1 Introduction

Astrophysical dark self-gravitating objects, such as black holes (BHs), with masses spanning from a few to several billion times that of the Sun, can be found in various systems, including stellar binaries and supermassive BHs (SMBHs) located in the galactic cores. To acquire a more profound comprehension of these dark configurations, it would be beneficial to delve further into their internal structure [1,2,3,4], either from a geometrical perspective or by exploring the type of matter that has contributed to their formation [5,6,7]. Considering that only 4% of the universe is normal matter (baryonic and leptonic), and dark matter (DM) concentrates towards galactic centers [8, 9], a compelling question arises: is there a connection between the central BH and DM in each galaxy? Although the no-hair theorem [10, 11] limits our ability to directly observe interior composition of BHs, theoretical models characterizing the internal features of BHs could shed light on a potential connection with DM [12,13,14,15,16,17]. Such models could not only explore the feasibility of this link but also potentially reveal new insights into both BHs and DM themselves. Building on this idea, a recent study [18] modeled the central galactic entity using a DM model derived from the outskirts of the galaxy. Further studies, such as those in [19, 20], investigate the fascinating possibility of a BHs growth fueled by DM absorption, adding another potential connection within the galactic bulge. These findings add to the growing body of evidence suggesting that BHs may play a key role in the evolution and configuration of galactic structures. Furthermore, these finding opens the door to the intriguing possibility of BHs being entirely composed of DM.

N-body simulations suggest that unique density profiles define DM haloes on both galaxy and cluster scales. The Einasto model has emerged as a widely favored choice for characterizing the density distribution of DM haloes. A variety of DM haloes can be fitted more accurately by nonsingular three-parameter models like the Einasto profile than by singular two-parameter models like the NFW, according to current high-resolution N-body cold-DM simulations. Several investigations have been conducted to explore the basic properties of the Einasto density model of DM. Retena et al. [21] investigated a general analytical expression for the surface density associated with Einasto density model, which is an important quantity that determines the gravitational lensing properties of DM haloes. Furthermore, specific formulas for the shear, first flexion, and second flexion of Einasto DM haloes have been explored through a Mellin-transform method, expressed in terms of Fox H and Meijer G functions [22]. These relations are applicable across a wide range of Einasto index values.

The Einasto DM profile is observed to be an extension of a Gaussian profile, based on non-commutativity. In this context, the de Sitter EOS plays a significant role in characterizing BH solutions. Consequently, addressing concerns related to perturbations or exotic spacetime configurations through the exploration of a broader framework would follow a similar approach. The assumption of de Sitter EOS is motivated by the need to avoid an overdetermined system of equations. This enabled us to describe that the Einasto density model may be employed to formulate either a fuzzy BH or a diffused stellar droplet, corresponding to different values of rescaled mass. We discuss another solution utilizing a different EOS, indicating that the regularity of a fuzzy stellar or fuzzy BH appears to necessitate that the \(P_{r}<0\). To achieve this, we must specify a particular EOS and a stress tensor. By taking into account the diffusive nature of Einasto density model, we anticipate that any alteration in the radial pressure must consider the effects of energy density variations across the entire volume. Therefore, it seems reasonable to assume that nonlocal effects may be relevant when dealing with such a distribution.

Drawing inspiration from the potential relationship between DM and BHs, we develop astrophysical solutions characterizing the central galactic objects referred to as “fuzzy BH” (or self-gravitating droplets). This model is reminiscent of BH or compact droplet models derived from non-commutative metric (\([\textbf{x}^{\mu },\textbf{x}^{\nu }]\ne 0\)) [23], emphasizing the significance of a Gaussian matter profile along with the de Sitter EOS. Within the realm of non-commutativity, the existence of quantum oscillations offers a possible answer to the problem of infinities, a long-standing problem encountered in field theories like gravity, which leads to problematic short-distance behavior. The notion of non-commutativity became interesting within the framework of high-energy physics with the emergence of string theory. Consequently, within non-relativistic limits where gravitational effects become negligible compared to matter, open strings give rise to the evolution of a quantum field theory on non-commutative metric, commonly referred to as non-commutative field theory. A key feature of loop quantum gravity is the emergence of non-commutative behavior within its quantum geometry [24, 25]. Regardless of whether non-commutative geometry is seen as a fundamental description of quantum spacetime or a tool that complements loop quantum gravity and string theory, it is currently being used to explore the idea of a spacetime “fuzziness” [26, 27]. This is achieved through the following relation

$$\begin{aligned} \left[ \textbf{x}^{\mu },\textbf{x}^{\nu }\right] =i\Phi ^{\mu \nu }, \end{aligned}$$

where \(\Phi ^{\mu \nu }\) represents an anti-symmetric, real-valued \(D\times D\) (D denotes the metric dimensions) matrix that defines the fundamental cell discretization of metric.

Expanding on the aforementioned approach to BH physics, we can further explore the idea that the Gaussian distribution is a particular manifestation of the Einasto DM halo profile. Building upon the concept of non-commutative black holes, we introduce a “fuzzy BH” model subject to the Einasto parameterization to investigate a potential relationship between the astrophysical DM and BHs. This study explores the potential connection between the central SMBH of our galaxy and DM, particularly within the current state of the galaxy. Several theoretical frameworks were proposed as potential alternatives before the widespread recognition of SgrA* [28, 29], the SMBH at the galactic center. These substitutes included dense-matter compact objects like boson-stars, gravastars [30, 31], burning disks [32], and ultra-dense cores of dark matter [33, 34], as well as naked gravitational singularities [35]. Additionally, the Ruffini–Argüelles–Rueda quantum model [36], based on a compact configuration with heavy fermions, suggested the potential existence of highly-dense quantum cores at the galactic center. When exploring alternative gravitational models to BHs, such as self-gravitating droplets or other ultra-dense exotic configurations, it is important to ensure their effective potential closely matches that of a classical BH, especially in the region where bound orbits exist [18]. In this context, our objective is to identify the suitable parameters within the Einasto DM parametrization to model a highly dense fuzzy BH or a compact self-gravitating DM droplet at the galactic core. To achieve our objective, we will consider a coupling of anisotropic stress-energy tensor with de Sitter-like EOS (\(\varepsilon (r)=-P_{r}\)) with Einasto DM parametrization. This specific pressure-density relationship is a common feature in the study of BHs [12,13,14,15,16]. The assumption of this particular EOS leads to the formation of different BH solutions corresponding to various values of the parameters \(\{\alpha ,\eta \}\) using the Einasto density parametrization at the galactic center. Moreover, the interior of the BH features no gravitational singularity for \(r = 0\); instead, it is substituted by a consistent de Sitter core. Furthermore, we also investigate the potential to create self-gravitating fuzzy droplets with an anisotropic fluid distribution governed by a non-local EOS. Again, the presence of negative pressure is inevitable in this case as well. Specifically, we build a consistent self-gravitating dark matter fuzzy droplet structure at the galactic center, where the effective potential allows for the presence of bound states for massive particles, all without encountering any gravitational singularity at \(r=0\).

