arXiv:2010.04807v1 [gr-qc] 9 Oct 2020 h atta tdmnsa xenlaeti re to order to in agent due external whole a an as demands standard universe it the The that to fact apply the not [6–9]. how- does mechanics, background ever, quantum of the quan- interpretation to in Copenhagen due effects occur bounces tum non-singular models, space a yielding case this phase, in preceding phase, universe. a contracting bouncing be- a be not must be can there does which singularity, expansion the If at ini- eliminated. gin Big-Bang is hot the singularity and tial scales. [1–5], approaches framework, small many this at under in emerge important universes beyond non-singular become Indeed, effects which quantum of GR, occurrence classical can the singularity by avoided the overcome be where be mechanics, can a quantum problem represents invoking this singularity However, initial problem. its major and (GR), Relativity ¶ ∗ § ‡ † [email protected] [email protected] [email protected] marcusbomfi[email protected] [email protected] ntefaeoko unu omlg nminisuper- in cosmology quantum of framework the In General in based is model cosmological standard The aygnssi omlgclmdl ihsmercadasy and symmetric with models cosmological in Baryogenesis BF-Cnr rsliod eqia ´scs airSiga F´ısicas, Xavier Pesquisas de Brasileiro Centro - CBPF BF-Cnr rsliod eqia ´scs airSiga F´ısicas, Xavier Pesquisas de Brasileiro Centro - CBPF pcr fcsooia etrain.Hne elsi b realistic Hence, perturbations. baryon constraint cosmological baryon-to-entrop observational constrain of observed other the spectra with the that compatible yield shown values is bounc the to It many imposed of bounce. context scenarios, the the around in nam with asymmetric investigated curvature, one are space-time mechanisms with current, Phy two other number Particle the baryon of and the Model Baryogenesis, involving Standard mechanism the couplings Two of radiationnew cosmology. extensions intense in as of puzzle proposed absence unsolved an the constitutes Background and Microwave Nucleosynthesis, Cosmic the dial from data observational by © h baryon The .INTRODUCTION I. 00Aeia hsclSceyDOI: Society Physical American 2020 A audd eTcooi,UR nvriaed Estado do Universidade - UERJ Tecnologia, de Faculdade - FAT − niaynaymtyi n ftetomcaim rpsdt proposed mechanisms two the of one if asymmetry antibaryon − niaynaymty(xeso atroe niatri o in antimatter over matter of (excess asymmetry antibaryon vnd ennoFrai54 i 97-1,Vtoi,Bra Vit´oria, 29075-910, zip 514, Ferrari Fernando Avenida o.Pe.Dta m28 i 73-0,Rsne Brazil. Resende, 27537-000, zip 298, km Dutra, Pres. Rod. PCso C nvriaeFdrld Esp´ırito Santo, do Federal Universidade - CCE PPGCosmo, .B Jesus, B. M. Dtd coe 3 2020) 13, October (Dated: † .C .Delgado M. C. P. .Pinto-Neto, N. .S Vicente S. G. bounces 10.1103/PhysRevD.102.063529 utepnigpae h rsneo h utfluid, obtain to dust order the in important of is matter, presence dark The be can which and phase. phase expanding expanding dust dust radiation bounce radiation, phase, evolution: by contracting following dominated radiation the phase, the has contracting where by which described case fluids, [15], the dust is perfect and example radiation are An contents matter 14]. [3, symmetric be expansion cosmology. and standard contraction the ef- by The of quantum described by phases are recovered. governed the is is whereas bounce limit fects, the classical around expan- the Friedmann physics when usual factor expan- the phase, scale of as sion phase recognize the a we until by which followed sion, last value, minimum which a reaches phase, contracting a [13]. many equation in Wheeler-DeWitt equation, the cosmology being quantum approaches a of solution tion hr h nvrei ecie yteqatmevolu- quantum factor, the scale by cosmology, the described of to is tion applied universe be the can where interpretation universe trajectories This (Bohmian) the real collapse that exist. and a so system theory, deterministic necessary, a this longer is In no is agent. inde- external assumption 12], de an [11, The of others among amplitude. pendent real- is, probability [10] a physical theory of a Broglie-Bohm collapse establish the from and ity measurement, a perform h vlto rmcnrcigt xadn hs can phase expanding to contracting from evolution The contains bounce quantum (dBB) Broglie-Bohm de The ds.10 i 29-8,Rod aer,Brazil. Janeiro, de Rio 22290-180, zip 150, st. ud ds.10 i 29-8,Rod aer,Brazil. Janeiro, de Rio 22290-180, zip 150, st. ud ‡ ¶ n .Mour˜ao T. and oigfo h etrso h power the of features the from coming s ucn oescnyedteobserved the yield can models ouncing ∗ ai,aeml,araycontaining already mild, are ratio, y dGaiainlBroeei.These Baryogenesis. Gravitational ed rmmte-niatrannihilation, matter-antimatter from isa iheege.Te eyon rely They energies. high at sics o aynaymtyhv been have asymmetry baryon for s nstois rdcin fprimor- of predictions anisotropies, so h reprmtr fthese of parameters free the on ts n cnro,ete ymti or symmetric either scenarios, ing clrfil,cle Spontaneous called field, scalar a kspaei nature. in place akes oRod Janeiro, de Rio do § rUies) indicated Universe), ur a zil. ( t mti quantum mmetric ,drvdfo aefunc- wave a from derived ), 2 an almost scale invariant spectrum of scalar cosmological