$k_{\text{peak}}$. The yellow-shaded area highlights the Planck $95\,\%$ C.L.
region, see (2.26). Right panel: the approximation (2.30) with $\alpha=0.005$
is plotted in grey together with the numerical results for the tensor-to-
scalar ratio against $\Delta N_{\text{CMB}}-\Delta N_{\text{peak}}$ (coloured
points) and $\Delta N_{\text{CMB}}-\Delta N_{\text{dip}}$ (empty triangles).
Each colour is associated with a specific $\phi_{\text{infl}}$, as detailed in
the legend. The yellow-shaded area represents the $95\,\%$ C.L. region from
the bound on $r_{\text{CMB}}$ (2.25).

In analogy with the analysis performed for the single-field inflection-point
model, we can compare the CMB observables listed in table 3 with the modified
universal predictions for $n_{s}$ (2.29), $r_{\text{CMB}}$ (2.30) and
$\alpha_{s}$ versus $n_{s}$ (2.36). In the left panel of figure 28 we plot eq.
(2.29) together with coloured points representing numerical results
$\left(\Delta N_{\text{CMB}}-\Delta N_{\text{peak}},\,n_{s}\right)$ for each
model considered. Similarly, the right panel shows eq. (2.30) with
$\alpha=0.005$ together with coloured points representing the numerical
results $\left(\Delta N_{\text{CMB}}-\Delta
N_{\text{peak}},\,r_{\text{CMB}}\right)$. We do not explicitly show the
results for $\alpha_{s}$ versus $\Delta N_{\text{CMB}}-\Delta N_{\text{peak}}$
as the comparison between the universal prediction (2.35) and the numerical
results is qualitatively the same as for the tilt $n_{s}$. Instead, we show
$\alpha_{s}$ versus $n_{s}$ in figure 29, alongside the $\alpha$–attractors
consistency relation (2.36).

As for the single-field case, the modified universal predictions describe well
the numerical results, with a small offset for the case of the tilt $n_{s}$
and its running $\alpha_{s}$. The match between the modified universal
predictions and the numerical results can be further improved by substituting
$\Delta N_{\text{CMB}}-\Delta N_{\text{peak}}\rightarrow\Delta
N_{\text{CMB}}-\Delta N_{\text{dip}}$ or, in other words, by taking as a
reference the scale associated with the local minimum in the power spectrum,
$k_{\text{dip}}$, instead of its local maximum, $k_{\text{peak}}$. We
demonstrate this in figure 28 by including the numerical results for $n_{s}$
and $r_{\text{CMB}}$ represented with empty triangles against $\Delta
N_{\text{CMB}}-\Delta N_{\text{dip}}$. By comparing triangles and circles (of
the same colour) with the grey lines, one can see that the CMB observables
(especially $n_{s}$) can be described even better by the position of the dip
in $P_{\zeta}(k)$. However, it is the peak in $P_{\zeta}(k)$ that could have
potentially observable consequences, such as PBH production and second-order
GW generation. Therefore we focus on the consequences of the modified
universal predictions for $k_{\text{peak}}>k_{\text{dip}}$ when it comes to
exploring the phenomenology.

Figure 29: The $\alpha$–attractors consistency relation (2.36) is plotted in
grey together with the numerical results for the tilt and its running for
multi-field models with $\\{\alpha=0.005,\,\gamma=10\\}$ and different
position of the inflection point $\phi_{\text{infl}}$, as detailed in the
legend. The yellow region highlights part of the $95\,\%$ C.L. region when
Planck data are compared with the
$\Lambda\text{CDM}+r_{\text{CMB}}+\alpha_{s}$ model, (2.22), while the purple
hatch-shaded area represents the lower $95\,\%$ C.L. region of $n_{s}$ for the
$\Lambda\text{CDM}$ model instead. The range of $\alpha_{s}$ shown is within
the observational bound (2.23).

The numerical results for the tilt and its running are well described by the
$\alpha$–attractors consistency relation (2.36), as shown in figure 29. As for
the single-field case, models which are compatible with the bound on the
spectral index (2.22) predict $\alpha_{s}$ about one order of magnitude
smaller than the current observational uncertainty in (2.23). We therefore
compare the model predictions with the CMB constraints for the
$\Lambda\text{CDM}$ model, in particular with the lower bound on $n_{s}$ given
in eq. (2.27) in the absence of running. Given the numerical results for
$n_{s}$ this implies that only configurations with $\phi_{\text{infl}}\lesssim
0.7$ are consistent with the CMB observations and the peak is located on
scales $k_{\text{peak}}\gtrsim 10^{20}\,\text{Mpc}^{-1}$.

Figure 30: Left panel: numerical results of $P_{\zeta}(k)$ for the multi-field
model with parameters $\\{\alpha=0.005,\,\gamma=10,\phi_{\text{infl}}=0.07\\}$
and initial condition $\theta_{\text{in}}=7\pi/10$. We display the results in
the region of the peak, with the pink line representing the scaling of the
infrared tail of it. Right panel: numerical results for the second-order GWs
produced during radiation domination by the enhanced scalar perturbations
whose power spectrum is displayed in the left panel. The pink line is obtained
using eq.(5.16) of [20] with $n_{\text{IR}}=12$ and
$n_{\text{UV}}\rightarrow\infty$ and well approximates the spectral shape of
the infrared tail of the numerical $\Omega_{\text{GW}}(k)$.

In the left panel of figure 30 we display the numerical results for the scalar
power spectrum in the region around the peak, for a model which is compatible
with large-scale (CMB) measurements of $n_{s}$, $\phi_{\text{infl}}=0.07$. The
pink line shows the scaling of the IR tail of the peak, $n_{\text{IR}}\simeq
12$, which by far exceeds the limits on the growth of the power spectrum
possible in single-field models [79, 126, 127]. In [15, 14, 26] it has been
shown that multi-field models with strong turns in field space evade the
single-field bound on the growth of $P_{\zeta}$, and we demonstrate here that
the same holds when the growth of the curvature perturbation is also due to
the strong curvature of the hyperbolic field space.

The peak in the scalar power spectrum displays a series of peaks which is due
to the strong and sharp turn in field space characterising the background
evolution, see e.g., the bottom-left panel of figure 16 for a model with
$\\{\alpha=0.1,\,\gamma=10,\,\phi_{\text{infl}}=0.542\\}$. In [23] it is shown
that a strong and sharp turn can lead to an exponentially enhanced amplitude
of $P_{\zeta}$, with an oscillatory modulation. In particular, in [14, 23] the
bending parameter $\eta_{\perp}$ is modelled with a Gaussian profile, see eq.
(4.38), and the Hubble rate is assumed to be smooth and slowly-varying. The
multi-field models discussed here are characterised by a more complicated
background evolution, see e.g., $\epsilon_{H}(N)$ in figure 26, and a profile
for the bending parameter which can only be partially described by a Gaussian
function, see e.g., figure 16. It is therefore not surprising that the
oscillations in $P_{\zeta}$ displayed in figure 30 cannot simply be identified
as either sharp or resonant features [128, 114, 129, 130], but are instead
more of a combination of the two, similar to models discussed e.g., in [26].

In the right panel of figure 30 we represent the numerical results for the
second-order GWs produced by the enhanced scalar fluctuations after horizon
re-entry during radiation domination. In particular, we display
$\Omega_{\text{GW}}(k)$ as numerically calculated using eq. (3.23) and the
scalar power spectrum displayed in the left panel of the same figure. The
sharpness of the peak of $P_{\zeta}$ (see the purple line in the left panel of
figure 27) results in a two-peak structure for $\Omega_{\text{GW}}(k)$, with a
broader and smaller peak followed by a dip and a narrower principal peak
located approximately at the scale $2/\sqrt{3}\,k_{\text{peak}}$. The
oscillatory modulation of the scalar power spectrum due to the sharp turn in
the background trajectory is imprinted in $\Omega_{\text{GW}}(k)$ as an
oscillatory pattern modulating the principal peak. The IR tail of the signal
can be understood in terms of the IR and UV scaling of $P_{\zeta}$ around the
first peak; substituting the values $n_{\text{IR}}=12$ and
$n_{\text{UV}}\rightarrow\infty$ into eq.(5.12) of [20], we get the pink line
shown on top of the numerical results. The UV tail of $\Omega_{\text{GW}}(k)$
(and $P_{\zeta}$) displays a more complicated scaling, which cannot be
understood in terms of such a simple approximation.

Figure 31: Numerical results for the second-order GWs produced within the
multi-field model with parameters
$\\{\alpha=0.005,\,\gamma=10,\,\phi_{\text{infl}}=0.07\\}$ and initial
condition $\theta_{\text{in}}=7\pi/10$.

