Big Bang Nucleosynthesis and Entropy Evolution in f(R,T) Gravity
EEPJ manuscript No. (will be inserted by the editor)
Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation
Snehasish Bhattacharjee and P.K. Sahoo Department of Astronomy, Osmania University, Hyderabad-500007, India, Email: [email protected] Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India,Email: [email protected]: 13th Jan 2020 / 24th March 2020
Abstract.
The present article is devoted to constrain the model parameter χ for the f ( R, T ) = R + χT gravity model by employing the constraints coming from big bang nucleosynthesis. We solve the fieldequations and constrain χ in the range − . κ ≤ χ ≤ . κ (where κ = πGc ) from the primordialabundances of light elements such as helium-4, deuterium and lithium-7. We found the abundances ofhelium-4 and deuterium agrees with theoretical predictions, however the lithium problem persists forthe f ( R, T ) gravity model. We also investigate the evolution of entropy for the constrained parameterspace of χ for the radiation and dust universe. We report that entropy is constant when χ = 0 for theradiation dominated universe, whereas for the dust universe, entropy increases with time. We finally usethe constraints to show that χ has negligible influence on the cold dark matter annihilation cross section. PACS.
The current accelerated expansion of the universe favors big bang cosmology. The model predicts the abundances ofseveral light elements of the primordial universe with great precision. The elements were produced as a result of nuclearfusion started seconds after the big bang and lasted for some minutes. Additionally, the model predicts inflation whichis a super exponential increase of the volume of the universe for a very short time (10 − sec). Inflation have beensuccessful in solving the flatness, horizon and homogeneity problems of the universe [1].However, many cosmological puzzles exist which hitherto cannot be explained by the standard big bang cosmologysuch as origin of dark matter and dark energy, cosmological constant problem, cosmic coincidence problem and theexact form of the inflation potential etc. [2,3,4]. To answer these problems, modifying GR have become a promisingalternative giving rise to a plethora of modified gravity theories. f ( R, T ) gravity is a widely studied modified gravity theory introduced in the literature in [5] and is a generalizationof f ( R ) gravity (see [6] for a review on modified gravity theories). In this theory, the Ricci scalar R in the action isreplaced by a combined function of R and T where T is the trace of the energy-momentum tensor. f ( R, T ) gravityhave been widely employed in various cosmological scenarios and have yielded interesting results in areas such as darkmatter [7] dark energy [8], super-Chandrasekhar white dwarfs [9], massive pulsars [10], wormholes [11], gravitationalwaves [12], baryogenesis [13], bouncing cosmology [14] and in varying speed of light scenarios [15].In this article we are interested in constraining the model parameter of f ( R, T ) gravity theory for the ansatz f ( R, T ) = R + χT , where χ is the model parameter. Constraining χ can help us to better understand the impact of χ incosmological models and also in the above mentioned astrophysical areas.Big Bang nucleosynthesis can be an excellent way to constrain the model parameters of any modified gravity theoryas the abundances of primordial light elements such as deuterium ( H ), helium ( He ) and lithium ( Li ) have beenobservationally constrained to great accuracy. These abundances are directly related to the Hubble