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

Rubab Manzoor, Abdul Jawad, Muhammad Adeel, Muhammad Saeed and Shamaila Rani, Collapsing stellar filament and exotic matter in Palatini f(R) gravity, The European Physical Journal C 79, article 831, 2019. Reproduced in full under the Creative Commons Attribution 4.0 International licence stated in the article; the published version is at doi.org/10.1140/epjc/s10052-019-7332-0.

(On this site, the other route to exotic matter through geometry rather than substance — mapping the requirement onto a quantum density — is Sijo K. Joseph’s /library/stm-04fab4674f, and the conformal-gravity version is at /library/stm-a967ad4e8e. The formal accounting of what a spacetime may and may not contain is Kontou and Sanders’ survey of the energy conditions at /library/stm-1bcd336342; the question of how little exotic matter a traversable geometry actually needs is at /library/stm-d54d953175, and the attempt to remove the requirement altogether is at /library/stm-3f01ca8468.)

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. It is found that the radial pressure related to baryonic matter remains non-zero at the boundary. We then discuss the stability criteria of the collapsing process in the framework of a three parametric model, in which f(R) is the Ricci scalar plus a coupling constant times a characteristic curvature, multiplied by one minus the quantity one plus the squared Ricci scalar over the squared characteristic curvature, raised to the power minus 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 and dark matter are striking discoveries of modern physics. Cosmic observations from Supernova Ia, the cosmic microwave background radiation and the Wilkinson Microwave Anisotropy Probe showed that our universe is expanding with an accelerating rate. There is a mysterious form of energy, called dark energy, having a gravitationally repulsive effect, which is responsible for this cosmic acceleration. Also, dark matter 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 existence and importance of dark matter in the stellar evolution. According to Planck data, the cosmic energy is distributed as follows: 68 percent is dark energy, 27 percent is dark matter and the rest of 5 percent is baryonic matter.

In order to understand the nature of dark energy and dark matter many models are introduced by various researchers. In this context, the cold dark matter model with a cosmological constant, the constant representing an energy density of vacuum energy, has been widely used in the theory of general relativity. But this proposal suffers two problems, the fine tuning problem and the cosmic coincidence problem. According to the fine tuning problem the predicted value of the cosmological constant is 120 orders of magnitude smaller than the value predicted by particle physics. The cosmic coincidence problem is the coincidence between densities of dark matter and dark energy. Quantum gravity fails to explain the dark energy issue because it requires vacuum energy to be greater than hundreds of orders of magnitude more than the observed value. There are different solutions for these problems, like the modification of the matter part of the Einstein-Hilbert action, such as quintessence, Chaplygin gases and K-essence. The other approach is modified gravity, which deals with the modified geometric part of the Einstein field equations. These alternatives to general relativity are considered viable if they provide admirable cosmological results as compared to general relativity.

The f(R) gravity theory is one of the viable and most explored examples of modified gravity, which involves a higher order curvature invariant term as compared to general relativity. This theory generalizes the Einstein-Hilbert action by replacing the linear scalar curvature term with a generic function of curvature. This gravity attains a lot of attention because its higher order curvature terms could interpret not only the dark energy issue admirably, explaining cosmic acceleration at low cosmic densities, but also the dark matter problem.

There are usually two different approaches to f(R) gravity: the metric approach and the Palatini approach. Between these approaches, the Palatini approach got a reasonable attention. Its formulation uses the metric as well as the Christoffel connection, the affine connection, as two independent geometric variables. 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. This version of f(R) gravity introduces a modified form of the Friedmann equations which is compatible with observational data. It has been shown that the initial-value Cauchy problem is well-formulated as well as well-posed in this approach. It is also indicated that Palatini f(R) gravity can be an interesting candidate for problems related to dark matter. Furthermore, this theory can reproduce the effective dynamics of loop quantum cosmology, which gives a link between the Palatini formalism and quantum gravity. The Palatini f(R) gravity is also applied in the study of astrophysical phenomena like neutron stars and black holes.

