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The of and Its Relation to the Number of according to the

Ioannis Haranas Wilfrid Laurier University, [email protected]

Ioannis Gkigkitzis East Carolina University, [email protected]

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Recommended Citation Ioannis Haranas and Ioannis Gkigkitzis, “The Mass of Graviton and Its Relation to the Number of Information according to the Holographic Principle,” International Scholarly Research Notices, vol. 2014, Article ID 718251, 8 pages, 2014. doi:10.1155/2014/718251

This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. Hindawi Publishing Corporation International Scholarly Research Notices Volume 2014, Article ID 718251, 8 pages http://dx.doi.org/10.1155/2014/718251

Research Article The Mass of Graviton and Its Relation to the Number of Information according to the Holographic Principle

Ioannis Haranas1 and Ioannis Gkigkitzis2

1 Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 2 Departments of Mathematics and Biomedical Physics, East Carolina University, 124 Austin Building, East Fifth Street, Greenville, NC 27858-4353, USA

Correspondence should be addressed to Ioannis Haranas; [email protected]

Received 2 June 2014; Accepted 27 July 2014; Published 29 October 2014

Academic Editor: Sergi Gallego

Copyright © 2014 I. Haranas and I. Gkigkitzis. This 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.

We investigate the relation of the mass of the graviton to the number of information 𝑁 in a flat universe. As a result we find that 𝑚 ∝1/√𝑁 the mass of the graviton scales as gr . Furthermore, we find that the number of contained inside the observable 𝑁 𝑁 ∝𝑁 horizon is directly proportional to the number of information ;thatis, gr . Similarly, the total mass of gravitons that exist 𝑁 𝑀 ∝ √𝑁 in the universe is proportional to the number of information ;thatis, gr .Inanefforttoestablisharelationbetween the graviton mass and the basic parameters of the universe, we find that the mass of the graviton is simply twice the Hubble mass 𝑚𝐻 as it is defined by Gerstein et al. (2003), times the square root of the quantity 𝑞−1/2,where𝑞 is the deceleration parameter of the universe. In relation to the geometry of the universe we find that the mass of the graviton varies according to the relation 𝑚 ∝ √𝑅 𝑚 gr sc, and therefore gr obviously controls the geometry of the time through a deviation of the geodesic spheres from the spheres of Euclidean metric.

1. Introduction and quantum field predict the existence of graviton . In graviton is the elementary In Einstein’s theory of , linearization of the that mediates the gravitational force and is expected field equations demonstrates that small perturbations of the to be massless, and that is because the gravitational force metric obey a wave equation [1]. These small disturbances, itself has an infinite range. Furthermore, graviton must bea referred to as gravitational waves, travel at the . spin-2 , which results from the fact that the source of However, some other predict a dispersive gravitation is the stress-energy tensor itself. Additionally, it propagation (see [2] for references). The most commonly can be shown that any massless spin-2 field could give rise considered form of dispersion supposes that the waves obey to a force that is indistinguishable from gravitation because a a Klein-Gordon-type equation that is given below: massless spin-2 field must couple to the stress-energy tensor 2 in the same way that the does3 [ ]. Therefore, 2 𝑚 𝑐2 1 𝜕 2 𝑔 ( −∇ −( ) )𝜙=0. (1) this result suggests that if a massless spin-2 particle is dis- 𝑐2 𝜕𝑡2 ℏ covered, it must be the graviton [1]. Thus graviton detection remains vital in the validation of the theories and also in Physically, the dispersive term is ascribed to the quantum of the research that strives to unify quantum with gravitation having a nonzero rest mass 𝑚𝑔,orequivalentlya general relativity. Although physicists normally speak as if noninfinite Compton 𝜆𝑔 =ℎ/𝑚𝑔𝑐,where𝜙 is the mediating the gravitational force exist, the extremely potential function of the gravitational field. weak character of the gravitational interaction makes the Moreover, several of today’s theories like theory, detection of graviton an extremely hard issue. Recently, superstringtheory,M-theoryandloopquantumgravity, Dyson has suggested that “the detection of a single graviton 2 International Scholarly Research Notices