The recent finding of the accelerated cosmic expansion has sparked a surge of research into the cosmological implications of extended, or modified, theories of gravity (ETG) models [37,38,39,40,41,42,43,44,45]. The ETG models are a new approach to understanding gravity at extreme scales, building on GR. They aim to solve new puzzles in astrophysics, cosmology, and high-energy physics that have emerged in recent times. The ETG models are essentially variations or extensions of Einstein’s GR. Exploring these models is driven by several factors, including limitations of general relativity at high energy scales, unexplained phenomena like cosmic inflation and dark matter/energy, and the desire for a deeper understanding of general relativity itself. In this direction numerous ETG models have been developed, acting as tools for comparison and testing against GR. The effects of these modifications are expected to show up in phenomena like the cosmic microwave background radiation, the structure of astrophysical entities, BHs, and neutron stars. Observations of these phenomena are crucial to distinguish between ETG models and GR. An alternative approach involves adjusting the fundamental field equations while maintaining specific mathematical properties and avoiding unwanted mathematical complications. Over the years, a wide array of ETG models have emerged that successfully reproduce the outcomes of Einstein’s theory while also producing the necessary cosmic evolution for key cosmological eras. One promising approach among these attempts are \(f(\textrm{R})\) theories. These theories modify GR by incorporating the Ricci scalar, \(\textrm{R}\), into an arbitrary function, allowing for more flexibility in explaining cosmic phenomena. In the \(\textrm{f}(\textrm{R})\) relativistic gravitational model, the action is expressed as [46, 47]

$$\begin{aligned} S=\frac{1}{16\pi }\int \textrm{f}(\textrm{R})\sqrt{|\textit{g}|}d^{4}x +S_{m}. \end{aligned}$$
(1)

Numerous models exist featuring various curvature terms, yet a successful model that induces accelerated expansion of the universe must align with both solar system dynamics and astrophysical observations on a large scale. A gravitational model that falls within the intersection of the above-mentioned extensions of GR is the \(\textrm{R} + \beta \textrm{R}^{2}\) model. This model is physically acceptable and free from ghostly-like modes, unlike scenarios where other higher-order corrections or complex functions of the scalar curvature are incorporated into the standard gravitational action. The \(\textrm{R} + \beta \textrm{R}^{2}\) gravitational model, referred to as the Starobinsky gravity, introduces an extra massive spin-0 state alongside the standard massless graviton. The presence of Starobinsky-like curvature corrections in the early universe gives rise to an accelerated stage known as inflation. This inflationary stage is supported by the presence of such a term in the weak energy limit of string theory, providing motivation for further study. This involves investigating how the higher-order curvature corrections, such as in the \(\textrm{R} + \beta \textrm{R}^{2}\) gravity, impacts aspects like the presence and characteristics of astrophysical dark stellar configurations [48,49,50].

Cooney et al. [51] explored the presence of relativistic stellar configurations within the \(\textrm{R} + \beta \textrm{R}^{2}\) gravity using a perturbation technique. Arapoglu et al. [52] applied a similar approach to constrain the value of the parameter \(\beta \), utilizing measurements of mass-radius relationship associated with neutron star and a range of realistic EOS. Astashenok et al. [53] investigated the structure of neutron stars with realistic EOS in perturbation Starobinsky theory. On the other hand, Yazadjiev et al. [54] employed a non-perturbation method, transforming the theory into a scalar-tensor theory and raised concerns about the perturbation method’s consistency. Staykov et al. [55] explored the oscillations of astrophysical stellar configurations using the non-perturbation formalism of \(\textrm{R} + \beta \textrm{R}^{2}\) gravity. Arapŏglu et al. [56] probed the stellar structure solutions characterizing the realistic self-gravitating perfect fluid spheres through the mechanism of matched asymptotic expansion in Starobinsky gravity. Astashenok et al. [57] and his coworkers examined the non-rotating static spherical compact distributions characterizing the neutron stars and quark stars parameterized by \(\beta \) for a class of fourth-order \(f(\textrm{R})\) model, whose gravitational Lagrangian is represented as \(f(\textrm{R})=\textrm{R}+\beta \textrm{R}^{2}\). Recently, the authors of [58] studied the Chandrasekhar limit pertaining to white dwarfs by considering a simple polytropic EOS using the Starobinsky model of gravity.

In this work, we develop astrophysical configurations, including self-gravitating fuzzy droplets or fuzzy BHs with spherically symmetric static structures. This investigation provides an extension of the work presented in [59] via \(f(\textrm{R})=\textrm{R} + \beta \textrm{R}^{2}\) gravity model. The remaining portion of article is organized as follows: in Sect. 2, we describe the fundamental structure of the Einasto parametrization together with key concepts. In Sect. 4, we establish the foundation for our analysis by presenting the basic formalism of the Starobinsky model of gravity. This section covers the relevant field equations, conservation equation, and the construction of dark matter configurations under the de Sitter-type EOS. We calculate the effective potential in Sect. 5 and construct the corresponding self-gravitating droplet configurations by considering non-local EOS within the context of Starobinsky gravity in Sect. 7. In Sect. 6, we summarize the key findings of this investigation.

2 The Einasto density model of dark matter

The N-body cold DM simulations [60,61,62] have revealed that specific three-parameter profiles offer exceptional accuracy in describing a wide range of DM distributions. Among these profiles is the Einasto profile [63], which extends the two-dimensional Sérsic parametrization [64], traditionally employed to characterize the surface brightness of early galaxies and the bulges of spiral galaxies, into three dimensions. The Einasto model of DM exhibits a characteristic power-law logarithmic slope in its density profile, defined as [22]

$$\begin{aligned} \gamma (r)\equiv \frac{d \ln \varepsilon }{d \ln r}(r)\propto r^{\frac{1}{\alpha }}, \end{aligned}$$
(2)

where \(\alpha \) symbolize the Einasto index. Integrating the above-mentioned expression, we have

$$\begin{aligned} \ln \left[ \frac{\varepsilon (r)}{\varepsilon _{s}}\right] =-k_{\alpha }\left[ \left( \frac{r}{r_{s}}\right) ^\frac{1}{\alpha }-1\right] . \end{aligned}$$
(3)

In this context, the symbol \(r_{s}\) corresponds to the radius of a sphere encompassing half of the total mass, \(\varepsilon _{s}=\varepsilon (r_{s})\), and \(d_{n}\) is a numerical constant employed to effectively establish \(r_{s}\) as the radius containing half of the mass. For DM haloes, an alternative form of (3) reads [65]

$$\begin{aligned} \ln \left[ \frac{\varepsilon (r)}{\varepsilon _{-2}}\right] =-2\alpha \left[ \left( \frac{r}{r_{-2}}\right) ^\frac{1}{\alpha }-1\right] , \end{aligned}$$
(4)

with \(\varepsilon _{-2}\) and \(r_{-2}\) characterizing the energy density and radius at which \(\varepsilon (r)\propto r^{-2}\). An alternative form of the above expression can be defined as

$$\begin{aligned} \varepsilon (r)=\varepsilon _{0}\exp \left[ -\left( \frac{r}{\eta }\right) ^{\frac{1}{\alpha }}\right] , \end{aligned}$$
(5)

where the scale length \(\eta \) is

$$\begin{aligned} \eta =\frac{r_{s}}{k^{\alpha }_{\alpha }}=\frac{r_{-2}}{(2\alpha )^{\alpha }}, \end{aligned}$$
(6)

while the central density

$$\begin{aligned} \varepsilon _{0}=\varepsilon _{s}\exp (k_{\alpha })=\varepsilon _{-2}\exp (2\alpha ). \end{aligned}$$
(7)

