A preliminary study about gravitational wave radiation and cosmic heat death
Jianming Zhang, Qiyue Qian, Yiqing Guo, Xin Wang, Xiao-Dong Li
aa r X i v : . [ a s t r o - ph . C O ] F e b MNRAS , 1– ?? (2020) Preprint 25 February 2021 Compiled using MNRAS L A TEX style file v3.0
A preliminary study about gravitational wave radiation andcosmic heat death
Jianming Zhang , Qiyue Qian † , Yiqing Guo , Xin Wang ‡ ,Xiao-Dong Li ∗ School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510297, P. R. China † Corresponding author: [email protected] ‡ Corresponding author: [email protected] ∗ Corresponding author: [email protected]
25 February 2021
ABSTRACT
We study the role of gravitational waves (GW) in the heat death of the universe. Due to the GW emission, in a verylong period, dynamical systems in the universe suffer from persistent mechanical energy dissipation, evolving to astate of universal rest and death. With N-body simulations, we adopt a simple yet representative scheme to calculatethe energy loss due to the GW emission. For current dark matter systems with mass ∼ − M ⊙ , we estimatetheir GW emission timescale as ∼ − years. This timescale is significantly larger than any baryon processesin the universe, but still ∼ times shorter than that of the Hawking radiation. We stress that our analysis couldbe invalid due to many unknowns such as the dynamical chaos, the quadrupole momentum of halos, the angularmomentum loss, the dynamic friction, the central black hole accretion, the dark matter decays or annihilations, theproperty of dark energy and the future evolution of the universe. Key words: gravitational waves, cosmology, astrophysics
The heat death fate of the universe, also known as the "BigChil" or "Big Freeze", is a conjecture suggesting that theuniverse would end up with a state of no thermodynamicfree energy. In such situation the universe would be unableto sustain any process that increases entropy.The idea of heat death was first studied by Kelvin, who ex-trapolated the second law of thermodynamics of mechanicalenergy dissipation to a cosmic scale in the 1850s, and foundit inevitably led to "a state of universal rest and death".Along with this conjecture, Helmholtz (Thomson 1857) envi-sioned a stable, thermodynamic equilibrium state in the end,stating that "the universe from that time forward would becondemned to a state of eternal rest." Similar viewpoint washeld by Clausius, who believed that the universe would evolveto "a state of unchanging death" once its entropy reaches themaximum.Discussions on the heat death hypothesis never stops sinceit was proposed. Jeans (1930); Eddington (1931) broughtthe notion of the heat death into relativistic cosmology.Barrow & Tipler (1978) demonstrated that the Hawkingblack hole evaporation process would cause a vortical insta-bility to develop in spatially homogeneous spacetimes, andfurthermore provided a novel picture of the universal heatdeath. Also, there were opposing viewpoints to the idea ofheat death, such as "rise and fall theory" by L.E.Boltzmanand the conjecture of "Maxwell’s demon" (Bennett 1982),etc. With the development of modern cosmology, it becomescommonly accepted that the ultimate fate of the universecrucially depends on its energy components. Heat death islikely to happen in a forever expanding universe, which isexpected to occur if the topology of the universe is open orflat (in a matter-dominated universe), or if the dark energycomponent keeps dominating. The latter possibility is cur-rently supported by multiple cosmological observations (seeAde et al. (2016) and the references therein).Dyson (1979) provided a comprehensive summary of phys-ical processes related to the heat death, which included thestellar evolution, the detachment of planets from stars andstars from galaxies, the decay of object orbits by gravita-tional radiation, the evaporation of black holes by the Hawk-ing process, the liquification of all matter at zero temper-ature, the decay of matter to iron, as well as the collapseof iron stars to neutron stars and the collapse of ordinarymatter to black holes. While most of these processes are ofgreat interest and worth more detailed investigation, in thispaper, we focus on the detailed investigation of the gravi-tational waves . As is mentioned in Barrow & Tipler (1978),“as time passes, clusters, galaxies and other stellar systemswill become increasingly bound by gravitational forces sincethey will radiate away their binding energy in the form of gravitational waves ”.Gravitational waves (GWs) generated by accelerated massare spacetime curvature disturbances which propagate aswaves outward from the source at the speed of light (Eistein1916, 1918). On September 14th, 2015, the Laser Interferom- © 2020 The Authors
