mmanuscript No. (will be inserted by the editor)
BBN Constraints on f ( Q, T ) Gravity
Snehasish Bhattacharjee ID a,1 Department of Physics, Indian Institute of Technology, Hyderabad 502285, IndiaFebruary 26, 2021
Abstract f ( Q, T ) gravity is a novel extension of the symmetric teleparallel grav-ity where the Lagrangian L is represented through an arbitrary function of thenonmetricity Q and the trace of the energy-momentum tensor T [23]. In this work,we have constrained a widely used f ( Q, T ) gravity model of the form f ( Q, T ) = Q n +1 + mT from the primordial abundances of the light elements to understandits viability in Cosmology. We report that the f ( Q, T ) gravity model can el-egantly explain the observed abundances of Helium and Deuterium while theLithium problem persists. From the constraint on the expansion factor in therange 0 . (cid:46) Z (cid:46) . m and n in the range − . (cid:46) n (cid:46) − .
08 and − . (cid:46) m (cid:46) .
52 respectively.
Keywords
Big-Bang Nucleosynthesis · f ( Q, T ) Gravity
PACS
The Λ CDM cosmological model has been remarkably successful in expoundingthe dynamics and evolution of the Universe from seconds after the big bang tothe present accelerated expansion. However, the model presumes the Universe isstatistically homogenous and isotropic at large scales, and that the two myste-rious entities termed the dark matter and the dark energy exists in substantialproportions, and that the law of gravity is well described by General Relativity(GR). Nevertheless, the model is incomplete, is evident from the fact that hith-erto, no conclusive evidence has been found to confirm the existence of dark matterand dark energy. Furthermore, the model cannot explain the existence of matter-antimatter asymmetry and it is becoming increasingly difficult to incorporate GR into quantum field theory to get a complete theory of reality [1,2,3,4,5,6,7,8,9,10]. Moreover, new models beyond the Λ CDM model are being continuously pro-posed to alleviate numerous tensions between observational data at different scales[11,12,13,14,15,16,17,18]. a e-mail: [email protected] a r X i v : . [ g r- q c ] F e b With this in mind, many alternatives have surfaced which aim to explain the ef-fects of dark matter and dark energy by modifying or extending GR. Extendedtheories of gravity are motivated by the fact that Einstein’s GR does not providea complete understanding of several gravitational effects at the UV and IR scales.In addition to introducing theories such as the f ( R ) gravity, which are essentiallysimple geometric extensions of GR, many alternative theories of gravity identifythe gravitational field to be best described by variable(s) other than the Ricciscalar. One of the cornerstones of GR and many extended theories of gravity isthe assumption of the Equivalence principle [1] which leads towards coinciding thecausal structure and the geodesic and secures the Levi-Civita connection [22].An approach to understanding gravity employing variables other than the Ricciscalar could turn out to be useful. For instance, the Ricci scalar could be replacedby the torsion scalar if the gravitational interactions are expressed in terms oftetrads. Such a theory of gravity is commonly termed teleparallel gravity, in whichthe affinities and not the Equivalence principle plays the most important role.The most successful formulation of a teleparallel gravity is the f ( T ) gravity whichreplaces the Ricci scalar R in the Einstein-Hilbert action with the Torsion scalar T .Recently, [23] proposed a novel extension of the symmetric teleparallel gravitycalled the f ( Q, T ) gravity for which the Lagrangian L is represented through anarbitrary function of the non-metricity Q and the trace of the energy-momentumtensor T . The gravitational equations are obtained through the variation of boththe metric and the connection with the coupling between matter and geometry ac-companying non-conservation of the energy-momentum tensor [23]. f ( Q, T ) grav-ity has been employed to successfully explain the matter-antimatter asymmetry[24] and the late-time acceleration [25].These alternative theories of gravity are formulated with the primary objective ofbeing self-consistent and furnish a complete picture of the evolution of the Uni-verse from the early phase of inflation to the period of structure formation tothe late-time acceleration (readers may refer to[19,20,21] a comprehensive discus-sion). Cosmography [27,28] and Big-Bang Nucleosynthesis (BBN) [29] are capableof providing powerful constraints on several of these alternate cosmological mod-els. In particular, since the chemical abundances of the primordial light elementssuch as Deuterium ( D ), Helium ( He ), and Lithium ( Li ) have been ascertained tohigh precision, BBN offers stringent constraints on alternate theories of gravitysince modified theories of gravity must explain the abundances of these metals tovalidate their applicability and efficiency.In this work, we plan to constrain f ( Q, T ) gravity from the primordial abundancesof the light elements to understand its viability in cosmology. The abundances ofthe aforementioned light elements have been estimated through various obser-vational techniques. The abundance of Deuterium has been estimated from theabsorption lines of gas clouds [30,31,32,33,34], while that for the Helium fromthe emission lines of the nearby ionized Hydrogen regions in metal-poor starburstgalaxies [35,36] and finally for Lithium from the atmospheres of very metal-poor stars [37,38]. This technique has been successfully exercised to constrain scalar-tensor gravity [39], f ( T ) gravity [29,22], f ( R ) gravity [40,41,42], f ( R, T ) gravity[26], Brans-Dicke cosmology with varying Λ [43], massive gravity [44] and higherdimensional dilation gravity [52].The manuscript is organized as follows: In Sec 2 we provide an overview of f ( Q, T ) gravity. In Sec 3, we delineate the concept of Big Bang Nucleosynthesis and con-strain the model parameters of the f ( Q, T ) gravity model, and in Sec 4 we presentthe conclusions. f ( Q, T ) Gravity