In figure 31 the numerical results for $\Omega_{\text{GW}}(k)$ are plotted
together with the sensitivity curves of upcoming space- and Earth-based GW
observatories. The amplitude of the GW signal is determined by the amplitude
of the peak of the scalar power spectrum, which in turn mainly depends on the
curvature of the hyperbolic field space, set by the parameter $\alpha$, and
the initial condition for the angular field, $\theta_{\text{in}}$. In
particular, after fixing all the other model parameters, reducing $\alpha$
enhances $P_{\zeta}(k_{\text{peak}})$ and therefore the amplitude of the
principal peak of $\Omega_{\text{GW}}(k)$. Our numerical results show that the
multi-field model described by
$\\{\alpha=0.005,\,\gamma=10,\,\phi_{\text{infl}}=0.07,\,\theta_{\text{in}}=7\pi/10\\}$
produces $\Omega_{\text{GW}}\simeq 10^{-10}$ at its peak.

The position of the principal peak in $\Omega_{\text{GW}}(k)$ is set by the
position of the largest peak in $P_{\zeta}$, which is the first peak displayed
in the left panel of figure 30. For models which are not in tension with the
large-scale measurements of $n_{s}$, the second-order GW principal peak is
located at $f_{\text{peak}}\gtrsim 50\,\text{kHz}$, which is obtained by
substituting the lower bound (2.34) in the position of the principal peak,
$2/\sqrt{3}\,k_{\text{peak}}$, and is consistent with our numerical results.
Given that the modified universal predictions (in particular, the one for
$n_{s}$ (2.29)) hold also for the multi-field inflection-point potential, the
considerations for PBH production discussed in section 3.2 for the single-
field case apply also for the multi-field extension.

In summary, compatibility with the CMB observations of $n_{s}$ place the peak
of the second-order GWs beyond the reach of current and upcoming GW
observatories both for the single- and multi-field inflection-point potentials
considered in this work, as well as bounding the mass of the PBHs that could
possibly be produced to values $M_{\text{PBH}}<10^{8}\,\text{g}$, which make
the PBHs too light to constitute candidates for dark matter in the Universe
today.

##  5 Discussion

In this work we explore the phenomenology of cosmological $\alpha$–attractor
models featuring an inflection point in the potential for scalar fields
evolving in a hyperbolic field space. We consider both single-field examples
and multi-field dynamics.

In all cases we show that the primordial perturbations generated on large
(CMB) scales can be described by a simple modification of the universal
predictions of $\alpha$–attractors for the scalar spectral index, eq. (2.29),
and the tensor-to-scalar ratio, eq. (2.30). A shift in the universal
predictions was previously noted by [11] for the single-field inflection-point
$\alpha$–attractor potential, and also in a different context by [44]. This
universal behaviour leads to a consistency condition relating the scalar
spectral tilt and its running, eq. (2.36). A consequence of the tight bounds
on the scalar spectral index from CMB observations is that the running of the
spectral index must be small, and any deviations from the standard single-
field $\alpha$-attractor dynamics is constrained to lie close to the end of
inflation. Hence any enhancement in the primordial power spectrum,
$P_{\zeta}$, is only allowed on small comoving scales, $k_{\text{peak}}\gtrsim
5\times 10^{18}\,\text{Mpc}^{-1}$. By adopting the tight observational bounds
on the spectral index obtained from CMB observations, in the absence of any
running of the spectral index, eq. (2.27), which we emphasise is constrained
by the consistency relation (2.36), we obtain a stronger constraints on the
scale of the peak of $P_{\zeta}$ in comparison to the previous work of
Dalianis et al. [11].

The lower bound on the comoving wavenumber, $k_{\text{peak}}$, implies that
any primordial black holes resulting from enhanced density perturbations on
small scales can only be produced with masses $M_{\text{PBH}}\lesssim
10^{8}\,\text{g}$. These PBHs have long since evaporated by the present time
so do not constitute a candidate for dark matter. Nonetheless they could yet
leave interesting signatures if stable Planck mass relics are left behind [86,
87, 88], or resulting from an early black-hole-dominated era [80, 81, 82, 83,
59].

Similarly the lower bound on $k_{\text{peak}}$ also has implications for the
induced gravitational waves produced at second order in perturbation theory
after inflation, see eq. (3.23). The peak of the GW signal is constrained to
be at very high frequencies, $f_{\text{peak}}\gtrsim 50\,\text{kHz}$, well
beyond the reach of current Earth-based or future space-based GW detectors.
Rather they could provide a target for future ultra-high frequency detectors
discussed in [131].

A key parameter for the inflationary dynamics is the curvature of the
hyperbolic field space, $\mathcal{R}_{\text{fs}}\equiv-4/(3\alpha)$. In
single-field models the small-scale $P_{\zeta}$ is enhanced by the presence of
the inflection point in the potential, leading to a phase of ultra-slow-roll
dynamics. The effect of $\alpha$ is primarily to set the relative amplitude of
the tensor modes with respect to scalar perturbations at CMB scales, see eq.
(2.30). In multi-field embeddings a far richer dynamical behaviour is
possible. We have seen that a large and negative curvature, $\alpha\ll 1$, can
cause a geometrical instability in the inflationary trajectory, as previously
studied in [106, 109]. A strongly-curved field space could also be accompanied
by strongly non-geodesic motion and a tachyonic instability in the
isocurvature perturbation, which gets transferred to the curvature
fluctuation, potentially leading to a small-scale growth of the scalar power
spectrum [14, 16, 17]. For these reasons we choose to focus on models
characterised by $\alpha\ll 1$, see also [45]. The behaviour we see in this
case differs from that seen in other multi-field $\alpha$-attractor models
studied, for example, in [43] where the authors consider a potential monotonic
in the radial direction, with field-space curvature $\alpha=1/3$. In that case
the single-field predictions, eqs. (1.6) and (1.7), were found to be stable
even in presence of a light angular direction. Here we introduce both a
feature in the radial potential (the inflection point) and consider larger
curvature ($\alpha\ll 1$), both of which amplify the geometrical
destabilisation of the background trajectory, breaking the slow-roll and slow-
turn approximations. While the radial field at early times follows the
standard evolution close to the boundary of the Poincaré disc ($r\rightarrow
1$), the dynamics of the angular field then leads to a second distinct phase
of inflation at late times. Nonetheless on large scales CMB observables can
still be explained by means of a simple modification of the standard universal
predictions of single-field $\alpha$–attractors, see eqs. (2.29) and (2.30).

We find marked differences in the spectral shape of the scalar power spectrum
found in the single- and multi-field cases. In the single-field case the peak
in $P_{\zeta}$ is broad ($n_{\text{IR}}\simeq 3.4,\,n_{\text{UV}}\simeq-4$),
while in the multi-field case it is narrower, with a much steeper infrared
growth ($n_{\text{IR}}\simeq 12$), and oscillations following the principal
peak. Figure 32 shows a comparison of the scalar power spectra obtained in a
single- and a multi-field inflection-point model. Multi-field effects can
explain the oscillatory behaviour of the green line after the peak, caused by
a sharp and strong turn in field space, providing an explicit realisation of
the mechanism discussed in [23, 24] (see also [26]).

Figure 32: Left panel: numerical results of $P_{\zeta}(k)$ for a single-field
inflection-point model, eq. (2.8), with parameters
$\\{\alpha=0.1,\,\phi_{\text{infl}}=0.5,\,\xi=0.0035108\\}$ (pink line) and
for a multi-field model, eq. (4.35), with parameters
$\\{\alpha=0.005,\,\gamma=10,\phi_{\text{infl}}=0.07\\}$ and initial condition
$\theta_{\text{in}}=7\pi/10$ (green line). These models predict $n_{s}=0.9568$
and $n_{s}=0.9569$ respectively and are both compatible with the CMB lower
bound on $n_{s}$, (2.27). Right panel: numerical results for the second-order
GWs produced during radiation domination by the enhanced scalar perturbations
whose power spectra are displayed in the left panel, together with the
sensitivity curves of current and upcoming GW observatories. The colour legend
is the same as on the left.

The differences in the spectral shape of the peak in $P_{\zeta}$ are reflected
in the power spectrum of the induced GWs, as shown in the right panel of
figure 32. In particular, the narrower, oscillatory peak in the multi-field
$P_{\zeta}$ leads to second-order GWs with a two-peak structure and an
oscillatory modulation of the second (principal) peak, see the green line in
the right panel of figure 32. In the single-field set-up, the two-peak
structure is almost wiped out as a consequence of the broadness of the scalar
power spectrum peak and the oscillations are absent, see the pink line in the
right panel of figure 32.

Strong enhancement of the scalar perturbations on small scales might be
expected to be associated with significant non-Gaussianity. This can have an
important impact on both the production of PBHs [132, 133, 134, 73, 135, 136,
137, 138, 139, 140, 12, 141, 142, 143] and second-order GWs induced [20, 144,
145, 146, 94, 147, 122], and it would be interesting to explore non-
Gaussianities in these models and their possible effect on the small-scale
phenomenology.

Note added: while this paper was under peer review, several related works
appeared on the arXiv [148, 149, 150, 151, 152, 153, 154].

