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Benchmark GWBs of primordial origin with large amplitude above kHz frequencies, compared to sensitivities of existing and planned experiments below the kHz \cite{LISA:2017pwj, LISACosmologyWorkingGroup:2022jok, Yagi:2011wg, AEDGE:2019nxb, KAGRA:2013rdx, LIGOScientific:2014qfs, LIGOScientific:2019vic, Hild:2010id, Punturo:2010zz, LIGOScientific:2016wof,Blas:2021mqw,Fedderke:2021kuy} as well as experiments sensitive at frequencies above the kHz from \cite{Aggarwal:2020olq} (in shaded gray). The green line is associated with a very strong first-order phase transition \cite{Caprini:2015zlo} ($\beta/H=7$, $\alpha=10$) at a temperature $T\sim 10^{10}$ GeV (compatible with a Peccei-Quinn phase transition with axion decay constant $f_a\sim 10^{10}$ GeV for instance \cite{VonHarling:2019rgb}). Interestingly, the irreducible background from inflation with inflationary scale $E_{\rm inf} \simeq 10^{16}$ GeV can be amplified if inflation is followed by kination (purple line) \cite{Gouttenoire:2021jhk} or if a kination era is induced much later by the rotating QCD axion DM field (blue line) \cite{Gouttenoire:2021wzu,Gouttenoire:2021jhk, Co:2021lkc}. Local cosmic strings can generate a signal (in red) as large as the BBN bound \eqref{eq:Neff_bound}, that also uniquely goes beyond $10^9$ Hz. The gray line shows the signal from preheating \cite{Figueroa:2022iho} corresponding to an inflaton mass $M\simeq M_{\rm pl}$ with a coupling $g = 10^{-3}$ to the thermal bath. Similar but suppressed GW spectra can come from the fragmentation of a scalar field, which is not the inflaton \cite{Kitajima:2018zco, Chatrchyan:2020pzh, Eroncel:2022vjg}. The lower gray shaded region is the spectrum from the Standard thermal plasma \cite{Ghiglieri:2015nfa, Ghiglieri:2020mhm, Ringwald:2020ist}, assuming a reheating temperature $T_{\rm reh}\simeq 6 \times 10^{15}$ GeV.

Local-string GWBs in the UHF range featuring different cutoffs. The solid lines show the stable-string GWB with the formation cutoff \eqref{eq:formation_cut_off}. The dashed, dotted, and dot-dashed lines correspond to the cutoffs from cusp \eqref{eq:freq_cusp}, kink \eqref{eq:freq_kink}, and friction \eqref{eq:freq_friction}, respectively, assuming the stable-string network. The four rectangular purple regions denote hypothetical sensitivities of four fictional UHF-GW experiments; see Eq.~\eqref{eq:UHF_experiment}. The ``thermal plasma" gray region is the GWB predicted in \cite{Ghiglieri:2015nfa, Ghiglieri:2020mhm, Ringwald:2020ist}, assuming the maximal reheating temperature $T_{\rm reh}^{\rm max} \simeq 6 \times 10^{15} ~ {\rm GeV}$ \cite{Ghiglieri:2015nfa, Ghiglieri:2020mhm, Ringwald:2020ist}, while the upper ``\textsc{bbn-gw}" gray region is excluded by the BBN bound \eqref{eq:Neff_bound}.

GWB spectra from metastable cosmic strings with the tension $G\mu$ and the metastability parameter $\kappa$ given by the three benchmark values in Fig.~\ref{fig:local_GW_GWB_contour}. The solid lines assume the GWB with the formation cutoff indicated by the vertical dashed gray line, while the dashed lines assume the GWB with cusp cutoffs. The IR tails of the spectra exhibit the transition from $f_{\rm GW}^2$ to $f_{\rm GW}^3$ in the low-frequency direction; see Eq.~\eqref{eq:causality_turning_point}. The top gray region denotes the BBN bound in Eq.~\eqref{eq:BBN_GW_CS}. The bottom gray region is the thermal plasma GWB. The purple rectangles denote hypothetical GW experiments operating at the UHF range [defined by Eq.~\eqref{eq:UHF_experiment}], while other colored regions are the power-law integrated sensitivity curves of current and future GW observatories, taken from \cite{Gouttenoire:2019kij}.