mayinfactberuledoutintherealuniverse”[4]. If researchers 𝑚𝑔.Similarly,inBousso[10] the author argues that the answer Dyson’s question in an affirmative way, concluding total observable is bounded by the inverse of the that graviton detection is impossible, this will immediately . This holds for all space-times with a raise issues in relation to the necessity of gravity quanti- positive cosmological constant, including dom- zation. However, attempts to extend the standard model inated by ordinary matter and recollapsing universes [10]. with gravitons have run into serious theoretical difficulties Boussos’ conjecture is largely influenced by Banks’ idea of at high energies (processes with energies close to or above cosmological constant [11]. Moreover, in Mongan [12], the the Planck scale) because of infinities arising due to quantum author examines a vacuum-dominated Friedmann universe effects (in other words, gravitation is nonrenormalizable). asymptotic to a de Sitter space, with a cosmological event Since classical general relativity and are horizon that its area in Planck’s units determines the maxi- incompatible at such energies, from a theoretical point of mum amount of information that will ever be available to any view, the present situation is not tenable. Some proposed observer. Moreover, in a recent paper by Mureika and Mann models of [5] attempt to address these issues, [13], the authors examine the idea of information transfer but these are speculative theories. between a test particle and the holographic screen in entropic There are only a few conceivable sources of graviton gravity. The transfer respects both the production, like decay, andcausality,andalowerlimitonthenumberofinformation of gravitons from neutral hydrogen, bremsstrahlung from bits in the universe relative to the mass may be derived. The -electron collisions in stellar interiors, and conserva- corresponding limits indicate that particles travelling with tions of to gravitons by interstellar magnetic fields. the speed of light— and/or graviton—have a nonzero −68 Here we will only briefly touch upon the graviton production mass 𝑚≥10 kg. Their result is found to be in excellent by black hole decay, which appears to be the most promising agreement with the current experimental mass bound of mechanism. Black holes of mass 𝑀 possess a Hawking photon and graviton, suggesting that might temperature that is equal to betheresultofarecentlocalsymmetrythatissoftlybroken [13]. In particular cosmological holography postulates that ℏ𝑐3 𝑇 = , all the in our universe is encoded on its BH (2) 8𝜋𝑘𝐵𝐺𝑀 cosmological horizon. This proposal has been put forward by Smoot [14]andinshortstatesthatallpossiblepastandfuture where 𝑘𝐵 is the , 𝐺 is the gravitational histories of our universe are encoded on its constant, ℏ is Planck’s constant, and 𝑐 is speed of light. A 19 and by that making a connection. Moreover, the authors with mass 𝑀≤10 kg can in principle 𝐸≥10 proceed by asking how much the universe as a whole, mass, evaporate gravitons of energy eV or higher and, with and information included can tell us about its parts or the no observational constraints on the mass of the primordial lightest possible mass of the elementary particles. The authors black holes applied [6],canconstitutemostoftheuniverse’s further claim that in entropic gravity there is a lower limit to . On the other hand, primordial black holes with 𝑀≤1012 thenumberofbitsofholographicscreenwhichprovidesthe mass kg that emit particles with energy higher information transfer between the test particle and the screen, 𝐸≥108 eVwouldhavealreadylongbeenevaporated. which obeys causality as well as the uncertainty principle. Recently, Finn and Sutton [7] have examined the binary Similarly, in Haranas and Gkigkitzis [15], the authors pulsar, PSR B1913 + 16, of which the observed decay rate coin- examine the of information number 𝑁 and cides with that expected from relativity to approximately 0.3% its relation to cosmological parameters in a universe, where [7]. A nonzero graviton mass would upset this remarkable in Gkigkitzis et al. [16] the authors use a recent result for the agreement altering the predicted orbital decay, implying an number of information 𝑁 derived from Landauer’s principle; upper limit on the graviton mass. The authors have obtained they obtain an expression for the cosmological constant Λ. a crude estimate on this bound using dimensional analysis. Finally, in Haranas and Gkigkitzis [17], the authors investigate For a system with characteristic frequency 𝜔 one expects the thenumberofinformation𝑁 as related to the minimum effects of a graviton mass to appear at second order in 𝑚/𝜔 quantum and gravitational in a vacuum-dominated [7]. For gravitational waves at twice the orbital frequency of 2 universe.Inthiscontributionweseektoinvestigateand PSR B1913 + 16, requiring (𝑚/𝜔) < 0.003 implies an upper understand the physics behind any possible relations that −20 2 limitoforder10 eV/c . This is comparable to the best limit mightresultfromtherelationofthegravitonmasstothe −21 2 from solar system observations, 𝑚 < 0.44 × 10 eV/c . surface area of the universe expressed in accord- Finn and Sutton [7] have examined an extension of linearized ing to the holographic principle. As a result the number of general relativity which includes a mass term for the graviton. information 𝑁 on the horizon of the universe enters our They have chosen the unique mass term for which the wave calculation, and therefore the relation of the graviton mass to equationofthelinearizedtheorytakesthestandardformwith thenumberofinformation𝑁 will be established, via the cos- an ℎ-independent [7]sourceandforwhichtheprepredictions mological constant lambda Λ dependence on the number of of massless general relativity are recovered by setting 𝑚→0 information. Moreover, we use Landauer’s principle of infor- at the end of the calculations. mationinrelatingthegravitonmasstothemassoftheuni- In recent papers by Novello and Neves [8]aswell verse and the mass of an . Finally, we fur- as Neves (2004) and Liao [9]theauthorsexpressalink ther investigate the relation of the graviton mass to the Ricci between the cosmological constant and the graviton mass scalar and basic cosmological parameters of the universe. International Scholarly Research Notices 3

2. The Mass of the Graviton theories assume that the universe began by a quantum fluc- tuation from nothing, underwent , and became so In a recent paper by Das (2014) the author uses the quantum largethatitislocallyalmostflatandthatsincetheinflationary Raychaudhuri equation to obtain the quantum corrected era the density of the universe is constant. Friedmann equation for the range of graviton/photon. Simi- This is the case of the existence of a nonzero cosmological larly, with reference to Wesson [18], Novello [19], Tajmar [20], constant Λ. More information of such a universe arising and Liao [9], we write that the proposed relation between the in quantum cosmological way is presented in Mongan [21]. graviton mass and the cosmological constant is Systems that dynamically evolve in time not only transform 2 2 but also process information. Using the relation given in 1 2Λ 𝑚 𝑐 = = gr , Haranas and Gkigkitzis [15, 17] we can write the cosmological 𝜆2 3 ℏ2 (3) gr constant as a function of the number of information to be 3𝜋 3𝜋 ℏ 𝑐 Λ= = Λ . where is Planck’s constant, is the speed of light, and 2 max (8) 𝜆 𝑁ℓ𝑃 ln 2 𝑁 ln 2 gr isthegravitonwavelength.Thisequationalsofollows from Einstein’s linearized field equations which include the cosmological constant lambda Λ andalsofromtheequations Substituting (8)into(2)weobtainthat of for a massive spin-2 field that propagates in a de ℏ 2𝜋 ℏ 2𝜋 𝑚 Sitter background. Using (1)weobtainthatthemassofthe 𝑚 = √ = √ =𝜉 Pl , gr 2 0 (9) gravitoncanbewrittenas 𝑐 𝑁ℓ𝑃 ln 2 𝑐ℓ𝑃 𝑁 ln 2 √𝑁

