Gaussian, so that the Fisher approximation recovers it. However, a remarkable
non-Gaussian correlation arises in the posterior of $\gamma_{\mathrm{ES}}$
with $h^{2}\Omega_{*}$ and $\omega_{\mathrm{ES}}$. In particular, the 2D
posteriors have 45-degree turns for small $\gamma_{\mathrm{ES}}$. This can be
explained as a ‘second order degeneracy’: simultaneously changing
$h^{2}\Omega_{*}$ and $\omega_{\mathrm{ES}}$ mimics a variation in
$\gamma_{\mathrm{ES}}$ (see fig. 23 in appendix A). In this case, as expected,
the Fisher approximation fails, and we can only set a mild lower bound at
$68$% C.L.. Both galactic and extragalactic foregrounds are reconstructed,
although constraints on the latter are prior-dominated, as they are masked by
the primordial signal within the LISA band.

Concerning ES-BNK_2 (see top right panel in fig. 13), the peak lies below the
galactic foreground, despite the ES-BNK_2’s having an amplitude larger than
ES-BNK_1. However, constraints on $\omega_{\mathrm{ES}}$ and
$\gamma_{\mathrm{ES}}$ are still quite good. In particular, the parameter
$\gamma_{\mathrm{ES}}$ can be detected, as the oscillatory part of the signal
falls within the LISA sensitivity thanks to its relatively large amplitude.
Some degeneracies among the parameters appear, but the non-Gaussianity of the
posterior is milder than for ES-BNK_1, and we can notice a good agreement with
the Fisher analysis, which captures quite well the shapes of the posteriors.
Noise and foregrounds are also well reconstructed.

Finally, the benchmark ES-BNK_3 peaks at $1\,\rm{mHz}$ (see bottom panel in
fig. 13) like ES-BNK_2, but has a smaller amplitude, so the peak is well below
the galactic foreground. Despite this fact, the overall peak’s amplitude and
position are well reconstructed, with their mean values within the 68 % CL
error contour. For their 2D posterior, there is good agreement between the
Fisher and the nested sampling analyses. As for ES-BNK_1, we notice a
degeneracy between $\omega_{\mathrm{ES}}$ and $\gamma_{\mathrm{ES}}$. The
oscillations fall outside the sensitivity of LISA, and, as a consequence, the
parameter $\gamma_{\mathrm{ES}}$ results unconstrained.

###  4.6 Forecasts for the linear oscillations template

We now focus on the LO template in eq. 20 where, for concreteness, we take the
log-normal bump for the envelope $\Omega_{\rm GW}^{\rm env}(f)$. Figure 14
shows the results obtained with the Fisher analysis in a parameter-space slice
crossing the benchmarks detailed underneath. The left panels show the (either
relative or absolute) error on the envelope parameters, while the right panels
show the errors on the LO parameters, namely the amplitude, frequency and
phase of the oscillations. In each panel, we vary the envelope amplitude
$h^{2}\Omega_{\rm GW}$ as well as the magnitude of the oscillations
$\mathcal{A}_{\rm lin}$, while for the other parameters we take $\\{f_{*}/{\rm
mHz},\,\rho,\,\omega_{\rm lin}\mathrm{Hz},\theta_{\rm lin}\\}=\\{1,0.08,5\cdot
10^{4},0\\}$.

Concerning the envelope parameters, the amplitude and the pivot frequency,
$h^{2}\Omega_{*}$ and $f_{*}$, have errors below $10\,\%$ for
$\log_{10}h^{2}\Omega_{*}\gtrsim-13$, whereas a similar error on the width
$\rho$ requires $\log_{10}h^{2}\Omega_{*}\gtrsim-11.5$, in agreement with the
analysis in section 4.2. Such errors turn out to be almost independent of the
amplitude of the oscillation $\mathcal{A}_{\rm lin}$. Concerning the LO
parameters, errors degrade as the amplitude $\mathcal{A}_{\rm lin}$ decreases.
Regarding the amplitude of the oscillations, an accuracy below $30\%$ (in the
considered parameter space slice) requires
$\log_{10}h^{2}\Omega_{*}\gtrsim-11$, with a slight dependence on the
amplitude of the oscillations, in particular the error deteriorates at
$\mathcal{A}_{\rm lin}\lesssim 0.2$. More surprisingly, pinpointing the
frequency $\omega_{\rm lin}$ to good accuracy is possible for signals with
smaller amplitude and until very small values of the oscillatory amplitudes.
In particular, an absolute error on $\log_{10}\omega_{\rm lin}$ below $0.3$ is
granted for $\log_{10}h^{2}\Omega_{*}\gtrsim-11.5$ even for very small
oscillatory amplitudes, _i.e.,_ $\mathcal{A}_{\rm lin}\lesssim 0.01$. The
accuracy on $\theta_{\rm lin}$ is similar to the one on $\mathcal{A}_{\rm
lin}$.

Constraints on all the parameters are clearly affected by astrophysical
foregrounds. The impact of the latter varies depending on the specific
parameter, but can degrade the accuracy of the error by up to an order of
magnitude.

Figure 14: Fisher forecasts for the LO template with the LN envelope. The
panels show the 68 % CL reconstruction error on $h^{2}\Omega_{*}$ (top left),
$\mathcal{A}_{\rm lin}$ (top right), $f_{*}$ (central left), $\omega_{\rm
lin}$ (central right), $\rho$ (bottom left) and $\theta_{\rm lin}$ (bottom
right) as a function of the injected values $h^{2}\Omega_{*}$ and
$\mathcal{A}_{\rm lin}$ and by fixing the other parameters to
$\\{f_{*}=1\,\mathrm{mHz},\,\rho=0.08,\,\omega_{\mathrm{lin}}\,\mathrm{Hz}=5\cdot
10^{4},\,\theta_{\mathrm{lin}}=0\\}$. SNR contour lines are plotted in white.
Depending on the panel, the pairs of dashed lines mark the absolute error
$\sigma=0.3$ ($\sigma=0.01$) and relative error 30% ($1\%$) contours,
respectively in the absence [white] and in the presence [black] of
foregrounds. The cross displays the benchmarks LO-BNK_1.

Benchmarks.— We consider benchmarks with overall amplitude
$h^{2}\Omega_{*}=10^{-11}$. We set the width of the lognormal envelope
$\rho=0.08$ to reproduce the second peak of the benchmarks chosen for the DP
template in fig. 11. We also fix the amplitude of the oscillations at
$\mathcal{A}_{\rm lin}=0.3$ as this value represents a proxy to discriminate
between standard radiation domination and a non-standard thermal history (see
section 2.6). Based on theoretical considerations, one expects
$\omega_{\mathrm{lin}}\sim 10/f_{*}$, which naturally provides various
oscillations within the range of frequency where the signal peaks. Concretely,
we fix it according to eq. 23 with the microphysical parameters
$\eta_{\perp}=14$ and $\delta=0.25$. This choice for $\eta_{\perp}$ and
$\delta$ exemplifies a sharp feature, induced by a strong turn in field space,
that gives an amplitude of the curvature power spectrum resulting, within the
Gaussian approximation, in an amount of PBHs in the range of the dark matter
abundance. At this point, only $f_{*}$ is still unfixed, and the two options
for its value, $f_{*}=1\,$mHz and $f_{*}=5\,$mHz, are the only difference in
our benchmarks LO-BNK_1 and LO-BNK_2. Specifically, LO-BNK_1 and LO-BNK_2 are
defined as the SGWB signals following eq. 20 with the LN envelope and the
parameters $\\{h^{2}\Omega_{*},\,f_{*}/{\rm mHz},\,\rho,\,{\cal A}_{\rm
lin},\,\omega_{\rm lin}{\rm mHz},\,\theta_{\rm lin}\\}$ equal to
$\\{-11,\,1,\,0.08,\,0.3,\,50,\,0\\}$ and
$\\{-11,\,5,\,0.08,\,0.3,\,10,\,0\\}$, respectively.

In fig. 15 (left plot) we show the corner plot of the reconstructed mean value
parameters for the LO-BNK_1. The oscillatory pattern turns out to be very well
reconstructed for an overall amplitude $\log_{10}(h^{2}\Omega_{*})=-11$,
although the main peak and oscillations are below the galactic foreground. In
particular, note the exquisite estimate (order of $0.01\,\%$ error at
$1$-sigma) for the frequency of the oscillations. Overall, all the mean values
for all the reconstructed parameters are within 68 % CL from the injected
values, while all 2D posteriors are Gaussian and exhibit percent accuracy in
very good agreement with the Fisher analysis. A very mild degeneracy appears
between the phase $\theta_{\mathrm{lin}}$ and the frequency
$\omega_{\textrm{lin}}$. This degenarecy is more manifest in the
reconstruction corner plot of LO-BNK_2 reported in the right plot of fig. 15.
In this case, the injected signal has the same overall amplitude as LO-BNK_1,
but its peak frequency $f_{*}$ is much larger and consequently, the signal is
shifted to higher frequencies and is not covered by the galactic foreground.
Compared to the previous case, the reconstruction accuracy is even better and
the overall oscillatory pattern is well captured, with relative errors on the
various parameters in agreement with the Fisher forecast in fig. 14. We also
find that the Fisher analysis captures very well the shape of the posterior
distribution. This is true even for the 2D contour between
$\theta_{\mathrm{lin}}$ and the frequency, which, in this case, shows a
significant degeneracy.