Zhang, Qian, Guo, Wang, & Li (2020) eter Gravitational Wave Observatory (LIGO) made the firstdirect detection of a GW event (Abbott et al. 2016), openinga new window for us to test the predictions of general relativ-ity and probe into the structures of the universe. GWs takeplace in all dynamical systems, causing persistent mechanicalenergy dissipation in the long-term evolution of our universe,and therefore are expected to become an essential process tothe heat death.GWs carry energy away from their sources and, in the caseof orbiting objects, lead to an in-spiral or the shrinking of theorbit’s radius. Before the detection made by LIGO, pulsartiming observations over decade had shown a gradual decayof the pulsar binary orbital period that matched the loss ofenergy and angular momentum in GW radiation predicted bygeneral relativity; this offered the first indirect evidence of theexistence of GWs (Hulse & Taylor 1975; Taylor & Weisberg1982). This process ends up with the merger of the objects,which would form a heavier compact object (e.g. a black hole)and produce strong GW signals detectable via experimentslike LIGO.The in-spiral caused by GW emission is a universal processof mechanical energy loss, regardless of the nature of the ob-jects. Therefore, it will become the key mechanism for darkmatter systems to reach heat death. Unlike the baryonic sys-tems such as gas, stars, galaxies and galaxy clusters, darkmatter can not lose mechanical energy via thermal radiationand (non-dynamical) friction. Therefore, the basic picture re-garding our thermal fate would be that dark matter systemsgradually lose mechanical energy, become more compact, andfinally collapse into black holes (BHs). Then BHs evaporate,and the universe eventually enters the "state of eternal rest".In this work we conduct a preliminary yet representativeinvestigation on the role of GW radiation in the cosmic death.In 2 we briefly introduce how GW emission affects the dynam-ical systems. In 3, we estimate the GW emission timescale ofdark matter systems in the current universe with the aid ofsimulations. In 4 we discuss the caveats of our analysis as wellas some closely related issues. We summarize and concludein 5.
By carrying energy away, the GW radiation causes a per-sistent loss of mechanic energy in a system. As a gravita-tional effect, this process only involves the mass and dynam-ical properties of the system. In the most simplified case of atwo-body system with orbital velocity v and mass of objects m and m , the rate at which energy is carried away by GWsis given as follows, ddt E GW = 325 Gc u r ω , (1)where we define u = m m /M , M = m + m , and ω be-ing the angular velocity. Here the constants G = 6 . × − m / s kg , c = 2 . × m / s . Notice that the mechanicorbital energy takes the form of E orb = − GMu r . (2) Considering the energy loss is only caused by GW radiation,we have ddt E orb = GMu r ˙ r = − ddt E Gw . (3)Now we are ready to estimate the timescale that the systemtakes to form a compact object, which we denote as ∆ t ≡ t s − t , where t is the current epoch and t s is the time whenthe system collapses. From 1, 2,3, we find ˙ r = − G c r ( m + m ) m m , (4)which yields to ∆ t = 5( r − r s ) c G Mm m , (5)where r , r s are the initial and final radius of the system,respectively. If we take r s as the Schwarzschild BH radiuswith mass of the system, i.e. r s = MGc , then the timescaleof evolution is completely determined. Notice that the timefor such BH to evaporate is ∆ t s = 5120 πG M ~ c , (6)where ~ is the reduced Planck constant. From the above estimation, we see that the timescale de-pends on the mass and orbital radius change. The cosmicstructure formation results in a series of collapsed dark mat-ter halos whose mass is distributed in a wide range. Theseobjects would become stable after entering the state of virialequilibrium, and are expected to stay stable for a long periodif no major merger happens.1 illustrates the distribution of dark matter halos atredshift z = 0 with M > M ⊙ in a small volume (256 h − Mpc) , created by dark matter simulation (Springel2005; Tassev et al. 2013; Klypin et al. 2016) with cosmolog-ical parameters of Ω m = 0 . , Ω b = 0 . , σ =0 . , n s = 0 . , and H = 67 .