the latter mechanism, which can naturally occur in an ef- perturbations. fective theory of gravity. Introducing a coupling between A more involved bounce dynamics is given in Ref [16], the derivative of the Ricci scalar and the baryon number where a single scalar field with exponential potential current, this interaction gives opposite signs for energy drives the bounce as a stiff matter fluid, behaves as a contributions to particles and antiparticles, also violat- dust fluid in the asymptotic past and future, and also ing CPT symmetry. This induces changes in the thermal presents a transient dark energy-type behavior occurring equilibrium distributions which result in a nonzero net only in the future of the expanding phase. This bounce baryon number. These two mechanisms are similar: the is asymmetric because the transient dark energy epoch new interaction terms violate CP and is CPT conserving occurs only in the expanding phase, not in the contract- in vacuo, but both dynamically break CPT in an expand- ing phase, avoiding problems related to the imposition of ing universe, where, in the Gravitational Baryogenesis vacuum state initial conditions in the contracting phase case, the curvature varies in time, or, in the Spontaneous if dark energy is present there, and overproduction of Baryogenesis case, where the scalar field, not being in gravitational waves, which are typical in bouncing mod- its vacuum state, drives the cosmological evolution, and els containing a canonical scalar field. Other asymmetric hence evolves in time. It is important to point out that bounces where obtained in Ref. [17], with either unitary a bounce solution is important for this mechanism to be and non-unitary evolution1. One particular interesting effective. From Ref. [33] in the context of Loop Quan- result was one solution describing an expanding cosmo- tum Cosmology, one can notice from its Eq. (14) that logical universe arising from an almost flat space-time. a radiation dominated universe cannot produce baryon This type of asymmetry is particularly relevant because asymmetry in the Einstein-Hilbert case (critical density these solutions may be used to account for non-negligible ρc ), whereas for a bouncing universe (finite ρc), it back-reaction due to quantum particle production around becomes→ ∞ possible. the bounce (see Refs. [20, 21]), which is important for the In this paper we present the baryogenesis scenario in study of baryogenesis. the context of dBB quantum cosmology, in both sym- The baryon antibaryon asymmetry (excess of matter metric and asymmetric non-unitary and unitary realiza- − over antimatter in our Universe) indicated by observa- tions, based on the results of Refs. [16, 17]. We consider tional data from Cosmic Microwave Background [22], both Spontaneous and Gravitational Baryogenesis mech- predictions of Big-Bang Nucleosynthesis [23], and the ab- anisms, which we call Baryogenesis with Scalar Coupling sence of intense radiation from matter-antimatter anni- and Baryogenesis with Curvature Coupling, respectively. hilation [24] constitutes an unsolved puzzle in cosmol- +0.6 The paper is outlined as follows. In Sec. II, we present ogy. The baryon-to-entropy ratio is nB/s = 9.2 0.4 the main aspects of standard cosmological baryogenesis, 11 − × 10− . The current view is based on the Sakharov con- which are subjected to the Sakharov conditions. The ditions [25], which should hold during the early hot Uni- mechanisms of spontaneous and gravitational baryoge- verse, yielding a net baryon asymmetry, and cease to nesis are introduced, stressing the fact that the third be satisfied as the Universe expands and cools. An Sakharov condition can be overcome in these frameworks. important theory on this subject is electroweak baryo- We work these conditions in detail for a hypothetical de- genesis [26–28], which satisfies all Sakharov conditions. cay, and we show that baryon asymmetry can take place However, it is unable to yield sufficient baryon asym- in thermal equilibrium. In Sec. III, we introduce the metry within the Standard Model of Particle Physics background bouncing models, i.e., the mini-superspace (SMPP). In order to solve this issue one needs to explore models in the dBB theory. Firstly, the standard symmet- physics beyond the SMPP, like in Ref. [29]. Another rele- ric quantum bouncing trajectories are obtained from ini- vant mechanism is the so called Spontaneous Baryogene- tial static Gaussian wave functions centered at the origin, sis [30, 31], which is based in the coupling of a scalar field i.e., without phase velocity. Secondly, we present asym- to the baryon number current. The main point is that metric quantum bounce trajectories for a non-unitary baryon asymmetry is generated while baryon violating wave function from an initial Gaussian wave function interactions are still in thermal equilibrium, which is not with nonzero phase velocity and for an unitary wave in contradiction with the Sakharov conditions because function from a superposition of Gaussian wave func- the scalar field coupling in a expanding universe violates tions multiplied by factors of the form exp(ip2χ2). These CPT invariance. A third mechanism, which is termed new quantum parameters are responsible for the