Fermi Degenerate Antineutrino Star Model of Dark Energy
Fermi Degenerate Antineutrino Star Model of Dark Energy
Neiser, Tom F.
2020-03-30 00:00:00
Hindawi Advances in Astronomy Volume 2020, Article ID 8654307, 11 pages https://doi.org/10.1155/2020/8654307 Research Article Tom F. Neiser Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA Correspondence should be addressed to Tom F. Neiser; tomneiser@physics.ucla.edu Received 1 December 2019; Accepted 26 February 2020; Published 30 March 2020 Academic Editor: Zdzislaw E. Musielak Copyright © 2020 Tom F. Neiser. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. When the Large Hadron Collider resumes operations in 2021, several experiments will directly measure the motion of anti- hydrogen in free fall for the first time. Our current understanding of the universe is not yet fully prepared for the possibility that antimatter has negative gravitational mass. +is paper proposes a model of cosmology, where the state of high energy density of the big bang is created by the collapse of an antineutrino star that has exceeded its Chandrasekhar limit. To allow the first neutrino stars and antineutrino stars to form naturally from an initial quantum vacuum state, it helps to assume that antimatter has negative gravitational mass. +is assumption may also be helpful to identify dark energy. +e degenerate remnant of an an- tineutrino star can today have an average mass density that is similar to the dark energy density of the ΛCDM model. When in hydrostatic equilibrium, this antineutrino star remnant can emit isothermal cosmic microwave background radiation and accelerate matter radially. +is model and the ΛCDM model are in similar quantitative agreement with supernova distance measurements. +erefore, this model is useful as a purely academic exercise and as preparation for possible future discoveries. major unsolved puzzle. Alternative models posit a negative 1. Introduction gravitational mass [7–10], negative inertial mass [11] or relic Type-Ia supernovae (SNe Ia) occur when an accreting white neutrino condensation [12, 13]. Despite these alternatives, dwarf star exceeds the Chandrasekhar limit and collapses the ΛCDM model is so far our most successful description of until the released potential energy detonates the star with the late universe. carbon fusion. Due to their similar initial conditions, SNe Ia While the ΛCDM model is the most widely accepted can serve as standard candles. Twenty years ago, this allowed model of the late universe, it does not address several key observations of SNe Ia to show an accelerating expansion of puzzles associated with the early universe. First, when we matter on cosmological scales [1–3]. +e big bang model trace the expansion of all observable matter (∼10 g) successfully describes this expansion with an expansion of backwards in time, we encounter a collective state of high the underlying metric. It assumes that the universe is ho- energy density that is associated with the big bang. Current mogeneous and isotropic on large scales, which is known as models treat the big bang either as the beginning of space the cosmological principle. Mathematically, the Fried- and time [14], or as part of a continuous bounce [15, 16]. man–Lemaˆıtre–Robertson–Walker (FLRW) metric upholds While it serves as initial condition for the ΛCDM model, the the cosmological principle by uniformly changing the metric origin of the big bang is a major unsolved puzzle. A second of space with a scale factor that varies in time. Today, a problem arises when we assume that this big bang state concordance from various observations defines the ΛCDM initially contained equal amounts of matter and antimatter, model, which divides the energy content of the universe into while unequal amounts are observed in the universe today. 31% matter and 69% dark energy, where dark energy is To account for this matter-antimatter asymmetry, it is commonly thought to be the constant energy density of the necessary to find a mechanism for baryogenesis [17, 18]. A quantum vacuum [4, 5]. +is translates to a dark energy third puzzle is associated with the cosmic microwave −30 3 120 density of ρ ≈6 ×10 g/cm , which is 10 times smaller background radiation (CMB). +e ΛCDM model assumes than expected [6]. +erefore the identity of dark energy is a that the CMB was emitted by initially hot and dense matter 2 Advances in Astronomy that has cooled sufficiently to become transparent to radiation occur, it becomes helpful to assume that antimatter has ∼10 years after the big bang [19]. However, the CMB is more negative gravitational mass. +is assumption will be tested at isotropic than expected, which is known as the horizon CERN and, if true, may also help identify dark energy. problem. +e theory of cosmological inflation addresses this by introducing a period of exponentially accelerating ex- −32 2.1. Chandrasekhar’s Equation of State. To characterize a pansion up to 10 s after the big bang [20–22]. +is could allow any two regions of the CMB in the early universe to have degenerate antineutrino gas in effective hydrostatic equilib- rium, we first make three assumptions. First, we ignore thermal been