77 km s − Mpc − .In total we select a sample of 56,549 halos, using the ROCKSTAR halo finder (Behroozi et al. 2013) that allowsfor robust tracking of substructure based on adaptive hier-archical refinement of friends-of-friends groups in six phase-space dimensions and one time dimension. In general, thehalos we obtain well obey the virial theorem, which statesthat a stabilized dynamical system should obey
GMR ≈ σ v ,with σ v being the velocity dispersion. The mass-velocity dis-tribution measured from the simulation is presented in 2.To simplify the situation and make everything calculable,in what follows we will equate the whole dark matter haloto a binary system of two objects with equal mass orbitingaround the center, and then the virial theorem leads to 7. Ifwe assume the distance between the two objects R = r ,where r being the initial radius of the system, then we have m (3 σ v ) ∼ GMm ( r ) , (7)where σ v is the − D velocity squared in the radial directionand M, m both equal to half of the halo mass. Then we have
MNRAS , 1– ????
MNRAS , 1– ???? (2020) W emission and cosmic heat death Figure 1.
Mass function of the halo samples used in this analysis.The samples include a set of 56,549 dark matter halos at redshift z = 0 in a volume of (256 h − Mpc) , created using N-body methodand the ROCKSTAR halo finder.
Figure 2.
Mass-velocity distribution of the halo samples used inour analysis. σ v = GM r , which is a good approximation for what we havemeasured from the N-body simulation.As consequences of GW emission, we believe that as timeelapses, the radius of the system r would shrink, the orbitalvelocity v would increase, and the magnitudes of potentialenergy V and kinetic energy T would both increase. Thegradual changes of the physical quantities are rather smallat the beginning, and dramatically increase at the end of theevolution.3 shows the evolution of three typical systems selected fromthe sample, having mass around , and M ⊙ ,respectively. Their circular velocity and kinetic energy arepresented in 1. We find that the results are consistent withwhat we expected. As demonstrated in 3, with time elapsing,the radius of the system r shrinks, velocity v and kineticenergy T increase, and the changes are rapid at the end ofthe evolution.From 5 and 6, we obtain the timescale distribution of halo-size binaries to evolve into BHs via GW radiation and BHsevaporation through Hawking radiation.The left panel of the 4 illustrates halo distribution with thetime period of GW radiation, which is estimated as ∼ − years. The right panel shows that the time for BHs to evaporate is much longer, which distributes in ∼ − years. In the above section, the typical timescale that dark mattersystems take to emit GWs before ceasing evolution is esti-mated using a two-body model. While the model is oversim-plified, our result is still representative and makes physicalsense due to two reasons. First, no matter how complicated amany-body system is, as long as the motions (velocities, ac-celerations) of the objects within are comparable to those ina two-body system with similar mass, size and kinetic energy,the evolution of these two systems should not differ from eachother significantly. Secondly, the two-body system is the onlycase of a gravitational system that is everlastingly stable. It ispossible that most complicated systems would finally evolveto a stage similar to a two-body system after a long enoughperiod.However, besides the two-body simplification we made inthe calculation, there are more caveats that could potentiallyinvalidate our results. We will summarize and discuss themas follows.