For both benchmarks, the noise as well as the foregrounds are well
reconstructed, with a slightly less accurate reconstruction of the
extragalactic foreground when its amplitude is below the cosmological signal.

Figure 15: 1D and 2D posterior distributions derived from the LO template
reconstruction of the benchmarks LO-BNK_1 (left) and LO-BNK_2 (right). Lines’
styles and color codes are as in fig. 3. LO-BNK_1 and LO-BNK_2 are defined as
in eq. 20 with
$\\{f_{*}/\mathrm{mHz},\,\log_{10}(\omega_{\mathrm{lin}}\,\mathrm{Hz})\\}=\\{5,\,4\\}$
and $\\{1,\,4.7\\}$ respectively, and fixed $\\{\rho,{\cal
A}_{\mathrm{lin}}\\}=\\{0.08,0.3\\}$.

###  4.7 Forecasts for the resonant oscillations template

The RO template is defined in eq. (2.7). For concreteness, in our RO forecasts
we focus on the case where the envelope is a flat spectrum, _i.e.,_
$h^{2}\Omega_{\rm GW}^{\rm env}=h^{2}\Omega_{*}$. In fig. 16 we forecast the
reconstruction errors obtained with the Fisher analysis as a function of
$\omega_{\textrm{log}}$ and the overall amplitude $h^{2}\Omega_{*}$, fixing
the amplitude of the logarithmic oscillations in the scalar power spectrum to
the representative value $A_{\textrm{log}}=0.5$. We recall that the parameter
$\omega_{\textrm{log}}$ not only controls the frequency of the two cosines in
eq. (2.7), but it also affects their amplitudes through the functions
$\mathcal{C}_{1,2}$, see fig. 1. The SNR and reconstruction-error contour
lines exhibit an oscillating behavior at $\omega_{\textrm{log}}\lesssim 10$.
In fact, in such a case a single period of oscillation spans more than the
LISA frequency band. This explains why the error on the reconstruction of
$\omega_{\textrm{log}}$ rapidly degrades for $\omega_{\textrm{log}}\lesssim
4$. Furthermore, a small variation in $\omega_{\textrm{log}}$ can lead to a
frequency shift that rapidly modifies the SNR when a maximum is replaced by a
minimum at $f\simeq 3\,$mHz, where LISA has its best sensitivity. This of
course leads to an important variation of the reconstruction errors. The same
feature is present independently of whether the foregrounds are also
reconstructed. On the other hand, for $\omega_{\textrm{log}}\gtrsim 15$ there
are many oscillations within the LISA frequency band and the SNR then becomes
independent of $\omega_{\textrm{log}}$. A relevant role is also played by the
function $\mathcal{C}_{1,2}$ (see section 2.7 for more details). The minimum
relative error is reached for $\omega_{\textrm{log}}\simeq 8$ for a given
amplitude, with an accuracy of $30\%$ (1$\%$) that requires SNR $\gtrsim 8$
(200) without foregrounds, and larger when foregrounds are taken into account.
This behaviour can be ascribed to the dependencies of $\mathcal{C}_{1,2}$ on
$\omega_{\textrm{log}}$. In particular, for these values of
$\omega_{\textrm{log}}$, $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are
comparable and the signal becomes the superposition of two harmonics with
about the same amplitude ${\cal A}_{1}\simeq{\cal A}_{2}$, a case which is
easier to identify. In general, the larger $\omega_{\textrm{log}}$, the
smaller the accuracy, simply because the amplitude of the oscillations gets
smaller (again see the $\mathcal{C}_{1,2}$ dependencies on
$\omega_{\textrm{log}}$) and eventually one would reach the limit where the
signal varies faster than the LISA frequency resolution, as can be argued from
the fact that the error rapidly increases as $\omega_{\rm log}$ grows.

Figure 16:  Fisher forecasts for the RO template with the flat envelope
$h^{2}\Omega_{\rm GW}=h^{2}\Omega_{*}$. The panels show the 68 % CL
reconstruction error on $\log_{10}(h^{2}\Omega_{*})$ (top left), $A_{\rm log}$
(top right), $\omega_{\rm log}$ (bottom left) and $\phi_{\rm log}$ (bottom
right) as a function of the injected values of $h^{2}\Omega_{*}$ and
$\omega_{\rm log}$ and by fixing $A_{\rm log}=0.5$ and $\phi_{\rm log}=0$. SNR
contour lines are plotted in white. Depending on the panel, the pairs of
dashed lines mark the absolute error $\sigma=0.3$ ($\sigma=0.01$) and relative
error 30% ($1\%$) contours, respectively in the absence [white] and in the
presence [black] of foregrounds. The cross display the benchmark RO-BNK_2.

Benchmarks.— As described in section 2.7, the template is characterised by two
harmonics only for $\omega_{\rm log}>\mathcal{O}(1)\omega_{c}$, with
$\omega_{c}\simeq 4.77$ and the order one parameter depending on $A_{\rm
log}$. Thus, we consider two benchmarks, with the rationale that
$\omega_{\log}$ provides a signal shape where the double oscillatory pattern
does appear, _i.e.,_ RO-BNK_1 with $\\{\log_{10}(h^{2}\Omega_{*})\,,\
A_{\textrm{log}}\,,\ \omega_{\textrm{log}}\,,\
\phi_{\textrm{log}}\,\\}=\\{-11,1,12,0\\}$, and does not appear, _i.e.,_ RO-
BNK_2 with $\\{\log_{10}(h^{2}\Omega_{*})\,,\ A_{\textrm{log}}\,,\
\omega_{\textrm{log}}\,,\ \phi_{\textrm{log}}\,\\}=\\{-12,0.5,6,0\\}$. The
left panel of fig. 17 shows the SGWBinner RO template-based reconstruction
applied to RO-BNK_1. The reconstruction is very accurate, with the injected
parameters well within 68 % CL contours. The peculiar shape of the template is
mainly reconstructed thanks to the part of signal that is above both
foregrounds and still in the LISA band. Given the small range of frequency
where this happens, we interpret it as the origin of the strong degeneracy
between the parameters $\omega_{\textrm{log}}$ and $\phi_{\textrm{log}}$. The
extragalactic foreground signal is not well reconstructed since, within the
LISA frequency band, it is orders of magnitude below RO-BNK_1.

The right panel of fig. 17 presents the SGWBinner reconstruction for RO-BNK_2.
With its envelope amplitude $h^{2}\Omega_{*}=10^{-12}$, RO-BNK_2 happens to be
right below both foregrounds, and is comparable to them around $f\sim{\rm
mHz}$. We are thus in an envelope amplitude regime where the error accuracy is
expected to degrade. However, it turns out that the amplitude is still high
enough to permit a clear reconstruction of the overall frequency shape. No
sizable biases emerge in the analysis as all the injected values are within
the 68 % CL contours. Notice also that a strong degeneracy appears between the
parameters $\omega_{\textrm{log}}$ and $\phi_{\textrm{log}}$. It is
interesting to note that LISA seems to constrain all the properties of this
benchmark, as can be seen from the reconstructed spectral shape of the SGWB.
This, however, is not translated into a constrain on the phase parameter
$\phi_{\rm log}$, which is unconstrained within its prior. This fact is likely
due to the implicit dependence of the amplitude of the signal on $\omega_{\rm
log}$ and $\phi_{\rm log}$, which enters in the amplitude of the two harmonics
in the RO template in eq. (2.7), introducing undesired degeneracies between
parameters. In both cases, for the two injected signals, we observe a good
agreement between the nested sampling and the Fisher analysis, which captures
quite well the shape of the posterior distribution.

Figure 17:  1D and 2D posterior distributions derived from the RO template
reconstruction with a flat envelope run on the benchmarks RO-BNK_1 (left) and
RO-BNK_2 (right). Lines’ styles and color codes are as in fig. 3. Benchmarks
RO-BNK_1 and RO-BNK_2 are defined as in eq. (2.7) with $\\{h^{2}\Omega_{*}\,,\
A_{\textrm{log}}\,,\ \omega_{\textrm{log}}\,,\ \phi_{\textrm{log}}\,\\}$ set
at $\\{-11,1,12,0\\}$ and $\\{-12,0.5,6,0\\}$ respectively.

##  5 Science interpretation highlights

The analysis in the previous sections shows the existence of several well-
motivated inflationary models within the LISA detectability reach.
Furthermore, for some specific parameter ranges, their signals can be
reconstructed with promising small statistical errors. In this section, we aim
at showcasing the scientific breakthroughs the community will achieve in the
case of a detection of an SGWB described by our templates.