ℏ 2Λ 𝜉 =(2𝜋/ 2)1/2 ≈ 3.010 𝑚 = (ℏ𝑐/𝐺)1/2 𝑚 = √ . (4) where 0 ln and Pl gr 𝑐 3 𝑚 ∝1/√𝑁 is the Planck mass; therefore we see that gr . Equation (4)isactuallythesameequationastheonegivenin Similarly, the total number of gravitons contained inside the observablehorizoncanbewrittenasafunctionofthenumber Wesson [18], in which 𝑚 =𝑚 /√3. The dependence of the gr wes of information 𝑁 in the following way: graviton mass on √Λ is a property that remains valid not only 2 for de Sitter but also for arbitrary background geometries as 𝑐3 𝑁ℓ ln 2 2 𝑁 = 𝑃 = ln 𝑁, (10) long as they stay in the setting of Einstein’s second description gr 𝐺ℏ 3𝜋 3𝜋 of the equations of motion [19]. We can also construct a mass ℏ 𝑁 ∝𝑁 associated with lambda by using Planck’s constant and also or in other words gr . Therefore, the total graviton mass the speed of light 𝑐 (ibid., [19]). Following Novello [19]we in the universe is equal to write that the total mass of gravitons that exist in the universe 𝑀 =𝑚 𝑁 =𝜂𝑚 √𝑁, is given by gr gr gr 0 Pl (11) 3 2 𝑐 ℏ 2Λ 𝑐 2 1/2 √ 𝑀 =𝑚 𝑁 = √ = √ , where 𝜂0 = (1/3)(2 ln 2/𝜋) and therefore 𝑀 ∝ 𝑁. gr gr gr (5) gr 𝐺ℏΛ 𝑐 3 𝐺 3Λ Similarly, using (9), we obtain that the number of information 𝑁 𝑁 is given by where gr is the total number of gravitons contained inside the observable horizon and it is equal [19]to 2 2𝜋 𝑚 𝑁= ( Pl ) . 𝑐3 1 Λ 2 𝑚 (12) 𝑁 = = = max , (6) ln gr gr 𝐺ℏΛ Λ Λ

3 Moreover, from (11), we obtain a second expression for the where ℓ𝑃 is the and Λ max =𝑐/𝐺ℏ is the number of information 𝑁 that reads maximum value of the cosmological constant as it is defined 2 in [15]. Therefore we write the total number of gravitons in 9𝜋 𝑀 𝑁= ( gr ) . (13) theuniverseinthefollowingway: 2 2 𝑚 ln Pl

Λ max 2 As a result we find that the number of information 𝑁 can 𝑁 = (√ )ℏ. (7) gr 𝑐 3Λ be expressed as the square of the ratio of two fundamental masses, namely, the Planck mass and the mass of the graviton 3. The Relation of the Graviton Mass to or the square of the total number of gravitons in the universe over the Planck mass. the Number of Information The holographic principle indicates a possible nonlocal- 4. Landauer’s Principle and itymechanisminanyvacuum-dominatedFriedmannuni- the Mass of the Graviton verse. To be more precise, a holographic nonlocal quantum mechanical description can be possible for a finite amount of In relation to the laws of physics we say that they determine information in a closed vacuum-dominated universe. Today’s the amount of information that a given system can register 4 International Scholarly Research Notices

(i.e., number of bits or nats) as well as the number of ele- In Alfonso-Faus and Fullana i Alfonso [31]theauthors mentary logic operations that the given system can perform claim that (3)isof“universalvalidity.”Theyfurthersaythat (i.e., number of operations). With reference to Landauer theunitofenergythatshouldbetakenastheminimum [16, 22, 23] we can say that information is physical, and quantum of energy is ℏ𝐻. This implies that the relativistic that also all information is registered and processed by energy of a mass 𝑀 has 𝑁 timesthisminimumquantumof physical systems. Physical systems can be described in terms energy 𝑁ℏ𝐻,where𝑁 is the number of information in nats. of information, and information processing is related to the In other words the product 𝑁ℏ𝐻 corresponds to the energy of system description by physical laws. Landauer’s principle is a all the information number 𝑁 carried by the system. Thus, the physical principle pertaining to the lower theoretical limit of expression for the cosmological constant lambda Λ obtained energy consumption of a computation. Landauer postulated in Gkigkitzis et al. [16]isusedbelow: that any “logically irreversible manipulation of information, 3𝜋 𝐻𝑐 such as the erasure of a bit or the merging of two computation Λ= ( ), 2 𝑀 𝐺 (17) paths, must be accompanied by a corresponding entropy ln 𝑢 increase in non-information-bearing degrees of freedom of where 𝑀𝑢 isthetotalmassoftheuniverseand𝐻 is the Hubble the information processing apparatus or its environment” parameter, and substituting (16)in(2) and simplifying we [24]. Landauer’s principle asserts that there is a minimum obtain that possible amount of energy required to change one bit of 1/2 information, known as the Landauer limit, and it is equal to ℏ2𝐻 𝑚3/2 𝑚3 𝑚 =𝜉√ =𝜉 ( 𝜋 )=𝜉( 𝜋 ) , gr 0 0 0 (18) [25] 𝐺𝑐𝑀𝑢 𝑀𝑢 𝑀𝑢 1/3 𝐸min =𝑘𝐵𝑇 ln 2, (14) 2 where 𝑚𝜋 =(ℏ𝐻/𝑐𝐺) is the mass of the pion as it is given by Weinberg [32], and therefore we can write the mass of the 𝑘 = 1.38 × 10−23 𝑇 where 𝐵 J/K is Boltzmann’s constant and universe 𝑀𝑢 in the following way: is the temperature of the circuit. Therefore Landauer’s energy 𝑚3 formula can be written in terms of the information number 𝑀 =𝜉2 ( 𝜋 ), 𝑁 in the following way: 𝑢 0 𝑚2 (19) gr

