On Negative Mass Cosmology in General Relativity

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AA_Final where H is the Hubble parameter, κ = 8π, a is the scale fac- Going a step forward and without assuming a specific form tor, k is the curvature parameter, ρ+ is the positive mass den- of Γ, we can say more about this NMC model by analysing the 2 sity and ρ− is the negative mass density associated with a pres- deceleration parameter defined by q ≔ −H˙ /H − 1. Thus, using sure p− through an equation of state w− = p−/ρ− for the nega- Eq. (2) the deceleration parameter for this model is given by tive masses. Throughout this Letter we leave w− general, unless specifically stated. According to the NMC model presented by ΩM+ |ΩM−| Γ q = + − 3 (1 + w−) + 2 − ΩΛ, (7) Farnes (2018), in the case where Λ , 0, there is a degeneracy 2 2 " H ! # between Ω and Ω , which is given by Ω = Ω +Ω , M− Λ degen M− Λ which is independent of the curvature parameter k. Current ob- and it is claimed that in a conventional ΛCDM cosmology we servations (Riess et al. 1998; Perlmutter et al. 1999) support that are measuring this degeneracy. Thus, by taking Ω to be zero, M− the Universe goes through a late cosmic acceleration expansion, we are falsely inferring a Λ instead of a negative mass density so we must have the correct combination of variables in Eq. (7) parameter. to get q(z = 0) < 0. For the NMC toy model, with Λ = 0 and The degeneracy previously shown can be discriminated by Γ= 3H,Eq. (7) gives the equation of state w− of the negative masses, which yields w− = 0 for non-relativistic matter. To resolve this degeneracy, the 1 q = Ω + |Ω |, (8) NMC model considers that negative matter is constantly created 2 M+ M− by adding a matter creation term to the Einstein field equations with Λ. This will result in an effective equation of state for the which is independent of w−. We notice in here that q > 0 is pre- served, therefore the NMC toy model does not allow an accel- negative mass fluid in which weff , 0. By adding the creation term, the Einstein field equations, erated expansion. This result leads us to reject this toy model as a viable cosmological scenario, since several observations Gab +Λgab = κTab, (3) apart from supernovae Type Ia as baryon acoustic oscillations (Bassett & Hlozek 2009), clusters of galaxies (Astier & Pain where Gab is the Einstein’s tensor, gab is the Robertson-Walker 2012) and gravitational waves as standard sirens (Abbott et al. metric, will be modified so that the energy-momentum tensor 2017) along with the standard cosmographyapproach at low red- Tab is shifts (Escamilla-Rivera & Capozziello 2019) confirm this cos- Tab = (ρ + p + Pc)uaub + (p + Pc) gab, (4) mic acceleration. Moreover, it is always possible to choose different values for with ρ the total mass density, p the total pressure, and Γ, ΩΛ and even w− (if we try to give a more exotic nature to Γ the negative masses), to get the required accelerated expansion. Pc ≔ − (ρ− + p−), (5) Nevertheless, the choices for these values might be non-physical 3H and they would significantly increase the complexity of the prob- aneffective pressure caused by the creation of negative mass par- lem and would erase the initial motivation of a NMC model as a ticles, where Γ is the creation rate which, in principle, could de- simple alternative explanation of the dark sector. pend on space and/or time. The associated continuity equation Following the above general ideas, a final comment can be for the negative masses is made for a negative-mass dominated cosmology (i.e. |ρ−| ≫ ρ+) with Λ = 0. From Eq. (1), these assumptions imply k = −1 and ρ˙− + 3H(ρ− + p−) = Γ(ρ− + p−). (6) thus it is not always possible to have physical solutions com- 2 See e.g. Pan et al. (2016) for additional details of the background patible with observations (see comments below), since H could creation of particles in cosmology. We note that the NMC model take negative values. In particular, in a negative dust Universe 2 with Γ = 0 refers to the conventional negative mass cosmology without creation of particles (w− = 0, Γ= 0), H changes sign in within GR without creation of negative mass particles. |Ω−,0| The effective equation of state parameter for a Universe a = , (9) Ω in which matter is constantly created resembles that of a cos- k,0 Γ = mological constant when 3H. Therefore, this NMC toy with Ωi,0 = Ωi|t=0, where the subindex i denotes each fluid model has what appears to be an effective cosmological constant component.The same pathology occurs in the case Λ > 0, where Λ− ≔ 8πGρ− < 0. If we take ρ+ =Λ= 0 in this toy model, we once again H2 could become negative for a value of a > 0. By can obtain the evolution of the scale factor in a negative mass- similar arguments, H2 could still be negative when considering dominated cosmology from Eq. (2). The solution corresponds to radiation and the creation of negative mass particles, because an anti-de Sitter (AdS) Universe that undergoes a cycle of ex- ρ− < 0 independently of w− and Γ,as canbe shownfromEq.