We consider the inflationary models motivating our template, and the mapping
between the template parameters and the inflationary model parameters (see
section 2). Using this mapping, we take the chains from the analysis of our
benchmarks and derive the posterior distribution for the fundamental
parameters of the inflationary models. We finally discuss the scientific
implications of the results. For simplicity, and to reduce the computational
costs of our analysis, we do not apply this method at such a level of detail
for every template. For the less computationally expensive templates and
fundamental models, we carry out a thorough study to show how the procedure
works, whereas for the others we just discuss how the bounds on template
parameters reflect on fundamental parameters of such models.

We stress that this approach has the crucial advantage that it can be carried
out in a post-processing phase. Due to the difference both in the number of
parameters and in the range of template priors and theoretical bounds, the
former usually being larger than the latter, performing parameter inference
directly scanning the model parameters within a Bayesian inference on the data
(or as a module of the global fit) is expected to be more precise. But it
would also be computationally more expensive than running the analysis
adopting the templates we analyze: it would require us to repeat the global
fit (or at least its final module interactions) for each fundamental model
compatible with the reconstructed signal. With the tools and templates
provided in this work, constraining fundamental parameters boils down, in
practice, to post-processing the outputs of our template-based
reconstructions.

As discussed in section 2.8, it is possible that non-standard phases in the
early universe after inflation, or the presence of relativistic species beyond
the standard model, would cause modifications of the frequency shape of all
the scenarios. While this could impact the accuracy of the template search,
assuming a priori the presence of a given deformation would require to run the
analysis for all the templates (and benchmarks) with a set of various
assumptions. We choose not to discuss such effects in this section and leave
this task to more targeted future work.

###  5.1 Inflationary scenarios predicting blue power-law spectra

We start our discussion by examining the implications of a detection of the
simplest template among those considered here, _i.e.,_ the PL template of
section 2.1. For illustration, we focus on the two benchmarks PL-BNK_1 and PL-
BNK_2 examined in the forecasts of section 4.1, and we analyse the physical
implications of our findings for the first two inflationary scenarios
discussed in section 2.1: axion inflation, and models with a massive graviton
during inflation. For both benchmarks, the constraints on template parameters
summarized in fig. 3 are translated into constraints on model parameters, and
represented in fig. 18. Due to significant model dependencies in the amplitude
of the massive-gravity SGWB spectrum (see section 2.1), our scientific
interpretation focuses solely on the tilt within that scenario.

PL-BNK_1. The parameter reconstruction associated with this benchmark would
set tight constraints on axion inflation and massive-gravity inflation. In the
context of axion inflation, under the assumptions of validity of eq. (6)
discussed in section 2.1, the results translate into accurate measurements on
the parameters $\xi_{*}=4.474\pm 0.003$ and $\epsilon_{*}=0.0450\pm 0.0003$
both at 68% C.L. (we assume $\eta_{*}=0$ to derive the second constraint). We
note that these constraints are derived assuming a value of $H_{*}/M_{\rm
pl}=10^{-5}$ (consistent with latest constraints on the tensor-to-scalar ratio
from large scale measurements) in order to invert the relation (9) between
$\xi_{*}$ and $\omega_{*}$. For scenarios with a massive graviton, we note
that our benchmark respects the so called Higuchi bound $m_{h}/H\geq\sqrt{2}$
Higuchi:1986py . Our results, presented in the top-right panel of fig. 18,
indicate very tight marginalized constraints on the graviton mass
$(m_{h}/H)^{2}=2.037\pm 0.005$.

PL-BNK_2. Similar to the previous case, the detection of this benchmark would
narrow down the viable parameter space of axion inflation, constraining its
parameters to $\xi_{*}=4.777^{+0.002}_{-0.001}$ and
$\epsilon_{*}(\eta_{*}=0)=0.01545^{+0.00005}_{-0.00007}$, still assuming
$H_{*}/M_{\rm pl}=10^{-5}$. Interestingly, the consequences would be more
intriguing for the massive graviton models. As we can learn from the bottom-
right panel of fig. 18, the posterior distribution would completely lie within
the red hatched region, excluding with high statistical significance the
viability of any massive graviton model that respects the Higuchi bound.
Interpreting this detection in terms of massive graviton would thus point to
the necessity of more complex theoretical constructions, as for example the
one developed in ref. Ricciardone:2017kre .

Figure 18:  Constraints on the parameters of axion inflation parameters (left
panels) and massive-graviton inflation (right panels) derived from the PL
template-based reconstruction of the benchmarks PL-BNK_1 (top panels) and PL-
BNK_2 (bottom panels). The hatched region is excluded by the Higuchi bound
Higuchi:1986py  ($m_{h}/H<\sqrt{2}$) in models respecting de Sitter
isometries.

###  5.2 Constraints on small-scale curvature power spectrum and primordial
black holes

One of the main motivations behind the second scenarios considered for the BPL
and the DP template is the enhancement of curvature perturbations at small
scales. We use our reconstruction forecasts in fig. 11 to estimate the future
bounds on the amplitude of ${\cal P}_{\zeta}$ at LISA scales.

The fundamental relation to consider is the one connecting the frequency $f$
of the SGWB to the comoving wavenumber $k=2\pi f$ characterising primordial
perturbations, which is

$\displaystyle f\simeq 15\,{\rm mHz}\left(\frac{k}{10^{13}{\rm
Mpc}^{-1}}\right).$ (55)

This shows that frequencies accessible by LISA correspond to small scales that
CMB and other large-scale observations cannot probe. Currently, the bound on
$\Delta N_{\text{\tiny eff}}$ coming from CMB data Planck:2018vyg  constrains
$\Omega_{\rm GW}\leq 1.6\cdot 10^{-6}$ at 95 % C.L., which corresponds to the
bound ${\cal A}_{s}\leq 0.69$. Primordial black hole overproduction (see,
_e.g.,_ refs. Saito:2008jc ; Carr:2020gox ; Yuan:2021qgz ; Ozsoy:2023ryl  for
reviews and references therein) sets stronger bounds. This forces the
amplitude to be below ${\cal A}_{s}\lesssim 10^{-2}$, depending on the
assumptions on spectral shape and curvature perturbation statistics (_e.g.,_
ref. Gow:2020bzo ). LISA will push this constraint to much smaller amplitudes
in the range of scales $k\in[10^{10},10^{14}]\,{\rm Mpc}^{-1}$.

Null observation of a BPL or DP SGWB. The null detection of a SGWB with broken
power-law or double-peak template can be converted into an upper bound on
${\cal P}_{\zeta}$. For presentation purposes, we estimate this upper bound
for the DP from the information available in fig. 9 which considers the ${\cal
P}_{\zeta}^{\rm ln}$ scenario.

In the parameter space where the precision on the signal amplitude
$h^{2}\Omega_{*}$ degrades below $30\%$, we are practically insensitive to the
presence of DP signal at 99.7 % CL (i.e. 3$\sigma$). In the case at hand, the
null detection would then lead to the upper bound ${\cal A}_{s}\leq 10^{-3.5}$
at 99.7 % CL in the ${\cal P}_{\zeta}^{\rm ln}$ parameter region with
$f_{*}\simeq 5$ mHz and $\Delta\lesssim 1$.232323Rigorous bounds in the low-
SNR limit would eventually be set adopting a Bayesian analysis. Analogously,
the upper bound on ${\cal A}_{s}$ at different $f_{*}$ can be obtained from
fig. 10, and becomes more severe by departing from $f_{*}=5\,{\rm mHz}\simeq
10^{-2.3}\,$Hz. Similar upper bounds are obtained assuming the null detection
of induced SGWB with a BPL shape (see fig. 6).

Detection of a BPL or DP SGWB background. Assuming the detection of a signal
compatible with the scalar-induced SGWB scenario, we can use the FIM forecast
errors to estimate the tight bounds LISA would set on the enhanced curvature
spectrum and the required inflationary scenario responsible for the
enhancement, see fig. 6 for the BPL case and fig. 9, 10 for the DP case.

Let us focus on BPL_BNK_1. The most informative constraints are set on: i) the
characteristic frequency $f_{*}$, which controls the location of the USR phase
leading to the enhancement of the spectrum. ii) the tilt $n_{{t},2}$, which
controls the plateau observed in $\Omega_{\rm GW}$ at high frequencies.
Assuming a second slow-roll phase following the USR enhancement (see section
2.3 for more details), this is related to the slow roll parameters
$n_{{t},2}\equiv{d\ln{\cal P}_{\zeta}}/{d\ln
k}\simeq-6\epsilon_{*}+2\eta_{*}$. Using the results reported in fig. 7 for
BPL_BNK_1, we find $\log_{10}(k_{*}{\rm Mpc})=11.6\pm 0.4$, which places the
USR phase at $\Delta N=29.7\pm 1.0$ e-folds after the slow-roll phase
controlling CMB perturbations, adopting $k_{\rm CMB}=0.05/{\rm Mpc}$ and
approximating the Hubble rate to a constant during that part of the
inflationary phase. Furthermore, if $\epsilon_{*}$ is negligible after the USR
phase (_e.g.,_ ref. Franciolini:2022pav ), we obtain the constraint
$\eta_{*}=0.01\pm 0.11$ for the parameters of the subsequent slow-roll phase.

