A higher order generalized uncertainty principle and Jeans instability
aa r X i v : . [ phy s i c s . g e n - ph ] J un A higher order generalized uncertainty principle and Jeans instability
Zhong-Wen Feng ∗ and Shu-Zheng Yang (Dated: June 15, 2020)The Jeans instability is regarded as important tool for analyzing dynamical of a self-gravitatingsystem. However, the astronomical observation data show that some Bok globules, which mass areless than the Jeans mass still have stars or at least experience the star formation process. In thispaper, we investigate the effects of higher order generalized uncertainty principle on the Jeans massof collapsing molecular cloud. It is found that the higher order generalized uncertainty principle hasa very significant effect on the canonical energy and gravitational potential of idea gas, and finalleads to modified Jeans mass lower than the original case, this result can be used to explain theproblem of Bok globules. Furthermore, we constrain the upper limits of the parameter of higherorder generalized uncertainty principle γ by using the different data of Bok globules. The resultsshow that the range from 10 to 10 , which are more stringent than most of those derived fromprevious works. I. INTRODUCTION
Einstein’s theory of the general relativity is regardedas the cornerstone of the development of modern physicsand astronomy. However, with the deepening of research,it is found that general relativity has flaws, which leadto many problems, such as to the black hole informa-tion paradox, naked singularity of spacetimes, and so on[1]. Now, one of most potential candidate to solve thoseproblems is the quantum gravity. Based on theories thestring theory, loop quantum gravity and noncommutativegeometry, people believed that there is the existence ofa minimum measurable length of the order of the Plancklength, which is supported by the Gedanken experiments[2–4].According the minimum measurable length, it is foundthat the Heisenberg uncertainty principle (HUP) can bemodified as Generalized uncertainty principle (GUP). In1995, Kempf, Mangano and Mann proposed a quadraticform of GUP, which we now call as “KMM model”with the expression as ∆ x ∆ p ≥ ~ (cid:2) γ ∆ p (cid:3) , where β = β (cid:14) M p c = β ℓ p (cid:14) ~ and β is the GUP parame-ter. It is easy to find that KMM model predicts a min-imal length ∆ x KMMmin ≈ ℓ p √ β [5]. Subsequently, incor-porating the idea of the maximal momentum, Ali, Dasand Vagenas constructed another GUP (ADV model),namely, ∆ x ∆ p ≥ ~ (cid:2) α ℓ p h p i / ~ + 4 (cid:10) p (cid:11) α ℓ p (cid:14) ~ (cid:3) ,where α is the GUP parameter. Moreover, this linerand quadratic GUP suggests the existence of a min-imal length ∆ x ADVmin ≈ α ℓ p and a maximal momen-tum ∆ p ADVmax ≈ ℓ p / α [6]. In recent years, the KMMmodel and ADV model play important roles in the re-searches of many physics contexts. For example, by us-ing the KMM model, people derived the GUP correctedHamilton-Jacobi equation and investigated the modifiedtunneling rate of particles with arbitrarily spins from ∗ Email: [email protected]; [email protected] event of curved spacetimes [7]. Moreover, the KMMmodel and ADV model can be also extended to Procaequation, which leads to the modified Hawking tempera-ture of black holes [8, 9]. Besides, Vagenas et al . studiedthe validity of the no-cloning theorem within the frame-work of GUP. They pointed out that the energy requiredto send information to a black hole is affected by quan-tum gravity [10]. In Ref. [11], the authors investigatedhow the KMM model affect the Casimir wormhole space-time, and obtain a class of asymptotically flat wormholesolutions.Despite KMM