arXiv:1709.03027v1 [gr-qc] 10 Sep 2017 eaiitcBlzaneuto n nrp urn,i is a it satisfies current, and modific equation Boltzmann nonadditive relativistic incorporating By framework. ativistic oyaisi bandol hte h entropic the whether only obtained is modynamics sdsrbdb h xeddBs-isen(Fermi-Dirac) Bose-Einstein prov extended also the is by it described fields, is gravitational of absence the in occurs ASnmes 41.a 66.c 25.75.-q 26.60.+c; 24.10.Pa; numbers: PACS ∗ § ‡ † [email protected] [email protected] raimundosilva@fisica.ufrn.br [email protected] h oetniekntcter o eeeaeqatmgas quantum degenerate for theory kinetic nonextensive The oetnieKntcTer n -hoe nGeneral in H-Theorem and Theory Kinetic Nonextensive H termi h rsneo rvttoa ed.Consisten fields. gravitational of presence the in -theorem .P Santos P. A. nvriaed ˜oPuo 50-9 oPuoS,Brasil SP, Paulo So 05508-090 S˜ao Paulo, de Universidade nvriaeFdrld mzns tcair-M Brasil Itacoatiara-AM, Amazonas, do Federal Universidade eatmnod ´sc,Ntl-R,50290 Brazil 59072-970, RN, - F´ısica, Natal de Departamento 4 oRoGad oNre osro 91-1,Brasil Mossor´o, 59610-210, Norte, do Grande Rio do eatmnod srnma Observat´orio Nacional, Astronomia, de Departamento 3 2 eatmnod ´sc,Uiesdd oEstado do F´ısica, Universidade de Departamento nvriaeFdrld i rned Norte, do Grande Rio do Federal Universidade 1 5 02-0 i eJnioR,Bai and Brasil RJ, Janeiro de Rio 20921-400 nttt eCeca xtseTecnologia, e Exatas Ciencias de Instituto eatmnod srnma(IAG-USP), Astronomia de Departamento 1 , ∗ .Silva R. Dtd coe 4 2018) 14, October (Dated: 2 , 3 Relativity , † Abstract .S Alcaniz S. J. 1 q prmtrle nteinterval the in lies -parameter q dta h oa olsoa equilibrium collisional local the that ed 4 hw htTalsetoi framework entropic Tsallis that shown -distributions. , ‡ tosi h olsoa emo the of term collisional the in ations n .A .Lima S. A. J. and si icse ntegnrlrel- general the in discussed is es ywt h n a fther- of law 2nd the with cy 5 § q ∈ [0 , ] As 2]. Some extensions of the orthodox Boltzmann-Gibbs-Shannon (BGS) entropy have been proposed in order to address the statistical and kinetic behavior of anomalous nonadditive systems [1, 2], among them, the Tsallis [3] and Kaniadakis [4] statistics (see also [5] for other possibilities). Here we focus our attention on the nonextensive Tsallis entropy which for a classical non-degenerated gas system of point particles reads (unless explicitly stated, in our units

kB = c = 1) S = f q ln fd3p, (1) q − q Z where q is a real number quantifying the degree of nonadditivity of Sq, and lnq(f) is the nonadditive q-logarithmic function whose inverse is the q-exponential. Both functions are defined by:

ln (f)= (1 q)−1(f 1−q 1), (f > 0), (2) q − − 1 e (f) = [1 + (1 q)f] 1−q , e (ln f)= f, (3) q − q q which reduce to the standard expressions in the limit q 1. The above formulas also imply → that for a gas system composed by two subsystems (A,B), the kinetic Tsallis measure verifies S (A + B)= S (A)+ S (B)+(1 q)S (A)S (B). Hence, for q = 1 the logarithm extensive q q q − q q measure associated to the GBS approach is recovered. The validity of the 2nd law of thermodynamics for the nonextensive path was kinetically investigated in the classical [6], special relativistic [7] and quantum-mechanical domains [8, 9] (see also [10] for a simplified kinetic derivation of the q-Maxwellian based on a nonadditive extension of the ansatz adopted by Maxwell in his seminal paper [11]). The thermodynamic consistency of all these short-range regimes provided the constraint q [0, 2], which was ∈ also obtained based on the convexity property of the extended relativistic entropy in the quantum regime [9] (cf. [12] for tighter bounds based on the third law of thermodynamics). More recently, relaxing the local Maxwellian equilibrium distribution, the classical Chan- drashekar result for dynamical friction in Newtonian gravity was investigated in the q- nonextensive context. It was found that the large timescale of globular clusters spiraling to the center of the Milky Way, as obtained from N-body simulations, can be interpreted as a departure from the extensive Gaussian result in agreement with Tsallis power-law approach [13].