(6). 2 pansion and contraction with a timescale of −3π /Λ−. This Therefore, within the NMC model we have a Universe which cyclic cosmology does not match with our currentp observations, could begin with a finite size. Nevertheless, we know from however Farnes (2018) argued that there are two possible expla- estimations of the early Universe in the standard ΛCDM model, nations for this: (i) either the Universe is so large such that the that the Universe must begin from a very compact and dense local geometry appears to be flat, or (ii) the presence of nega- state (Weinberg 2008; Ellis et al. 2012) to be able to predict tive matter and its creation would modify the CMB anisotropies. cosmological observables, such as the abundances of primordial Both arguments would make the NMC toy model compatible nuclei (Cyburt et al. 2016; Steigman 2007; Iocco et al. 2009) with early time observations. A few examples that employ this and other early universe observations (Aghanim et al. 2020; NMC toy model are included in dark matter halos formation Cruz & Escamilla-Rivera 2021). In this way, we must fine-tune simulations, structure formation simulations and also, in galaxy the nature of ρ− and Γ in order to make the NMC scale factor rotation curves, by means of cosmological N-body simulations. compatible with these observables. Outside from this toy model, more research is needed in order to understandthe origin, the properties and the value of the creation Runaway motion.- Considering comoving coordinates and a rate Γ. smooth one-parameter family of geodesics γs(t), we denote the

Article number, page 2 of 4 Sebastián Nájera et al.: On Negative Mass Cosmology in General Relativity unit timelike tangent vector field to the family of geodesics as These results can be corroborated with Ellis & Van Elst (1999) a a a T = (∂/∂t) , such that T Ta = −1, and the separation vec- for positive masses and a FRW geometry.Thus, observersnearby tor between nearby geodesics as Xa = (∂/∂r)a, which satisfies to the positive particles will be attracted while observers in a a the orthogonality property TaX = 0. In this way, the geodesic neighbourhood of the negative particles will be repelled. Phys- deviation equation is ically, this represents the so-called runaway motion. Moreover, using the NMC’s effective energy–momentum tensor in Eq. (4), Aa = −R aXbT cT d, (10) a cbd AM reads as a c a c b a where A = T ∇cv = T ∇c T ∇bX is the relative accelera- a κ Γ a a AM = −ρ+ + |ρ−| (1 + w−) + 3w− + 1 X , (18) tion between nearby geodesics and R bcd is the Riemann tensor. 6 ( " H #) In particular, it is well known that for fundamental observers in RW space–times, Aa = −qH2(∂/∂r)a. Since q = q(t) and and considering w− = 0 the latter equation can be reduced to H = H(t), the relative acceleration must be common to all ob- a κ Γ a servers, and due to the symmetries of the Robertson–Walker A = −ρ+ + |ρ−| + 1 X , (19) metric we must deal with other geometries in order to estimate M 6 " H !# the effects of runaway motion. We now consider the particular case of negative and positive therefore,we obtain analogousresults to the dust case. We notice particles in an arbitrary space–time. Using the expansion of the that, unlike the fundamental observers in RW space–times, the relative acceleration in Eqs. (17)-(19), is not homogeneous nor Riemann tensor Γ isotropic, hence in a NMC model with |ρ−| H + 1 > |ρ+|, we 1 Rabcd = Cabcd + ga[cRd]b − gb[cRd]a − Rga[cgd]b, (11) have a positive relative acceleration between nearby geodesics, 3 which could cause non-linear effects in the velocity pertur- bations. Thus, structure formation in this kind of cosmology where Cabcd is the Weyl tensor, and Rab is the Ricci tensor we can rewrite Eq. (10) as becomes a fine-tuning problem.