On large scales, 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 dark matter haloes is predicted by N-body simulation. The cold dark matter model indicated a network of low-density filaments that connects dark matter haloes, that is, dark matter filaments. Numerous researchers have studied dark matter filaments connecting massive clusters. Higuchi and colleagues predicted a filament connecting the massive galaxy clusters RXJ0018.3 plus 1618 and CL0015.9 plus 1609. Some people predicted galaxy filaments that have a typical cylindrical radius of about two megaparsecs divided by the reduced Hubble constant.

The phenomenon of gravitational collapse is responsible for the birth of stellar structures — stars, planets and clusters of galaxies — and for radiation and gravitational waves. During the collapse process a stable astronomical body, like a star, or a stable system of bodies, such as clusters or super-clusters of stars, turns into an unstable one due to its own gravity. That is why a great number of researchers explored characteristics of gravitational collapse for the study of evolving stellar distributions. Herrera and Santos described the dynamics of a non-dissipative collapsing cylindrically symmetric anisotropic filamentary structure based on baryonic matter only. They used the Darmois junction conditions to frame the collapse process and found that the radial pressure remains non-zero throughout the collapse. That collapse discussion is based on baryonic matter only and does not involve the role of dark matter or dark energy.

The study of stellar evolution in the presence of exotic terms may provide modified dynamics of stellar evolution, as compared to general relativity, which in turn may reveal the modification hidden in the phenomenon of structure formation of the universe. In this paper, we investigate the dynamics of a 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 relates to the dynamics of baryonic matter, radiation and gravitational waves. For this purpose, we use higher order curvature invariant based 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 parametric Palatini f(R) model to the collapsing process to determine the effects of exotic matter on the stability criteria and gravitational waves. Finally, Section 5 summarizes the results.

2 Basic scenario

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

2.1 Palatini f(R) gravity

The notion of f(R) theory for a given time moderation in the gravitational scheme of general relativity 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 the Ricci scalar in place of the cosmological constant in the standard Einstein-Hilbert action. Equation 1 in the source writes that action as the integral over four-dimensional spacetime of the square root of minus the metric determinant times the function of the Ricci scalar, divided by twice the coupling constant, plus the matter action.

The variation of the action with respect to the metric and with respect to the connection provides the equations of motion for the Palatini formalism. Equation 2 in the source states that the derivative of the function with respect to the Ricci scalar, multiplied by the Ricci tensor, minus one half of the metric times the function, equals the coupling constant times the matter stress-energy tensor. Equation 3 in the source states that the covariant derivative of the inverse metric times the square root of minus the determinant times that derivative of the function vanishes.

The trace of the second equation gives a direct relation between the Ricci scalar built from the connection and the trace of the stress-energy tensor, given as Equation 4 in the source: the Ricci scalar times the derivative of the function, minus twice the function, equals the coupling constant times the trace of the stress-energy tensor.

The Palatini f(R) gravity is consistent with the classical theory of gravity for those observational cases which provide roots of the above equation. The vacuum case of that trace equation gives a constant Ricci scalar, which in turn, along with the connection equation, provides conservation of the metric tensor as well as fixing the connection as the Levi-Civita connection. Consequently the field equation reduces, for the vacuum case, to Equation 5 in the source: the metric Ricci tensor minus one quarter of the constant Ricci scalar times the metric equals zero. We can notice that this theory can reduce to general relativity, with and without the cosmological constant, according to the chosen f(R) model.

Solving the connection equation, inserting the obtained result in the field equation and describing it in terms of the metric, we may express the single formulation of the Palatini f(R) field equation as Equation 6 in the source, which collects: the second covariant derivatives of the derivative of the function, minus the metric times its d’Alembertian, divided by that derivative; one half of the metric times the constant Ricci scalar; the coupling constant times the matter stress-energy tensor divided by the derivative of the function; one half of the metric times the ratio of the function to its derivative, less the Ricci scalar; a term in three over twice the squared derivative of the function, carrying one half of the metric times the squared gradient of that derivative, less the product of its gradients; and finally minus the metric Ricci tensor, all summing to zero.

