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\small The fractions $\Delta^{\rm no-CT}_{\rm 1-loop}$ (blue) and $\Delta^{\rm no-CT}_{\rm 2-loop}$ (red), computed using no$-$CT (often called SPT) kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. The UV contribution to the loop corrections is significant, especially at two-loops. The sharp peaks at $k \sim 0.082 \, \um$ and $k\sim 0.48 \, \um$ are due to the loop correction vanishing.

\small The fractions $\Delta^{\rm no-CT}_{\rm 1-loop}$ (blue) and $\Delta^{\rm no-CT}_{\rm 2-loop}$ (red), computed using no$-$CT (often called SPT) kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. The UV contribution to the loop corrections is significant, especially at two-loops. The sharp peaks at $k \sim 0.082 \, \um$ and $k\sim 0.48 \, \um$ are due to the loop correction vanishing. :

\small The fractions $\Delta^{\rm no-CT}_{\rm 1-loop}$ (blue) and $\Delta^{\rm no-CT}_{\rm 2-loop}$ (red), computed using no$-$CT (often called SPT) kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. The UV contribution to the loop corrections is significant, especially at two-loops. The sharp peaks at $k \sim 0.082 \, \um$ and $k\sim 0.48 \, \um$ are due to the loop correction vanishing. : \small $\Delta^{{\rm no}-{\rm CT}}_{\rm 1-loop}(\nu)$ (\ref{fig:Delta_1L_noCT_nu}) and $\Delta^{{\rm no}-{\rm CT}}_{\rm 2-loop}(\nu)$ (\ref{fig:Delta_2L_noCT_nu}) at $k=0.25 \, \um$. We show that the infrared part of the loop corrections, without the counterterms, are sensitive to the tilt $\nu$ of the linear power spectrum in the UV.\small The fractions $\Delta^{\rm no-CT}_{\rm 1-loop}$ (blue) and $\Delta^{\rm no-CT}_{\rm 2-loop}$ (red), computed using no$-$CT (often called SPT) kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. The UV contribution to the loop corrections is significant, especially at two-loops. The sharp peaks at $k \sim 0.082 \, \um$ and $k\sim 0.48 \, \um$ are due to the loop correction vanishing. : Caption not extracted

: \small Singular ultraviolet regions for the one-loop diagrams of standard perturbation theory in $\Pcal_{\rm 1-loop}$, assuming $\Plin(q) \sim 1/q$ in the ultraviolet.

: \small Singular ultraviolet regions for the two-loop diagrams of standard perturbation theory in $\Pcal_{\rm 2-loop}$, assuming $\Plin(q) \sim 1/q^\nu$ in the ultraviolet. In brackets, we include the degree of divergence of the integrals in the singular regions.

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\small The fractions $\Delta^{\rm UV-reg.}_{\rm 1-loop}$ (blue) and $\Delta^{\rm UV-reg.}_{\rm 2-loop}$ (red), computed using UV-reg. kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. Compared to the no-CT case, the UV contribution is significantly lower. The sharp peak for the two-loop at $k \sim 0.3 \, \um$ is due to the loop correction vanishing.

\small The fractions $\Delta^{\rm UV-reg.}_{\rm 1-loop}$ (blue) and $\Delta^{\rm UV-reg.}_{\rm 2-loop}$ (red), computed using UV-reg. kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. Compared to the no-CT case, the UV contribution is significantly lower. The sharp peak for the two-loop at $k \sim 0.3 \, \um$ is due to the loop correction vanishing. :

\small The fractions $\Delta^{\rm UV-reg.}_{\rm 1-loop}$ (blue) and $\Delta^{\rm UV-reg.}_{\rm 2-loop}$ (red), computed using UV-reg. kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. Compared to the no-CT case, the UV contribution is significantly lower. The sharp peak for the two-loop at $k \sim 0.3 \, \um$ is due to the loop correction vanishing. : \small {$\Delta^{\rm UV-reg.}_{\rm 1-loop}(\nu)$~(\ref{fig:Delta_1L_UVreg_nu})~and} $\Delta^{\rm UV-reg.}_{\rm 2-loop}(\nu)$ (\ref{fig:Delta_2L_UVreg_nu}) at ${k=0.25~\um}$. We show that UV contribution to the loop corrections is now much less sensitive, compared to no-CT, to the tilt $\nu$ of the linear power spectrum. Furthermore, we are able to compute the loop integrands at $\nu=1$, where the no-CT expressions were singular.\small The fractions $\Delta^{\rm UV-reg.}_{\rm 1-loop}$ (blue) and $\Delta^{\rm UV-reg.}_{\rm 2-loop}$ (red), computed using UV-reg. kernels and the Planck linear power spectrum up to $k=10^4$~$h$/Mpc. Compared to the no-CT case, the UV contribution is significantly lower. The sharp peak for the two-loop at $k \sim 0.3 \, \um$ is due to the loop correction vanishing. : Caption not extracted