Constraints on Primordial Black Hole population. Enhanced curvature power
spectra are associated with the formation of PBHs. Therefore, we can forecast
the constraining power of LISA on this dark matter candidate. We are
considering test case scenarios in which ${\cal P}_{\zeta}$ features a narrow
bump in the mHz range, corresponding to a PBH population with asteroid typical
mass. Uncertainties on template parameters translate into two observables: i)
$f_{*}$ controls the typical mass of the PBHs population; ii) the curvature
power spectral amplitude ${\cal A}_{s}$ translates (highly non-linearly) into
uncertainties on $f_{\text{\tiny PBH}}\equiv\Omega_{\text{\tiny
PBH}}/\Omega_{\text{\tiny DM}}$ which gives the PBH abundance in terms of the
dark matter abundance. For narrow spectra, the typical PBH mass is related to
the mass contained in the Hubble scale $M_{H}$ at the time of mode crossing as
$\langle m_{\text{\tiny PBH}}\rangle\simeq 0.6M_{H}$ Musco:2023dak , where the
angular brackets indicate average PBH mass computed accounting for critical
collapse, and $M_{H}$ is related to the curvature wavenumber by
$M_{H}/M_{\odot}\simeq 7\times 10^{-11}[k/(10^{12}{\rm Mpc}^{-1})]$.

Let us consider the case of a DP detection in more detail. From the results in
the upper-left panel of fig. 11, i.e. considering DP-BNK_1, we find that the
mean PBH mass would be measured with exquisite precision to be
$\log_{10}\langle m_{\text{\tiny PBH}}/M_{\odot}\rangle\approx-11.407\pm
0.003$. Finally, considering the same benchmark scenario, the overall
amplitude translates into an uncertainty on the PBH abundance
$\log_{10}(f_{\text{\tiny PBH}})=0.01\pm 0.06$. For the computation of the
abundance we assumed threshold statistics and Gaussian curvature
perturbations, see e.g. Young:2019yug ; DeLuca:2019qsy ; Ferrante:2022mui ;
Gow:2022jfb . Similar conclusions are reached in the case of DP-BNK_3, also
associated with the production of large fraction of the dark matter in the
form of asteroidal mass PBHs. Regardless of the well-known exponential
dependence of the PBH abundance on the power spectral amplitude, assuming
emission in a radiation-dominated universe and Gaussian perturbations, LISA
observations would be able to constrain PBH dark matter abundance with better
than $10\%$ precision, provided $f_{\text{\tiny PBH}}$ saturates the upper
bound from PBH overproduction. Therefore, LISA constraints will likely be
limited by the large model dependence of the PBH abundance
LISACosmologyWorkingGroup:2023njw  as well as remaining systematic
uncertainties on their computation (see, _e.g.,_ ref. DeLuca:2023tun ).
Importantly, in case the LISA reconstruction precision is degraded compared to
the one forecasted in this work, it would drastically decrease the
capabilities to pin-down the PBH abundance, due to the strong dependence of
$f_{\rm PBH}$ on the amplitude ${\cal A}_{s}$. In the cases of DP-BNK_2 and
DP-BNK_4, the assumed amplitude of perturbations is small and associated with
a negligible abundance of PBHs. Upper bounds derived from LISA data would be
strong enough to rule out a large contribution to the dark matter from this
PBH formation scenario.

A more thorough derivation of constraints on the scalar-induced SGWB scenarios
requires more complex post-processing of the forecast presented in the
previous sections, based on solving the inverse problem accounting for the
full fundamental-parameter degeneracies and model dependence. This is part of
an ongoing LISA cosmology working group activities and its results will be
reported elsewhere.

###  5.3 Small-scale primordial features

Gravitational-wave cosmology has the potential to shed light on primordial
features during inflation. These are characterised by various types of
oscillations in the SGWB frequency profile that are distinctive of specific
mechanisms active during inflation. LISA has the ability to reconstruct
parameters associated with primordial features with a striking precision and
thus opens the possibility to detect them at scales much shorter than the CMB-
LSS ones, where current constraints are based.

Excited states. The most informative parameter to retrieve from the detection
of the excited states template (see section 2.5) is $\omega_{\rm ES}$. This
parameter determines the location of the primary peak of the signal as well as
the periodicity of the higher-frequency peaks. Measuring it enables us to
deduce the existence of a dynamically generated excited state during
inflation, and to identify when this particle production event occurred, or
equivalently the momentum scale of modes crossing the Hubble radius at that
time, $k_{\rm ES}=4\pi/\omega_{\rm ES}$ Fumagalli:2021mpc .

More concretely, let us call $\Delta N$ the number of e-folds between the time
the CMB pivot scale exits the Hubble radius and the time particle are
copiously produced. Using the previous expression for $k_{\rm ES}$ and the
relation (55), we obtain

$\Delta N=\log\left[\left(\frac{4}{3}\cdot
10^{15}\right)\left(\frac{1}{\omega_{\rm
ES}\mathrm{Hz}}\right)\left(\frac{\mathrm{Mpc}^{-1}}{k_{\mathrm{CMB}}}\right)\left(\frac{H_{\mathrm{CMB}}}{H_{\mathrm{f}}}\right)\right].$
(56)

Thus, by using error propagation in eq. 56, we find that the detection of
either ES-BNK_1, ES-BNK_2 or ES-BNK_3 with the uncertainties on $\omega_{\rm
ES}$ reported in fig. 15 would respectively lead to $\Delta N=30.7\pm 0.1$,
$\Delta N=29.12\pm 0.01$ and $\Delta N=29.13\pm 0.07$ with the (mild)
assumption $H_{\mathrm{CMB}}=H_{\mathrm{f}}$ and using
$k_{\mathrm{CMB}}=0.05\,\mathrm{Mpc}^{-1}$.

The second parameter of interest is $\gamma_{\rm ES}$. It controls the high-
frequency behaviour of the signal, in particular the range of frequencies
where the signal dies off. Its knowledge teaches us how much inside the
horizon particle production took place or, equivalently, the energy scale
$E_{\rm ES}\equiv\gamma_{\rm ES}H$ (in Hubble units) of the produced
particles. Unfortunately, as we have learned from section 4.5, this parameter
is intrinsically related to the low-signal part of the template, and thus is
hard to reconstruct, unless the signal is loud, _i.e.,_
$\log_{10}h^{2}\Omega_{*}\gtrsim-11$. Thus, among the three benchmarks chosen
in fig. 13, only with the detection of ES-BNK_2 (with an injected $\gamma_{\rm
ES}=10$) can LISA retrieve relevant information on $E_{\rm ES}$ thanks to the
reconstruction $\gamma_{\rm ES}=12.6\pm 1.3$.

Note that a sharp feature phenomenon generating an excited state with copious
particle production generates both the signal described by the excited state
template (17) and the one described by the linear oscillations template (20)
discussed below. As the relative amplitudes of the two signals is model-
dependent, and as their peak frequencies differ, we treated the two templates
separately. It is nonetheless instructive to compare the two. The time of the
sharp feature is the most robust physical information, in the sense that it
can be identified well from each template. By contrast, how much inside the
Hubble radius the particle production was efficient is retrieved very easily
from the linear oscillations template, as it is then related to the peak
frequency (see below). On the other hand, in the case of the excited state
template, this information is retrieved only if the signal is very loud as the
information is then contained in the high-frequency, but low amplitude part of
the signal.

Sharp features, i.e. linear oscillations. Let us now discuss what can be
learned in a relatively model-independent manner from the detection of linear
oscillations in the SGWB, using the LO template in eq. 20. The most
informative parameter is the oscillation frequency $\omega_{\rm lin}$.
Measuring it enables us to prove the existence of a sharp feature during
inflation, and to identify when it occurred. Sharp features refer to localized
events — of duration smaller than one $e$-fold — independently of their
precise realisation like, _e.g.,_ a step in the inflationary potential or a
sharp turn in the inflaton trajectory. They generate density fluctuations with
a power spectrum characterised by linear oscillations (see eq. (21)) with
frequency $\omega=2/f_{\textrm{f}}$ directly related to the scale crossing the
Hubble radius at the time of occurence of the feature, $k_{\textrm{f}}=2\pi
f_{\textrm{f}}$ (setting the scale factor today and using natural units).
These oscillations are then transferred to the scalar-induced GW density
profile if their amplitude is sufficently large, _i.e.,_ for a significant
particle production generated by the sharp feature, with occupation numbers of
order one or larger Fumagalli:2020nvq . Hence, pinpointing the frequency of
the template $\omega_{\textrm{lin}}=c_{s}^{-1}\omega$ gives us
$k_{\textrm{f}}$ or, equivalently, the time of the feature. There is a
theoretical degeneracy in the mapping between the measured
$\omega_{\textrm{lin}}$ and the inferred $k_{\textrm{f}}$ due to the
uncertainty about the propagation speed of density fluctuations $c_{s}$, equal
to $1/\sqrt{3}$ in the conventional scenario of radiation domination at
horizon reentry. This degeneracy may be lifted, however, as we comment below.