model and ADV model are two of moststudied GUP, they still have the some defects, such asthe minimal length and the maximal momentum are onlyvalid for small GUP parameters, and does not imply non-commutative geometry [12]. Therefore, for overcomingthose difficulties, Pedram introduced a nonperturbativehigher order GUP, which agree with various proposalsof quantum gravity [13]. Subsequently, this higher orderGUP was used to corrected the blackbody radiation spec-trum and predicts the cosmological constant. Based onthis heuristic work, many new form of higher order GUPwere presented [14–16]. Recently, in Ref. [17], Shababiand Chung proposed a new generalized Heisenberg alge-bra as follows: [ x, p ] = iγ − γp , (1)which leads to a higher order GUP:∆ x ∆ p ≥ ~ − γ ∆ p ) , (2)where γ = γ (cid:14) M p c = γ ℓ p (cid:14) ~ with the dimension-less GUP parameter γ , the Planck mass M p and thePlanck length ℓ p . Meanwhile, Eq. (2) guarantees theexistence of an absolute smallest uncertainty in position∆ x min = 4 ~ √ γ (cid:14) p max = 1 (cid:14) √ γ , which never appears in the frame-work of Heisenberg uncertainty principle (HUP). How-ever, when γ →
0, the expression of Eq. (2) reduces tothat of HUP.According to the higher order GUP (2), people investi-gated the modified eigenfunctions and eigenvalues for theparticle in a box and one-dimensional hydrogen atom, re-spectively. Moreover, one may also find that the higherorder GUP corrected behavior of Bohr-Sommerfeld quan-tization, which can be used to calculate energy spectraof quantum harmonic oscillator and quantum bouncer[16]. In the light of the previous work, it is believed thatthe higher order GUP can not only be used to study theproperties of the quantum systems, but also to solve theproblems of astrophysics, black hole physics, cosmology,and so on. For example, a lot of astronomical data showthat there are some Bok globules have stars or at least ex-perience the star formation process [18]. However, theirmass are less than the Jeans mass [19]. For solving thisproblem, Moradpour, et al . used the KMM model andADV model to correct the limited of Jeans mass [20].Motivated by the above arguments, we try to extend thehigher order GUP into the Jeans instability and calcu-late the GUP corrected limited of Jeans mass of thoseespecial Bok globules in the present article.The paper is organized as follows: Section II is de-voted to a review of the Jeans instability limit in clas-sical case. In Section III, according to the new higherorder GUP (2), we compute the corrections to the po-tential energy and canonical energy of a molecular cloud,respectively. Then, we discuss the modified Jeans grav-itational instability, and then derive the GUP correctedJeans mass for explaining the problem of Bok globules.By using the GUP corrected Jeans mass, the parameterof the new higher order GUP is constrained in Section IV.Finally, conclusions can be found in Section V.
II. THE JEANS INSTABILITY LIMIT INCLASSICAL CASE
In this section, we briefly review the Jeans instabilitylimit and derive the Jeans mass in classical case. Basedon the argument that relies upon the virial theorem inRef. [20], the authors pointed out that the collapse ofmolecular cloud happen if the gravitational potential en-ergy E p is larger than the canonical energy U , namely U < − E p /2 . (3)Since the gravitational potential energy and the canonicalenergy are related to the mass and free energy of a molec-ular cloud, respectively. For obtaining the Jeans mass, itis necessary to derive the expressions of gravitational po-tential energy and the canonical energy. Since the space-time is flat in the classical case, the Newton’s law of