The R-axion and non-Gaussianity

We study cosmological implications of an R-axion, a pseudo Nambu-Goldstone boson associated with spontaneous breaking of an U(1)_R symmetry, focusing on its quantum fluctuations generated during inflation. We show that, in the anomaly mediation, the R-axion decays into a pair of gravitinos, which eventually decay into the visible particles producing the neutralino LSP. As a result, the quantum fluctuations of the R-axion are inherited by the cold dark matter isocurvature density perturbation with potentially large non-Gaussianity. The constraints on the inflation scale and the initial misalignment are derived.


Introduction
Supersymmetry (SUSY) stabilizes the weak scale against radiative corrections and offers a solution to the gauge hierarchy problem. To explain the smallness of the weak scale compared to the ultraviolet scale such as the Planck scale, however, SUSY must be broken dynamically. It is well known that constructing a viable dynamical SUSY breaking (DSB) model is a highly non-trivial task. In particular, it was shown that, in calculable models with a generic superpotential, the existence of an R symmetry is a necessary condition for having a SUSY breaking vacuum and that a spontaneously broken R symmetry is a sufficient condition [1].
The spontaneously broken continuous R symmetry leads to a Nambu-Goldstrone boson, the so called R-axion. The R-axion acquires a mass from an R-symmetry breaking. In particular, the continuous R symmetry must be explicitly broken by a constant term in a superpotential, W = C 0 , in order for the cosmological constant to vanish. Therefore the R-axion becomes necessarily massive.
The cosmology of the R-axion was studied in Ref. [2], assuming that the constant C 0 is the main source of the R-axion mass. The initial misalignment of the R-axion from the potential minimum gives rise to coherent oscillations, which may affect the standard cosmology such as the big bang nucleosynthesis (BBN). Their main conclusion was that, in renormalizable hidden sector models where the SUSY breaking in a DSB sector is transmitted to the visible sector via Planck-suppressed operators, the R-axion is cosmologically harmless since it can be as heavy as O(10 7 ) GeV and decays before BBN. It was also pointed out that the R-axion decay may produce a large amount of the gravitinos, which in turn sets an upper bound on the reheating temperature to avoid conflicts with BBN.
In this letter we study the cosmology of the R-axion, focusing on its quantum fluctuations generated during inflation. As discussed recently in Ref. [3], a light scalar such as the R-axion may generate sizable non-Gaussianity in the isocurvature fluctuation, as in the case of the QCD axion. As we will see in the following sections, the quantum fluctuations of the R-axion are inherited by the neutralino dark matter (DM) non-thermaly produced by the gravitino decay. A sizable fraction of the parameter space in the inflation scale and the initial misalignment turns out to be already excluded by the current observations. The isocurvature perturbation and its associated non-Gaussianity, therefore, put a tight constraint on the R-axion cosmology.

