Cosmological imprints of string axions in plateau

We initiate a study on various cosmological imprints of string axions whose scalar potentials have plateau regions. In such cases, we show that a delayed onset of oscillation generically leads to a parametric resonance instability. In particular, for ultralight axions, the parametric resonance can enhance the power spectrum slightly below the Jeans scale, alleviating the tension with the Lyman $\alpha$ forest observations. We also argue that a long-lasting resonance can lead to an emission of gravitational waves at the frequency bands which are detectable by gravitational wave interferometers and pulsar timing arrays.

Introduction.-Compactifications in string theory generically predict various axions in 4D low energy effective field theory. Exploring imprints of axions in cosmology provides us with an important tool to probe extra dimensions predicted in string theory [1]. Phenomenological impacts of axions have been mostly studied by considering the quadratic potential. However, once an axion is away from the potential minimum, in general, it deviates from the quadratic form. In particular, when the dilute instanton gas approximation does not hold, the scalar potential can be more flatten than the conventional cosine form [2,3] as in α-attractor models [4]. Therefore, it is worth investigating phenomenological consequences of axions with such plateau regions in their scalar potentials.
Distinctively, the axions with such plateau regions generically undergo a parametric resonance, which exponentially enhances the modes in the resonance bands and potentially leaves various phenomenological impacts. This parametric resonance instability has been studied in the context of inflation with a shallower potential than the quadratic form. In Refs. [5], it was shown that the instability leads to a fragmented configuration of the oscillating scalar field, the so called oscillon (see also Refs. [6,7]). In Ref. [8], it was shown that the oscillating axion can induce resonance phenomena also in gravitational waves.
In string theory, there appear axions in a wide mass range. For example, the large volume scenario predicts the presence of extremely light axions (see e.g., Refs. [9]). The onset time of the oscillation varies, depending on the mass scale of the axion. In particular, the ultra-light axion (ULA) whose mass is of O(10 −22 eV) starts to oscillate before the matter radiation equality and behaves as a fuzzy dark matter. The ULA has been discussed, relating to the small scale issues in ΛCDM [10]. Meanwhile, it was argued that the ULA with m ≃ 10 −22 eV is marginally incompatible with the Lyman α forest observations, since the ULA smooths out the small scale structures too much [11]. It is interesting to see whether the parametric resonance instability can relax the tension with the Lynman α forest observations or not. * Electronic address: jiro˙at˙phys.sci.kobe-u.ac.jp † Electronic address: urakawa.yuko˙at˙h.mbox.nagoya-u.ac.jp Setup of problem.-In order to study dynamics of string axions whose potentials have a shallower region than the quadratic potential, we consider a canonical scalar field φ with a scalar potential V (φ) given by Since the axion is a pseudo-scalar, it is reasonable to impose additionally Z 2 symmetry on the potential. This class of potential also includes the conventional cosine potentialṼ (φ) = 1 − cosφ. The parameter f agrees with the decay constant in the case with the cosine potential. Roughly speaking, m determines the onset time of the oscillation (under a certain initial condition) and f determines the energy density for a given m.
In this letter, we will investigate phenomenological consequences of the axions with a potential which satisfies the conditions i) and ii). Here, we neglect a backreaction of the axions on the geometry of the universe, deferring more detailed analyses for future studies. Under this approximation, we solve the Klein-Gordon (KG) equation ✷φ − V φ = 0 in a fixed geometry. In this analysis, whether φ dominates the universe or not is not essential. As a special case, our analysis includes cases where φ is an inflaton, which starts to oscillate at the end of inflation.
