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Enhanced Gravitational Waves from Inflaton Oscillons

In broad classes of inflationary models the period of accelerated expansion is followed by fragmentation of the inflaton scalar field into localized, long-lived and massive oscillon excitations. We demonstrate that matter-dominance of oscillons, followed by their rapid decay, significantly enhances the primordial gravitational wave (GW) spectrum. These oscillon-induced GWs are distinct and could be orders of magnitude lower in frequency than the previously considered GWs associated with oscillon formation. We show that detectable oscillon-induced GW signatures establish direct tests independent from cosmic microwave background radiation (CMB) for monodromy, logarithmic and pure natural (plateau) potential classes of inflationary models, among others. We demonstrate that oscillon-induced GWs in a model based on pure natural inflation could be directly observable with the Einstein Telescope, Cosmic Explorer and DECIGO. These signatures offer a new route for probing the underlying inflationary physics.

Introduction.-Understanding the earliest stages of the history of the Universe is a fundamental quest of physics. Since the dynamics of the early Universe are associated with high energy scales far beyond the reach of terrestrial laboratories, they are inherently challenging to probe.
An early period of rapid cosmic inflationary expansion resolves variety of conceptual problems and provides an attractive framework for explaining the observed Universe from initial conditions [1][2][3][4][5][6], such as formation of galaxies. Inflation predicts a characteristic nearly scaleinvariant power spectrum, described by a spectral index n s , of the angular anisotropies in the cosmic microwave background (CMB) temperature as well as potentially a non-negligible B-mode polarization and a related ratio of tensor-to-scalar fluctuations r [7]. Recent CMB observations established accurate measurements of the scale dependence of the angular power spectrum and a tight upper bound on r, strongly disfavoring single-parameter single-field models of the slow-roll inflation [8]. Nevertheless, many classes of two-parameter models are fully consistent with the observational constraints. However, lacking variety of additional signatures these models are difficult to test with the CMB alone in the near future.
Similarly, the possibilities for probing the subsequent stage of post-inflationary reheating [9][10][11] are restricted. For most reheating scenarios, the only signature is a high-frequency stochastic gravitational wave (GW) background, sourced by the non-linear evolution of matter on sub-horizon scales [12,13]. Such signatures, however, reside outside the detection range of the current and planned GW observatories [14,15].
We demonstrate that with subsequent rapid oscillon decay the epochs of single-field two-parameter inflation models associated with oscillons will result in dramatically enhanced scale-invariant scalar perturbations and source novel OIGWs signatures. These OIGW signatures are distinct from and can be orders of magnitude lower in frequency than GWs associated with oscillon-formation. The OIGW signal could lie in the observational range of upcoming GW experiments. Further, we demonstrate that the OIGWs provide a complementary probe of inflationary models that are challenging to test by CMB measurements.
Inflation.-Consider single-field models of inflation in which the inflaton, φ, is minimally coupled to gravity, i.e., where R is the Ricci scalar, g is the determinant of the metric and M Pl = 2.435×10 18 GeV is the reduced Planck mass. The scalar inflaton field φ drives the inflationary expansion of the Universe during its slow-roll phase.