gravi-tation is F = GM m (cid:14) r , and the gravitational potentialbecomes V ( r ) = − GM / r , the corresponding potential energy is given by E p = Z M V ( r ) dr = Z R πρ ( r ) rGM dr = − GM R . (4)It should be noted that, the density of dark cloud ρ ( r )in Eq. (4) is assumed as a constant ρ for the sake ofsimplicity.Next, the classical fundamental commutation relationin the flat spacetime can be expressed as [ x, p ] = i ~ ,which leads to HUP ∆ x ∆ p ≥ ~ /2. When consider-ing the Liouville theorem, the density of states in mo-mentum space in the spherical coordinate systems is D ( p ) dp = 4 πV p dp (cid:14) h , which leads to the original par-tition function of the ideal gas as follows Z = Z N N ! = 4 πVN ! Z p h exp (cid:18) − p µk B T (cid:19) dp = 4 πVN ! (2 πµk B T ) / h , (5)where N , µ and T are the numbers of non-interactingparticles,mass and temperature of the ideal gas, respec-tively [21]. Besides, considering mass of each ideal gasmolecule is µ , the total mass of the ideal gas satisfiesthe relationship M = µN . According to Eq. (5), onecan straightforward to derive the equation of state of theideal gas P V = N k B T , and the canonical energy can beexpressed as U = N k B T (cid:18) ∂ ln Z∂T (cid:19)
N,V = 32
N k B T. (6)Obviously, the original canonical energy is depended thenumber of non-interacting particles N and their tem-perature T . Now, substituting Eq. (4) and Eq. (6)into inequality (3) and considering the radius as R =(3 M /4 πρ ) , the result is M > (cid:18) k B TGµ (cid:19) (cid:18) πρ (cid:19) . (7)Hence, the abovemention equation indicates a lowerbound of the cloud mass to collapse, which is known asthe Jeans mass M J = (cid:18) k B TGµ (cid:19) (cid:18) πρ (cid:19) . (8)Despite the formation of most stars can be examined bythe original Jeans instability and Jeans mass, people stillobserved some Bok globules, such as CB 84, CB 110, theymass are less than M J but still have stars or at least ex-perience the star formation process [19]. For explant thatproblem that the astronomical observation brought out,we investigate the modified Jeans mass in the frameworkof higher order GUP. III. THE JEANS INSTABILITY LIMIT INHIGHER ORDER GUP CASE
The GUP have various implications for a wide rangeof physical systems. Meanwhile, they are regarded as apowerful tool to solve various difficult problems in theseresearch fields. Therefore, in order to obtain the modifiedJeans gravitational instability, we discuss how higher or-der affects the gravitation potential energy and canonicalenergy of idea gases, respectively.
A. GUP corrected gravitation potential energy ofidea gases
For deriving the GUP corrected gravitation potentialenergy, we need to use the theory of entropic force. Itis an intriguing explanation for Newton’s law of grav-ity, which is based on the holographic principle and anequipartition rule. Now, by taking ~ = 1 and usingEq. (2), one can easily find the following inequality∆ p ≥ / ∆ x (cid:16) − x γ + p − x γ + 81∆ x γ (cid:17) / + (cid:16) − x γ + p − x γ + 81∆ x γ (cid:17) / / ∆ xγ . (9)Expanding RHS of Eq. (9) with respect to γ and con-sidering the absolute smallest uncertainty in position √ γ = 9∆ x /4 [23], one has∆ p ≥ − x (cid:20) − x √ γ + 3 √ γ x + 3 γ x + O (cid:0) γ (cid:1)(cid:21) ≥ − x (cid:20) √ γ x + O (cid:0) γ (cid:1)(cid:21) ≈ − x (cid:20) x γ + O (cid:0) γ (cid:1)(cid:21) . (10)In Ref. [24], it was proposed that the uncertainty mo-mentum ∆ p can be defined as the energy ω . Hence, theabovementioned inequality can be rewritten as ω ≥ − x (cid:20) x γ + O (cid:0) γ (cid:1)(cid:21) . (11)From the Verlinde’s entropy force theory, for calculat-ing the modified gravitational force, one