parameter H , whichultimately involve the model parameters of any chosen modified gravity theory. This method have been successfullyemployed to constrain the model parameters of f ( R ) gravity [17,18], f ( T ) gravity [19], scalar-tensor gravity models[20] and to test the viabilities of Brans Dicke cosmology with varying Λ [21], Higher Dimensional Dilaton GravityTheory of Steady-State Cosmological (HDGS) model in the context of string theory [2] and massive gravity theory[16]. The discrepancy between predicted and observed abundance of lithium (’lithium problem’) is investigated in [22](and in references therein). a r X i v : . [ phy s i c s . g e n - ph ] A p r S. Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation
The paper is organized as follows: In Section 2 we provide an overview of f ( R, T ) gravity. In Section 3 we summarizebig bang nucleosynthesis and present a through analysis to constrain χ . In Section 4, we investigate the evolution ofentropy for the radiation and matter filled universes for the constrained range of χ . In Section 5, we investigate whether χ influences the cold dark matter annihilation cross section and Section 6 is devoted to discussions and conclusions. f ( R, T ) Gravity
The action in f ( R, T ) gravity is given by S = (cid:90) √− g (cid:20) κ f ( R, T ) + L m (cid:21) d x (1)where L m represent matter Lagrangian and κ = πGc .Stress-energy-momentum tensor for the matter fields is given as T µν = − √− g δ ( √− g L m ) δg µν = g µν L m − δ L m δg µν (2)varying the action (1) with respect to the metric yields f ,R ( R, T ) R µν + Π µν f ,R ( R, T ) − g µν f ( R, T ) = κ T µν − ( T µν + Θ µν ) f ,T ( R, T ) (3)where −∇ µ ∇ ν + g µν (cid:3) = Π µν (4) g αβ δT αβ δg µν ≡ Θ µν (5)and f i,X ≡ d i fdX i . Upon contraction (3) with g µν , the trace of the field equations is obtained as f ,R ( R, T ) R − f ( R, T ) + 3 (cid:3) f ,R ( R, T ) = − ( Θ + T ) f ,T ( R, T ) + κ T (6)We now consider a flat FLRW background metric as ds = dt − a ( t ) [ dx + dy + dz ] (7)where a ( t ) denote the scale factor. For a universe dominated by a perfect fluid the matter Lagrangian density is givenas L m = − p . Upon employing this to (3) and (6) yields1 f ,R ( R, T ) (cid:20) − RHf ,R ( R, T ) + pf ,T ( R, T ) − Rf ,R ( R, T ) + 12 ( f ( R, T )) (cid:21) + f ,T ( R, T ) + κ f ,R ( R, T ) ρ = 3 H (8)1 f ,R ( R, T ) (cid:20) − (cid:16) f ( R, T ) + ˙ R f ,R ( R, T ) + ¨ Rf ,R ( R, T ) − Rf ,R ( R, T ) (cid:17) − pf ,T ( R, T ) + 2 H ˙ Rf ,R ( R, T ) (cid:21) + f ,T ( R, T ) + κ f ,R ( R, T ) p = − H − H (9)where H denote the Hubble parameter, overhead dots denote derivative with respect to time, p represents pressureand ρ represents density with T = ρ − p .setting f ( R, T ) functional form to be f ( R, T ) = R + χT. (10)Substituting (10) in (8) and solving for Hubble parameter ( H f ( R,T ) ), we obtain H f ( R,T ) = (cid:15)t , (11) . Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation 3 where (cid:15) = − (cid:20) − κ + ( ω − χ ( ω + 1) ( χ + κ ) (cid:21) . (12)where ω = p/ρ denote the EoS parameter. The scale factor a ( t ) takes the form a ∼ t (cid:15) (13)The expression of density ρ reads ρ = (cid:0) κ − χ ( ω − (cid:1) (cid:0) χ (3 + 8 χ − ω ) + 2 κ (1 + χ (3 + ω )) (cid:1) t ( κ + χ ) (1 + 6 χ + 8 χ ) (1 + ω ) (14)For a radiation dominated universe ( ω = 1 / H f ( R,T ) = (cid:20) χ/ κ κ + χ ) (cid:21) /t (15)In Einstein’s GR, the expression of Hubble parameter in radiation dominated universe reads H = 12 t (16) f ( R, T ) gravity In this method we are interested in finding a suitable value or range of χ which can suffice the