# Collapsing stellar filament and exotic matter in Palatini *f*(*R*) gravity

## Abstract

We explore the dynamics of collapsing stellar filament in the presence of exotic material like dark matter. We use Palatini *f*(*R*) theory to include exotic substance in the collapsing process. We derive a collapse equation by applying Darmois junction conditions on collapsing surface boundary \(\Sigma \). It is found that the radial pressure related to baryonic matter remains non-zero at \(\Sigma \). We then discuss the stability criteria of the collapsing process in the framework of three parameteric model, \(f(R)=R+\lambda R_{c}[ 1-(1+\frac{R^{2}}{R^{2}_{c}})^{-n}]\). It is concluded that the stability of collapsing filament depends upon a directly proportional relation of gravitational effects of exotic terms with the radial pressure of seen matter. Stability criteria of family of polytropic filamentary structures are also discussed. For all stable polytropic filaments, it is found that the density of seen material is exponentially related to the exotic forces. Finally, we explore theoretical relation between gravitational waves and dark terms. It is theoretically predicted that the presence of exotic material can affect the propagation of gravitational waves.

## 1 Introduction

The concepts of dark energy (DE) and dark matter (DM) are striking discoveries of modern physics. Cosmic observations from Supernova Ia, CMBR (cosmic microwave background radiation) and Wilkinson Microwave Anisotropy Probe showed that our universe is expanding with an accelerating rate. There is a mysterious form of energy (called DE) having gravitationally repulsive effect is responsible for this cosmic acceleration [1, 2]. Also, DM is a form of non-baryonic matter which neither emits nor absorbs electromagnetic radiations and is detectable by its gravitational effects on the baryonic matter. The observational studies related to mass discrepancy in galactic clusters and galactic rotational curves problems indicated the existences and importance of DM in the stellar evolution [3, 4, 5]. According to Planck data, the cosmic energy is distributed as follows: \(68\%\) is DE, \(27\%\) is DM and rest of \(5\%\) is baryonic matter [6].

In order to understand the nature of DE and DM many models are introduced by various researchers. In this context, the \(\Lambda \)CDM model, with cosmological constant \((\Lambda )\) representing a energy density of vacuum energy, has been widely used in the theory of general relativity (GR). But this proposal suffers two problems like the “fine tuning problem” [7] and the “cosmic coincidence problem” [8, 9]. According to fine tuning problem the predicted value of \(\Lambda \) is 120 order of magnitude smaller than the value predicted by the particle physics. The cosmic coincidence problem is the coincidence between densities of dark matter and dark energy. The quantum gravity fails to explain DE issue because it requires vacuum energy to be greater than hundreds of order magnitude more than observed value. There are different solutions for these problems like the modification of matter-part of the Einstein Hilbert action such as quintessence [10], Chaplygin gases [11, 12, 13], K-essence [14], etc. The another approach is the modified gravity which deals with the modified geometric part of the Einstein field equations [15]. These alternative of GR are considered viable if they provide admirable cosmological results as compare to GR.

The *f*(*R*) gravity theory is one of the viable and most explored example of modified gravity which involves higher order curvature invariant term as compare to GR. This theory generalizes the Einstein-Hilbert action by replacing linear scalar curvature term (*R*) with a generic function of curvature “*f*(*R*)” [16, 17, 18, 19, 20, 21, 22]. This gravity attains a lot of attention because of its higher order curvature terms (generic function *f*(*R*)) could interpret not only DE issue admirably (explain cosmic acceleration at low cosmic densities) [23] but also DM problem [24, 25].

There are usually two different approaches of *f*(*R*) gravity: the metric approach [23] and the Palatini approach [26, 27]. Between these approaches, the Palatini approach got a reasonable attention. Its formulation uses metric as well as christoffel connection (affine connection, \(\Gamma _{\mu \nu }^{\rho }\)) as two independent geometric variables. The metric *f*(*R*) gravity deals with fourth-order differential equations which are not easy to handle. In contrast, the Palatini *f*(*R*) gravity provides second order singularity-free equations to represent exotic quantities [28]. This version of *f*(*R*) gravity introduces modified form of Friedmann equations [29, 30, 31] which is compatible with observational data [32, 33]. It has been shown that initial-value Cauchy problem is well-formulated as well as well-posed in this approach [34, 35]. It is also indicated that Palatini *f*(*R*) gravity can be an intersecting candidate for problems related DM [36]. Furthermore, this theory can reproduce the effective dynamics of Loop Quantum which gives a link between Palatini formalism and the Quantum gravity [37]. The Palatini *f*(*R*) gravity is also applied in the study of astrophysical phenomena like neutron stars [38, 39] black holes [40] etc.