should considera spherically symmetric gravitational system, which al-lows the quantum particles entry or exit its horizon [25].Accordingly, when the system absorbs or releases parti-cles, the minimal change in the horizon area is ∆ A min ≥ πωRℓ p , where R is the size of a quantum particle. Basedon the arguments of Ref. [26], the quantum particle hasa lower boundary R ≥ ∆ x , which leads to ∆ A min ≥ πω ∆ xℓ p . Substituting this relation into Eq. (11),one has ∆ A min ≥ − πℓ p (cid:2) γ /28∆ x + O (cid:0) γ (cid:1)(cid:3)(cid:14) x = 2 r with the radius of the gravitational sys-tem r . Furthermore, considering the geometric propertiesof the gravitational system, the relation between ∆ x and A is given by ∆ x = 4 πr = A / π [27, 28]. Therefore, theminimal change in the horizon area of the gravitationalsystem goes to∆ A min ≥ λℓ p (cid:20) A π γ + O (cid:0) γ (cid:1)(cid:21) , (12)where λ is an undetermined coefficient. In Ref. [26], theauthors pointed out that the information of the gravita-tional system is reflected in its area. On the other hand,based on the information theory, it is believed that thearea of one system can affect its smallest increase in en-tropy [29]. Since the fundamental unit of entropy as onebit of information is ∆ S min = b = ln 2, one obtains dSdA = ∆ S min ∆ A min = bλℓ p (cid:2) A π γ + O ( γ ) (cid:3) , (13)which turns out to be S = Ak B ℓ p (cid:20) − π A γ ln A + O (cid:0) γ (cid:1)(cid:21) , (14)where we fix b / λ = k B /4 since when γ → S = Ak B /4 should be recovered [30]. FromEq. (14), it is realize that the entropy of gravitationalsystem has been corrected by the higher orders of GUP.When ignore the effect of GUP, the modified entropyreduces to the original case. The correction term inbrackets are determined by the parameter γ and the log-arithmic term ln A , which is coincident with previousworks [31–34]. According to the holographic principleand entropy-area law, when assuming the gravitationalsystem has N -bits information, the number of bits canbe expressed as follows: N = 4 Sk B = Aℓ p (cid:20) − π A γ ln A + O (cid:0) γ (cid:1)(cid:21) . (15)Next, one can further denotes the total energy of thegravitational system as E , which is average distributionis in N bits. Hence, each bit contains k B T /2 energy,then, following the equipartition rule, the total energytakes the form E = N T k B /2 . (16)On the other hand, in Ref. [25], Verlinde demonstratedthe Entropic force is more fundamental than gravity, andentropic force of a gravitational system is F ∆ x = T ∆ S, (17)where F is the entropy force, T is the temperature, ∆ S is the change of entropy of the gravitational system, and∆ x represents the displacement of the particle with themass m from the gravitational system, which satisfy therelation ∆ S = 2 πmc ∆ x [35, 36]. Now, substituting theexpression of change of entropy, Eq. (14)-Eq. (16) intoEq. (17), and considering E = M c and A = 4 πr , theNewton’s law of gravitation should be corrected as fol-lows: F GUP = F (cid:20) γr ln (cid:0) πr (cid:1) + O (cid:0) γ (cid:1)(cid:21) . (18)Obviously, when γ = 0, Eq. (18) reduce to the origi-nal Newton’s gravitation F = GM m (cid:14) r . Based on themodified Newton’s law of gravitation, the correspondingGUP corrected gravitational potential is V GUP = Z F GUP m dr = − GMr (cid:26) γ r (cid:20) (cid:0) πr (cid:1)(cid:21) + O (cid:0) γ (cid:1)(cid:27) , (19)where the terms of O (cid:0) γ (cid:1) are ignored since their contri-bution to gravitational potential is very small. ApplyingEq. (19) to a molecular cloud with radius R , mass M andthe almost uniform density ρ , the following expressionfor the modified potential energy is obtained E p (GUP) = Z R V GUP ( r ) dM = − GM R (cid:20) γ R ln (cid:0) πR (cid:1) + O (cid:0) γ (cid:1)(cid:21) . (20)It is obvious that this field equation is affected by themass M , the radius R , and the GUP parameter γ . Whentaking γ = 0, the modified potential energy reduces tothe original case original potential energy Eq. (4). B. GUP corrected canonical energy of idea gases
