Affleck-Dine inflation

The Affleck-Dine mechanism in its simplest form provides baryogenesis from the out-of-equilibrium evolution of a complex scalar field with a simple renormalizable potential. We show that such a model, supplemented by nonminimal coupling to gravity, can also provide inflation, consistent with Planck constraints, simultaneously with the generation of the baryon asymmetry. The predictions of the model include significant tensor-to-scalar ratio and possibly baryon isocurvature fluctuations. The reheating temperature is calculable, making the model fully predictive. We require color triplet scalars for reheating and transfering the primordial baryon asymmetry to quarks; these could be observable at colliders. They can also be probed at higher scales by searches for quark compositeness in dijet angular distributions, and flavor-changing neutral current effects.


I. INTRODUCTION
Theoretical mechanisms for baryogenesis abound and take many very different forms, but one common attribute is that they occur at some cosmological epoch following inflation. This seems like a necessity, since exponential expansion should dilute any preexisting baryon asymmetry. In this work we present an exception, showing that it is possible to generate the baryon asymmetry of the universe (BAU) during the course of inflation, using the inflaton as an essential element.
One of the earliest proposed baryogenesis mechanisms was that of Affleck and Dine (AD) [1] in which a complex scalar field carrying baryon number can spontaneously create the BAU starting from field values displaced from the minimum of the potential. A baryon-violating coupling is required to satisfy Sakharov's requirements [2]. Although the AD mechanism is most commonly implemented in supersymmetric models whose potentials have nearly flat directions, it was originally demonstrated using a simple renormalizable potential of the form in the seminal reference [1]. (We use the subscript to denote the Jordan frame since a change of frames will be invoked below.) When λ = 0, the potential has a U(1) global symmetry, that we will identify with baryon number. A generic initial condition such that φ = 0 spontaneously breaks CP and thermal equilibrium, as also required by Sakharov. The field winds around, at first generating baryon number, until Hubble damping of φ(t) makes the λ interaction negligible, and baryon number becomes conserved. The same kind of potential could be used for a twofield version of chaotic inflation [3]. Constraints from the Planck experiment now disfavor chaotic inflation with φ 2 or φ 4 potentials [4] since they predict too high tensorto-scalar ratio r, given the measured value of the scalar perturbation spectral index n s = 0.965 ± 0.004. However this problem can be cured by adding a nonminimal coupling to gravity (we write 2ξ rather than ξ to agree with the usual convention for inflation along a single compo-nent of the complex scalar), where m P = 2.44×10 18 GeV is the reduced Planck mass, that we set to 1 unless explicitly shown. This introduces a noncanonical kinetic term for φ upon Weyl-transforming to the Einstein frame, and it flattens the potential at large field values to make the predictions of the model compatible with Planck observations, in the case of a real scalar field inflaton [5,6]. Our goal is to determine whether this can still hold true for the two-field model, while at the same time generating the observed baryon asymmetry. A potential issue is that isocurvature perturbations can be produced in two-field models, and these are constrained by the Planck observations. A similar idea was explored in ref. [7], using the complex sneutrino field in a supergravity model as both the inflaton and the AD field for leptogenesis. The model is rather special and intricate, whereas ours could be considered simpler and more generic. The predictions of r versus n s illustrated there are in significant tension with the latest data. Moreover isocurvature fluctuations were not considered in ref. [7].