## Acknowledgments

The authors are grateful to Matteo Braglia and Andrew Gow for numerous
discussions, as well as Dani Figueroa, Jacopo Fumagalli, Juan García-Bellido,
Sébastien Renaux-Petel, Angelo Ricciardone and Lukas Witkowski for very useful
comments. The work of LI is supported by UK STFC grant ST/R505018/1 and
ST/S505651/1. DW and HA are supported by UK STFC grant ST/S000550/1. M.F.
would like to acknowledge support from the “Atracción de Talento” grant
2019-T1/TIC15784, his work is partially supported by the Spanish Research
Agency (Agencia Estatal de Investigación) through the Grant IFT Centro de
Excelencia Severo Ochoa No CEX2020-001007-S, funded by
MCIN/AEI/10.13039/501100011033. LI acknowledges the kind hospitality of Jean
Allison and her family while this work was completed. For the purpose of open
access the authors have applied a Creative Commons Attribution (CC-BY) licence
to any Author Accepted Manuscript version arising. Supporting research data
are available on reasonable request from the corresponding author.

##  Appendix A Universality of $\bm{\alpha}$–attractors

Cosmological $\alpha$–attractors correspond to a class of inflationary models
which provide robust observational predictions despite having apparently
different formulations (see [27] and references therein). In particular, the
large-scale CMB spectral index and tensor-to-scalar ratio are given by eqs.
(1.6) and (1.7) at leading order in the expansion in terms of $1/\Delta
N_{\text{CMB}}$. We review here how the $\alpha$–attractor models form a
universality class and derive the observables
$(n_{s},\;r_{\text{CMB}},\;\alpha_{s})$ for single-field $\alpha$–attractor
models given by a monomial potential in terms of the radial distance from the
centre of the Poincaré disc (1.4)

$V(\phi)=V_{0}\tanh^{p}{\left(\frac{\phi}{\sqrt{6\alpha}}\right)}\;.$ (A.1)

In canonical single-field slow-roll inflation, $(n_{s},\;r,\;\alpha_{s})$ can
be given in terms of potential slow-roll parameters as [155]

$n_{s}=1-6\epsilon_{V}+2\eta_{V}\;,\;\;\;r_{\text{CMB}}=16\epsilon_{V}\;,\;\;\;\alpha_{s}=16\epsilon_{V}\eta_{V}-24{\epsilon_{V}}^{2}-2{\xi_{V}}^{2}\;,$
(A.2)

where $\epsilon_{V}$, $\eta_{V}$ and $\xi_{V}$ are expressed in terms of
derivatives of the inflaton potential,

$\epsilon_{V}=\frac{1}{2}\left(\frac{V_{\phi}}{V}\right)^{2}\;,\;\;\;\eta_{V}=\frac{V_{\phi\phi}}{V}\;,\;\;\;{\xi_{V}}^{2}=\frac{V_{\phi}V_{\phi\phi\phi}}{V^{2}}\;.$
(A.3)

Given the explicit form of the potential in eq. (A.1), it is possible to write
these potential slow-roll parameters in terms of $\phi$,

$\displaystyle\epsilon_{V}$ $\displaystyle=$
$\displaystyle\frac{p^{2}}{3\alpha}\operatorname{csch}^{2}{\left(\frac{2\phi}{\sqrt{6\alpha}}\right)}\>,$
(A.4) $\displaystyle\eta_{V}$ $\displaystyle=$
$\displaystyle\frac{2p}{3\alpha}\operatorname{csch}^{2}{\left(\frac{2\phi}{\sqrt{6\alpha}}\right)}\left(p-\cosh{\left(\frac{2\phi}{\sqrt{6\alpha}}\right)}\right)\;,$
(A.5) $\displaystyle\xi_{V}^{2}$ $\displaystyle=$
$\displaystyle\frac{2p^{2}}{9\alpha^{2}}\operatorname{csch}^{4}{\left(\frac{2\phi}{\sqrt{6\alpha}}\right)}\left(3+2p^{2}-6p\cosh{\left(\frac{2\phi}{\sqrt{6\alpha}}\right)}+\cosh{\left(\frac{4\phi}{\sqrt{6\alpha}}\right)}\right)\;.$
(A.6)

Substituting the above into eq. (A.2) yields the large scales observables in
terms of $\phi$. They are evaluated when the CMB scales left the horizon,
i.e., at $\phi=\phi_{\text{CMB}}$. In order to do so, we use the inflaton
equation of motion, eq. (2.9), which in the slow-roll approximation can be
simplified to give

$\frac{\mathrm{d}\phi}{\mathrm{d}N}\simeq-\frac{V_{\phi}}{V}\;.$ (A.7)

Integrating the equation above yields the number of e-folds elapsed between
the two field values $\phi_{\text{CMB}}$ and $\phi_{\text{end}}$,

$\Delta
N_{\text{CMB}}\simeq\int_{\phi_{\text{end}}}^{\phi_{\text{CMB}}}\mathrm{d}\phi\,\frac{V}{V_{\phi}}\;.$
(A.8)

Performing the integration above for the potential in eq. (A.1) yields

$\Delta
N_{\text{CMB}}\simeq\frac{3\alpha}{2p}\left[\cosh{\left(\frac{2\phi_{\text{CMB}}}{\sqrt{6\alpha}}\right)}-\cosh{\left(\frac{2\phi_{\text{end}}}{\sqrt{6\alpha}}\right)}\right]\;,$
(A.9)

where the value of $\phi_{\text{end}}$ is fixed by the condition
$\epsilon(\phi_{\text{end}})=1$, corresponding to

$\sinh^{2}{\left(\frac{2\phi_{\text{end}}}{\sqrt{6\alpha}}\right)}\simeq\frac{p^{2}}{3\alpha}\;.$
(A.10)

Substituting eq. (A.10) into eq. (A.9) and expressing the equation in terms of
$\phi_{\text{CMB}}$ yields

$\sinh^{2}{\left(\frac{\phi_{\text{CMB}}}{\sqrt{6\alpha}}\right)}\simeq\frac{p\Delta
N_{\text{CMB}}}{3\alpha}+\frac{\sqrt{3\alpha+p^{2}}}{2\sqrt{3\alpha}}-\frac{1}{2}\;.$
(A.11)

In this way the CMB observables in eq. (A.2) can be written as

$\displaystyle n_{s}$ $\displaystyle\simeq 1-\frac{2\Delta
N_{\text{CMB}}+\frac{1}{p}\sqrt{3\alpha(3\alpha+p^{2})}+\frac{3\alpha}{2}}{\Delta
N_{\text{CMB}}^{2}+\frac{\Delta
N_{\text{CMB}}}{p}\sqrt{3\alpha(3\alpha+p^{2})}+\frac{3\alpha}{4}}\;,$ (A.12)
$\displaystyle r_{\text{CMB}}$ $\displaystyle\simeq\frac{12\alpha}{\Delta
N_{\text{CMB}}^{2}+\frac{\Delta
N_{\text{CMB}}}{p}\sqrt{3\alpha(3\alpha+p^{2})}+\frac{3\alpha}{4}}\;,$ (A.13)
$\displaystyle\alpha_{s}$
$\displaystyle\simeq-8\frac{18\alpha^{2}+p(3\alpha+4\Delta
N_{\text{CMB}})\sqrt{3\alpha(3\alpha+p^{2})}+p^{2}[4\Delta
N_{\text{CMB}}^{2}+3\alpha(1+2\Delta N_{\text{CMB}})]}{(3p\alpha+4\Delta
N_{\text{CMB}}\sqrt{3\alpha(3\alpha+p^{2})}+4p\Delta
N_{\text{CMB}}^{2})^{2}}\;.$ (A.14)

In the large $\Delta N_{\text{CMB}}$ expansion, these expressions reduce to
eqs. (1.6) and (1.7) for the observables $n_{s}$ and $r_{\text{CMB}}$
respectively on large (CMB) scales, while we can relate the running of the
spectral index to the spectral index itself

$\alpha_{s}\approx-\frac{2}{\Delta
N_{\text{CMB}}^{2}}\approx-\frac{(n_{s}-1)^{2}}{2}\,,$ (A.15)

regardless of the parameters of $V_{0}$ and $p$ appearing in the potential
(A.1). The spectral index $n_{s}$ and running $\alpha_{s}$ are dependent only
on $\Delta N_{\text{CMB}}$, while $r_{\text{CMB}}$ depends only on $\Delta
N_{\text{CMB}}$ and $\alpha$, such that

$r_{\text{CMB}}\approx 3\alpha(n_{s}-1)^{2}\,.$ (A.16)

This is due to the potential (A.1) remaining finite at the boundary of the
moduli space ($r\equiv\tanh{\left(\phi/\sqrt{6\alpha}\right)}\rightarrow 1$),
which is a key feature of $\alpha$–attractor models. The transformation to the
canonical field $\phi$ (1.4), renders the potential a function of
$\tanh{\left(\phi/\sqrt{6\alpha}\right)}$, which ensures the flatness of the
potential for large field values ($\phi\rightarrow\infty$) and makes
observational predictions on large scales approximately independent of the
precise form of the function describing the potential dependence on
$\tanh{\left(\phi/\sqrt{6\alpha}\right)}$.