Evolution of the homogeneous mode.-First, we consider the evolution of the homogeneous mode of φ. In the following, we assume the background expansion law as a ∝ t p with p > 0. When the field φ does not dominate the universe, the power p cannot be determined only from the dynamics of φ. Then, the KG equation for the homogeneous mode is given by with x ≡ m/H = mt/p. Notice that all the dimensionful parameters dropped out from the equation and the onset time of oscillation, x osc , is solely determined by the initial conditions φ i ≡φ(x i ) andφ x,i ≡ dφ/dx(x i ). The Hubble parameter at x osc for a given m is determined as H osc = m/x osc . When the axion stays at a plateau, it behaves as a cosmological constant. In fact, when the potential gradient term is negligible, the equation of motion (1) can be solved analytically as φ(x) = C 1 + C 2 x 1−3p . Meanwhile, around the bottom of the potential with |φ| ≪ 1, Eq. (1) can be also solved analytically. However, in general, Eq. (1) can be solved only numerically in the intermediate range. Notice that when φ is located at the plateau region at the onset of oscillation, the oscillation does not necessarily take place around x ≃ 1 or H ≃ m. As an example, we consider an α attractor type potential given byṼ It should be, however, noted that the resonance instability generically takes place, as long as the scalar potential satisfies the conditions i) and ii). This potential is shown in Figure 1 for different values of c. Forφ < 1, the second derivative of the potential is . The curvature of the potential becomes smaller for a larger value of c around the bottom of the potential. Figure 2 shows that the oscillation starts at x osc ≫ 1, when we choose the initial conditioñ φ i = 5 andφ x,i = −1, starting at the plateau region. (When the plateau is wide enough, the evolution does not much depend on the initial velocity because of the over damping.) The orange dotted line shows the time evolution for the conventional cosine potentialṼ (φ) = 1 − cosφ withφ i = π and φ x,i = −10 −4 . Even with this fine-tuned initial condition, the oscillation starts much earlier than the plateau case. Let us consider the case where φ is a ULA. Assuming that φ starts to oscillate before the matter radiation equality, we can estimate the decay constant by equating the energy density of the radiation ρ eq γ with that of dark matter ρ eq m = ρ eq φ /β φ , where β φ denotes the fraction of the ULA among the total dark matter, as where the quantities with the index eq denote those at the equal time and the quantities with the index osc denote those at the onset of the oscillation. Here, taking into account that the kinetic energy and the potential energy are comparable at the onset of the oscillation, we used ρ osc φ ≃ 2V osc ≃ (mf ) 2 . Thus, once x osc is determined by solving Eq. (1) and m is given, the decay constant f should be given by Eq. (2).
Parametric resonance instability.-Next, we study the evolution of the inhomogeneous modes. The perturbed KG equation for the axion is given byφ+3Hφ+(k/a) 2 ϕ+V φφ ϕ = 0, where ϕ is the perturbed axion field and we neglected the metric perturbations. According to our numerical analysis, the metric perturbation does not play a crucial role in the early stage of the resonance instability, where the linear analysis can apply. Again, we can rewrite the perturbed KG equation Depending on the choice of the initial time, the corresponding wavenumberk varies, while k/a is independent of the choice of the initial time. Here, we choose x i = 1/10.  Figure 3 shows the time evolution ofφ k for the same potentials as in Fig. 2 during RD. Figure 4 shows the time evolution ofφ k for different wavenumbersk during RD (up) and MD (bottom), whenṼ is given by the α-attractor type potential with c = 0. The modes withk = 10 −1 andk = 10 −1/2 got slightly enhanced just after the onset of the oscillation due to the tachyonic instability. However, this is not very efficient, because the second derivative of the potential oscillates taking both positive and negative values. (In Ref. [7], a role of the tachyonic instability was discussed in detail for another model.) In order to understand the instability more intuitively, here let us analyze the equation (3), neglecting the Hubble friction. Around the bottom of the potential, i.e.,φ < 1, the homogeneous mode of the axion oscillates with the frequency |Ṽφφ| 1/2 ≃ 1 and the solution is given byφ =φ * cos x. Using this solution, Eq. (3) is given by Mathieu equation as where A and q are defined as Here, a osc denotes the scale factor at around the onset of the oscillation. The parametric resonance takes place for the narrow band A ≃ n 2 , where n is an integer. The width of the resonance band is proportional to (q/A) n . The dominant growing mode is the n = 1 mode and the growth rate γ with ϕ ∝ e γx is given by γ ≃ q/2 = (2 + 3c)φ 2 * /16. Notice also that the resonance band becomes wider for a larger c as shown in Fig. 3. The resonance wavenumbers can be predicted from the first resonance band of Mathieu equation, leaving aside factor deviations.