asym- Gravitational Baryogenesis [32], is a natural extension of metry. In Secs. IV and V, we analyse the gravitational and spontaneous baryogenesis mechanisms for the mod- els presented in Sec. III, and the one presented in Ref [16]. Some analytical results are obtained for the baryon-to- 1 In dBB quantum cosmology, it is not necessary to impose a entropy ratio, and constraints on the physical parameters probabilistic interpretation to the wave function of the whole of the theory are obtained. In the Conclusion, section VI, system. Only for the so called conditional wave functions, which apply to sub-systems of the whole system, and satisfy an effective we summarize and comment the results, and discuss fu- Schr¨odinger equation with unitary evolution, does a probability ture perspectives. notion emerges. See Ref. [18, 19] for details. 3
II. COSMOLOGICAL BARYOGENESIS the CP transformation implies that CP [Γ(X q q )] = Γ(X¯ q¯ q¯ ), (9) Proposals of baryogenesis mechanisms are tradition- → L L → R R CP [Γ(X q q )] = Γ(X¯ q¯ q¯ ). (10) ally concerned with satisfying the three Sakharov’s con- → R R → L L ditions [25], which are: A) violation of the baryon number If we assume that CP is a symmetry of this decay, we B, B) violation of C and CP and C) thermal equilibrium get deviation. The understanding of these conditions is easily per- CP [Γ(X qLqL)] = Γ(X qLqL), (11) → → ceived by analyzing a hypothetical decay. Suppose that CP [Γ(X qRqR)] = Γ(X qRqR), (12) a particle X decays only in two channels, which produces → → the baryon numbers B1 and B2, where the respective de- so it is clear that cay rates are Γ(X q1q1) and Γ(X q2q2). Then, the Γ(X qLqL)+Γ(X qRqR) X total decay rate→ is of the form: → r r¯ = → → − ΓX ¯ ¯ Γ = Γ(X q q )+Γ(X q q ). (1) Γ(X q¯Lq¯L)+Γ(X q¯Rq¯R) X → 1 1 → 2 2 → → − ΓX Therefore, the probability that X will decay on the chan- =0, (13) nel producing the number B1 is given by: leading that the difference in the amount of baryons also r = Γ(X q q )/Γ , (2) → 1 1 X to be null in Eq. (3). Finally, to explain the third Sakharov’s criterion, it is where the channel associated with baryon number B2 has a complementary probability of occurrence, i.e., 1 r. enough to calculate the average of the baryon number ˆ The decay of the antiparticle of X, X¯, yields the− bary- operator B in thermal equilibrium at a temperature T = 1/β, which reads: onic numbers B¯1 = B1 and B¯2 = B2, which, in turn, have probabilities− respectively given− byr ¯ andr ¯ 1. βH − Bˆ = T r e− Bˆ Therefore, the baryon number difference is simply h iT h i 1 βH ∆B = rB + (1 r)B r¯B¯ (1 r¯)B¯ = T r (CPT )(CPT )− e− Bˆ 1 − 2 − 1 − − 2 h i = T r e βH (CPT ) 1Bˆ(CPT ) = (r r¯)(B B ). (3) − − − 1 − 2 h βH i = T r e− Bˆ The total baryon number variation requires that B1 and − B2 to be different baryon numbers, hence, the necessity of = Bˆ h . i (14) condition A. However, the decay probabilities of particles −h iT and their antiparticles should also be different, leading to We have considered the fact that the Hamiltonian H condition B, as we will now see. commutes with the CPT operator. Thus, in thermal The CPT symmetry imposes that the total particle equilibrium, there is no mean baryon generation, i.e., decay rates and their associated antiparticles to be equal: Bˆ = 0. h iT ΓX =ΓX¯ . When we inspect a simple decay channel, such Concluding, the usual approaches to baryogenesis rely as Γ(X qq), the necessity of C violation in this context → on finding situations in the Universe where the three becomes clear. Indeed, Sakharov conditions are satisfied. The spontaneous and Γ(X qq) gravitational baryogenesis scenarios [30–32], however, r = → , (4) take another route. The new couplings with the baryon Γ X current they propose, either with a scalar field or the cur- Γ(X¯ q¯q¯) r¯ = → . (5) vature of space-time, lead to a violation of CPT invari- ΓX ance in a time dependent spacetime, as the Friedmann Therefore, if charge conjugation is a valid symmetry, model. Hence, baryon generation takes place in ther- mal equilibrium, given that the Sakharov first condition r = Cr =r ¯ r r¯ =0, (6) holds. ⇒ − In this case, in high temperatures, where we expect hence C must be violated. Suppose now that CP symme- the mechanisms which do not conserve baryon number try, which consists of charge and parity transformations, take place, the difference between the number density of is valid, even if C is violated. Hence, for the hypothetical baryons and antibaryons reads (see, e.g. Ref. [34]) probability decay channel, 3 gBT µB Γ(X qLqL)+Γ(X qRqR) n n ¯ = , (15) r = → → , (7) B − B 3 T ΓX Γ(X¯ q¯ q¯ )+Γ(X¯ q¯ q¯ ) where gB is the number of degrees of freedom of baryons, r¯ = → L L → R R , (8) ΓX and µB is the chemical potential associated with the 4 baryon number density through the new coupling with Ψ is the wave function of the universe, a is the scale the baryon current proposed in these two scenarios. The factor, and T is a parameter related to the perfect fluid, total entropy density, in turn, is given by [34], which plays the role of time through dt = a3ωdT . Note that for ω =1/3, the parameter T becomes equal to the 4π2g T 3 conformal time η. s = ∗ , (16) 45 Imposing the Gaussian initial wave function at T =0 where g is the total multiplicity