2 On the other hand, we recall that the foundations of the kinetic counterpart of BGS statistics in the framework of general relativity theory (GRT) has been investigated and understood since long ago [14–17]. In the following decades, this approach was applied to relevant problems in astrophysics and cosmology, like stellar and cosmological nucleossyn- thesis, baryogenesis, and fluctuations of the cosmic background radiation [18–21]). However, as far as we know, the same does not occurs with the nonextensive Tsallis proposal. In prin- ciple, this is a clear obstacle to enlarge the field of applications for testing the nonextensive kinetic formalism which is considered more convenient to systems endowed with long range forces. In this letter we fill this gap providing an analytical proof of the H-theorem for degenerate nonextensive quantum gases in curved spacetimes. To this end, we combine two new in- gredients: (i) nonadditive entropy current, and (ii) possible effects of statistical correlations on the collisional term of the general relativistic kinetic equation (q-extension of the Boltz- mann “Stozssahlansatz” hypothesis). As we shall see, the second law of thermodynamics (the positiveness of the source of entropy) also constrains the nonextensive parameter on the interval 0 q 2. As an extra bonus, the results for the nonextensive general relativistic ≤ ≤ Juttner gas are also obtained as a particular case. In order to understand how nonextensive effects alters the standard approach, let us now consider a relativistic rarified quantum gas in a curved space-time which is out but not too far from equilibrium. The gas particles are supposed to interact through short and long- range forces. The former act only during the instantaneous binary collisions while the latter

are described by the metric tensor gµν. Following standard lines, the evolution of the one particle distribution function in phase space, f(x, p) f(xµ,pµ), is governed by the kinetic ≡ in the presence of gravitation [14–17]

∂f ∂f L[f] pµ Γµ pνpα = C[f] , (4) ≡ ∂xµ − να ∂pµ where L[f] is the Liouville operator and C[f] is the invariant phase space density of collisions. µ In the above equation is also implicit that f(x, p)p uµdΣdP is the particle number whose world lines intersect the volume element uµdΣ around x and a 4-momentum defined in the range (p,p + dp), such that µ, ν, β, ... =0, 1, 2, 3. As the quantum gas has particles with rest mass m, their 4-momentum must be restricted the part of the space of moments, therefore

3 an element of volume for this portion of phase space is defined by

dP := A(p)δ(p pµ + m2)√ gdp0dp1dp2dp3 , (5) µ −

where A(p) = 2 if p0 > 0, and A(p) = 0 for all other cases. The adopted signature is (-,+,+,+) so that p pµ = m2. µ − Following standard lines, the unified invariant collisional term (for all three different statistics) can be written as

′ ′ ′ ′ ′ ′ ′ ′ C[f]= [fˆfˆ1f f1 ff1fˆ fˆ1]W (pp1 p p1)dP1dP dP1 . (6) − → Z Z Z 3 Here, the particle distribution is given by fˆ 1+ κ h f, where h is the Planck constant and ≡ gs gs is the degeneracy factor of particles with spin s. In particular, the number κ = +1 refers to bosons (due to the statistics of indistinguishable particles), κ = 1 are fermions (due to − Pauli exclusion principle), while κ = 0 stand for particles without spin. The scalar collision ′ ′ frequency, W (pp1 p p1), depends only of the momentum variables and remains invariant → ′ ′ ′ ′ under space and time reflexions, W (pp1 p p1)= W (p p1 pp1). → → The macroscopic variables and the balance equations describing the thermodynamic states of the gas particles are microscopically calculated through the relativistic distribution function f(x, p), which here also depends on the κ parameter. The key macroscopic variable to the proof of the H-theorem is the entropy current defined by

g Sµ(x)= pµ[f ln f s fˆln fˆ]dP, (7) − − κh3 Z from which one obtains an expression for the second law of thermodynamics

