Enhancement of small-scale induced gravitational waves from the soliton/oscillon domination

At the linear order, the perturbed metric in the Newtonian gauge reads

where the $a(\eta)$ is the scale factor, $\eta$ is the conformal time. We neglect vector perturbations, first-order GWs and anisotropic stress. According to the Friedmann equations and the continuity equation, in the Universe dominated by a perfect fluid with a general equation of state parameter $\omega$, the scale factor can be expressed as

In the early Universe, effective self-resonance resulted in the amplification of perturbations of a real or complex scalar field $\phi$. As a consequence, the homogeneous scalar condensate fragments into lumps, leading to the formation of solitons, the long-lived, spatially localized structures. The solitons are distributed in space by Poisson statistics, with a mean physical separation $d$. Solitons form rapidly from fragmented scalar field lumps, allowing their formation to be approximated as instantaneous, occurring at time $\eta=\eta_{\text{f}}$. Before the formation of solitons, the Universe could have been dominated by either matter or radiation. To address this, we consider both scenarios separately. For convenience, we define the energy density of solitons as $\rho_{\text{s}}$ and the total energy density of the Universe as $\rho_{\text{tot}}$, and we introduce the fractional energy denisity of solitons at $\eta=\eta_{\text{f}}$ as $\Omega_{\text{s,f}}=\langle\rho_{\text{s,f}}\rangle/\rho_{\text{tot,f}}$. If the solitons are formed during the eMD era, then $\Omega_{\text{s,f}}=1$. Conversely, if the solitons are formed during the eRD era, we have $\Omega_{\text{s,f}}\leq 1$. In this paper, we primarily focus on the two cases where $\Omega_{\text{s,f}}=1$ and $\Omega_{\text{s,f}}\ll 1$.

After formation, solitons gradually lose energy by emitting relativistic radiation at a slow rate and behave as non-relativistic matter. The conserved charge of solitons, $Q$, also decreases with time, and solitons rapidly decay once the charge reaches the critical value $Q_{\text{cr}}$, a parameter that depends on the specific soliton model. For solitons formed during the eRD era, their energy density is negligible at $\eta_{\text{f}}$. As the Universe expands, the energy density of radiation scales as $\rho_{\text{r}}\propto a^{-4}$ while the energy density of matter scales as $\rho_{\text{m}}\propto a^{-3}$, respectively. Therefore, for solitons with sufficiently long lifetimes, they could eventually dominate the Universe before decaying. Since the decay process of solitons is also rapid, we can also assume instantaneous decay. Therefore, we define the time when the solitons dominate the Universe as $\eta=\eta_{d}$ and the decay time as $\eta=\eta_{r}$. The Universe is dominated by matter at $\eta=\eta_{\text{m}}$, and we find the relationship $\eta_{\text{r}}>\eta_{\text{m}}=\eta_{\text{d}}>\eta_{\text{f}}$.

Oscillons can also form in the eMD era. The post-inflationary Universe can host a long range of strongly nonlinear processes due to parametric resonance, which is referred to as preheating. The resonance fragments the inflaton condensate and the Universe is strongly inhomogeneous on sub-horizon scales. The energy density is transferred from the oscillating scalar field to the oscillons. Before oscillon formation, the inflaton performs rapid, damped oscillations about a quadratic minimum of the potential after inflation with the frequency much higher than the expansion rate of the Universe. During the oscillation period, the energy density already scales as $a^{-3}$. In this scenario, the Universe to be dominated by matter at $\eta=\eta_{\text{m}}$, then solitons form at $\eta=\eta_{\text{f}}$ and dominate the Universe at $\eta_{\text{d}}$, where $\eta_{\text{f}}=\eta_{\text{d}}>\eta_{\text{m}}$, and finally decay at $\eta=\eta_{\text{r}}$.