primordial abundancesof light elements. Specifically we will be studying the ratio of Hubble parameter in f ( R, T ) gravity to the Hubbleparameter of standard big bang cosmology for the radiation dominated universe. The ratio is represented as Z = H f ( R,T ) H SBBN (17)where H f ( R,T ) is given by (15) and H SBBN is given by (16) and SBBN stands for Standard Big Bang nucleosynthesis.The primordial abundances of the light elements ( D , He , Li ) depend on the expansion rate of the universe and onthe baryon density [29,24]. The baryon density parameter reads η ≡ η B ≡ η B η γ (18)Where η (cid:39) η B represents the baryon to photon ratio [25]. Z (cid:54) = 1 correspond to non-standard expansion factor. This can arise due to GR modification or due to the presence ofadditional light particles such as neutrinos which would make the ratio to be, Z = (cid:0) ( N ν − (cid:1) / [2]. However,we are interested for the case where the value of ( Z −
1) comes from GR modification and hence we shall assume N ν = 3. He abundance in f ( R, T ) Gravity
The first step in producing helium ( He ) starts with producing H from a neutron ( n ) and a proton ( p ). After that,Deuterium is converted into He and Tritium ( T ). n + p → H + γ ; H + H → He + n ; H + H → H + p (19) He is finally produced from the combination of H with H and He ; H + H → He + n ; H + He → He + p (20)The simplest way to ascertain the He abundance is from the numerical best fit given in [26,27] Y p = 0 . ± . . η −
6) + 100 ( Z − Z = 1, we recover the SBBN He fraction, which reads ( Y p ) | SBBN = 0 . ± . He abundance to be 0 . ± . . ± . . ± . . Z − η = 6. This constrains Z in the range 1 . ± . S. Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation H abundance in f ( R, T ) Gravity
Deuterium H is produced from the reaction n + p → H + γ . Deuterium abundance can be ascertained from thenumerical best fit given in [29] y Dp = 2 . ± . (cid:18) η − Z − (cid:19) . (23)For Z = 1& η = 6, y Dp | SBBN = 2 . ± .
16. Observational constraint on deuterium abundance is y Dp = 2 . ± . . ± .
03 = 2 . ± . (cid:18) η − Z − (cid:19) . (24)This constraints Z in the range Z = 1 . ± . Z for the deuterium abundance partially overlapswith that of the helium abundance. Thus χ can be fine tuned to fit the abundances for both H and He . Li abundance in f ( R, T ) Gravity
The lithium abundance is puzzling in the sense that the η parameter which precisely fits the abundances of otherelements successfully does not fit the observations of Li and the ratio of the expected SBBN value of Li abundanceto the observed one is between 2 . − . Li . This is known as the Lithium problem [2].The numerical best fit expression for Li abundance reads [29] y Lip = 4 . ± . (cid:20) η − Z − (cid:21) (25)Observational constraint on lithium abundance is y Lip = 1 . ± . Z to fit the Li abundanceis Z = 1 . ± . From Table 1 it is clear that f ( R, T ) gravity yields excellent estimates for the abundances of helium and deuteriumwhich match better to observations than the SBBN model. However, the abundance of lithium is still a problemfor both the models (SBBN and f ( R, T ) gravity). In Figure 1, we show χ as a function of Z . For Z in the range0 . ≤ Z ≤ . χ in the range − . κ (cid:46) χ (cid:46) . κ . Table 1.
The abundances He-4, Deuterium and Li-7 for different models
Models and data/Abundances Y p y Dp y Lip
SBBN model 0 . ± . . ± .
16 4 . ± . f ( R, T ) Gravity 0 . ± . . ± . . ± . . ± . . ± .
03 [28] 1 . ± . . Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation 5 - χ Fig. 1. χ as a function of Z . Vertical lines are drawn at Z = 0 . Z = 1 . χ = − .