##  Appendix B Numerical computation of the single-field scalar power spectrum

In this appendix we present in detail how we compute the scalar power
spectrum, $\mathcal{P}_{\zeta}(k)$, for the single-field inflation models
considered in section 2. Our procedure is closely related to similar
strategies described in the literature, see for example [9, 156].

The inflaton potential in eq. (2.8) features an approximate stationary
inflection point located at $\phi_{\text{infl}}$ whose effect is to introduce
a transient ultra-slow-roll phase. The slow-roll approximation breaks down and
a full numerical analysis of the Mukhanov–Sasaki equation is needed. As an
example, we refer to the power spectrum resulting from the specific model
$\\{\alpha=0.1,\;\phi_{\text{infl}}=0.5,\;\xi=0.0035108\\}$, whose background
evolution is displayed in figure 2. For illustrative purposes we consider here
the case of instant reheating, which implies $\Delta N_{\text{CMB}}\simeq 55$
(see table 1).

In the slow-roll approximation, valid on large scales, far from the inflection
point, it is possible to estimate the scalar power spectrum using the
expression for $P_{\zeta}(N)$ given in (2.17), by substituting in the
background quantities $H(N)$ and $\epsilon_{H}(N)$. $P_{\zeta}(N)$ can be
transformed into $P_{\zeta}(k)$ by means of eq. (2.32) for $k(N)$. We
normalise the amplitude of the scalar power spectrum at
$k_{\text{CMB}}=0.05\,\text{Mpc}^{-1}$ by using the Planck 2018 measurement of
$\mathcal{A}_{s}$ [35], which in turns identifies the amplitude of the
potential as $V_{0}=7.7\times 10^{-10}$. Eqs. (2.19)–(2.21) predict for this
configuration the large-scale observables

$n_{s}=0.9569\;,\;\;\;\alpha_{s}=-0.00092\;,\;\;\;r_{0.002}=4.956\times
10^{-4}\;,$ (B.1)

consistent with the latest Planck data release [35], where the spectral tilt
and its running are evaluated at $k_{\text{CMB}}$.

While eq. (2.17) is reliable when the evolution is well described by slow
roll, a full numerical analysis of the perturbations is necessary in order to
compute the exact scalar power spectrum including non-slow-roll evolution, as
occurs through the inflection point and approaching the end of inflation. The
Mukhanov–Sasaki equation describes the evolution of the scalar curvature
perturbation associated with the comoving wavenumber $k$, $\zeta_{k}$. We
define $\zeta_{k}\equiv v_{k}/z$, where $z\equiv a\,\phi^{\prime}$, and a
prime denotes derivatives with respect to e-folds, $N$. The mode $v_{k}$ is
given by the solution of the Mukhanov–Sasaki equation [157]

$v_{k}^{\prime\prime}+(1-\epsilon_{H})v_{k}^{\prime}+\left[\frac{k^{2}}{a^{2}H^{2}}+(1+\epsilon_{H}-\eta_{H})(\eta_{H}-2)-(\epsilon_{H}-\eta_{H})^{\prime}\right]v_{k}=0\;,$
(B.2)

where the Hubble slow-roll parameters, $\epsilon_{H}$ and $\eta_{H}$, are
defined in eqs. (2.13) and (2.14). Note however that this equation does not
assume slow roll.

We solve eq. (B.2) for modes ranging from
$k_{\text{CMB}}=0.05\,\text{Mpc}^{-1}$ to $k_{\text{end}}=2.6\times
10^{22}\,\text{Mpc}^{-1}$ and follow the evolution of $v_{k}$ for each
wavemode from the sub-horizon regime ($k\gg aH$, where canonical quantum
commutation relations give the normalisation for the mode function,
$|v_{k}^{2}|=1/2k$ [157]) to super-horizon scales ($k\ll aH$). These solutions
then enable us to calculate the scalar power spectrum on super-horizon scales
as

$P_{\zeta}(k)=\frac{k^{3}}{2\pi^{2}}\Big{|}\frac{v_{k}}{z}\Big{|}^{2}\;,$
(B.3)

where each mode is evaluated well after horizon crossing ($k\ll aH$), when
$\zeta_{k}\equiv v_{k}/z$ approaches a constant value.

There are a few practical considerations regarding the strategy used to solve
eq. (B.2):

  * •

when numerically solving eq. (B.2), it is useful to express $\epsilon_{H}$ and
$\eta_{H}$ in terms of derivatives of the field $\phi(N)$, instead of using
their definitions in terms of derivatives of $H(N)$. In particular, the
relevant terms in eq. (B.2) reduce to [158]

$\displaystyle 1-\epsilon_{H}$ $\displaystyle=1-\frac{\phi^{\prime 2}}{2}\;,$
(B.4)
$\displaystyle(1+\epsilon_{H}-\eta_{H})(\eta_{H}-2)-(\epsilon_{h}-\eta_{H})^{\prime}$
$\displaystyle=-2-3\frac{\phi^{\prime\prime}}{\phi^{\prime}}-\frac{\phi^{\prime\prime\prime}}{\phi^{\prime}}+\frac{\phi^{\prime
2}}{2}+\frac{\phi^{\prime}\phi^{\prime\prime}}{2}\;,$ (B.5)

where the second and third derivatives of the field are given by the
Klein–Gordon equation (2.9) for the field, re-written in terms of $\phi(N)$,

$H^{2}\phi^{\prime\prime}+HH^{\prime}\phi^{\prime}+3H^{2}\phi^{\prime}+V_{\phi}=0\,,$
(B.6)

and its derivative;

  * •

the factor $k/aH$ can be normalised by noting that the scale
$k_{\text{CMB}}=0.05\,\text{Mpc}^{-1}$ crossed the horizon $\Delta
N_{\text{CMB}}$ e-folds before the end of inflation;

  * •

in order to minimise numerical errors, one should evolve the background
solution for a sufficiently long time before the relevant scales cross the
horizon. This can be achieved by choosing a large enough initial value of
$\phi$;

  * •

instead of solving directly for the complex perturbation, $v_{k}$, it is
simpler to solve separately for its real and imaginary parts [159]. For each
mode, the integration of eq. (B.2) is started 5 e-folds before horizon
crossing, where Bunch–Davies initial conditions are applied [157], and is
integrated up until the end of inflation. In terms of the real and imaginary
part of $v_{k}$, the initial conditions are

$\text{Re}\\{v_{k}\\}=\frac{1}{\sqrt{2k}}\;,\;\text{Re}\\{v_{k}^{\prime}\\}=0\;,\;\text{Im}\\{v_{k}\\}=0\;,\;\text{Im}\\{v_{k}^{\prime}\\}=-i\frac{\sqrt{k}}{\sqrt{2}k_{\text{in}}}\;,$
(B.7)

where $k_{\text{in}}$ is the mode that crossed the horizon when the
integration is started;

  * •

the correct normalisation on CMB scales for the power spectrum in eq. (B.3) at
$k=k_{\text{CMB}}$ is set by fixing $V_{0}$ and hence the Hubble scale when
$k_{\text{CMB}}$ leaves the horizon.

Figure 33: Mode evolution, $v_{k}(N)/z$, for the CMB scale, $k_{\text{CMB}}$.
The real and imaginary parts, and modulus correspond to the pink- and green-
dashed lines, and the black-solid line respectively. The vertical thin line
signals the moment in which the CMB scale crosses the horizon $\Delta
N_{\text{CMB}}\sim 55$ e-folds before the end of inflation.

As an example, we solve eq. (B.2) for $k_{\text{CMB}}$ and plot in figure 33
the mode evolution, $v_{k}(N)/z$. The mode starts off in the Bunch–Davies
vacuum, oscillates in the sub-horizon regime and freezes to a constant value
after crossing the horizon.

In the left panel of figure 34, the evolution of three different modes is
represented for comparison in the last 10 e-folds before the end of inflation.
The continuous line is associated with the mode $k_{\text{CMB}}$, which for
the range of e-folds represented is well outside of the horizon and frozen at
a constant value. The dashed line describes the scale
$k_{\text{dip}}=10^{17}\,\text{Mpc}^{-1}$, which corresponds to the dip in the
scalar power spectrum. Finally, the mode corresponding to the peak in the
scalar power spectrum, $k_{\text{peak}}=9\times 10^{19}\,\text{Mpc}^{-1}$, is
plotted with the dotted line. The mode associated to $k_{\text{peak}}$
experiences the largest growth as it crosses the horizon close to the onset of
the ultra-slow-roll phase. See [156] for a detailed discussion of the
mechanism of growth (suppression) which shapes the modes’ evolution and the
scalar power spectrum.

Figure 34: Left panel: comparison between different modes’ evolution in the
last 10 e-folds of inflation. The region between the two thin vertical lines
is characterised by $\eta_{H}>3$. Right panel: numerical results for the
scalar power spectrum, $P_{\zeta}(k)$, compared with the slow-roll
approximation in eq. (2.17).