The above equation can be rewritten in Einstein-type field equations as Equation 7 in the source: the Einstein tensor built from the metric equals the coupling constant divided by the derivative of the function, multiplied by the sum of the matter stress-energy tensor and a dark stress-energy tensor. Equation 8 in the source gives that dark stress-energy tensor: the second covariant derivatives of the derivative of the function less the metric times its d’Alembertian, divided by the coupling constant, plus the derivative of the function times the metric times one half over the coupling constant, multiplied by the ratio of the function to its derivative less the Ricci scalar, plus a term in one over twice the coupling constant and the squared derivative of the function, carrying three halves of the metric times the squared gradient of that derivative less the product of its gradients.

This tensor shows the stress-energy tensor associated to the Palatini f(R) gravity. The d’Alembertian is the double covariant derivative contracted with the inverse metric, the Einstein tensor is the Ricci tensor less one half of the metric times the Ricci scalar, and the covariant derivative is the one related to the Levi-Civita connection of the metric tensor. Moreover, the function and its derivative are functions of the Ricci scalar built from the connection.

2.2 Galaxy filament

Here, we describe the geometry of a collapsing filament containing clusters of galaxies and dark terms. For this, we assume a cylindrically symmetric configuration that is bounded by a cylindrical surface and represented by a line element given as Equation 9 in the source: minus the squared first metric function times the squared time differential, plus the same squared function times the squared radial differential, plus the squared second metric function times the squared axial differential, plus the squared third metric function times the squared angular differential. Equation 10 in the source gives the ranges of the coordinates: the time coordinate runs over the whole real line, the axial coordinate runs over the whole real line, and the angular coordinate runs from zero to two pi.

Consider that the collapsing filament is composed of an anisotropic dissipative baryonic fluid whose energy-momentum tensor is given as Equation 11 in the source: the density plus the radial pressure, times the product of the four-velocities, plus the radial pressure times the metric, plus the difference between the angular and radial pressures times the product of the angular unit vectors, plus the difference between the axial and radial pressures times the product of the axial unit vectors, plus the heat flux times the symmetrised product of the radial unit vector and the four-velocity. Equation 12 in the source fixes those vectors: the radial vector is orthogonal to the four-velocity, the four-velocity is minus the first metric function along the time direction, the radial vector is the first metric function along the radial direction, the axial vector is the second metric function along the axial direction, and the angular vector is the third metric function along the angular direction.

The field equations, the dark stress-energy tensor and the fluid tensor provide five non-zero components of the Einstein tensor, but we need only two of them to discuss the dynamics of the collapsing structures. Equation 13 in the source is the radial-radial component, built from second time derivatives of the second and third metric functions and from products of the first derivatives of all three, and set equal to the coupling constant times the radial pressure times the squared first metric function, plus the radial-radial component of the dark tensor times that same square. Equation 14 in the source is the mixed time-radial component, built from mixed derivatives of the second and third metric functions and from products of their first derivatives with the derivatives of the first, and set equal to the heat flux times the squared first metric function plus the time-radial component of the dark tensor times that same square.

3 Collapse of stellar filament

It is well known that a collapsing star is always bounded by an exterior configuration. In this study, we assume Einstein-Rosen coordinates to represent the exterior vacuum distribution corresponding to the hypersurface, so the exterior geometry is given as Equation 15 in the source: minus the exponential of twice the difference of the two exterior functions, multiplied by the difference between the squared exterior time differential and the squared exterior radial differential, plus the exponential of twice the second exterior function times the squared axial differential, plus the exponential of minus twice that function times the squared exterior radius times the squared angular differential. Both exterior functions depend on the exterior time and radial coordinates.

The vanishing of the exterior Ricci tensor provides the gravitational wave equation, given as Equation 16 in the source: the second exterior time derivative of the wave function, minus its second exterior radial derivative, minus its first radial derivative divided by the radius, equals zero. Equation 17 in the source gives the companion relation: the radial derivative of the first exterior function equals the radius times the sum of the squared time derivative and the squared radial derivative of the wave function.