of relativistic degrees of 1 8 4 χ2 freedom,∗ containing all degrees of freedom of bosons, g , Ψ (χ)= exp , (21) b 0 σ2π −σ2 and fermions, gf , which has the form: 7 and implementing an unitary evolution, we obtain the g = gb,i + gf,j . (17) wave function solution ∗ 8 i j 1 X X 8σ2 4 σ2χ2 Ψ(χ,T )= exp The baryon-to-entropy ratio is then given by π(σ4 + T 2) −σ4 + T 2 Tχ2 1 σ2 π nB 15gBµB = . (18) exp i 4 2 + arctan . (22) s 4π2g T × − σ + T 2 T − 4 ∗ The mechanisms of gravitational and spontaneous Using the guidance equation of the dBB formalism, baryogenesis will be discussed in Sections IV and V, sep- dχ 1 ∂S arately. They lead to different µ , but the whole calcu- = , (23) B dT −2 ∂χ lation also depends on the particular time evolution of the background Friedmann model which is being consid- where S is the phase of the wave function, the correspon- ered. In the next section, we describe the models we will dent Bohmian trajectory for the scale factor a reads investigate. 1 −ω T 2 3(1 ) a(T )= a 1+ , (24) b σ2 III. THE BACKGROUND BOUNCING MODELS " #
We consider cosmological models that arise through where ab is the value of the scale factor a at the mo- the Wheeler-DeWitt quantization of the background, ment of the bounce T = 0. The expression (24) describes taking into account the dBB interpretation of Quantum a symmetric bounce, which corresponds to the classical Mechanics [10]. The latter solves the measurement prob- solution for large values of T , and is plotted in Fig. 1. lem through an effective collapse of the wave function, The value considered for the equation of state parame- which is a consequence of the deterministic character of ter, ω =1/3, represents an universe filled with radiation this interpretation. Thus, an external classical domain fluid, as we expect for early times. is not required in order to describe the measurement a process, and we are able to quantize the entire universe 10
2 [3, 19]. The procedure of quantization and the bounce ab=1, ¡ =1
2
solutions are detailed in [17], and we will only mention 8 ab=2, =1 the most relevant results for the present work. The other 2
a =1, ¢ =2 bounce solution given in Ref. [16] will be discussed in b section V. 6 The Wheeler-DeWitt equation to be satisfied for a flat, homogeneous and isotropic universe filled with a perfect 4 fluid with equation of state P = ωρ, where P is the pressure, ρ the energy density and ω the equation of state parameter, for a particular choice of ordering in a (in 2 which the Wheeler-DeWitt equation is covariant under redefinitions of the scale factor) reads T -4 -2 2 4 ∂Ψ(χ,T ) 1 ∂2Ψ(χ,T ) FIG. 1. a vs. T for ω = 1/3. The curves are obtained for i = 2 , (19) ∂T 4 ∂χ some representative values of ab and σ. where In order to relate the wave function parameters to ob- 2 3(1−ω) servables, we obtain the Hubble parameter H =a/a ˙ , χ = a 2 , (20) 3(1 ω) where dot denotes derivative with respect to the physical − 5
a cosmic time. For large values of T, the squared Hubble 10 parameter reads p=0.5
a2 a4 p=1.0 8 H2 = b = H2Ω 0 , (25) a4σ4 0 r0 a4 p=0 6 where the subscript 0 in all quantities indicates their cur- rent values. We then identify the dimensionless density 4 parameter for radiation today as
2 2 ab Ωr0 = 4 2 4 . (26) a0H0 σ T -10 -5 5 10 Performing the change of variables given by xb = a0/ab andσ ¯ = σ√a0H0, we obtain 1 FIG. 2. a+ vs T for σ = 1, ab = 1, ω = 3 .
1 a σ¯2 = . (27) 10 xb√Ωr0 p=1.0
The curvature scale at the bounce is given by 8 p=3.0 p=0 1 1 Lb = = , (28) 6 √ x2H √6Ω R T =0 b 0 r0
where R is the Ricci scalar. It allows us to find lower 4 and upper bounds to xb by requiring that the curva- ture scale at the bounce is some few orders of magnitude 2 larger than the Planck scale (in order to ensure that the Wheeler-DeWitt equation is a valid approximation of a T more fundamental theory of quantum gravity [35]), and -10 -5 5 10 smaller than the nucleosyntesis scale. As a result, we 1 have FIG. 3. a− vs T for σ = 1, ab = 1, ω = 3 .
1011 x < 1031. (29) ≪ b In this case, we perform the following change of vari- ables: We will also consider asymmetric solutions arising from a0 the following initial wave function xb = , (32) ab χ2 σ¯ = σ a0H0, (33) Ψ0(χ) = exp 2 ipχ . (30) p −σ ∓ p¯ = p2 , (34) a0H0 T Implementing a non-unitary evolution, which is detailed η¯ = , (35) in [17], we obtain the resulting trajectory for the scale σ2 factor a, which is given by x p¯σ¯2 y2 = b , (36) 2 2 3p(1 ω) 3(1−ω) T which leads to a squared Hubble parameter in the ex- a (T )= − T + a 2 1+ ± ± 4 b σ2 panding phase given by ( " 1 2 2 2 2 3(1−ω) 3p(1 ω) (T 2 + σ4) 2 4 2 2 2 + , (31) y + 1+ y ab H0 a0 − 3(1 ω) 2 ± 4 a − # ) H = . (37) b p σ¯4a4 where ab is the scale factor at the moment of the bounce Thus the dimensionless density parameter of radiation pσ4 today reads Tb = 2χb and χb is related to ab through Eq. (20). The ∓ 2 solutions a+ and a in Eq. (31) are plotted in Figs. 2 − y2 + 1+ y4 and 3, respectively, for ω = 1/3. The classical solution ± Ωr0 = 4 2 , (38) also arises for large values of T for both cases. σ¯pxb 6 while the wave function parameterσ ¯ reads a 4