According to Einasto, in order to model real galactic structures accurately, the associated descriptive functions must meet several conditions [63]. The assumption of a suitable descriptive function has a significant role in realistic modeling of galactic structures. In this respect, the assumption of density profile (\(\varepsilon (r)\)) seems to be the most beneficial choice. This preference stems from the fact that key descriptive functions including the gravitational potential, cumulative mass distribution and the surface mass distribution can be derived as integrals of the energy density. Additionally, a physical model must meet several conditions:

  1. (i)

    \(0<\varepsilon (r)<\infty \),  \(\forall ~r>0\).

  2. (ii)

    \(\varepsilon (r)\) should smoothly decrease, approaching zero for large values of r.

  3. (iii)

    the descriptive functions should not exhibit jump discontinuities.

  4. (iv)

    particular moments, including those defining the total mass, central gravitational potential, and effective radius associated with the mass function should be finite.

Einasto proposed various descriptive functions to describe galactic structures. Among these, Einasto profile, a special case, aligns most closely with observations. In his pioneering work, Einasto employed the Einasto DM profile to model the Andromeda Galaxy (M31). Furthermore, Einasto extended his work by applying the Einasto profile to model a broader range of nearby galaxies. This included not only the Andromeda Galaxy and our Milky Way, but also dwarf galaxies like M32, M87, Fornax, and Sculptor. Einasto’s models were not limited to single structures; instead, they were constructed as multi-component systems, where each component represented a physically distinct population of stars. These components were defined by a unique set of three parameters: \(\varepsilon (r)\) (density scale), \(\eta \) (scale length), and \(\alpha \) (shape parameter). Numerous physically homogeneous astrophysical configurations characterized by values of Einasto index \(\alpha \) are discussed in the literature. For example,

  • \(4.54\lesssim \alpha \lesssim 8.33\) corresponds to the haloes with masses form dwarf to clusters, with an average value of \(\alpha =5.88\) [60].

  • In the Millennium Run, \(\alpha \sim 4.35\) for the cluster-sized [66] and \(\alpha \sim 5.88\) for galaxy sized haloes [62]. Further investigations with similar findings for the galaxy-sized haloes within the Aquarius simulation have been discussed in [67].

  • \(\alpha \sim 3.33\) for the most massive haloes in the Millennium Run [61].

For an Einasto profile describing the DM distribution, the total mass, M, is given by

$$\begin{aligned} M=4\pi \int _{0}^{\infty }z^{2}\varepsilon (z)dz. \end{aligned}$$
(8)

Combining Eqs. (4) and (8), we get

$$\begin{aligned} M&=4\pi \varepsilon _{0}\eta ^{3}\alpha \Gamma (3\alpha ), \quad where \nonumber \\ \Gamma (3\alpha )&=\int _{0}^{\infty }\exp (-z)z^{3\alpha -1}dz. \end{aligned}$$
(9)

Thus, the density profile (5) turns out to be

$$\begin{aligned} \varepsilon (r)=\frac{M}{4\pi \alpha \eta ^{3}\Gamma (3\alpha )}\exp \left[ -\left( \frac{r}{\eta }\right) ^{\frac{1}{\alpha }}\right] . \end{aligned}$$
(10)

3 Corrections proposed by Starobinsky and Einasto’s fuzzy black holes

Motivated by attempts to renormalize GR, the inclusion of quadratic corrections to the Ricci scalar represents a direct extension of GR. These corrections have gained significant importance in cosmological models, particularly in the context of inflation [68]. The action for Starobinsky gravity reads [37]

$$\begin{aligned} S\equiv S_{\mathcal {G}}+ S_{m}&=\frac{1}{16\pi }\int \left( \textrm{R}+\beta \textrm{R}^{2} \right) \sqrt{|\textit{g}|}d^{4}x\nonumber \\&\quad +\int \mathscr {L}_{m}\sqrt{|\textit{g}|}d^{4}x, \end{aligned}$$
(11)

where \(\textrm{R}\) denotes the curvature scalar, which is obtained by the contraction of Ricci tensor \(R_{\mu \nu }\), while \(\mathscr {L}_{m}\) signifies the ordinary matter Lagrangian. This action represents the simplest extension of the standard GR-action, where the assumption \(\beta =0\) is employed. The variation of the action (11) provides the following filed equations

$$\begin{aligned}&\left( 1+2\beta \textrm{R}\right) G_{\mu \nu }+\frac{1}{2}\textit{g}_{\mu \nu }\beta \textrm{R}^{2}+2\beta \left( \textit{g}_{\mu \nu }\Box -\nabla _{\mu }\nabla _{\nu }\right) \textrm{R}\nonumber \\&\qquad =8\pi \Theta _{\mu \nu }, \end{aligned}$$
(12)

with \(G_{\mu \nu }\equiv R_{\mu \nu }-\frac{1}{2}\textrm{R}\textrm{g}_{\mu \nu }\) symbolizes the usual Einstein tensor, \(\Box \equiv \textit{g}^{\mu \nu }\nabla _{\mu }\nabla _{\nu }\) is the D’ Alembertian, \(\nabla _{\mu }\) signifies the covariant derivative. The trace of Eq. (12) reads

$$\begin{aligned} 6\beta \Box \textrm{R}-\textrm{R}=8\pi \Theta , \end{aligned}$$
(13)

where

$$\begin{aligned} \frac{1}{f}\Box =-\frac{1}{f^{2}}\frac{d^{2}}{d t^{2}}+\left( \frac{2}{r}+\frac{1}{f}\frac{df}{dr}\right) +\frac{d^{2}}{dr^{2}}. \end{aligned}$$
(14)

Furthermore, \(\Theta _{\mu \nu }\) symbolizes the stress-energy tensor associated with the anisotropic matter configuration reads

$$\begin{aligned} \left( \Theta ^{~\mu }_{\nu }\right) =dia \textit{g}\left( \varepsilon ,-P_{r},-P_{\bot },-P_{\bot }\right) , \quad P_{r}\ne P_{\bot }. \end{aligned}$$
(15)