µ f µS = ln( )L(f)dP 0, (8) ∇ − fˆ ≥ Z where denotes the covariant derivative. ∇µ By rewritten the above expression in terms of (6), one may check that the r.h.s. of the above equality is necessarily nonnegative thereby proving the standard H-theorem [17]. The collisional equilibrium states are obtained when ln(f/fˆ) is a summational invariant. In this case, the integral vanishes identically and the unified form of the equilibrium distribution function is derived gs µ −1 f0(x, p)= [exp(˜α β p ) κ] , (9) h3 − µ −

4 whereα ˜ = ln(g /h3) α. Note that for κ = 0 the relativistic form of the Maxwell-Boltzmann- s − Juttner distribution is recovered. At this point, it is worth mentioning that the collisional operator (6) is based on the Boltzmann“Stossahlansatz” hypothesis, which means that colliding particles are uncorre- lated. This celebrated nonmechanical assumption implies that the two point correlation functions are factorizable [f(x,p,p1)= f(x, p)f(x, p1)]. Let us now discuss the impact of the nonextensive entropic approach in the covariant

kinetic theory, and how the Hq-theorem for a quantum dilute gas is harmonized with the curved spacetime description. In principle, the Boltzmann “Stossahlansatz”, as well as the own measure of entropy will be modified [6–8]. The new hypothesis combined with changes in the entropy flux and collisional term, encode all the statistical properties of the nonextensive microscopic description of a dilute quantum gas. To begin with, we propose to the entropy current (7) the following nonadditive expression:

g Sµ(x)= pµ[f q ln f s fˆq ln fˆ]dP, (10) q − q − κh3 q Z 3 where the function ln has been defined in (2), and fˆq = (1+ κ h f)q [see definition below q gs (6)]. Operationally, it means that not only the logarithm functions are replaced by power laws (q-ln), but also that the collisional term must be suitably modified. By introducing the useful notation

g G (f) f q ln f s fˆq ln f,ˆ (11) q ≡ q − κh3 q it is easy to check that ′ Sµ = G (f)L[f]dP, (12) ∇µ q − q ′ Z where Gq(f) is the derivative with respect to f which is given by

′ G (f)= q[f q−1 ln f fˆq−1 ln fˆ]. (13) q q − q and using the duality property q∗ 2 q, the above expression becomes → −

′ G (f)= q[ln ∗ f ln ∗ fˆ]. (14) q q − q

At this point it is interesting to consider two basic identities defining a deformed algebra involving q-logarithm and q-exponential functions which will be used to simplify several

5 expressions. The q-difference and q-product is defined by [22]:

x y 1 x y := − y = , ⊖q∗ 1+(1 q∗)y ∀ 6 1 q∗ − 1 − 1 1 x y := x −q∗ + y −q∗ 1 1−q∗ x,y > 0, ⊗q∗ − which allow to write the q-ln and q-exp in a more compact form:

lnq∗ (f) lnq∗ (fˆ) lnq∗ (f/fˆ) lnq∗ (f) q∗ lnq∗ (fˆ)= − , (16a) ≡ ⊖ 1+(1 q∗) ln ∗ fˆ − q implies that (13) can be recast as

′ 1 ∗ ˆ −q ∗ ˆ Gq(f)= qf lnq (f/f), (17)

with expression (12) taking the form

µ 1−q∗ S = q fˆ ln ∗ (f/fˆ)L[f]dP, (18) ∇µ q − q Z which should be compared with (8). Actually, all the above expressions reduce to the standard ones in the mathematical limit q, q∗ 1. → On the other hand, nonextensive effects can be incorporated in the general relativistic Boltzmann equation (4) only through the collisonal term because the Liouville operator is not modified. Therefore, let us consider the following transport equation

L[f]= Cq[f], (19)

where beyond the standard requirements the structure of Cq(f) does not obey the “Stossahlansatz”

and limq→1 Cq(f)= C(f) [6, 7]. It is natural to consider that Boltzmann’s “Stosshalansatz” is not valid because the nonextensive effects give rise to statistical correlations requiring an extension of the molecular chaos hypothesis, an assumption underlying the Boltzmann approach which still remains controversial nowadays [23]. In order to capture correctly the effects of gravitational field in what follows we consider the following expression:

′ ′ ∗ ′ ′ f f1 ˆq ˆ ˆ ˆ ∗ Cq[f]= f f1f f1 q ′ fˆ′ ⊗ fˆ Z Z Z "  1 ! f f1 ′ ′ ∗ WdP1dP dP1. (20) − ˆ ⊗q ˆ f  f1  which reduces to (6) in the appropriate limit.