The cosmic expansion makes the parametric resonance instability more inefficient mainly due to the following two effects: first, the amplitudeφ * decreases due to the Hubble friction, reducing the growth rate and second, more importantly here, the wavenumbers in the resonance band(s) are red shifted. When the gradient of the potential is shallower, the onset of the oscillation gets more delayed, i.e., x osc ≫ 1. Then, when the parametric resonance instability sets in, the redshift of the resonant modes due to the cosmic expansion is not effectively important any more. This leads to the sustainable resonant growth without being disturbed by the cosmic expansion. Because of that, an efficient resonance instability requires a shallower potential than the quadratic potential region, where the oscillation takes place at x osc ≃ 1. As is shown in Fig. 3, for a larger c, the resonance instability proceeds more efficiently, since the oscillation starts later. To visualize this aspect more clearly, we also plotted the time evolution ofφ andφ for the cosine potential in Fig. 2 and Fig. 3, respectively. (This mild resonance in the cosine example was pointed out in Refs. [12].) When φ is the inflaton, it has to decay until the end of reheating for a successful transition to RD. On the other hand, when φ is not a dominant component of the universe at the onset of the instability, e.g. the ULA, which starts the oscillation during RD, the unstable growth can continue longer time. In Fig. 3 and Fig. 4, where we choose the initial field valuẽ φ i = 5, the parametric resonance persists sufficiently long. On the other hand, when we choose a smaller value of |φ i | as an initial condition, the parametric resonance can persist only in a shorter period, leaving only a milder enhancement of the fluctuation.
Jeans scale.-It is known that the ULA has an emergent pressure on small scales and the Jeans wavenumber is given by k J (a) ≃ √ mH a , where we used c s ≃ k/(2ma) [10]. When the ULA dark matter becomes the dominant component of the universe for a > a eq , the structures below the Jeans length are smoothed out. As is shown in Fig. 4, the parametric resonance instability takes place for the wavenumbers slightly above the Jeans wavenumberk J (a). This can be understood as follows. The resonance wavenumber in the first band k r satisfies k r /(ma osc ) ≃ O(1). Therefore, using k J /(ma) ≃ 1/ √ x, we find a universal relation k r ≃ √ x osc k J (a eq ). When the scalar potential of the ULA dark matter has a plateau region, the parametric resonance which takes place for the smaller scales than the Jeans scale can enhance the perturbation of the ULA before the matter-radiation equality. This may supply the missing small scale structures, asserted in Ref. [11]. Related to this point, in Ref. [13], it was argued that the enhancement around k J can lead to a significant enhancement of the low-mass halo abundance even for the conventional cosine potential, where the resonance is much more moderate than potentials with plateau regions.
Oscillon formation and GWs emission.-When the parametric resonance instability persists sufficiently long, the linear perturbation ceases to be a good approximation, even if we start with an almost homogeneous initial condition. In Refs. [5], it was numerically shown that the long-lasting instability can lead to a fragmented configuration of the oscillating scalar field. Since the cosmic expansion is not crucial at the oscillon formation, clumps of the oscillon will be formed also during RD and the later MD.