In the right panel of figure 34 we show a comparison of the numerical results
for $P_{\zeta}(k)$ compared with the slow-roll approximation in eq. (2.17). On
large scales the numerical results agree very well with the slow-roll
approximation, showing that the slow-roll CMB predictions in (B.1) are
reliable on these scales. On the other hand, on small scales the exact power
spectrum differs substantially from the slow-roll approximation, both in terms
of the position and the height of the peak. For the configuration under
analysis, the power spectrum features a peak of $\mathcal{O}(0.01)$ at the
comoving scale $k_{\text{peak}}$.

##  Appendix C Limiting behaviour of the single-field potential

In sections 2 and 3 we considered the phenomenology of the inflection-point
potential in eq. (2.8) with the aim of realising an ultra-slow-roll phase and
enhancing the scalar perturbations on small scales. For completeness we
discuss here the limiting behaviour of the same potential when the inflection
point is located at small or large $\phi$ values. For simplicity we restrict
our discussion to the case $\xi=0$ and choose $\alpha=0.1$.

In the limit $\phi_{\text{infl}}\rightarrow 0$, the dominant contribution to
the potential in eq. (2.8) comes from the term proportional to
$\tanh^{3}{(\phi/\sqrt{6\alpha})}$,

$\lim_{\phi_{\text{infl}}\rightarrow 0}V(\phi)\simeq
V_{0}\left[\frac{1}{3}\coth{\left(\frac{\phi_{\text{infl}}}{\sqrt{6\alpha}}\right)}\right]^{2}\,\tanh^{6}{\left(\frac{\phi}{\sqrt{6\alpha}}\right)}\;.$
(C.1)

It is therefore interesting to analyse the inflationary predictions of the
inflection-point potential with $\phi_{\text{infl}}$ small and compare them
with those obtained from the $\alpha$–attractor T-model potential [36]

$U(\phi)=U_{0}\tanh^{6}{\left(\frac{\phi}{\sqrt{6\alpha}}\right)}\;.$ (C.2)

On the other hand, when $\phi_{\text{infl}}$ is large
$\tanh{(\phi_{\text{infl}}/\sqrt{6\alpha})}\to 1$ and there is no simple
limiting behaviour for the inflection-point potential, as illustrated in
figure 35. For large $\phi$ values ($\phi\gtrsim 2$ for the configuration
plotted in figure 35) the dominant contribution comes from the
$\tanh^{3}{(\phi/\sqrt{6\alpha})}$ term (since $f_{1}\approx-f_{2}$ in (2.7)
and the $\tanh{(\phi/\sqrt{6\alpha})}$ and $\tanh^{2}{(\phi/\sqrt{6\alpha})}$
terms approximately cancel), but for smaller $\phi$ the potential receives
contributions from all the terms.

Figure 35: Behaviour of the inflection-point potential in the large
$\phi_{\text{infl}}$ limit. The plot is produced with the parameters
$\\{\alpha=0.1,\,\phi_{\text{infl}}=10,\,\xi=0\\}$. The potential
$V(\phi)/V_{0}$ in eq. (2.8) (black-dashed line) is plotted together with the
contributions coming from the single terms.

We consider two benchmark values, $\phi_{\text{infl}}=\\{0.1,\,10\\}$ in eq.
(2.8), in order to study their background evolution and compare it with that
obtained from the potential in eq. (C.2). The evolution of the Hubble slow-
roll parameter, $\epsilon_{H}$, and the inflaton field, $\phi$, are plotted in
the top row of figure 36 against the number of e-folds to the end of
inflation, $\Delta N\equiv N_{\text{end}}-N$.

Figure 36: Top Row: Comparison between the background evolution of the
inflection-point model with $\phi_{\text{infl}}=\\{0.1,\,10\\}$ and the
$\alpha$–attractor T-model in eq. (C.2). The bullet points in the right-hand
plot indicate the field values when the CMB scale crossed the horizon. Bottom
Row: In the left panel the potentials are plotted together with bullet points
indicating the field values corresponding to the CMB scale (right bullet) and
the end of inflation (left bullet). The legend identifying each line is the
same as in the top-left panel. In the right panel the predictions of the CMB
observables are represented together with the marginalised joint $68\,\%$ and
$95\,\%$ C.L. regions in the $(n_{s},\,r)$ plane at $k=0.002\,\text{Mpc}^{-1}$
as obtained from $\text{{Planck}}+\text{BK}15+\text{BAO}$ data assuming the
$\Lambda\text{CDM}+r_{\text{CMB}}$ cosmological model [35]. The black star
corresponds to the $\alpha$–attractor T-model potential (C.2), the pink
triangle corresponds to the inflection-point potential with
$\phi_{\text{infl}}=10$ and the remaining coloured circles correspond to
different choices of small $\phi_{\text{infl}}$ indicated in the legend.

As expected, the background evolution for the inflection-point potential with
$\phi_{\text{infl}}=0.1$ is almost identical to that produced by the
$\alpha$–attractor T-model potential (C.2). The evolution corresponding to
$\phi_{\text{infl}}=10$ is instead quite different and the reason why this is
the case is clear from the bottom-left panel of figure 36, where the
potentials and the field values corresponding to scales observed in the CMB
are shown. For the model with $\phi_{\text{infl}}=10$, the inflationary
evolution observable in the CMB is located at $\phi<2$ where the potential is
not well-approximated by the function $\tanh^{6}{(\phi/\sqrt{6\alpha})}$, as
discussed above and illustrated in figure 35. The bottom-right panel in figure
36 shows the CMB observables $(n_{s},\,r_{\text{CMB}})$ at the scale
$0.002\,\text{Mpc}^{-1}$ predicted by each potential. One sees that the
predictions obtained with $\phi_{\text{infl}}=0.1$ are not distinguishable
from those produced by the T-model for $\phi_{\text{infl}}\leq 0.4$. The
slightly lower value of $n_{s}$ predicted by $\phi_{\text{infl}}=0.5$ could in
future allow us to distinguish it form the $\alpha$–attractor T-model. As
expected, the predictions of the model with $\phi_{\text{infl}}=10$ differ
from the T-model ones and the two could potentially be distinguished by the
predicted value of $r_{0.002}$, lower for the inflection-point potential with
$\phi_{\text{infl}}=10$.

##  Appendix D Parameter study of the multi-field potential

The multi-field potential (4.35) is parametrised by
$\\{\alpha,\,\phi_{\text{infl}},\,\gamma\\}$. While in the main text we
discuss the effect of varying $\alpha$, we consider here the dependence on
$\phi_{\text{infl}}$ and $\gamma$, for trajectories with a fixed initial value
$\theta_{\text{in}}=7\pi/10$. The models discussed in this appendix are not
necessarily compatible with the CMB measurements on large scales, but instead
they are selected because they demonstrate the impact of changing
$\phi_{\text{infl}}$ and $\gamma$.

Figure 37: Left panel: effect of varying the position of the inflection point
in the multi-field potential (4.35) with parameters
$\\{\alpha=0.1,\,\gamma=10\\}$ and initial conditions
$\\{\theta_{\text{in}}=7\pi/10,\,\theta^{\prime}_{\text{in}}=0,\,\phi_{\text{in}}=3.1,\,\phi^{\prime}_{\text{in}}=0\\}$.
Right panel: effect of changing the parameter $\gamma$ in the potential (4.35)
with parameters $\\{\alpha=0.1,\,\phi_{\text{infl}}=0.542\\}$ and same initial
conditions as in the left panel.

First we numerically solve the background equations (4.22)–(4.24) for a model
with $\\{\alpha=0.1,\,\gamma=10\\}$, testing
$\phi_{\text{infl}}=\\{0.54,\,0.542,\,0.544\\}$, and show the resulting
$\epsilon_{H}(N)$ against $\Delta N\equiv N_{\text{end}}-N$ in the left panel
of figure 37. We see that the value of $\phi_{\text{infl}}$ affects the time
the fields spend around the inflection point and therefore the duration of the
second slow-roll phase (which is itself an attractor solution regardless of
the value $\phi_{\text{infl}}$). In other words, $\phi_{\text{infl}}$
determines the time at which the transition between the two inflationary
phases happens.

Next, we test $\gamma=\\{5,\,10,\,100,\,500\\}$ for the multi-field potential
(4.35) with parameters $\\{\alpha=0.1,\,\phi_{\text{infl}}=0.542\\}$. We
consider $\gamma>3$ so that the turn in field space happens within the last
$60$ e-folds of inflationary evolution, i.e., so that the second slow-roll
phase lasts less than $60$ e-folds. The resulting profile for
$\epsilon_{H}(N)$ is displayed in the right panel of figure 37 and
demonstrates that the value of $\gamma$ affects the duration of the second
phase of inflation, similar to the effect of varying $\phi_{\text{infl}}$.
Also, increasing $\gamma$ beyond $\gamma\sim 100$ does not significantly
change the background evolution.