thermalized. +is also addresses the question of why our expanding metric appears to be spatially flat, known as the or radiation pressure of the antineutrino gas by assuming it is highly degenerate with temperature T/T ≪ 1, where T is the flatness problem. However, inflation suffers from problems F F such as the entropy problem or the multiverse problem Fermi temperature. Second, we assume that the neutrino is a Dirac fermion, which means that it is not its own antiparticle. [23, 24]. Moreover, the standard model interpretation of the CMB appears to be in tension with recent cosmology-inde- +is standard model assumption is being tested by the search for neutrinoless double-beta decay [34, 35]. +ird, we assume pendent measurements of the expansion rate at both low redshifts (z<0.15) and high redshifts (1.4< z<5.1) at the∼4σ that all neutrinos in the star are electron flavored neutrinos level [25–27]. Lastly, a dipolar anisotropy in the expansion with effective inertial mass m . Note that due to neutrino oscillations [36, 37], free electron neutrinos have an effective rate of nearby galaxies (z∼0.1) violates the assumptions of homogeneity and isotropy of the ΛCDM model at the 4σ level mass m � |U | m , where U are the Ponte- ] i ei i ei corvo–Maki–Nakagawa–Sakata leptonic mixing matrix ele- [28]. +ese puzzles motivate a search for new models, with the ΛCDM model as the benchmark. ments and m are eigenstates of definite mass (i=1, 2, 3, respectively). We will show below that this third assumption is In the present work, a model of cosmology is proposed that attempts to address the origin of the big bang and dark reasonable in the context of the Schwinger mechanism (see equation (16)) if the electron neutrino mass m is much less energy. A degenerate self-gravitating gas of antineutrinos, which we will call an antineutrino star, collapses when its than the muon (m ) or tau (m ) neutrino masses. ] ] μ τ With these assumptions we can use Chandrasekhar’s mass exceeds the Chandrasekhar limit, M ∝1/m [29, 30]. ] ] e e +e small neutrino mass (m ) motivates a model of cos- equation of state for degenerate matter, derived from hy- drostatic equilibrium of gravitational and degeneracy mology, where the collapse of an antineutrino star creates the state of high energy density of the big bang with a pressures [29, 30, 38]. We thus get the equations of density, minimum of new physics. As initial condition of the pro- √����� √����� 2 2 ρ � x 1 + x 1 + 2x − lnx + 1 + x , (1) posed model, we choose a quantum vacuum state due to its minimal entropy. +is state is gravitationally unstable and and pressure, organically forms spatially separated neutrino stars and √����� √����� antineutrino stars when we assume that antimatter has 2 2 (2) P � Kx 1 + x x − 1 + lnx + 1 + x , negative gravitational mass. After collapse in an energetic event as described below, a fraction of the antineutrino gas where x is dimensionless and proportional to the Fermi eventually returns to effective hydrostatic equilibrium. If momentum p , viewed from the core, a degenerate antineutrino star rem- nant could today emit isothermal cosmic microwave x � , background radiation and radially accelerate matter. (3) m c +e main finding is that both the new model and the ΛCDM model describe redshift-distance measurements and K has dimensions of energy density, with comparable quantitative accuracy. +e best-fit pa- 4 5 m c rameters of the new model give a density of the antineutrino e (4) K � , star that is similar to the dark energy density of the ΛCDM 8π Z model, and constrain the electron neutrino mass to high and the other constants take their usual meaning. We want statistical precision. +e new model is qualitatively consis- to solve the following equation for hydrostatic equilibrium: tent with CMB anisotropies [31] and large-scale structures dP Gm(r) [32, 33], which presently challenge the ΛCDM model’s (5) � −ρ , assumptions of homogeneity and isotropy. +ese results dr r encourage future work to further develop and test the where m(r) is the gravitational mass enclosed inside a radius presented model. r and is given by dm 2. Antineutrino Star Model of the Early Universe (6) � 4πr ρ(r). dr Similar to a white dwarf star, a degenerate gas of antineutrinos +e spherically symmetric gravitational potential φ(r) is collapses when its mass exceeds the Chandrasekhar limit. In given by this section we will show that this collapse can transform dφ Gm(r) energy at the scale of the mass-energy content of the known (7) � . dr r universe. In order to create the conditions for this event to Advances in Astronomy 3 For regions outside an antineutrino star (r> R) the 0.4 2 2 potential simplifies to a = 25.6 Gly (5 meV/m c ) 56 2 2 1 1 b = 3.25 × 10 g × (5 meV/m c ) (8) φ(r) � φ(R) − GM − , 0.3 r R where M<0 is the gravitational mass of the antineutrino star. +e radius R of the star can be found where P(R) �0, 0.2 while the total mass of the star is given by M � m(R), (9) M � 4π ρ(r)r dr. 0 0.1 Chandrasekhar’s equation of state above does not take into account general relativistic effects, which we can ignore 0.0 for simplicity if the radius of the star is much larger than its Schwarzschild radius, R /R≪1. Note that we assume that R/a inertial masses (m �|m|) of matter and antimatter are equal Figure 1: Mass-radius relationship for a neutrino star. +e peak and positive, and that the gravitational mass (m � m) of occurs at R/a �1.06 and M/b �0.356, which for neutrinos of mass matter is positive (m /m �1) and of antimatter is negative g i m � 5.0meV/c corresponds to R �27 billion lightyears (Gly) and (m /m � −1); for generality, we do not use overbar notation. 