1/2 p1=3.5, p2=1.0 2 2 − σ¯ = xb Ωr0(1 p/¯ Ωr0) . (39) ∓ p1=5.5, p2=1.0 h i 3 p p1=1.0, p2=3.5 As a consequence, for the solution a+, the relationp ¯ < √Ωr0 must be satisfied. As argued in [17], this solution p1=1.0, p2=5.5 is of special interest, once it can represent a bounce so- 2 p1=0, p2=0 lution with an almost Minkowski contracting phase asp ¯ p1=1.0, p2=1.0 approaches √Ωr0. For this asymmetric solution, the minimum curva- 1 ture scale does not occur at the bounce, but atη ¯min = √1+y4 1 − . It reads ∓ 2 q 1 T L = -2 -1 1 2 min √ R η¯min 1 3 FIG. 4. a vs T for σ = 1.0, ai = 1.0, Ti = 1.0 ω = 3 . R 1+ 1 p¯ H0 √Ωr = ∓ 0 . (40) q 2 2 p¯ p¯ 8√3Ωr0xb 1 2 ∓ √Ωr0 ∓ √Ωr0 allowing us to identify the dimensionless density param- r eter for radiation today as Note that, for the asymmetric solutions, the presence of extra parameter(s) related to asymmetry makes the phys- a2 Ω = i . (45) ical bounds to be on Lmin instead of xb. They are given r0 a4H2(T 2 + σ4) by 0 0 i Replacing the initial values T and a = a (T ) by T and 58 Lmin 20 i i i i b 10− < 10− . (41) 2 2 4 ≪ R ab = a(Tb), where Tb = (p1 p2)σ /2, and performing H0 the following change of variables− Another asymmetric solution is obtained by consider- ing the following initial wave function a0 xb = , (46) ab 2 χ 2 2 Ψ (χ)= C exp + ip χ σ¯ = σ a0H0, (47) 0 −σ2 1 p2 2 2 pi χ 2 2 p¯i = , (48) + exp ip χ , (42) a0H0 −σ2 − 2 2 where where i =1, 2, we obtain the parameterσ ¯ as
1 √ − 2 1/2 2 2 2 2 2 2 2 2 − C = 1 i(p1 + p2)+ 2 2 xb Ωr0 (¯p1 p¯2) π 4 − σ σ¯ = 1+ 1+ 2− . (49) 2 x Ωr0 1 1/2 ( " s b #) 2 − 2 − + i(p2 + p2)+ + √2σ . (43) 1 2 σ2 Note that Tb appears in Ωr0 squared, thus p1σ and p σ appear in fourth order. Disregarding these terms, Implementing an unitary evolution and following the 2 Eq. (49) reduces to Eq. (27) of the symmetric case. dBB procedure, we obtain a differential equation, which is shown in [17] and can be solved numerically with initial Since we are considering a limit in which the param- eters related to asymmetry are small, the difference be- condition ai = a(Ti). The numerical solutions are plot- ted in Fig. 4. The classical limit arises for large values tween the curvature scale at the bounce Lb and the mini- of T . This solution also encompasses multiple bounces. mum curvature scale Lmin is not relevant. The expression However, for our purpose in this work, we consider only for Lb is given by Eq. (28), since p1σ and p2σ appear in the single bounce solutions. fourth order, and are disregarded. As a consequence, the lower and upper bounds For the limit where p1σ 1 and p2σ 1, it is possible to relate the wave function≪ parameters≪ to observables. The squared Hubble parameter reads 58 Lb 20 10− < 10− , (50) ≪ RH0 2 2 ai H = 4 2 4 , (44) a (Ti + σ ) reduce to Eq. (29). 7
2 IV. BARYOGENESIS WITH CURVATURE whereη ¯ η/σ is a dimensionless conformal time, ′ ≡ ≡ COUPLING d/dη¯, and A(¯η) a(¯η)/ab. The derivative of Ricci scalar, Eq. (55), in cosmic≡ time as a function of the conformal In recent decades, many suggestions have been made time then yields, regarding the production of baryonic matter in the early ′′′ Universe. In this section, we explore the gravitational ˙ 6 A(¯η)A (¯η) 3A′(¯η)A′′(¯η) R(¯η)= 3 6 −5 . (56) baryogenesis proposal of Ref. [32], implemented through ab σ A (¯η) a coupling term between the derivative of the Ricci cur- µ vature scalar, ∂µR, and the baryonic current, J , in the The next step is to obtain a relation of the typeη ¯ = cases of symmetric and asymmetric bounces of section η¯(T ) far from the bounce, which is necessary in order to III. We assume the coupling term to be a CP-violating evaluate nB/s at T = TD. This can be accomplished interaction with the form, as proposed in Ref. [32], by evaluating the relations t = t(T ) and t = t(¯η), both far from the bounce. Combining theses relations, one 1 4 √ µ obtainsη ¯ =η ¯(T ). To obtain the first relation, we use 2 d x g(∂µR)J , (51) M − the fact that, for reasonable values of TD, we expect the ∗ Z universe to still be dominated by radiation, and we can where M is the cutoff energy scale of the effective theory. write the energy density as both This term,∗ in an expanding universe, also dynamically breaks CPT, and favors a net asymmetry towards the 2 3Mp production of baryons over antibaryons. This happens ρ(t)= 2 , (57) 0 32πt because the J (= nB) term has a different sign for mat- ter versus antimatter, and as we will demonstrate here, and it can be used to calculate the net asymmetry of matter and antimatter once the universe reaches a decoupling g π2T 4 ρ(T )= ∗ , (58) temperature for this effective theory. Then, if we also 30 assume that the characteristic timescale τ of the interac- tion runs faster than the expansion rate of the universe, where Mp is the Planck mass. From the equivalence be- that is, tween Eqs. (57) and (58), we can get t(T ), which results in 1 τ < A. Symmetric bounce For the symmetric case, we consider the bounce gen- erated by the initial Gaussian wave function given by Eq. (21), which produces the scale factor given by Eq. (24). For a radiation dominated bounce (ω = 1/3), in conformal dimensionless time this reads: 2 a(¯η)= ab 1+¯η , (63) whereη ¯ = η/σ2. Far from thep bounce, we can relate the cosmic time t withη ¯ as: a σ2η¯2 t(¯η)= b , (64) 2 1/2 where, from Eq. (27), σ =σ ¯(a0H0)− can be related to the Hubble radius RH0 =1/H0 and the radiation density today Ωr0 as R 2 H0 FIG. 5. Parameter space of xB, TD, M∗ that that gives σ = . (65) −11 a x2√Ω nB /s 9 10 . These are parameterized by X = log(xb), b b r0 ≈ ∗ D = log T¯D and M = log M¯ ∗, respectively. 