As in the previous section, let us call $\Delta N$ the number of e-folds
between the time the CMB pivot scale exits the Hubble radius and the time of
the feature. Using $\omega_{\rm lin}=2c_{s}^{-1}/f_{\rm f}$ and the relation
(55), we obtain the analogous of eq. 56. This yields $\Delta
N=\log\left(\frac{4\cdot 10^{17}}{15}c_{s}^{-1}/(\omega_{\rm
lin}\mathrm{Hz})\right)$ for $H_{\mathrm{CMB}}=H_{\mathrm{f}}$ and
$k_{\mathrm{CMB}}=0.05\,\mathrm{Mpc}^{-1}$. Thus, by using error propagation,
we find that the detection of either LO-BNK_1 or LO-BNK_2 with the
uncertainties on $\omega_{\rm lin}$ reported in fig. 15 would respectively
lead to $\Delta N=27.55\pm 0.02$ and $\Delta N=29.161\pm 0.002$ for the
typical radiation domination scenarios with $c_{s}=1/\sqrt{3}$.

Further information can be derived from the precise measurement of the peak
frequency $f_{*}$. Combined with $\omega_{\textrm{lin}}$, it establishes the
quantity $E_{\rm f}=f_{*}\omega_{\textrm{lin}}/2$, which gives the
characteristic energy scale of the feature in Hubble-scale units. LISA would
measure it as $E_{\rm f}=25\pm 1$ for LO-BNK_1 and $E_{\rm f}=25.05\pm 0.05$
for LO-BNK_2. Intriguingly, recasting this bound on two-field inflation setups
with strong and sharp turns yields a measurement of the rate $\eta_{\perp}$ at
which the inflaton abruptly changes direction in the exponential expansion of
the universe close to the Big Bang.242424In these setups, one finds
$\eta_{\perp}\simeq E_{\rm f}/2$ from eq. 23. In turn, the measurement of
$\eta_{\perp}$ sheds light on the number of e-folds $\delta$ during which the
feature was active. In fact, from a given value ${\cal P}_{0}$ of the scalar
power spectrum away from the feature, measured or assumed at _e.g.,_ the CMB
scale, one can extrapolate the product $\eta_{\perp}\delta$ via eq. (22) and,
subsequently, infer $\delta$ from $\eta_{\perp}$.

Finally, in a wide parameter space of our forecast, LISA can reconstruct
${\cal A}_{\textrm{lin}}$ with an accuracy that permits imposing a lower bound
above $0.2$. With such a lower bound on ${\cal A}_{\textrm{lin}}$, LISA would
prove not only the existence of an inflationary sharp transition but also of a
non-standard thermal history at the time of Hubble reentry Witkowski:2021raz .
Indeed, for a universe dominated by a perfect fluid, a measurement ${\cal
A}_{\textrm{lin}}>0.2$ indicates that the fluid’s equation of state at the
Hubble reentry time was stiffer than the one of radiation, $w>1/3$. From the
analysis in section 4.6, it is rather remarkable that identifying the
oscillation frequency requires signals with an amplitude only an order of
magnitude above the detection threshold of the overall peak itself, and that
this identification can be performed with oscillations of relative amplitude
${\cal A}_{\textrm{lin}}$ as tiny as $0.01$.

Resonant features, i.e. logarithmic oscillations. Detecting a signal
fulfilling the RO template of (2.7) would prove the existence of periodic
modulations of the inflationary Lagrangian with frequency
$\omega_{\textrm{log}}$ in e-fold unit. Based on our forecast, LISA has a
resolution on $\omega_{\textrm{log}}$ that permits ruling out the presence of
oscillations in a wide parameter region of the RO template (c.f. the parameter
region with a robust upper value on $\omega_{\textrm{log}}$ in fig. 16).

As explained in section 2.7, the literature has started investigating explicit
setups leading to resonant oscillations in the SGWB only recently. A precise
relationship between $\omega_{\rm log}$ and the microphysics of inflation thus
requires further investigation. On cosmological scales, oscillations resulting
from a heavy field of mass $M$ being displaced from its minimum have been
studied in detail Slosar:2019gvt . Considering such a mechanism for
definiteness, the physics interpretation of the detection of the benchmarks in
fig. 17 would bound $\omega_{\mathrm{log}}=M/H$ within $12\pm 0.2$ for RO-
BNK_1 (injected $\omega_{\rm log}=12$) and $5.89\pm 0.24$ for RO-BNK_2
(injected $\omega_{\rm log}=6$) at 68 % CL , thus pinpointing with high
precision the mass of new particles well above what can be detected with
terrestrial colliders. Actually, the accuracy of the reconstruction of
$\omega_{\textrm{log}}$ mildly depends on the frequency for
$\omega_{\textrm{log}}\gtrsim 6$: from $\omega_{\textrm{log}}=6$ to
$\omega_{\textrm{log}}=100$, the same relative error on
$\omega_{\textrm{log}}$ is attained by having a signal louder by
$\log_{10}(h^{2}\Omega_{*})\simeq 0.7$ only, for all relevant values of the
overall signal amplitude $h^{2}\Omega_{*}$. This is quite remarkable, as the
relative amplitude of the oscillations in the template roughly goes from 100%
to 1% between these two values of $\omega_{\textrm{log}}$ (for a fixed order-
one value of $A_{\textrm{log}}$) Fumagalli:2021cel . This highlights the
striking discovery potential of LISA, which can identify resonant features
during inflation with characteristic energy scales at least two orders of
magnitude above the Hubble scale. This result prompts further studies by
theorists, as this quantitative analysis reveals that LISA will be sensitive
to much smaller effects than is hitherto acknowledged Calcagni:2023vxg .

##  6 Conclusions

In this work, we demonstrated the scientific potential of a template-based
analysis of cosmological SGWB with the LISA detector. Our focus is on
inflationary scenarios producing an SGWB in the milli-Hertz regime. We showed
that LISA will provide very accurate constraints on parameters of templates
which characterise several inflationary scenarios.

The simplest vanilla single-field models of inflation generate a slightly red-
tilted SGWB spectrum. Within this minimal framework, extrapolating to milli-
Hertz LISA frequencies the current bounds on the non-observation of primordial
B-modes from BICEP-Keck and Planck Planck:2018vyg ; BICEP:2021xfz ;
Galloni:2022mok ; Paoletti:2022anb , obtained at femto-Hertz scales, leads to
a spectrum which is well below the sensitivity of LISA and other existing and
planned interferometers. However, there are several theoretical realisations
of inflation which go beyond the simplest setup described above, and which
change its predictions towards large frequencies. GW observations thus provide
a unique avenue to obtain direct experimental information on frequency scales
much larger than the CMB ones, which map to the inflationary dynamics closer
to the final stages of inflation. With the SGWB measurement, LISA then holds
the potential to unveil the physics of inflation and, at worst, in the event
of a non-observation of a primordial SGWB, provide constraints ruling out many
inflationary setups and helping prioritize the theoretical efforts towards the
surviving models.

As a first step towards testing inflation with LISA observations, we
identified seven representative templates, which encompass the spectral shapes
of several spectra of SGWBs motivated by inflationary models. The collection
of such a template set, together with corresponding ones developed in
companion papers for early universe phase-transitions Caprini:2024hue  and
cosmic strings Blanco-Pillado:2024aca  scenarios, is designed to start a
centralized LISA repository of cosmological signals, enabling extensions,
updates and applications for future LISA studies. Furthermore, to enhance the
capabilities of our template repository, we have prototyped a template-based
data analysis pipeline integrated with the SGWBinner code Caprini:2019pxz ;
Flauger:2020qyi .

As a second step, we performed a Fisher matrix analysis on each of our
templates. We identified the parameter regions that can be probed by LISA, and
forecasted the accuracy at which each template parameter can be constrained.
Furthermore, we developed a full template-based reconstruction, adapting the
SGWBinner code to simulate the data which include instrumental noise,
foreground signals and the inflationary signals, and performed parameter
inference based on the full likelihood.252525The implementation of the
template-based approach within SGWBinner goes beyond the mere validation of
the Fisher results. Rather, it represents a significant advance toward the
development of a prototype SGWB module for the comprehensive LISA global fit.
For each template, we discussed appropriate theoretical priors for its
parameters, and we identified a few representative injection points, or
benchmarks, for which we simulated LISA data. Then, we ran the full template-
based reconstruction of the SGWBinner, which uses a nested sampling algorithm
to explore the likelihood, and we computed the posterior distribution of the
template parameters. Such analysis helped to clarify the degree of realisation
dependence of our data and eventual degeneracies as well as non-Gaussianities
of the posterior distributions that cannot be fully captured by the Fisher
analysis, especially for low SNR injections. As such, we found the two
approaches, which we compared against each other for each benchmark, highly
complementary.