II. MODEL
Eqs. (1,2) are sufficient to determine the inflationary trajectory until the epoch of reheating. It is convenient to make a Weyl rescaling of the metric, g µν → Ω 2 g µν , with Ω 2 = 1/(1 + 2ξ|φ| 2 ). The Lagrangian in the Einstein frame, including gravity, is then Writing the complex scalar as φ = (X + iY )/ √ 2 and ignoring spatial gradients, the scalar kinetic term takes the form with Ω 2 = 1/(1 + ξ(X 2 + Y 2 )). Thus X and Y are not canonically normalized fields. Instead of reexpressing them in terms of such fields, we will numerically solve the equations of motion for X and Y to determine the predictions for inflation and baryogenesis. Details of deriving the first-order equations convenient for numerical integration can be found for example in ref. [8] (see eqs. (2.100-2.101)). We choose initial conditions close to the inflationary attractor solution, by setting the derivatives of the canonical momenta Π X = dL/dẊ, Π Y = dL/dẎ initially to zero. More is needed in order to get reheating and transfer of the baryon asymmetry, initially stored in φ, into quarks. A natural option for reheating is the Higgs portal coupling λ φh |φ| 2 |H| 2 . However since we also need a coupling to quarks, it is simpler to use the same interactions both for reheating and for transfer of the baryon asymmetry. This can be accomplished by introducing three QCD triplet scalars Φ i carrying baryon number 2/3, with couplings where a, b, d are color indices, the quarks are righthanded (SU(2) L singlets) and d c R denotes the conjugate down-type quark. For simplicity we omit generation indices on the quarks and the Yukawa couplings y i . These interactions allow for the decay φ → uudddd via virtual Φ i exchange, and imply that φ carries baryon number 2. The same conclusion holds if we choose Φ 1ūR u c R and Φ 2,3dR d c R couplings instead of (5). For small values of the λ coupling, we can view reheating as occurring through the perturbative decays φ → Φ 1 Φ 2 Φ 3 , which rapidly thermalize with the quarks and thereby the rest of the standard model degrees of freedom. Assuming that φ is much heavier than Φ i , the decay rate is

III. INFLATION + BARYOGENESIS
An interesting aspect of our model is that the same parameters that influence inflationary observables can also affect the magnitude of the baryon asymmetry. Thus, although we describe the two processes separately, a fully viable model depends upon the interplay between the two.

III.1. Slow-roll parameters
Although we can solve for the inflaton trajectories without reference to the canonically normalized fields, that we will denote by (U, V ), it is necessary to know them for computing inflationary perturbations. It is straightforward to diagonalize the kinetic term (4) at a given point in field space to find with e 1 = Ω −1 (1 + 6Ω 2 ξ 2 (X 2 + Y 2 )) −1/2 , e 2 = Ω −1 , θ = tan −1 (Y /X). Then L kin = 1 2 (U 2 +V 2 ). The matrix Z allows us to transform slow roll parameters computed in the original field basis (indices i, j) to those in the canonical basis (indices m, n): where V E = Ω 4 V J is the Einstein frame potential. The extra rotation R ψ in eq. (7) is not necessary for diagonalizing the kinetic term, but it is required in order to be able to interpret Z as the Jacobian matrix ∂(X, Y )/∂(U, V ). If we omit R ψ so that Z = Z 0 , such an interpretation is not generally consistent since then the relations may not be satisfied. We are free to set ψ = 0 at a given point in field space, such as the point of horizon crossing, but not its derivatives. Eqs.
This has the solution The consistent identification of Z with a Jacobian matrix insures that η mn is symmetric in mn, even though the second term in (8) is not explicitly symmetric. Then we can write the second term in eq. (8) as To compute the adiabatic perturbation spectrum and the tensor-to-scalar ratio, we use the slow-roll formalism of ref. [9], evaluating the slow-roll parameters (8) Table I. Parameters and initial values for two benchmark models, including the total number of e-foldings of inflation Ntot, number of e-foldings before horizon crossing N * , spectral index ns (evaluated at k * = 0.05 Mpc −1 ), tensor-to-scalar ratio r and off-diagonal transfer matrix element TRS, which is a measure of the correlation between adiabatic and isocurvature perturbations. to the σ, s basis of adiabatic/entropy directions, defined by with α = tan −1 (V /U ). The rotated slow roll parameters are given by [10] Then to leading order in the slow-roll expansion, the scalar spectral index and tensor-to-scalar ratio are [9] where c ∆ = −2C η σs , s ∆ = + 1 − c 2 ∆ , C = 2 − ln 2 − γ ∼ = 0.73 (γ is the Euler constant), and the derivatives of V E with respect to U and V are computed similarly to eq. (8). Including the effect of isocurvature modes (T RS ), which we will explain below, the scalar amplitude is with V * = V E evaluated at horizon crossing and we have neglected terms of order T 2 RS . We searched the parameter space via Markov Chain Monte Carlo (MCMC) to find models in agreement with Planck constraints on A s , n s , r and the baryon asymmetry (discussed below). Two benchmark models are identified in table I. The correlation of r with n s is shown over the interval N * = (50, 60) e-foldings, for several values of ξ and fixed values of the potential parameters corresponding to the two benchmark models in fig. 1.
On each curve a heavy dot is indicated to show the prediction of the model, for the chosen value of λ , that determines the reheating temperature and thus the number of e-foldings N * between horizon crossing and the end of inflation. The value of N * is determined by solving eq. (47) of ref. [4] (see also ref. [11]), where H 0 is the Hubble constant today, ρ end is the energy density at the end of inflation, g * = 106.75+18 (counting the extra degrees of freedom from the colored scalars), and the reference scale k * = 0.002 Mpc −1 for comparison with the Planck preferred regions in the n s -r plane. The energy density at the time of reheating is ρ rh = 4 3 Γ 2 φ m 2 P , as explained below-see eq. (24); this makes N * depend upon λ as N * ∼ 1 3 ln λ .
Since V * appears in eq. (16) but also depends upon N * , we rescale the parameters of the potential while iteratively determining N * , keeping A s = e 3.044 × 10 −10 fixed to the observed central value [4]. To illustrate the dependence on λ , we indicate two other horizon-crossing positions on the ξ = 0.07 curve of model 1, for larger and smaller values of λ . The relation between λ and the reheat temperature will be discussed in section III.3.
The strong correlation between the tensor ratio r and the nonminimal coupling ξ is also clearly seen in the larger sample of models from two MCMC chains, fig. 2 (left). The points shown have a total χ 2 < 10, defining χ 2 in the usual way in terms of the observables r, n s and η B , summed over observables x i with central valuex i and experimental error δx i . The black points come from a chain where the experimental limit on r was somewhat relaxed. The correlation between r and n s within the chains is also notable, as shown in fig. 2 (right). In both plots, one can notice a population of models scattered away from the main trends. These are special cases in which the total number of e-foldings of inflation are not much greater than the minimum required, N e ∼ 60. We will discuss these cases in more detail below.