𝐸=𝑁𝑘𝐵𝑇 ln 2. (15) or in other words as the ratio of two fundamental particle masses in the universe. Recent experimental studies provide further evidence that Landauer’s prediction is true. For example, the equivalence 5. Time Dependence of the Graviton Mass, between information and energy can be interpreted using the results obtained in recent experiment by Funo et al. [26]. Ricci Scalar, and Information Number In their experiment the authors have shown that entan- In order to investigate a possible relation of the graviton mass glementcanproduceanincreaseorgainofthermodynamic to time 𝑡 let us now consider a flat universe, that is, 𝑘=0, work, where the gain is determined by the change of the where 𝑘 is the curvature constant and density parameter Ω= information content. Similarly, Berut´ et al. [27]haveshown 1. Following Gkigkitzis et al. [16]wecanwritethatinaflat that there is a link between and thermo- universe the number of information 𝑁 evolvesasafunction dynamics. Landauer’s principle is a simple consequence and of time in the following way: its logic emanates directly from the second law of thermo- dynamics. The law states that the entropy of a closed system 4𝜋 𝑐 2 4𝜋 𝑡 2 𝑁 (𝑡) = ( ) 𝑡2 =( )( ) . cannotdecreaseatthesametimewiththecorresponding (20) ln 2 ℓ𝑃 ln 2 𝑡𝑃 temperature. Therefore, if one nat of information is lost during a computation, the amount of entropy generated Theboundisnotfixedbutrathergrowsastimeprogresses is at least 𝑘𝐵 ln 2, and therefore the energy emitted in the and the horizon expands and encompasses more particles environment is 𝐸≥𝑘𝐵𝑇 ln 2. Landauer’s principle has been [33]. The expression above is in agreement with the work of accepted as a physical law, but it has also been challenged by Lloyd [23]andDavies[33] where the authors predict that in 2 Shenker [28]andNorton[29] and defended by Bennett [24] aflatuniverse𝑁∝𝑡.Substituting(20)in(9)weobtainthat and Ladyman et al. [30]. 1/2 ln 2 𝑡 Next, with reference to Alfonso-Faus and Fullana i 𝑚 =( ) 𝑚 ( Pl ), (21) Alfonso [31] and Gkigkitzis et al. [16], the authors say that “all gr 8𝜋 Pl 𝑡 𝑀 𝑀𝑐2 physical systems of mass and energy are equivalent to andthereforewefindthatthemassofthegravitoninaflat an amount of information in number of bits” of the order of 𝑚 ∝1/𝑡 universe evolves as gr .Using(21)wefindthatina 𝑡=𝑡 𝑡 = 4.346 × 𝑀𝑐2 flatuniverseatthemomentwhere 0 where 0 𝑁≅( ), 1017 ℏ𝐻 (16) s[24] is the age of the universe the mass of the graviton becomes 1/2 where 𝑀 is the mass of the system, 𝑐 is the speed of light, ℏ is ln 2 −62 𝑚 =( ) 𝑚 = 8.777 × 10 𝑚 [ ]. (22) Planck’s constant, and 𝐻 is the Hubble constant. gr 8𝜋 Pl Pl kg International Scholarly Research Notices 5