Our results show that LISA has the potential to test primordial SGWBs to a
high level of precision. Reducing the experimental and theoretical
uncertainties below the statistical errors quantified in our work is a
challenge that the community will have to face in order to avoid undermining
the exquisite reconstruction precision that LISA offers us. For example, if
primordial SGWB described by a power-law with amplitude $\Omega_{\rm GW}\sim
10^{-11}$ does exist at mHz frequencies, corresponding to the minimum of its
sensitivity, LISA will constrain the template parameters with an accuracy
better than percent level. Remarkably, a similar accuracy is achieved also for
richer frequency shapes, such as those with broken power-law, log-normal and
double-peak profiles. If the SGWB spectrum have distinctive oscillatory
features, the accuracy can even improve below few percent. Such impressive
precision will enable the distinction between inflationary mechanisms that
produce the signal, paving the way for groundbreaking discoveries.

Finally, we discussed the results of our template-based analysis and
translated them into constraints on the fundamental parameters of inflationary
models that produce the template. We presented the analysis using fundamental
parameters for several illustrative inflationary scenarios where a
(semi-)analytic relationship between the template and the fundamental model
parameters was available. We found that the qualitative interpretation of the
reconstructed signals is valuable and informative. We demonstrated that with
the projected reconstruction accuracy, LISA would be capable of testing
various aspects of inflationary physics, such as the couplings of inflationary
axions to gauge fields, the graviton mass during inflation, the primordial
fluctuations that could have given rise to primordial black holes, small-scale
primordial features like linear or logarithmic oscillations, and the effects
of excited states during inflation.

The potential of LISA to detect primordial SGWBs has been evident for some
time. Our results demonstrate that such a detection can be achieved with
remarkable accuracy. This hinges on accurately characterising the noise
sources for the LISA detector, precisely modelling astrophysical contributions
to the foreground, and ensuring that binary waveforms have residuals that do
not mimic the SGWB signal. Our paper may serve as a call to action for the
global scientific community, urging collaborative efforts to address these
challenges.

## Acknowledgments

We acknowledge the LISA Cosmology Working Group members for seminal
discussions. We especially thank the authors of refs. Caprini:2024hue ;
Blanco-Pillado:2024aca  for the collaborative developments of the SGWBinner
code shared among this and those works. We are also grateful to Ema
Dimastrogiovanni, Matteo Fasiello, Sachiko Kuroyanagi and Lukas Witkowski for
initial contributions and hints on some inflationary models motivating our
templates. GC is supported by grant PID2020-118159GB-C41 funded by the Spanish
Ministry of Science, Innovation and Universities
MCIN/AEI/10.13039/501100011033. JF is supported by the grant
PID2022-136224NB-C22, funded by MCIN/AEI/10.13039/5011000110 33/FEDER, UE, and
by the grant 2021-SGR00872. GN is partly supported by the grant Project No.
302640 of the Research Council of Norway. MPe acknowledges support from
Istituto Nazionale di Fisica Nucleare (INFN) through the Theoretical
Astroparticle Physics (TAsP) project, and from the MIUR Progetti di Ricerca di
Rilevante Interesse Nazionale (PRIN) Bando 2022 - grant 20228RMX4A. MPi
acknowledges the hospitality of Imperial College London, which provided office
space during some parts of this project. SRP is supported by the European
Research Council under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 758792, Starting Grant project
GEODESI). AR acknowledges financial support from the Supporting TAlent in
ReSearch@University of Padova (STARS@UNIPD) for the project “Constraining
Cosmology and Astrophysics with Gravitational Waves, Cosmic Microwave
Background and Large-Scale Structure cross-correlations”. GT is partially
funded by the STFC grants ST/T000813/1 and ST/X000648/1. VV is supported by
the European Union’s Horizon Europe research and innovation program under the
Marie Skłodowska-Curie grant agreement No. 101065736, and by the Estonian
Research Council grants PRG803, RVTT3 and RVTT7 and the Center of Excellence
program TK202.

##  Appendix A Visualising the effects of the template parameters

In this appendix, we show how variations of the parameters of each template
affect the SGWB spectral shape. This is intended to provide a visual
interpretation of the dependence of the template to the various parameters. In
a few cases, we exaggerate the range of variation of some parameters, to
better display their effect. For presentation purposes, in all plots contained
in this appendix, we arbitrarily fixed $f_{*}=1\,$mHz (remembering that a
change in $f_{*}$ simply shifts the spectrum $\Omega_{*}(f/f_{*})$
horizontally in frequency).

  * •

Power law template. In fig. 19, we show the effect of the amplitude
$h^{2}\Omega_{*}$ (left panel) and of the tilt $n_{t}$ (right panel) on the
power-law template of eq. (2). Positive (negative) values of $n_{t}$
correspond to blue (red) tilted spectra. Notice that the pivot frequency is
completely degenerate with the amplitude in this template, and therefore it is
removed from the set of parameters.

Figure 19:  Effect on the SGWB spectrum of the power law parameters
$\Omega_{*}$ (left) and $n_{t}$ (right). For reference, we indicate with a
gray dashed line the LISA power-law integrated sensitivity curve Babak:2021mhe
; Colpi:2024xhw . The black solid line corresponds to PL-BNK_2, whose
parameters are $\\{\log_{10}\Omega_{*},n_{t}\\}=\\{-11,\,0.77\\}$.

  * •

Broken power law. In fig. 20, we show the effect of the two tilts $n_{t,1}$
and $n_{t,2}$ as well as the transition parameter $\delta$, as defined in eq.
(13). The two tilts control the slope of the spectrum at small and large
frequencies, respectively. If the two tilts are equal to each other, the peak
of the spectrum is located at $f_{*}$, while the peak shifts to smaller
(larger) frequencies, when $n_{t,1}<n_{t,2}$ ($n_{t,1}>n_{t,2})$. The
transition parameter $\delta$ controls the sharpness of the peak and how
quickly the power law behavior is attained in the tails.

Figure 20: Effect on the SGWB spectrum of the broken power law template
parameters. The black line indicates the spectrum assuming
$\\{\log_{10}\Omega_{*},n_{t,1},n_{t,2},\delta\\}=\\{-10,\,2,\,-2,\,1\\}$.

  * •

Log-normal bump. In fig. 21 we show the spectral variations obtained by
modifying the log-normal template parameters as defined in eq. (12). The shape
of this template is controlled by $\rho$ that determines the width of the
spectrum. In the limit $\rho\gg 1$, this approaches a flat spectrum.

Figure 21: Effect on the SGWB spectrum of the log-normal template parameters.
The black line indicates the spectrum assuming
$\\{\log_{10}\Omega_{*},\,f_{*}\,[{\rm
Hz}],\,\rho\\}=\\{-10,\,10^{-3},\,0.5\\}$.

Figure 22: Effect on the SGWB spectrum of the double peak template parameters.
The black line indicates the spectrum assuming
$\\{\log_{10}\Omega_{*},\beta,\kappa_{1},\kappa_{2},\rho,\gamma\\}=\\{-10,0.242,0.456,1.234,0.08,6.96\\}$
(corresponding to a log-normal primordial power spectrum with $\Delta=0.2$ as
the benchmarks in the main text).

  * •

Double peak template. In fig. 22, we show the effect of varying the parameters
of the double peak template of eq. (14). A first set of parameters
$\\{\beta,\kappa_{1}\\}$ controls the properties of the first (low-frequency)
peak: $\beta$ parameterises its the amplitude compared to the dominant peak,
while $\kappa_{1}$ shifts it compared to $f_{*}$ (smaller values of
$\kappa_{1}$ would place the peak at smaller frequencies). A second set of
parameters $\\{\kappa_{2},\rho,\gamma\\}$ controls the shape of the dominant
peak: $\kappa_{2}$ shifts its frequency, $\rho$ controls its width and
$\gamma$ parameterises its skewness by controlling the large frequency cut-
off.

  * •

Excited states. In fig. 23, we show the spectral variations obtained by
modifying the excited states template parameters as defined in eq. (17). The
parameter $\omega_{\rm ES}$ controls the shift in frequencies of the entire
template (analogously to $f_{*}$). The shape is affected only by varying
$\gamma_{\rm ES}$. In particular, larger values of $\gamma_{\rm ES}$ move the
cut-off of the template to larger frequencies, allowing more oscillation
cycles to appear close to the dominant peak.

Figure 23: Effect on the SGWB spectrum of the excited states template
parameters. The black line indicates the spectrum assuming
$\\{\log_{10}h^{2}\Omega_{*},\log_{10}\gamma_{\rm ES},\log_{10}\omega_{\rm
ES}\\}=\\{-9,1,3.778\\}$. (The latter corresponding to
$\omega_{\mathrm{ES}}=6\cdot 10^{3}\,\mathrm{Hz}^{-1}$).