Here, \(P_{r}\) and \(P_{\bot }\) denote the radial and tangential pressures, respectively. Now, we construct astrophysical compact models characterizing the fuzzy BHs using the Einasto DM parametrization. In this respect, we assume the self-gravitating astrophysical static sphere having mass M featuring with a density profile given in Eq. (10). It is important to note that this profile corresponds to the Gaussian profile by considering \(\alpha =1/2\) and \(\eta =\sqrt{\theta }\) [25]. Consider the line element characterizing the interior of self-gravitating compact static distribution with spherical symmetric in co-moving coordinates \(\{t,r,\theta ,\phi \}\) as

$$\begin{aligned} ds^{2}&=\textit{g}_{\mu \nu }dx^{\mu }dx^{\nu },\quad (\textit{g}_{\mu \nu })\nonumber \\&=diag \Big (f(r),-\frac{1}{f(r)},-r^{2},-r^{2}\sin ^{2}\theta \Big ). \end{aligned}$$
(16)

Then, the combination of Eqs. (13) and (16) produces

$$\begin{aligned} 6\beta \frac{d \textrm{R}}{dr} f&=8\pi \Theta +2\beta \textrm{R}^{2}+(1-2\beta \textrm{R})\textrm{R}\nonumber \\&\quad +3\alpha \frac{d \textrm{R}}{dr}\frac{d f}{dr}\left( f^{2}-\frac{f}{r}+1\right) , \end{aligned}$$
(17)

where

$$\begin{aligned} \textrm{R}=\frac{1}{r^{2}}\left( \frac{d^{2} f}{dr^{2}}r^{2}+4\frac{d f}{dr}r+2f-2\right) . \end{aligned}$$
(18)

Then the “tt” and “rr” and “\(\theta \theta \)” components of the equation of motion corresponding to the Starobinsky model read

$$\begin{aligned} 8\pi \varepsilon&=\frac{1}{r^{2}}\frac{d}{dr}\Big [r\left( 1-f\right) \Big ](1+2\beta \textrm{R})\nonumber \\&\quad -2\beta f\left[ \left( \frac{2}{r}+\frac{1}{2f}\frac{df}{dr}\right) \frac{d\textrm{R}}{dr}+\frac{d^{2} \textrm{R}}{dr^{2}}\right] -\frac{1}{2}\beta \textrm{R}^{2}, \end{aligned}$$
(19)
$$\begin{aligned} 8\pi P_{r}&=\frac{1}{r}\left[ \frac{df}{dr}-\frac{1}{r}\left( 1-f\right) \right] (1+2\beta \textrm{R})\nonumber \\&\quad -2\beta f\left( \frac{2}{r}+\frac{1}{2f}\frac{df}{dr}\right) \frac{d\textrm{R}}{dr}-\frac{1}{2}\beta \textrm{R}^{2}, \end{aligned}$$
(20)
$$\begin{aligned} 8\pi P_{\bot }&=\frac{1}{r}\left( \frac{df}{dr}+\frac{r}{2}\frac{d^{2}f}{dr^{2}}\right) (1+2\beta \textrm{R})\nonumber \\&\quad -2\beta f\left[ \left( \frac{1}{r}+\frac{1}{f}\frac{df}{dr}\right) \frac{d \textrm{R}}{dr}+\frac{d^{2} \textrm{R}}{dr^{2}}\right] -\frac{1}{2}\beta \textrm{R}^{2}, \end{aligned}$$
(21)

The left-hand side of Eq. (12) divergence-free (generalized Bianchi identity), which implies

$$\begin{aligned} 0\equiv \Theta ^{\mu \nu }_{~~~;\nu }=\partial _{\nu }\Theta ^{\mu \nu }+\Gamma ^{\mu }_{\nu \alpha }\Theta ^{\alpha \nu } +\Gamma ^{\nu }_{\nu \alpha }\Theta ^{\mu \alpha }, \end{aligned}$$
(22)

which gives

$$\begin{aligned} \frac{d\Theta ^{~r}_{r}}{dr}&=-\frac{1}{2}\textit{g}^{tt} \frac{d\textit{g}_{tt}}{dr}\left( \Theta ^{~t}_{t}-\Theta ^{~t}_{t}\right) -\frac{1}{2}\textit{g}^{\theta \theta } \frac{d\textit{g}_{\theta \theta }}{dr}\left( \Theta ^{~r}_{r}-\Theta ^{~\theta }_{\theta }\right) \nonumber \\&\quad -\frac{1}{2}\textit{g}^{\phi \phi } \frac{d\textit{g}_{\phi \phi }}{dr}\left( \Theta ^{~r}_{r}-\Theta ^{~\phi }_{\phi }\right) . \end{aligned}$$
(23)

Then, according to the stress-energy momentum (14), the above equation becomes

$$\begin{aligned} \frac{d P_{r}}{d r}=-\frac{1}{2f}\frac{d f}{d r}(\varepsilon +P_{r}) -\frac{2}{r}(P_{r}-P_{\bot }), \end{aligned}$$
(24)

which is the standard form of hydrostatic equilibrium expression for anisotropic relativistic stellar distribution with spherical symmetry. The expression for relativistic mass (m(r)) associated with the non-rotating and static astrophysical configuration as suggested by Misner and Sharp [69] is

$$\begin{aligned} f(r)\equiv 1-\frac{8\pi }{r}\int ^{r}_{0}z^{2}\Theta ^{~t}_{t}(z)dz=1-\frac{2m}{r}. \end{aligned}$$
(25)

Feeding back Eq. (25) into Eq. (14), we have

$$\begin{aligned}&(1+2\beta \textrm{R})\frac{dm}{dr}=4\pi \varepsilon r^{2}+\frac{r^{2}}{2}\beta \textrm{R}^{2}\nonumber \\&\quad +\beta r^{2}\left( 1-\frac{2m}{r}\right) \left\{ \left[ \frac{2}{r}+\left( \frac{1}{r-2m}\right) \left( \frac{m}{r} -\frac{dm}{dr}\right) \right] \right. \nonumber \\ {}&\quad \left. \frac{d\textrm{R}}{dr}+\frac{d^{2}\textrm{R}}{dr^{2}}\right\} , \end{aligned}$$
(26)

while \(\frac{df}{dr}\) is obtained from rr-component of the field equation

$$\begin{aligned}&\frac{1}{2f}\frac{df}{dr}=\frac{2m}{r(r-2m)}+\frac{r}{(r-2m)(1+2\beta r)}\nonumber \\&\quad \times \left\{ 8\pi P_{r}+\frac{\beta }{2} \textrm{R}^{2}{+}\beta \left[ 2(r-2m)+\left( m{-}r\frac{dm}{dr}\right) \frac{d\textrm{R}}{dr}\right] \right\} . \end{aligned}$$
(27)

The value of curvature scalar \(\textrm{R}\) using mass m(r) reads

$$\begin{aligned} \textrm{R}(r)=-\frac{2}{r^{2}}\left( r\frac{d^{2}m}{dr^{2}}+2\frac{dm}{dr}\right) . \end{aligned}$$
(28)

Now, by imposing the condition \(\Theta ^{~r}_{r}=\Theta ^{~t}_{t}=-\varepsilon \), the conservation equation gives the following result

$$\begin{aligned} \Theta ^{~\theta }_{\theta }=-\Theta ^{~t}_{t}-\frac{r}{2}\frac{d\varepsilon }{dr}. \end{aligned}$$
(29)

