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A New Scale of Particle Mass in a Fractal Universe with A A new scale of particle mass in a fractal universe with a cosmological constant Scott Funkhouser(1) and Nicola Pugno(2) (1)Dept of Physics, the Citadel, 171 Moultrie St. Charleston, SC, 29409 (2)Department of Structural Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy ABSTRACT Considerations of the total action of a fractal universe with a cosmological constant lead to a new microscopic mass scale that is a function of the fractal dimension and fundamental parameters. For a universe with a fractal dimension D=2 the model predicts a mass that is of order near the nucleon mass. A scaling law between the cosmological constant and the mass of the nucleon follows that is identical to a known scaling law originating with Zel’dovich. This analysis also provides new limitations on the possible nature of dark matter in a fractal universe with a cosmological constant. 1. A new scale of mass Consider a system that is well represented by a fractal structure with dimension D. Let the total mass M of the system be dominated by some number N of a certain species of particle whose mass is m, so that N≈M/m. The total action A of the fractal system must be related to the Planck quantum of action h by [1,2] M ( D +1)/ D A ≈ hN (D +1)/ D ≈ h . (1) m Certain prominent features of the universe exhibit strong signatures of a fractal structure. Insomuch as the universe may be represented by a fractal, (1) should relate the total cosmic action to the parameters of the dominant species of massive particles in the universe [2]. Furthermore,€ if there exists a cosmological constant then there would exist also an associated fundamental scale of cosmic action and, from (1), a new, fundamental, microscopic scale of mass. In this cosmic epoch the average density energy due to the matter in the universe is nearly equal to the apparent vacuum density that may be the result of a cosmological constant. That alignment is known as the “cosmic coincidence”. The occurrence of the cosmic coincidence is significant per se, but it is also important in that it implies the existence of a cosmological constant, and associated with a cosmological constant are certain fundamental scales of energy, time, mass and distance. According to detailed observations the value of the cosmological constant, Λ, is of the order 10-35s-2 [3]. The cosmological scale of distance associated with Λ is approximately −1/ 2 RΛ ~ cΛ , (2) where c is the vacuum-speed of light and Λ~10-35s-2. The term on the right side of (2) is of the order 10-26m and represents essentially the De Sitter horizon c(3/Λ)1/2, which is the largest possible event horizon in a universe with a positive cosmological constant. As a result of the cosmic coincidence€ and the standard cosmological model both the cosmic particle horizon and event horizon are currently scaled to RΛ [3]. The fundamental cosmological mass MΛ that follows from Λ is c 3 M ~ , (3) Λ G Λ 53 where G is the Newtonian gravitational coupling and MΛ~10 kg [4]. In this current epoch the mass contained within a sphere whose radius is the particle horizon and the mass contained in the sphere whose radius is the event horizon are both of order near, and € scaled physically to, MΛ [3]. The mass associated with the particle horizon is asymptotically approaching a maximum value near MΛ. The mass associated with the event horizon, however, was of the order MΛ in early and recent times but will decrease dramatically in future ages since vacuum-dominance has begun [5]. The fundamental, cosmological scale of energy EΛ that follows from the existence of a cosmological constant is obtained by multiplying the mass in (3) by c2, c 5 E ~ . (4) Λ G Λ The energy in (4) is also roughly the characteristic gravitational energy of the 2 fundamental cosmic mass since GMΛ/RΛ~c . Due to the cosmic coincidence the mass- energy of the observable universe and the gravitational potential energy of the observable € mass are both currently of order near EΛ since the current mass and particle horizon of the observable universe are scaled to MΛ and RΛ, respectively. The characteristic time scale tΛ associated with the cosmological constant is roughly −1/ 2 tΛ ~ Λ . (5) It is essentially the time required for light to traverse the De Sitter horizon, and thus it is that largest physically significant characteristic time. Due to the cosmic coincidence the current age of the universe is roughly equal to tΛ [3]. € The characteristic action AΛ of a universe with a cosmological constant may be stated approximately by c 5 A ~ E t ~ . (6) Λ Λ Λ GΛ The fundamental action in (6) is also the maximum characteristic action of the cosmos because the largest possible mass-energy of the cosmos is roughly EΛ and the largest possible characteristic time that could be associated with our universe is roughly t . As a € Λ result of the cosmic coincidence the current cosmic action A0~E0t0, where E0 is the current characteristic cosmic energy and t0 is the age of the universe, is near the fundamental, maximum value in (6). Associated with the fundamental, characteristic action AΛ of the universe is, according to (1), a characteristic scale of particle mass for a fractal universe. Equating the action A in (1) to the characteristic action AΛ of the universe and letting MΛ represent the total mass M, the particle mass m in (1) would represent a new, fundamental mass scale mΛ(D), where D /(D +1) D /(D 1) h c 3 GhΛ + mΛ (D) ~ MΛ ~ 1/ 2 5 . (7) AΛ GΛ c Note that there is some uncertainty in the specification of the mass in (7). The ±1 fundamental action AΛ defined in (6) may lack a geometrical coefficient of order 10 , ±D/(D+1) thus generating a uncertainty of order 10 . The cosmic mass MΛ as approximated in (3) may also€ lack a geometrical coefficient, thus introducing an additional factor of order 10±1 in (7). Such ambiguities may be resolved in a complete theory of the putative fractal nature of the universe, and the associated uncertainties are small in comparison to the orders of magnitude that separate the cosmic and particle scales of mass. In connection with the mass in (7) there exists a fundamental pure number BΛ(D) given by 5 D /(D +1) MΛ c BΛ (D) ≡ ~ , (8) mΛ (D) GhΛ that represents the quantity of the fundamental particle species. Eqs. (7) and (8) are particularly significant since the term c5/(GhΛ)~10122 is roughly the maximum number of degrees of freedom allowed to our universe following from the Bekenstein-Hawking bound applied to €the De Sitter horizon [3]. 2. Applications of the new mass scale Certain observations suggest that the universe exhibits structural signatures consistent with a fractal whose dimension D is near 2 [1]. There is also substantial theoretical motivation for expecting a fractal universe with D=2 [6]. The associated particle mass mΛ(2) would be 1/ 6 h 4 Λ mΛ (2) ~ . (9) G2c 2 The mass in (9) is of the order 10-28kg, which is roughly the nucleon mass scale. Accordingly, the associated quantity BΛ(2) of dominant particles in the cosmos would be of order near (10122)2/3~1081, which is of order near the baryon number of the observable universe. € Moreover, if the mass in (9) were indeed scaled to the nucleon mass u then a known scaling law would follow that has been proposed independently for a variety of physical reasons. Inverting (9) to produce a relation for the cosmological constant with mΛ(2)~u leads to G2c 2u6 Λ ~ . (10) h 4 Zel’dovich first proposed that Λ should be proportional to u6 based on considerations of quantum field theory [7]. Carneiro and Mena Marugan have proposed that (10) follows from the holographic conjecture [8]. The relationship in (10) also would resolve a number of problematic large-number€ coincidences among the parameters of nature [3]. Furthermore, (10) follows from applying the Bekenstein-Hawking bound to a quantum- cosmological model in which three dimensions inflated from the collapse of seven extra dimensions [9]. The derivation of (10) presented here is based on independent physics and could be mutually consistent with all of the other arguments in favor of a scaling between the cosmological constant and the sixth-power of the nucleon. If the dominant mass component of the universe is baryonic then the relationship in (7) could be used to place limits on the putative fractal dimension D. If D were different than 2 then the associated particle parameters would be substantially different from the nucleon’s parameters. However, if some yet-unidentified particle dominates the cosmic mass then (7) could be used to place limits on the mass of the putative new particle as a function of the fractal dimension. If D were nearly equal to 2 then the mass of the unidentified, dominant particle must be of order near the nucleon mass. Alternatively, some observations suggest that D could be as small as 1.6 on the largest scales [1]. In that case the mass of the dominant particle would be larger than the nucleon mass by a factor 0.051 mΛ (1.6) BΛ (2) = ~ NΛ , (11) mΛ (2) BΛ (1.6) 6 since is mΛ(2) of order near the nucleon mass. The term in (11) is of the order 10 , which would require a new particle whose mass-energy is of order near 1015eV.
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