56 g i M �1.16 ×10 g. At smaller radii, the mass-radius curve corre- Note that the gauge is fixed to zero at the star’s center, φ(0) � sponds to unstable configurations. +erefore neutrino stars may be 0, so that φ(r)<0 everywhere else. relevant for cosmology. +e above equations can be solved numerically for a given central density ρ and effective neutrino mass m . To facilitate this, we recast equations (5)–(7) above using di- +ese large-scale factors suggest that neutrino stars and mensionless units for radial position η � r/a, mass μ � m/b, antineutrino stars may be relevant for cosmology [39]. and potential ϕ � φ/c , where 1/2 2 2πZ 1 2.2. Chandrasekhar Limit. +e estimated mass-energy a � , 55 content of the universe at the time of the big bang is∼10 g Gc m [5]. In the early universe, this mass-energy was concentrated (10) 3/2 in a state of high energy density. Possible origins of the big √�� � Zc 1 b � 2π . bang have been discussed by several authors [14–16]. One G m possibility, which seems to have been overlooked in earlier discussions, is that the big bang was created by the gravi- +is gives for hydrostatic equilibrium √����� tational collapse of an antineutrino star. When a white dwarf 3 5 dx 3 x + 3x + 2x − 1 + x arcsinhx μ star exceeds the Chandrasekhar limit, its own gravitational � − , (11) 4 2 dη 8x η pressure overwhelms the degeneracy pressure of electrons and it collapses in a supernova [29]. Similarly, the limiting and for the mass enclosed mass of a degenerate gas of electron antineutrinos occurs at √����� √����� dμ the dimensionless values of η �1.06 and μ �0.356, which 2 2 2 2 � η x 1 + x 1 + 2x − lnx + 1 + x , (12) corresponds respectively to dη 5meV while the potential becomes � 27.1Gly × , m c dϕ μ � . (13) (15) dη η 5meV M � 1.16 × 10 g × . e 2 Using the boundary conditions μ(0) �0, central density m c x(0) � x , and fixing the gauge ϕ(0) �0 allows us to solve the above three equations. For illustrative purposes, the mass- +e electron neutrino mass was recently constrained to radius relationship for an antineutrino star is plotted in m <1.1eV/c by the KATRIN experiment [40, 41]. +is Figure 1. +e constants a and b depend on the neutrino mass gives a lower limit of M > 2.39 × 10 g, which is only four as follows: orders of magnitude below the estimated mass-energy content of the universe at the time of the big bang (∼10 g). 5meV Moreover, an effective electron neutrino mass of a � 25.6billionlightyears(Gly) × , 2 m � 5meV/c would correspond to a Chandrasekhar limit m c of M � 1.16 × 10 g. Due to these cosmological length and (14) mass scales, the study of neutrino stars and antineutrino 5meV b � 3.25 × 10 g × . stars falls within the realm of cosmology. A small neutrino m c mass motivates the ansatz of this paper that the collapse of an M/b 4 Advances in Astronomy antineutrino star created the state of high energy density of During the subsequent expansion of material, baryonic the big bang with a minimum of new physics. matter and antimatter start to annihilate faster than they are created. We assume that a small remnant of baryonic matter survives due to baryogenesis [17, 18]. While the details of 2.3. Quantum Vacuum Instability. +e hypothesis that the baryogenesis remain an active area of research, negative collapse of an antineutrino star created the state of high gravitational mass has been proposed as a possible origin of energy density of the big bang immediately raises the CP violation [44], which is one of the requirements for question on the origin of the first neutrino stars and anti- baryogenesis. Similar to a supernova bounce, a fraction of neutrino stars. We attempt to address this qualitatively in the the original antineutrino star is accelerated to escape ve- following. Since the nature of the quantum vacuum is still locities from the shock wave associated with the bounce. +e poorly understood, we warn the reader that this subsection is remaining antineutrinos and matter expand as two adiabatic more speculative than the others. Before the first neutrino ideal gases initially in thermal equilibrium. During adiabatic and antineutrino stars are formed, it is reasonable to assume expansion, the temperature (T) decreases with an increase in −1/3 −2/3 that the initial condition of the universe was an infinite volume (V) as T∝ V for a relativistic gas and T∝ V volume of quantum vacuum due to its low number of de- for a nonrelativistic gas. +us, baryonic matter undergoes grees of freedom. +e quantum vacuum contains a sea of nucleosynthesis until density and temperature decrease virtual particle-antiparticle pairs going into and out of ex- sufficiently to “freeze out” certain reactions. +is process istence. +e