Using the N-body methods, one can start with a certain dis-tribution of mass points, and trace their evolution by inte-grating the equations of motion (Aarseth & Binney 1978).Yet we are not clear whether this kind of stability can sus-tain within a timescale of − years due to many well-known difficulties, e.g., singular gravitational potential whentwo particles become too close to each other, and chaotic be-havior of the N-body problem when N > . Although somestudies (Voglis et al. 2002; Kalapotharakos & Voglis 2005;Kalapotharakos et al. 2008; Muzzio 2012) have shown thatit is possible to obtain models of elliptical galaxies whichare highly stable over time intervals of the order of a Hub-ble time (even with a black hole growing in the center; seeMerritt & Quinlan (1998); Poon & Merritt (2002, 2004)), sofar it is impossible to trace the evolution of the system withinthe very long period of the GW radiation. Possibility exists that dark matter consists of particles thatactually decay on a very long time scale. Pandey et al. (2019);Mukherjee et al. (2019) showed that dark matter with de-cay lifetime ∼ − seconds can help to explain somedetected excesses (Chen & Takahashi 2009; Yin et al. 2009;Ishiwata et al. 2009; Ibarra & Tran 2009; Chen et al. 2009;Nardi et al. 2009; Arvanitaki et al. 2009), while some otherwork (Cohen et al. 2017; Cadena et al. 2020) suggested thatdark matter should be stable within a long timescale of ∼ − seconds. These timescales are already close tothe GW emission timescale ( ∼ − in unit of second). There have been scenarios envisaged in which dark mat-ter could give a significant contribution to the mass
MNRAS , 1– ?? (2020) Zhang, Qian, Guo, Wang, & Li (2020)
Table 1.
Dynamical properties of the three selected halos at the present time i Mass M ⊙ Radius of the system (km)
Circular Velocity (km / s) Kinetic Energy ( MJ ) . × . × . × . × . × . × . × . × . × t (year) r ( k m ) halo 1halo 2halo 3 t (year) v ( k m / s ) t (year) T ( M J ) Figure 3.
The evolution of three dynamical systems selected from the halo sample (see 1 for detailed data). With time elapsing, theirradius r shrinks, while the velocity v and kinetic energy T increase. The evolution becomes rapid at the end of the evolution. GW radiation timescale (year) −1 N Hawking radiation timescale (year) −1 N Figure 4.
The left panel shows the time needed for the halos to evolve to BHs through GW radiation. Time for these BHs to evaporatevia Hawking radiation is demonstrated in the right panel. accretion of the central black hole of the galaxies.Balberg & Shapiro (2002) showed that black holes withmass > M ⊙ could be formed directly as a conse-quence of relativistic core collapse of halos if halos embed-ding galaxies are constituted of self-interacting dark mat-ter (Dave et al. 2001), while Zelnikov & Vasiliev (2005) andMunyaneza & Biermann (2005) showed that a significant ac-cretion may happen in the case when dark matter particlesare scattered by stars in molecular clouds near the centralblack hole. Although in general the accretion of collision-less dark matter particles into black holes are thought to beless efficient than that expected from the dissipative bary-onic fluid (Peirani & de Freitas Pacheco 2008), current stud-ies can not rule out the possibility that the dynamical system would be significantly affected or even terminated due to theblack hole accretion in the long timescale of GW emission. To be precise, the rate of GW emission depends on the chang-ing rate of the quadrupole moment (Eistein 1916, 1918) of thesystem, i.e. ddt E GW = 325 Gc I e ω , (8)where I is the moment of inertia of the halo, e is the eccentric-ity of the halo, and ω is the angular velocity. Using the aboveformula, we re-calculate the GW emission efficiency of thehalo sample using the positions and velocities of their mem- MNRAS , 1– ????
