasons, the BH shadow for a spherically symmetric space-time is a circle of radius r sh on the plane of a distant observer, irrespective of the inclination angle.

Note, that the computation we have outlined is strictly valid for asymptotically flat space-times only. For other metrics, e.g. those matching on to the cosmological accelerated expansion at large distances hence asymptotically de Sitter, the size of the BH shadow can explicitly depend on the radial coordinate of the distant observer (see e.g. Refs. [25, 157] ). An example of such a situation occurs with the Kottler metric. In addition, the computation outlined does not (necessarily) hold in theories with electromagnetic Lagrangian other than the Maxwell one (for instance non-linear electrodynamics), since in this context photons do not necessarily move along null geodesics of the metric tensor, but along null geodesics of an effective geometry, a fact which has only been recently appreciated in the literature. We will nonetheless consider certain classes of non-linear electrodynamics theories in this work, carefully accounting for this effect, in Sec. III L and Sec. III E.

In what follows, we will compute the shadow radius of various metrics of interest, and then compare them to the image of SgrA * by imposing the bounds of Eqs. (2,3. These, as discussed earlier, already take into account various observational, theoretical, and modeling uncertainties, including the potential multiplicative offset between the radius of the bright ring of emission and the underlying shadow radius. Before moving on to actually performing this analysis, we will give a very brief overview of the space-times we consider, along with motivation for moving beyond the Schwarzschild space-time.

We will consider a wide range of gravity theories, fundamental physics scenarios, and space-times beyond those of the Schwarzschild BH, for which r sh = 3 √ 3M . The no-hair theorem (NHT) [41] [42] [43] [44] [45] states that the only possible stationary, axisymmetric, and asymptotically flat BH solutions of the vacuum Einstein-Maxwell equations are described by the Kerr-Newman family of metrics [158, 159] . Recall that the latter describes electrically charged, rotating BHs, and reduces to the Kerr metric in the absence of electric charge, to the Reissner-Nordström metric in the absence of rotation, and finally to the Schwarzschild metric in the absence of both charge and rotation. As John Archibald Wheeler phrased it, "black holes have no hair ", with the term "hair" referring generically to parameters other than the BH mass M , spin J, and electric charge Q, which are required for a complete description of the BH solution. Throughout the rest of this paper, by "mass" M we shall refer to the Arnowitt-Deser-Misner (ADM) mass as defined for an asymptotically-flat space-time [160, 161] .

At present, there is no sign of a tension between the NHT and the observations of astrophysical BHs. Why is then going beyond the Kerr-Newman family of metrics a well-motivated endeavor? The first reason is tied to the fact that a wide variety of theoretical and observational issues hint towards the possibility of our understanding of gravity as provided by General Relativity (GR) is likely to be incomplete. These range from the quest for unifying gravity and quantum mechanics (QM), to the conflict between unitary evolution in QM and Hawking BH radiation as encapsulated in the BH information paradox [162] , and finally to cosmological observations requiring the existence of dark matter and dark energy, and possibly a phase of cosmic inflation and/or early dark energy. In the presence of new physics which eventually will address these issues, it is perfectly reasonable to assume that the NHT may only be an approximation, valid to the current level of precision. Moving beyond astrophysical systems, the study of controlled violations of the NHT is well-motivated by developments in our understanding of the gauge/gravity duality [163] , with important applications to various condensed matter systems [164] , including holographic superconductors [165] and quantum liquids [166] .

The second reason is possibly even more fundamental, and is related to the well-known Penrose-Hawking singularity theorems, i.e. the fact that in GR continuous gravitational collapse leads to the inevitable, but at the same time arguably undesirable, appearance of singularities [167] [168] [169] . For instance, the Kerr-Newman family of metrics possesses a well-known physical (noncoordinate) singularity at r = 0. The cosmic censorship conjecture [170, 171] notwithstanding, the mere existence of singularities has prompted a long-standing search for "regular" BH solutions, which regularize the central singularity (see e.g. Refs. for examples of works in these directions). Therefore, testing the metrics of regular BHs, or BH mimickers which address at least in part the existence of singularities, is a very well-motivated direction, particularly at present time with the availability of horizon-scale BH images. Finally, as the nature of dark matter and dark energy is not understood at present, one should be open-minded to the possibility that either or both of these phenomena may be connected to potential modifications to GR.

With these considerations in mind, in this work we will consider a wide range of space-times beyond the Schwarzschild BH, testing them against the EHT horizon-scale image of SgrA * . The space-times we consider broadly speaking fall within these (by no means mutually exclusive) categories:

1. Regular BHs, whether arising from specific theories or constructed in a phenomenological setting;

2. BHs in modified theories of gravity and/or modified electrodynamics;

3. BHs in theories with additional matter fields (typically scalar fields), typically bringing about violations of the no-hair theorem;

4. BH mimickers such as wormholes (or effective wormhole geometries) and naked singularities; 5. Modifications induced by novel fundamental physics frameworks (e.g. the generalized uncertainty principle, or Barrow entropy).

Of course, several space-times we will consider fall within more than one of the above categories at the same time (especially 1. and 2., or 1. and 3.). Most of these spacetimes are described by one or more extra parameters, which in most cases we shall generically refer to as "hair", "hair parameters", or "charges", in the latter case following the language adopted by Refs. [16, 112] .