universe today is no longer in this low-entropy could produce light elements comparable to big bang nu- quantum vacuum state. To explain this, we assume that cleosynthesis (BBN) [45]. Note that a neutrinonova is not matter and antimatter gravitationally repel, which causes the associated with an expansion of the underlying metric, but quantum vacuum to be gravitationally unstable by the an expansion of material (as in a supernova). Since baryonic following mechanism. Short-lived perturbations in the particles are much more massive than antineutrinos, they particle-antiparticle density create a weak and fluctuating become nonrelativistic at a much higher temperature than gravitational field on small scales (this field is established by antineutrinos, and thus begin structure formation in a much the particles themselves). By the Schwinger mechanism, this smaller volume. +e antineutrinos that have not been field has a nonzero probability of creating real particles by accelerated to escape velocities eventually form a degenerate separating virtual ones before they can annihilate [9, 42]. For self-gravitating gas with sub-Chandrasekhar mass example, the pair creation rate per unit volume and time in a (M<M ), and reestablish thermal equilibrium a sufficient constant local gravitational field gradient, g, is time afterwards (t≫ R/c). Newly formed galaxies are sub- 4 5 3 sequently radially accelerated from initial proximity to the dN m c Z|g| 1 πmc � exp −n . (16) center of the antineutrino star remnant. Observers in the rest 4 3 2 dtdV πmc n Z|g| 4πZ n�1 frame and inside of this antineutrino star in hydrostatic equilibrium would detect isotropic black body radiation, According to the above equation, even the weak and which we identify as the CMB. We call this the ATLAS fluctuating gravitational field of particles in the quantum (AnTineutrino Lepton gAS) model (in Greek mythology vacuum can create real particles with nonzero probability. Atlas is a titan who is able to lift up the celestial spheres), and +e exponential dependence on the effective mass (m) refer to the antineutrino star as ATLAS-1 (see Figure 2 for a strongly favors creation of neutrino-antineutrino pairs summary). compared to more massive particles of the Standard Model To summarize this section, the collapse of an antineu- [9, 43]. It also favors cold neutrinos over hot neutrinos, trino star can explain the energetic event commonly known which allows them to bind gravitationally. +us, the as- as the big bang. It also suggests that an antineutrino star sumption of negative gravitational mass of antimatter en- remnant in hydrostatic equilibrium may exist in the universe ables the gradual formation of mutually repulsive neutrino today. +is degenerate antineutrino star is detectable by stars and antineutrino stars. isothermal background radiation and the large-scale motion of galaxies. In the following section, we quantitatively show 2.4. Qualitative Description of the Collapse of an Antineutrino that this antineutrino star remnant could account for the Star. Whenever an accreting antineutrino star exceeds its large-scale motion of galaxies with similar precision as the mass limit, it collapses in a “neutrinonova.” At sufficiently ΛCDM model. high temperatures and densities, antineutrinos could transform most of their kinetic energy via high-energy 3. Antineutrino Star Model of the Late Universe collisions into equal quantities of baryonic matter and an- timatter. +is is qualitatively similar to the conversion of +e ΛCDM model can mathematically describe the large- large amounts of kinetic energy into equal quantities of scale motion of galaxies with only two parameters, the matter and antimatter in experiments such as the Large fractional matter density (Ω ) and dark energy density (Ω ). m Λ Hadron Collider at CERN. +e collapse could subsequently As we will show below, an antineutrino star model can be reversed in a big bounce. While the details of the bounce competitively describe the large-scale motion of galaxies, mechanism are unknown at this point, they would likely while also accounting for the physical origin of the big bang involve nuclear fusion and other physical processes similar and dark energy. +e equation of state of an antineutrino to the bounce event of supernovae. star is defined by two unknown parameters, namely, the Advances in Astronomy 5 M < M υ υ e e CMB Vacuum instability – – υ υ e e M ≥ M (a) (b) (c) Figure 2: Summary schematic of the ATLAS model. (a) Assuming that antimatter has negative gravitational mass, the quantum vacuum is unstable and forms spatially separated neutrino stars and antineutrino stars. An antineutrino star collapses in a neutrinonova when its mass exceeds the Chandrasekhar limit. A large fraction of the kinetic energy released in the collapse converts to baryons and antibaryons. After baryogenesis and nucleosynthesis, the surviving baryons and antineutrinos expand adiabatically. (b) Structure formation begins in a much smaller volume for baryons (inner sphere) than for antineutrinos (outer sphere). (c) +e antineutrino star remnant returns to hydrostatic equilibrium, emits isothermal cosmic microwave background radiation