6 Once the basic nonextensive modifications on the 4-entropy and collisional term have been explicited the proof of the covariant version of the H -theorem ( Sµ 0) in the q ∇µ q ≥ context of general relativity can be performed. By replacing the equation (20) in (18), we obtain

′ ′ ′ ′ f f f1 µ ˆˆ ˆ ˆ ∗ ∗ µSq = q ff1f f1 lnq ′ q ′ ∇ fˆ fˆ ⊗ fˆ Z Z Z   "  1 ! f f1 ′ ′ ∗ WdP1dP dP1. (21) − ˆ ⊗q ˆ f  f1  ′ ′ By using the invariance of term W (pp1 p p1) in the above relation, as well as by considering → ′ ′ ′ the permutations of the type (p p1) (f f1), (p p ) (f f ) and (p1 p1) ′ → ⇒ → → ⇒ → → ⇒ (f1 f1), it is possible to show that →

µ q ′ ′ S = fˆfˆ1fˆ fˆ1 (22) ∇µ q 4 × Z Z Z ′ ′ f f1 f f1 ln ∗ + ln ∗ ln ∗ ln ∗ q ˆ q ˆ − q ˆ′ − q ˆ′ " f  f1  f  f1 !# ′ ′ f f1 f f1 ′ ′ ∗ ∗ WdP1dP dP1. ˆ′ ⊗q ˆ′ − ˆ ⊗q ˆ "f  f1 ! f  f1 #

Now, in order to simplify the above expression, let us use the following identity based upon

extended algebra, i.e ln ∗ (a ∗ b) = ln ∗ (a) + ln ∗ (b). Thereby, we obtain q ⊗q q q

µ q ′ ′ ′ ′ S = τ = fˆfˆ1fˆ fˆ1[ln ∗ Z ln ∗ Y ](Z Y )WdP1dP dP1, (23) ∇µ q q 4 q − q − Z Z Z where ′ ′ f f1 f f1 Z = ∗ ,Y = ∗ , (24) ˆ′ ⊗q ˆ′ ˆ ⊗q ˆ f  f1 ! f  f1  and τq is the entropy source which must be positive due the second law of thermodynamics.

This positiveness of the entropy source is related with the inequality (Z Y )[ln ∗ Z ln ∗ Y ] − q − q ≥ 0. It shows that the generally covariant H-theorem is alo consistent with nonadditive effects present in the Tsallis framework because

Sµ = τ 0. (25) ∇µ q q ≥ 7 Note that, for, q < 0 or q > 2, the equation (25) is a decreasing function of time, being thermodynamically forbidden. It thus follows that the parameter q is kinetically restricted on the interval [0, 2]. Such constraint is compatible with those which have been calculated through the Hq-theorem in the classical, relativistic and quantum-mechanics regimes [6– 8], as well as through an approach of quantum Clausius’s inequality [24]. The inequality (25) shows that the q-entropy source must be positive or zero thereby furnishing a kinetic argument for the second law of thermodynamics in the nonextensive formalism. Naturally, like in the standard Boltzmann approach, this argument does not provide a complete kinetic proof of the second law since an extra statistical assumption is also required. By imposing that the source of entropy vanishes (τq = 0) we obtain the necessary and sufficient condition stablishing the local equilibrium states. It also implies that L(f) 0 which means that f ≡ is a constant of motion. The simplest case happens in the absence of collisions (W = 0). In the collisional equilibrium case, the distribution function follows from the condition of detailed balance:

′ ′ f f1 f f1 ln ∗ + ln ∗ = ln ∗ + ln ∗ . (26) q ˆ q ˆ q ˆ′ q ˆ′ f  f1  f  f1 !

The above sum of lnq-terms remains constant during a collision (invariant summational). It is well known that the most general invariant collisional is a linear combination involving a scalar and the 4-momentum pµ (see for instance [15]). Consequently, the distribution function is given by the condition

f µ ln ∗ = α + β p , (27) q ˆ µ f  where α(x) and βµ = βµ(x) are quantities conserved in the collisions. By solving for f we obtain