The first geometry we consider beyond the Schwarzschild one, and arguably the simplest, is the Reissner-Nordström (RN) metric describing an electrically charged, non-rotating BH. Recalling that we are setting 4π 0 = = c = G = 1, and explicitly introducing the BH mass (which we will later set to M = 1 for convenience), the metric function for a BH of electric charge Q is given by [222] [223] [224] [225] 

Setting M = 1, 7 , it is easy to show that this space-time possesses an event horizon only for Q ≤ 1. However, even for larger values of the electric charge the spacetime, which is now a naked singularity, can still possess a photon sphere and thereby cast a shadow, provided 1 < Q ≤ 9/8 ≈ 1.06: this region of parameter space describes what is commonly referred to as the "Reissner-Nordström naked singularity". Using Eqs. (5,6), we straightforwardly obtain the following expression for the radius of the shadow cast by the Reissner-Nordström space-time, which is valid within both the BH and naked singularity regimes:

It is at first glance not obvious that Eq. (8) does indeed reduce to the Schwarzschild result of r sh = 3 √ 3 in the Q → 0 limit, due to the apparent divergence in the denominator. However, the proper way of taking this limit is to Taylor-expand the denominator: when doing so, one indeed finds that the denominator of Eq. (8) tends to 8/3, whereas the numerator straightforwardly tends to 6 √ 2, leading overall to the correct limit of 3 √ 3. The evolution of the shadow radius as a function of the electric charge is shown in Fig. 2 , for both the BH and naked singularity regimes of the Reissner-Nordström space-time, alongside the observational constraints imposed by the EHT image of SgrA * [Eqs. (2, 3) ]. We see that as the electric charge increases, the shadow radius decreases, which can be understood by studying how the electric charge affects the effective potential felt by test particles. This is a feature which is common to many extensions of the Schwarzschild metric, as we shall see throughout the paper: most extensions will lead to a shadow radius which decreases with increasing charge/hair parameter (although notable exceptions exist, which we shall also discuss in this paper). From Fig. 2 , we observe that the EHT observations set the 68% C.L. upper limit Q 0.8, and the 95% C.L. upper limit Q 0.9. Therefore, within 2 standard deviations, the EHT observations rule out the possibility of SgrA * being an extremal Reissner-Nordström BH (Q = 1). Of course, the naked singularity regime (1 < Q ≤ 9/8) is completely ruled out: therefore, the EHT observations rule out the possibility of SgrA * being one of the simplest possible naked singularities (although other naked singularity space-times are allowed as we shall see later). Finally, re-inserting SI units, we see that the previous 2σ upper limit translates to Q 0.9 √ 4π 0 GM ∼ 6.5 × 10 26 C. In practice, this limit is significantly weaker than the limit Q 3 × 10 8 C, or equivalently Q O(10 −19 ) in units of BH mass, obtained from astrophysical considerations in Refs. [226, 227] .

We now consider regular BH solutions, which are free of the r = 0 core singularity present within the Kerr-Newman family of metrics. Several approaches towards constructing regular BH metrics replace the Kerr- Newman core with a different type of core, such as a de Sitter or Minkowski core, or phenomenologically smearing the core over a larger surface. The Bardeen spacetime is one of the first regular BH solutions to the Einstein equations ever proposed [228] , and describes the space-time of a singularity-free BH with a de Sitter core [229] . 8 The Bardeen space-time carries a magnetic charge q m , and can be thought of as a magnetic monopole [230] satisfying the weak energy condition. The metric function can be written as:

with the magnetic charge satisfying the condition q m ≤ 16/27 ≈ 0.77. We compute the corresponding shadow radius numerically, showing its evolution as a function of the normalized magnetic charge q m /M in Fig. 3 . As with the Reissner-Nordström BH, we see that increasing the charge parameter decreases the shadow radius. Within 2σ, the EHT observations are consistent with SgrA * being a Bardeen BH for any (allowed) value of the magnetic charge, including the extremal value q m = 16/27.

Another well-known regular BH solution has been proposed by Hayward [231] . This space-time replaces the r = 0 GR singularity by a de Sitter core with cosmological constant Λ = 3 2 . In other words, one assumes that an effective cosmological constant plays an important role at short distances, with Hubble length . Such a behavior, while introduced at a phenomenological level, has been justified within the context of the equation of state of matter at high density [229, 232] , or an upper limit on density or curvature [233] [234] [235] , the latter expected to emerge within a quantum theory of gravity [236] . The metric function can be written as:

which the Hubble length associated to the effective cosmological constant satisfying ≤ 16/27. We compute the corresponding shadow radius numerically, showing its evolution as a function of the parameter /M in Fig. 4 . Again, we see that increasing decreases the shadow radius. Within the present constraints, we see that the EHT observations are consistent with SgrA * being a Hayward BH for any (allowed) value of the Hubble length associated to the effective cosmological constant, including the extremal value = 16/27.

A well-known generalization of the Hayward BH, with the same quantum gravity-inspired theoretical motivation and featuring an additional charge parameter, has been proposed by Frolov in Ref. [237] . The metric function is given by:

where the additional charge parameter q satisfies 0 < q ≤ 1, and the length below which quantum gravity effects become important satisfies ≤ 16/27 as with the Hayward BH. In principle, q admits an interpretation in terms of electric charge measured by an observer situated at infinity, where the metric is asymptotically flat.