61 Considering the values H0 = 1.22 10− Mp, Mp = 1.22 1022 MeV, from Eqs. (59) and× (64) one obtains × B. Asymmetric bounce 10 1.0 10− x η¯(T¯)= × b . (66) T¯ Let us now consider the case of asymmetric bounces, and look for the effects of asymmetry. The calculations For the the symmetric scale factor given by Eq. (63), are similar to those of the symmetric case. We also using Eq. (66) in Eq. (54), the result for nB/s far from address the role of unitarity in the quantum evolution. the bounce (¯η 1) reads: ≫ In the following, we choose two different cases, a non- unitary and a unitary asymmetric bounce. n T¯7 B =6.2 10 86 D , (67) − 2 ¯ 2 s × xb M ∗ 1. Non-Unitary Asymmetric Bounce 5 where we used Ω =8 10− . Finally, n /s is given in r0 × B terms of the parameters xb, T¯D and M¯ . ∗ For the non-unitary asymmetric case, we consider the One must notice that Eq. (66) was obtained through bounce generated by the initial Gaussian wave function an expansion assuming thatη ¯ 1, which imposes the 2 ≫ given by Eq. (30), which produces the two classes of scale condition : factors, given by Eq. (31). For a radiation dominated xb bounce in conformal dimensionless time, this reads: & 1.0 1011. (68) T¯ × 2 4 2 a (¯η)= ab y η¯ + 1+ y 1+¯η , (69) Hence, this inequality must be considered together with ± ± 2 2 p 2 p conditions (60) - (62) when we look for the regions of whereη ¯ = T/σ , y = xb p¯σ¯ /2,σ ¯ = σ√a0 H0 and 2 interest in the parameter space. p¯ = p/(a H0), assuming p > 0, are dimensionless vari- ¯ 0 In Figure 5, we present a region plot for TD xB and ables and the signals account for two possible bounce ¯ × ± 2 lines for constant values of M . The region of parame- solutions. The bounce occurs atη ¯b = y . 11∗ ∓ ters that give nB/s 9 10− are the values of the gray As before, using the classical Friedmann equations in ≈ ∗ region that are crossed by the constant M¯ lines. In the the expanding era far from the bounce, when quantum ∗ following, we present analogous results for the asymmet- effects are negligible, we can obtain the relevant relations ric bounce cases, and we compare the results. for our purpose. The scale factor, Eq. (69), results: a η¯ a(¯η)= b , (η 1). (70) 1 p¯ ≫ ∓ √Ωr0 2 To apply the conditionη ¯ ≫ 1, we assumedη ¯ > 10 as sufficient 2 q because we expand terms 1 +η ¯ . We use the same assumption The radiation density parameter Ωr0 is given by Eq. (38), for the asymmetric cases in the following sections. which is used in order to expressσ ¯ in the form of Eq. 9 (39). Also, due to the asymmetry of the scale factor, we When (1) = 1, L¯ reduces to the symmetric result (see can define a density parameter for the contracting phase, Ref. [20]C for details.) Ωcr (the ratio between the radiation energy density and In terms of L¯, T¯D, M¯ and λ, the baryon-to-entropy the critical density when the Hubble parameter at con- ratio results ∗ traction has the same value as today), in terms of Ωr0, 7 nB 87 TD which reads: =1.4 10− (λ), (79) ¯ 2 ¯ 2 s × M L H p¯ ∗ Ωcr = 1 Ωr0. (71) ∓ √Ω where r0 For simplicity, we define the following parameter: 2√2(1 + λ2)3/2 1+ λ2 (λ)= 3 = 2 . (80) p¯ H (1 + λ) 2λ (λ) λ = 1 , (72) C ∓ √Ω r r0 When (1) = 1 and, using Eq. (77) in Eq. (79), we recoverH the symmetric result of Eq. (67). observing the restrictionp< ¯ √Ω for the ( ) (a ) case. r0 + Now we look for the region in parameter space which In terms of this parameter, Eqs. (70) - (71) can− be rewrit- gives n /s 9 10 11. Defining S = log L¯, D = log T¯ ten as: B − D and M = log≈M¯×, from Eq. (79) one obtains: 4 4 ∗ abη¯ 1+ λ (1 λ ) a(¯η)= , Ωr0 = ±2 2 −4 , D(S,M,λ)=11+0.14 (S +2M log (λ)) . (81) λ 2λ xb σ¯ − H 2 1 4 We need to apply the condition (61) to this equality and σ¯ = , Ωcr = λ Ωr0, xb√Ωr0λ also consider conditions (62) and (76). In addition to where λ includes both cases. Using this unified descrip- these conditions, we also consider that η 1, since our results are valid far from the bounce. Imposing≫ it to tion, we look for analytical± constraints on the free param- eters of the theory, which yield the observed baryon-to- Eq. (74), and using Eq. (77) to eliminate xb, yields, entropy ratio. S > 2D + 20.4 0.434 log (λ2 (λ)). (82) Far from the bounce, we can relate the cosmic time t − C withη ¯ as: Therefore, the complete set of conditions on the param- eters are: a σ2η¯2 t(¯η)= b . (73) 2λ 7 D 22, (83) ≤ ≤ From Eqs. (59) and (73), and using the parameters of the 16 M 0, (84) − ≤ ∗ ≤ symmetric case, we obtain 20 < S< 58, (85) 10 2 1.0 10− x λ S > 2D + 20.4 0.434 log (λ (λ)), (86) η¯(T¯)= × b . (74) − C T¯ D(S,M,λ)=11+0.14 (S +2M log (λ)) . (87) − H From the latter equation, our approximation demands we However, we still want to use xb in order to compare consider η 1, which results: ≫ to the results of the symmetric case. From the allowed x 1.0 1011 values for L¯, one can obtain the correspondent values for b & × (75) T¯ λ xb. Writing X = log xb, from Eq. (77) one obtains, As it was explained in section III, we have to use con- S =2X 3.8 log C(λ). (88) − − dition (40) on Lmin = 1/R(¯ηmax) directly. Defining