EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Baryogenesis in f ( Q, T ) Gravity
Snehasish Bhattacharjee a,1 , P.K. Sahoo b,2 Department of Astronomy, Osmania University, Hyderabad-500007, India Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078,IndiaReceived: 12 February 2020 / Accepted: 13 March 2020
Abstract
The article communicates exploration of grav-itational baryogenesis in presence of f ( Q, T ) gravitywhere Q denote the nonmetricity and T the trace ofthe energy momentum tensor. We study various baryo-genesis interactions proportional to ˙ Q and ˙ Qf Q for the f ( Q, T ) gravity model f ( Q, T ) = αQ n +1 + β T , where α , β and n are model parameters. Additionally we re-port the viable parameter spaces for which an observa-tionally consistent baryon-to-entropy can be generated.Our results indicate that f ( Q, T ) gravity can contributesignificantly and consistently to the phenomenon of grav-itational baryognesis. Keywords f ( Q, T ) Gravity · Modified Gravity · Baryogenesis · Early Universe · Equation of State
PACS
Our universe favors matter over antimatter for somemysterious reasons. Observations from Cosmic MicrowaveBackground [1], coupled with successful predictions fromthe Big Bang Nucleosynthesis [2], recommend an over-whelming supremacy of matter over antimatter.Cosmological theories that aim at resolving this fun-damental issue falls under the domain of Baryogenesis.Theories such as GUT baryogenesis, Thermal Baryoge-nesis, Affleck-Dine Baryogenesis, Electroweak Baryoge-nesis, Black hole evaporation Baryogenesis and Spon-taneous Baryogenesis propose interactions which goesbeyond the standard model to explain this profounddominance of matter in the universe [3]. These mecha-nisms were further developed in [4] a e-mail: [email protected] b e-mail: [email protected] Gravitational Baryogenesis is one such theory proposedin [5] and further developed and extended to many mod-ified gravity theories [6]. This particular theory employsone of the Sakharov criterion [7] which assures a baryon-antibaryon asymmetry from the existence of a CP- vi-olating interaction, which reads1 M ∗ (cid:90) √ g ( ∂ i R ) J i d x (1)where M ∗ is the mass parameter of the underlying ef-fective theory, g , J i and R denote respectively the met-ric scalar, baryon current and Ricci scalar. Hence, fora flat FLRW background, the baryon to entropy ratio η B /s is proportional to time derivative of Ricci scalar˙ R . For a radiation dominated universe with EoS pa-rameter ω = p/ρ = 1 /
3, the net baryon asymmetryproduced by (1) is zero.The paper aims at investigating gravitational Baryoge-nesis through other curvature invariants and specificallythrough the nonmetricity Q . For the f ( Q, T ) gravity,the CP-violating interaction is given by1 M ∗ (cid:90) √ g ( ∂ i ( Q + T )) J i d x (2)where T denote the trace of energy momentum tensorand the nonmetricity Q is defined as [8] Q = 6 H N (3)where H ( t ) represents Hubble parameter and N ( t ) thelapse function. A remarkable difference between (1) with(2) is that the latter yields a nonzero baryon asymmetryeven for a radiation dominated universe ( ω = 1 / ∂ i Q and ∂ i f ( Q ) and compare our re-sults with