In order to investigate the relation of the graviton mass with scalar as a function of graviton range and the cosmological fundamental cosmological parameters let us look at Haranas parameter 𝑞 to be and Gkigkitzis [34]. In this paper the authors derive the Ricci 3 (𝑞 − Ω) scalar that is defined by [35, 36] 𝑅 = . sc 4𝜆2 (𝑞 − 1/2) (30) 6𝑅̈ 6𝑅̇2 6𝑘 gr 𝑅 =−( + + ) . (23) sc 𝑐2𝑅 𝑐2𝑅2 𝑅2 Next, following Gershtein et al. [37] we write the expression for the time that the universe expands from a maximal density 𝑅 is the scale factor of the universe. In a flat universe the to a minimal density dominance and is determined by the resulting Ricci scalar can be written as a function of the stage of the nonrelativistic matter dominance to be number of information number 𝑁 in the following way to be [34] 2 𝜋ℏ 𝑡 ≅ √ . max 3 𝑚 𝑐2 (31) 6𝜋 (𝑞 − Ω) gr 𝑅 = , sc 2 ℓ2 𝑁 (24) ln 𝑃 Therefore, (31)canalsoberelatedtothenumberofinforma- 𝑁 𝑞 Ω=1 tion via the relation of the graviton mass to the number of where and are the deceleration and density param- information, and therefore we find that eters of the flat universe. Solving (24)forthedeceleration parameter of the universe we obtain 𝜋 2 𝑡 ≅ √ ln 𝑡 √𝑁. (32) max Pl ln 2 2 3 𝑞=Ω+ ℓ 𝑅 𝑁. (25) 6𝜋 𝑃 sc Solving (32) for the number of information 𝑁 we obtain that Next, in a flat universe filled with nonrelativistic matter, that 2 is, 𝑝=0, corrected for the graviton contribution, we have that 3 𝑡 𝑁= ( ) . (33) the deceleration parameter given by [37] 𝜋 2 𝑡 ln Pl 2 𝑅̈ 1 1 𝑚 𝑐2 OurresultisinagreementwiththatofLloyd[23]and 𝑞=− = + ( gr ) . (26) 𝑅𝐻2 2 4𝐻2 ℏ Davies [33] where the author predicts that this is equal to the maximum number of bits registered by the universe using Therefore, equating (25)and(26), we obtain that the mass of matter, energy, and gravity, and it is found with the help of the graviton is given by the Bekenstein bound and the holographic principle to the universe as a whole. It is given by the square of the ratio of 2 2 2 4ℏ 𝐻 1 ln 2 2 theageoftheuniversetothatofPlancktime.Similarly,using 𝑚 = [Ω − + ℓ 𝑅 𝑁] , (27) gr 𝑐4 2 6𝜋 𝑃 sc (9) for the total mass of the gravitons, we find that the number of information 𝑁 canalsobewritteninthefollowingway: ℏ𝐻/𝑐2 =𝑚 where the quantity Hub is the Hubble mass defined 𝑀 2 𝑚0 =ℏ𝐻/𝑐2 = 9𝜋 gr in [38].Inthepresenterawehavethat Hub 0 𝑁= ( ) . (34) −66 −66 2 2 𝑚 ℎ(3.8 × 10 ) g ≅ 2.7 × 10 gwhenℎ = 0.71 [38]. Equation ln Pl (27) gives the mass of the graviton in terms of the Humble 2 parameters 𝐻, the density parameter Ω,thePlancklengthℓ𝑃, 6. Discussion and Numerical Results 𝑅 the Ricci scalar sc, and finally the number of information number 𝑁.Using(5) in Haranas and Gkigkitzis [34]and In this paper we have considered an expression for the eliminating the number of information 𝑁,thatis,𝑁 = 6𝜋(𝑞− massofthegravitonasitisgivenbyNovello[19]. Novello’s 2 Λ Ω)/ ln 2ℓ 𝑅 , we obtain the following expression for the mass expression depends on the cosmological constant lambda . 𝑃 sc Using the result in Haranas and Gkigkitzis [15, 17]wefinda of the graviton to be 𝑚 relation of graviton mass gr to the number of information 2ℏ𝐻 1 1/2 1 1/2 𝑁. As a result we find that the mass of the graviton is 𝑚 = (𝑞− ) =2𝑚 (𝑞− ) . (28) 𝑁 gr 𝑐2 2 Hub 2 inversely proportional to the number of information ;that 𝑚 ∝1/√𝑁 is, gr . Furthermore we find that the number of Similarly, we find that the mass of the graviton is related to gravitons contained inside the observable horizon is directly 𝑅 𝑁 𝑁 ∝ the Ricci scalar sc in the following way: proportional to the number of information ;thatis, gr 𝑁. Similarly, the total mass of gravitons that exist in the 1/2 2ℏ (𝑞 − 1/2) universe is proportional to the number of information 𝑁; 𝑚 = [ 𝑅 ] , gr sc (29) 𝑀 ∝ √𝑁 𝑐 3(𝑞−Ω) that is, gr . We find two different expressions for the number of information 𝑁,onethatisgivenasthesquareof where 𝑅𝐻 is the universe horizon, as well as to deceleration the ratio of the Planck mass to the mass of the graviton, that 𝑞 𝑁∝(𝑚/𝑚 )2 parameter . Using the fact that the graviton range is given is, 𝑃 gr , and another that is equal to the square 𝜆 =ℏ/𝑚 𝑐 by gr gr [38]weobtainanexpressionfortheRicci of the ratio of the total graviton mass in the universe over the 6 International Scholarly Research Notices

