A Model for Massless Gravitons in Radiation and Matter-Dominated Universes

J. Astrophys. Astr. (2020) 41:30 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12036-020-09646-7Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

Letter

A model for massless gravitons in radiation and matter-dominated universes

ELI CAVAN1,* , IOANNIS HARANAS1 and IOANNIS GKIGKITZIS2

1Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada. 2Department of Mathematics, Northern Virginia Community College, 8333 Little River Turnpike, Annandale, VA 22003, USA. *Corresponding Author. E-mail: [email protected]

MS received 20 April 2020; accepted 25 August 2020

Abstract. A toy model of a graviton is explored by considering the minimum amount of information gravitons could carry. The total entropy of the universe is calculated and compared to estimates from super massive black holes and massive models of the graviton. The running cosmological constant is calculated using the entropy relation previously computed and compared to its experimentally accepted value. Both results are quantiﬁed considering radiation and matter-dominated universes.

Keywords. Graviton—entropy—cosmology—cosmological constant.

1. Introduction authors developed equations relating the number of gravitons in the universe to the cosmological constant, Many of the fundamental physical forces are mediated and furthermore they related this to the total amount by a massless propagator; for example, the photon for of information in the universe. Mureika and Mann the electromagnetic force or the gluon for the strong (2010) bounded the properties of any massless prop- nuclear force. For gravity, the graviton is the predicted agator using Heisenberg’s Uncertainty Relation propagator. In the electromagnetic case the propagator (HUR) and holography arguments. Using these argu- is a spin 0 particle, whereas the gluon and graviton ments, they calculated that the mass of any quanta carry spin 1 and spin 2 labels respectively. The nature satisfies of the graviton is not completely known because a 16p2M suitable renormalizable theory quantum gravity has m u ; ð2Þ not yet been discovered. Some authors such as Mur- Nu eika and Mann (2010) and Novello (2005) predicted where Mu is the mass of the universe and Nu is the that these propagators will ultimately have experi- total amount of information contained within the mentally determinable mass, while still a few other universe. Similarly, the speed of information transfer considered effective field theory (EFT) approaches is constrained by that lend these propagators massless (Balseanu 2019). For example, Haranas and Gkigkitzis (2014) supposed 2kDE v ; ð3Þ that gravitational waves follow a Klein–Gordon like I h equation: ! where k is the Compton wavelength of the particle and 1 o2 m c2 2 DE in the allowed quantum energy fluctuation. The Àr2 À g w ¼ 0; ð1Þ c2 ot2 h authors came to the conclusion that these arguments failed for the graviton (depending on the holographic where mg is the predicted mass of the propagator, the source). Theories have constrained the mass of the graviton. It is also well-known that photons follow a graviton to be as large as 10À49 kg (de Rham et al. -55 -69 massless version of the Klein–Gordon equation. The 2017) and as small as 10 –10 kg (Mureika & 30 Page 2 of 5 J. Astrophys. Astr. (2020) 41:30