In order to understand the numerical results displayed in figure 37 it is
useful to calculate the effective squared-mass of the angular field,
${m_{\theta}}^{2}\equiv\partial^{2}U(r,\,\theta)/\partial\theta^{2}$. Using
the multi-field potential (4.35), written in terms of the (non-canonical)
radial field $r$, and assuming $H^{2}\simeq U(r,\,\theta)/3$, yields

$\frac{{m_{\theta}}^{2}}{H^{2}}=18\,r\,r_{\text{infl}}\,\frac{\gamma}{1+\gamma}\frac{\left(r^{2}+3r_{\text{infl}}^{2}-\frac{3\,r\,r_{\text{infl}}}{1+\gamma}\right)\cos{\theta}-\frac{3\,\gamma\,r\,r_{\text{infl}}}{1+\gamma}\,\cos{2\theta}}{\left(r^{2}+3r_{\text{infl}}^{2}-3\,r\,r_{\text{infl}}\,\frac{1+\gamma\,\cos{\theta}}{1+\gamma}\right)^{2}}\;,$
(D.1)

where
$r_{\text{infl}}\equiv\tanh{\left(\phi_{\text{infl}}/\sqrt{6\alpha}\right)}$.
In the large $\gamma$ limit, the mass (D.1) for a given $r$ is independent of
$\gamma$,

$\begin{split}\lim_{\gamma\rightarrow\infty}\frac{{m_{\theta}}^{2}}{H^{2}}=18\,r\,r_{\text{infl}}\frac{\left(r^{2}+3r_{\text{infl}}^{2}\right)\cos{\theta}-3\,r\,r_{\text{infl}}\,\cos{2\theta}}{\left(r^{2}+3r_{\text{infl}}^{2}-3\,r\,r_{\text{infl}}\cos{\theta}\right)^{2}}\;,\end{split}$
(D.2)

which explains why the background evolution remains unchanged for sufficiently
large values of $\gamma$, as observed in the right panel of figure 37.

During the first phase of inflation the angular field is frozen,
$\theta\simeq\theta_{\text{in}}$, and the radial field satisfies $r\simeq 1$,
see eq. (1.4), therefore the effective squared-mass of the angular field (D.1)
is completely determined by $\gamma$ and $\phi_{\text{infl}}$. This explains
why variations of the position of the inflection point or changes in $\gamma$
have the same effect on the background evolution, i.e., both change the time
at which the transition between the first and second phases of inflation
happens. In this sense, the potential parameters $\phi_{\text{infl}}$ and
$\gamma$ are degenerate and, for fixed $\alpha$ and $\theta_{\text{in}}$, what
really determines the duration of the second slow-roll phase overall is the
effective mass of the angular field.

Figure 38: Comparison between two models with same hyperbolic field-space
curvature and the same initial value of the squared-mass of the angular field.
The models parameters are listed in (D.3) and (D.4) and the same initial
conditions $\\{\theta_{\text{in}}=7/10\,\pi,\,\phi_{\text{in}}=1.05\\}$ are
considered in both cases. We show the numerical evolution of the slow-roll
parameter on the left and the squared-mass (D.1) on the right. The two multi-
field models display the same background evolution, which is due to selecting
the values of $\gamma$ and $\phi_{\text{infl}}$ such that the angular mass is
the same at the beginning of the evolution.

We demonstrate this by considering two different models formulated on the same
hyperbolic field space, but whose potential parameters $\gamma$ and
$\phi_{\text{infl}}$ are selected such that the initial mass of the angular
field (D.1) is the same, ${m_{\theta}}^{2}/H^{2}\simeq-0.762108$. The two
models we consider are described by the parameters

$\displaystyle\text{model}_{1}$
$\displaystyle\rightarrow\\{\alpha=0.005,\,\phi_{\text{infl}}=0.07,\,\gamma=10\\},$
(D.3) $\displaystyle\text{model}_{2}$
$\displaystyle\rightarrow\\{\alpha=0.005,\,\phi_{\text{infl}}=0.066682,\,\gamma=100\\}\;.$
(D.4)

The slow-roll parameter $\epsilon_{H}(N)$ and the angular field mass
${m_{\theta}}^{2}/H^{2}$ are represented respectively in the left and right
panels of figure 38. The comparison between the lines shows that the
background evolution stemming from the same initial angular mass is the same
in the two models and that $\gamma$ and $\phi_{\text{infl}}$ are not
independent parameters.

Given the degeneracy between $\gamma$ and $\phi_{\text{infl}}$, in section 4.5
we choose to fix $\gamma$ and study the effect of changing the position of the
inflection point and thus the initial mass of the angular field. The value of
$\phi_{\text{infl}}$ affects the position of the transition between the first
and second phases of evolution, and therefore it affects the CMB predictions
of the model, so needs to be adjusted in order to produce a model which is not
in tension with the CMB measurements, as we study in section 4.5.

##  Appendix E 2D hyperbolic field space: polar vs planar coordinates

Different coordinate maps can be chosen to describe the (non-trivial) field
space of multi-field models, where in this field space the coordinates are the
fields themselves. The kinetic Lagrangian of $\alpha$–attractor models
displayed in eq. (1.5) employs polar coordinates on the hyperbolic field
space, with radial and angular fields, $(\phi,\,\theta)$, and the curvature of
field space is $\mathcal{R}_{\text{fs}}=-4/(3\alpha)$.

Other coordinate maps have been used in the literature to describe hyperbolic
field spaces with constant, negative curvature, see e.g., [160, 161, 162]
where planar coordinates have been selected. In this case, the kinetic
Lagrangian reads

$\mathcal{L}_{\text{planar}}=-\frac{1}{2}(\partial
u)^{2}-\frac{1}{2}\text{e}^{2bu}(\partial v)^{2}\;,$ (E.1)

where we label the set of planar coordinates as $(u,\,v)$ and the curvature of
field space is $\mathcal{R}_{\text{fs}}=-2b^{2}$. Provided the curvature is
the same, i.e., $b=\sqrt{2/(3\alpha)}$, the field-space geometry is the same
as in eq. (1.5), while the coordinate map selected is different.

Planar coordinates were used in [16] to show how the hyperbolic geometry of
field space could play a key role in enhancing the scalar power spectrum on
small scales. In [16] the fields $(u,\,v)$ have a separable potential

$U(u,\,v)=U_{0}\frac{u^{2}}{u_{0}^{2}+u^{2}}+\frac{1}{2}m_{v}^{2}v^{2}\;,$
(E.2)

where $u$ has a plateau-type potential at large values of $u$, and the second
field, $v$, has an apparently simple mass term. The authors of [16]
demonstrate that the background evolution and the geometry of field space
following from (E.1) and (E.2) could result in a transient tachyonic
instability of the isocurvature perturbation,
${m_{s,\,\text{eff}}}^{2}/H^{2}<0$, which can lead to an enhancement of the
scalar perturbation (see the discussion in section 4.1).

Among all the possible combinations of model parameters discussed in [16], we
focus here on the inflationary potential described by
$\\{u_{0}=\sqrt{6},\,m_{v}^{2}=U_{0}/500\\}$, with $b=7.84$ and initial
conditions $\\{u_{\text{in}}=7,\,v_{\text{in}}=7.31\\}$, and refer the reader
to the original work [16] for the equations describing the background
evolution in planar coordinates. As demonstrated in [16], this set of
parameters and initial conditions produces a peak in the scalar power spectrum
$P_{\zeta}=\mathcal{O}(10^{-2})$ located at the scales where the Laser
Interferometer Space Antenna (LISA) operates. In this case the PBHs generated
could potentially account for all of the dark matter in our Universe.

Figure 39: Background evolution of the fields $u$ and $v$ represented on top
of the potential profile, eq. (E.2). We are here reproducing the background
evolution of a model originally discussed in [16]. The model parameters and
initial conditions considered are
$\\{u_{0}=\sqrt{6},\,m_{v}^{2}=U_{0}/500,\,b=7.84,\,u_{\text{in}}=7,\,v_{\text{in}}=7.31\\}$.

In figure 39 the fields evolution is superimposed on top of the potential
(E.2). The field $u$ drives a first stage of inflation, while $v$ is
effectively frozen, with $v^{\prime}$ suppressed by the geometrical factor
$\text{e}^{-2bu}$ as long as $u\gg b^{-1}$, i.e., $u$ takes values larger than
the curvature length of the field space. When $u\sim b^{-1}$ the suppression
is lifted and $v$ starts evolving, driving a second stage of inflation as $u$
settles into its effective minimum. At the transition between the two
inflationary stages, slow roll is violated and the effective squared-mass of
the isocurvature perturbation briefly becomes negative, see the top-right
panel of figure 1 in [16].