GWB spectra from metastable cosmic strings with the tension $G\mu$ and the metastability parameter $\kappa$ given by the three benchmark values in Fig.~\ref{fig:local_GW_GWB_contour}. The solid lines assume the GWB with the formation cutoff indicated by the vertical dashed gray line, while the dashed lines assume the GWB with cusp cutoffs. The IR tails of the spectra exhibit the transition from $f_{\rm GW}^2$ to $f_{\rm GW}^3$ in the low-frequency direction; see Eq.~\eqref{eq:causality_turning_point}. The top gray region denotes the BBN bound in Eq.~\eqref{eq:BBN_GW_CS}. The bottom gray region is the thermal plasma GWB. The purple rectangles denote hypothetical GW experiments operating at the UHF range [defined by Eq.~\eqref{eq:UHF_experiment}], while other colored regions are the power-law integrated sensitivity curves of current and future GW observatories, taken from \cite{Gouttenoire:2019kij}.

Local-string GWBs in blue with the metastability and cusp cutoffs, at fixed $G\mu=10^{-5}$, for different $\kappa$ where the dashed orange lines assume the cutoff from formation instead of the cusp. The spectrum exhibits a flat plateau shape when $f_{\rm GW}^{\rm meta} < f_{\rm GW} < f_{\rm GW}^{\rm cusp}$. For $f_{\rm GW}^{\rm meta} > f_{\rm GW}^{\rm cusp}$, the spectrum has a peak shape with the suppressed amplitude depending on the ratio $f_{\rm GW}^{\rm meta}/f_{\rm GW}^{\rm cusp}$ [see Eq.~\eqref{eq:GW_suppressed_two_cutoffs}].

Local-string GWBs in blue with the metastability and cusp cutoffs, at fixed $G\mu=10^{-5}$, for different $\kappa$ where the dashed orange lines assume the cutoff from formation instead of the cusp. The spectrum exhibits a flat plateau shape when $f_{\rm GW}^{\rm meta} < f_{\rm GW} < f_{\rm GW}^{\rm cusp}$. For $f_{\rm GW}^{\rm meta} > f_{\rm GW}^{\rm cusp}$, the spectrum has a peak shape with the suppressed amplitude depending on the ratio $f_{\rm GW}^{\rm meta}/f_{\rm GW}^{\rm cusp}$ [see Eq.~\eqref{eq:GW_suppressed_two_cutoffs}].

Future detectability regions of metastable local-string GWB by several experiments (only the gray region from the BBN bound and the dark blue from LIGO O2 are excluded). The criterion for detection and constraints is any part of the spectrum lying in the sensitivity and constrained regions. The purple solid, dashed, long-dashed, and dot-dashed lines illustrate the extended detectable region from hypothetical UHF experiments operating at different $\{\Omega_{\rm GW}^{\tt sens}h^2,f_{\tt sens}\}$. The red symbols are associated with the benchmark GW spectra in Fig.~\ref{fig:local_GW_GWB}. Assuming loops also produce particles through cusps, the dashed gray line marks the region where $f_{\rm GW}^{\rm brk}>f_{\rm GW}^{\rm cusp}$ and the GW is suppressed according to Eq.~\eqref{eq:GW_suppressed_two_cutoffs} [see also Fig.~\ref{fig:local_GW_GWB_combined_cutoff}]. Below the gray dotted line ($T_{\rm brk} > T_{\rm cusp}$), all loops produced before the network decay release energy via particle production and thus do not generate GW signal. An earlier version of this plot appeared in \cite{Buchmuller:2019gfy}; we extended it by displaying the gained reach from UHF detectors.

By varying the string tension $G\mu$, positions of the cutoffs---cusp, kink, and friction on the local-string GWB [see Eqs.~\eqref{eq:kink_cutoff_position}, \eqref{eq:cusp_cutoff_position}, and \eqref{eq:friction_cutoff_position}]. For cusp and kink, the solid, dashed, and dotted lines correspond respectively to $\beta_c,\beta_k$ = 1,10,100. The friction cutoff with $\beta_{\rm fric} = 0.01,0.1,1$ is shown as solid, dashed, and dotted lines, respectively. The yellow spectra are those of the stable-string network with the formation cutoff.

Colored region in each plot shows the detectable local-string GWB spectrum, including cusp (left-column), kink (middle-column), and friction (right-column) cutoffs. The detectable region splits into three parts depending on the feature of the GWB spectrum: the plateau (flat slope $\Omega_{\rm GW} \propto f_{\rm GW}^{0}$), the cutoff (where the slope changes), and the UV slope ($\Omega_{\rm GW} \propto f_{\rm GW}^{-1/3}$). We assume the hypothetical UHF experiments operating at $\Omega_{\rm GW}^{\tt sens} = 10^{-10}$ with central frequencies: (i) 10 kHz, (ii) 0.1 GHz, and (iii) 10 GHz. The yellow region shows where these UV cutoffs are higher than the formation cutoff. The gray contours are the characteristic strain $h_c$ that would be detected in such experiments.