Similarly, we can find the relationship of the comoving wavenumbers with $k\sim aH$ corresponding to different time nodes: the wavenumber at the time of soliton formation, $k_{\text{f}}$; the wavenumber when the Universe begins to be dominated by matter, $k_{\text{m}}$; the wavenumber when solitons start to dominate the Universe, $k_{\text{d}}$; and the wavenumber at the time of soliton decay, $k_{\text{r}}$. Additionally, we define the characteristic wavenumber of solitons as $k_{\text{UV}}\equiv\frac{a}{d}$, beyond which the fluid description of solitons becomes invalid; the average number of solitons within a Hubble event horizon at the time of their formation as $n^{3}$. The lifetime of solitons can be described in terms of $N$, where $N$ is defined as $N\equiv\frac{k_{\text{f}}}{k_{\text{r}}}$. If solitons were formed during eRD era, the hierarchy of the relevant scales reads

and the relationship between them can be expressed through the parameters of the soliton model as follows

where the middle equation implies that the matter domination begins when the energy density of radiation and solitons are equal.

Similarly, if solitons form during the eMD era, we have

where we assume that $k_{\text{UV}}>k_{\text{m}}$ since the formation of solitons is a sub-horizon process. The relationship between them at this time is

We also obtain the evolution of the scale factor $a(\eta)$ and conformal Hubble parameter $\mathcal{H}$ as conformal time $\eta$ in different cases. For the solitons formed during the eRD era, the results can be written as

For the solitons formed during the eMD era,

Since solitons are rapidly diluted by cosmic expansion after their formation, while their physical size remains unchanged, they become well-separated and can be treated as non-interacting, point-like entities. Therefore, considering that the solitons are randomly distributed in space according to Poisson statistics with a mean physical separation of $d$, the power spectrum of soliton density perturbations can be expressed as [55]:

where $\delta\rho_{\text{s}}$ represents the density perturbations of $\rho_{\text{s}}$. The smoothed spectrum of density fluctuations is valid up to $k_{\text{UV}}$. For smaller scales, the complex effects of granularity become important and we set $k_{\text{UV}}$ as the cutoff scale.

The Fourier expansion of soliton density perturbations is

By plugging eq. (12) into eq. (11), the power spectrum of $\delta_{\mathbf{k}}$ can be expressed as

If solitons form during the eRD era, i.e., $\Omega_{\text{s,f}}\ll 1$, we define the density contrast of solitons, $\delta_{s}=\frac{\delta\rho_{\text{s}}}{\rho_{\text{s}}}$. In this case, isocurvature perturbations are given by

where we have used $\Omega_{s}\ll 1$. Thus, isocurvature fluctuations can be wiritten as

and the dimensionless isocurvature power spectrum can then be expressed as

During the eRD era, such isocurvature perturbations will convert into adiabatic perturbations. At the beginning of the sRD era, $\eta=\eta_{\text{r}}$, we can write the Bardeen potential, $\Phi_{k}$, as a summation of the two components,

where the first component comes from quantum fluctuations during inflation, while the second component is due to the isocurvature perturbations introduced by solitons. At large scale, curvature perturbations $\zeta$ are related to $\Phi$ as

As predicted by most well-motivated inflationary models, the power spectrum of $\zeta_{k,\text{inf}}$ is approximately scale-invariant,

From Planck-2018 data, we apply the scalar spectrum amplitude $A_{s}=2.1\times 10^{-9}$, the pivot scale $k_{*}=0.05$ $\mathrm{Mpc}^{-1}$, and the scalar spectral index $n_{\text{s}}=0.965$ [56].

Then, we investigate the evolution of isocurvature perturbations in the eRD era and eMD era. At the linear order, scalar perturbations $\Phi$ satisfies

where the prime denotes the deviative with respect to conformal time and $c_{s}^{2}$ is the sound speed of the Universe. Isocurvature perturbations act as a source term in the evolution of scalar perturbations and transform into adiabatic perturbations when the solitons dominate the Universe. In the eMD era, $\Phi_{k}$ does not evolve with time for the subhorizon modes. For $k<k_{\text{s}}$, the mode enters the horizon during the eMD era and becomes constant. For $k>k_{\text{s}}$, the mode enter the horizon during the eRD era and decreases until the eMD era begins. In addition, isocurvature perturbations will convert into adiabatic perturbations and we can obtain the power spectrum $\Phi_{k,\mathrm{s}}$ by solving eq. (20)

If solitons form during the eMD era, i.e., $\Omega_{\text{s,f}}=1$, we can obtain curvature perturbations directly from energy density perturbations of solitons using the relationship

Considering the fact that $\Phi$ is a constant during the eMD era, we obtain that

where $\Phi_{k}$ and $\delta_{k}$ are the Fourier forms of $\Phi$ and $\frac{\delta\rho_{\text{s}}}{\rho_{\text{s}}}$. In this way, we can get the power spectrum of scalar perturbations induced by solitons as