Here, we focus on the charge q, while fixing = 0.3. This is similar to the choice reported in Ref. [16] , which instead fixed = 0.4, and can be motivated by the fact that the shadow radius changes slowly with , as is clear from Fig. 4 . Therefore, the limits on q we will report can be interpreted as constraints along a particular slice of the full parameter space. We compute the corresponding shadow radius numerically, showing its evolution as a function of the charge q/M in Fig. 5 . We see that, for = 0.3, the EHT observations set the upper limits q 0.8M (1σ) and q 0.9M (2σ), qualitatively similar to the constraints on the electric charge of the Reissner-Nordström metric Q we obtained in Sec. III A. Recall once more that these constraints are valid for = 0.3. From the results reported in Sec. III C and Fig. 4 for the Hayward BH, we can expect that fixing to a smaller/larger value would result in weaker/tighter constraints on q, as increasing leads to a slight decrease in the shadow size: in other words, we can expect a negative correlation/parameter degeneracy between and q, which could easily be confirmed by a 2-dimensional scan of the parameter space.

Non-linear electrodynamics (NLED) theories are wellmotivated extensions of Maxwell's Lagrangian in the high-intensity regime [238] , and appear in the low-energy limit of various well-known theories, including Born-Infeld electrodynamics [239] , as well as several string or supersymmetric constructions [240] [241] [242] [243] [244] . Another important class of regular BHs arises in the context of GR coupled to Bronnikov NLED, where the Lagrangian of the latter is given by [245] :

We recall that the relativistic invariants U and W are constructed from the electromagnetic field-strength tensor F αβ and its dual F αβ as follows:

where , αβµν , and A µ denote respectively the Hodge dual operator, the Levi-Civita symbol, and the electromagnetic gauge field. Einstein-Bronnikov gravity admits regular BHs only if these carry magnetic and not electric charge, i.e. choosing a purely magnetic configuration for the gauge field:

where q m is the magnetic charge. In this case, the central singularity is removed if the parameter a = q 3/2 m /M is different from zero. The metric function can be written explicitly as [245] :

As anticipated earlier, however, within the context of NLED theories photons do not move along null geodesics of the metric tensor, but of an effective geometry which depends on the details of the NLED theory. Therefore, knowledge of the metric function alone is in principle not sufficient to compute the shadow cast by BHs within NLED theories.

To proceed, we follow Novello's method [246, 247] to compute the effective geometry. We compute the shadow radius numerically, closely following the procedure outlined in Ref. [72] . In Fig. 6 , we show the evolution of the shadow radius as a function of the only free parameter, q m . We see once more that increasing the magnetic charge q m acts to decrease the shadow radius. We also see that the EHT observations restrict the magnetic charge to q m 0.8M (1σ) and q m M (2σ), while noting that within Bronnikov NLED the magnetic charge is in principle allowed to take values q m > M . the associated energy density and pressure asymptote to zero rather than to a final value determined by the effective cosmological constant of the de Sitter case. The authors of Ref. [250] argued that the GCSV BH is mathematically interesting due to its tractability (the curvature tensors and invariants take forms which are much simpler than those of the Bardeen, Hayward, and Frolov BHs), and physically interesting because of its non-standard asymptotically Minkowski core. The metric function can be written as:

where the hair parameter satisfies g > 0, in order for the singularity to be spread out across a Minkowski core. We compute the shadow size numerically, and show its evolution against g in Fig. 7 . As for all solutions considered so far, we see that increasing the hair parameter decreases the shadow size. The EHT observations set the limit g 0.8M (1σ) and g M (2σ). We are not aware of any a priori theoretical restriction on the value of g (besides from the requirement g > 0), given that the metric has been introduced as a toy model.

As a further example of regular BH, we consider the Kazakov-Solodukhin (KS) BH, which arises within a string-inspired model where spherically symmetric quantum fluctuations of the metric and the matter fields are governed by the 2D dilaton-gravity action [251] . The KS BH then provides a well-motivated example of quantum deformation of a Schwarzschild BH. The metric function for this space-time is given by: where the hair parameter > 0 sets the scale over which quantum deformations of the Schwarzschild BH shift the central singularity to a finite radius. Physically speaking, what effectively occurs is that the singularity is smeared out over a two-dimensional sphere of area 4π 2 . While is expected to be of order the Planck length, in a phenomenological approach it can in principle take any positive value, as done for instance in Ref. [252] . We compute the shadow size numerically, and show its evolution against in Fig. 8 . Although in principle all values > 0 are allowed, we restrict the plot to < 2, since larger values are well excluded by the EHT observations. The KZ BH provides the first example, among those considered, of a space-time where the size of the BH shadow increases with the hair parameter. We find that the KZ space-time is marginally consistent with the EHT observations at 1σ, while at 2σ we find the upper limit M . The physical meaning of this limit is that the singularity cannot be spread out across a scale larger than the gravitational radius, and in any case has to be confined within the event horizon. This example also highlights how the EHT near-horizon image of SgrA * places extremely tight constraints on fundamental physics scenarios which lead to a larger shadow compared to the Schwarzschild BH.

We now start moving away from regular BH metrics, considering a metric encompassing both a regular BH as well as a traversable wormhole (WH) regime. Recall that a WH is a tunnel-like structure which can connect two distance points of space-time [253, 254] , and typically requires a violation of the null energy condition in order for its geometry to be sustained [255] , either in the of exotic matter or a modification to Einstein's gravity. 10 In Ref. [286] , Simpson and Visser proposed a one-parameter extension of the Schwarzschild geometry which interpolates between the latter and a traversable WH, while passing through a regular BH geometry. The Simpson-Visser metric function is given by:

where the additional parameter is required to satisfy a > 0. For a = 0, Eq. (18) reduces to the Schwarzschild space-time. On the other hand, for 0 < a < 2M , the metric describes the space-time of a regular BH with a one-way space-like throat. For a = 2M , the space-time corresponds to that of a one-way WH with an extremal null throat. Finally, for a > 2M , Eq. (18) describes a traversable WH with a two-way time-like throat. The size of the shadow associated to the space-time in Eq. (18) is easy to compute analytically. First of all, we note that the location of the photon sphere is r ph = √ 9 − a 2 , so the Simpson-Visser geometry supports a photon sphere (and therefore casts a shadow) only for a < 3. The shadow radius is then given by r sh = √ 27 − 3a 2 . As a consistency check, we easily see that in the limit a → 0 we recover the Schwarzschild results r ph = 3 and r sh = 3 √ 3. We show the shadow radius evolution against a in Fig. 9 , and see that the size of the shadow decreases with increasing a. Moreover, we find that the EHT observations set the limit a 1.5M (1σ) and a 1.7M (2σ). These limits are consistent with most of the region of parameter space describing a regular BH, but exclude both the one-way 

We now consider what is arguably the simplest example of non-rotating, asymptotically flat, traversable wormhole, described by the well-known Morris-Thorne (MS) metric [254] :

where Φ(r) is referred to as the redshift function, and Ψ(r) as the shape function. One of the simplest choices for these two in principle free functions is to set Φ = −r 0 /r and Ψ(r) = (r/r 0 ) γ [254, 287, 288] , where r 0 is the radial coordinate of the wormhole throat, and γ is a constant. Using this definition, it can be easily shown that the ADM mass is M = r 0 , so the metric function can be rewritten as:

Importantly, the size of the wormhole shadow in units of the ADM mass is then uniquely determined and does not depend on additional parameters, since the only free parameter in the theory (γ) only enters g rr , which does not affect the computation of the shadow. For the choice of redshift function we have adopted, the equations governing the location of the photon sphere and shadow radius simplify considerably. The equation for r ph takes the form r ph (1 − r ph ) = 0, for which the physical solution is r ph = 1. Then, the shadow radius Fig. 2 for the Morris-Thorne traversable wormhole described by the metric in Eq. (19) . Also shown are the 1σ, 2σ, 3σ, 4σ, and 5σ constraints derived from the EHT observations shown in different shades of gray. The red regions are excluded by the same observations at more than 5σ. For comparison, the size of the shadow of a Schwarzschild BH with the same ADM mass is also shown (dashed black curve).

is r sh = 1/ √ e −2 = e. In other words, the size of the wormhole shadow in units of the ADM mass is exactly equal to Euler's number, r sh /M = e ≈ 2.718, i.e. almost a factor of 2 smaller than the size of the shadow of a Schwarzschild BH with the same ADM mass. By direct comparison, and as is visually clear in Fig. 10 , the EHT measurements rule out the possibility of SgrA * being a Morris-Thorne traversable wormhole, for the particular well-motivated choice of redshift function Φ = −r 0 /r. Note that this result has been also been discussed by the EHT collaboration in Ref. [16] .

We now move on to BH solutions which violate the nohair theorem in a controlled way. As is well known, one of the most straightforward but also best motivated ways of violating the NHT is through the presence of additional matter fields, such as a scalar field. We consider a real scalar field φ, conformally coupled to gravity, through the following Lagrangian terms:

where R is the Ricci scalar, and the 1/6 factor ensures that the coupling is conformal (in 4 dimensions). BH solutions within this theory have been obtained by Astorino [289] (see also Refs. [290] [291] [292] [293] [294] [295] [296] [297] for BH and black brane solutions within related models), and contain a hair parameter S which is related to the scalar field pro-file (and is directly connected to the value of the scalar field at the horizon). The hair parameter, or scalar charge, emerges as an integration constant and describes primary hair, i.e. it cannot be expressed as a function of the other hair parameters. The metric function reads:

where the scalar charge can be either positive or negative. The space-time described by Eq. (22) is closely related to the well-known "BBMB black hole" [298, 299] , and appears very similar to the Reissner-Nordström metric [Eq. (7)]. However, there is a crucial difference in that the scalar hair parameter can take negative values, which are instead effectively precluded within the Reissner-Nordström metric as Q 2 > 0. The S < 0 regime is in fact better known as mutated Reissner-Nordström BH, and effectively describes the geometry of a traversable wormhole [300] . Formally speaking, it corresponds to an analytic continuation of the Reissner-Nordström metric for imaginary values of the electric charge (and hence negative values of Q 2 ), although we caution against taking this interpretation too far. We compute the shadow size numerically (see also Ref. [96] , where the shadow associated to this space-time was studied), and show its evolution against the scalar charge S in Fig. 11 . We see that within the regime S > 0 the shadow size decreases with increasing scalar charge, analogously to the case with the electric charge in the Reissner-Nordström metric. In this regime, the EHT observations set the limit S 0.6M (1σ) and S 0.8M (2σ). In the mutated Reissner-Nordström BH/wormhole regime S < 0, the shadow size instead increases when increasing the absolute value of the scalar charge, quickly bringing the shadow size outside the range allowed by the EHT observations. The latter in fact virtually exclude S < 0 at 1σ, while allowing S −0.4M at 2σ. Among all the wormhole metrics we have considered so far, this is the only one which can be brought into marginal agreement with the EHT observations.