the ¯ This means that for each allowed values of S, there is a new parameter L = RH0 /Lmin, one gets corresponding one for X. 20 ¯ 58 10 < L< 10 , (76) From Eqs. (86), (87) and (88), one can note that asym- which replaces (60). The lower limit is due to imposing metry manifests itself due to the presence of the functions 3 (λ) and (λ) on these conditions. Inspecting the be- Lmin > 10 lp, where lp is the Planck length, whereas the C H upper limit comes from the requirement that the bounce havior of these functions, we can qualitatively realize that energy scale should be much bigger than the nucleosyn- larger values of λ imply smaller values of xb, and greater values of D and M¯ as compared to the symmetric case, thesis energy scale, Lmin FIG. 7. Parameter space of xB, TD, M∗ forp ¯ = 0.90√Ωr0 −11 that that give nB /s 9 10 . These are parameterized by ≈ ∗ X = log(xB), D = log T¯D and M = log M¯ ∗, respectively, for scale factor a+. FIG. 6. Parameter space of xB, TD, M∗ forp ¯ = 0.50√Ωr0 −11 that that give nB /s 9 10 . These are parameterized by ≈ ∗ X = log(xB), D = log T¯D and M = log M¯ ∗, respectively, for scale factor a+. In the (-) case, corresponding to the choice a in Eq. (69), we note in Figs. 9 -12 that increasingp ¯− en- large the allowed region in parameter space. The right side of this region does not change, whereas its left side grows to the left. Asp ¯ increases, smaller X (or S), and greater D and M are allowed in order to baryogenesis to 11 occur with the right value. Forp ¯ & 10 √Ωr0, the al- lowed region stabilizes: X, M and D reach their broader ranges. In the following we consider another asymmetric FIG. 8. Parameter space of xB, TD, M∗ forp ¯ = 0.99√Ωr0 −11 bounce case, where there are two asymmetry parameters that that give nB /s 9 10 . These are parameterized by ≈ ∗ instead of one. X = log(xB), D = log T¯D and M = log M¯ ∗, respectively, for scale factor a+. 2. Unitary Asymmetric Bounce σ¯ and the dimensionless energy density parameter of ra- We now consider the unitary asymmetric bounce given diation today Ωr0, given by Eq. (49), is obtained in the by the initial wave function of Eq. (42), from which we limit p1σ 1 and p2σ 1, the analysis of baryogenesis can obtain a solution for the trajectory of the scale fac- with curvature≪ coupling≪ in this section also refers to this tor a(T ). The differential equation for a(T ) is rather expansion. In this limit, it is possible to obtain an ana- involved (see Eq. (54) of Ref. [17]), so that we perform lytical expression for the scale factor by considering the some approximations on a(T ). Since the relation between terms p1σ, p2σ and √p1 p2 σ only up to second order. It 11 2 4 FIG. 9. Parameter space of xB, TD, M∗ forp ¯ = 10 √Ωr0 FIG. 11. Parameter space of xB, TD, M∗ forp ¯ = 10 √Ωr0 −11 −11 that that give nB /s 9 10 . These are parameterized by that that give nB /s 9 10 . These are parameterized by ≈ ∗ ≈ ∗ X = log(xB), D = log T¯D and M = log M¯ ∗, respectively, X = log(xB), D = log T¯D and M = log M¯ ∗, respectively, for scale factor a−. for scale factor a−. 3 FIG. 10. Parameter space of xB, TD, M∗ forp ¯ = 10 √Ωr0 −11 that that give nB /s 9 10 . These are parameterized by 11 ≈ ¯∗ ¯ FIG. 12. Parameter space of xB, TD, M∗ forp ¯ = 10 √Ωr0 X = log(xB), D = log TD and M = log M∗, respectively, −11 that that give nB /s 9 10 . These are parameterized by for scale factor a−. ≈ ∗ X = log(xB), D = log T¯D and M = log M¯ ∗, respectively, for scale factor a−. reads (p2 p2)σ2η¯ a(¯η)= a 1 1 − 2 1+¯η2. (89) b − 2(1+η ¯2) Far from the bounce, the scale factor reads: p From now on we apply the transformation of variables 2 2 xb = a0/ab,σ ¯ = σ√a0H0 andp ¯i = pi /a0H0, i =1, 2. a(¯η)= abη,¯ (90) 12 such that the cosmic time t in terms ofη ¯ gives The chemical potential in this case is given by a σ2η¯2 φ˙ t(¯η)= b , (91) µ = , (97) 2 B M ∗ which are both identical to the symmetric case. As we did where M is the energy scale of the coupling. Again, for the previous cases, matching Eqs. (57) and (58), we in a dynamical∗ universe where φ evolves in time, CPT- are able to find a relation betweenη ¯ and the temperature invariance is broken, as in the curvature coupling dis- T¯. Using Eq. (91), we obtain cussed in the previous section, and baryons can be cre- ated even in thermal equilibrium. 10 1.0 10− x η¯(T¯)= × b , (92) In this framework, the bounce background dynamics T¯ is given in Ref. [16], where a scalar field with exponen- tial potential drives the bounce as a stiff matter fluid, which is also identical to the symmetric bounce. As men- behaves as a dust fluid in the asymptotic past and future tioned before, disregarding the terms (¯p σ¯)4, (¯p σ¯)4 and 1 2 (which guarantees an almost scale invariant spectrum of (¯p p¯ )2σ¯4, we obtain thatσ ¯, given by Eq. (49), also re- 1 2 scalar perturbations, see Ref. [16]), and also presents a duces to Eq. (27) of the symmetric case. transient dark energy-type behavior occurring only in the As we did for the other bounce solutions, using future of the expanding phase. This bounce is asymmet- Eqs. (89) and (92) and the constants defined in the sym- ric because the transient dark energy epoch occurs only metric case, we obtain n /s equal to symmetric result, B in the expanding phase, not in the