Mann 2010). Gravitons are expected to be seen 2. Formulation of the model propagating in gravitational waves; and as a mediator between massive particles at the quantum gravity All massless bosons move at the speed of light; level. Recent experiments (LIGO and VIRGO) have therefore, all massless bosons obey the same equation already confirmed the existence of gravitational relating wavelength and frequency as the photon, waves. namely c ¼ kf , where f is the frequency, k is the Throughout this contribution, we compare results wavelength of the boson and c ¼ 3 Â 108 m/s is the from Egan and Lineweaver (2010), who measured the speed of light. Now we introduce a massless equation entropy of the Universe using measurements from that relates these fundamental constants to the amount binary pulsars (estimated to be the largest source of of information. In Alfonso Faus and Fullana i Alfonso entropy in the universe). The authors estimated the (2013), they argued a similar formula for massive entropy budget for weakly interacting dark matter. In entities; the equation seen in that contribution is this contribution, we have suggested a correction to Mc2 the equation for the minimum amount of information N ¼ ; ð4Þ carried by any massive entity which was theorized in Hh Alfonso Faus and Fullana i Alfonso (2013). The cor- h where H is Hubble’s constant and h ¼ 2p is the rection allows the equation to be valid for massless reduced Planck’s constant. Equation (4) is the ratio of particles. The authors used this equation to consider a the energy due the total mass energy of the universe 2 theory of Cosmic Background Bose Condensation EM = Mc over the gravitational energy of the uni- (CBBC) which may be a candidate to explain the verse Egr ¼ Hh. They associated N as the minimum current expansion of the universe. Their equation for amount of information (1 Nat) that a massive entity the minimum amount of information carried by any must carry, i.e., it is the amount of information asso- massive particle is used in the calculations by ciated with any system of mass, M. This follows from Gkigkitzis et al.(2013) and is compared to the Landauer’s principle that physical systems carry experimental results from Egan and Lineweaver information. For a massless particle, we must replace (2010). Gkigkitzis et al.(2013) used the results from the rest energy by the particles momentum Lloyd (2002) and Landauer to calculate the time E ¼ pc ¼ hc, where h ¼ 6:62607004 Á 10À34 Js is the varying entropy in both a flat and curved universe. k Planck’s constant. Hence, in (4), we replace Mc2 ! hc Within our model, we explore a time varying, or k which is valid for a radiation-dominated universe, and running cosmological constant, which is compared to take as an assumption the smallest amount of infor- results from other models (Haranas & Gkigkitzis mation that a massless particle carries to be 2013; Lopez & Nanopoulos 1996; Singh 2010; Liu & c Wesson 2001). Throughout this contribution, we N ¼ 2p ; ð5Þ relate our model to two of the three epochs of the Hk universe: radiation-dominated and matter-dominated where k, the wavelength of the particle can also be universe; a dark energy-dominated universe is the associated with its interaction range. This equation subject of future research. The motivation for this cannot be used for a matter-dominated universe research is threefold: because there is no evidence that the universe can expand with speed c in a matter-dominated universe. (a) To show that a toy model of a massless graviton For a radiation-dominated universe, where the prin- can reproduce results observed in massive models ciple particles are massless, this equation is valid. (with the application being a bound of the total However, we will find a resultant equation which is entropy budget of the universe in matter and valid for both matter- and radiation-dominated uni- radiation dominated epochs). verses, where we must make a different substitution (b) To study a toy model in which the time varying that is valid for matter-dominated universes. We cosmological constant is a function of informa- assume the graviton to have an interaction range as tion (through its dependence on Hubble’s large as the universe, hence are motivated to take constant). h (c) To show how the classical solutions for matter and k ¼ ; ð6Þ radiation dominated epochs can be used to study Muc more complex cosmological models such as to be the wavelength in (5) (i.e., the Compton wave- entropy, gravitons, cosmological constant, etc. length of the universe). It is clear that it would be J. Astrophys. Astr. (2020) 41:30 Page 3 of 5 30 impossible to experimentally verify that the graviton a_ which is given by H ¼ a, where a is the scale factor of has such a large interaction range. Our model here is a the universe. A review of these definitions can be toy model that aims to investigate the mathematical found in Islam (1992). Considering a matter-domi- consequences of taking the interaction range to be as nated universe constrains the scale factor, a,tobe large as the observable universe. M is the total mass 2 u proportional to t3 (Liddle 2015) and substituting the in the observable universe which takes the form definition of Hubble’s constant into (11), our relation c3 for the scale factor gives Mu ¼ ; ð7Þ GH 9k ln 2 S ¼ B t2: ð12Þ where G is the Newton’s constant. Using (6) and (7), 2 4tp we can rewrite equation (5)tobe 17 c5 Substituting t ¼ 4:35 Â 10 s (13.8 billion years) to N ¼ : ð8Þ GH2h be the age of the universe implies that a bound for the entropy at the current age of the universe is Similarly, replacing M ! Mu in (4) we arrive at (8), S ¼ 1:40 Â 1099 J/K, matching the results of Gkigk- which is valid for matter-dominated universe. In this itzis et al.(2013) and Egan and Lineweaver (2010)up way, we have found an equation relating the total to an order of magnitude. information budget of the universe due to gravitons in For a radiation-dominated universe, we have that both matter- and radiation-dominated universes. Rec- 1 2 5 the scale factor is proportional to t (Liddle 2015); in a ognizing that t2 ¼ c , we can further rewrite (8)as pGh similar way we obtain 2 2 1 t 4k ln 2 N ¼ ¼ ; ð9Þ S ¼ B t2: ð13Þ Ht t 2 p p tp which mirrors the relation seen in Gkigkitzis et al. So, a constraint on the entropy during the early (2013). Note that this expression also agrees with universe (say 30000 years from the Big Bang) is expression (5) in Lloyd (2002) where the author states S ¼ 1:18 Â 1088 J/K. Our model predicts that the that the total number of bits in the universe is pro- entropy has increased by 11 orders of magnitude since 2 portional to t in a matter-dominated universe. the radiation-dominated period of the universe. tp