The left panel shows the time needed for the halos to evolve to BHs through GW radiation. Time for these BHs to evaporatevia Hawking radiation is demonstrated in the right panel. accretion of the central black hole of the galaxies.Balberg & Shapiro (2002) showed that black holes withmass > M ⊙ could be formed directly as a conse-quence of relativistic core collapse of halos if halos embed-ding galaxies are constituted of self-interacting dark mat-ter (Dave et al. 2001), while Zelnikov & Vasiliev (2005) andMunyaneza & Biermann (2005) showed that a significant ac-cretion may happen in the case when dark matter particlesare scattered by stars in molecular clouds near the centralblack hole. Although in general the accretion of collision-less dark matter particles into black holes are thought to beless efficient than that expected from the dissipative bary-onic fluid (Peirani & de Freitas Pacheco 2008), current stud-ies can not rule out the possibility that the dynamical system would be significantly affected or even terminated due to theblack hole accretion in the long timescale of GW emission. To be precise, the rate of GW emission depends on the chang-ing rate of the quadrupole moment (Eistein 1916, 1918) of thesystem, i.e. ddt E GW = 325 Gc I e ω , (8)where I is the moment of inertia of the halo, e is the eccentric-ity of the halo, and ω is the angular velocity. Using the aboveformula, we re-calculate the GW emission efficiency of thehalo sample using the positions and velocities of their mem- MNRAS , 1– ???? (2020) W emission and cosmic heat death ber particles, and find the result is actually ± timeslarger than the result obtained using the simple two-bodymodel adopted in this analysis. In any case, these two resultsdo not differ for many orders of magnitude. Therefore ourmain conclusion remains valid.The calculation of the quadrupole moment is based on thecurrent mass distribution within the halo samples, and it isvery likely that this mass distribution will change dramati-cally within the long period of GW emission. Thus, in thispaper we still adopt the simple two-body model to estimatethe power of the GW emission. Although halos may lose their angular momentum throughinteractions with the other objects, we believe that this willnot have significant effect on our main conclusion. As theuniverse evolves, the shrinking of the horizon size will isolatehalos, and prohibit their interactions with each other. Theabove process is expected to happen in a timescale muchshorter than the GW emission timescale.In particular, the future event horizon, i.e., the region thatlight emitted from an object can reach in the infinite future,takes the form of r FEH ≡ Z + ∞ t = t emit cdta . (9)If the Λ CDM is adopted as the theoretical model, the fu-ture event horizon would shrink to Mpc within ∼ years, which is much shorter compared with the GW emis-sion timescale.Another process that may contribute to the collapse is thedynamic friction. In general, its timescale depends inverselyon the mass of the system (Chandrasekhar 1942). Once a clus-ter is collapsed, violent relaxation is ineffective, and furtherrelaxation occurs through two-body interactions. Schechter(1976); Schechter & P.J.E.Peebles (1976) calculated the dragforce and the dynamic friction timescale of a system com-posed by homogeneous, isotropic, Maxwellian distributedparticles. Following their calculation, we find that the dy-namic friction of typical galaxy clusters is about × year,which is also far less than the GW emission timescale.Finally, one may wonder whether our results is related withthe final parsec problem . For that, our opinions are as follows.On one hand, if there exists any mechanism that may fastenthe merger of supermassive binary black holes (SBBHs) whenthey are extremely close to each other, such mechanism mayalso fasten the final stage of