𝑁∝(𝑀 /𝑚 )2 Λ × Planck mass, that is, gr 𝑃 . Next, using a recent the cosmological constant lambda and is equal to 4.661 122 definition for the number of information 𝑁 resulting from 10 , a number in agreement with that given in Funkhouser Landauer’s principle, we find that the mass of the graviton [39]. This is still two orders of magnitude smaller than the can be expressed as the square root of third power of the pion 3 number of information that can fit in the universe to which a mass 𝑚𝜋 over the mass of the universe 𝑀𝑢, which implies that physical meaning can be attributed. This is a huge number but the mass of the universe is further equal to the third power still small to compare various numbers, one of which is called 𝑚3 𝑚2 ofthepionmass 𝜋 overthesquareofthePlanckmass Pl. Graham number [40].Finallytheentropyofthegraviton Moreoverwefindthatinaflatuniversethegravitonmass mass can be estimated using the formula 𝑚 ∝1/𝑡 varies according to gr , and furthermore we find that 𝑡=𝑡 𝑡 𝑆 =𝑘 (2𝑁) at time 0 (where 0 is age of the universe) the mass of gr 𝐵log2 𝑚 = 8.777 × 10−62𝑚 = 1.909 × 10−69 the graviton is gr Pl kg. 2𝜋 𝑚 2 This order of magnitude is in agreement with the result given Pl 𝑚 = =( )( ) 𝑘𝐵 in Gershtein et al. [38] where the authors predict that gr ln 2 𝑚 −66 gr (36) 3.2 × 10 g to 95% confidence level. Similarly DGP (Dvali- 122 Gabadadze-Porrati) constraints predict that the mass of the ≈ 4.191 × 10 𝑘𝐵 𝑚 ≅10−67 −10−69 graviton falls in the range gr kg [13]. 99 In an effort to establish a relation of the mass of graviton ≈ 6.0 × 10 J/K. to basic parameters of the universe, wefind that the mass of the graviton is simply twice the Hubble mass 𝑚𝐻 as it is The calculated entropy is of the same order of magnitude defined by Gershtein et al. [38], times the square root of the as the entropy within the cosmic horizon calculated when quantity 𝑞−1/2,where𝑞 is the deceleration parameter of matter within is taken into account and has been recently given in a paper by Egan and Lineweaver [41]tobeequalto theuniverse.Inreferencetothegeometryoftheuniverse 122 99 we find that the mass of the graviton varies according to (2.88 ± 0.16)×10 𝑘𝐵 ≅ 4.0×10 J/K. Following Novello [19] 2 3 2 𝑚 ∝ √𝑅 𝑚 ∝𝑅 𝑚 we have that 𝑁 =𝑐/𝐺ℏΛ = 1/ℓ Λ=Λmax/Λ,where𝑁 the relation gr sc or gr sc and therefore gr gr 𝑃 gr obviously controls the geometry of the space time through is the total number of gravitons in the universe, and therefore a deviation of the geodesic spheres from the spheres of the total entropy due to the total number of gravitons is given Euclidean metric. In general relativity, the scalar curvature by is the Lagrangian density for the Einstein-Hilbert action. 4 𝑚 2𝜋 𝑘𝐵𝑐 Therefore, the graviton mass gr determines the curvature 𝑆 (tot) = ( ) . (37) 2 gr 2 Λ𝐺2𝑚2 𝑅 𝑅 ∝𝑚 ln gr sc since sc gr and therefore the Einstein-Hilbert action itself depends on the graviton mass. Similarly, we obtain that 2 3 2 Using (6)and(7) along with the fact that ℓ =𝐺ℏ/𝑐 in (32) the Ricci scalar scales as 𝑅 ∝1/𝜆 , and in a similar way the 𝑃 sc gr we find that the entropy due to the total number of gravitons action itself can also depend upon the range of the graviton 𝑁 𝜆 in the universe as a function of the number of information gr. Furthermore, the time interval taken for the universe is given by the following simplified expression: expansion from a maximum to a minimum density varies as 𝑡 ∝ the square root of the number of information, that is, max ln 2 2 −24 2 √ 𝑆 ( ) =( )𝑘 𝑁 = 1.015 × 10 𝑁 . 𝑁, resulting in an expression for the number of information gr tot 3𝜋 𝐵 (38) as a function of time 𝑡 that agrees with that derived by Lloyd 𝑁=(3/𝜋 2)𝑁 [23]; namely, ln Lloyd.Using(2)and(6)wefind Moreover in a flat universe using18 ( )weobtainthatthe that number of information 𝑁 associated with the mass of the entropy due to the total number of gravitons in the universe graviton is given by evolves in time in the following way:

4 2 2 16𝜋 𝑡 3𝜋ℏ 2𝜋 𝑚 122 𝑁=( )= ( Pl ) ≈ 4.20 × 10 , 𝑆 (tot) =( )𝑘𝐵( ) . (39) 2 2 2 (35) gr 3 2 𝑡 2𝑐 ℓ 𝑚 2 𝑚 ln Pl ln Pl gr ln gr 𝑡=𝑡 𝑚 = 3.2 × During a very early era and in particular when Pl the total where the mass of the graviton is taken to be gr −66 entropy due to the total number of gravitons takes the values 10 g[38]. This is a rather curious result which suggests that the number of information bits 𝑁 associated with the 16𝜋 −22 𝑆 (tot) =( )𝑘𝐵 = 3.335 × 10 J/K, (40) graviton mass is of the same order of magnitude as the gr 3 ln 2 numbers appearing in . For example, Funkhouser 𝑘 = 1.38 × 10−23 𝑡 = √ℏ𝐺/𝑐5 = 5.391 × [39]findsanewlargenumberofcoincidencesaswellasa where 𝐵 J/K and 𝑝 −44 scaling law for the cosmological constant and other quantities 10 s. Similarly, in the present era when 𝑡 is the age of the 𝑡=𝑡 = 13.798 × which are also investigated. As a result the author claims universe, age by =4.387 1017 s [24], we obtain that the pure numbers originate naturally from basic ratios that the total entropy due to the total number of gravitons is of fundamental parameters and they do not require arbitrary given by powers of coefficients39 [ ]. In a similar paper [17]wehave 𝑁 𝑆 ( ) = 3.336 × 10222 / . shown that the number of information is associated with gr tot J K (41) International Scholarly Research Notices 7