Measurements of the angular anisotropies in the CMB place significant constraints on the shape of the inflaton potential, V (φ). Observables related to CMB are directly linked to the inflaton potential [8], including the scalar power spectrum amplitude, A s , scalar spectral index, n s , tensor-to-scalar power ratio, r, and the number of expansion e-folds before the end of the slow-roll inflation, N , when the pivot scale k =0.05 Mpc −1 exited the horizon where V ≡ V (φ ), etc and φ is the value of the inflaton field when the pivot scale crossed outside the Hubble sphere, k = k = a H with scale factor a and Hubble parameter H, and φ end is the value of the inflaton at the end of inflation, whenä = 0. CMB observations [8] favor inflaton potentials, V (φ), having a plateau region away from the minimum (along which the slow-roll inflation occurs), and a power-law minimum (about which the inflaton condensate oscillates at the beginning of reheating). Motivated target potentials consistent with observations include: • Monodromy -string and supergravity scenarios often lead to power-law models, central realization being axion monodromy [29,30] • Pure Natural (Plateau) -broad classes of axion models, incited by considerations from fundamental theory, lead to universal predictions in (n s , r) plane described by a converging attractor [27,28] • Log-normal -logarithmic terms in the potential can naturally appear from radiative corrections and supersymmetric theories (e.g. [50]) In Fig. 1 (left panel) we illustrate the behavior in field space of the potentials described by Eq. (2), Eq. (4) and Eq. (3). In Supplemental Material we explicitly derive the observables A s , n s , r and N for these potentials. Inflaton oscillons.-Inflationary potentials shallower than quadratic, such as in classes of models based on monodromy inflation Eq. (2), log-normal inflation Eq. (4) and plateau inflation Eq. (3), can naturally result in scalar field fragmentation and oscillon formation after inflation (e.g. [40,41]). Oscillons are quasi-stable, approximately spherical localized and massive field configurations. Formation of oscillons during non-linear behavior of the field has been explicitly confirmed by numerous lattice simulations (e.g. [42,43,47]).
After the end of inflation, the inflaton oscillates around the minimum of its potential V . The non-linearities associated with the behavior of the potential lead to an instability in the fluctuations and non-adiabatic production of inflaton particles of definite co-moving momentum, preheating [9]. The field oscillations by the homogeneous condensate φ can be described by the linearized equations of motions for the perturbations φ(t, x) = φ(t) + δφ(t, x), which in the static Universe approximation and ignoring metric perturbations are given in Fourier space as Since The individual momentum modes and their resonant stability can be analyzed with Floquet theory (e.g. [40,42]), with the general solution to Eq. (5) being where P ± (t) are periodic functions determined by the initial conditions, µ k is the Floquet exponent. The magnitude of the real part of the Floquet exponent, | (µ k )|, characterises the exponential mode growth, provided it is sufficiently faster than the Hubble expansion rate H ∼ t −1 , after inflation. Efficient resonance occurs when | (µ k )|/H O(10). Momentum modes δφ k have physical wavenumber k/a, where a is the expansion scale factor. They scan the Floquet bands as the scale factor changes with time. In Fig. 2 we depict resulting Floquet charts with stability bands, which we have numerically computed as in Refs. [42,[52][53][54], for field fluctuations associated with monodromy inflation Eq. (2), log-normal inflation Eq. (4) and plateau inflation Eq. (3).
Unstable modes lead to parametric resonance and copious formation of oscillons, for potentials shallower than quadratic with self-interactions (e.g. [42,55,56]). The resulting oscillons have high density contrast δ = ρ/ρ > 1 and comoving number density a 3 n sol ∼ (k nl /2π) 3 determined by the dominant non-linear wave mode k nl . They oscillate at the approximate frequency of the effective mass m, with typical size R ∼ 10/m that is much smaller than the relevant horizon scale.
Although oscillons are not protected by a topological invariant and a conserved charge they can be remarkably long lived, which from semi-analytic studies can be attributed to an approximate adiabatic invariance of the oscillating field [37,57,58]. Oscillon lifetime has been recently systematically explored by detailed numerical calculations for broad classes of potentials [51,59,60], including large amplitude oscillons, confirming that oscillons are generically long-lived, with a lifetime in some models (including the ones considered here) exceeding ∆t 6 × 10 8 /m followed by a rapid decay. Since oscillons behave as presureless dust, they can naturally lead to long periods of (oscillon)matter-domination.
Oscillons form soon after the end of slow-roll inflation provided M 10 −2 M Pl (see Fig. 2 and discussion after Eq. (6)). The mean energy density of the Universe at the time of formation is ρ f ∼ m 2 M 2 , and at the time of decay ρ d ∼ (m/n) 2 M 2 Pl . n = m∆t/(2π) is the number of oscillations it takes for the oscillons to decay and we assume H d ∼ m/n. Looking ahead, the ratio ρ f /ρ d ∼ n 2 M 2 /M 2 Pl determines the duration of production and thereby the magnitude of the induced GW signal.