and radially accelerates matter. If we are close to the core, this model could explain the overall expansion of matter. central density (ρ ) and the effective electron neutrino mass center. When allowing for negative gravitational mass, we need to define two different metrics for matter and for (m ). An initial average expansion velocity of baryonic matter (v ) as a third parameter improves the physical basis photons. of our model at low redshifts (z<0.04) and could qualita- By symmetry, we assume that photons in a gravitational tively explain the low redshift contribution to the 4.4σ potential undergo blueshift or redshift independently of the Hubble tension [25, 26]. matter or antimatter nature of the gravitational source. We will thus use a Schwarzschild metric for photons that is agnostic to the type of matter, 3.1. Definition of Our Observational Frame. An antineutrino 2φ(r) M star establishes a background potential that is the dominant 2 2 dτ � 1 + dt c c contribution to the apparent velocities of galaxies. +erefore, c |M| we can infer our approximate position empirically from our (17) 2 2 2 velocities relative to other galaxies and relative to the rest dr r dΩ c c − − , frame of the antineutrino star, which is the rest frame of the 2 2 c 1 + 2φ(r)/c )(M/|M|) c CMB. +e velocities of galaxies relative to us scale approxi- mately isotropically with distance at a rate of where M/|M| is the sign of the gravitational mass of the −1 −1 H ≈70km·s Mpc [25, 26]. +e velocity of the Local Group 2 2 2 2 source, t is coordinate time, dΩ =sin θ dϕ +dθ is the −1 relative to the CMB is comparably small at v ≈ 627km·s CMB angular path element in spherical coordinates, and φ(r) is the [46]. +ese velocities suggest we are approximately at rest and gravitational potential of the antineutrino star given by near (but not at) the center of the star. +is can be physically equations (7) and (8). Note that the above equation is only explained by the small potential gradient near the center, an approximation of the metric inside the star (r< R), since which causes matter initially close to the center to remain we used Chandrasekhar’s Newtonian model for simplicity to effectively at rest, while matter far from the center accelerates find φ(r). It is worth repeating here that this approximation down the gravitational potential hill. +is empirical inference is accurate when changes in gravitational potential inside the of our observational frame guides our derivation of the star satisfy 2|φ(r)|/c ≪1. Outside of the star (r> R), the distance-redshift relationship below. above Schwarzschild metric accurately captures all further changes in gravitational potential relative to the central observer up to 2|φ(r)|/c ≲1. 3.2. Derivation of a Distance-Redshift Relationship. We will Since proper time is zero for photons, the velocity v � use the Schwarzschild metric to determine the redshift of (dr/dt) of distant photons moving radially (dΩ =0) is light emitted by a galaxy in free fall from initial proximity to the center of an antineutrino star. We may identify our- 2φ(r) M selves as Schwarzschild observers, who are by definition at � 1 + . (18) c c |M| rest where the gauge is fixed to zero, namely at the star’s 6 Advances in Astronomy To summarize the above result, photons are assumed to 1.0 behave in a way that is agnostic to the matter or antimatter nature of the gravitational source. We assume that matter experiences a Schwarzschild 0.8 metric that has been minimally extended to capture the interaction of negative and positive gravitational masses 0.6 2φ(r) m dr 2 2 dτ � 1 + dt − 2 2 c |m| c 0.4 (19) 1 r dΩ 0.2 · − , 2 2 1 + 2φ(r)/c (m/|m|) c where m/|m| is the sign of the gravitational mass of a test 0.01 0.1 1 10 particle. Here we have introduced a sign-dependence that transforms the metric’s dependence on gravitational po- ρ– /ρ v 0 tential (φ) into a dependence on potential energy, V(r) v /c = mφ(r). Note that the above expression allows matter and 2|φ(z)|/c antimatter to experience two different space-time metrics that are equal where the gauge is fixed to zero. Figure 3: Density-redshift profile, and relative contributions to the We can now use energy conservation [47] to find the redshift by gravitational potential and radial velocity. In the ATLAS model, redshifts at low-z are mainly due to radial velocity (dashed time dilation factor for galaxies with average initial velocity curve) and at high-z due to time dilation of free-falling sources in v undergoing free fall from the center of the star, ����� the gravitational potential φ (solid curve). +e density (dotted curve) acts as a form of dark energy density and vanishes at the dτ 2φ(r) m v (20) � 1 + 1 − . antineutrino star’s radius, R �21.1Gly (or z �2.49). +is plot is 2 2 dt c |m| c based on calculations using the best-fit parameters found in the following section. +e coordinate velocity v �dr/dt is found similarly, ������������������� � 2 2 v 2φ(r) m v v 2φ(r) m s 0 0 � 1 + − 1 − . (21) catalog (http://supernova.lbl.gov/union/; date accessed: 08/ 2 2 2 2 c c |m| c c c |m| 09/2017) of 580 SNe Ia is used [48]. +e best-fit parameter 2 −29 3 values are m � 6.70meV/c , ρ =1.60 ×10 