On large-scale, galaxies are present in the form of bunches called clusters or super-clusters. These bunches of galaxies are connected by a bridge-like arrangement called galaxy filaments. A rich web of galaxy filaments containing galaxies and their clusters along with DM haloes are predicted by the N-body simulation phenomenon [41, 42, 43, 44]. The cold DM model indicated a network of low-density filaments that connects DM haloes (DM filaments) [45]. Numerous researchers have studied DM filaments connecting massive clusters. Higuchi et al. [46] predicted filament connecting massive galaxy clusters \(RXJ0018.3+1618\) and \(CL0015.9+1609\). Some people predicted [43] galaxy filaments that have a typical cylindrical radius \(\sim 2h^{-1}Mpc\).

The phenomenon of gravitational collapse is responsible to the birth of stellar structures (stars, planets and cluster of galaxies), radiations and gravitational waves. During collapse process a stable astronomical body (like star) or a stable system of bodies (clusters or super clusters of stars) turns into unstable one due to its own gravity. That’s why a great numbers of researchers explored characteristics of gravitational collapse for the study of evolving stellar distributions. Herrera and Santos [47] described dynamics of non-dissipative collapsing cylindrically symmetric anisotropic filamentary structure based on baryonic matter only. They used Darmois junction conditions to framework the collapse process and found that the radial pressure remains non zero throughout the collapse. This collapse discussion is based on baryonic matter only and doesn’t involve the role of DM or DE.

The study of stellar evolution in the presence of exotic terms (DE and DM) may provide modified dynamics of stellar evolution, as compare to GR, which in turns may reveal the modification hidden in the phenomenon of structure formation of the universe [48, 49, 50, 51, 52, 53, 54, 55, 56]. In this paper, we investigate the dynamics of collapsing cylindrically symmetric stellar filament in the presence of baryonic and non-baryonic matter. It is an attempt to find how the dark part of the universe (exotic terms) relates with the dynamics of baryonic matter, radiations and gravitational waves. For this purpose, we use higher order curvature invariant base modified gravity “ the Palatini *f*(*R*) gravity”. The paper is formulated as follows: the next section describes the Palatini *f*(*R*) gravity and the galaxy filament. Section 3 discusses collapse of cylindrically symmetric filamentary structures by using Darmois junction conditions. Section 4 applies the three parameteric Palatini *f*(*R*) model to the collapsing process to determine the effects of exotic matter on the stability criteria and gravitational waves. Finally, Sect. 5 summarizes the results.

## 2 Basic scenario

In this section, we dicuss the basic scenario regarding galaxy filament in the Palatini *f*(*R*) gravity.

### 2.1 Palatini *f*(*R*) gravity

*f*(

*R*) theory for a given time moderation in the gravitational scheme of GR is one of the successful approaches. It has given very useful results in the field of cosmology as well as physics which explain the cosmic expansion. The main concern of this theory is to substitute an algebraic general function of Ricci scalar “

*f*(

*R*)” instead of the cosmological constant in the standard Einstein-Hilbert action [57]. It can be written as

*f*(

*R*) gravity is consistent with classical theory of gravity for those observational cases which provide roots of above equation. The vacuum case of Eq. (4) gives \(R={\bar{R}}=constant\) which in turns along with Eq. (3) provide conservation of metric tensor as well as fixing \((\Gamma _{\mu \nu }^{\rho })\) as Levi-Civita connection. Consequently, Eq. (2) reduces for the vacuum case to

*f*(

*R*) model. Solving Eq. (3) for \(\Gamma _{\mu \nu }^{\rho }\), inserting the obtain result in Eq. (2) and describing it in terms of \(g_{\mu \nu }\), we may express the single formulation of the Palatini

*f*(

*R*) field equation given by

*f*(

*R*) gravity, \({\bar{\Box }} ={\bar{\nabla }}_{\mu }{\bar{\nabla }}_{\nu }g^{\mu \nu },~ {\bar{G}}_{\mu \nu }\equiv {\bar{R}}_{\mu \nu }-\frac{1}{2} g_{\mu \nu }{\bar{R}}\), while \({\bar{\nabla }}_{\mu }\) indicates covariant derivative related to the Levi-Civita connection of metric tensor. Moreover, \(f_{R}\) and

*f*are functions of \(R(\Gamma ) \equiv g^{\mu \nu } R_{\mu \nu } (\Gamma )\).

### 2.2 Galaxy Filament

## 3 Collapse of Stellar Filament

*T*and \({\tilde{R}}\). The equations \(R_{\beta \gamma }=0\) provides the gravitational wave equation as

## 4 Galaxy filament and the Palatini *f*(*R*) gravity

Observations indicate that galaxies are attached together through cosmic webs which is actually a network of galaxy filaments connected via dark terms. It is believed that there exist filamentary structure made up of dark material among galaxies which behave like a bridge between galaxies. In this section, we will present model of galaxy filament in the Palatini *f*(*R*) gravity and discuss the features of exotic material in the phenomenon of stability of filaments as well as gravitational waves.