The classes of motivated potentials above allow for theories with significantly prolonged oscillon lifetime. A realization of this can be seen within the modified pure natural (plateau) model given by ) . Oscillon-induced gravitational waves.-Primordial scalar perturbations induce GWs at second order. While often suppressed [20], sudden transition from an early matter-dominated era (eMD) to radiation-dominated (RD) era significantly enhances induced GWs [19,61]. Such amplification of GWs has been extensively explored in the context of evaporating primordial black holes, whose "gas" matter-dominates the Universe before suddenly transitioning to radiation due to Hawking evaporation (e.g. [20-22, 25, 26, 62, 63]).
After the inflaton fragments into oscillons, they matter-dominate the Universe before rapidly decaying, leading to an onset of reheating. The length of matterdominated era is set by the oscillon lifetime. We demonstrate that due to the rapid decay of the oscillons at the end of their lifetime, these processes result in significant induced GW production 1 . We stress, however, that in our scenario there is direct connection of induced GWs with the inflaton's dynamics. Oscillon isocurvature perturbations could lead to yet another source of induced GWs (e.g. [65,66]), study of which we leave for future work [67]. Hence, GW signal estimates in this study are conservative.
We consider that inflation results in a generic primordial curvature (scalar) power spectrum described by where k max is the cutoff scale. For the calculation of the induced GWs we follow [19,72]. We consider that that eMD of the Universe rapidly transitioned to RD, as is the case for oscillon-dominated phase, at a conformal time η = η R .  [70] as well as LIGO O5 [71] are also displayed.

The energy density of induced GWs is
where η is conformal time and P h (η, k) is the timeaveraged power spectrum of GWs related to the curvature perturbations as [72,73] where the kernel function I(u, v, k, η, η R ) describes dynamics of scalar and tensor (i.e. GW) perturbations that depends on (using Green's function methods) a source function described by the gravitational potential Φ and its conformal derivative Φ (see Supplemental Material). As the oscillons suddenly decay at the end of the lifetime, Φ jumps from Φ = 0 in eMD to Φ = 0 in RD phase. The resulting significant amplitude increase of Φ dramatically enhances induced GWs through I ∼ (Φ /H) 2 (e.g. [19]). Physically, the enhancement results from rapid oscillations of perturbations with unsuppressed amplitudes in Φ, constant before η R , on timescales shorter than the decay.
The approximate general expressions for the induced GWs from the power spectrum of Eq. (10) are reviewed in the Supplemental Material. For the case of spectral index n s 1 the GW spectrum can be obtained from simplified analytic expressions [19] After cosmological evolution considerations, the resulting GW spectrum at present time is given by (e.g. [43]) where Ω r,0 h 2 4.2 × 10 −5 is the current radiation energy density and η c is the moment when Ω GW becomes constant after gravitational potential has sufficiently decayed soon after modes re-enter the horizon, well before matter-radiation equality, x R = kη R . Note however, that for a sudden eMD-RD transition η c ≈ η R . We display our results for conservative realization of x max,R = k max η R ≤ 450 limiting the regimes to linear analysis (e.g. [20]), as well as those slightly extrapolated to non-linear regime via x max,R = k max η R ≤ 10 3 (e.g [74]). We leave detailed numerical and simulation analyses of these effects for future work.
