$\bar{f}_{\mathrm{NL}}^{\Delta_{1,2}}=0$. For BOSS, we can observe in Fig 10
that this correlation is close to unity, even for widely separated values of
$\Delta$. We see in Fig. 17

Figure 17: Correlation matrix for measurements of galaxy power spectra with
different values of the non-Gaussian scaling exponent $\Delta$ as defined in
(4.2) for the billion-object survey and four biasing models. The lower right
panel is equivalent to Fig. 10, which shows the correlations for BOSS. As in
that figure, we hold the $\Lambda$CDM parameters fixed, take the fiducial non-
Gaussian amplitudes to be zero, $\bar{f}_{\mathrm{NL}}^{\Delta_{i}}=0$, and
indicate the coverage of the local and equilateral templates with a
correlation coefficient of at least $0.9$ by the orange lines. We can clearly
observe the impact of the biasing model and the different experimental
specifications compared to BOSS.

that this statement still holds for the billion-object survey. In fact, the
range of $\Delta$ which are highly correlated with the local shape further
expanded. Naively, this suggests there is little information about the shape
of the signal, described by $\Delta$, encoded in these observables.

To first approximation, the fiducial value of
$\bar{f}_{\mathrm{NL}}^{\Delta}=0$ is the reason why we see little ability to
distinguish different values of $\Delta$. Since there is no signal to measure,
it is not surprising that the precise shape is not very important. If we were
instead to detect $f_{\mathrm{NL}}^{\Delta}$, we would expect that the signal
to noise would be a fairly sensitive function of $\Delta$ and we could
therefore measure the true value of $\Delta$ precisely. This is exactly what
is found in Fig. 18,

Figure 18: Forecasts for $\sigma(f_{\mathrm{NL}}^{\Delta})$ (top) and
$\sigma(\Delta)$ (bottom) as a function of $\Delta$ if the scaling exponent is
either fixed (left) or varied (right) for the billion-object survey. To
compute the constraint on $\Delta$, we take the fiducial value of the non-
Gaussian amplitude, $\bar{f}_{\mathrm{NL}}^{\Delta}$, to be the value
necessary to achieve a $5\sigma$ detection in the respective forecast. We fix
the $\Lambda$CDM parameters and marginalize over four different biasing
models.

in which we show the forecasted constraints for measuring $\Delta$,
$\sigma(\Delta)$, assuming that $f_{\mathrm{NL}}^{\Delta}$ is detected at a
significance of $5\sigma$ in the billion-object survey. The key result is that
if we detect $f_{\mathrm{NL}}^{\Delta}$ while holding $\Delta$ fixed, we
expect to have enough sensitivity to measure the scaling exponent with
$\sigma(\Delta)\sim 0.05-0.5$. In this sense, the strategy of searching for
$f_{\mathrm{NL}}^{\Delta}$ at fixed $\Delta$ is a reliable strategy to search
for new physics, but would yield additional information about the origin of
the signal even at the threshold of a detection. When we instead marginalize
over $\Delta$, we require a significantly larger non-Gaussian amplitude
$f_{\mathrm{NL}}^{\Delta}$ to be detected. In this case, the corresponding
uncertainties on $\Delta$ would however be generally smaller, with
$\sigma(\Delta)<0.1$.

At first sight, it might seem surprising that the curves of $\sigma(\Delta)$
in Fig. 18 have minima at the values of $\Delta$ where the curves of
$\sigma(f_{\mathrm{NL}}^{\Delta})$ have maxima or downward bends. By comparing
the curves for different biasing models, we can however see that these
features are the result of marginalizing over the bias parameters. For
example, we noted in the main text that the maximum in
$\sigma(f_{\mathrm{NL}}^{\Delta})$ near $\Delta=1$ when marginalizing over
$b_{1}$ arises because the linear bias $b_{1}$ and
$f_{\mathrm{NL}}^{\Delta=1}$ are degenerate at high $k$. The fact that the
measurement of $f_{\mathrm{NL}}^{\Delta}$ is less constraining at this point
then also implies that we are very sensitive to the exact value of the scaling
exponent $\Delta$. When we hold
$f_{\mathrm{NL}}^{\Delta}/\sigma(f_{\mathrm{NL}}^{\Delta})$ fixed, we should
therefore expect that $\sigma(\Delta)$ is smaller in regions of $\Delta$ where
$f_{\mathrm{NL}}^{\Delta}$ is degenerate with bias parameters.

Let us finally return to the correlation coefficients between scale-dependent
biases with different scaling exponent $\Delta$. In order to be consistent,
the impact of marginalization over the bias parameters must also appear in
$\cos(f_{\mathrm{NL}}^{\Delta_{1}},f_{\mathrm{NL}}^{\Delta_{2}})$.
Specifically, the impact of the marginalization has a large impact on
$\sigma(f_{\mathrm{NL}}^{\Delta})$ for larger values of $\Delta$. For the
inferred constraints to be consistent with our forecasts, this has to imply
that the correlation between large and small $\Delta$ decreases by a similar
factor. If this was not the case, our constraints inferred from
$f_{\mathrm{NL}}^{\mathrm{loc}}$ at large $\Delta$ would be stronger than
directly measuring $f_{\mathrm{NL}}^{\Delta}$.

This expectation is precisely what occurs, as shown in Fig. 17. The
correlation coefficients for larger $\Delta$ are affected by the number of
bias parameters that are marginalized over. With more marginalized bias
parameters, the range over which $\Delta<2$ and $\Delta=2$ are degenerate
decreases significantly. This is expected since the (equilateral) signal for
$\Delta=2$ gets most of its information from large wavenumbers, where the
biases play a significant role (cf. §3.2). In contrast, the increase of the
range of strong overlap with $\Delta=0$ is likely due to the increasing
difficulty to distinguish similar shapes of $\Delta$ with the inclusion of
more bias parameters. Finally, it is also worth noticing that the non-Gaussian
signal with $\Delta=0$ (local PNG) appears to be anti-correlated with the
scale-dependent bias signal for $\Delta\sim 2$ in the last three panels of
Fig. 17 in which we marginalize over the biasing model.

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