3 −1 gs h µ f = exp ∗ ( α(x) β p (x)) κ , (28) h3 g q − − µ −  s  It is possible to show that the moments from f are finite whether βµ(x) is a time-like vector directed for the future, i.e., β (x) = β(x)u , where u uµ = 1, u0 > 0 and β > 0. µ µ µ − By replacing (28) into Boltzmann’s equation (19) with vanishing collision integral, it follows that ∂ α(x)pµ + (β )pγpν =0. (29) µ ∇ν γ 8 If m = 0 and β is along the geodesic and pµ is parallel transported, we have 6 µ (β )=0, (30) ∇ν γ and therefore α is a constant. βµ is a constant as well as a time-like vector, thus we obtain the expressions

−1 gs µ f = exp ∗ (˜α β˜ p ) κ , (31) h3 q − µ − 3 3 q∗−1 h 3 q∗−1 i whereα ˜ = ln ∗ (g /h ) (g /h ) α and β˜ =(g /h ) β , which reduces to the extensive q s − s µ s µ result in the limit q, q 1 (cf. (9)). ∗→ For κ = 1, the above expression represents the Bose-Einstein and Fermi-Dirac relativis- ± tic distributions in the Tsallis statistical formalism. When quantum effects are negligible (κ = 0), it reduces to the relativistic nonextensive distribution [7]

g µ f0(x, p)= exp ∗ (˜α β˜ p ), (32) h3 q − µ Finally, in the nonrelativistic limit the nonextensive Maxwell distribution, which is the basis of nonextensive is properly calculated [10]. In particular, in the extensive (or additive entropy) limit q 1 the standard cases are recovered. → Broadly speaking, the relativistic nonextensive q-distribution in the presence of gravi- tation is the most general approach when nonadditive effects upon the statistical physics are taken into account. Special cases like the special relativistic, classical and quantum limits [6–8] are all covered here. Even the nonrelativistic distribution is also obtained as a limiting case of expression (31). Physically, it is also very compelling that the heart of the H-theorem, that is, the fact that the entropy source is non-negative, as required by the sec- ond law of thermodynamics, constrains the nonextensive q-parameter on the same interval [0, 2] irrespective of the physical regime [6–8]. It should be also stressed that a simplifying and more direct deduction could be obtained by starting with the Boltzmann equation in the comoving frame. This argument provides a more direct proof of Hq-theorem. However, the approach based on the q-calculus and q-algebra as discussed in Ref. [22] and adopted here, may become an useful tool for different problems involving the q-statistical framework. Furthermore, there are trivial and nontrivial kinetic effects in the general relativistic (GR) framework. For instance, in the calculations of the transport coefficients [16, 38] the local

9 expressions are equivalent in both special relativity and GR context. Also, the same must be true for the q-relativistic kinetic approach. However, non-trivial GR kinetic effects are also ordinarily expected in several situations, as for instance, in the cosmological domain. In this concern, current calculations of the primordial power spectrum of matter perturbations and the temperature anisotropies of the cosmic background radiation are calculated through the Boltzmann equation and nontrivial effects appear there due to specific gravity contributions coming from the local relativistic gravitational potential [20]. Indeed, the same must happen when the relativistic q-distribution is adopted. In principle, it should be interesting to investigate how the q-kinetic approach as discussed here modify the standard results. Last but not least, the nonextensive general relativistic distribution (28) derived here must also be valid in the context of more general metric gravitational theories.

ACKNOWLEDGMENTS

APS wishes to thank FAPEAM (Funda¸c˜ao de Apoio a Pesquisa do Amazonas) for the DCR fellowship. This work was also partially supported by the Brazilian agency CNPq, CAPES (PROCAD 2013), FAPERJ (Funda¸c˜ao de Apoio a Pesquisa do Estado do Rio de Janeiro) and FAPESP (Funda¸c˜ao de Apoio a Pesquisa do Estado de S˜ao Paulo).

[1] M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy: Interdisciplinary Applications, Ox- ford University Press, New York, 2004. [2] G. Kaniadakis, Eur. Phys. J. B 70 (2009) (special number). [3] C. Tsallis, J. Stat. Phys. 52, 479 (1988). [4] G. Kaniadakis, Phys. Rev. E 66, 056125 (2002). [5] C. E. Shannon, A Mathematical Theory of Communication, Bell Syst. Tech. J. 27, 379 (1948); ibidem 27, 623 (1948). A. Renyi, On Measures of Entropy and Information, Proc. 4th Berk. Symp. Math. Statist. and Probl., University of California Press 1, 547(1961); B. D. Sharma, and I. J. Taneja, Three Generalized Additive Measures of Entropy, Elec. Inform. Kybern. 13, 419 (1977). [6] J. A. S. Lima, R. Silva, and A. R. Plastino, Phys. Rev. Lett. 86, 2938 (2001),