We use Darmois junction conditions for the matching of the interior collapsing boundary with the exterior one. So, we assume continuity of the first fundamental form, that is, continuity of the interior and exterior spacetimes, and continuity of the second fundamental form, that is, continuity of the extrinsic curvatures. To apply the junction conditions, the derived equations of the collapsing boundary surface are given as Equations 18 and 19 in the source: on the interior side, the radial coordinate minus its boundary value is zero; on the exterior side, the exterior radius minus its value as a function of exterior time is zero. The interior radial term is a constant because the boundary describes a comoving collapsing boundary of the fluid. For the application of the junction conditions, we arrange that the boundary has the same parametrisation whether it is assumed as embedded in the interior or the exterior spacetime.

Using the interior boundary equation in the interior line element, we get the interior metric on the surface as Equation 20 in the source: minus the squared proper time differential, plus the squared second metric function times the squared axial differential, plus the squared third metric function times the squared angular differential, where the proper time on the boundary is defined by Equation 21 in the source, the first metric function times the coordinate time differential. We shall use the proper time, the axial coordinate and the angular coordinate as parameters on the boundary surface.

The exterior line element together with the exterior boundary equation gives the exterior metric on the boundary as Equation 22 in the source. According to continuity of the first fundamental form, the interior metric is the same as the exterior metric on the boundary if three conditions hold, given as Equations 23, 24 and 25 in the source: the exponential of minus twice the difference of the exterior functions, times the square root of one minus the squared rate of change of the exterior radius with exterior time, times the exterior time differential, equals the proper time differential; the exponential of the second exterior function equals the second interior metric function; and the exponential of minus that function, times the exterior radius, equals the third interior metric function. Here we consider, as Equation 26 in the source, that one minus that squared rate of change is positive on the boundary, so that the exterior time behaves as a timelike coordinate.

The second fundamental form on the boundary describes continuity of the extrinsic curvatures on the interior as well as the exterior geometry, and is given as Equation 27 in the source, the extrinsic curvature contracted with the boundary parameter differentials. Equation 28 in the source defines the extrinsic curvature as minus the outward unit normal contracted with the second derivative of the embedding plus the Christoffel symbols of the appropriate spacetime contracted with its first derivatives. From the interior and exterior boundary equations, the calculated unit normals are given as Equations 29 and 30 in the source: the interior normal points along the radial direction with magnitude the first metric function; the exterior normal is built from the exponential of the difference of the exterior functions, the reciprocal square root of one less the squared rate of change of the exterior radius, and the components minus that rate of change and one. Here the dot indicates differentiation with respect to the proper time. The derived normal vectors are spacelike under the positivity constraint. The obtained non-zero components of the extrinsic curvature on each side are given as Equations 31 to 36 in the source: on the interior side, minus the radial derivative of the first metric function over its square, minus the second metric function times its radial derivative over the first, and minus the third metric function times its radial derivative over the first; on the exterior side, three expressions built from the second proper-time derivatives of the exterior radius and time, the gradients of the exterior functions, and the exponentials of the second exterior function.

The proper-time relation and the continuity conditions, together with continuity of the extrinsic curvature, represent the complete junction conditions at the boundary.

Next, we obtain useful outcomes of the junction conditions in the scenario of a collapsing galactic filament. In order to do that we use the field equations to describe boundary conditions in a brief form and derive some useful results. From the first junction condition we have Equation 37 in the source: the exponential of minus twice the difference of the exterior functions, times the difference between the squared proper-time derivatives of the exterior time and the exterior radius, equals one. The second and third conditions give Equation 38 in the source: the exterior radius equals the product of the second and third interior metric functions.

Differentiation of that product relation, together with the proper-time definition, provides Equation 39 in the source: the proper-time derivative of the exterior radius equals the coordinate-time derivative of the product of the second and third metric functions, divided by the first. The continuity of the two angular extrinsic curvature components, together with the two matching conditions, implies Equation 40 in the source: the proper-time derivative of the exterior time equals the radial derivative of that same product, divided by the first metric function.