Note that if the solitons decay is not a perfectly instantaneous process, the finite duration will affect the modes whose period is larger than the rate of transition. This gives rise to an additional $k$-dependent suppression factor, which depends on the specific decay behavior of the soliton, in general, can be expressed as [57]:

where $\Phi^{\text{instant}}_{\text{sRD}}$ refers to the instantaneous transition value of scalar perturbations [58], $S\sim\mathcal{O}(1)$ and $m$ describes the decay behavior of the soliton which is dependent on the soliton model, and generally satisfies $-1<m<0$¹¹1 In certain soliton models, such suppression does not occur, represented effectively by setting $S=1$ and $m=0$..

Scalar and tensor perturbations of the metric coupled with each other at second order expansion of the Einstein equation. The equation of motion for tensor perturbations in the Fourier space can be written as

where $S_{\textbf{k}}$ represents the source term arising from first-order scalar perturbations. Using the Green’s function method, the power spectrum of tensor perturbations can be expressed as

where the overline denotes the oscillation average and we have defined dimensionless parameter $x=k\eta$. $I(v,u,x)$ is the kernel of induced GWs, which takes into account the time evolution of scalar perturbations,

The Green’s function $G_{\textbf{k}}(x,\tilde{x})$ can be obtained by solving the equation

In the following, we define $\Phi_{\textbf{k}}(\eta)\equiv T(k\eta)\phi_{\textbf{k}}$, where $T(k\eta)$ is the transfer function and $\phi_{\textbf{k}}$ is the primordial value. Since the velocities of solitons are negligible, we can calculate $f(u,v,\tilde{x},x_{i})$ in terms of the transfer function $T(k\eta)$

The GW energy spectrum is defined as the fraction of the GW energy density per logarithmic wavelength

The gravitational wave (GW) source term peaks around $\eta_{\mathrm{r}}$ and rapidly diminishes thereafter. Once generated, GWs hardly interact between GW and the background plasma is negligible, and the GW energy density scales with the expansion of the Universe as $a^{-4}$. Utilizing entropy conservation, the GW energy spectrum today, $\Omega_{\text{GW},0}$, can be expressed in terms of $\Omega_{\text{GW}}$ in the sRD era as [59, 60]

where $\Omega_{\text{r},0}h^{2}\sim 4.18\times 10^{-5}$ is the energy density fraction of radiation evaluated today [53]. In this paper, we use $g$ to denote the effective relativistic degrees of freedom , and $g_{c}\equiv g(\eta=\eta_{c})$.

In the eMD Universe, we can divide the contribution of $I(v,u,x)$ into three components, $I_{\text{eRD}}$, $I_{\text{eMD}}$ and $I_{\text{sRD}}$, which correspond to the GW production in different eras: the eRD era at first, followed by the eMD era and finally the sRD era. However, we only focus on GWs generated during the sRD era, because for both $\mathcal{P}_{\Phi_{\text{inf}}}$ and $\mathcal{P}_{\Phi_{\text{s}}}$, the dominant contribution to GWs comes from $I_{\text{sRD}}$, and the GW energy spectrum can be expressed as

As mentioned above, the scalar power spectrum $\mathcal{P}_{\Phi}$ can be written as

If solitons form during the eRD era, the inflationary scalar perturbation modes, $\Phi_{k,\mathrm{inf}}$, that reenter the horizon during the eRD era, are significantly suppressed so that we can set the cutoff scale of $\Phi_{\text{inf}}$ at $k=k_{\text{m}}$. If solitons form during the eMD era, primarily referring to oscillons in this paper, $\Phi_{k,\mathrm{inf}}$ with $k>k_{\text{m}}$ correspond to perturbations that remained inside the horizon during inflation, which provide the initial conditions for the self-resonance of inflaton perturbations. Thus, the cutoff condition $k<k_{\text{m}}$ is also valid in this case. Taking into account that eq. (21) and eq. (24) are valid only for $k<k_{\text{UV}}$, we can obtain