Thus, instead of being a structure-less point, a source is revealed to be a anisotropic self-gravitating with energy density \(\varepsilon (r)\), radial pressure \(P_{r}=-\varepsilon (r)\), and tangential pressure

$$\begin{aligned} P_{\bot }(r)=-\varepsilon (r)-\frac{r}{2}\frac{d\varepsilon }{dr}. \end{aligned}$$
(30)
Fig. 1
figure 1

Behavior of radial pressure \(P_{r}\) (left panel) and tangential pressure \(P_{\bot }\) (right panel) versus r subject to the de Sitter EOS corresponding to different values of \(\alpha \)

Full size image

We notice that both the stress components (\(P_{r},P_{\bot }\)) satisfy the expected physical constraints at the horizon(s), with respect to an origin, and asymptotically far away. Furthermore, \(P_{r}\) is a strictly increasing function that approaches zero as \(r\rightarrow \infty \). Also, \(P_{\bot }\) approaches zero as \(r\rightarrow \infty \) as shown in Fig. 1. This approach effectively shows that the density distribution ((10)) can be used to model a non-rotating, spherical droplet of self-gravitating fluid with unequal principle stresses. Now, we solve Starobinsky field equations, conservation expression (24) and the values of \(P_{r}\) (29) and \(P_{\bot }\) (30), along with incorporating the condition that the metric transitions to the Minkowski metric at infinity, we obtain the metric (16) with

$$\begin{aligned} \textit{g}_{tt}\equiv f(r)=1-\frac{2m}{r}. \end{aligned}$$
(31)

To examine the regularity of the relativistic metric potential f(r), we introduce a scaled mass parameter, \(\xi =M/\eta \). Moreover, considering f as a function of \(\frac{r}{\eta }\) reveals the existence of a critical value for the scaled mass parameter, \(\xi _{0}\). At this specific value, f possesses a double root at \(y_{0} = r_{0}/\eta \). Furthermore, for \(\xi \) exceeding the critical value \(\xi _{0}\), two distinct horizons emerge at radii \(r_{I}\) and \(r_{II}\). Conversely, no horizons form within the metric when \(\xi \in (0,\xi _{0})\). Figure 2 displays the behavior of the metric potential f, showcasing the existence of zero, one, or two horizons based on the Einasto index \(\xi \). It is significant to mention that, unlike the Schwarzschild BH with its singularity at \(r=0\), our findings indicate a regular de Sitter core in the central region. This suggests that the Einasto parametrization, with an anisotropic fluid distribution, resolves the central singularity issue. However, this solution relies on assuming an EOS for radial pressure of the form \(\Theta ^{~r}_{r}=-\Theta ^{~t}_{t}\). Crucially, for \(\xi <\xi _{0}\), the interior region surrounding the origin exhibits a de Sitter-like metric, thereby eliminating the possibility of a naked singularity.

Fig. 2
figure 2

Behavior of metric function f versus \(u=(\frac{r}{\eta })^{1/\alpha }\) with \(\xi =0.5\) subject to the de Sitter EOS corresponding to various values of \(\alpha \). When \(\xi =1.1 >\xi _{0}\) (indicated by the cyan-colored line), two horizons are observed. Conversely, in the case of \(\xi =0.8<\xi _{0}\), there is no horizon (the same applies to other \(\xi \)-values). This case is interesting because it deals with a self-gravitating compact droplet where the fluid features unequal principal stresses

Full size image

Now, we will determine the Hawking temperature \(T_{H}\) of BHs consisting of DM for Starobinsky model of gravity. Therefore, the temperature of the BHs can be computed using the following formula (see [25, 59, 70] and references therein)

$$\begin{aligned} T_{H}=\frac{1}{4\pi }\left( \frac{df}{dr}\right) _{r=r_{H}}, \end{aligned}$$
(32)

where the event horizon of the astrophysical configuration is locate at \(r_{H}\).

Fig. 3
figure 3

Behavior of Hawking temperature \(T_{H}\) versus \(x=(\frac{r}{\eta })^{1/\alpha }\) subject to the de Sitter EOS corresponding to \(\alpha =3\)

Full size image

Figure 3 describes that the temperature \(T_{H}\) of the Starobinsky BH increases with decrease in horizon radius. The temperature initially increases with a shrinking horizon. However, this rise reaches a maximum value before dropping sharply and ultimately vanishing at the radius of the extremal BH (\(r_{H}=r_{0}\)). Moreover, for an extremal BH, the value of \(T_{H}\) must be zero as the metric potential f possesses two coincident roots at \(r=r_{0}\). As opposed to witnessing a divergent pattern in the BH temperature, we discover that the evaporation process culminates in a stable extremal BH that has a final state of zero temperature. The characteristics of this final configuration are determined by the triplet \(\{\alpha ,\eta ,\xi \}\). As highlighted in reference [71], a finite final temperature would prevent significant back-reaction. This absence of back-reaction suggests the stability of our solution throughout the BH lifespan, encompassing its entire evaporation process until reaching the final configuration. It is worth noting the nomenclature at this stage. An entity without a horizon is denoted as a fuzzy self-gravitating droplet, while the presence of atleast one horizon leads to its classification as a fuzzy BH.

4 The effective potential

This section investigates whether the central BH in our galaxy, with a mass of \(4.1\times 10^{6}\) \(M_{\odot }\) and a Schwarzschild radius \(R_{BH}=3.92\times 10^{7}\) pc [28, 29], can be modeled using the scheme of Starobinsky DM-BHs that was developed in Sect. 3. Next, we need to figure out the specific values of the important factors in these self-gravitating models. This process can be divided into two steps. Initially, we ensure that the value of M, incorporated into the line element (16) via the metric potential f, aligns with the BH-mass \(M_{BH}\). We then constrain the mass m to closely match the BH mass \(M_{BH}\). This matching is achieved by evaluating m at the minimum point (\(x_{min}\)) of the Schwarzschild effective potential, which describes the effective potential experienced by a massive particle due to the presence of the BH. To achieve this, we need

$$\begin{aligned} 1-\frac{m(x_{min})}{M_{BH}}\le 10^{-2}. \end{aligned}$$
(33)

Our analysis will show that this requirement ensures two things:

  1. 1.

    The minimum value of the regular BH’s effective potential (Schwarzschild potential) is the same as the minimum value of the effective potential for our self-gravitating fuzzy object.

  2. 2.

    These two effective potentials are very similar not only at their minimum points but also in a broad region around them (see Fig. 4, right panel).