the evolution of a halo to a BH.In this case, the thermal death may occur a bit earlier. On theother hand, considering that the timescale of the GW emis-sion is − years, it is extremely difficult to predict thestatus of the dynamic system after such a long time evolution.One possible situation is that, since two-body system is theonly stable gravitational system, by the time all halos wouldexist in a form of two-body system. Without a third objectinteracting with them, there maybe no mechanism that canaccelerate the evolution of the system. In the case that the nature of dark energy is phantom like( w < − , ˙ ρ > ), its density will reach infinity in a finitetime, disrupt any bounded system, and cause a "big rip" fateof the universe (Caldwell et al. 2003; Li et al. 2012), makingour analysis completely meaningless. Similarly, our analysisis also invalid in the scenario of bouncing cosmology (see Cai(2014) and the references therein). In this paper, we study the role of GW in the heat death.GWs happen in almost all dynamical systems in the universe,causing persistent mechanical energy dissipation in the long-term evolution of the universe and driving the universe to "astate of universal rest and death". With the N-body simula-tions, we adopt a simple yet representative scheme to com-pute the GW emission process, and estimate its timescale as ∼ − years, which depends on the mass of the sys-tem. This timescale is significantly larger than any baryonprocess in the universe, but still ∼ times shorter thanthe timescale of Hawking radiation.By taking the GW emission into consideration, our workextends the scope of heat death. This means that, similarto the baryon systems, dark matter has to lose mechanicalenergy via radiation and therefore cannot persist forever.In this work, we study typical halos with mass ∼ − M ⊙ . But the results may be different if using a simulationwith higher resolution and bigger size, or if the nature of darkmatter is not as assumed in the standard cold dark matterscheme.Base on our current knowledge and technology, it is un-likely to obtain a comprehensive understand of the physicsprocess of dynamical chaos, dark matter decays or annihi-lations, central black hole accretion, and the nature of darkenergy. Neither do we know the roles they are going to play inthe GW emission process. Thus, our analysis would be invalidunder certain circumstances.We adopt a rather simplified model to estimate thetimescale of the GW emission. A further study may requirerunning a N-body simulation with the GW emission consid-ered, which is of great challenge and, not that necessary, giventhat there are so many unknown factors that may affect theprocess.Several issues about the gravitational wave and thermaldeath are not discussed in this short work. In a genericanisotropic universe without positive cosmological constant,it is possible to generate an infinite amount of entropy by tak-ing advantage of cosmological shear and curvature anisotropy,allowing civilizations to exist forever (Ellis & Barrow 2002;Barrow & Hervik 2003). This provides a novel method toavoid thermal death by utilizing the gravitational waves.Neither do we discuss the "gravitational field entropy"(Barrow et al. 1988; M.Patel & Lineweaver 2017), which isassociated with the increase of irregularity in the universeand provides a novel picture of the universal heat death.In all, heat death is an interesting conjecture about the fateof the universe, and we expect more studies on this issue withthe progress of our understanding about the physics processesin the universe. MNRAS , 1– ?? (2020) Zhang, Qian, Guo, Wang, & Li (2020)
The data used to support the findings of this study are avail-able from the corresponding authors upon request.
ACKNOWLEDGEMENTS
We thank Prof. Rongxin Miao for helpful discussions. Weacknowledge the use of