Now, differentiation of those two relations, along with the two Einstein tensor components and the proper-time definition, gives Equation 41 in the source, an expression for the combination of second proper-time derivatives of the exterior time and radius in terms of the interior metric functions, the coupling constant, the radial pressure and the radial-radial dark pressure, the heat flux and the time-radial dark term.

From the continuity of the time-time and axial extrinsic curvatures, together with the wave equation and the preceding relations, we have Equation 42 in the source, which sets a combination of the interior metric derivatives, the coupling constant, the radial pressure with its dark counterpart, and the heat flux with its dark counterpart, equal to the squared difference of the squared proper-time derivatives multiplied by the squared radial derivative of the wave function.

Differentiation of the two matching conditions with respect to proper time and coordinate time gives Equations 43 and 44 in the source, relating the time and radial derivatives of the wave function to the coordinate-time derivatives of the second and third metric functions. Those relations, along with the continuity of the two angular extrinsic curvatures, give Equation 45 in the source.

Finally, substituting that result into the earlier relation and using the wave equation, we have the collapse equation, Equation 46 in the source: the radial-radial dark pressure term, plus the coupling constant times the radial pressure, minus the sum of the heat flux and the time-radial dark term divided by the squared radial derivative of the metric product, equals the exponential of twice the difference of the exterior functions, multiplied by the squared radial derivative of the wave function, and by the quantity twice the ratio of the time to the radial derivative of the wave function times the boundary velocity, less the squared boundary velocity over one minus that square. Here the boundary velocity is the rate of change of the exterior radius with exterior time, that is, the radial velocity of the collapsing boundary surface.

The result denotes that the radial pressure on the surface is non-zero. The reason is that it may be due to the flux of momentum of the gravitational wave which is emitted from the cylinder. The above result can be used as a collapse equation of a filamentary structure in the presence of exotic terms.

4 Galaxy filament and the Palatini f(R) gravity

Observations indicate that galaxies are attached together through cosmic webs, which are actually a network of galaxy filaments connected via dark terms. It is believed that there exist filamentary structures made up of dark material among galaxies which behave like a bridge between galaxies. In this section, we will present a model of a 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 the Palatini f(R) formalism

In this section, we will turn our attention to the physical perspective of Palatini f(R) gravity. We consider a well-known three parametric form of Palatini f(R) gravity for the explanation of the physical aspect of the galaxy filament. The model is described by Equation 47 in the source: the function equals the Ricci scalar plus a coupling parameter times a characteristic curvature, multiplied by one minus the quantity one plus the squared Ricci scalar over the squared characteristic curvature, raised to the power minus n. Here the coupling parameter and the exponent are positive real numbers, and the characteristic curvature is a constant term having a value of the order of the present effective Ricci scalar.

It can be noticed that this model represents the absence of a cosmological constant in flat spacetime at null contribution, that is, at zero Ricci scalar. Moreover, a constant Ricci scalar equal to a positive multiple of a reference curvature provides a de Sitter model for the value of the coupling parameter given as Equation 48 in the source: the quantity one plus the squared multiplier, raised to the power n plus one, times the multiplier, all divided by twice the quantity that same power less n plus one times the squared multiplier less one.

Now, to obtain useful results, we consider a specific choice of the multiplier and then calculate the value of the coupling parameter. It can be noticed from the above equation that twice the coupling parameter exceeds the multiplier, which places the de Sitter point below the asymptotic value. If we fix the multiplier with the exponent much greater than one, and fix the exponent with the multiplier much greater than one, we get the multiplier approaching twice the coupling parameter. In this situation, this model is consistent with the cold dark matter model with a cosmological constant. Einstein gravity can be obtained under the limit in which the function reduces to the Ricci scalar.

In this context, Equation 49 in the source identifies the radial-radial component of the dark stress-energy tensor with the radial pressure exerted by the exotic terms, assembled from the derivative of the function, its first and second time and radial derivatives, the metric functions and their derivatives, and the ratio of the function to its derivative. Equation 50 in the source identifies the time-radial component with the dissipation of the exotic material in the collapsing system, built from the proper-time derivative of the derivative of the function, that derivative itself, the third metric function and its time derivative. Their full values are given in the Appendix.