According to Eq. (3) and Liouville theorem,one can obtain the weighted phase space volume (cid:0) − γp (cid:1) (cid:0) − γp (cid:1) d ~xd ~p remains invariant underthe time evolution, and the density of states in momen-tum space in the spherical coordinate systems reads D ( p ) dp = 4 πVh p (cid:0) − γp (cid:1) (cid:0) − γp (cid:1) dp. (21)Plugging Eq. (21) into the partition function of a darkcloud, which approximated by an ideal gas containing N non-interacting particles with mass m at temperature T ,one reaches Z GUP = 4 πVN ! Z p h (cid:0) − γp (cid:1) (cid:0) − γp (cid:1) exp (cid:18) − p µk B T (cid:19) dp = 4 πVN ! (2 µk B T ) / I ( γ ) h , (22) where I ( γ ) = √ π (cid:2) − k B µT γ + O (cid:0) γ (cid:1)(cid:3) . the GUPcorrected canonical energy is presented as U GUP = N k B T (cid:18) ∂ ln Z GUP ∂T (cid:19) N,V = U (cid:2) − µk B T γ + O (cid:0) γ (cid:1)(cid:3) , (23)where original canonical energy U = 3 N k B T /2. Obvi-ously, the modified canonical energy dose not only relatedto the original canonical energy U , but also to parameter γ , mass of a non-interacting particle m and temperature T . C. GUP corrected Jeans mass
Now, by retaining the terms up to O (cid:0) γ (cid:1) , and substi-tuting Eq. (20) and Eq. (23) into Eq. (3), one yields N k B T (1 − k B µT γ ) < GM R (cid:20) γ R ln (cid:0) πR (cid:1)(cid:21) . (24)Considering the radius R = (3 M /4 πρ ) , Eq. (24) canbe rewritten as follows:5 k B T (1 − k B µT γ ) µ (4 πρ /3) < M χ, (25)where χ = G (cid:26) γ M /4 πρ ) / ln (cid:20) π (cid:16) M πρ (cid:17) / (cid:21)(cid:27) .It is clear from Eq. (25) that if γ = 0, the up-per bound of mass goes to the Jeans mass M J =(5 k B T / Gm ) (3/4 πρ ) . On the other hand, due to M ≫ γ is finite, the GUP corrected Jeans mass isobtained by saturating Eq. (25), with the result M J GUP = M J (1 − µk B T γ ) . (26)One may see that the GUP corrected Jeans mass M J GUP is related to the original Jeans mass M J , the idea gasmass m , and the GUP parameter γ . Meanwhile, it shouldbe noting that the mass of non-interaction particle andthe temperature of gravity system must be real num-bers greater than zero, if γ >
0, the M J GUP is positivewhen µk B T γ < M J GUP is lower than M J . Up to now, the astronomical observations show thatsome Bok globules’ mass are less than their correspond-ing Jeans mass. The GUP corrections to the Jeans masscan be a candidate to explain those observational facts.On the other hand, Ong claimed that the negative GUPparameter is physically meaningful since it restore theChandrasekhar limit in Ref. [37]. So, it is interesting toinvestigate how modified Jeans mass change for γ < IV. CONSTRAINTS FOR PARAMETER γ In many pervious works, the GUP parameter γ is al-ways assumed to be 1, so that modified results are neg-ligible unless energy approaches Planck scale. However,if the assumptions regarding the GUP parameters is notbe considered, the bound of GUP parameters can be ob-tained by previous experimental and observational data.For example, the astronomical observations show thatsome Bok globules’ mass is smaller than their originalJeans mass, however, gravitational collapse still occurredand produce the stars. Those facts can be explained byour work since the effect of GUP is able reduce the Jeansmass. Now, substituting the mass M , temperature