g/cm , and ] 0 +e above equations allow us to calculate the redshift −3 v =6.18 ×10 c. +ese give χ /dof � 1.03, where dof stands 0 ] seen by a Schwarzschild observer [47], for a model’s degrees of freedom. For reference, the ΛCDM dt v model has a comparable fit of χ /dof � 1.09 [5]. In Figure 4 z � 1 + − 1. (22) dτ v the theoretical distance modulus of the ATLAS model with the above best-fit parameters is plotted together with the +erefore, redshift is caused by a combination of ap- distance-redshift data in a Hubble diagram. parent radial velocity v and time dilation of free-falling For comparison, the theoretical distance modulus of the sources in the gravitational potential φ. Specifically, con- concordance ΛCDM model [5] is shown with −1 −1 tributions of v to redshift dominate at low-z and contri- H =67.74km·s Mpc . Note that the central density of the butions from φ(r) via time dilation dominate at high-z (see antineutrino star ρ is comparable to the dark energy density Figure 3). +e distance modulus of distant SNe Ia in re- of the ΛCDM model. +erefore the ATLAS model can ceding galaxies is account for the overall expansion of matter, with the density profile of an antineutrino star in equilibrium effectively μ (z) � 25 + log [r(z)(1 + z)], (23) th 10 acting as a dark energy density (see Figure 4). where r(z) is the distance from the center of the antineutrino Following Riess et al. [1], the probability density function (PDF) for a given cosmological parameter is quantified with star in Megaparsec [1]. Bayes’ theorem, which gives a PDF for the electron neutrino mass of 3.3. Cosmological Parameters. We can compare the theo- c ∞ dv exp −χ /2 dρ 0 0 0 0 retical distance modulus to observed distance moduli for a pm μ � , (25) ] 0 c ∞ ∞ given set of cosmological parameters, θ � (m , ρ , v ). dv dρ exp −χ /2 dm ] 0 0 0 ] e 0 0 0 e Goodness of fit is determined with a χ statistic, where μ represents all measured distance moduli, which are 2 0 μ z ; θ − μ z th,i i 0,i i assumed to be independent and normally distributed. +is (24) χ (θ) � , gives a mass to one standard deviation of 0,i m � 6.70 ± 0.23meV/c (see Figure 5). where σ is the measurement uncertainty in the observed Note that this uncertainty is purely due to statistical error 0,i distance modulus μ . As observational data, the “Union2.1” and does not include possible systematic errors, which can 0,i Advances in Astronomy 7 2.0 1.0 0.0 –1.0 –2.0 0.1 1 0.1 1 Ω = 0, Ω = 0 m Λ SN Ia Ω = 0.31, Ω = 0.69 m Λ 2 –29 3 m = 6.70 meV/c , ρ = 1.60 × 10 g/cm , v = 0.00618c v 0 0 (a) (b) Figure 4: (a) Hubble diagram of 580 SNe Ia (circles) from the Union catalog [48] with comparable fits by the ΛCDM model (dashed line) and the ATLAS model (solid line). (b) Hubble diagram relative to the empty universe model for a better comparison. +e difference between the two models at z<0.04 is created by the assumption of an initial average expansion velocity of galaxies (v ), which could qualitatively explain the 4.4σ Hubble tension [25, 26]. Note that the central density ρ is quantitatively comparable to the dark energy density of the −30 3 ΛCDM model (ρ ≈6 ×10 g/cm [5]). be caused by model assumptions or instrument calibration Note that Chandrasekhar’s equation of state for the [48]. +is mass is consistent with the present experimental above best-fit parameters naturally gives rise to these cos- upper limit of m <1.1eV/c [41], and may be tested by mological scales. Due to its large size, a degenerate anti- future or ongoing experiments. Measurements of neutrino neutrino star can be detected by the large-scale motion of mixing angles can give the muon and tau neutrino masses, galaxies and an isothermal microwave background [19]. for example, the most recent (NuFIT 3.2 (2018), http://www. +e best-fit value for the initial velocity, −3 nu-fit.org/?q�node/166; date accessed: 06/07/2018) best-fit v �6.18±0.55 ×10 , can be identified as the root-mean- values give [m , m ] ≈ [30.1,26.4]meV/c [49]. square speed of thermalized protons, since protons are likely ] ] μ τ to be the most abundant matter particle by mass in the early universe. +is would correspond to a proton temperature of 4. Discussion T �11.9±2.1keV, which is consistent with temperatures at which nucleosynthesis ends due to a decrease in fusion +e antineutrino star parameters are approximately con- reaction rate with a decrease in density [50, 51]. Moreover, sistent with the assumptions of the ATLAS model. +e best- this initial average expansion velocity is responsible for the fit muon and tau neutrino masses are larger than the electron difference between the ΛCDM model and the ATLAS model neutrino mass by a factor of >3. +is is consistent with the at low redshifts (z<0.04). +is difference corresponds to a simplifying assumption that the antineutrino star consists faster average expansion rate at low redshifts than predicted only of electron neutrinos due to the Schwinger mechanism by the ΛCDM model and could qualitatively explain the 4.4σ (see section IIA). +e best-fit electron neutrino mass (m ) Hubble tension reported by Riess et al. [25, 26] (see gives a Chandrasekhar limit of Figure 4). +e best-fit parameters are also