3. Calculation of the entropy of the universe 4. Relation to the cosmological constant

The entropy of a system is given by Boltzmann’s Next, we will use the equation for entropy we have equation. However, Lloyd (2002) derived a different derived to explore the dynamics of a time-varying equation for the entropy carried by 1 Nat of infor- (running) cosmological constant. We would now like mation which satisﬁes Landauer’s principle, i.e., that to relate Equation (11) to the one seen in Gkigkitzis information is physical and is registered and carried et al.(2013), which calculates the entropy of a uni- by physical systems (Lloyd 2002). We are also verse with a cosmological constant. The relation seen motivated by Mureika and Mann (2010) that gravity in that contribution was may be expressed as an entropic force. For one mas- 3pk sive entity, the entropy associated with 1 Nat is given S ¼ B ; ð14Þ Kl2 by S ¼ kB ln 2 (Gkigkitzis et al. 2013). For N bits of p information, this equation becomes K being the cosmological constant and l is the Planck S Nk : p ¼ B ln 2 ð10Þ length. Given that we calculated the same value for

Here kB is the Boltzmann’s constant. Substituting the entropy of the universe in the preceding section, from (9), we have we are motivated to equate (11) and (14). Solving this equation for the cosmological constant leads to kB ln 2 S ¼ : ð11Þ 3pH2 HtðÞ2t2 K ¼ ; ð15Þ p c2 ln 2 Let H ¼ HtðÞ, i.e., a time-varying Hubble’s constant. which matches the result found in Haranas and To solve this equation, we recall the definition of H, Gkigkitzis (2013). If we want to explore the dynamics 30 Page 4 of 5 J. Astrophys. Astr. (2020) 41:30 of a time-varying cosmological constant, we can allow hc relation as the photon (E ¼ k ); and that the graviton both K and H to have explicit time dependence. carries physical information. We compare our calcu- Substituting the definition of H(t) from the previous lation of the total entropy of the universe to results 2 a_ 3 section, H ¼ a and using the fact that atðÞt for a from Gkigkitzis et al.(2013) and Egan and Line- matter-dominated universe, we arrive at weaver (2010). Similarly, we calculate the cosmo- 12p logical constant within our model and compare to K ¼ tÀ2; ð16Þ other models (Haranas & Gkigkitzis 2013; Lopez & 9c2 ln 2 Nanopoulos 1996; Singh 2010; Liu & Wesson 2001). which shows that for this simple model, the cosmo- Collectively the results of our model can be summa- logical constant gets smaller as time evolves. Substi- 1 1 rized as K / t2 and S / K for both matter- and radia- tuting t ¼ 4:35 Â 1017s for the current age of the tion-dominated universes. We have shown that the universe (as done in the preceding section) yields the time-dependence of the cosmological constant can be numerical value of the cosmological constant to be calculated through the dynamics of the Hubble func- K ¼ 3:55 Â 10À52, very close to accepted value of tion. This contrasts with most models, for example, À52 Ktrue ¼ 1:1056 Â 10 . Similarly, for the radiation- Overduin and Cooperstock (1998), where the time- dominated period we have dependence of K is chosen a priori. Because our 3p results for the entropy of the universe and for the K ¼ tÀ2: ð17Þ cosmological constant are similar to models with a 4c2 ln 2 massive graviton, it may be that we exist in a universe Substituting t ¼ 30000 years leads to K ¼ 4:22 Â in which it is impossible to distinguish (from macro- 10À41 and so the model predicts that the cosmological scopic measurements) whether we live in a universe constant has increased by 11 orders of magnitude from with a massive or massless graviton. This point could the radiation period to the matter-dominated period (in be further explored through experimental evidence of the preceding section, we saw that this is also true for the full model which properly accounts for the the total entropy of the Universe) and so it is clear for graviton’s interaction range. This points to the fact 1 that the question of whether the graviton has mass will our model S / K for both matter- and radiation-dominated eras. In Lopez and Nanopoulos (1996), Singh ultimately be answered by a theory of quantum (2010) and Liu and Wesson (2001), a running cos- gravity, i.e., a renormalizable theory describing sub- 1 atomic interactions is required to further constrain the mological constant which scales as K / t2 is calculated using different approaches. Note that Overduin mass of the graviton, and ultimately determine if it is and Cooperstock (1998) chose a relationship between massless or not. In future contributions, we will apply K and time a priori, whereas in this contribution the our model to dark energy dominated universes; in explicit time-dependence of the cosmological constant particular, we will explore how different inflationary on time is derived from the time-dependent Hubble models behave under consideration of a massless function, HtðÞ. Gkigkitzis et al. (2013) has shown the graviton. Hubble function to depend on the total amount of information, N. This shows that the time-dependence of the cosmological constant is intricately related to References the total amount of information in the Universe. Alfonso Faus A., Fullana i Alfonso M. J. 2013, Cosmic background Bose condensation (CBBC), Astrophys. 5. Conclusion Space Sci., 347, 193, https://doi.org/10.1007/s10509- 013-1500-08 We have considered a mathematical toy model in Balseanu M. 2019, Graviton Phys., 74, 1002, https://doi. org/10.13140/rg.2.2.10316.87686 which the graviton has zero mass and have calculated de Rham C., Deskins J. T., Tolley A. J., Zhou S. Y. the consequences for matter-dominated and radiation- 2017, Graviton Mass Bounds, Rev. Mod. Phys., 89, dominated universes. A full, experimentally verifiable 025004 model would account for the fact that the interaction Egan C., Lineweaver C. 2010, A Larger Estimate of the range of the graviton is smaller than the observable Entropy of the Universe, 710, 1825, arXiv:0909.3983 universe. The essential assumptions of the model are Gkigkitzis I., Haranas I., Kirk S. 2013, Number of that the graviton obeys the same wave dispersion information and its relation to the cosmological constant J. Astrophys. Astr. (2020) 41:30 Page 5 of 5 30