a 1 Concluding remarks.- In this letter we thoroughly analysed A = − Ccbde + gc[dRe]b − gb[dRe]c − Rgc[dge]b " 3 # a negative mass cosmological (NMC) model which arise as an ea b c d alternative, simpler and unified explanation of the dark sector of g X T T . (12) the Universe, within the context of General Relativity, and with a Since the Weyl tensor is related to free space and the Ricci scalar possible background of continuously created negative mass par- is related to the energy–momentum tensor through the Einstein ticles. We found that this model requires fine-tuning in order field equations, we can divide the geodesic deviation equation to predict a late cosmological accelerated expansion, to have a a Hubble parameter defined in the whole cosmic redshift range ex- into an acceleration due to free space AFS, and the acceleration due to matter Aa , defined by pected from early Universe observations, and to allow structure M formation. Moreover, we discarded the reviewed model with a Aa ≔ −C aXbT cT d, (13) negative mass creation rate of Γ = 3H, because it does not pre- FS cbd dict late accelerated expansion; this particular choice of Γ had a 1 been used to explain and numerically model the effects of dark A ≔ − gc[dRe]b − gb[dRe]c − Rgc[dge]b M " 3 # energy and dark matter. In its current state, the NMC model re- geaXbT cT d. (14) quires a more concise theoretical formulation before being con- sidered as a viable model of the Universe which can be tested If we consider that the acceleration is due solely to matter and with cosmological surveys. 1 use the Einstein field equations Rab = κ(Tab − 2 Tgab) and R = −κT, then Eq. (14) reads as Acknowledgments.- AA, AG and SN acknowledge financial support from CONACYT postgraduate grants program. CE-R a κ a b b a ρ a acknowledges the Royal Astronomical Society as FRAS 10147 AM = T b X − QbX T − + 2p X , (15) 2 3 and by DGAPA-PAPIIT-UNAM Project IA100220. This article is also based upon work from COST action CA18108, supported where we have used that T = gabT = −ρ + 3p, with the energy ab by COST (European Cooperation in Science and Technology). density ρ = uaubT , the isotropic pressure p = (gab + uaub)T ab ab The ideas treated in this Letter were derived from a discussion in and the definition of the spatial energyflux vector Q ≔ −T bT . a ab the Lecture entitled “Aplicaciones Astrofísicas y Cosmológicas So far, the matter content is completely general. We will re- de la Relatividad General” at ICN-UNAM. strict to a special case. We consider the matter content as two compact fluids, the first composed of matter with positive density and the second with negative density. These fluids are sep- References arated such that we can neglect gravitational effects due to the opposite fluid and thus we can consider both as comoving. With Abbott, B. P. et al. 2017, Nature, 551, 85 Aghanim, N., Akrami, Y., Ashdown, M., et al. 2020, Astronomy & Astrophysics, these assumptions, we can rewrite the energy-momentum tensor 641, A6 as Astier, P. & Pain, R. 2012, Comptes Rendus Physique, 13, 521–538 Barcelo, C. & Visser, M. 2002, Int. J. Mod. Phys. D, 11, 1553 Tab = (ρ+ −|ρ−|)TaTb, (16) Bassett, B. A. & Hlozek, R. 2009 [arXiv:0910.5224] Bondarenko, S. 2019, Modern Physics Letters A, 34, 1950084 and from Eq. (15), we get Bondi, H. 1957, Reviews of Modern Physics, 29, 423 Bonnor, W. B. 1989, General Relativity and Gravitation, 21, 1143 κ Cruz, N. J. & Escamilla-Rivera, C. 2021, The European Physical Journal Plus, Aa = −ρ + |ρ | Xa. (17) M 6 + − 136, 1 Article number, page 3 of 4 A&A proofs: manuscript no. AA_Final

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