consistent with an an- M � 6.45 × 10 g, (26) tineutrino star that is degenerate, giving a central Fermi temperature T �34.1K> T �2.73K [52]. However, with a limiting radius of F,0 CMB,0 the above values for R /R and T /T suggest that the S CMB F systematic error due to model assumptions is ∼20% and R � 15.1Gly, (27) dominates over the statistical error. Future work could which is consistent with the model of the early universe (see account for general relativistic effects O[(R /R) ] and the Section 2.2). +e re-formed antineutrino star has mass temperature ratio T/T to improve the accuracy of the M/M � 0.855 and radius R /R �0.410 or S antineutrino star model. Since these effects are small and dominate at high-z, where supernova data is still sparse, they R � 21.1Gly. (28) are ignored in the present model for simplicity. μ (mag) μ – μ (mag) empty 1σ 3σ 3σ 2σ 2σ 1σ 8 Advances in Astronomy 0.02 0.02 0.01 0.01 0.00 0.00 1.55 1.60 1.65 0.008 0.02 0.02 0.006 0.01 0.01 0.004 0.00 0.00 1.55 1.60 1.65 0.004 0.006 0.008 7.5 7.5 0.02 0.02 7.0 7.0 6.5 6.5 0.01 0.01 6.0 6.0 0.00 0.00 1.55 1.60 1.65 0.004 0.006 0.008 6.0 6.5 7.0 7.5 –29 3 v /c m (meV/c ) ρ (10 g/cm ) 0 v Figure 5: Likelihood contours for the central density (ρ ) of the antineutrino star, the initial radial expansion velocity of matter (v ), and the effective electron neutrino mass (m ). +e SNe Ia data constrain the effective electron neutrino mass to high statistical precision, m � 6.70 ± 0.23meV/c . However, the systematic uncertainty associated with model assumptions likely dominates over the statistical uncertainty. +e ATLAS model relies on testable assumptions, in gravitational mass, the ATLAS model provides a helpful particular on the assumption of negative gravitational mass. alternative to the ΛCDM model, which is not optimized for While there are strong theoretical arguments against neg- this possibility. ative gravitational mass (for a review, see [53]), there have +e ATLAS model can be empirically distinguished from not yet been any conclusive direct experimental tests. +e the ΛCDM model with a variety of other empirical tests, assumption of negative gravitational mass is in principle such as observations that challenge the ΛCDM model’s assumption of isotropy and homogeneity on cosmological testable on cosmological scales, galactic scales, and labora- tory scales. On cosmological scales, this assumption is scales. For instance, observations of the largely isotropic consistent with observations as described above: (i) it allows Hubble expansion and small CMB dipole suggest that we are for the formation of the first neutrino and antineutrino stars relatively close to, but not at, the exact center of the star. +is from a quantum vacuum state, and (ii) it accounts for the off-center location could qualitatively explain anisotropies, overall expansion of matter in the late universe. On galactic such as hemispherical asymmetries in the power spectrum, scales, it predicts detectable antimatter emission from hot detected at the∼3σ level in the CMB [31]. +ese observations accreting matter close to the event horizon of black holes challenge the isotropy assumption of the ΛCDM model. In [54, 55]. For example, there exists a positron excess in the an effort to explain these observations, many CMB an- galactic center [56] and in cosmic rays [57–60], which could isotropies have been found to become less significant with be emitted by accreting compact objects. On laboratory the introduction of rotation (specifically, by using a Bianchi VII scales, several experiments are currently undergoing up- model, albeit one that is decoupled from the standard grades at the GBAR, ALPHA-g, and AEGIS laboratories at ΛCDM model) [31, 64]. Future work could investigate CERN [61–63]. Using antiprotons from Large Hadron whether this rotational axis in the CMB corresponds to the Collider (LHC) operations resuming in 2021, these exper- rotational axis of an antineutrino star. iments will produce antihydrogen and directly observe its +e ATLAS model predicts that any hemispherical motion in free fall. +erefore, laboratory experiments are asymmetries in the CMB [31] and any anomalies due to soon able to test a key assumption of the ATLAS model. In rotation of the antineutrino star disappear when viewed the event that experiments at CERN discover negative from rest at the exact center of the star. +erefore, it may be 2σ 3σ 1σ v /c m (meV/c ) e Advances in Astronomy 9 possible to locate this central position with the study of CMB anisotropies alone. For example, the intersection of the axis 0.0 of rotation found in the CMB with the plane on which hemispherical asymmetries are minimized could help locate –0.2 the central region of the antineutrino star. Independently, this central region could coincide with an underdensity of –0.4 matter in our cosmic neighborhood such as the so-called “dipole repeller” at z≈0.05 [65]. +e dipole repeller is a –0.6 void that appears to have contributed to roughly half of –0.8 our velocity (v ) relative to the CMB [65]. If the center CMB of the antineutrino star were indeed in our cosmic neighborhood (z ≲0.05) and contributing significantly to –20 v , then one would