Finally,usingtherelationgiveninMureikaandMann[13], we Conflict of Interests write that the graviton mass satisfies the following relation: The authors declare that there is no conflict of interests 2 𝑀𝑢 𝑚 ≥ 16𝜋 ( ), (42) regarding the publication of this paper. gr 𝑁 𝑀 𝑁=𝑁 where 𝑢 is the mass of the universe and 𝑢 is the References number of events or operations that could have occurred in 𝑡=𝑡 the age of the universe age.Usingthat,themaximum [1]C.W.Misner,K.S.Thorne,andJ.A.Wheeler,Gravitation,W. number of bits using gravitational degrees of freedom as H. Freeman, 1973. well as conventional matter and energy is equal to the [2]C.M.WillandN.Yunes,“Testingalternativetheoriesofgravity maximumnumberofelementaryoperations[23]andusing using LISA,” Classical and Quantum Gravity,vol.21,no.18,pp. an expression given in Gkigkitzis et al. [16]wewritethat 4367–4381, 2004. 4𝜋 𝑡 2 [3] A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky, 𝑁 (𝑡) ≅𝑁 = ( ) , (43) Problem12.16.ProblemBookinRelativityandGravitation, ops 2 𝑡 ln Pl Press, 1975. and after substituting in (42)weobtainthat [4]F.J.Dyson,“‘Theworldonastring’,reviewofTheFabricof the Cosmos: Space, Time, and the Texture of Reality by Brian 𝑚 ≥4𝜋 2(𝑡2 𝐻2)𝑀 . gr ln Pl 𝑢 (44) Greene,” New York Review of Books,vol.51,no.8,2004. [5] D. Colosi, L. Doplicher, W. Fairbairn, L. Modesto, K. Noui, 𝑡 = (𝐺ℏ/𝑐5)1/2 Next substituting for the Planck time Pl and and C. Rovelli, “Background independence in a nutshell: the forthemassoftheuniverseasitisgiveninHaranasand dynamics of a tetrahedron,” Classical and Quantum Gravity,vol. 3 Gkigkitzis [15]andalsoValev[42], namely, 𝑀𝑢 =𝑐/𝐺𝐻, 22, no. 14, pp. 2971–2989, 2005. and simplifying we obtain that [6] B. J. Carr, “Primordial black holes—recent developments,” in ℏ𝐻 Proceedings of the 22nd Texas Symposium,Stanford,Calif,USA, 𝑚 ≥4𝜋 2( )≥4𝜋 2𝑚 , 2004, http://arxiv.org/abs/astro-ph/0504034. gr ln 𝑐2 ln Hub (45) [7] L. S. Finn and P. J. Sutton, “Bounding the mass of the graviton 𝑚 =ℏ𝐻/𝑐2 using binary pulsar observations,” D,vol.65,no. where Hub is defined in Gershtein et al. [38], to be the Hubble mass. Following Sivaram [43] we define the 4, Article ID 044022, 2002. 𝐸 =ℏ𝐻 [8]M.NovelloandR.P.Neves,“Themassofthegravitonandthe gravitational self-energy of the graviton to be GSE (45) which can be further written as cosmological constant,” Classical and Quantum Gravity,vol.20, 𝐸 no. 6, pp. L67–L73, 2003. 𝑚 ≥4𝜋 2 ( GSE ) ≤4𝜋 2𝑚 , (46) [9] L. Liao, “On the in de Sitter ,” gr ln 𝑐2 ln GSE Submitted, http://arxiv.org/abs/gr-qc/0411122. 𝑚 where GSE is the corresponding gravitational self-mass of [10] R. Bousso, “Positive vacuum energy and the 𝑁-bound,” The the graviton. In Valev [44] the author gives a theoretical JournalofHighEnergyPhysics, no. 11, 2000. estimation of the graviton mass based on the assumption [11] T. Banks, “Cosmological breaking of or little that the Compton wavelength of the graviton is close to the lambda goes back to the future II,” submitted, http://arxiv.org/ Hubble distance 𝑐/𝐻, which produces a value of the graviton abs/hep-th/0007146. 𝑚 ≈ℏ𝐻/𝑐2 mass that is given by the relation gr . [12] T. R. Mongan, “Holography and non-locality in a closed vacuum-dominated universe,” International Journal of Theoret- 7. Conclusions ical Physics, vol. 46, no. 2, pp. 399–404, 2007. [13] J. R. Mureika and R. B. Mann, “Does entropic gravity bound the In this paper we investigate the relation of the graviton mass masses of the photon and graviton?” Modern Physics Letters A, −1/2 to the number of information 𝑁 in the universe, and an 𝑁 vol. 26, no. 3, pp. 171–181, 2011. dependence has been found. Similarly, we find that the total [14] G. F. Smoot, “Go with the flow, average holographic universe,” number of gravitons inside the horizon of the universe is International Journal of Modern Physics D,vol.19,no.14,pp. proportional to the number of information 𝑁.Furthermore, 2247–2258, 2010. [15] I. Haranas and I. Gkigkitzis, “Bekenstein bound of information using Landauer’s principle, we obtain that the mass of the 𝑁 graviton can also be expressed as the one-third power of the number and its relation to cosmological parameters in a universe with and without cosmological constant,” Modern ratio of the mass of the pion over the mass of the universe, Physics Letters A: Particles and Fields, Gravitation, Cosmology, fromwhichweobtainthatthemassoftheuniverseisrelated ,vol.28,no.19,ArticleID1350077,8pages,2013. to the cube of the mass of the pion over the square of the [16] I. Gkigkitzis, I. Haranas, and S. Kirk, “Number of information graviton mass. Moreover, we find that the evolution of the 𝑚 ∝1/𝑡 and its relation to the cosmological constant resulting from graviton has an inverse time dependence; that is, gr . Landauer’s principle,” Astrophysics and Space Science,vol.348, Finally, in relation to the geometry of the universe, we find pp. 553–557, 2013. that the mass of the graviton is related to the Ricci scalar in the 𝑚 ∝ √𝑅 [17] I. Haranas and I. Gkigkitzis, “The number of information bits following way: gr sc,whereatthesametimetheRicci related to the minimum quantum and gravitational masses in scalar depends on the inverse of the graviton range according a vacuum dominated universe,” Astrophysics and Space Science, 𝑅 ∝1/𝜆2 to the relation sc gr. vol. 346, no. 1, pp. 213–218, 2013. 8 International Scholarly Research Notices