We now consider one of the most well-motivated highenergy physics models: the Randall-Sundrum II (RSII) brane-world model, which has been tested over a wide range of scales [301] [302] [303] [304] [305] [306] [307] [308] [309] [310] . This is an AdS 5 brane-world model where the extra dimension has an infinite size and negative bulk cosmological constant [311] . When projecting the full 5-dimensional BH solution of the theory onto our 4D metric, BHs effectively inherit a tidal charge q which carries projected information about 5D bulk (and in particular AdS 5 curvature radius ), while characterizing the effect of the bulk on the brane. More specifically, the tidal charge controls the strength of projected Weyl tensor transmitting tidal charge stresses from the bulk to the brane, which takes the form [312] :

where M p is the 5-dimensional Planck mass, u µ is the velocity 4-vector, r µ is an unit radial vector, and h µν = g µν + u µ u ν . Physically speaking, a negative tidal charge therefore amplifies the gravitational effects of the bulk on the brane are amplified. BH solutions within the RSII model have been studied in a larger number of works (see e.g. Refs. [313] [314] [315] [316] [317] [318] [319] [320] [321] [322] ). The metric function for spherically symmetric solutions takes the following form:

which is identical to the metric given in Eq. (22) for the BH/wormhole with conformally coupled scalar hair, with S → q/4. The only difference is therefore in the physical interpretation of the associated charge. In Fig. 12 we plot the size of the shadow radius as a function of the tidal charge: the plot is of course analogous to Fig. 11 . It is clear that the EHT observations place very tight limits on the tidal charge, q 0.15M at 1σ and q 0.2M at 2σ. The same observations virtually exclude negative tidal charge values at 1σ, while allowing q −0.1M at 2σ. This is in contrast with the shadow of M87 * , which instead exhibited a very slight preference for negative tidal charge, as noted earlier in Refs. [69, 92] . This fact again highlights how SgrA * 's shadow places extremely tight constraints on scenarios which tend to enlarge the shadow radius compared to the Schwarzschild BH. We note that the negative tidal charge regime of the RSII BH would appear to admit an interpretation in terms of mutated Reissner-Nordström BH/wormhole, though to the best of our knowledge we are not aware of this possibility having being discussed in the literature on BHs in brane-world models.

Earlier in Sec. III E we discussed regular magnetically charged BHs arising from Einstein-Bronnikov NLED. Another well-motivated NLED theory is Euler-Heisenberg NLED [323] , which appears in the low-energy limit of Born-Infeld electrodynamics, and is described by the following Lagrangian:

Choosing the same gauge field configuration as in Eq. (14) , and carefully accounting for the NLED-induced effective geometry, BH shadows within the Einstein-Euler-Heisenberg model were studied in Ref. [72] , whose procedure we follow. For completeness, we report the metric function for magnetically charged BH solutions within this theory, given by:

where q m is the BH magnetic charge and µ is the NLED coupling appearing in Eq. (25) . We note that as long as µ = 0, values of the magnetic charge q m > 1 are allowed. We compute the shadow radius numerically, following Ref. [72] . Note that we now have 2 free parameters: q m and µ. However, for simplicity, we fix the Lagrangian coupling to µ = 0.3 The reason is two-fold: a) as noted in Ref. [72] , most of the effect on the shadow size come from the Q m rather than µ, and b) µ ∼ 0.3 is approximately the largest allowed coupling before the perturbative approach of the theory around the Maxwell Lagrangian for µ → 0 ceases to be meaningful. In Fig. 13 we plot the evolution of the shadow radius against q m , at fixed µ = 0.3. We immediately note that the behavior of the shadow radius, and therefore the constraints from the EHT observations, depend on whether we are in the q m < 1 or q m > 1 regime. For q m < 1, we find the upper limit q m 0.7M (1σ) and q m 0.8M (2σ). On the other hand, for q m > 1, we observe a sharp discontinuity in the shadow radius, with the shadow size apparently diverging. This is due to a singularity in the NLED-induced effective geometry, which passes through a singularity: see Ref. [72] for detailed explanations.

Leaving aside the issue of whether the q m > 1 regime is physical in first place, given this singularity, in this regime we find that the shadow size varies drastically as q m changes. For µ = 0.3, consistency with the EHT observations requires q m ∼ 1.25M . This number depends to some extent on the value to which µ is fixed (see Ref. [72] for further discussions), although for values of µ 0.3, following the results of Ref. [72] , we can expect consistency with the EHT observations for values of the magnetic charge 1.1M q m 1.5M . We stress once again, however, that it is unclear whether we can consider this range of parameter space physical.

The Sen BH is a non-regular space-time appearing in the low-energy limit of heterotic string theory, whose field content includes both electromagnetic fields and the dilaton field Φ [324] . The metric function derived in this configuration reads:

where q m is an effective charge (including both the electric charge and an effective charge associated with the dilaton field) measured by a static observer at the infinity, and is bound by the requirement q m ≤ √ 2. We compute the shadow radius numerically and plot its evolution against q m in Fig. 14 , from which we see that the shadow radius quickly decreases with increasing charge. We also see that the EHT observations set the upper limit q m 0.6M (1σ) and q m 0.75M (2σ), ruling out the possibility of SgrA * being an extremal Sen BH with q m = √ 2M .

One of the best motivated string-inspired models among the many that predict deviations from GR is Einstein-Maxwell dilaton gravity (EMDG), which arises as an effective theory of the heterotic string at low energies once the additional six dimensions of the tendimensional manifold are compactified onto a torus [324] , see also Sec. III M. In EMDG, a dynamical dilaton field Φ evolving under a potential V (Φ) is included in the GR Lagrangian as:

where α regulates the coupling of the dilaton with photons. Given a BH of dilaton charge Q, the corresponding metric function of the so-called Einstein-Maxwelldilaton-1 BH is given by [325] [326] [327] : with the requirement that the parameter q = Q α /8 associated to the dilaton charge satisfies q ≤ √ 2. We compute the shadow radius numerically, and show its evolution as a function of the charge parameter q in Fig. 15 . We see that the EHT observations set the upper limit q 0.8M (1σ) and q M (2σ), ruling out the extremal Einstein-Maxwell-dilaton-1 BH (q = √ 2M ).