also expect to see an asymmetry in CMB –10 –10 the expansion rate of nearby galaxies. Specifically, one –20 x (Gly) would expect an asymmetry that is aligned with v . +is CMB is consistent with observations at the ∼2σ level of a dipole Figure 6: Surface plot of the gravitational potential in the equatorial anisotropy in the distribution of nearby (z ≲0.1) radio plane of the antineutrino star. Solving Chandrasekhar’s equation of galaxies that is aligned with v [66, 67]. +is is also state for the best-fit parameters gives a radius of cosmological scale, CMB R �21.1 Gly (shaded area). While baryonic test masses far from the consistent with recent observations of a similarly aligned core accelerate down the gravitational potential hill, observers ini- dipole anisotropy in the expansion rate of nearby tially at rest and close to the core will remain effectively at rest due to supernovae (z∼0.1) [28]. the relatively flat potential gradient here. +us, the effect of an Moreover, one would expect the gravitational interaction antineutrino star on galaxies viewed by central observers is com- between galaxy clusters and the antineutrino gas to generate parable to the effect of a uniformly expanding metric. correlations between the distribution of large-scale structures and temperature fluctuations in the CMB. +is could qual- attempt these fits yet; they are beyond the scope of the itatively explain the observed correlations between anisot- present work. +is work provides a viable cosmological ropies in the CMB and anisotropies in the matter distribution, model for the possibility that experiments at CERN discover known as baryon acoustic oscillations [4, 68]. negative gravitational mass. +e present work encourages Furthermore, recall that structure formation in the future work to further develop and test the ATLAS model presented model of the early universe begins in a smaller while relying on the ΛCDM model as a benchmark. volume for baryons than for antineutrinos due to their large mass difference (see Figure 2). +e initial baryonic structures will be relatively weakly bound and on a supercluster mass 5. Conclusions scale due to their initially high temperature. As they con- tinue their formation, these structures are dispersed by their +e entropy of the universe increases relative to an initial state acceleration in the gravitational potential of the antineutrino of low entropy. A natural candidate for this initial state is the star (see Figure 6). +is places evolved structures, whose old quantum vacuum. With the assumption that antimatter has age is difficult to reconcile with the ΛCDM model, at high negative gravitational mass, which will be tested at CERN, this redshifts. +is is also qualitatively consistent with the ex- vacuum gradually decays into neutrinos and antineutrinos. istence of large supercluster-size structures in the distant +e collapse of an antineutrino star is a possible explanation universe that are no longer gravitationally bound. For ex- for the energetic event commonly known as the big bang. ample, the Hercules–Corona Borealis Great Wall has been After collapse and subsequent net creation of matter via detected at the 1 −3σ level with a size of 7–10 Gly [32, 33]. baryogenesis, an antineutrino gas adiabatically expands and +is challenges the homogeneity assumption of the ΛCDM partially re-forms into an antineutrino star remnant in hy- model, which predicts structures not larger than ∼1.21Gly drostatic equilibrium. If viewed from its core, this star can [69, 70]. +erefore, the ATLAS model appears qualitatively today emit the isothermal CMB radiation and radially ac- consistent with observations that challenge the fundamental celerate matter. +is addresses the nature of dark energy, and assumption of homogeneity and isotropy of the ΛCDM removes the horizon and flatness problems without invoking model. cosmological inflation. +e above ATLAS model is in good Lastly, the ATLAS model proposes a possible origin of quantitative agreement (χ /dof � 1.03) with distance-red- the big bang and dark energy, which are two major puzzles of shift measurements. +e electron neutrino mass is con- the ΛCDM model. However, this work relies primarily on strained as a cosmological parameter to m � 6.70 SNe Ia data to constrain its respective cosmological pa- ± 0.23(stat.)meV/c . +e model is qualitatively consistent rameters (m , ρ , v ). +is work is still at an early stage, and with existing observations of large structures in the distant ] 0 0 comparable to early work on the ΛCDM model [1–3]. universe and anisotropies in the CMB. +e presented alter- During the twenty years since the discovery of dark energy, native cosmological model is helpful for the possibility that the ΛCDM model interpretation of SNe Ia data has been in experiments at CERN discover negative gravitational mass. general concordance with additional fits to observations Moreover, the apparent consistency of the presented model such as baryon acoustic oscillations [4, 68]. We do not with observational data motivates future work to further y (Gly) 2φ/c 10 Advances in Astronomy [14] J. B. Hartle and S. W. 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