The action in f ( Q, T ) gravity reads [23] S = 116 π (cid:90) √− g [ f ( Q, T ) + L M ] d x (1)where g denote the metric scalar, and L M denote the matter Lagrangian.Varying (1) with respect to the metric tensor generates the following field equation[23]8 πT (cid:15)ε = f T ( T (cid:15)ε + Θ (cid:15)ε )+ f Q (cid:16) Q αβ(cid:15) P αβε − P (cid:15)αβ Q αβε (cid:17) − √− g (cid:53) α (cid:0) f Q √− gP α(cid:15)ε (cid:1) − f g (cid:15)ε (2)where, T (cid:15)ε = − √− g δ ( √− g L M ) δg (cid:15)ε , f (cid:15) = ∂f∂(cid:15) , Θ (cid:15)ε = g (cid:15)ε δT (cid:15)ε δg (cid:15)ε (3)and the superpotential P α(cid:15)ε is given as [23] P α(cid:15)ε = 14 (cid:104) Q α g (cid:15)ε + 2 Q α ( (cid:15)ε ) − δ α ( (cid:15)Qε ) − Q α(cid:15)ε − ˜ Q α g (cid:15)ε (cid:105) (4)where Q α = Q εαε , and ˜ Q α = Q (cid:15)α(cid:15) . (5)Let us consider a FLRW geometry of the form ds = − N ( t ) dt + a ( t ) (cid:88) j =1 , , (cid:16) dx j (cid:17) (6)where N ( t ) denote the lapse function, and a ( t ) denote the scale factor. It may benoted that the lapse function is unity for a FRW background.Substituting (6) in (2), the Friedman equations in f ( Q, T ) gravity reads [23]8 πρ = − G G (cid:16) ˙ F H + F ˙ H (cid:17) + f − F H , (7)and 8 πp = 2 (cid:16) ˙ F H + F ˙ H (cid:17) − f F H , (8) where, ˜ G = f T π , F = f Q . (9)For this work, we shall set the functional form of f ( Q, T ) to the following [23] f ( Q, T ) = Q n +1 + mT (10) where n , and m are free parameters. Using (10), (7), and (8), the expression ofHubble parameter H ( t ) is given as [23] H ( t ) = H (16 π − m ( γ − n + 1)3 γH ( t − t )( m + 8 π ) − ( mγ − β + 4 π )) ( n + 1) , (11)where H , and t denote respectively the present value of the Hubble parameterand the current age of the Universe, and γ is the barotropic EoS parameter. f ( Q, T ) Gravity
We shall now attempt to constrain the model parameters m and n of the f ( Q, T )gravity model from the primordial abundances of the light elements (i.e, D , He ,and Li ). We are restricting the analysis to a radiation-dominated Universe ( i.e, γ =4 /
3) since the phenomena of BBN transpired when the Universe was dominated byradiation. The main idea behind this technique is to obtain a suitable parameterrange of m , and n for which the theoretical primordial abundances could be con-sistent with observations. To be more precise, we are concerned with the ratio ofthe Hubble parameter 11 obtained for the f ( Q, T ) gravity model ( H f ( Q,T ) ) to theHubble parameter for the standard cosmological model (i.e, the Hubble parameterof GR ( H GR )) in the radiation dominated Universe. We shall define the ratio as Z = H f ( Q,T ) H GR . (12)It may be noted that the primordial abundances of the light elements are highlysensitive to the baryon density and on the rate of the expansion of the Universe(i.e, on the Hubble parameter) [46,47]. The baryon density is expressed as η = 10 η B = 10 η B η γ , (13)where, η B η γ denote the baryon-to-photon ratio and η (cid:39) Z (cid:54) = 1 symbolizes the expansion of the Universe to be governed by a non-standardcosmological model and indicates a non-standard expansion factor. Such casescould transpire if GR turns out to be an incorrect theory of gravity or if thereexists additional species of neutrinos other than the three standard types con-firmed observationally. However, in this work, we are interested in exploring GRmodifications and therefore we shall set the total neutrino species to be equal tothree.3.1 Helium ( He ) abundance in f ( Q, T ) gravity
The production of Helium is a three step process. In the first step, a Deuterium( De ) atom is produced through a neutron ( n ) and a proton ( p ). This is followedby the production of a lighter Helium isotope ( He ) atom along with a Tritium( T ). the equations can be illustrated as follows: n + p → De + γ ; De + De → He + n ; De + De → T + p (14) Finally, the the heavier helium atom He is produced through T , De , and He as follows: T + De → He + n ; He + De → He + p. (15)The numerical best fit equation to estimate the primordial abundance of He canbe expressed as [49,50] Y p = 0 . ± . . Z −
1) + ( η − . (16)For a standard expansion factor ( i.e, Z = 1), the primordial abundance of He reads Y p = 0 . ± . . ± . η = 6 we can write,0 . ± . . ± . . Z − . (17)Upon solving this, the constraint on the expansion factor reads Z = 1 . ± . H ) abundance in f ( Q, T ) gravityThe production of a Deuterium atom takes place via a neutron and a proton asfollows n + p → D + γ. (18)The primordial abundance of Deuterium is estimated from the following numericalbest fit equation [46] y Dp = 2 . ± . (cid:20) η − Z − (cid:21) . . (19)Similar to the previous case, we set η = 6. For standard expansion factor Z = 1,we end up with a theoretical estimate of y Dp = 2 . ± . y Dp = 2 . ± . . ± .