We compare here the planar and polar coordinates description of an hyperbolic
field space. Although different coordinate choices in field space are
physically equivalent, once the form of the kinetic Lagrangian of the model is
fixed, a specific choice for the potential, such as the one in eq. (E.2),
distinguishes between different physical models. Our aim is to place the
inflationary model discussed in [16] (and therefore potentially other models
formulated using planar coordinates, eq. (E.1)) in the context of multi-field
$\alpha$–attractors described using polar coordinates, and hence understand
better the mechanism that allows for the enhancement of the curvature
perturbation in that model.

In section E.1 we derive a coordinate transformation which allows us to
transform from one coordinate map to the other. We re-analyse the model
described in [16] employing polar coordinates in section E.2. We find that the
potential is singular in polar coordinates, sharing the same singularity as
the kinetic Lagrangian, and initial conditions close to the singularity are
necessary in order to enhance the scalar perturbation. This explains why the
second field $v$ can lead to observable effects in this model, contrary to
what was found in [43] for $\alpha$–attractor models, i.e., for models with
non-singular potentials.

###  E.1 Mapping between polar and planar coordinates

2D hyperbolic spaces with constant negative curvature (H2) can be identified
with spacelike hyperboloids embedded in a 3D Minkowski spacetime [163]. This
embedding procedure provides an intermediate step to map between polar and
planar coordinates in the field space. Using the 3D Minkowski spacetime line
element

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}\;,$ (E.3)

surfaces with a fixed timelike displacement from the origin, are given by

$t^{2}-x^{2}-y^{2}=R^{2}\;.$ (E.4)

These surfaces have hyperbolic (H2) geometry, while hyperboloids with a fixed
spacelike displacement from the origin have dS2 geometry [163, 164].

Using polar coordinates, the line element of the hyperbolic field space is

$\mathrm{d}s_{\text{polar}}^{2}=\mathrm{d}\phi^{2}+R^{2}\sinh^{2}(\phi/R)\mathrm{d}\theta^{2}\;\;\;\;\;\;(0\leq\phi<+\infty,\,0\leq\theta<2\pi)\;,$
(E.5)

where the curvature length of field space is $R\equiv\sqrt{3\alpha/2}$.
Choosing coordinates $(\phi,\,\theta)$ on the hyperboloid (E.4) such that

$\displaystyle t$ $\displaystyle=R\cosh(\phi/R)\;,$ (E.6) $\displaystyle x$
$\displaystyle=R\sinh(\phi/R)\cos\theta\;,$ (E.7) $\displaystyle y$
$\displaystyle=R\sinh(\phi/R)\sin\theta\;,$ (E.8)

the line element (E.3) yields the H2 line element in polar coordinates, eq.
(E.5). The polar coordinates cover the whole upper ($t>0$) hyperboloid for
$0\leq\phi<+\infty$.

The line element of the hyperbolic field space using planar coordinates is

$\mathrm{d}s_{\text{planar}}^{2}=\mathrm{d}u^{2}+e^{2bu}\mathrm{d}v^{2}\;\;\;\;\;\;(-\infty<u<+\infty,\,-\infty<v<+\infty)\;,$
(E.9)

where the curvature length of field space is $R=b^{-1}$. It is useful to first
rewrite the line element (E.9) in a conformally-flat form,

$\mathrm{d}s_{\text{planar,
conf}}^{2}=\Omega^{2}\left(\mathrm{d}w^{2}+\mathrm{d}v^{2}\right)\;.$ (E.10)

This is achieved by means of the transformation

$w=-\frac{\text{e}^{-bu}}{b}\;,$ (E.11)

which leads to $\Omega^{2}\equiv 1/(-bw)^{2}$. Note that we have chosen
integration constants such that for $-\infty<u<\infty$ we have $-\infty<w<0$.
Choosing coordinates $(w,v)$ on the hyperboloid (E.4) such that

$\displaystyle t$
$\displaystyle=-\frac{R^{2}}{2w}\left(1+\frac{v^{2}}{R^{2}}+\frac{w^{2}}{R^{2}}\right)\;,$
(E.12) $\displaystyle x$
$\displaystyle=-\frac{R^{2}}{2w}\left(1-\frac{v^{2}}{R^{2}}-\frac{w^{2}}{R^{2}}\right)\;,$
(E.13) $\displaystyle y$ $\displaystyle=-\frac{Rv}{w}\;,$ (E.14)

the line element (E.3) yields the H2 line element (E.10), with $R=b^{-1}$.

Using (E.12)–(E.14) to express the conformal planar coordinates $(w,\,v)$ in
terms of those in Minkowski spacetime gives

$\displaystyle w$ $\displaystyle=-\frac{R^{2}}{t+x}\;,$ (E.15) $\displaystyle
v$ $\displaystyle=\frac{Ry}{t+x}\;.$ (E.16)

Substituting (E.6)–(E.8) in the above expressions gives the conformal planar
coordinates in terms of the polar coordinates. Finally expressing the
conformal planar coordinates $(w,\,v)$ in terms of the planar coordinates
$(u,\,v)$ yields

$\displaystyle u$
$\displaystyle=R\ln\left[\cosh(\phi/R)+\sinh(\phi/R)\cos\theta\right]\,,$
(E.17) $\displaystyle v$
$\displaystyle=\frac{R\sinh(\phi/R)\sin\theta}{\cosh(\phi/R)+\sinh(\phi/R)\cos\theta}\,.$
(E.18)

###  E.2 A hyperbolic model with a singular potential

In order to analyse the model of [16] using polar coordinates, we use the
kinetic Lagrangian (1.5) and we express the potential (E.2) in terms of polar
coordinates $(\phi,\,\theta)$ by means of the coordinate map (E.17)–(E.18)

$U(\phi,\,\theta)=U_{0}\,\Big{\\{}\frac{\left(R\ln\left[\cosh(\phi/R)+\sinh(\phi/R)\cos\theta\right]\right)^{2}}{6+\left(R\ln\left[\cosh(\phi/R)+\sinh(\phi/R)\cos\theta\right]\right)^{2}}\\\
+\frac{1}{2\times
500}\left(\frac{R\sinh(\phi/R)\sin\theta}{\cosh(\phi/R)+\sinh(\phi/R)\cos\theta}\right)^{2}\Big{\\}}\;.$
(E.19)

In equation (E.19), $R=\sqrt{3\alpha/2}$ and the model parameters have been
substituted according to the parameters chosen in figure 39. In particular,
for the hyperbolic field space to be the same, $\mathcal{R}_{\text{fs,
polar}}=\mathcal{R}_{\text{fs, planar}}\simeq-123$, we set
$\alpha=2/(3b^{2})\simeq 0.01$.

Figure 40: Left panel: profile of the potential (E.19) as a function of the
angular field $\theta$ for different fixed values of $\phi$. The selected
range of $\phi$ values shows the potential’s divergence at $\theta=\pi$. Black
dots show the background evolution of the fields superimposed on the
potential. Earlier stages of inflation correspond to larger values of $\phi$.
Right panel: profile of the potential (E.19) for fixed values of $\theta$ as a
function of the radial field $\phi$. The range of $\theta$ corresponds to the
values taken by the field during the background evolution displayed in figure
41 and discussed in the main text.

While being fairly simple in planar coordinates, the potential looks much more
complicated when transformed to polar coordinates. The second term in eq.
(E.19) corresponds to the mass term for $v$ in the original potential (E.2)
and is singular at $\theta\to\pi$ for large values of $\phi$. We visualise the
two-field potential as a function of $\phi$ and $\theta$ in figure 40; from
the left panel one can see that the potential diverges at $\theta=\pi$ for
$\phi\gtrsim 1$.

Figure 41: Left panel: evolution of the background fields over the final 60
e-folds of inflation driven by the potential (E.19) in polar coordinates.
Right panel: evolution of $\epsilon_{H}$ over the last 60 e-folds of inflation
for the same potential. The black line shows $\epsilon_{H}$, while the
coloured lines show the contributions from the radial (green-dashed line) and
angular (pink-dotted line) fields.

We numerically solve eqs. (4.22)–(4.24) to obtain the background evolution for
$\phi$ and $\theta$. We select the initial conditions
$\\{\phi_{\text{in}}=7.1504,\,\theta_{\text{in}}=3.1067\\}$ and slow-roll
initial conditions for the velocities of the fields. We choose this set of
initial conditions as they produce the same background evolution in terms of
$u$ and $v$ shown in figure 39. The corresponding evolution of the fields in
polar coordinates, $\phi$ and $\theta$, and the slow-roll parameter,
$\epsilon_{H}$, is shown in figure 41. In particular, in the left panel we
plot the trajectory in field space, showing how $\phi$ drives a first stage of
inflation, after which there is a turn in field space and $\theta$, previously
frozen, starts evolving. In the right panel of the same figure, the evolution
of the slow-roll parameter $\epsilon_{H}$ and its components are shown against
the number of e-folds to the end of inflation, $\Delta N\equiv
N_{\text{end}}-N$. As expected, the major contribution to $\epsilon_{H}$ in
the first phase of inflation comes from the kinetic energy of $\phi$, while
the evolution of $\theta$ dominates a second stage of inflation. Between the
two phases, the slow-roll approximation is violated ($\epsilon_{H}\simeq 1$).

