1. Introduction

The presence of enigmatic phenomena such as dark matter (DM) and dark energy (DE) has been unveiled by cosmological discoveries such as cosmic accelerated expansion and the rotation curves of galaxies. The DE is a peculiar form of energy that is responsible for the expansion of the universe, while DM is a form of nonbaryonic matter that does not emit or absorb electromagnetic radiation and is assumed to exist by its gravitational influence on baryonic matter. Scientific investigations of galaxy rotational curves and differences in mass between galactic clusters provide the existence and importance of DM in star formation [1, 2]. According to Planck statistics [3], approximately 27% of the universe is composed of DM, while baryonic matter makes up only 5%, and the remaining 68% is made up of DE.

Several researchers have proposed various models to investigate the characteristics of DE and DM. The Λ CDM model, which includes a (Λ) cosmological constant to represent vacuum energy within the framework of general relativity (GR), has been utilized for this purpose. However, this approach has failed due to issues such as “fine-tuning” [4] and “cosmic coincidence” [5]. As a result, modified theories have garnered significant attention [6]. One of the simplest extensions of GR is called f(R) gravity [7], which involves replacing the R scalar curvature in the Einstein action with a f(R) generic function. Different models of f(R) have been investigated to determine their compatibility with various cosmological events, including the inflationary phase [8], late-time cosmic evolution [9], and their ability to pass solar system testing [10] and astrophysical constraints [11]. Sharif and Saleem [12] have also examined the stability of Einstein universe with anisotropic homogeneous perturbations for various f(R) models.

Einstein’s theory of GR can be derived using two distinct approaches: the metric formalism and the Palatini formalism, which was introduced by Albert Einstein. The Palatini formalism uses both the metric tensor and the affine connection components, while the metric formalism only uses the metric tensor. In modified gravity theories, such as f(R) modified gravity, the gravitational field equations obtained through the two approaches can differ, particularly in terms of the order of the field equations. The Palatini approach typically yields second-order partial differential equations, while the metric approach often produces higher-order derivative field equations. Additionally, the Palatini variational formulation introduces several new algebraic relationships that depict the complicated relation between the matter fields and the affine connection. This connection can be determined through a group of equations that link it not only to the metric but also to the matter fields. The use of minimally coupled gravity theories, particularly (R, T) gravity, in the context of gravitational collapse, is motivated by their ability to generalize GR, incorporate dark matter and energy, modify collapse dynamics, align with observational data, offer theoretical flexibility, influence gravitational wave emission, and explore energy dissipation mechanisms. These comprehensive advantages make (R, T) gravity an invaluable tool for advancing our understanding of gravitational collapse and the broader dynamics of the universe [13, 14]. The Palatini formulation of f(R, T) gravity [15] in astrophysics and cosmology has also been extensively studied. Wojnar and Velten [16] examined the neutron stars within the context of Palatini f(R) theory and analyzed a few viable solutions for stability. In the same context, Wojnar [17] further investigated the stability of neutron stars. Bhatti et al. [18] also explored the stability scenario of neutron stars by considering the statically symmetric geometry consisting of an isotropic perfect fluid in the context of Palatini f(R, T) theory. In the same framework, Barrientos and Rubilar [19] studied the curvature singularities and examined the possibility to remove them using specific models of f(R, T) models.

The emergence of astronomical bodies like galaxies, planets, and star clusters and the production of gravitational waves (GWs) and radiation are consequences of gravitational collapse. As an outcome of this collapse, a cosmic object that was once stable can become unstable when influenced by its own gravitational pull. Scientists have studied the effects of gravitational collapse on the evolution of stellar distributions. Ostriker [20] investigated the behavior of compressible filaments with cylindrical symmetry, while Breysse et al. [21] studied the stability of self-gravitating fluid structures with cylindrical symmetry. Several other researchers [22–25] have also explored filamentary objects using cylindrical symmetry in both analytical and numerical contexts. The stability analysis of filamentary structures has been studied in various gravitational theories. Sharif and Manzoor [26] explored Brans–Dicke theory, while Zubair et al. [27] analyzed cylindrically symmetric collapsing objects in f(R, T) theory. Guha and Ghosh [28] explored the dynamical instability of gravitational collapse in the same gravity. Hoemann et al. [29] studied the collapsing behavior of filament structures in a two-phase process and calculated the time it takes for the objects to collapse.