resulting from Landauer’s principle, Astrophys. Space Lloyd S. 2002, Computational capacity of the Universe, Sci., 348, 553 Phys. Rev. Lett., 88, 237901 Haranas I., Gkigkitzis I. 2013, Ricci scalar and its relation Lopez J. L., Nanopoulos D. V. 1996, A new cosmological in the evolution of the entropy and information of in the constant model, Mod. Phys. Lett. A, 11, 01 universe, Mod. Phys. Lett. A, 2013, 3 Mureika J., Mann R. 2010, Does entropic gravity bound the Haranas I., Gkigkitzis I. 2014, The Mass of Graviton and Its masses of the photon and graviton? Mod. Phys. Lett. A, Relation to the Number of Information According to the 26, https://doi.org/10.1142/S0217732311034840 Holographic Principle, International Scholarly Research Novello M. 2005, The mass of the graviton and the Notices, 2014, 8, Article ID: 718251 cosmological constant puzzle, Class. Quantum Gravit., Islam J. N. 1992, An Introduction to Mathematical 0504505v1 Cosmology, Cambridge University Press, Cambridge, Overduin J. M., Cooperstock F. I. 1998, Evolution of the p. 32 scale factor with a variable cosmological term, Phys. Liddle A. 2015, An Introduction to Modern Cosmology, Rev. D, 58, 043506 John Wiley & Sons Singh K. 2010, On Robertson–Walker universe model with Liu H., Wesson P. 2001, Universe Models with a Variable variable cosmological term and gravitational constant in Cosmological ‘‘Constant’’ and a ‘‘Big Bounce’’, Astro- cosmological relativity theory, Turkish J. Phys., 34, phys. J., 562, 1, https://doi.org/10.1086/323525 https://doi.org/10.3906/ﬁz-1007-2