R-axion mass and abundance
We consider the dynamical SUSY breaking in which SUSY is broken by the strong gauge dynamics in the SUSY breaking sector. Let us denote the dynamical scale by Λ. We also assume that the R symmetry is spontaneously broken as a result of the SUSY breaking, and f denotes the breaking scale of the R symmetry. Then the R-axion arises as a Nambu-Goldstone boson. It is known that the constant term in the superpotential, which is needed to cancel the cosmological constant, explicitly breaks the U(1) R symmetry, generating a non-vanishing mass of the R-axion. Assuming that the constant term is the dominant source of the explicit R -breaking, we can estimate the R-axion mass as [2] where m 3/2 denotes the gravitino mass, and M P ≃ 2.4 × 10 18 GeV is the reduced Planck mass. In the following we assume that f is constant during and after inflation. We will come back to this issue in Sec. 4. Let us assume that the R symmetry is already spontaneously broken during inflation. To this end we require that inflation scale satisfy the following inequality: where H inf denotes the Hubble scale during inflation. In most part of this letter, we assume f ∼ Λ ∼ m 3/2 M P , although we will show a constraint in the case of f ≫ Λ [5] later. For the R-axion to develop quantum fluctuation that extends beyond the horizon, the R-axion mass must be lighter than the Hubble parameter during inflation: We focus on the parameter space where the above inequalities are satisfied. Indeed, as long as the gravity or anomaly mediation [4] is considered, the above conditions (2) and (3) are met for plentiful inflation models.
Let us now estimate the R-axion abundance. The R-axion starts to oscillate about the potential minimum when the Hubble parameter becomes comparable to its mass, H ≃ m a . The number density of R-axion n a devided by the entropy density s is given by where T R is the reheating temperature after inflation. We have assumed that the R-axion starts oscillate before the reheating is completed, since otherwise too many gravitinos would be generated by thermal scatterings (see Eq. (9)). Here we have defined an effective initial displacement, where θ i (= 0 ∼ π/2) denotes the (dimensionless) initial misalignment of the R-axion from its potential minimum.
The decay of the R-axion depends on the SUSY breaking mediation mechanism. In the gauge mediation [6], since the SUSY breaking field is more strongly coupled to the visible sector, the R-axion tends to mainly decay into the supersymmetric standard-model (SSM) particles [7,8]. On the other hand, in the gravity and anomaly mediation [4], we expect that the R-axion decays into a pair of the gravitinos with a non-negligible branching ratio.
We consider the anomaly mediation in this letter, since the two conditions (2) and (3) are satisfied for most of the known inflation models and the R-axion will mainly decay into the gravitinos which makes our analysis simple and robust.
The abundance of the gravitino produced from the R-axion decay is where B 3/2 is the branching fraction of the R-axion decay into gravitinos. We have for f θ i > H inf /(2π), and for f θ i < H inf /(2π), respectively. On the other hand, the gravitino is also produced by scattering of particles in thermal plasma and the abundance is given by [9] where mg denotes the gluino mass. Thus the gravtino abundance produced by the R-axion decay can be comparable with that from thermal production for B 3/2 ∼ θ i ∼ 1.
In the anomaly-mediation, the gravitino is so heavy that it decays into the SSM particles well before BBN. Therefore there is no BBN constraint on the gravitino abundance. However, the lightest supersymmetric particle (LSP), which we assume to be the Winolike neutralino, is produced by the gravitino decay. We obtain the upper bound on the reheating temperature, by requiring that the neutralino abundance should not exceed the observed cold dark matter (CDM) abundance: where mW is the Wino mass and the gravitino abundance (9) is assumed. 1 As we will see below, an additional constraint on the R-axion cosmology is obtained by taking account of the quantum fluctuations of the R-axion. In particular, the Hubble scale during inflation is constrained from above by the WMAP data, even if the abundance of the neutralino originated from the R-axion is a subdominant component of the dark matter.

Isocurvature Perturbation and Non-Gaussianity from R-axion
If the R symmetry is spontaneously broken during inflation with the condition (3) satisfied, the R-axion obtains quantum fluctuations. Since the energy density of the R-axion is negligibly small compared with that of the inflaton, its fluctuation can be regarded as an isocurvature perturbation. This becomes fluctuation in the R-axion abundance after it begins to oscillate. The R-axion decays into gravitinos, which eventually decay into the SSM particles producing the neutralino LSP. Since some fraction of the relic dark matter comes from the R-axion, this gives rise to the CDM isocurvature perturbation. The CDM isocurvature perturbation in this model is given by where a i = f θ i is the classical deviation from the potential minimum, δa is the quantum fluctuation of the R-axion, whose Fourier modes satisfy the following condition, and r denotes the fraction of the DM produced by the R-axion decay to the total DM abundance. The second term in Eq. (11) represents the non-Gaussian fluctuation, and hence, non-Gaussianity resides in the CDM isocurvature perturbation in this model. The effects of such an isocurvature-type non-Gaussianity was recently studied in detail by Refs. [3,11,12,13] (see also Refs. [14,15,16] for early attempts). The WMAP5 constraint on the (uncorrelated) isocurvature perturbation reads P S /P ζ < 0.19 at 95% C.L. where P S and P ζ are power spectra of isocurvature and curvature perturbations at a pivot scale k 0 = 0.002 Mpc −1 [17]. Now let us estimate the non-Gaussiniaty. First, the curvature perturbation ζ can be expanded as follows according to the δN formalism [18,19,20], where φ a represent scalar fields which is light during inflation including the inflaton itself, and N a = ∂N/∂φ a and so on, with N representing the local e-folding number measured along the world line x =const. In a similar way, the CDM isocurvature perturbation can be expanded as where S a = ∂S m /∂φ a and so on. Then the non-linearity parameter of the isocurvature type, f NL , is given by [3] where we have introduced an infrared cutoff L taken to be the present Hubble scale [21], and ∆ 2 δφ = (H inf /2π) 2 . In the present model, there exist two light scalar fields : the inflaton and R-axion. Since the former is responsible for the curvature perturbation and the latter for the isocurvature perturbation, the non-linearity parameter f (iso) NL is calculated using Eq. (11) and the WMAP normalization condition |ζ| ≃ 5 × 10 −5 [17]. Numerically, it is evaluated as for ( for f θ i H inf /(2π), where we have used the relation mW = (g 2 /16π 2 )m 3/2 in the anomalymediation with g being the SU(2) gauge coupling constant.
We have shown the constraints on the inflation scale and the initial misalignment in Fig. 1 for f = 10 12 GeV and in Fig. 2 for f = 10 14 GeV. In both figures we have set m 3/2 = 100 TeV and the reheating temperature T R is taken to be ∼ 8 × 10 9 GeV so that the Wino LSP from the gravitinos produced by thermal scatterings account for the observed CDM abundance. Below the blue dashed line, the isocurvature constraint is satisfied. In the shaded regions neither isocurvature perturbation nor non-Gaussianity will arise since R symmetry may be restored during inflation (H inf > f ) or R axion is heavier than the Hubble scale during inflation (H inf < m a ). In these region, our arguments are not applied. Above the red solid line, the neutralino LSP is overproduced by the decay of R-axion-induced gravitino. In these figures contours of f (iso) NL = 1, 10 and 100 (from left to right) are plotted. Ref. [22] put a limit on f (iso) NL as f (iso) NL < 15 using Minkowski functional method, and hence the region with f (iso) NL ≫ 15 is disfavored. It is seen that even if the initial misalignment is chosen so as to suppress the Raxion abundance, the constraints from isocurvature and/or non-Gaussianity exclude large parameter region once the quantum fluctuation of the R-axion is taken into account. Remarkably, high scale inflation models with H inf = 10 10 ∼ 10 12 GeV are disfavored.