Alongside the conformally coupled scalar case considered in Sec. III J, we now study another scenario featuring a scalar field, which leads to a controlled violation of the no-hair theorem. We consider a minimally coupled real scalar field with a potential, whose dynamics are governed by the following Lagrangian:

Following Ref. [96, 328, 329] , we take the following form for the potential:

where g(φ) is given explicitly in Ref. [96] . The specific choice of potential is not particularly motivated from a high-energy theory perspective. However, in the context of BH physics, it is particularly interesting because it provides a toy example of a controlled violation of the no-hair theorem violation, which proceeds by explicitly introducing a new energy scale in the scalar sector. More specifically, the scalar field profile is given by: where the scale ν > 0 governs the scalar field fall-off, and effectively acts as a hair parameter. We compute the shadow radius of BHs within this theory numerically, following Ref. [96] . We show the evolution of the shadow radius against the hair parameter ν in Fig. 16 , finding that the shadow size increases quickly with an increasing hair parameter. As in other similar examples we have already seen, the EHT observations are marginally consistent with this space-time at 1σ, and set the very tight 2σ limit ν 0.4M . This effectively constrains the energy scale at which violations of the no-hair theorem become strong.

Einstein-aether gravity is a theory of gravity that dynamically violates Lorentz symmetry by means of a unit norm time-like vector field (the "aether"), which defines a preferred time-like direction at each point of space-time. The Einstein-aether gravity Lagrangian is governed by 4 free parameters c i with i = 1, 2, 3, 4, and is given by the following [330] [331] [332] :

where u µ is the aether field, λ is a Lagrange multiplier enforcing the condition u µ u µ = −1, and the tensor h αβ µν is given by:

In what follows, we will make use of the widely used shorthand notation c ij = c i + c j , c ijk = c i + c j + c k .

Two exact static spherically symmetric BH solutions within Einstein-aether gravity are known [333, 334] . The first solution, called Einstein-aether type 1, holds for c 14 = 0 and c 123 = 0. The metric function is given by: with the requirement 0 < c 13 < 1. We compute the shadow radius numerically, plotting it as a function of the Lagrangian parameter c 13 in Fig. 17 . We see yet again an example of shadow radius increasing with increasing fundamental parameter, which leads to this scenario being extremely tightly constrained by the ETH observations. The latter are only marginally consistent with this space-time within 1σ, while setting the 2σ upper limit c 13 0.7 (note that the c i s are dimensionless). While this is an independent constraint on c 13 obtained within the very strong-field regime, we note that it is still significantly weaker than constraints arising from GWs. For instance, the multi-messenger GW event GW170817 sets the limit |c 13 | 10 −15 [335] . The second exact BH solution in this theory holds when c 14 = 0 and c 123 = 0, and is called called Einsteinaether type 2. The metric function is given by:

with the requirement 0 ≤ c 14 ≤ 2c 13 < 2. In this case, the space-time is described by 2 free parameters. For simplicity and illustrative purposes, we fix c 13 = 0.99, which as argued earlier is excluded in the case of the Einsteinaether type 1 BH. The purpose is simply to illustrate how a non-zero value of c 14 can in principle help bring a previously excluded range of c 13 values into an agreement with the EHT observations. We compute the shadow radius numerically, plotting it as a function of the Lagrangian parameter c 13 in Fig. 18 . For illustrative purposes, we fix c 13 = 0.99, close to the maximum value allowed, which earlier we showed was excluded for the Einstein-aether-1 BH. In this case we see that in principle a value of 1. values of c 14 are excluded by GW observations, with the GW170817 event setting the limit c 14 2.5 × 10 −5 [335] .

Even though the cosmic censorship conjecture would appear to forbid the existence of naked singularities [167] , these are nonetheless known to appear as a result of gravitational collapse given suitable initial conditions [336] [337] [338] [339] [340] [341] [342] [343] [344] [345] . The Janis-Newman-Winicour naked singularity is a solution of the Einstein-Klein-Gordon equations for a massless scalar field [346] (see also Refs. [347] [348] [349] [350] ). Despite being a naked singularity, it possesses a photon sphere and therefore casts a shadow.

The metric function cannot be written in a closed form in terms of the Boyer-Lindquist coordinate r, but this can be achieved using the auxiliary variable ρ:

where ν = 1 − 1 + q/M , q is the scalar charge of the naked singularity, and the scalar field radial profile is given by the following:

The auxiliary variable is related to the Boyer-Lindquist coordinate r through the following transcendental coordinate transformation:

This space-time can only cast a shadow for 0 ≤ ν ≤ 0.5. We compute the shadow radius numerically, and plot its evolution against the parameter ν in Fig. 19 . There is a wide range of parameter space where SgrA * 's shadow is consistent with being a JNW naked singularity. In fact, the EHT observations set rather weak limits on ν, with ν 0.4 at 1σ and ν 0.45M at 2σ, barely excluding the extremal JNW naked singularity (ν = 0.5). These limits can of course be translated into rather weak limits on the strength of the scalar field. More importantly, among all the examples of BH mimickers we have considered so far, the JNW naked singularity is the first one which is in excellent agreement with the EHT observations, which instead exclude the wormhole solutions we have considered earlier in Sec. III H, Sec. III I, and Sec. III J.