10 cond-mat/0101030 [7] R. Silva and J. A. S. Lima, Phys. Rev. E 72, 057101 (2005); Z. B. B. de Oliveira and R. Silva Annals of Physics 375, 227 (2016). [8] R. Silva, D. H. L. Anselmo, and J. S. Alcaniz, Europhys. Lett. 89, 10004 (2010). [9] S. Abe and A. K. Rajagopal, Phys. Rev. Lett. 91, 120601 (2003). [10] R. Silva, A. R. Plastino and J. A. S. Lima, Phys. Lett. A 249, 401 (1998). [11] J. C. Maxwell, Philos. Mag. Ser. 4, 20, 21 (1860). [12] E. P. Bento, G. M. Viswanathan, M. G. E. da Luz, and R. Silva, Phys. Rev. E 91, 022105 (2015). [13] J. M. Silva, et al., JCAP 05, (2016) 021, arXiv:1604.02034 [astro-ph.GA]. [14] G. E. Tauber and J. W. Weinberg, Phys. Rev. 122, 1342 (1961). [15] J. Ehlers, Abh. Math. Naturw. 11, 1, K1 (1961). [16] W. Israel, Journ. Math. Phys. 4, 9, 1163 (1963). [17] J. M. Stewart, Non-Equilibrium Relativistic Kinetic Theory, Springer-Verlag (1971). [18] E. W. Kolb and M. Turner, The Early Universe, Addison-Wesley (1994). [19] J. Bernstein, Kinetic Theory in the Expanding Universe, Cambridge (1989). [20] S. Dodelson, Modern Cosmology, Academic Press (2003). [21] S. Weinberg, Cosmology, Cambridge (2008). [22] E. P. Borges, Physica A 340, 95 (2004). [23] H. D. Zee, The Physical Basis of the Direction of Time, Spring-Verlag (1992). [24] S. Abe, A. K. and Rajagopal, Phys. Rev. Lett. 91, 120601 (2003). [25] K. Huang, “Statistical Mechanics, John Wiley & Sons (1987). [26] C. Y. Wong and G. Wilk, C. Tsallis Phys. Rev. D 87, 114007 (2013). [27] C. Y. Wong, G. Wilk, L. J. L. Cirto and C. Tsallis, Phys. Rev. D 91, 114027 (2015). [28] B. C. Li, Y. Z. Wang, F. H. Liu, X. J. Wen and Y. E. Dong, Phys. Rev. D 89, 054014 (2014). [29] Z. Tang, Y. Xu, L. Ruan, G. van Buren, F. Wang and Z. Xu, Phys. Rev. C 79, 051901 (2009). [30] J. A. S. Lima, R. Silva, J. Santos, Astron. and Astrophys. 396, 309 (2002), astro-ph/0109474 [31] M. Nauenberg, Phys. Rev. E 67, 036114 (2003). [32] T. S. Biro and P. Van, Phys. Rev. E 83, 061147 (2011). [33] R. Hanel, S. Thurner, and M. Gell-Mann, Proc. Natl. Acad. Sci. USA 111, 6905 (2014). [34] J. S. Andrade Jr., G. F. T. da Silva, A. A. Moreira, F. D. Nobre and E. M. F. Curado, Phys.

11 Rev. Lett. 105, 260601 (2010). [35] J. S. Andrade Jr., G. F. T. da Silva, A. A. Moreira, F. D. Nobre and E. M. F. Curado, Phys. Rev. Lett. 107, 088902 (2011). [36] C. Beck and E. G. D. Cohen, Physica A 322, 267 (2003). [37] P. Douglas, S. Bergamini, and F. Renzoni, Phys. Rev. Lett. 96, 110601 (2006); B. Liu and J. Goree, Phys. Rev. Lett. 100, 055003 (2008); R. M. Pickup, R. Cywinski, C. Pappas, B. Farago, and P. Fouquet, Phys. Rev. Lett. 102, 097202 (2009); V. Khachatryan et al. (CMS Collaboration), Phys. Rev. Lett. 105, 022002 (2010); G. Combe, V. Richefeu, M. Stasiak, A. P. F. Atman, Phys. Rev. Lett. 115, 238301 (2015). [38] S. Weinberg, Astrophys. J. 168, 175 (1971).

12