In our scenarios, at the time of oscillon formation all sub-horizon density perturbations, k H f , collapse into oscillons. During the ensuing oscillon matterdomination era, the modes which undergo gravitational instability are the ones which re-enter the horizon before the oscillon decay, H f > k > H d . Thus, we set k max = H f . Since matter-radiation equality coincides with the time of oscillon decay, η R = η d , one can show that x max,R = (ρ f /ρ d ) 1/6 ∼ (nM/M Pl ) 1/3 . According to Eq. (11), the induced GW signal has a pronounced peak whose magnitude and frequency today are given by Ω GW,0 ∼ few × 10 −29 (nM/M Pl ) 7/3 and f GW,0 ∼ few×10 10 (m 3 M 2 /(nM 5 Pl )) 1/6 Hz, where to compute the former we used Eq. (12) and to estimate the latter we used f GW,0 ∼ 10 10 k max /(a d ρ 1/4 d )Hz [43]. For the monodromy, plateau and log-normal oscillons, n is only bounded from below (n 10 8 ) [59], whereas M is bounded from above 2 (M 10 −2 M Pl ), see Fig. 2. In order to remain in the linear approximation, we use the 2 The classicality of the resonant inflaton dynamics imposes an additional constraint on the model parameters, leading to a lower bound on M [43]. Oscillon formation, ρ f ρ grad (k nl /a f ), is classical, when the backreacting gradient energy is of modes upper bound on M and set n = 10 10 , whereas we extrapolate in the nonlinear regime when n = 10 11 . In Fig. 3 we display computed induced GWs from inflaton oscillons for the monodromy, log-normal, pure natural (plateau)-inflation as well as modified pure natural (plateau)-inflation model potentials along with existing constraints. The values of m and M are in agreement with CMB bounds on slow-roll inflation (see Supplemental Material). In the monodromy, log-normal and pure natural-inflation models oscillons form when φ ∼ M , whereas in the modified pure natural inflation whenφ ∼ M . We demonstrate that the induced GWs are orders above in amplitude compared to GWs associated with oscillon-formation from preheating-type resonant instabilities [43], f GW,0 ∼ (M Pl /M ) 1/2 10 8 Hz, Ω GW,0 ∼ 10 −6 (M/M Pl ) 2 , (see also [44][45][46][47]).
Pure natural inflation is viable with current CMB constraints at the level of one standard deviation and monodromy as well as log-normal models are at few standard deviations. Thus, an independent test is essential to establish their status. Induced GWs from oscillons could allow to probe these models with CMB-S4 [69] through constraints on N eff , in addition to other preheating scenarios [76,77]. Further, modified pure natural (plateau) model can be directly testable with Einstein Telescope, Cosmic Explorer and DECIGO at GW frequencies significantly smaller than conventionally associated with oscillon formation (for a review of high frequency GWs see [78]). Our results and limits can be readily applied to GW strains, using h c (f ) = 1.26 × 10 −18 (f /Hz) −1 Ω GW,0 h 2 [79].
Conclusions.-Generic models of inflation support production of oscillons through inflaton field fragmentation. We have demonstrated that oscillon matterdomination phase followed by rapid oscillon decay results in dramatic production of induced GWs. These signatures are significantly enhanced at lower frequencies compared to GWs associated with oscillon formation previously studied. We have shown that these induced GW signatures can provide a crucial independent test for broad classes of inflationary models, such as pure natural (plateau), log-normal and monodromy potentials. Induced GWs associated with modified pure natural inflation could be directly observed with the Einstein Telescope, Cosmic Explorer and DECIGO. These results establish a new window into the early Universe dynamics.
of occupancy greater than one, ρ grad (k nl /a f ) ρ BD (k nl /a f ) ∼ (k nl /a f ) 4 , where ρ BD is the energy density in the Bunch-Davies vacuum [75]. For k nl /a f ∼ m, see Fig. 2, M m, implying M 5.7, 13, 30 × 10 −4 M Pl for monodromy, plateau and log-normal inflation, respectively. Relaxing the assumptions can modify the allowed range in M .

SUPPLEMENTAL MATERIAL "Enhanced Gravitational Waves from Inflaton Oscillons"
Kaloian D. Lozanov, Volodymyr Takhistov In this Supplemental Material we provide detailed expressions for the inflationary model parameters of the scenarios considered in the main text. Further, we give an overview of the analytic expressions for induced gravitational waves associated with sudden matter-to-radiation epoch transition for the generic power spectrum of the main text.

I. INFLATIONARY MODEL PARAMETERS
Below we derive expressions for inflationary observables of potentials described in the main text, assuming slow-roll and |φ | |φ end | for simplicity. For monodromy potential of Eq. (2) these are given by For log-normal potential of Eq. (4) these are given by For pure natural (plateau) potentials of Eq. (3)