The Gauss-Bonnet invariant G is a very important quantity, which in more than 4 dimensions plays a role in determining the allowed Lagrangian terms (Lovelock invariants) of diffeomorphism-invariant, metric theories of gravity with equations of motion of second order [351] , and is given by:

Crucially, in 4 dimensions G amounts to a total derivative, and therefore cannot contribute to the gravitational dynamics. However, in Ref. [352] Glavan and Lin proposed what effectively amounts to a dimensional regularization procedure applied to the following action:

by rescaling the coupling constant as α → α/(D − 4) (see also the generalization of Ref. [353] ). This procedure is shown to lead to non-trivial contributions even in With all the caveats discussed in Refs. [355] [356] [357] [358] [359] [360] [361] in mind, static spherically symmetric solutions within 4DEGB gravity have been studied, and the metric function is given by the following (see e.g. Refs. [83-85, 248, 352, 362-368] ):

which requires α ≤ 1/16π ∼ 0.02. We need to take the negative branch of Eq. (42), as it is the only one which recovers the Schwarzschild metric function A(r) = 1 − 2M/r in the correct limit. We compute the shadow radius numerically, and show its evolution as a function of α in Fig. 20 . We see that the shadow radius decreases with increasing α. However, for all values of α ≤ 1/16π, the shadow radius is always consistent with the EHT observations within better than 1σ. Therefore, the size of SgrA * 's shadow does not place meaningful constraints on the 4DEGB coupling α, which is instead constrained by alternative astrophysical probes, such as binary BH systems, which limit α 10 8 m 2 [369] .

The manifest incompatibility between GR and quantum mechanics has led to attempts at revisiting Heisenberg's uncertainty principle to incorporate quantum gravitational effects [370] [371] [372] . Among the simplest approaches is the inclusion of a deformation parameter β that regulates the non-linear dependence of the positionmomentum commutator [x, p] on the magnitude of the momentum. A more general approach leads to a correction to Heisenberg's uncertainty principle as [373] [374] [375] :

where β can take any positive value. Such a modification leads to a formulation of "deformed" quantum mechanics and ultimately to the introduction of discreteness in the nature of space-time. The value of β can be predicted both in low-energy formulations of string theory [376, 377] as well as by comparing the corrections to Hawking's theorem for the BH temperature obtained from two different approaches [378] , and is expected to be of order unity from these approaches.

Owing to the quantum correction to the Hawking temperature associated to a BH, the generalized uncertainty principle (GUP) modified the ADM mass of a BH [379, 380] . It is generally expected that the GUP modifies the ADM mass of a BH as M = M +βM 2 Pl /M 2 , so that the associated metric function reads:

Unsurprisingly as β increases so does the shadow size, because increasing the ADM mass increases the radial coordinate of the photon sphere. We compute the shadow size and show its evolution as a function of β in Fig. 21 . We see that the EHT observations are only marginally consistent with β = 0 at 1σ, setting the limit β 0.01M 2 , while the 2σ limit β 0.1M 2 is an order of magnitude weaker. Restoring the Planck mass to obtain the dimensionless GUP parameter β, we require β 0.1(M/M Pl ) 2 ∼ 1.5 × 10 88 . Unsurprisingly this is approximately 6 orders of magnitude stronger than the limit obtained from M87* in Refs. [67, 381] , as Sgr * 's mass is about 3 orders of magnitude smaller. However, this same limit is still weaker by several orders of magnitude compared to independent tests such as perihelion precession, pulsar periastron shift, light deflection, gravitational waves, gravitational redshift, Shapiro time delay, geodetic precession, and quasiperiodic oscillations (see e.g. Refs. [381] [382] [383] ).

A modification to the Bekenstein-Hawking relation between the area of a BH and its entropy [384, 385] has been considered by Barrow [386] , in relation to possible effects from space-time foam, and leading to a fractal structure for the horizon with finite volume (but larger than in the standard case) and either finite or infinite area. This entropy model has later been considered as a model for holographic dark energy [387, 388] (see also Refs. [389] [390] [391] [392] [393] [394] [395] [396] [397] [398] [399] [400] [401] [402] [403] [404] [405] [406] [407] [408] ). In the toy model considered, a three-dimensional fractal is built around the BH event horizon, modifying its total surface and volume. 11 In this picture, the entropy-area law of a BH is modified as:

where 0 ≤ ∆ ≤ 1 is related to the dimension of the fractal, with ∆ = 0 resulting in the usual Bekenstein-Hawking theorem S = A/4 and with ∆ = 1 expressing the largest deviation of the BH temperature from the expected value. The metric function associated to this modification has been derived in Ref. [123] and is given by:

Starting from the metric function in Eq. (46) , the shadow radius can be computed analytically using Eq. (6), and resulting in r sh = 3 √ 3(∆ + 2)2 ∆−1 M 1+∆ π ∆/2 . For consistency, one can see that in the limit ∆ → 0, we recover the Schwarzschild results r sh = 3 √ 3M , while in the opposite limit ∆ → 1 the shadow depends on an additional factor of two in the exponent ∝ M 2 due to the effects of the fractal volume. In general, the shadow size increases with increasing ∆, reflecting the fact that the BH surface area increases due to quantum gravity effects.