cosmological observations. The paper is or-ganized as follows: In Section 2 we provide a summary a r X i v : . [ phy s i c s . g e n - ph ] M a r of f ( Q, T ) gravity and obtain the field equations. InSection 3 we explain in detail the gravitational baryo-genesis in f ( Q, T ) gravity and infer the viability of a f ( Q, T ) gravity model in producing observationally ac-ceptable baryon to entropy ratio and finally in Section4 we present our conclusions. f ( Q, T ) Gravity The action in f ( Q, T ) gravity is given as [8] S = (cid:90) (cid:20) π f ( Q, T ) + L M (cid:21) √− gd x (4)where g ≡ det ( g ij ) denote the metric scalar.Variation of action (4) with respect to metric tensorcomponents yields the field equations in f ( Q, T ) gravityas [8]8 π T ij = − √− g (cid:53) α (cid:0) f Q √− gP αij (cid:1) + f T ( T ij + Θ ij ) − f g ij + f Q (cid:16) Q αβi P αβj − P iαβ Q αβj (cid:17) (5) f i = ∂f∂i , T ij = − √− g δ ( √− g L M ) δg ij , Θ ij = g ij δ T ij δg ij (6)and P αij is called superpotential and is defined as [8] P αij = 14 (cid:104) Q α ( ij ) − Q αij + Q α g ij − δ α ( iQj ) − ˜ Q α g ij (cid:105) (7)where Q α = Q jαj , ˜ Q α = Q iαi (8)We now consider a flat FLRW spacetime of the form ds = − N ( t ) dt + a ( t ) (cid:88) i =1 , , (cid:0) dx i (cid:1) (9)where a ( t ) represent the scale factor and the lapse func-tion N ( t ) = 1 for a flat spacetime.Employing (9) in (5), we finally obtain the modifiedFriedman equations with N = 1 as [8]8 πρ = − F H + f − G G (cid:16) ˙ F H + F ˙ H (cid:17) (10)8 πp = 6 F H − f (cid:16) ˙ F H + F ˙ H (cid:17) (11)where F = f Q , ˜ G = f T π (12)Combining equations (10) and (11), we obtain the equa-tion for Hubble parameter H as˙ H + ˙ FF H = 4 πF (cid:16) G (cid:17) ( ρ + p ) (13) f ( Q, T ) Baryogenesis According to cosmological observations such as CMB[1] and BBN [2], the observed baryon to entropy ratioreads η B s (cid:39) × − (14)Sakharov reported three conditions for a net baryonasymmetry to occur through baryon number violation,C and CP violation and processes occurring outside ofthermal equilibrium [7].When the temperature T falls below a critical value T D through the evolution of the Universe, the baryon toentropy ratio can be written as [5] η B s (cid:39) − g b g ∗ s ˙ RM ∗ T D (15)where g b represent the total number of intrinsic de-grees of freedom of the baryons, g ∗ s represent the totalnumber of degrees of freedom of the massless particlesand the critical temperature T D is the temperature ofthe cosmos when all the interactions producing baryonasymmetry comes to a halt.We shall presume that a thermal equilibrium prevailswith energy density being associated with temperature T as ρ ( T ) = π g ∗ s T (16)Hence, for a CP violating interaction of (2), the result-ing baryon to entropy ratio in f ( Q, T ) gravity reads η B s (cid:39) − g b π g ∗ s ( ˙ Q + ˙ T ) M ∗ T D (17) We shall assume cosmological pressure and density obeysa barotropic equation of state of the form p = ( γ − ρ ,where γ is a constant and 1 ≤ γ ≤ ρ = f − F H π (cid:16) γ ˜ G (cid:17) . (18)For relativistic matter p = ρ/ T = 0.Thus, the baryon to entropy ratio in f ( Q, T ) gravityfor radiation dominated universe reduces to η B s (cid:39) − g b π g ∗ s ˙ QM ∗ T D (19)We shall now compute the baryon-to-entropy ratio for aCP-violating interaction proportional to the nonmetric-ity Q for two cases: First, with the Universe comprisingpredominantly of a perfect fluid with cosmological pres-sure p and matter density ρ following a barotropic equa-tion of state and second, when the Universe is filled witha perfect fluid and the cosmic dynamics is governed bythe f ( Q, T ) theory of gravity.3.1 The perfect fluid CaseFor a perfect fluid following barotropic equation of state,the Ricci scalar reads R = − πG (1 − γ − ρ (20)Thus, for a radiation dominated universe, γ = 4 / R = 0 which further implies (15) acquire a nullvalue. Nonetheless, we shall show that when the baryon-to-entropy ratio is proportional to ∂ i Q (Eq. (2)), the re-sultant baryon to entropy ratio is non-zero even for γ =4 /
3. Assuming a flat FLRW background with ( − ,+,+,+)metric signature and with the following expressions ofscale factor a ( t ) and energy density ρ ( t ) in a radiationdominated universe as a ( t ) = a t / (21) ρ = ρ a ( t ) − = ρ t − (22) the baryon to entropy ratio (19) reads η B s (cid:39) . g b πT D M ∗ ρ (cid:113) ρ g ∗ s (23)where we have used ˙ Q = 12 H ˙ H and the decouplingtime t D is written in terms of critical temperature T D by equating (16) with (22) as t D (cid:39) (cid:114) ρ π g ∗ s (cid:18) T D (cid:19) (24)Substituting g ∗ s = 106, g b ∼ ρ = 3 × GeV , T D = 2 × GeV and M ∗ = 2 × GeV , the resultantbaryon to entropy ratio reads η B s (cid:39) . × − (25)which is close to the observational value (14). Thus, theproblem of baryogenesis can be resolved in Einstein’sgravity if the CP-violating interactions are made pro-portional to the nonmetricity Q instead of R .3.2 The perfect fluid with f ( Q, T ) gravity caseWe shall now compute baryon to entropy ratio for thecase when the Universe is filled with a perfect fluid andthe evolution of the Universe is governed by the f ( Q, T )theory of gravity.We consider the f ( Q, T ) functional form to be [8] f ( Q, T ) = αQ n +1 + β T (26)where α , n and β are model parameters. Substituting(26) in Eqs. (10) and (11), the expression of Hubbleparameter H ( t ) and density ρ ( t ) for this model reads[8] H ( t ) = H ( n + 1) [16 π − β ( γ − γ ( β + 8 π ) H ( t − t ) − ( n + 1) [ βγ − β + 4 π )](27) ρ ( t ) = α ( n +1) (2 n + 1) H ( t ) n +1) β ( γ − − π (28)Equating (16) and (28), the coupling time t D can bewritten as t D = t + (1 + n ) (cid:18) − H + 5 / (2+2 n ) π − / (1+ n ) (cid:16) − (2+ n ) g ∗ s T D (16 π − β ( γ − α +2 nα (cid:17) − / (2+2 n ) (cid:19) [16 π − β ( γ − β + 8 π ) γ (29) where H ( t ) is the present value of the Hubble parameter.Time derivative of Hubble parameter (27) reads˙ H = − (cid:20) H (1 + n )( β + 8 π )[16 π − β ( γ − γ (1 + n )[16 π − β ( γ − H ( t − t )( β + 8 π ) γ (cid:21) (30)Substituting (27), (29) and (30) in (19), the baryon to entropy ratio for γ = 4 / η B s (cid:39) × [1 − (3 / n ))] g b π [2+(3 / (1+ n ))] ( β + 8 π ) (cid:16) ( − − n ) g ∗ s T D [16 π + β ] α (1+2 n ) (cid:17) [3 / (2(1+ n ))] g ∗ s M ∗ (1 + n ) T D [16 π + β ] (31)By choosing M ∗ , g ∗ s , g b , T D as before, α = 10 − , β = − . π and n = 2 .