Figure 42: Numerical solutions for the slow-roll parameter $\epsilon_{H}$ for
the same model evolved using polar coordinates (black line) or planar
coordinates (red-dashed line). The numerical solutions, using the
corresponding potentials in polar (E.19) or planar (E.2) coordinates, are
identical.

The numerical solutions obtained with the polar coordinates description with
the potential (E.19) is identical to that employing planar coordinates with
the potential (E.2), as expected given the one-to-one correspondence between
the two models. To show this, we compare the slow-roll parameter
$\epsilon_{H}$. Using polar coordinates, $\epsilon_{H}(N)$ is

$\epsilon_{H}(N)=\frac{1}{2}\left(\phi^{\prime
2}+\frac{3\alpha}{2}\sinh^{2}{\left(\sqrt{\frac{2}{3\alpha}}\phi\right)}\theta^{\prime
2}\right)\;,$ (E.20)

see eq. (4.22). When employing planar coordinates, we have instead

$\epsilon_{H}(N)=\frac{1}{2}\left(u^{\prime 2}+\text{e}^{2bu}v^{\prime
2}\right)\;.$ (E.21)

Substituting in the corresponding numerical solutions for the fields, we show
$\epsilon_{H}$ obtained from eq. (E.20) and (E.21) in figure 42. As expected
the two lines coincide exactly.

While we have been focusing on a configuration which was chosen in [16] to
produce a peak in the scalar power spectrum (and consequently in the induced
second-order GWs) at LISA scales, a range of different initial conditions in
field space are discussed in [16] (see table 1 therein). In particular,
varying the initial condition $v_{\text{in}}$ allows them to move the peak in
the scalar power spectrum to scales where other future GW detectors could
operate, e.g., SKA, BBO and ET. Inverting (E.17) and (E.18) enables us to
convert a set of initial conditions $(u_{\text{in}},\,v_{\text{in}})$ into the
corresponding set in polar coordinates
$(\phi_{\text{in}},\,\theta_{\text{in}})$. We have checked that the initial
conditions listed in table 1 of [16] are all within $1.5\%$ of
$\theta_{\text{in}}=\pi$. Thus we see that the configurations associated with
enhanced scalar fluctuations on small scales stem from initial conditions very
close to the singularity in the potential at $\theta=\pi$. As already pointed
out in [109], when the potential and the kinetic Lagrangian share the same
singularity, large-scale observables are sensitive to the specific shape of
the potential and to the initial conditions.

It is straightforward to show that the kinetic Lagrangian (1.3) and the
potential (E.19) share the same pole in the conformal polar coordinates
$(r,\,\theta)$, where $r$ is defined in (1.4). By following a similar
procedure to what was done in deriving eqs. (E.17) and (E.18), we obtain the
coordinate transformation

$\displaystyle u$
$\displaystyle=R\,\ln\left[\frac{1+r^{2}+2r\cos\theta}{1-r^{2}}\right]\;,$
(E.22) $\displaystyle v$
$\displaystyle=\frac{2R\,r\,\sin\theta}{1+r^{2}+2r\cos\theta}\;.$ (E.23)

In order to assess the behaviour of the fields close to $\theta=\pi$, we
define $\delta\equiv\pi-\theta$ and expand eqs. (E.22) and (E.23) to obtain

$\displaystyle\lim_{\delta\rightarrow 0}u$
$\displaystyle=R\,\ln\left[\frac{1-r}{1+r}+\frac{r\delta^{2}}{1-r^{2}}\right]\;,$
(E.24) $\displaystyle\lim_{\delta\rightarrow 0}v$
$\displaystyle=\frac{2R\,r\,\delta}{(1-r)^{2}+r\delta^{2}}\;.$ (E.25)

From the expression above it is clear that as $\delta\to 0$ the term
$m^{2}v^{2}/2$ in the potential (E.2) has a pole at $r=1$, as does the kinetic
Lagrangian (1.3).

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with $T$ the temperature (in Kelvin), and the corresponding parameters being the pre-exponential factor $A_r$ = 3.57$\times$10$^{-16}$ cm$^3$ s$^{-1}$, the temperature-dependent exponent $B_r$ = 1.52, and the activation energy $C_r$ = 1740 K. Using the NTRS-NASA thermodynamic data McBride2001tdff.rept.....M
and assuming thermodynamic equilibrium, the Gibbs free energy of formation of the forward reaction, and the corresponding equilibrium constant can be calculated Gail2013pccd.book.....G. This allows the calculation of the forward reaction rate $k_f$. Fitting these results with the Arrhenius form yields $A_f$ = 1.54$\times$10$^{-14}$ cm$^3$ s$^{-1}$, $B_f$ = 1.245, and a high energy barrier of $C_f$ = 9468 K. It can be seen that at the temperatures relevant for planet atmospheres, the forward reaction rate is much lower than the reverse rate.

[OH]/[H] ratio for equilibrium and disequilibrium predictions. Panel (a): Proxy for the [OH]/[H] ratio assuming thermodynamic equilibrium and that most of the O is in H$_2$O at solar metallicity (blue) and at $Z=10\,Z_\odot$ (orange). The prediction by Ref. tsai_photochemically_2023 is shown as a green line for comparison. Panel (b): The molar fraction of [OH]/[H] under two conditions: solar metallicity (blue) and 10$\times$ solar metallicity (orange), is depicted in both chemical equilibrium (dashed line) and disequilibrium (solid line) calculations.

Similar to Ref. tsai_photochemically_2023, we then can calculate the [OH]/[H] ratio assuming thermodynamic equilibrium and that most of the O is in H$_2$O. This yields panel (a) in Suppl. Inf.  Figure <ref>, to be compared with Extended Data Figure 10 of Ref. tsai_photochemically_2023. At temperatures below $\sim$750 K, the [OH]/[H] ratio drops below $\sim$2$\times$10$^{-6}$ for $Z$ = 10 $Z_\odot$, and hence a factor 10 lower at solar metallicity. This scarcity of OH has been used as an argument for the lack of  SO$_2$ formation at equilibrium temperatures below approximately $\sim$1,000 K for a planet with WASP-39b parameters, favouring instead the prevalence of sulphur allotropes S$_x$ tsai_photochemically_2023.

The central inquiry that emerges is how SO$_2$ can be created  within the atmosphere of WASP-107b if the aforementioned argument stands. The solution becomes evident through Suppl. Inf. Figure <ref>, where it is demonstrated that the photodissociation of various specific abundant molecules sparks the generation of SO$_2$. This assertion is exemplified in panel (a) of Suppl. Inf.  Figure <ref>, where the photodissociation of either only H$_2$O (or only N$_2$ or NH$_3$) leads to the emergence of SO$_2$. But the photodissociation of only CH$_4$ (only acting at wavelengths $\la$140 nm; see Suppl. Inf. Figure <ref>)
yields negligible amounts of SO$_2$, while the photodissociation of the rare molecule HNO$_2$ yields outcomes consistent with chemical equilibrium predictions, wherein all photodissociation is thus excluded (see dotted line in panel (b) of Figure <ref>).
This phenomenon arises from the fact that the photodissociation of various specific abundant molecules releases atoms and radicals that induce a very active photochemistry even down to pressure layers of approximately 1 bar. Reactions involving the liberated atoms and radicals often display temperature-independent behaviour without energy barriers (i.e., $B$ = $C$ = 0) and possess pre-exponential factors typically on the order of 10$^{-11}$ – 10$^{-7}$ cm$^3$ s$^{-1}$. Consequently, a significant amount of H atoms and OH radicals is formed, leading to the oxidisation of sulphur into SO$_2$. Hence, although the photolysis of H$_2$O can initiate the production of SO$_2$ in WASP-107b, it is not the sole molecule whose photodissociation holds the potential to induce SO$_2$ formation.

In summary, the overarching scenario that unfolds reveals that the primary pathways initiating the formation of SO$_2$ in the low-density atmosphere of WASP-107b are twofold. Firstly, through the photodissociation of H$_2$O in the upper atmospheric layers at pressures below a few times 10$^{-6}$ bar, yielding atomic H and OH radicals. These OH radicals are key for oxidising sulphur that is liberated from H$_2$S. Secondly, in the pressure range of 10$^{-5}$ – 1 bar, the prevailing determinant of the chemical composition is the interplay of photochemical processes acting upon various abundant molecules, not limited to H$_2$O. These processes can generate a sufficiently substantial quantity of free atoms and radicals, that can be redistributed through eddy diffusion. This initiates  a cascade of barrierless thermochemical reactions that progressively culminate in the formation of SO$_2$. Given the fact that a large ensemble of those reactions are temperature-independent, the equilibrium temperature stands as just one among several factors dictating the formation (or not) of SO$_2$. As long as the UV irradiation and FUV/NUV ratio remain moderate and the gravity is low, these processes will lead to the formation of SO$_2$ in sufficient amounts to be detectable even within a $\sim$740 K temperature planet.