Since $\eta_{\text{r}}\gg\eta_{\text{m}}$, whether solitons were produced in the eMD era or the eRD era, the conmoving Hubble parameter $\mathcal{H}$ in the sRD era can be approximately expressed as $\mathcal{H}_{\text{sRD}}\approx\frac{2}{2\eta-\eta_{\text{r}}}$. We introduce a dimensionless variable $x_{\text{r}}=k\eta_{\text{r}}$, which corresponds to the beginning of the sRD era. Based on the background evolution given by eq. (7) and eq. (9), the GW kernel can be obtained by solving the Green’s equation eq. (29) during the sRD era

For simplicity, we define

The general solution for the transfer function in the sRD era, preceded by an eMD era can be written as

where $j_{1}$ and $y_{1}$ are the first and second spherical Bessel functions, and $A,B$ are constants which depend on $x_{r}$. Demanding the continuity of the transfer function and its time dericative at $\eta=\eta_{\text{r}}$, we can determine $A$ and $B$ as

In principle, using eq. (39) and eq. (30), we can directly obtain the GW energy density $\Omega_{\text{GW}}$. However, our aim is to establish a relationship between GWs and the parameters of the soliton model, which will further allow us to impose constraints on the soliton model parameters. Therefore, we seek to derive approximate analytical formulas for induced GWs. We then obtain the approximate form of $\mathcal{I}_{\text{sRD}}$ on the scales of $k\gg k_{\text{r}}$ and $k\ll k_{\text{r}}$, and use this approximate form to calculate induced GWs.

In the small-scale approximation, $k\gg k_{r}$ and $x\gg x_{\text{r}}$, the primary contributions to the kernel term $\mathcal{I}_{sRD}$ come from the sum of trigonometric terms. These include the terms such as $A(u,v,x_{\text{r}})x_{\text{r}}^{4}\sin\left(D_{\pm\pm}x_{\text{r}}\right)$, $B(u,v,x_{\text{r}})x_{\text{r}}^{4}\cos\left(D_{\pm\pm}x_{\text{r}}\right)$, cosine integral terms like $C(u,v,x_{\text{r}})x_{\text{r}}^{4}\mathrm{Ci}\left(D_{\pm\pm}x_{\text{r}}\right)$, and sine integral terms $D(u,v,x_{\text{r}})x_{\text{r}}^{4}\mathrm{Si}\left(D_{\pm\pm}\right)$, where $A(u,v,x_{\text{r}})$, $B(u,v,x_{\text{r}})$, $C(u,v,x_{\text{r}})$ and $D(u,v,x_{\text{r}})$ represent the coefficients that depend on $u$, $v$ and $x_{\text{r}}$. Additionally, we define $D_{\pm\pm}=\frac{u\pm v\pm\sqrt{3}}{3\sqrt{3}}$ [6]. Furthermore, we introduce the cosine integral and sine integral functions, defined as:

In this case, we are interested only in the resonant part of induced GWs. Following the calculation of the kernel term of $\mathcal{I}_{\text{sRD}}$, the terms containing $\mathrm{Ci}$ contribute the main part [37]

For the large-scale approximation, considering the cutoff conditions of $\mathcal{P}_{\Phi,\text{inf}}$ and $\mathcal{P}_{\Phi,\text{s}}$, the integrals over $u$ and $v$ are primarily dominated by the region where $u+v-1$ is large and $u-v$ is small. Under these approximations, we find that

Therefore, the energy spectrum of induced GWs can be expressed as

These four terms represent the contribution of $\mathcal{P}_{\Phi,\text{inf}}$ and $\mathcal{P}_{\Phi,\text{s}}$ on small and large scales, respectively. Using eq. (44) and eq. (45), we can derive the following expression

Here, the lower and upper limits of the integral are described by $s_{\text{m}}$ and $s_{\text{UV}}$, which come from the cutoff condition of $\Theta(k_{\text{m}}-k)$ and $\Theta(k_{\text{UV}}-k)$. The upper limit $s_{\text{UV}}$ is a function of $k$

The result of eq. (46) also contains contributions from the cross terms between $\mathcal{P}_{\Phi,\text{inf}}$ and $\mathcal{P}_{\Phi,\text{s}}$. In most cases either $\mathcal{P}_{\Phi,\text{inf}}$ or $\mathcal{P}_{\Phi,\text{s}}$ dominates the total $\mathcal{P}_{\Phi}$, $\Omega_{\mathrm{GW}}$ is mainly contributed from the dominant term and the cross term is negligible.