Parameter space of the heavy axion below the solid black line gives rise to the axion MD era with duration $a_{\rm end}/a_{\rm dom}$ [marked by the dashed black line], which decays before the BBN (otherwise, it is constrained as in the dark gray region.) The axions are produced from string-network decay as a subdominant energy density at $T_{\rm dec}$ (vertical dotted orange lines in the top panel). When they dominate the Universe, the axion MD era starts at $T_{\rm dom}$ (blue dashed lines) and later ends when they decay at $T_{a \gamma}$ (green dotted lines). The lighter gray region has the string network decaying before the string formation at temperature $f_a$. The perturbativity of the axion effective-field-theory description breaks down in the red region.

Parameter space of the heavy axion below the solid black line gives rise to the axion MD era with duration $a_{\rm end}/a_{\rm dom}$ [marked by the dashed black line], which decays before the BBN (otherwise, it is constrained as in the dark gray region.) The axions are produced from string-network decay as a subdominant energy density at $T_{\rm dec}$ (vertical dotted orange lines in the top panel). When they dominate the Universe, the axion MD era starts at $T_{\rm dom}$ (blue dashed lines) and later ends when they decay at $T_{a \gamma}$ (green dotted lines). The lighter gray region has the string network decaying before the string formation at temperature $f_a$. The perturbativity of the axion effective-field-theory description breaks down in the red region.

Top panel: GWB spectra in the energy-density fraction $\Omega_{\rm GW}h^2$ (left) and in the characteristic strain $h_c$ (right) corresponding to the benchmark $\{m_a,f_a\}$ values in Fig.~\ref{fig:global_spectrum_gw}-bottom. The black dotted lines show the peak position of GWB spectrum for a constant $f_a$ while varying $m_a$ within the range where $m_a$ is allowed by $T_{a\gamma}$ above BBN and $T_{\rm dec} < T_{\rm reh}^{\rm max}$; see the bottom panel. The scaling of these lines follow $\Omega_{\rm GW}^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{8/5}$ and $h_c^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{-1/5}$ [see main text for derivation]. Bottom panel: Contours of the peak amplitude (left) and peak frequency (right) of the GW spectra, i.e., at $f_{\rm GW}(T_{\rm dec})$. The GW from heavy-axion strings cannot generate an observable GW signal even for large $f_a$ due to the heavy dilution from the axion matter-dominated era. ($\star,*$ for $m_a = 5 \cdot 10^{13}$ GeV, $+,\times$ for $m_a = 10^{8}$ GeV)

Top panel: GWB spectra in the energy-density fraction $\Omega_{\rm GW}h^2$ (left) and in the characteristic strain $h_c$ (right) corresponding to the benchmark $\{m_a,f_a\}$ values in Fig.~\ref{fig:global_spectrum_gw}-bottom. The black dotted lines show the peak position of GWB spectrum for a constant $f_a$ while varying $m_a$ within the range where $m_a$ is allowed by $T_{a\gamma}$ above BBN and $T_{\rm dec} < T_{\rm reh}^{\rm max}$; see the bottom panel. The scaling of these lines follow $\Omega_{\rm GW}^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{8/5}$ and $h_c^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{-1/5}$ [see main text for derivation]. Bottom panel: Contours of the peak amplitude (left) and peak frequency (right) of the GW spectra, i.e., at $f_{\rm GW}(T_{\rm dec})$. The GW from heavy-axion strings cannot generate an observable GW signal even for large $f_a$ due to the heavy dilution from the axion matter-dominated era. ($\star,*$ for $m_a = 5 \cdot 10^{13}$ GeV, $+,\times$ for $m_a = 10^{8}$ GeV)

Top panel: GWB spectra in the energy-density fraction $\Omega_{\rm GW}h^2$ (left) and in the characteristic strain $h_c$ (right) corresponding to the benchmark $\{m_a,f_a\}$ values in Fig.~\ref{fig:global_spectrum_gw}-bottom. The black dotted lines show the peak position of GWB spectrum for a constant $f_a$ while varying $m_a$ within the range where $m_a$ is allowed by $T_{a\gamma}$ above BBN and $T_{\rm dec} < T_{\rm reh}^{\rm max}$; see the bottom panel. The scaling of these lines follow $\Omega_{\rm GW}^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{8/5}$ and $h_c^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{-1/5}$ [see main text for derivation]. Bottom panel: Contours of the peak amplitude (left) and peak frequency (right) of the GW spectra, i.e., at $f_{\rm GW}(T_{\rm dec})$. The GW from heavy-axion strings cannot generate an observable GW signal even for large $f_a$ due to the heavy dilution from the axion matter-dominated era. ($\star,*$ for $m_a = 5 \cdot 10^{13}$ GeV, $+,\times$ for $m_a = 10^{8}$ GeV)