T and the original Jeans mass M J of a Bok globules intoEq. (26), and considering the GUP corrected Jeans massequals to the mass of Bok globules, namely, M J GUP = M ,the upper bound of the dimensionless GUP parameter γ is as follows: γ = c M p k B µT " − (cid:18) M M J (cid:19) / . (27)Remarkably, the different data of Bok globules leads tothe different upper limit for GUP. Hence, according toRef. [18], the upper bounds on γ set by different dataof Bok globules are presented in Table. I. TABLE I. Name, temperature, mass, Jeans mass and the up-per bounds on GUP parameter γ of Bok globulesName T M M J γ (K) (M ⊙ ) (M ⊙ )CB 87 11.4 2.73 9.6 4 . × CB 110 21.8 7.21 8.5 4 . × CB 131 25.1 7.83 8.1 8 . × CB 161 12.5 2.79 5.4 2 . × CB 188 19.0 7.19 7.7 2 . × FeSt 1-457 10.9 1.12 1.4 1 . × Lynds 495 12.6 2.95 6.6 3 . × Lynds 498 11.0 1.42 5.7 5 . × From Table. I, we set with M p = 2 . × − kg, c =2 . × m · s − , k B = 1 . × − J · k − , and M ⊙ denotes the Sun mass. The final results are determined by the mass of non-interaction particles µ . For gettingthe exact value of the upper bounds on γ , we furtherassume that the non-interacting particles are composedof the hydrogen atoms, which is most abundant elementin the universe with the mass µ H = 1 . × − kg. Inthis case, one can find that the range of the order of GUPparameter γ from 10 to 10 .Since it is the first time to get the upper limit of theparameter of GUP (4), we can not compare the GUP pa-rameter γ horizontally. However, one can compare ourresults with those from KMM model and ADV model.According to a series of experiments, such as Electrontunneling, Gravitational bar detectors, Rb cold-atom-recoil experiment, Late-time cosmology et al . the up-per bound range of the GUP parameter of KMM modelfrom 10 to 10 [38–47], and the upper bound range ofthe GUP parameter of ADV model from 10 to 10 [38, 42, 43, 45, 47–49], respectively. As we stated ear-lier in the introduction, the higher order GUP overcomesthe problems of KMM model and ADV model, and con-siders more factors, which leads to our results are morestringent than most of those derived from previous ones.Therefore, GUP (4) is a more effective quantum gravityphenomenon mode, it can help researchers better solvethe problems in physics in the future work V. DISCUSSION
In this paper, by incorporating a new higher order ofGUP with the virial equilibrium and Verlinde’s entropyforce theory, we investigated the modified canonical en-ergy U GUP and the modified gravitational potential en-ergy E p (GUP) of a molecular cloud, respectively. Subse-quently, according to those modifications, the GUP cor-rected Jeans mass M J GUP is obtained. It is found thatthe GUP can effectively increase the gravitational po-tential, and reduce the canonical energy and the Jeansmass. This leads to collapse of Bok globules with massesless than the standard value M J , which is consistent withthe facts of astronomical observations. Finally, using thedifferent data of Bok globules, we constrain the upperbound of the GUP parameter γ . The results showedthat its range of upper bound from 10 to 10 , whichare more stringent than most of those derived from pre-vious works. [1] P. Chen, Y. C. Ong, D.