In this respect, it is worth recalling that for a metric ansatz (14), the radial geodesic takes the form of an energy conservation equation, defined as (see [72] for details)

$$\begin{aligned} \frac{1}{2}\left( \frac{d r}{d\tau }\right) ^{2}+U_{eff}(r)=constant . \end{aligned}$$
(34)

Here, the parameter \(\tau \) serves a dual role. It signifies proper time when describing the world-line associated with massive particle, and it acts as an affine parameter for massless particles. An equation for the effective potential associated with the metric (16) reads as follows [72]

$$\begin{aligned}&U_{eff}=\frac{\Omega ^{2}}{2r^{2}}\left( 1- \frac{2m}{r}\right) - \mathcal {X}\left( \frac{m}{r}\right) , \quad with \nonumber \\ {}&\quad {\displaystyle \mathcal {X}={{\left\{ \begin{array}{ll}+1&{}{\text {if }}m_{p}\ne 0\\ -1&{}{\text {if }}m_{p}=0\\ \end{array}\right. }}}. \end{aligned}$$
(35)

Here, we use \(\Omega \) to symbolize the total angular momentum and \(m_{p}\) to signify the test particle’s mass. It is crucial to highlight that the effective potential for regular BH can be obtained from Eq. (35) by simply replacing m with \(M_{BH}\). It is important to note that the effective potential for the Schwarzschild metric can be derived directly from Eq. (35) by simply substituting \(M_{BH}\) for the mass function m. By introducing the dimensionless variables, we can redefine the angular momentum and the radial variable as \({\widetilde{L}}=\frac{\Omega }{r_{s}}\) and \(x^{\star }=\frac{r}{r_{s}}\), respectively with \(r_{s}=2M_{BH}\). The Schwarzschild effective potential reads

$$\begin{aligned} U_{eff(S)}(x^{\star })=-\frac{{\widetilde{L}}^{2}}{2{x^{\star }}^{3}}+\frac{{\widetilde{L}}^{2}}{2{x^{\star }}^{2}}-\frac{\mathcal {X}}{2x^{\star }}. \end{aligned}$$
(36)

The event horizon is presently situated at \(x^{\star }=1\). The maximum and minimum values associated with \(U_{eff(S)}(x^{\star })\) are given as

$$\begin{aligned} x^{\star }_{max}&=\frac{{\widetilde{L}}^{2}}{2}\left[ 1-\left( 1-\frac{3}{{\widetilde{L}}^{2}}\right) ^{1/2}\right] , \nonumber \\ x^{\star }_{min}&=\frac{{\widetilde{L}}^{2}}{2}\left[ 1+\left( 1-\frac{3}{{\widetilde{L}}^{2}}\right) ^{1/2}\right] , \end{aligned}$$
(37)

such that \({\widetilde{L}}>\sqrt{3}\). Rescaling the effective potential (35) identically in the massive case leads to

$$\begin{aligned} U_{eff}=\frac{{\widetilde{L}}^{2}}{2{x^{\star }}^{2}}-\left( \frac{1}{x^{\star }}+\frac{{\widetilde{L}}^{2}}{{x^{\star }}^{3}}\right) \frac{m}{r_{s}}, \end{aligned}$$
(38)

where the value of m in terms of \(M_{BH}\) can be obtained through the expression (26). To verify the existence of solutions (non-empty solution set) associated with the inequality (33), we will initially consider various choices of \(\mathbb {H}\). This initial step aims to find values for \(\mathbb {H}\) that lead to related scale factors, \(\eta = r_{s}\mathbb {H}\), possessing the similar order of magnitude as presented in [63, 65, 73]. We will solve the inequality (33) for the parameter \(\alpha \), considering each possible value of \(\mathbb {H}\). The outlined procedure requires us to additionally fix the value of \({\widetilde{L}}\), as \(x^{\star }_{min}\) relies on the rescaled value of angular momentum. As a reference point, studies in [63, 73] report a scaling factor of \(\eta _{e} = 2.121\times 10^{9}\) kpc for a DM halo. In our model, choosing \(\mathbb {H}=10\) yields a corresponding scaling factor of \(\eta = 3.92\times 10^{9}\) kpc. To determine the allowed values of the parameter \(\alpha \) in inequality (33), we will explore various combinations of \({\widetilde{L}}\) and \(x^{\star }_{min}\). For example, if \({\widetilde{L}}=5\) and \(x^{\star }_{min} = 25 + 5\sqrt{22}\), then \(\alpha <0.80\). However, for a larger system with \({\widetilde{L}} = 100\) and \(x^{\star }_{min}= 104 + 10\sqrt{9997}\), \(\alpha < 2.73\) is achieved. According to our model, will the system form a fuzzy self-gravitating BH or a compact self-gravitating droplet when \(\mathbb {H}=10\) and \(\alpha \) is selected such that it fulfills inequality (33)? To address this question, we begin by recognizing a key relationship in geometric units: \(r_{s}=2M_{BH}\). This allows us to rewrite \(M_{BH}=\frac{r_{s}}{2}\). Consequently, the rescaled mass \(\xi \) for this solution will be determined by

$$\begin{aligned} \xi =\frac{M_{BH}}{\eta }=\frac{r_{s}}{2\eta }=\frac{1}{2\mathbb {H}} \quad with \quad \eta =r_{s}\mathbb {H}, \end{aligned}$$
(39)

which determines the metric potential f in terms of \(\mathbb {H}\) (Figs. 5, 6).

Fig. 4
figure 4

Behavior of effective potentials \(U_{eff(S)}\) (pink line) and \(U_{eff}\) (gold line) in the massless case (left panel), and \(U_{eff(S)}\) (gold asterisk-line) and \(U_{eff}\) (red line) in the massless case (right panel), with \(\mathbb {H}=6\), \({\widetilde{L}}=8\), and \(\xi =0.2\)

Full size image

5 Fuzzy dark matter self-gravitating droplets

In Sect. 4, we assumed a de Sitter-type EOS (\(\varepsilon =-P_{r}\)) subject to the anisotropic distribution of matter within the formalism of \(\textrm{R}+\beta \textrm{R}^{2}\) gravity model. This is because using a isotropic (equal principle stresses) with the generalized relativistic conservation expression would result in a system with more equations than unknowns. By considering the Einasto parametrization as our starting point, we were able to demonstrate that the eventual result either a “fuzzy BH” or a “diffused, static self-gravitating droplet” is contingent upon the specific value of the parameter \(\xi \). This section provides another example using a different EOS. This example reinforces the idea that a key feature for a “fuzzy BH” or a “fuzzy, static self-gravitating droplet” to have a regular (non-singular) center seems to be a negative radial pressure, at least in some region. To achieve this, we need to choose a specific EOS and a corresponding stress-energy tensor. The diffuse nature of these profiles raises the question of non-locality. Given that the pressure is distributed across the system, it is reasonable to expect that changes in radial pressure would not solely rely on local variations in energy density. This suggests that the entire volume might be involved, hinting at possible interactions beyond immediate surroundings, which is characteristic of nonlocal phenomena. To incorporate non-locality, we could follow the approach outlined in studies [74,75,76]. This method suggests that the stress-energy tensor at a specific point in metric does not just depend on the local conditions there. Instead, it is also influenced by the average energy density distribution across the entire object (enclosed configuration). Specifically, we examine an anisotropic fluid characterized by a non-local EOS, defined as

$$\begin{aligned} P_{r}=\varepsilon -\frac{2}{r^{3}}\int ^{r}_{0}\rho (z)z^{2}dz. \end{aligned}$$
(40)