Sharif and Manzoor [30] studied the stellar objects in the presence of mysterious elements (DM and DE) and discussed the dynamics of the evolutionary process which may expose the hidden change within the structure formation of the cosmos. In the framework of GR, Herrera and Santos [31] analyzed the matching conditions for a collapsing anisotropic cylindrical fluid and found that the pressure at the cylinder’s surface is nonzero. In the context of f(G) gravity, Sharif and Fatima [32] discussed the dynamical behavior of stellar filaments and found that the presence of hidden elements influences both the expansion growth and collapse of the universe. Rubab at el. explored the collapse of stellar filaments in both the metric formalism [33] and Palatini formalism [34] of f(R) theory. Saleem et al. [35] also discussed the collapsing dynamics of filamentary structures under the influence of exotic terms.

Inspired by previous research, we have investigated the behavior of filamentary objects in the presence of mysterious terms within the context of Palatini f(R, T) theory. Our study is structured as follows: Section 2 provides an overview of the Palatini formalism of f(R, T) theory, while Section 3 employs the Darmois junction conditions (DJC) to illustrate how filamentary objects collapse. Section 4 covers the results derived from the junction conditions and analyzes the collapse dynamics. To examine the connection between GWs and DM, Section 5 introduces a particular f(R, T) model into the collapse mechanism. The outcomes are summarized in Section 6.

2. The Palatini f(R, T) Theory

The concept of f(R, T) gravity, which was proposed by Harko, has led to various intriguing findings in the fields of astrophysics and cosmology [36]. This theory arises from the inclusion of a general trace term in the Einstein action. The respective action for this theory can be observed as

()

where S_m represents the matter components, κ stands for the coupling parameter, g denotes the metric tensor’s determinant, R manifests the Ricci scalar, and T illustrates the trace of energy-momentum tensor. For the matter configuration, the tensor is defined as

()

The respective field equations to equation (1) are given as

()

where the derivative terms f_T and f_R are

()

Here, ◻ = g^ξχ∇_ξ∇_χ, ∇_ξ exhibits the covariant divergence and Θ_ξχ is defined as

()

For matter content, we take S_m = ρ [3] which reduces equation (5) as

()

By differentiating equation (1) with respect to the  connection and considering that the stress-energy tensor’s trace is independent of , we can observe that

()

where the covariant divergence ∇ defined with  represents the independent connection, and

()

Here, ξχ denotes the symmetrization over ξ and χ. After contracted with δ, the equation (8) is

()

Note that if we consider f(R, T) = f(R), then f_R(R, T) = 0 and we recover the field equations for Palatini f(R) gravity. Solving equation (9) for  and substituting the obtained result in equation (2), we may obtain the single formalism of the Palatini f(R, T) field equations in terms of  defined as follows:

()

After some manipulations, we can rewrite the above field equations as effective Einstein equations as

()

where  illustrates the effective stress-energy tensor defined as

()

The divergence of equation (12) is presented as

()

3. Filamentary Objects

Here, we take the spacetime of collapsing filament that is surrounded by a cylindrical surface and is given by [31]

()

The following limits are used to ensure cylindrical symmetry: −∞ ≤ t≤∞, 0 ≤ ϕ ≤ 2Π, −∞ < z < ∞. The coordinates are defined as x⁰ = t, x¹ = r, x² = z, and x³ = ϕ. The stress-energy tensor composed of dissipative and anisotropic fluid is presented as

()

where the symbol ρ represents the energy density, while P_z, P_ϕ, and P_r stand for matter stresses, and the symbols V_ξ, K_ξ, S_ξ, and l_ξ represent the four vectors. The term V_ξ represents a four-velocity vector, while K_ξ and S_ξ are spacelike vectors. Additionally, l_ξ is a null-like vector that is perpendicular to V_ξ. The following identities are satisfied by these vectors:

()

By using equations (14)–(16), the field (11) produces five nonzero components of Einstein tensor. However, to analyze the behavior of the collapsing structures, we require the specific components listed as follows:

()

The associated partial derivatives are provided as

()

3.1. Collapse of Cosmic Filaments

We are aware that a collapsing star is always surrounded by an exterior configuration. For this research, the Einstein–Rosen coordinates are used to demonstrate the exterior vacuum distribution associated with the hypersurface . As a result, the line element for exterior geometry is described as [37]

()

where coefficients η and φ depend on radial  and temporal  components. The equation R_ξχ = 0 provides the gravitational field wave equations as follows:

()

We are applying DJCs [38] instead of Israel junction conditions because the geometry is simple and does not involve any thin shell. This involves the continuity in both the first fundamental form (i.e., the interior and exterior geometries) and the second fundamental form (i.e., the extrinsic curvatures). The equation for collapsing boundary  for exterior and interior coordinates is presented as

()

()

The exterior spacetime is denoted by O₊, while the interior spacetime is represented by O₋. The hypersurface  acts as the comoving boundary of the collapsing surface, and thus, the value of  remains constant. To apply the junction conditions correctly, it is necessary to ensure that the surface boundary has the same parameters, regardless of whether it is embedded in O₊ or O₋.

To maintain the continuity of the fundamental form, we apply the exterior and interior geometry on . To accomplish this, we employ equation (21) on (14), to obtain interior spacetime on the boundary expressed as

()

where τ the time coordinate on  is described as follows:

()

where Ω⁰ = τ, Ω² = z, and Ω³ = ϕ will be taken into account as parameters on the surface boundary . By using equation (22), we can simplify the exterior spacetime equation (19) on  given by

()

The continuity of interior spacetime equation (23) corresponding to exterior spacetime equation (25) provides

()

with

()

The other fundamental form of boundary condition depending on (K_θψ) extrinsic curvature is defined as

()

where

()

Here,  represents the affine connections that should be calculated using the interior equation (19) or exterior equation (14) spacetimes. The symbol  manifests the unit normals to the  related to exterior and interior metrics while x^η illustrates the equation of  associated with O₋ or O₊. The unit normals of surface  related to O₋ or O₊ derived from from equations (21) and (22) are given as

()

()

The dot notation represents differentiation with respect to the time coordinate τ, as mentioned in equation (25). When equation (27) is fulfilled, both unit vector equations (30) and (31) are spacelike. The extrinsic curvature  has the following nonzero components:

()

()

()

()

()

()

Equations (24) and (26) collectively generate all the Darmois conditions and ensure the continuity of K_θψ across the .

4. Outcomes of Boundary Conditions

In the context of collapsing cosmic filaments, the significant results of boundary conditions are derived in this section. In order to simplify the boundary conditions and create some useful formulae, we take the Einstein equations connected to exterior and interior geometries [31, 39]. Equation (25) yields the following outcome:

()

and using equations (26) and (27), we have

()

Following the differentiation of the previous equation using equation (32),

()

Also, with equations (26) and (27), the continuity of curvature K₃₃ and K₂₂ yields the following result:

()

Differentiating equations (17) and (32) along with equations (40) and (41) yield

()

The curvatures K₂₂, K₀₀ and equations (18), (23), (26), (38), (39), (41) as well as (42) give the following expression:

()

Differentiating equations (26) and (27) with equations (24) and (25)results in

()

()

Now, the connection between the extrinsic curvatures K₂₂, K₃₃ and equations (44) and (45) yields the following result:

()

Finally, utilizing equations (20) and (41) and inserting equation (46) into (43), we obtain

()

Here,  represents the radial velocity of the collapsed surface . The collapse equation for the stellar objects in the presence of exotic terms is represented by equation (47). It is demonstrated by the above equation that the radial pressure at boundary  stays nonzero due to dissipation and DM.