III.2. Isocurvature fluctuations
During inflation, the components of the canonically normalized fields U, V can fluctuate by order H/(2π), where H is the Hubble parameter. Fluctuations dσ normal to the inflaton trajectory are entropy modes, and they could become observable isocurvature fluctuations if they decay into different species than the adiabatic fluctuations, that are parallel to the trajectory. The relation between adiabatic/entropy perturbations and the canonical field fluctuations is given in eq. (12).
To find the observable entropy fluctuations, we need to compare (dσ, ds) to the directions in field space that correspond to baryon number fluctuations dB, and the orthogonal direction, that will be related to (dU, dV ) through some different rotation angle β. Numerically we find that β ∼ = 0 during inflation, implying that the entropy perturbations are purely in the baryon number (compensated by radiation) to a good approximation, known as BDI (baryon density isocurvature). This can be seen starting from the definition of baryon density from the zeroth component of the baryon current carried by φ, leading to the fluctuation where the omitted terms are subleading in the slow roll approximation. The direction of the fluctuation (19) turns out numerically to be very nearly orthogonal to the inflaton trajectory in field space. Although both σ and s decay into quarks during reheating, only s decays encode the baryon asymmetry, whereas σ decays equally into quarks and antiquarks, that thermalize with the rest of the SM degrees of freedom. We closely follow the formalism of ref. [9] (see also ref. [10]) to compute the power in isocurvature. The main task is to numerically solve the equations for the evolution of the perturbations dU, dV between horizon crossing and the end of inflation, and to relate them to the adiabatic/isocurvature perturbations dσ, ds using eq. 12). Here primes denote d/dN e , C 1 = 3+H /H, and dC = C 1 (U dU +V dV ). The barred parametersη ij are defined as in eq. (8), except that we divide by the total energy density ρ = 3H 2 instead of V E , so that the equations remain valid even when the slow-roll approximation is not. The transfer function for the curvature (adiabatic) and entropy perturbations is a matrix that relates the amplitudes of (dσ, ds) at horizon crossing to those at a later time, after inflation. We can get the matrix elements by solving the system (20) from the respective initial conditions (dσ, ds) = (1, 0) and (0, 1). The results are shown for the two benchmark models in fig. 3. The adiabatic perturbation is conserved, resulting in T RR = 1, and the T SR element is always very small, in accordance with the slow-roll prediction T SR = 0 [9], meaning that there is negligible conversion of entropy to adiabatic modes. For all cases in our MCMC, the entropy autocorrelation T SS 0.1 is too small to be observable, but in some cases like in model 1, the cross-correlation T RS is significant. It is related to the correlation angle defined by cos ∆ = T RS which is constrained by Planck as | cos ∆| 0.1-0.3, depending upon pivot scale k * and which datasets are combined. (Ref. [4] notes that the constraints on BDI correlation are the same as for cold dark matter isocurvature, CDI.) Therefore model 1 is an example where the predicted BDI correlation is close to the experimental sensitivity.
The models with large BDI require somewhat special initial conditions, in which the total duration of inflation is not more than ∼ 80 e-foldings. This is because significant curvature of the inflaton path in field space is needed during horizon crossing for generating isocurvature. Models with long periods of inflation tend to have such curvature earlier than horizon crossing, subsequently becoming nearly linear and thus resembling single-field inflation. This is illustrated for the two benchmark models in fig. 4, that shows the field trajectories and horizon crossing points. It is further borne out by fig. 5, showing the correlation between |T RS | and total number of e-foldings N tot for models within an MCMC chain satisfying χ 2 < 10. On the other hand, models like our benchmark model 2, having longer periods of inflation, lead to predictions that are relatively insensitive to the initial conditions, since the field trajectory settles into a unique trough in the potential.