We now explore an alternative metric ansatz depicting a non-rotating, spherical stellar configuration governed by the Einasto parameterization, defined as

$$\begin{aligned}&ds^{2}=\textit{g}_{\mu \nu }dx^{\mu }dx^{\nu },\nonumber \\&(\textit{g}_{\mu \nu })=diag \Big (S^{2}(r),-\frac{1}{T(r)},-r^{2},-r^{2}\sin ^{2}\theta \Big ). \end{aligned}$$
(41)
Fig. 5
figure 5

Behavior of effective potentials \(U_{eff(S)}\) (gray line) and \(U_{eff}\) (red line) in the massive case with \(\mathbb {H}=0.1\), \({\widetilde{L}}=4\), and \(\xi =1.0\)

Full size image
Fig. 6
figure 6

Behavior of metric potential f versus \(x^{\star }\) with \(\mathbb {H}=0.1\), \(\xi =5\) (left panel) and \(\mathbb {H}=10\), \(\xi =0.05\) (right panel) corresponding to different values of \(\alpha \)

Full size image

Therefore, the non-null components associated with the Starobinsky gravity and the aniostropic matter distribution (15), read as

$$\begin{aligned} 8\pi \varepsilon&=\frac{1}{r^{2}}\frac{d}{dr}\Big [r\left( 1-T\right) \Big ](1+2\beta \textrm{R})\nonumber \\&\quad -2\beta T\left[ \left( \frac{2}{r}+\frac{1}{2T}\frac{dT}{dr}\right) \frac{d\textrm{R}}{dr}+\frac{d^{2} \textrm{R}}{dr^{2}}\right] -\frac{1}{2}\beta \textrm{R}^{2}, \end{aligned}$$
(42)
$$\begin{aligned} 8\pi P_{r}&=\frac{1}{r}\left[ \frac{2T}{S}\frac{dS}{dr}-\frac{1}{r}\left( 1-T\right) \right] (1+2\beta \textrm{R})\nonumber \\&\quad -2\beta T\left( \frac{2}{r}+\frac{1}{S}\frac{dS}{dr}\right) \frac{d\textrm{R}}{dr}-\frac{1}{2}\beta \textrm{R}^{2}, \end{aligned}$$
(43)
$$\begin{aligned} 8\pi P_{\bot }&=\frac{1}{2rS}\left[ 2T\frac{dS}{dr}+2rT\frac{d^{2}S}{dr^{2}}+\left( S+r\frac{dS}{dr}\right) \frac{dT}{dr}\right] \nonumber \\ {}&\quad \times (1+2\beta \textrm{R})\nonumber \\&\quad -2\beta T\left[ \left( \frac{1}{r}+\frac{1}{S}\frac{dS}{dr}+\frac{1}{2T}\frac{dT}{dr}\right) \frac{d \textrm{R}}{dr}+\frac{d^{2} \textrm{R}}{dr^{2}}\right] \nonumber \\&\quad -\frac{1}{2}\beta \textrm{R}^{2}. \end{aligned}$$
(44)
Fig. 7
figure 7

Behavior of radial pressure \(P_{r}\) (left panel) and tangential pressure \(P_{\bot }\) (right panel) versus \(x=(\frac{r}{\eta })^{1/\alpha }\) subject to the non-local EOS corresponding to different values of \(\alpha \)

Full size image

Applying the divergence-free condition, also known as the conservation relation (\(\nabla _{\nu }T^{\mu \nu }=0\)), yields the following form of the hydrostatic equilibrium equation

$$\begin{aligned} \frac{d P_{r}}{d r}=-\frac{1}{S}\frac{d S}{d r}\left( \varepsilon +P_{r}\right) -\frac{2}{r}\left( P_{r}-P_{\bot }\right) . \end{aligned}$$
(45)

The mass parameter m(r) associated with the geometry (41) is defined by the relation

$$\begin{aligned} T(r)=1-\frac{2m}{r}. \end{aligned}$$
(46)

Substituting (46) into (42), we obtain

$$\begin{aligned}&\frac{dm}{dr}=\frac{1}{\left\{ 2(1+2\beta \textrm{R})+\frac{2\beta r^{2}T}{(r-2m)}\frac{d\textrm{R}}{dr}\right\} }\nonumber \\&\quad \times \left[ 8\pi \varepsilon r^{2}+2\beta r^{2}T\left\{ \frac{d^{2} \textrm{R}}{dr^{2}}+\left( \frac{2r-3m}{r(r-2m)}\right) \right\} +\frac{\beta }{2}r^{2} \textrm{R}^{2}\right] .\nonumber \\ \end{aligned}$$
(47)

On the other hand, the expression for \(\frac{dS}{dr}\) is obtained from (43) as

$$\begin{aligned}&\frac{1}{S}\frac{dS}{dr}=\frac{1}{\left\{ \frac{2}{r^{2}}(r-2m)(1+2\beta \textrm{R})-2\beta T\frac{d\textrm{R}}{dr}\right\} }\nonumber \\&\quad \times \left[ 8\pi P_{r}+\frac{2m}{r^{3}}(1+2\beta \textrm{R})+\frac{4\beta }{r}\frac{d\textrm{R}}{dr}T+\frac{\beta }{2}\textrm{R}^{2}\right] . \end{aligned}$$
(48)

The integration of the above expression provides

$$\begin{aligned} S^{2}(r)=e^{\varphi (r)}, \quad with \quad \varphi (r)=\int \Phi (r)dr, \end{aligned}$$
(49)

and

$$\begin{aligned}&\Phi (r)=\frac{1}{\left\{ \frac{1}{r^{2}}(r-2m)(1+2\beta \textrm{R})-\beta T\frac{d\textrm{R}}{dr}\right\} }\nonumber \\&\quad \left[ 8\pi P_{r}+\frac{2m}{r^{3}}(1+2\beta \textrm{R})+\frac{4\beta }{r}\frac{d\textrm{R}}{dr}T+\frac{\beta }{2}\textrm{R}^{2}\right] . \end{aligned}$$
(50)

Then, using Eqs. (45) and (48), we obtain

$$\begin{aligned}&P_{\bot }=P_{r}+\frac{r}{2}\left[ \frac{dP_{r}}{dr}\right. \nonumber \\ {}&\quad \quad \left. +\frac{\varepsilon +P_{r}}{T}\frac{1}{\left\{ \frac{2}{r^{2}}(r-2m)(1+2\beta \textrm{R})-2\beta T\frac{d\textrm{R}}{dr}\right\} }\right. \nonumber \\&\quad \left. \left\{ 8\pi P_{r}+\frac{2m}{r^{3}}(1+2\beta \textrm{R})+\frac{4\beta }{r}\frac{d\textrm{R}}{dr}T+\frac{\beta }{2}\textrm{R}^{2}\right\} \right] . \end{aligned}$$
(51)

Here, it is crucial to emphasize two observations. Firstly, the metric potential T(r) is identical to \(\textit{g}_{rr} = -\textit{g}^{-1}_{tt}(r)\) in the spherically symmetric metric (14). This indicates that the analysis of the zeros of \(\textit{g}_{rr}\) conducted in Sec. III is applicable to the current scenario. Furthermore, T(r) appears in the denominator of (51). Furthermore, the metric potential T(r) is featured in (51) in the denominator, leading to the \(P_{\bot }\) becoming singular at the zeros of T(r). Alternatively, the static metric ansatz (41) can be cast into the form

$$\begin{aligned}&ds^{2}=\textit{g}_{\mu \nu }dx^{\mu }dx^{\nu },\nonumber \\&(\textit{g}_{\mu \nu })=diag \left( e^{\varphi (r)},-\left( 1-\frac{2m}{r}\right) ^{-1},-r^{2},-r^{2}\sin ^{2}\theta \right) . \end{aligned}$$
(52)

which describes the geometry of wormholes [23, 77, 78]. The function \(e^{\varphi (r)}\) determines the gravitational redshift profile, while m(r) defines the spatial configuration of the wormhole. These functions are regarded as the redshift function and shape function, respectively. Assuming an anisotropic stress-energy tensor imposes two key requirements.