The formed oscillon can emit the gravitational waves (GWs) [6,7,14]. In contrast to the GWs from the oscillon formed during the reheating, the peak frequency of the GW spectrum emitted later times can be in sensitivity bands of GW detectors. (Another imprint of an oscillon formed near the present epoch was discussed in Ref. [15].) Here, we roughly evaluate the peak frequency of the GW emitted either during RD or MD (the later MD) as f 0 ≃ m/(1 + z * ), where z * denotes the redshift at the emission. When we identify the Hubble parameter at the emission as the one at the onset of the oscillation, i.e., H(z * ) ≃ m/x osc , assuming that the oscillon formation proceeds quite rapidly in the cosmological time scale, the frequency of the GWs emitted during RD can be given by where we used (1 + z * ) ≃ (H(z * )/H eq ) 1/2 (1 + z eq ). Similarly, using (1 + z * ) ≃ (H(z * )/H eq ) 2/3 (1 + z eq ), we obtain the frequency of the GWs emitted during MD as When we avoid choosing a hierarchically large value of |φ i |, . For a direct detection, the GWs emitted from the oscillon during RD is more promising, e.g., for m ≃ 10 −6 eV and x osc ≃ 10 4 , the frequency f 0 is in the band of space interferometers [16] and for m ≃ 10 3 eV and x osc ≃ 10 4 , f 0 is in the band of ground based interferometers [17]. (When axions have unsuppressed interactions with the electromagnetic field, the larger mass range should be excluded because of the photon decay process [1,10].) Meanwhile, for m ≃ 10 −16 eV, f 0 is in the detectable range by pulsar timing arrays [18].
In order to compute the amplitude of GWs from the oscillon, we introduce a parameter ǫ(≤ 1) which denotes the ratio between the relative spectral energy density of the emitted GWs and that of the homogeneous axion at the onset of the oscillation, which is of O((mf ) 2 ). Assuming that GWs were emitted just after the onset of the oscillation (in cosmological time scales), we obtain Ω gw as [19] Ω gw ≃ ǫΩ r (mf ) 2 ρ eq a osc a eq In particular, for m 10 −27 eV, the axion starts to oscillate before the equal time, behaving as a dark matter component. Then, using Eq.(2), we obtain Ω gw ≃ 10 −4 ǫβ φ x osc 10 4 10 −22 eV m .
When the axion is the dominant component of dark matter, i.e., β φ ≃ 1 and x osc = O(10 4 ), we obtain Ω gw ≃ 10 −7 ǫ for m ≃ 10 −16 eV. Meanwhile, we obtain Ω gw ≃ 10 −12 ǫ for m ≃ 10 −6 eV and Ω gw ≃ 10 −16 ǫ for m ≃ 10 3 eV. Therefore, for ǫ 10 −6 (this value can be achieved for GWs emitted from the oscillon formed during the reheating [7]), we can expect a detection of GWs emitted by the resonantly oscillating modes of the string axions, using the pulsar timing arrays. An accurate estimation of ǫ requires a further numerical analysis and this will be reported in our future publication.
Summary: New window in string axiverse.-In this letter, we initiated a study on phenomenological imprints of string axions with plateau regions in their scalar potentials. We found that for such axions, the resonance instability can last considerably long, because the delayed onset of the oscillation makes the redshift of the resonant modes insignificant. This instability takes place slightly below the Jeans scale, suggesting various implications on the structure formation of ULA dark matter. A long-lasting resonance instability leads to the oscillon formation. In contrast to the GWs from the oscillon formed during the reheating, the GWs from the oscillon formed at later times can be emitted in the directly detectable ranges.
If the oscillating scalar field dominates the universe, even if the resonance instability terminates without forming the fragmented oscillon, the instability can still leave detectable imprints. In particular, for the ULA with m ≃ 10 −22 eV, it will be intriguing to study whether the resonance instability can relax the tension with the small scale measurements. Notice that the resonance instability finishes, once the amplitude of φ is damped away due to the Hubble friction. Therefore, when the ULA starts the oscillation much earlier than the equal time, it will not enhance the small scale structures at the time of galaxy formations. This may suggest that the ULA with the plateau may be able to explain the small scale issues of ΛCDM, being compatible with Lyman α forest observations. This will be examined in our future study. In our setup, the scalar field behaves as a cosmological constant before starting the oscillation. Therefore, when it existed also during inflation, the isocurvature constraint should be also taken into account likewise usual ULAs.