14 and χ = 0 . f ( R, T ) Gravity
Baryon to entropy ratio is a useful parameter characterizing the over abundance of matter over anti-matter in theuniverse. Since the law of conservation of energy momentum is not maintained in f ( R, T ) gravity, we investigate howthis affects adiabaticity [31,32]. In SBBN model, the entropy of the universe is a conserved quantity throughout itsevolution and this is due to the fact that at low energies, baryon number is neither created nor destroyed since thereare no decays and consequently the baryon to entropy ratio η S is a constant [32]. Equivalently, once the large scaleannihilation processes have concluded, the baryon to photon ratio η B is also a constant, and both quantities can beconnected easily [32].From the first law of thermodynamics, we obtain dE + pdV = T dS (26)where S = s ( a ) and E = ρ ( a ) are the entropy and internal energy of the universe respectively. This gives [32] d ( ρa ) + pd ( a ) = T dS → Ta ˙ S = ˙ ρ + 4 Hρ (27)From statistical mechanics, density ρ is related to temperature T as [33] ρ = π g ∗ s T (28)where g ∗ s = 107 is the effective number of relativistic degrees of freedom contributing to the entropy of the universe[32].Substituting all the values, we obtain˙ S = 1 . t − ( χ χ ) χ (cid:0) . . χ + 1 . χ + 0 . χ (cid:1) (1 + χ ) (0 .
125 + 0 . χ + χ ) (cid:104) (3+4 χ )(3+14 χ +12 χ ) t (1+ χ ) (1+6 χ +8 χ ) (cid:105) . (29)In Figure 2, we observe that ˙ S is positive for χ > χ < S = 0 for χ = 0 (GR) for the radiation universe. However, Figure 3shows that ˙ S > S increases as χ increases. It is also evident that ˙ S decreases slowly with time for the dust universe. S. Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation
Fig. 2.
Time evolution of ˙ S in radiation universe for − . κ (cid:46) χ (cid:46) . κ . Fig. 3.
Time evolution of ˙ S in dust universe for − . κ (cid:46) χ (cid:46) . κ . Table 2.
Rate of change of entropy ( ˙ S ) for different models Models Rate of change of entropy ( ˙ S )Radiation universe ( ω = 1 /
3) Dust universe ( ω = 0)GR 0 (cid:20) . t . t (cid:21) f ( R, T ) Gravity (cid:20) − . t . t . (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) χ = − . (cid:20) . t . t . (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) χ =0 . (cid:20) . t . t . (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) χ = − . (cid:20) . t . t . (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) χ =0 . f ( R, T ) Gravity
Recent cosmological observations have constrained the normalized cold dark matter density in the range [34]0 . (cid:46) Ω cdm h (cid:46) .
126 (30)In this section we shall assume dark matter to be composed of weakly-interacting massive particles (WIMPs). In [17]the authors derived an analytical expression where the WIMP cross section ¯ σ is written in terms of the relic densityof dark matter, its mass m and on the power n for the power law f ( R ) gravity model of the form f ( R ) ∼ R n . We . Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation 7 shall now investigate the role of χ in dark matter annihilation cross section for a given WIMP mass.The expression relating the dark matter relic density, its mass, dark matter annihilation cross section and parametersof a modified gravity model reads [17] Ω cdm h = 1 . × ( ¯ m + 1) x ( ¯ m +1) GeV − f ( h ∗ /g / ∗ s ) M p ¯ σ (31)where ¯ m = m + (1 − n ) (32)where ¯ m = m for GR and m = 0 & 1 correspond to s-wave and p-wave polarizations respectively and for n = 1, GRis recovered. x f is the freeze-out temperature and given as [33,17] x f = ln[0 . m + 1)( g/g / ∗ s ) M p m ¯ σ ] − ( ¯ m + 1) ln[ln[0 . m + 1)( g/g / ∗ s ) M p m ¯ σ ]] (33)where g = 2 is the spin polarizations of the dark matter particle [17] and M p is the Planck mass.In [17], the authors found substantial influence of n in ¯ σ although n had very little deviation from GR ( n − (cid:46) . m for our f ( R, T ) gravity model and check the influence of χ on ¯ σ .For ¯ m of the form ¯ m = m + χ (34)we get ¯ m = m when χ = 0. Now from BBN, χ is constrained in the range − . κ (cid:46) χ (cid:46) . κ which is O − .Hence our f ( R, T ) gravity model produces ¯ σ very close to that predicted from GR. Nonetheless, it would be interestingto do the same analysis with f ( R, T ) gravity models with a power-law dependence on T . Modified gravity theories are becoming popular owing to the failures of GR in explaining the current acceleration ofthe universe. In modified gravity theories, the model parameters are fine tuned to obtain the desired results whichsometimes differ significantly from GR. In this work we investigate the viability of the most widely studied and simplestminimal matter-geometry coupled f ( R, T ) gravity model of the form f ( R, T ) = R + χT in cosmological models andin many astrophysical areas.The present manuscript uses the constraints of abundances of light elements such as helium-4, deuterium and lithium-7to constrain the model parameter χ to unprecedented accuracy. From the analysis, we report a tight constraint on χ in the range − . κ (cid:46) χ (cid:46) . κ .We also study the evolution of entropy for the constrained parameter space of χ for the radiation and dust universe.We report that entropy ( S ) is constant for χ = 0 for the radiation dominated universe, whereas for the dust universe,˙ S > χ .We also found that χ has negligible influence on dark matter annihilation cross section (¯ σ ) and produces ¯ σ very closeto that predicted by GR.The constraints on χ obtained from the present analysis makes it clear-cut that the parameter χ has negligibleinfluence in cosmological models and in above mentioned astrophysical areas. It would certainly be interesting toapply the method to constrain the model parameters for other f ( R, T ) gravity models and to check their viability inrepresenting the current state of the universe.As a final note we add that in [35], the authors reported that the gravitational energy-momentum pseudotensorcan also be an important tool in distinguishing and constraining different theories of gravity. Specifically, in [36], theauthors reported that the gravitational pseudotensor is useful to identify the dissimilarities in quadrupolar gravitationalradiation coming from Einstein’s gravity and f ( R ) gravity. This idea was further extended to teleparallel gravity in [35].Since gravitational waves differ substantially from one theory of gravity to another [37], detection of the polarizationmodes of the gravitational radiation can be promising to constrain extended theories of gravity [35]. Acknowledgments
SB thank Biswajit Pandey for constant support and motivation. SB also thank Suman Sarkar and Biswajit Das forhelpful discussions. PKS acknowledges CSIR, New Delhi, India for financial support to carry out the Research project[No.03(1454)/19/EMR-II Dt.02/08/2019]. We are very much grateful to the honorable referee and the editor for theilluminating suggestions that have significantly improved our work in terms of research quality and presentation.
S. Bhattacharjee, P.K. Sahoo: Big Bang Nucleosynthesis and Entropy Evolution in f ( R, T ) Gravitation
References