-H. Yeom, Phys. Rep. (2015)1. arXiv:1412.8366[2] M. Maggiore, Phys. Lett. B (1993) 83.arXiv:hep-th/9309034[3] G. Amelino-Camelia, Int. J. Mod. Phys. D (2002) 35.arXiv:gr-qc/0012051[4] F. Scardigli, Phys. Lett. B (1999) 39.arXiv:hep-th/9904025 [5] A. Kempf, G. Mangano, R. B. Mann, Phys. Rev. D (1995) 1108. arXiv:hep-th/9412167[6] A. F. Ali, S. Das, E. C. Vagenas, Phys. Lett. B (2009) 497. DOI: 10.1016/j.physletb.2009.06.061[7] Z. W. Feng, H. L. Li, X. Z. Tao, S. Z. Yang, Eur. Phys.J. C (2016) 212. arXiv:1604.04702[8] I. Sakalli, A. ¨Ovg¨un, Europhys. Lett. (2017) 60006.arXiv:1702.04636 [9] A. ¨Ovg¨un, K. Jusufi, Eur. Phys. J. Plus (2017) 298.arXiv:1703.08073[10] E. C. Vagenas, A. F. Ali, H. Alshal, Eur. Phys. J. C (2019) 276. arXiv:1811.06614[11] K. Jusufi, P. Channuie, M. Jamil, Eur. Phys. J. C (2020) 127. arXiv:2002.01341[12] P. Pedram, Phys. Lett. B (2012) 317.arXiv:1110.2999[13] P. Pedram, Phys. Lett. B (2012)638.DOI: 10.1016/j.physletb.2012.10.059[14] W. S. Chung, H. Hassanabadi, Eur. Phys. J. C (2019)213. DOI: 10.1140/epjc/s10052-019-6718-3[15] H. Hassanabadi, E. Maghsoodi, W. S.Chung, Eur. Phys. J. C (2019) 358.DOI: 10.1140/epjc/s10052-019-6871-8[16] H. Shababi, W. S. Chung, Mod. Phys. Lett. A (2018)1850068. DOI: 10.1142/S0217732320500182[17] H. Shababi, W. S. Chung, Phys. Lett. B (2017) 445.DOI: 10.1016/j.physletb.2017.05.015[18] J. Vainio, I. Vilja, Gen. Relativ. Gravit. (2016) 129.arXiv:1512.04220[19] R. Kandori, Y. Nakajima, M. Tamura, K. Tatematsu, Y.Aikawa, T. Naoi, K. Sugitani, H. Nakaya, T. Nagayama,T. Nagata, M. Kurita, D. Kato, C. Nagashima, S. Sato,Astron. J. (2005) 2166. arXiv: astro-ph/0506205[20] H. Moradpour, A. H. Ziaie, S. Ghaffari, F. Feleppa, Mon.Not. R Astron. Soc. (2019) L69. arXiv:1907.12940[21] W. S. Chung, H. Hassanabadi, Int. J. Mod. Phys. A (2019) 1950041. DOI: 10.1142/S0217751X19500416[22] P.Wang, H. Yang, X. Zhang, J. High Energy Phys. (2010) 043. arXiv:1006.5362[23] Z.-W. Feng, S.-Z. Yang, H.-L. Li, X.-T. Zu, Adv. HighEnergy Phys. (2016) 2341879. arXiv:1607.04114[24] R. J. Adler, P. Chen, D. I. Santiago, Gen. Relativ. Gravit. (2001) 2101. DOI: 10.1023/A:101528143041[25] E. Verlinde, J. High Energy Phys. (2011) 29.arXiv:1001.0785[26] J. D. Bekenstein, Phys. Rev. D (1973) 2333.DOI: 10.1103/PhysRevD.7.2333[27] G. Amelino-Camelia, M. Arzano, A. Procaccini, Phys.Rev. D (2004) 107501. arXiv:gr-qc/0405084[28] B. Majumder, Phys. Lett. B (2011) 402.arXiv:1106.0715 [29] C. Adami, arXiv:quant-ph/0405005[30] A. Awad, A. F. Ali, Central Eur. J. Phys. (2014) 245.arXiv:1403.5319[31] P. Bargue¯no, E. C. Vagenas, Phys. Lett. B, (2015)15. arXiv:1501.03256[32] N. M.-Dur´an, A. F. Vargas, P. Hoyos-Restrepo, P. Bar-gue¯no, Eur. Phys. J. C (2016) 559. arXiv:1606.06635[33] Z.-W. Feng, S.-Z. Yang, Phys. Lett. B (2017) 737.arXiv:1501.03256[34] Z.-Y. Fu, H.-L. Li, Y. Li, D.-W. Song, Eur. Phys. J. Plus (2020) 125. DOI: 10.1140/epjp/s13360-020-00190-5[35] I. Sakalli, Int. J. Theor. Phys. (2011) 2426.arXiv:1103.1728[36] A. Sheykhi, H. Moradpour, N. Riazi, Gen. Relativ.Gravit. (2013) 1033. arXiv:1109.3631[37] Y. C. Ong, J. High Energy Phys. (2018) 195.arXiv:1806.03691[38] S. Das, E. C. Vagenas, Phys. Rev. Lett. (2008)221301. DOI: 10.1103/PhysRevLett.101.221301[39] F. Marin, F. Marino, M. Bonaldi, M. Cerdonio, L. Conti,P. Falferi, R. Mezzena, A. Ortolan, G. A. Prodi, L. Taf-farello, G. Vedovato, A. Vinante, J.-P. Zendri , Nat. Phys. (2013) 71. DOI: 10.1038/nphys2503[40] S. Ghosh, Class. Quant. Grav. (2014) 025025.arXiv:1303.1256[41] F. Scardigli, R. Casadio, Eur. Phys. J. C (2015) 425.arXiv:1407.0113[42] D. Gao, M. Zhan, Phys. Rev. A (2016) 013607.arXiv:1607.04353[43] Z.-W. Feng, S.-Z. Yang, H.-L. Li, X.-T. Zu, Phys. Lett.B (2017) 81. arXiv:1610.08549[44] S. Kouwn, Phys. Dark Univ. Volume (2018) 76.arXiv:1805.07278[45] P. Bushev, J. Bourhill, M. Goryachev, N. Kukharchyk,E.Ivanov, S. Galliou, M. Tobar, and S. Danilishin, Phys.Rev. D (2019) 066020. arXiv: 1903.03346[46] S. Giardino, V. Salzano, arXiv:2006.01580[47] J. C. S. Neves, Eur. Phys. J. C (2020) 343.arXiv:1906.11735[48] S. Das and R. Mann, Phys. Lett. B (2011) 596.arXiv:1109.3258[49] A. F. Ali, S. Das, E. C. Vagenas, Phys. Rev. D84