  * •

Linear oscillations. In fig. 24 we show the effect of linear oscillations
applied to a log-normal envelope as a function of the parameters of the LO
template in eq. (20). The parameter $\rho$ determines the with of the log-
normal envelope while $\mathcal{A}_{\mathrm{lin}}$ controls the amplitude of
the oscillations, $\omega_{\mathrm{lin}}$ their frequency in linear scale, and
$\theta_{\mathrm{lin}}$ their phase.

Figure 24: Effect on the SGWB spectrum of the linear oscillations template
parameters. The black line indicates the spectrum assuming
$\\{\mathcal{A}_{\mathrm{lin}},\log_{10}(\omega_{\mathrm{lin}}\,\mathrm{Hz}),\theta_{\mathrm{lin}}\\}=\\{0.3,4.7,0\\}$,
where the envelope is assumed to be the log-normal with $\vec{\theta}_{\rm
env}=\\{\log_{10}\Omega_{*},\rho\\}=\\{-9,0.08\\}$. In these plots, the range
of frequencies was zoomed in close to the peak of the signal, to highlight
spectral oscillations. For this reason only a small portion of the LISA power-
law integrated sensitivity is visible (gray dashed line).  Figure 25:  Effect
on the SGWB spectrum of the logarithmic resonant oscillation template
parameters. The black lines corresponds to
$\\{\log_{10}\Omega_{*},A_{\textrm{log}},\omega_{\textrm{log}},\phi_{\textrm{log}}\\}=\\{-9,0.5,12,0\\}$
and a flat envelope.

  * •

Logarithmic resonant oscillations. In fig. 25, we show the effect of changing
the parameters of the logarithmic resonant oscillations template of eq. (2.7),
assuming a flat underlying envelope. The parameters $A_{\textrm{log}}$ and
$\phi_{\textrm{log}}$ affect, respectively, the amplitude and phase of the
oscillations, and $\omega_{\textrm{log}}$ modifies nontrivially the behaviour
of the spectrum; while for sufficiently large values of
$\omega_{\textrm{log}}$ a regular double feature is observed, only a
monochromatic oscillation is observed in the opposite limit, with much larger
overall amplitude. This is induced by the complex dependencies of the
coefficients $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ defined in eq. 25.

##  Appendix B Induced GWs at second order: from $\mathcal{P_{\zeta}}(k)$ to
$\Omega_{\rm GW}(f)$

Within single- and multi-field models of inflation featuring an enhancement of
the scalar fluctuations by a short period of nonattractor evolution, one
expects the generation of a potentially sizeable SGWB at frequencies $f$ when
perturbations of comoving wavenumber $k\simeq 6.47\times 10^{14}\,{\rm
Mpc}^{-1}\left({f}/{{\rm Hz}}\right)$ re-enter the Hubble sphere (see ref.
Domenech:2021ztg  for a recent review). These scales are related to the
formation of primordial black holes with asteroidal masses, which attracted
ample attention due to the possibility that they could provide candidates for
(part of) dark matter Sasaki:2018dmp ; Carr:2020gox . LISA will be able to
test and constrain this scenario Saito:2008jc ; Bartolo:2018evs .

One can compute the GW signal sourced by scalar perturbations Tomita:1975kj ;
Matarrese:1993zf ; Matarrese:1997ay ; Acquaviva:2002ud ; Mollerach:2003nq ;
Nakamura:2004rm ; Ananda:2006af ; Baumann:2007zm ; Espinosa:2018eve ;
Kohri:2018awv ; Inomata:2019yww ; Domenech:2019quo  by solving the Einstein
equations at second order in perturbation theory. Assuming the emission takes
place in a radiation-dominated universe, as is the case for those scales in
standard cosmologies, one can derive the transfer functions

$\displaystyle{\cal I}_{c}(d,s)$ $\displaystyle=$
$\displaystyle-36\pi\frac{(s^{2}+d^{2}-2)^{2}}{(s^{2}-d^{2})^{3}}\Theta(s-1)\,,$
(57) $\displaystyle{\cal I}_{s}(d,s)$ $\displaystyle=$
$\displaystyle-36\frac{(s^{2}+d^{2}-2)}{(s^{2}-d^{2})^{2}}\left[\frac{(s^{2}+d^{2}-2)}{(s^{2}-d^{2})}\ln\frac{(1-d^{2})}{|s^{2}-1|+\epsilon}+2\right]\,,$
(58)

by integrating the equation of motion for the emission of GWs at second order
with the Green’s function method. In order to find concise analytical
solutions, such as the one presented in eq. (57), in the literature the
observer is typically located at the asymptotic future, $\eta_{\rm
obs}\rightarrow\infty$. This, however, generates a fictitious divergence in
the limit of $s\rightarrow 1$, which is also retained in the spectrum produced
by monochromatic scalar perturbations (see also discussions in ref.
Ananda:2006af ). The regulator $\epsilon$ is introduced in order to avoid this
divergence. A value of order $\epsilon=10^{-13}$ is sufficient to recover the
physically regularized solution assuming the emission does not continue after
the matter-radiation equality for modes with frequency around the LISA peak
sensitivity ($f\simeq 10^{-3}$Hz). Given the logarithmic nature of the
divergent term, the observables related to $\Omega_{\rm GW}$ are, in practice,
insensitive to slightly different choices of the regulator. For concreteness,
we will adopt $\epsilon=10^{-13}$. The main peak in the scalar-induced GW
spectrum roughly coincides with the peak in the curvature power spectrum, and
its amplitude is $\Omega_{\rm GW}=\mathcal{O}(10^{-5}){\cal A}_{s}^{2}$, where
${\cal A}_{s}$ is the characteristic scalar spectral amplitude.

Different models of inflation can give rise to different kinds of peaks in the
curvature power spectrum, depending on the detailed shape of the curvature
spectrum.262626Also, post-inflationary physics, as an unconventional phase of
expansion prior to radiation domination epoch, can change the slope of the
induced spectrum, see, _e.g._ , refs. Domenech:2019quo ; Domenech:2020kqm . In
this sense, induced SGWB can probe the early thermal history of our universe.
We do not attempt to explore the full model dependence. Instead, in the
following subsections we motivate general shapes of the curvature power
spectrum that are expected in a general class of inflationary dynamics giving
rise to enhanced spectra at small scales, and derive what values of the
parameters characterise the template (14).

###  B.1 The spectral index in the infrared

Although it is known that, for the sources studied in this section, we have a
universal $\Omega_{\rm GW}\propto f^{3}$ scaling in the deep infrared
Caprini:2009fx ; Cai:2019cdl ; Hook:2020phx , the slope of the spectrum
changes when approaching the first peak. Interestingly, it still has a
universal behavior, with a spectral index $n_{p}\equiv{{\rm d}\ln\Omega_{\rm
GW}}/{{\rm d}\ln f}\simeq 2.5.$ We confirm this statement by following ref.
Pi:2020otn  in the case of narrow curvature peaks (see also refs. Yuan:2019wwo
; Adshead:2021hnm ; Atal:2021jyo ; Franciolini:2023pbf ; Yuan:2023ofl ;
Xie:2024cwp  for an alternative derivation).

We assume that $\mathcal{P}_{\zeta}(k)$ has a pronounced peak with maximum at
$\kappa\equiv k/k_{*}=1$. We assume that the peak has a narrow width
controlled by a dimensionless quantity dubbed $\Delta$, much smaller than one.
We Taylor expand $\ln\mathcal{P}_{\zeta}$ around the maximum as

$\displaystyle\ln\mathcal{P}_{\zeta}(\kappa)\,=\,\ln\left(\frac{{\cal
A}_{\zeta}}{\sqrt{2\pi}\,\Delta}\right)-\frac{\ln^{2}\kappa}{2\,\Delta^{2}}+\frac{\alpha_{3}}{3!\,\Delta^{3}}\,\ln^{3}\kappa+\frac{\alpha_{4}}{4!\,\Delta^{4}}\,\ln^{4}\kappa+\dots,$
(59)

where, for convenience, in this section we define ${\cal
A}_{\zeta}=\sqrt{2\pi}\,\Delta{\cal A}_{s}$, following the notation introduced
in ref. Pi:2020otn . The coefficients $\alpha_{i}$, $i\geq 3$ in the Taylor
expansion are real numbers, generally expected to be of ${\cal O}(1)$. The
pure log-normal case corresponds to $\alpha_{i}=0$. We expand
$\Omega_{\text{GW}}(\kappa)$ in the limit of small $\kappa$, at leading order
in $\Delta$ and $\kappa$, which corresponds to the infrared limit, going
beyond the limit of ref. Pi:2020otn . With some manipulations, we can write
the SGWB spectrum as