Top panel: GWB spectra in the energy-density fraction $\Omega_{\rm GW}h^2$ (left) and in the characteristic strain $h_c$ (right) corresponding to the benchmark $\{m_a,f_a\}$ values in Fig.~\ref{fig:global_spectrum_gw}-bottom. The black dotted lines show the peak position of GWB spectrum for a constant $f_a$ while varying $m_a$ within the range where $m_a$ is allowed by $T_{a\gamma}$ above BBN and $T_{\rm dec} < T_{\rm reh}^{\rm max}$; see the bottom panel. The scaling of these lines follow $\Omega_{\rm GW}^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{8/5}$ and $h_c^{\rm peak} \propto (f_{\rm GW}^{\rm peak})^{-1/5}$ [see main text for derivation]. Bottom panel: Contours of the peak amplitude (left) and peak frequency (right) of the GW spectra, i.e., at $f_{\rm GW}(T_{\rm dec})$. The GW from heavy-axion strings cannot generate an observable GW signal even for large $f_a$ due to the heavy dilution from the axion matter-dominated era. ($\star,*$ for $m_a = 5 \cdot 10^{13}$ GeV, $+,\times$ for $m_a = 10^{8}$ GeV)

Maximal GWBs expected from cosmic strings (local in blue, global in red/orange) that can arise at ultra-high frequencies without leading to any observable signal in the frequency range of existing or planned interferometers. The spectra from global axionic strings can be large and detectable at lower frequencies for light axions \cite{ Ramberg:2019dgi,Chang:2019mza,Gouttenoire:2019kij,Gelmini:2021yzu,Chang:2021afa,Gorghetto:2021fsn,Gelmini:2022nim,Servant:2023mwt}. However, when requiring a signal that arises only beyond the kHz, this fixes the axion mass to be above the GeV scale to avoid constraints from late decays after BBN. Such heavy axions induce an early temporary matter-domination era that suppresses the GWB.

Top panel: The solid lines show GWB spectrum from each $k$-mode of loop oscillations, taking into account the limit on maximum emission energy; see Eq.~\eqref{eq:higher_mode_cutoff}. We compare this to the GWB spectra (dashed lines), assuming only the formation cutoff. Bottom panel: the total GW spectra, summing up to the first million modes.

Top panel: The solid lines show GWB spectrum from each $k$-mode of loop oscillations, taking into account the limit on maximum emission energy; see Eq.~\eqref{eq:higher_mode_cutoff}. We compare this to the GWB spectra (dashed lines), assuming only the formation cutoff. Bottom panel: the total GW spectra, summing up to the first million modes.

Local-string GWB spectra varying $G\mu$ in blue with the metastability ($\kappa = 30$) and cusp cutoffs (in red). The spectrum exhibits a flat plateau shape when $f_{\rm GW}^{\rm meta} < f_{\rm GW} < f_{\rm GW}^{\rm cusp}$. For $f_{\rm GW}^{\rm meta} > f_{\rm GW}^{\rm cusp}$, the spectrum becomes a peak shape with the suppressed amplitude [see Eq.~\eqref{eq:GW_suppressed_two_cutoffs}].

The GWB from metastable local strings consists of three contributions: (i) loops (solid lines) discussed in section~\ref{sec:local_metastable_network}, (ii) segments from loops (dot-dashed lines), and (iii) segments from long strings (dashed lines). Assuming $\Gamma_{\rm seg} = \Gamma = 50$, the left plot shows the spectra of the same benchmark scenarios as in Fig.~\eqref{fig:local_GW_GWB}, where we plot again the loops' GWB for comparison. The right plot illustrates the effect of $\Gamma_{\rm seg}$ on the GWB from segments for the benchmark scenario ``cross".

The GWB from metastable local strings consists of three contributions: (i) loops (solid lines) discussed in section~\ref{sec:local_metastable_network}, (ii) segments from loops (dot-dashed lines), and (iii) segments from long strings (dashed lines). Assuming $\Gamma_{\rm seg} = \Gamma = 50$, the left plot shows the spectra of the same benchmark scenarios as in Fig.~\eqref{fig:local_GW_GWB}, where we plot again the loops' GWB for comparison. The right plot illustrates the effect of $\Gamma_{\rm seg}$ on the GWB from segments for the benchmark scenario ``cross".

In the axion parameter space, the contour lines show amplitudes and frequencies of the modified causality tail of the axion-string GWB due to the axion MD era.

In the axion parameter space, the contour lines show amplitudes and frequencies of the modified causality tail of the axion-string GWB due to the axion MD era.