Kunlun cluster located in Schoolof Physics and Astronomy, Sun Yat-Sen University. Thiswork is supported by National SKA Program of ChinaNo. 2020SKA0110401. XDL acknowledges support from theNSFC grant (No. 11803094), the Science and Technology Pro-gram of Guangzhou, China (No. 202002030360).
REFERENCES
Aarseth, S. J., & Binney, J. 1978, Mon. Not. Roy. Astron. Soc.,185, 227Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, Phys. Rev.Lett., 116, 061102Ade, P. A., Aghanim, N., Arnaud, M., et al. 2016, Astronomy &Astrophysics, 594, A13Arvanitaki, A., Dimopoulos, S., Dubovsky, S., et al. 2009, Phys.Rev. D, 79, 105022Balberg, S., & Shapiro, S. L. 2002, Phys. Rev. Lett., 88, 101301Barrow, J., & Hervik, S. 2003, Physics Letters B, 566, 1Barrow, J. D., & Tipler, F. J. 1978, Nature, 276, 453Barrow, J. D., Tipler, F. J., & Anderson, J. L. 1988, The AnthropicCosmological Principle (Oxford University press), 613–658Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109Bennett, C. H. 1982, International Journal of Theoretical physics,21, 905Cadena, S. H., Franco, J. S., Alfaro Molina, R., et al. 2020, PoS,ICRC2019, 520Cai, Y.-F. 2014, Sci. China Phys. Mech. Astron., 57, 1414Caldwell, R. R., Kamionkowski, M., & Weinberg, N. N. 2003, Phys.Rev. Lett., 91, 071301Chandrasekhar, S. 1942, university of chicago, 231Chen, C.-R., Nojiri, M. M., Takahashi, F., & Yanagida, T. 2009,Prog. Theor. Phys., 122, 553Chen, C.-R., & Takahashi, F. 2009, JCAP, 02, 004Cohen, T., Murase, K., Rodd, N. L., Safdi, B. R., & Soreq, Y.2017, Phys. Rev. Lett., 119, 021102Dave, R., Spergel, D. N., Steinhardt, P. J., & Wandelt, B. D. 2001,The Astrophysical Journal, 547, 574Dyson, F. J. 1979, Rev. Mod. Phys., 51, 447Eddington, A. S. 1931, Nature, 127, 447Eistein, A. 1916, Sitzungsber. Preuss. Akad. Wiss., 1, 688—. 1918, Sitzungsber. Preuss. Akad. Wiss., 1, 154Ellis, E. G. F. R., & Barrow, J. D. 2002, The Far-Future Universe:The Far, Far Future (Pontifical Academy of Sciences, the Vat-ican), 23Hulse, R. A., & Taylor, J. H. 1975, The Astrophysical Journal,195, L51Ibarra, A., & Tran, D. 2009, JCAP, 02, 021Ishiwata, K., Matsumoto, S., & Moroi, T. 2009, Phys. Lett. B, 675,446Jeans, J. 1930, Cambridge University PressKalapotharakos, C., Efthymiopoulos, C., & Voglis, N. 2008, Mon.Not. Roy. Astron. Soc., 383, 971Kalapotharakos, C., & Voglis, N. 2005, Celest. Mech. & Dynam.Astron., 92, 157Klypin, A., Yepes, G., Gottlöber, S., Prada, F., & Heß, S. 2016,Mon. Not. Roy. Astron. Soc., 457, 4340 Li, X.-D., Wang, S., Huang, Q.-G., Zhang, X., & Li, M. 2012, Sci.China Phys. Mech. Astron., 55, 1330Merritt, D., & Quinlan, G. D. 1998, The Astrophysical Journal,498, 625M.Patel, V., & Lineweaver, C. H. 2017, Physics, Computer Science.Entropy, 19, 411Mukherjee, T., Pandey, M., Majumdar, D., & Halder, A. 2019,arXiv:1911.10148Munyaneza, F., & Biermann, P. L. 2005, Astron. Astrophys., 436,805Muzzio, J. 2012, arXiv:1204.0709Nardi, E., Sannino, F., & Strumia, A. 2009, JCAP, 01, 043Pandey, M., Majumdar, D., Halder, A., & Banerjee, S. 2019, Phys.Lett. B, 797, 134910Peirani, S., & de Freitas Pacheco, J. 2008, Phys. Rev. D, 77, 064023Poon, M., & Merritt, D. 2002, The Astrophysical Journal, 568, 89—. 2004, The Astrophysical Journal, 606, 774Schechter, P. L. 1976, Astrophys.J, 209, 297Schechter, P. L., & P.J.E.Peebles. 1976, Astrophys.J, 203, 670Springel, V. 2005, Mon. Not. Roy. Astron. Soc., 364, 1105Tassev, S., Zaldarriaga, M., & Eisenstein, D. J. 2013, Journal ofCosmology and Astroparticle Physics, 6, 036Taylor, J. H., & Weisberg, J. M. 1982, Annu. Rev. Astron. Astro-phys., 253, 908Thomson, W. 1857, Proceedings of the Royal Society of Edinburgh,3, 139Voglis, N., Kalapotharakos, C., & Stavropoulos, I. 2002, Mon. Not.Roy. Astron. Soc., 337, 619Yin, P.-f., Yuan, Q., Liu, J., et al. 2009, Phys. Rev. D, 79, 023512Zelnikov, M., & Vasiliev, E. A. 2005, Int. J. Mod. Phys. A, 20,4217MNRAS , 1– ????