III.3. Baryogenesis and reheating
To compute the baryon asymmetry, we use the baryon density stored in φ, eq. (18). It is convenient to compare this to the number density of φ particles, prior to  reheating, since the ratio η = n B /n φ reaches a constant value that we denote as η e at the end of inflation, during the period of φ oscillations around the minimum of the potential. The time evolution of η is illustrated for model 1 in fig.  6. Reheating occurs at the time t rh = 1/Γ φ where Γ φ is the decay width of φ. Defining n φ = n φ,e at the end of inflation (t = t e ), n φ at the time of reheating will be n φ,rh = n φ,e a e a rh 3 = n φ,e where we used the fact that the φ oscillations matterdominate the universe until reheating, and t rh t e . The value of n φ,rh is independent of n φ,e , so long as the latter is large enough to provide sufficient expansion of the universe prior to reheating. This will be true if the energy density at the end of inflation is much greater than that at reheating.
The baryon-to-entropy ratio at reheating is given by with s = (2π 2 /45)g * T 3 rh and reheat temperature [12] T rh = 90 Including a factor of 36/111 [13] for the reduction of baryon number by redistribution into lepton number by sphalerons, it follows that which is conserved into the late universe. The measured value is η B = 8.6 × 10 −11 [14]. The coupling λ should be small in order to justify the perturbative reheating assumption, but from the point of view of technical naturalness, it need not be very small. A three-loop diagram involving λ renormalizes the λ|φ| 4 interaction, giving the estimate to avoid destabilizing the inflationary potential by quantum corrections. The baryon asymmetry generated during inflation depends sensitively on the value of the B-violating coupling λ . In fig. 7 we again show how η B evolves with N e from the beginning of inflation until shortly after it ends, but for a range of different values of the baryon-violating coupling λ . The effect of the φ oscillations can be seen briefly around N e = 65, but these are quickly Hubbledamped and η B settles to a constant value that we have identified as η e in eq. (25). The dependence of the final baryon asymmetry is not monotonic. At first this may be surprising, since one can derive the time-dependence of n B from the inflaton field equations, However one finds that λ has an important effect on the background inflaton trajectory, which explains the nonlinear dependence. This is illustrated in fig. 8. Hence the processes of inflation and baryogenesis are nontrivially intertwined in our model: adjusting λ can affect not only η B but also the inflationary observables. The effects of B violation after inflation are negligible. At low energies, integrating out φ and Φ i leads to a dimension-36 operator involving 24 quarks. It could induce conversion of four neutrons into their antiparticles in a neutron star, but the rate is far too small to be significant. In the early universe, we must check that the ∆B = 8, Φ 12 i operator induced by φ exchange is out of equilibrium, to avoid washing out the B asymmetry. The rate can be estimated as By demanding that the decoupling temperature exceed the reheat temperature T rh in eq. (26), we find a constraint which is more lenient than the consistency requirement (28).