12 the resultant baryon to en-tropy ratio reads ∼ . × − which is in excellentagreement with observations. In Fig. (1a), (1b) and (1c)we show η B /s as a function of α , β and n respectively. × - × - × - × - × - × - × - × - × - α η B S (a) - - - - - - - × - - × - - × - × - × - × - β η B S (b) - × - × - × - × - × - × - × - n η B S (c) Fig. 1: (1a) shows η B /s as a function of α , (1b) shows η B /s as a function of β and (1c) shows η B /s as afunction of n . We choose g ∗ s = 106, g b ∼ T D (cid:39) M ∗ = 2 × GeV . Interestingly, in Fig. (1b), the baryon to entropy ra-tio becomes negative for β (cid:46) − π which is unphysicalas it implies an overabundance of antimatter over or-dinary matter. Also note that for n (cid:38) . η B /s = 0which indicate no asymmetry between antimatter andmatter and therefore not acceptable. We shall now define a more complete and generalizedbaryogenesis interaction proportional to ∂ i f ( Q, T ). TheCP-violating interaction then reads1 M ∗ (cid:90) √ g ( ∂ i f ( Q, T )) J i d x (32) For (32), the resulting baryon to entropy ratio reads η B s (cid:39) − g b π g ∗ s ( ˙ Qf Q + ˙ T f T ) M ∗ T D (33)As discussed in the previous section that for a radiationdominated universe T = 0, we finally obtain η B s (cid:39) − g b π g ∗ s ˙ Qf Q M ∗ T D (34)Substituting (27), (29) and (30) in (34), the baryonto entropy ratio for γ = 4 / η B s (cid:39) (cid:34) (2 n ) (3+2 n ) [1 − (3 / n ))] g b π [2+(3 / (1+ n ))] α ( β + 8 π ) A [3 / (2(1+ n ))] (cid:0) [ − / n )] π [1 / (1+ n )] A (cid:1) (2 n ) g ∗ s M ∗ T D [16 π + β β ] (cid:35) (35)where A = (cid:32) − (2+ n ) g ∗ s T D [16 π + β ] α (1 + 2 n ) (cid:33) (36)Substituting M ∗ , g ∗ s , g b , T D as before, α = 0 . β = 8 . π and n = − . η B /s ∼ . × − which is very close toobservational constraints. In Fig. (2a), (2b) and (2c) weshow η B /s for the generalized baryogenesis interactionas a function of α , β and n respectively.Thus, the problem of baryogenesis can be resolved in f ( Q, T ) gravity if the CP-violating interactions are madeproportional to the nonmetricity Q instead of R . × - × - × - × - × - α η B S (a) × - × - × - × - × - β η B S (b) - - - - - - × - × - × - × - × - × - n η B S (c) Fig. 2: (2a) shows η B /s as a function of α , (2b) shows η B /s as a function of β and (2c) shows η B /s as afunction of n . We choose g ∗ s = 106, g b ∼ T D (cid:39) M ∗ = 2 × GeV . The article presented a thorough investigation of grav-itational baryogenesis interactions in the framework of f ( Q, T ) gravity where Q denote the nonmetricity and T the trace of the energy momentum tensor. For this typeof modified gravity we find the baryon-to-entropy ratioto be proportional to ˙ Q , since for a radiation dominateduniverse T = 0. We ascertained the baryon-to-entropyratio proportional to nonmetricity Q two different sce-narios, First, with the Universe comprising predomi-nantly of a perfect fluid with cosmological pressure p and matter density ρ following a barotropic equationof state and second, when the Universe is filled with aperfect fluid and the cosmic dynamics is governed bythe f ( Q, T ) theory of gravity. We choose the functionalform of f ( Q, T ) gravity to be f ( Q, T ) = αQ n +1 + β T ,where α , β and n are model parameters. For the perfectfluid case, the obtained baryon-to-entropy ratio η B /s (cid:39) . × − while for the f ( Q, T ) gravity model we ob-tained η B /s ∼ . × − , both of which are in excel-lent agreement with observational value of (cid:39) × − .Next, for the f ( Q, T ) gravity model, we explored a morecomplete and generalized baryogenesis interaction pro-portional to ˙ Qf Q . For this baryogenesis interaction, wefound the baryon-to-entropy ratio η B /s ∼ . × − which is very close to the observational value. Acknowledgments:
We are very much grateful tothe honorable referee and the editor for the illuminatingsuggestions that have significantly improved our workin terms of research quality and presentation. SB thanksBiswajit Pandey for helpful discussions. PKS acknowl- edges CSIR, New Delhi, India for financial support tocarry out the Research project [No.03(1454)/19/EMR-II Dt.02/08/2019].
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