We show the evolution of the shadow radius as a function of ∆ in Fig. 22 . We clearly see that the EHT observations set extremely strong constraints on the parameter ∆. In particular, we find the upper limit ∆ 0.001 at 1σ, tighter than cosmological constraints requiring ∆ 0.1888 [409, 410] , but weaker than Big Bang Nucleosynthesis constraints requiring ∆ 1.4 × 10 −4 [411] . At 2σ, we have the weaker constraint ∆ 0.035, which is nonetheless still stronger than the cosmological one. With taking these limits into consideration, the behavior of a holographic DE component based on Barrow entropy is prevented from deviating significantly from the cosmological constant Λ, neither in the quintessence-like or phantom regime, and therefore cannot play a significant role in the context of the Hubble tension [412] [413] [414] .

Horizon-scale images of supermassive black holes and their shadows have opened a new unparalleled window onto tests of gravity and fundamental physics in the very strong-field regime, including the possibility that astrophysical BHs may be described by alternatives to the Kerr metric. In this work, we have used the horizonscale images of SgrA * provided by the Event Horizon Telescope [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] to test some of the most popular and well-motivated scenarios deviating from the Kerr metric. Compared to horizon-scale images of M87 * , there are significant advantages in the use of images of SgrA * , as we discussed towards the end of Sec. I, with the most notable advantage being that SgrA * 's proximity to us results in an exquisite calibration of its mass-to-distance ratio, essential for connecting the angular size of its shadow to theoretical predictions within a given model. Our tests have been performed by connecting the angular size of the bright ring of emission to that of the underlying BH shadow, and utilizing prior information on SgrA * 's exquisitely calibrated mass-to-distance ratio. This is a robust and well-tested methodology, adopted for instance by the EHT collaboration themselves [16, 112] .

We have studied a very wide range of well-motivated scenarios, including both fundamental theoretical scenarios and more phenomenological ones. In particular, we have considered: various regular BH space-times (among which are the well-known Bardeen, Hayward, and Frolov BHs); string-inspired space-times (such as the Kazakov-Solodukhin and Sen BHs, as well as Einstein-Maxwelldilaton gravity); metrics arising within non-linear electrodynamics theories; space-times violating the no-hair theorem due to the presence of additional (minimally or conformally coupled) scalar fields; alternative theories such as the Einstein-aether theory, Randall-Sundrum II model, and regularized 4-dimensional Einstein-Gauss-Bonnet gravity; fundamental frameworks leading to new ingredients such as the generalized uncertainty principle and Barrow entropy's modification to the Bekenstein-Hawking entropy; and finally BH mimickers including three classes of wormholes (Simpson-Visser, Morris Thorne, and the mutated Reissner-Nordström wormhole) and two classes of naked singularities (Janis-Newman-Winicour and Reissner-Nordström) . For almost all of the space-times considered, we have found that the EHT observations set limits on fundamental dimensional parameters of order O(0.1) times some power of mass M , which is not surprising considering that EHT observations probe the BH space-time on horizon scales (i.e. on scales comparable to the gravitational radius), so constraints of this order should be expected. For dimensionless parameters, e.g. the Einstein-aether coupling c 13 or the Barrow entropy parameter ∆, we have found constraints of the order or O(0.1) or stronger: in the case of Barrow entropy, we have found constraints which are actually stronger than the cosmological ones. Overall, we have shown that the EHT horizon-scale images of SgrA * place particularly tight constraints on fundamental physics scenarios which predict shadows larger than that of a Schwarzschild BH of the same ADM mass, as the EHT observations on average prefer a shadow size which is slightly smaller than the latter, given the Keck and VLTI priors on the mass-to-distance ratio (this had also been noted in Ref. [16] ): examples of such scenarios include the Kazakov-Solodukhin spacetime, the mutated Reissner-Nordström wormhole and the closely related negative tidal charge regime of the Randall-Sundrum II model, the BH with scalar hair considered in Sec. III O, Einstein-aether gravity, the generalized uncertainty principle, and Barrow's modification to the Bekenstein-Hawking entropy.

There is certainly ample opportunity for further work in this very promising direction. First of all, for obvious reasons we were forced to make a selection when considering which fundamental scenarios to study, and many interesting models were inevitably left behind: we plan to return to these in future works, or possibly in an updated version of this paper. These scenarios could include viable models for the dark matter and dark energy perme-ating the Universe, as well as well-motivated extensions of General Relativity: in this sense, a detailed investigation of the synergy and between BH shadows, gravitational waves, cosmological and astrophysical probes, and laboratory tests of gravity and fundamental physics is certainly a direction worth pursuing (see e.g. Ref. [141] ), as one can expect a very strong complementarity across these probes. A future study could also include the effect of rotation on all the space-times considered. In closing, horizon-scale images of BHs and their shadows have granted us an unprecedented probe of gravity and fundamental physics in the era of multi-messenger astrophysics, and we have only just begun exploring the potential carried by such an exciting probe.

We had independently started this work in preparation for the Event Horizon Telescope press conference announcing on May 12, 2022 the new imaging concerning SgrA * . Some of the models we tested (most notably the Reissner-Nordström, Bardeen, Hayward, Frolov, Janis-Newman-Winicour, Kazakov-Solodukhin, and Einstein-Maxwell-dilaton space-times) have also been inevitably tested by the EHT collaboration in Ref. [16] . However, in this work we have considered several other models and scenarios beyond those considered by the EHT group. We have verified that, for the overlapping models, both results agree, therefore providing a valuable and independent cross-check.

Event Horizon Telescope), Galaxies

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope), Astrophys

Event Horizon Telescope)

Event Horizon Telescope)

Koninklijke Nederlandse Akademie van

Proceedings, 5th International Conference on Gravitation and the theory of relativity

Heisenberg-Euler effective Lagrangians: Basics and extensions

Starkman

We thank Yifan Chen for useful discussions. S.V.