IV. PARTICLE PHYSICS IMPLICATIONS
The colored scalars Φ i can have observable effects at low energies. If sufficiently light, they can be pairproduced at LHC. The Yukawa interactions in eq. (5) have the same form as R-parity violating coupling of squarks to quarks in supersymmetric models, leading to various mass exclusions in the range 80-525 GeV [15] or 100-600 GeV [16], depending upon the flavor structure of the couplings.
However heavier colored scalars can be probed indirectly, using an effective field theory description where they are integrated out to give dimension-6, four-quark operators. For baryogenesis, the flavor structure of the new Yukawa couplings was not important, but at low energies it can have an observable effect on the angular distributions of jets at LHC, or flavor-changing neutralcurrents like meson-antimeson oscillations. Using chiral Fierz identities [17], the effective Lagrangian is where a, b, d, e are color indices, i, j, k, l label flavor, and P R projects onto right-handed chirality. In the bottom line we have specialized to the case where i = j and the operator contributes to meson-antimeson oscillations, since these combinations are much more severely constrained than the flavor-diagonal ones, or those connecting mesons of different masses. From dijet angular distributions, CMS finds a limit of [18]  for flavor-diagonal operators, presumably of the first generation (since the limit on higher generation quarks will be somewhat weakened by parton distribution functions). However K 0 -K 0 , B 0 -B 0 and B 0 s -B 0 s mixing give more stringent constraints [19], shown in table II.

V. CONCLUSIONS
We have studied a new model of inflation with the novel feature that the inflaton carries baryon number, and it can produce the baryon asymmetry via the Affleck-Dine mechanism, mostly during inflation, with relatively small evolution over the few e-foldings after inflation ends. It is a simple but complete model, including a calculable perturbative reheating mechanism that allows one to make definite predictions for the inflationary observables, given a set of input parameters. One testable prediction is that the tensor-to-scalar ratio r is likely to be observable, depending upon the value of the nonminimal coupling of the inflaton to gravity. For the values ξ ∼ 0.01 − 1 considered in this work, we find r > 0.04, which is within the sensitivity of upcoming CMB experiments. For example LiteBIRD will probe values down to r ∼ 10 −3 [20].
Since ours is a two-field inflation model, another possible signal is correlated baryon isocurvature-adiabatic fluctuations that have been constrained by the Planck collaboration. We find that these can occur at an observable level if the total duration of inflation did not greatly exceed the canonical minimum number of efoldings, N tot ∼ 60. In this case the inflaton trajectory can turn significantly in field space around the time of horizon crossing. We are not aware of other models in the literature that predict baryon isocurvature perturbations.
The model relies upon new colored scalar particles in order to transfer the baryon asymmetry from the inflaton to the standard model quarks. These could have observable effects in laboratory experiments if sufficiently light, even at the scale of 10 4 TeV for K 0 -K 0 oscillations. The colored scalars could also mediate purely hadronic rare flavor changing decays, that we have not considered here. The new source of baryon violation needed for baryogenesis is however hidden at the high scale the inflaton mass ∼ 10 −7 m P , out of reach of laboratory probes.
We have considered only the simplest scenario for reheating. It is possible that sufficiently large values of λ could lead to more efficient reheating through parametric resonance [12]. To our knowledge, this has not been previously studied for couplings of the form φΦ 3 such as are present in our model. Moreover we ignored the Higgs portal coupling |φ| 2 |H| 2 which could reduce the baryon asymmetry by producing extra radiation. We leave these issues for future study.