4.2 Stability of collapsing filament with exotic material

The stability of a stellar body is of great importance. Any stable compact model is considered incompetent if it is unstable against fluctuations. The restabilisation of the collapsing process leads to the formation of stellar structures in the universe. Here, we obtain the stability criteria of a collapsing galaxy filament in the presence of exotic material. We consider that the collapsing of the filament stops and the stellar body becomes stable. In this situation the collapsing velocity vanishes and the dissipation effects related to baryonic material become negligible. In this case the collapse equation yields Equation 51 in the source: the coupling constant times the radial pressure equals the dark dissipation term divided by the squared radial derivative of the metric product, minus the dark radial pressure. This shows a relation between exotic material and baryonic material in the stability of the collapsing filament in the Palatini f(R) gravity.

To discuss necessary and sufficient conditions of the stability criteria, we assume the inverse situation. Let there be a non-dissipative case satisfying Equation 52 in the source: the coupling constant times the radial pressure, minus the dark dissipation term over that squared radial derivative, plus the dark radial pressure, equals zero. Under these conditions the collapse equation reduces to Equation 53 in the source, in which the exponential prefactor times the squared radial derivative of the wave function multiplies the bracket containing the boundary velocity, and the whole expression equals zero. This gives Equation 54 in the source: the boundary velocity equals minus one plus the square root of one plus the squared ratio, divided by that ratio, where the ratio is the time derivative of the wave function over its radial derivative.

Now, if the condition is sufficient for stability then the boundary velocity vanishes and we get the ratio equal to zero, so the time derivative of the wave function vanishes and the wave function depends on the exterior radius alone. In this case the matching conditions, the proper-time relation and the continuity of the axial extrinsic curvature imply that the second and third metric functions depend on the radial coordinate only at the boundary. As the radial coordinate is constant at the boundary, those metric functions become constant there. In this situation the product relation implies that the exterior radius is constant at the hypersurface, which in turn shows that the collapsing process has stopped. Hence our supposition is correct: the condition plays the role of a sufficient condition in the stability analysis of a stellar filamentary structure.

In order to discuss the stability criteria of a family of filamentary structures, we assume a polytropic type baryonic matter distribution, that is, the radial pressure equals a constant times the density raised to the polytropic exponent, where the exponent is n plus one over n. Here n indicates the polytropic index. Under this consideration, the stability condition implies Equation 55 in the source: the coupling constant times the constant times the density raised to the polytropic exponent equals the dark dissipation term over the squared radial derivative of the metric product, plus the dark radial pressure. Equivalently, Equation 56 in the source gives the density as that same right-hand side divided by the coupling constant times the constant, all raised to the power n over n plus one.

The result indicates that in the stable condition the density of baryonic matter is exponentially related to the effects due to exotic material. For polytropic models the filamentary structure becomes unstable if the density falls on the other side of that same expression, stated as Equation 57 in the source. It is well known that different values of the polytropic index represent various stellar configurations, so the polytropic equation along with the stability condition gives instability criteria for a family of filamentary structures in the Palatini f(R) gravity.

4.3 Gravitational waves and exotic material

The collapsing process of stellar bodies is one of the sources of gravitational waves. Recently, it has been studied that the propagation of gravitational waves can be affected by the presence of dark matter, just as light waves are disturbed by different media of propagation. But this effect is so small that it would be far below the sensitivity of current detectors. Polarization modes of gravitational waves are also discussed in f(R) gravity, which provides the existence of gravitational waves in this theory. In the present study of a collapsing filament, we can describe the relation of gravitational waves with higher order curvature invariant terms, the exotic terms, by using the collapse equation.