030 = 2 . ± . (cid:20) η − Z − (cid:21) . . (20)The solution to this equation imposes a strict constraint on Z as Z = 1 . ± . Li ) abundance in f ( Q, T ) gravityThe primordial abundance of Lithium is inconsistent with the theoretical predic- tions of the Λ CDM model. The abundance of Lithium is estimated to lie between2.4 to 4.3 times the theoretical predictions [52,53]. The baryon density parameter η which describes elegantly the abundances of both Helium and Deuterium, failsto predict the same for Lithium. This is sometimes termed the Lithium problemand hints at the existence of new physics beyond the standard model [52]. The numerical best-fit equation describing the Lithium abundance can be ex-pressed as [46] y Lip = 4 . ± . (cid:20) { − Z ) + η } (cid:21) . (21)The observational abundance of Lithium lie in the range y Lip = 1 . ± .
300 [51].Plugging this into the above equation constraints Z in the range Z = 1 . ± . m and n to beconsistent with the observations of these elements. However, the f ( Q, T ) gravitymodel cannot explain the Lithium abundance given the large discrepancy betweenobservations and theoretical predictions. From Table 1, we find that the theoreticalestimate for the abundances of Helium and Deuterium for the Λ CDM and the f ( Q, T ) gravity model are fairly close and falls well within the current observationalconstraints and therefore allows the parameters m and n to be constrained strictly.We find that for 0 . (cid:46) Z (cid:46) . m and n are − . (cid:46) n (cid:46) − .
08 and − . (cid:46) m (cid:46) .
52 respectively.Table 1: The theoretical predictions for the abundances of He , H and Li in Λ CDM model and in f ( Q, T ) gravity model along with observational constraints.Models/Observations Y p y Dp y Lip
Observational data 0 . ± . . ± .
030 [51] 1 . ± .
300 [51] f ( Q, T ) Gravity 0 . ± . . ± . . ± . Λ CDM 0 . ± . . ± . . ± . - m Fig. 1: m as a function of Z for a fixed n = 1 .
11. The plot is drawn for 0 . (cid:46) Z (cid:46) . - - - - - n Fig. 2: n as a function of Z for a fixed m = 0 .
1. The plot is drawn for 0 . (cid:46) Z (cid:46) . f ( Q, T ) gravity is a novel extension of the symmetric teleparallel gravity wherethe Lagrangian L is represented through an arbitrary function of the nonmetric-ity Q and the trace of the energy-momentum tensor T [23]. f ( Q, T ) gravity hasbeen very successful in explaining the matter-antimatter asymmetry [24] and thelate-time acceleration [25].
In this work, we constrained a widely used f ( Q, T ) gravity model of the form f ( Q, T ) = Q n +1 + mT from the primordial abundances of the light elements tounderstand its viability in cosmology. We report that the f ( Q, T ) gravity modelexplains gracefully the observed abundances of Helium and Deuterium while theLithium problem persists. From the constraint on the expansion factor in the range . (cid:46) Z (cid:46) . m and n in the range − . (cid:46) n (cid:46) − .
08 and − . (cid:46) m (cid:46) .
52 respectively. Therefore,in addition to explaining the matter-antimatter asymmetry and the late time ac-celeration, f ( Q, T ) gravity also explains the abundances of Helium and Deuteriumand therefore is turning out to be a viable alternative to the Λ CDM cosmologicalmodel.In future work, we shall try to investigate the growth of density fluctuations andconfigurational entropy in f ( Q, T ) gravity to understand the efficiency and applica-bility in explaining the growth of cosmic structures and the accelerated expansionof the Universe in greater detail.
Acknowledgments
I thank Shantanu Desai for reading the manuscript and for the fruitful discussions.I also thank DST, New-Delhi, Government of India for the provisional INSPIREfellowship selection [Code: DST/INSPIRE/03/2019/003141].