5. f(R, T) Palatini Formalism and Cosmic Filaments

Based on cosmological studies, it has been examined that galaxies throughout the cosmos are connected by a network of filaments made up of DM. This system of filaments has long been believed to create a superstructure or web linking galaxies. In this work, we will explore the impact of DM using a specific model of f(R, T) gravity.

5.1. Minimally Coupled Model of f(R, T) Theory

To investigate how DM affects the collapsing filament of stars, we employ a particular model of f(R, T) formalism that is defined as

()

The α(−T) term introduces a coupling between matter and geometry, distinguishing it from standard f(R) models. This coupling allows for an integrated description of the universe’s large-scale structure by accounting for interactions between dark matter and dark energy. Additionally, the T term introduces unique dynamics to cosmic structures, such as the collapse of filamentary objects. The symbol α shows the coupling constant, while the parameter l represents the influence of matter. When l is equal to zero, the model becomes identical to GR. When l is between 0 and 1, the model can be considered a candidate for DM on a galactic scale [40]. Utilizing equation (48), we can obtain the following differentials:

()

By utilizing the particular model equation (48) of f(R, T) theory, the collapsed equation (47) simplifies to the following equation:

()

It can be observed from the above equations that the radial pressure, components of the exotic matter, and the velocity of the collapsing boundary are not zero on the collapsing border due to dissipation. It is thought that the gravitational pull of DM causes ordinary matter to condense into cosmic clusters and galaxies in filament structures. Our collapsing model predicts that the gravitational attraction of mysterious matter on the collapsing filament is similar to that of DM.

During the process of collapse, the body’s natural ability to gravitate towards itself makes its gravitational force more significant than its internal pressure. As the body collapses, it dissipates energy, resulting in a momentum flux of GWs. Equation (49) can be used to examine the relationships between DM and different phenomena during the collapsing scenario.

5.2. Dark Matter and Gravitational Waves

The discovery of gravitational waves in 2015 by LIGO and Virgo [41] initiated a new and interesting field of investigation in gravitational theories. These waves result from different occurrences like the big bang and the gravitational collapse of stellar clusters. The current study suggests that the presence of DM could impact the transmission of GWs in a similar way to how different mediums influence the transmission of light waves [42]. However, this impact is very small and falls well below the sensitivity level of current detectors.

The significance of studying gravitational waves has increased as they provide a means of observing and understanding the kinematics of the universe and the development of cosmic structures resulting from various phenomena. During the study of these waves, their direction can be determined by analyzing their polarization modes [43]. In the framework of GR, a GW exhibits two distinct polarized modes. However, in alternative gravitational theories, such as f(R) and f(R, T) gravity, research has revealed the presence of two additional modes for gravitational waves. These additional modes differ from those observed in GR [44].

Equation (49) for collapse can be utilized to investigate how exotic terms of filament collapse are connected to GWs. Suppose there is a cylindrical source that remains motionless for some time before it starts to shrink and generates a powerful radiation pulse that emanates from its axis. In this respect, the relevant function can be written as [20]

()

The function , which varies with time, can be written as  to indicate the strength of the wave path. The static Levi–Civita solution is denoted by φ_st. In this situation, g₀ is a fixed value and  refers to the delta function. Physically, this equation implies that as the cylindrical structure collapses, it emits a GW pulse from its axis. The function  encapsulates the dynamics of this emission, with g₀ indicating the initial pulse strength and  representing the time evolution of this strength. The term  in the denominator accounts for the wavefront propagation from the source to the observer, illustrating how the GWs radiate from the collapsing cylinder over time. Equation (51) and the wave equation (20) give

()

Equations (20) and (51) reveal how GW and DM are interrelated when there is visible matter dissipation and pressure at the . We can obtain a connection between GW and DM from equation (49), if assume that the pressure and dissipation of baryonic matter are negligible at , meaning q ≈ 0 and p_r ≈ 0. This relationship is given by

()

Equations (49)–(53) suggest that the existence of exotic terms can disrupt the transmission medium of GWs during the collapse process.

6. Conclusion