### 4.1 Three parametric model of Palatini *f*(*R*) formalism

*f*(

*R*) gravity. We consider a well-known three parametric form of Palatini

*f*(

*R*) gravity for the explanation of physical expect of the galaxy filament. The model is described by [61, 62]

*n*are positive real numbers, \(R_{c}\) represents a constant term having values of the order of the present effective Ricci scalar. It can be notice that this model represents absence of cosmological constant in flat spacetime at null contribution (\(R=0\)). Moreover, \(R=constant={\bar{R}}=R_{0}x_{1},~x_{1}>0,\) provides de-Sitter model for the following values

*n*with \(x_{1}>>1\) , we get \(x_{1}\rightarrow 2\lambda \). In this situation, this model is consistent with \(\Lambda CDM\) model. The Einstein gravity can be obtained under the limit \(f(R)\rightarrow R\). In this context, we have

### 4.2 Stability of collapsing filament with exotic material

*f*(

*R*) gravity. To discuss necessary and sufficient condition of the stability criteria, we assume inverse situation, let there is a non-dissipative case satisfying

Now, if the condition (52) is sufficient for stability then \(\upsilon = 0\) and we get \(y = \frac{\varpi _{,T}}{\varpi _{,{\tilde{R}}}}=0\), which yield \(\varpi = \varpi ({\tilde{R}})\). In this case, Eqs.(24), (40) and the continuity of \(K_{22}\) imply \(B=B(r)\) and \(C=C(r)\) at \(\Sigma \). As *r* is constant at the boundary \(\Sigma \), the metric terms \(B=B(r)\) and \(C=C(r)\) become constant at \(\Sigma \). In this situation, Eq. (38) implies \({\tilde{R}}=constant\) at hypersurface which inturn shows that collapsing process is stop. Hence our supposition is correct, the condition (52) play the role of sufficient condition in the stability analysis of stellar filamentary structure.

*k*is a constant and

*n*indicates polytropic index. Under this consideration, Eq. (52) implies

*f*(

*R*) gravity.

### 4.3 Gravitational waves and exotic material

The collapsing process of stellar bodies is one of the source of gravitational waves [64]. Recently, it is studied that the propagation of gravitational waves can be affected by the presence of dark matter just like light waves are disturbed by different medium of propagation. But this effects is so small that it would be far below the sensitivity of current detector [65]. Polarization modes of gravitational waves are also discussed in *f*(*R*) gravity [66] which provide the existence of gravitational waves in this theory. In the present study of collapsing filament, we can describe the relation of gravitational waves with higher order curvature invariant terms (exotic terms) by using collapse equation (46).

*h*(

*T*) represents a time dependent function that shows the strength of the wave source and it is describes by \(h(T)=h_{0}\delta (T)\). Here, \(h_{0}\) indicates a constant and \(\delta (T)\) is the dirac delta function. Equation (58) satisfies the wave equation (9) which provides

## 5 Conclusion

Observational surveys indicate the DM as one of a fundamental constituent of filamentary structure on the galactic scales. We have studied a dissipative collapsing cylindrically symmetric filamentary structure in the presence of exotic matter. For this purpose, we have incorporate dark terms by using higher order curvature invariant principle (the Palatini approach of *f*(*R*) gravity). The Darmois junction conditions is used to model the collapsing filament and to derive a collapse equation at the boundary surface \(\Sigma \). From the collapse equation, it is found that the higher order curvature invariant terms (representing DM) along with dissipation related to baryonic matter preserve the radial pressure on the collapsing boundary. This result is consistent with model discussed in GR in the absence of DM [47]. Galactic observations predicted that the the existence and evolution of galaxies or their clusters rely on a big amount of unseen material like DM [3, 4, 5, 45]. This indicates that the presence of exotic terms can affect the dynamics of evolving filamentary structures. We have used the Palatini *f*(*R*) gravity formalism by considering three parameter model, \(f(R)=R+\lambda R_{c}[ 1-(1+\frac{R^{2}}{R^{2}_{c}})^{-n}]\) as a candidate of exotic terms in the stability of collapsing filament. It is concluded that the stability criteria of collapsing filament rely upon the relation of higher order curvature invariant terms (exotic matter) and radial pressure (due to baryonic material). For stable configuration, radial pressure are directly proportional to the gravitational effects of dark terms. To discuss stability criteria for family of filamentary structures, we have applied polytropic equation of state for baryonic matter content. It is found that instability criteria of polytropic filaments provides a exponential relation between density of baryonic material and the gravitational effects of exotic terms which depends upon the polytropic index *n*.

In current phase, it is suspected that the phenomenon of gravitational waves can resolve the secret of exotic terms in the universe. The feature of dark terms as a medium of propagation of gravitational waves can reveal the secret of hidden terms [65, 69]. In this study, we have investigated relation of gravitational waves emerging from collapsing filament with exotic materials. From the collapse equation a connection between gravitational waves and dark terms has been acquired which shows that the existence of dark terms can affect the propagation of gravitational waves.

It is mentioned here that further study on the stability criteria can help to find the ratio of baryonic matter and dark material in the filamentary structure.

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