Discussion and Conclusions
We have so far assumed that the U(1) R breaking scale f is constant during and after inflation. This is the case if the bosonic partner of the R-axion, namely R-saxion s, is stabilized at ∼ f during inflation. We are concerned with a class of SUSY breaking models in which the SUSY breaking field is stabilized without taking account of supergravity effects. Therefore, the R-saxion is heavier than the R-axion, and its mass is expected to be of O(m 3/2 M P /f ). Therefore, for an inflation models with H inf smaller than O(m 3/2 M P /f ), this assumption is justified. Otherwise, the R-saxion potential generically receives sizable Figure 1: Contours of f (iso) NL = 1, 10 and 100 are shown by black solid lines from left to right. Below the blue dashed line, the isocurvature constraint is satisfied. In the shaded regions neither isocurvature perturbation nor non-Gaussianity will arise since R symmetry may be restored during inflation (H inf > f ) or R axion is heavier than the Hubble scale during inflation (H inf < m a ). Above the red solid line, the neutralino LSP is overproduced. In this figure we have taken f = 10 12 GeV. corrections during inflation, and the R-saxion may be deviated from the true minimum. This can affect our analysis in two ways. First, the isocurvature perturbation is modified. For a larger value of s , the isocurvature perturbation becomes smaller and the constraint gets relaxed correspondingly, where s is the expectation value of |s| during inflation. In particular, the Gaussian part of the isocurvature perturbations are suppressed by f / s . Second, the R-saxion will start coherent oscillations, and mainly decays into two R-axions. As long as the oscillation amplitude is of O(f ), the produced R-axions do not affect our analysis on the isocurvature perturbation. This is not only because the R-saxion is heavier and the number density is suppressed, but also because, in contrast to the R-axion, the quantum fluctuations of the R-saxion can be suppressed due to the Hubble-induced mass term.
We have studied the cosmological impact of the R-axion which arises as a result of the spontaneous breaking of the R symmetry in a DSB model. The initial misalignment during inflation gives rise to a coherent background R-axion that decays into gravitinos. In the anomaly mediation, the gravitino of mass heavier than O(10) TeV decays before BBN, and so, the R-axion seems cosmologically harmless in this framework. However, the Raxion acquires quantum fluctuations during inflation if the R symmetry is spontaneously Figure 2: Same as Fig. 1, but for f = 10 14 GeV. broken during inflation. Consequently, the Wino LSP DM has an isocurvature fluctuation with potentially large non-Gaussianity. We have derived a constraint on the inflation scale and the initial misalignment to satisfy the current observations. Our main conclusion is that the R-axion becomes cosmologically harmless for a sufficiently low inflation scale.