$\displaystyle\Omega_{\text{GW}}$
$\displaystyle=\frac{3}{\pi}\frac{\mathcal{A}_{\zeta}^{2}}{\Delta^{2}}\kappa^{2}e^{\Delta^{2}}\int^{\infty}_{-\infty}\,ds\,\int^{\xi(s)}_{\chi(s)}\,dt\,\mathcal{T}\left(s,t\right)\exp\left[-\frac{F_{0}(s,t)}{2\Delta^{2}}\right],$
(60)

with

$\displaystyle F_{0}(s,t)$
$\displaystyle=\left(s+\sqrt{2}(\ln\kappa+\Delta^{2})\right)^{2}+t^{2}$
$\displaystyle-\frac{\alpha_{3}}{6\Delta}\left[\sqrt{2}s(s^{2}+3t^{2})+6(s^{2}+t^{2})\ln\kappa+6\sqrt{2}s\ln^{2}\kappa+4\ln^{4}\kappa\right]+\alpha_{4}(\dots)$
(61)

and

$\displaystyle\mathcal{T}(s,t)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\cosh(\sqrt{2}t)-\frac{1}{4}e^{-\sqrt{2}s}-e^{\sqrt{2}s}\sinh^{2}(\sqrt{2}t)\right)^{2}\left(\cosh(\sqrt{2}t)-\frac{3}{2}e^{-\sqrt{2}s}\right)^{4}$
(62)
$\displaystyle\times\left\\{\left[\ln\left|\frac{3-4e^{\sqrt{2}s}\cosh^{2}\left(t/\sqrt{2}\right)}{3-4e^{\sqrt{2}s}\sinh^{2}\left(t/\sqrt{2}\right)}\right|-\frac{2}{\cosh(\sqrt{2}t)-\frac{3}{2}e^{-\sqrt{2}s}}\right]^{2}\right.$
$\displaystyle\qquad\left.+\pi^{2}\Theta\left(2e^{\frac{s}{\sqrt{2}}}\cosh\frac{t}{\sqrt{2}}-\sqrt{3}\right)\vphantom{\ln\left|\frac{3-4e^{\sqrt{2}s}\cosh^{2}\left(t/\sqrt{2}\right)}{3-4e^{\sqrt{2}s}\sinh^{2}\left(t/\sqrt{2}\right)}\right|^{2}}\right\\}.$

The extrema of integration along the coordinate $t$ are defined as

$\displaystyle\chi(s)$
$\displaystyle=\mathbf{Re}\left[\sqrt{2}\text{arccosh}\left(\frac{e^{-s/\sqrt{2}}}{2}\right)\right],\qquad\xi(s)=\sqrt{2}\text{arcsinh}\left(\frac{e^{-s/\sqrt{2}}}{2}\right),$
(63)

where the real part is taken to ensure that $\chi(s)=0$ for $s>-\sqrt{2}\ln
2$. Within the working assumption of small $\Delta$, we make use of the
saddle-point approximation to evaluate the integral. We substitute
$t=\tau\,\Delta$ and
$s=-\sqrt{2}\,\left(\ln\kappa+\Delta^{2}\right)+\sigma\,\Delta$. We keep only
the leading contributions in $\Delta$, to obtain

$\frac{-F_{0}(\sigma,\tau)}{2\Delta^{2}}=-\frac{1}{2}\sigma^{2}-\frac{1}{2}\tau^{2}+\frac{\alpha_{3}}{6\,\sqrt{2}}\sigma\left(\sigma^{2}+3\tau^{2}\right)+\frac{\alpha_{4}}{48}\left(\sigma^{4}+6\sigma^{2}\tau^{2}+\tau^{4}\right)+\Delta(\dots).$
(64)

Finally, in the IR limit $\kappa/\Delta\ll 1$, we can expand the integral over
$\tau$ in eq. 60, to find

$\displaystyle\frac{\Omega_{\text{GW}}(k)}{\mathcal{A}_{\zeta}^{2}}=$
$\displaystyle\frac{3\kappa^{3}}{{2\pi}\Delta}\,\left[\frac{\pi^{2}}{4}+\ln^{2}\left(\frac{\sqrt{3}\,\kappa\,e^{\Delta^{2}}}{2}\right)\right]\int^{\infty}_{-\infty}\,d\sigma\,\exp^{\left[-\frac{1}{2}\sigma^{2}+\frac{\alpha_{3}}{6\,\sqrt{2}}\sigma^{3}+\frac{\alpha_{4}}{48}\sigma^{4}+\dots\right]}.$
(65)

Importantly, the integral along $\sigma$ is independent of $\kappa$, hence it
only affects the overall normalization and not the scale dependence in the IR.
At small $\kappa$, the spectrum scales as $\kappa^{3}$ (as expected) but there
are log corrections that modify the slope. We can compute the spectral index
associated with eq. (65), to obtain

$\displaystyle
n_{\Omega}\,=\,3+\frac{8\,\ln\left({\sqrt{3}\,\kappa\,e^{\Delta^{2}}}/{2}\right)}{\pi^{2}+4\ln^{2}\left({\sqrt{3}\,\kappa\,e^{\Delta^{2}}}/{2}\right)}\,.$
(66)

For very small $\kappa$, the second term is negligible, but as $\kappa$
increases the log corrections change $n_{\Omega}$ and bring it nearby
$n_{\Omega}=2.5$. See fig. 26 for a plot of formula (66).

Figure 26: Plot of eq. (66), choosing $\Delta=0.2$. The spectral index of the
SGWB approaches $n_{\Omega}=3$ for $\kappa\to 0$, but at larger $\kappa$ it is
more like $2.5$, in agreement with the numerical results.

These results justify fixing spectral index appearing in eq. (14) to
$n_{p}=2.5$ at the intermediate frequencies before reaching the far-infrared
tail.

###  B.2 DP template from log-normal curvature spectrum

We discuss here details of the connection between the DP template (14) and the
log-normal scalar power spectrum scenario. For definiteness, we take the log-
normal primordial curvature power spectrum as

$\mathcal{P}_{\zeta}(k)\\!=\\!{\cal
A}_{s}\exp\\!\left[-\frac{1}{2\Delta^{2}}\ln^{2}\\!\left(\frac{k}{k_{*}}\right)\right],$
(67)

where ${\cal A}_{s}$ parameterises the peak amplitude while $\Delta>0$
describes the width of the peak. In the left panel of fig. 27 we show how the
template presented in eq. (14) fits an example case with $\Delta=0.3$. We
report the scaling of the parameters of the template, as a function of
$\Delta$, in the right panel of fig. 27. These fits, alongside the ones
derived in the following section, motivate the choice of prior ranges for the
parameters that characterise the DP template.

Figure 27: Left panel: The red points indicate the scalar-induced GW spectrum
from a log-normal curvature power spectrum and the solid curves show the fit
by the DP template to these points. Right panel: Best-fit parameters
characterising the DP template when matched with the GW spectrum induced from
a log-normal curvature power spectrum of the form (67) as a function of
$\Delta$.

###  B.3 DP template from a broken power-law curvature spectrum

Following the same structure of the previous section, we report here details
of the connection between the DP template (14) and the broken power-law scalar
power spectrum defined in eq. (16). In the left panel of fig. 28 we show how
the template presented in eq. (14) fits an example case with $p_{1}=p_{2}=4$.
We report the scaling of the parameters of the template, as a function of
$p_{2}$, in the right panel of fig. 28, where we have fixed $p_{1}=4$ as
motivated by USR models of inflation (see also related discussion in the main
text).

Figure 28: Left panel: The red points indicate the scalar-induced GW spectrum
from a broken power-law curvature power spectrum and the solid curves show the
fit by the DP template to these points. Right panel: Fit of the the DP
template to the scalar-induced GW spectrum from a broken power-law curvature
power spectrum as a function of $p_{2}$.

A transient nonattractor phase in single-field inflationary scenarios leads to
a spectral growth which is bounded from above. In particular, ref.
Byrnes:2018txb  (see also ref. Ozsoy:2019lyy ) showed that for all single-
field models which include a short nonattractor phase the curvature spectrum
cannot grow faster than ${\cal P}_{\zeta}(k)\,\propto\,k^{4}$. However,
including a further prolonged phase of non-slow-roll evolution, the spectrum
can grow as ${\cal P}_{\zeta}(k)\,\propto\,k^{5}\,(\ln k)^{2}$
Carrilho:2019oqg . If we consider multiple phases of non-attractor evolution,
then ${\cal P}_{\zeta}(k)\,\propto\,k^{8}$ can be reached Tasinato:2020vdk ;
Davies:2021loj . Instead, after the peak, the decay of the spectrum can occur
with an arbitrarily steep (negative) slope.

In light of these results, we verified that the DP template is able to capture
also the SGWB spectral features produced by models with a very steep spectral
growth. In fig. 29, we show the corresponding best-fit template parameters as
a function of $p_{1}$, where, for simplicity, we have fixed the spectral drop
after the peak $p_{2}$ to very large values.

Figure 29: Fit of the the DP template to the scalar-induced GW spectrum from a
broken power-law curvature power spectrum as a function of $p_{1}$ in the
limit $p_{2}\to\infty$.

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