  • The tangential pressure, \(P_{\bot }(r)\), must remain finite.

  • The metric potential T(r) needs to have two zeroes.

We will explore the possibility of this wormhole existing in future work, and will sidestep the latter condition by requiring \(\xi < \xi _{0}\). It is worth mentioning that this requirement ensures that the metric variable \(\varphi (r)\) remains well-behaved throughout its domain. This is because it prevents the function T(r) (appearing in Eq. (50)) from having any real number solutions (roots). As a consequence, the metric described by Eq. (52) represents a fuzzy, self-gravitating DM droplet. Figure 7 (right panel) displays the behavior of the \(P_{\bot }\), which shows that it it remains regular for all values of r, as long as the condition \(\xi < \xi _{0}\) holds. Furthermore, Fig. 7 (left panel) depicts the behaviors of radial pressures using the various \(\alpha \)-values. Interestingly, under a non-local EOS, the self-gravitating droplet features no singularity at \(r=0\). This can be straightforwardly verified using the formula for the Kretschmann scalar, a quantity used to measure the curvature of spacetime, for the metric (52) as

$$\begin{aligned} \mathcal {K}\equiv R^{\mu \nu \gamma \delta }R_{\mu \nu \gamma \delta }&=\frac{1}{4}\left[ \Phi \frac{dT}{dr}+2T\frac{d\Phi }{dr}+ \Phi ^{2}T\right] ^{2}\nonumber \\ {}&\quad +\frac{2}{r^{2}}\left[ \Phi ^{2}T^{2}+\left( \frac{dT}{dr}\right) ^{2}\right] . \end{aligned}$$
(53)

It is worth noting that due to the inclusion of the term \(1/r^{2}\) in Eq. (53), it is not immediately evident whether the \(\mathcal {K}\) is singularity-free at \(r=0\) in the model under consideration. Using Maple, we confirmed that the Einasto model does not feature a central curvature singularity (for details see [59, 70]).

6 Summary and discussions

The ETG models, featuring higher-order corrections, provide effective approaches for modeling astronomically dark self-gravitating systems, such as neutron stars, BHs, and white dwarfs [38, 43, 79,80,81,82,83]. In addition to providing additional insights into cosmic evolution, the ETG models also hold the potential to recover the outcomes of Einstein’s GR [42]. Among the increasingly favored modified cosmological theories is the quadratic \(f(\textrm{R})\) gravity model, which is also known as Starobinsky gravity. In this direction, the abundance of DM in the galactic structures led us to explore a possible correspondence between DM and BHs, using the Einasto parametrization to describe the distribution of DM within the context of Starobinsky gravity. We investigated how coupling the Einasto DM parametrization with an anisotropic fluid distribution governed by the chosen EOS allows for obtaining different BH solutions against the backdrop of \(\textrm{R}+\beta \textrm{R}^{2}\) modified corrections. These objects can be classified as horizonless (droplets) or possessing one or two horizons (black holes). The choice of de Sitter-like EOS (\(\varepsilon =-P_{r}\)) gives rise to the astronomically dark stellar configurations, such as BHs or self-gravitating droplets, depending on the specific values of the mass parameter \(\xi \). We stress that these configurations are solutions to the field equations associated with the Starobinsky model of gravity and correspond to regular, stable fuzzy self-gravitating structures. The nature of these astrophysical objects is determined by the presence and number of event horizons. Objects with one or two horizons are classified as fuzzy BHs, while those lacking a horizon are categorized as self-gravitating droplets. These DM configurations exhibit characteristics that differ from the DM clump models presented in [18]. Although both approaches investigate a link between the central galactic object and DM, the nature of the DM itself differs significantly.

We discovered a non-monotonic relationship between the Hawking temperature and horizon radius for Einasto-inspired BHs. The temperature initially exhibits a positive correlation with a shrinking horizon, reaching a maximum point. However, it then undergoes a rapid decline and reaches zero precisely at the radius of an extremal BH. Our approach allows for the construction of an effective potential for both fuzzy self-gravitating droplets and fuzzy BHs. This potential governs the equations of motion for particles orbiting these objects, resulting in trajectories similar to those observed around standard galactic BHs. The choice of EOS plays a crucial role in the resulting configuration. Our prior investigation with de sitter-like EOS demonstrated the possibility of forming either self-gravitating droplets or BHs. However, employing a nonlocal EOS allows us to construct solely self-gravitating droplets. Unfortunately, it seems that a negative \(P_{r}\) might be an inherent feature of this approach. Investigation of the effective potential uncovers two key features. First, it allows for bound states to form, signifying the trapping of particles. Second, it reveals the presence of an unstable orbit for massive particles when their angular momentum is low. We highlight that the Einasto parametrization generalizes the Gaussian distribution. It is worth noting that BH solutions based on similar density parameterizations have been previously explored in the literature. Our findings suggest that analyzing perturbations [84] or exotic spacetime structures using a more general framework [85] (like a different coordinate system) would likely follow similar principles.

This study enables us to bring together concepts from fuzzy DM, pressure anisotropy, and modified gravity theories to propose spherically symmetric solutions that can account for both BHs and self-gravitational droplets found in galactic centers. Besides modeling the supermassive central galactic objects, these astrophysical solutions could be beneficial for reproducing the motion of the S-stars, which are significant tools for understanding the features of BHs. In this respect, the authors of [86] examined the DM nature of the SgrA* through the motion of the S-stars. This study is pertinent to our work as it arrives at comparable conclusions using a distinct methodology. They demonstrated that the astrometry data of S2 and G2 are more accurately matched by geodesics within the geometry of a self-gravitating distribution of DM consisting of 56 keV-fermions, termed “darkinos”. This model also accounts for the rotation curves of the outer halo of the Milky Way galaxy. By analyzing the astrometry data of the 17 most well-resolved S-stars, they further supported the proposition that SgrA* serves as a dense core composed of darkinos. By employing numerical simulations, they demonstrated that a DM mass can not only replicate the observed kinematics of S-stars but also potentially explain the G2 anomaly [59, 87].