Let the collapsing of the stellar filament emit gravitational radiation directed outward from the axis. The waveform of the radiation is divided into three forms: a precursor described by the quadrupole moment formalism; a sharp pulse; and the oscillatory tails. The quadrupole moment formalism takes place at very early time. The sharp pulse of the radiation carries much of the energy of the wave and lasts longer than the quadrupole moment phase. In the current study, we consider the sharp pulse of the gravitational radiation directed outward from the axis. In this condition the wave function can be described as Equation 58 in the source: one over two pi, times the integral from minus infinity to the exterior time less the exterior radius of the source strength function, divided by the square root of the difference between the retarded time and the squared exterior radius, plus the static Levi-Civita solution. Here the source strength is a time-dependent function that shows the strength of the wave source, written as a constant times the Dirac delta function of exterior time.

That expression satisfies the wave equation and provides two regimes: where the exterior radius exceeds the exterior time, the wave function is just the static solution; where the exterior time exceeds the exterior radius, it is the constant divided by two pi times the square root of the difference between the squared exterior time and the squared exterior radius, plus the static solution.

The wave function together with the collapse equation describes the relation of gravitational waves with exotic material in the presence of baryonic matter pressure and dissipation at the boundary. If we consider that the baryonic matter stress and dissipation are negligible at the boundary, then from the collapse equation we get a direct relation between the dark terms and the gravitational waves, given as Equation 59 in the source: the radial-radial dark term, plus the time-radial dark term divided by the squared radial derivative of the metric product, equals the exponential prefactor times the squared radial derivative of the wave function, multiplied by the bracket in the boundary velocity.

The collapse equation, the waveform and this last relation show that the presence of higher order curvature invariant terms, representing dark matter, can disturb the propagation of gravitational waves in the collapse of a stellar filament.

5 Conclusion

Observational surveys indicate dark matter as one of the fundamental constituents of filamentary structure on galactic scales. We have studied a dissipative collapsing cylindrically symmetric filamentary structure in the presence of exotic matter. For this purpose, we have incorporated dark terms by using the higher order curvature invariant principle, the Palatini approach of f(R) gravity. The Darmois junction conditions are used to model the collapsing filament and to derive a collapse equation at the boundary surface. From the collapse equation, it is found that the higher order curvature invariant terms, representing dark matter, along with dissipation related to baryonic matter, preserve the radial pressure on the collapsing boundary. This result is consistent with the model discussed in general relativity in the absence of dark matter.

Galactic observations predicted that the existence and evolution of galaxies or their clusters rely on a large amount of unseen material like dark matter. 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 the three parameter model as a candidate for exotic terms in the stability of the collapsing filament. It is concluded that the stability criteria of a collapsing filament rely upon the relation of higher order curvature invariant terms, that is, exotic matter, with the radial pressure due to baryonic material. For a stable configuration, the radial pressure is directly proportional to the gravitational effects of the dark terms. To discuss stability criteria for a family of filamentary structures, we have applied a polytropic equation of state for the baryonic matter content. It is found that the instability criteria of polytropic filaments provide an exponential relation between the density of baryonic material and the gravitational effects of exotic terms, which depends upon the polytropic index.

In the 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. In this study, we have investigated the relation of gravitational waves emerging from a 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.

Data availability statement

This manuscript has no associated data or the data will not be deposited. Authors’ comment: this work is done on a theoretical basis, not on an experimental basis. All the information has been given in the manuscript.

Appendix A

The appendix gives the two dark quantities in full: the radial pressure exerted by the exotic terms and the dissipation of the exotic material. Each is a single expression running to most of a printed page, assembled from the coupling parameter, the exponent, the characteristic curvature, the Ricci scalar with its first and second time and radial derivatives, and the three metric functions with their derivatives, in which the recurring factor is one plus the squared Ricci scalar over the squared characteristic curvature raised to successive negative powers. The extracted characters are not reliable enough to reprint here; the two expressions in their original form are at the source.

Rubab Manzoor and Muhammad Adeel and Muhammad Saeed, Department of Mathematics, University of Management and Technology, Johar Town Campus, Lahore, Pakistan. Abdul Jawad and Shamaila Rani, Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.

(Reference-number markers, running heads and page furniture have been dropped, and display equations rendered in words with their source equation numbers; the sixty-nine references and the equations in their original form are at the source.)