contracting phase, Eq. (67), up to seventh order, plus an extra term in the avoiding problems related to the imposition of vacuum eighth order: state initial conditions in the contracting phase if dark ¯8 nB nB 73 TD 2 2 energy is present there, and overproduction of gravita- = +2.8 10− p¯ p¯ . (93) s s × M¯ 2x4 | 1 − 2 | tional waves, which are typical in bouncing models con- sym ⋆ b taining a canonical scalar field. Note that the maximum value of the new term corre- The pressure and energy density associated with φ are, 2 2 sponds to the maximum value of the difference p¯1 p¯2 . respectively, > & | − | In order to assureη ¯ 1 (¯η 10), we find the same 1 condition obtained for the symmetric bounce: P = φ˙2 V (φ), (98) φ 2 − xb & 1.0 1011. (94) 1 ˙2 ¯ ρφ = φ + V (φ), (99) T × 2 Finally, considering reasonable values of xb, TD and and the potential reads M⋆, given by conditions (60) - (62), we obtain the region 11 λκφ of parameters that allows nB/s =9 10− . Considering V (φ)= V0e− , (100) n /s up to eighth order, we obtain× the same plot of the B where the constant V has units of mass to the fourth symmetric case, given by Figure 5, for any value of the 0 power, and λ is dimensionless, chosen to satisfy λ √3 in parametersp ¯ andp ¯ that satisfyp ¯ σ¯ 1andp ¯ σ¯ 1. 1 2 1 2 order to get an almost scale invariant spectrum of≈ scalar This means that the eighth order does≪ not bring≪ new perturbations. The background dynamics can be made possibilities of parameters allowed. Hence, the gravita- simpler through a choice of dimensionless variables, tional baryogenesis of the unitary asymmetric bounce in this limit is equal to the gravitational baryogenesis of the √8πφ˙ √8πV symmetric case. x = , y = . (101) √6MP H √3MP H In these new variables, the Friedmann constraint and the V. BARYOGENESIS WITH SCALAR effective equation of state parameter, w = P/ρ, read, COUPLING x2 + y2 =1, w =2x2 1. (102) − The action for a canonical scalar field φ in a curved The above definitions lead to the planar system: space-time with metric gµν reads dx 3 1 = 3x(1 x2)+ λ y2, (103) S = d4x√ g (∂ φ)2 V (φ) . (95) dα − − 2 − 2 µ − r Z dy 3 = xy 3x λ , (104) As discussed in Refs. [30, 31], spontaneous baryogenesis dα − r 2! can be driven by the coupling of the baryonic current with the derivative of the scalar field ∂µφ through where α ln(a). This system is supplemented by the equations≡ 1 4 µ d x√ g(∂µφ)J . (96) ˙ 2 2 M − α˙ = H, H = 3H x . (105) ∗ Z − 13 In the expanding phase, the variable x is close to 1 near and a net baryon number can emerge, even in thermal the bounce (the kinetic part dominates, hence w 1, equilibrium. ≈ as stiff matter), and decrease to 0 (an effective w 1, Making use of these proposals, we analyzed baryon ≈− as dark energy), passing in between through x = 2/3, production in background bouncing models, extending or w 1/3, where it is connected with the standard investigations already realized in the context of loop cosmological≈ evolution before nucleosynthesis. Baryonp quantum cosmology [33] to bouncing models coming from production should terminate in this epoch. Afterwards Wheeler-DeWitt quantum cosmology in the framework of the baryon number ”freezes” in the current value of the dBB quantum theory [16, 17]. We investigated many 11 nB/s 9 10− [31]. Hence, the decoupling takes place possible bouncing solutions, symmetric and asymmetric when ≈ × around the bounce. The free parameters are the energy scale of the coupling, the curvature scale at the bounce, 1 Pφ ρφ, (106) and the decoupling temperature. ≈ 3 In the case of gravitational baryogenesis, the results for which, from Eqs. (98) and (99), yields the symmetric bounce shows that a broad region of phys- ical parameters fulfil the observed value of the baryon-to- 3 2 entropy ratio. It is allowed by almost all possible values ρφ φ˙ . (107) ≈ 4 of coupling energy scales, with preference for the lower The radiation density, in turn, is given by the Stefan ones, a region of seven orders of magnitude for the curva- Boltzmann’s law, ture scale at the bounce, with preference for the deeper ones, and a narrower interval for possible decoupling tem- 2 peratures (four orders of magnitude), with preference for π g 4 ρ = ∗ T , (108) r 30 the lower ones. Note that the bounce background is not a necessary which, together with Eq. (107), implies that condition for a net baryon number within gravitational and spontaneous baryogenesis, but it allows a larger 2 2π g 2 φ˙ ∗ T . (109) range of parameters. In the case of gravitational baryo- ≈ r 45 genesis, the decoupling temperature must be attained when R˙ is not negligible. In classical cosmology, dom- Hence, the chemical potential (97) reads inated by fields satisfying an effective equation of state p = wρ, one has R˙ = 24πGH(1 3w)(1 + w)ρ, hence 2π2g T 2 − − µ ∗ . (110) it is negligible in a radiation dominated phase, or even B ≈ 45 M r ∗ during inflation. Of course, in the primordial Universe The baryon-photon (entropy) ratio, given in terms of the w is not exactly 1/3, but it is quite close, and hence the range of parameters necessary to yield sufficient baryo- decoupling temperature TD and the coupling parameter M , now reads genesis is more constrained. 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