1. For a review, see e.g., A. Linde, Lect.Notes Phys. , 1-54 (2008).2. S. Boran and E. O. Kahya, Adv. High Energy Phys. , 282675 (2014); arXiv:1310.6145.3. P. J. E. Pebbles and B. Ratra, Rev.Mod.Phys. , 559606 (2003).4. S. Perlmutter et al., Astrophys. J. , 565-586 (1999).5. T. Harko et al., Phys. Rev. D, , 024020 (2011).6. S. Capozziello and M. D. Laurentis, Phys. Rept. , 167 (2011); S. Capozziello, Int. J. Mod. Phys. D11 , 483 (2002).7. R. Zaregonbadi, et al., Phys. Rev. D , 084052 (2016).8. G. Sun and Y.-C. Huang, Int. J. Mod. Phys. D, , 1650038 (2016).9. F. Rocha et al. arXiv:1911.08894 (2019)10. S.I. dos Santos, G.A. Carvalho, P.H.R.S. Moraes, C.H. Lenzi and M. Malheiro, Eur. Phys. J. Plus, , 398 (2019); P.H.R.S.Moraes, J.D.V. Arbanil and M. Malheiro, J. Cosm. Astrop. Phys. , 005 (2016).11. P.H.R.S. Moraes and P.K. Sahoo, Eur. Phys. J. C , 677 (2019); E. Elizalde and M. Khurshudyan, Phys. Rev. D, ,024051 (2019); P.H.R.S. Moraes, W. de Paula and R.A.C. Correa, Int. J. Mod. Phys. D, , 1950098 (2019); E. Elizaldeand M. Khurshudyan, Phys. Rev. D, , 123525 (2018); P.H.R.S. Moraes and P.K. Sahoo, Phys. Rev. D, , 024007 (2018);P.K. Sahoo, P.H.R.S. Moraes and P. Sahoo, Eur. Phys. J. C, , 46 (2018); P.K. Sahoo, P.H.R.S. Moraes, P. Sahoo andG. Ribeiro, Int. J. Mod. Phys. D, , 1950004 (2018); P.H.R.S. Moraes and P.K. Sahoo, Phys. Rev. D, , 044038 (2017);P.H.R.S. Moraes, R.A.C. Correa and R.V. Lobato, J. Cosm. Astrop. Phys., , 029 (2017); T. Azizi, Int. J. Theor. Phys. , 3486 (2013).12. M. Sharif and A. Siddiqa, Gen. Rel. Grav., , 74 (2019); M.E.S. Alves, P.H.R.S. Moraes, J.C.N. de Araujo and M. Malheiro,Phys. Rev. D, , 024032 (2016).13. P.K. Sahoo and S. Bhattacharjee, Int. J. Theor. Phys, DOI: 10.1007/s10773-020-04414-3 (2020) [arXiv: 1907.13460].14. P. Sahoo et al., Mod. Phys. Lett. A, DOI: 10.1142/S0217732320500959 (2020) [arXiv: 1907.08682].15. S. Bhattacharjee and P. K. Sahoo, Eur. Phys. J. Plus, , 86 (2020) [arXiv:2001.06569].16. G. Lambiase, JCAP, , 028 (2012).17. J. U. Kang and G. Panotopoulos, Phys Lett B, , 6 (2009).18. R. P. L. Azevedo and P. P. Avelion, Phys. Rev. D, , 064045 (2018); M Kusakabe, et al., Phys. Rev. D, , 104023 (2015).19. S. Capozzielo, G. Lambiase and E. N. Saridakis, Eur. Phys. J. C, , 576 (2017)20. A. Coc and K. A. Olive, Phys. Rev. D, , 083525 (2006).21. R. Nakamura, M. Hashimoto, S. Gamow and K. Arai, A& A , 23 (2006).22. J. Larena, J. M. Alami and A. Serna, Astrophys. J, , 1 (2007); T. R. Makki and M. F. E. Eid, Mod. Phys. Lett. A, ,24, 1950194 (2019).23. G. Steigman, Adv. High Energy Phys. , 268321 (2012)24. V. Simha, G. Steigman, JCAP , 016 (2008)25. WMAP Collaboration (E. Komatsu et al.), Astrophys. J. Suppl. , 18 (2011).26. J.P. Kneller, G. Steigman, New J. Phys. , 117 (2004)27. G. Steigman, Annu. Rev. Nucl. Part. Sci. , 463 (2007)28. Brian D. Fields et al, JCAP , 010 (2020).29. G. Steigman, Adv. High Energy Phys. , 268321 (2012)30. B. D. Fields, Annual Review of Nuclear and Particle Science , 47-68 (2011).31. T. Harko, Phys. Rev. D , 044067 (2014); I. Prigogine, J. Geheniau, E. Gunzig, and P. Nardone, Proc. Natl. Acad. Sci.U.S.A. , 7428 (1988).32. M. P. L. P. Ramos and J. Paramos, Phys. Rev. D , 104024 (2017).33. E. W. Kolb, M. S. Turner, The Early Universe, Addison-Wesley Publishing Company, Redwood City, California (1989).34. D. N. Spergel, et al., WMAP Collaboration, Astrophys. J. Suppl. ,377 (2007).35. S. Capozziello, M. Capriolo and M. Transirico, Int. J. Geom. Meth. Mod. Phys, , 1850164 (2018).36. M. De Laurentis and S. Capozziello, Astropart. Phys. 35 (2011) 257.37. K. Bamba, S. Capozziello, M. De Laurentis, S. Nojiri and D. Sez-Gmez, Phys. Lett. B, , 194 (2013); H. Abedi and S.Capozziello, Eur. Phys. J. C,78