[18] P.S. Wesson, “Is mass quantized,” Modern Physics Letters A,vol. [40] D. Wells, The Penguin Dictionary of Curious and Interesting 19,no.26,p.1995,2004. Numbers, Penguin Adult, 1997. [19] M. Novello, “The mass of the graviton and the cosmologi- [41] C. A. Egan and C. H. Lineweaver, “A larger estimate of the cal constant puzzle,” submitted, http://arxiv.org/abs/astro-ph/ entropy of the universe,” The Astrophysical Journal,vol.710,no. 0504505. 2, p. 1825, 2010. [20] M. Tajmar, “A note on the local cosmological constant and [42] D. Valev, “Estimations of total mass and energy of the universe,” the coincidence problem,” Classical and Quantum Physics International,vol.5,no.1,pp.15–20,2014. Gravity,vol.23,no.15,pp.5079–5083,2006. [43] C. Sivaram, The Early Universe, Instituto per la Ricerca di Base, [21] T. Mongan, General Relativity and Gravitation,vol.33,pp.1415– Hardonic Press, Molise, Italy, 1996. 1424, 2001, 33,. [44] D. Valev, “Neutrino and graviton rest mass estimations by a [22] R. Landauer, “Dissipation and noise immunity in computation phenomenological approach,” arXiv:hep-ph/0507255v6. and communication,” ,vol.335,no.6193,pp.779–784, 1988. [23] S. Lloyd, “Computational capacity of the universe,” Physical Review Letters, vol. 88, Article ID 237901, 2002. [24] C. H. Bennett, “Notes on Landauer’s principle, reversible computation, and Maxwell’s demon,” Studies in History and Philosophy of Modern Physics,vol.34,no.3,pp.501–510,2003. [25] P.Faist,F.Dupuis,J.Oppenheim,andR.Renner,“Aquantitative landauer’s principle,” http://arxiv.org/abs/1211.1037. [26] K. Funo, Y. Watanabe, and M. Ueda, “Thermodynamic work gain from entanglement,” Physical Review A, vol. 88, Article ID 052319, 2013. [27] A. Berut,A.Arakelyan,A.Petrosyan,S.Ciliberto,R.Dillen-´ schneider, and E. Lutz, “Experimental verification of Landauer’s principle linking information and ,” Nature, vol.483,no.7388,pp.187–189,2012. [28] R. Shenker, Logic and Entropy, 2000, http://philsci-archive.pitt .edu/115/. [29] J. D. Norton, “Eaters of the lotus: Landauer’s principle and the return of Maxwell’s demon,” http://philsci-archive.pitt .edu/1729/. [30] J. Ladyman, S. Presnell, and A. J. Short, “The use of the information-theoretic entropy in thermodynamics,” Studies in History and Philosophy of Science. Part B: Studies in History and Philosophy of Modern Physics,vol.39,no.2,pp.315–324,2008. [31] A. Alfonso-Faus and M. J. Fullana i Alfonso, “Cosmic back- ground bose condensation (CBBC),” Astrophysics and Space Science,vol.347,no.1,pp.193–196,2013. [32] S. Weinberg, Gravitation and Cosmology,NewYork,NY,USA, John Wiley and Sons, 1972. [33] P. C. W. Davies, “The implications of a cosmological informa- tion bound for complexity, and the nature of physical law,” http://arxiv.org/abs/quantph/0703041. [34] I. Haranas and I. Gkigkitzis, “Geometry of the universe and its relation to entropy and information,” Advances in Astronomy, vol. 2013, Article ID 809695, 3 pages, 2013. [35] J. N. Islam, An Introduction to Mathematical Cosmology,Cam- bridge University Press, 1992. [36] D. F. Lawden, Introduction to Tensor Calculus, Relativity and Cosmology, Dover Publications, 2002. [37] S. S. Gershtein, A. A. Logunov, and M. A. Mestvirishvili, “The upper limit on the graviton mass,” Rossiyskaya Akademiya Nauk. Doklady Akademii Nauk,vol.360,no.3,pp.332–334, 1998. [38] S. S. Gershtein, A. A. A. Logunov, and M. A. Mestvirishvili, “Graviton mass and the total relative mass density Ω𝑡𝑜𝑡 in the universe,” Doklady Physics,vol.48,no.6,pp.282–284,2003. [39] S. Funkhouser, “The large number coincidence, the cosmic coincidence and the critical acceleration,” Proceedings of the Royal Society A,vol.462,no.2076,pp.3657–3661,2006. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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