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: \small The power spectrum (i) $\Plo^{\rm Planck}$ computed at leading order (linear contribution) (blue), (ii) $\Pnlo^{\rm Planck}$ computed at NLO which includes the one-loop correction (red), and (iii) $\Pnnlo^{\rm Planck}$ computed at NNLO which includes both the one-loop and two-loop corrections (green). In Fig.~(a) we show the scaling of the full power spectrum across the range $[10^{-3}, 1]$ $h/$Mpc, while in Fig.~(b) we focus on the $k$ range $[0.1, 0.6]$ $h/$Mpc, where the effect of the loop correction becomes more significant. Both figures are at redshift $z = 0.57$. : Caption not extracted

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: \small The K-factors for the power spectrum. The NLO K-factor $\Knlo^{\rm Planck} = \frac{\Pnlo^{\rm Planck}}{\Plo^{\rm Planck}}$ (red), and NNLO K-factor $\Knnlo^{\rm Planck} = \frac{\Pnnlo^{\rm Planck}}{\Plo^{\rm Planck}}$ (green) are shown. Both figures are at redshift $z = 0.57$. : Caption not extracted

: The Planck linear power spectrum $\Plin^{[0]}$ rescaled by ${\cal N}^{[j]}$ and the linear power spectrum $\Plin^{[j]}$ for the $j=1$ model which is expected to have quite substantial differences. The power spectrum $\Plin^{[1]}$ is evaluated at redshift $z = 0$. : The part of the linear power spectrum $\DeltaP_{\rm lin.}^{[1]}(k)$, defined in Eq.~\eqref{eq:DeltaP_definition}, which is not accounted for by rescaling the Planck linear power spectrum. The value of $\DeltaP_{\rm lin.}^{[1]}(k)$ contribution is plotted in the upper part. In the lower part we plot the relative contribution $\vert \DeltaP_{\rm lin.}^{[1]}(k)/ \Plin^{[1]}(k) \vert$. We evaluate $\DeltaP_{\rm lin.}^{[1]}$ at the same redshift as the $j=1$ cosmology, here $z=0$. The ratio in the bottom panel is redshift independent.

: \small The relative contribution of $\DeltaP_{\rm lin.}^{[1]}$ to $\Plin^{[1]}$ is small in the range of interest for our loop computation. : Caption not extracted

\small The ratios $\frac{\l \mathcal{N}^{[1]} \r \Delta_1 \Pcal_{\rm 1-loop}^{[1]}}{ \Pcal_{\rm 1-loop}^{[1]}}$ and {$\frac{\Delta_2 \Pcal_{\rm 1-loop}^{[1]}}{ \Pcal_{\rm 1-loop}^{[1]}}$. We see that the $\Delta_2 \mathcal{P}^{[1]}_{1-\rm loop}$ contributes only a small correction relative to the full one-loop result across all $k$'s. The large spike near $k\sim0.1$ $h/$Mpc is due to the $\mathcal{P}^{[1]}_{1-\rm loop}$ correction vanishing.}

\small The ratios { $\frac{\l \mathcal{N}^{[j]}\r^2 \Delta_1 \Pcal_{\rm 2-loop}^{[1]}}{ \Pcal_{\rm 2-loop}^{[1]}}$}, {$\frac{ \l \mathcal{N}^{[j]}\r \Delta_2 \Pcal_{\rm 2-loop}^{[1]}}{ \Pcal_{\rm 2-loop}^{[1]}}$} and { $\frac{\Delta_3 \Pcal_{\rm 2-loop}^{[1]} }{\Pcal_{\rm 2-loop}^{[1]}}$}. We see that the $\Delta_2 \mathcal{P}^{[1]}_{2-\rm loop}$ and $\Delta_3 \mathcal{P}^{[1]}_{2-\rm loop}$ terms contribute only a small correction relative to the full two-loop integral across all $k$'s. In particular, we confirm that $\Delta_3 \mathcal{P}^{[1]}_{2-\rm loop}$ is negligible over most of the displayed range. The sharp spike near $k\sim0.4$ $h/$Mpc arises from the vanishing of $\mathcal{P}^{[1]}_{2-\rm loop}$.

\small Comparison between the exact and fitted differences in the linear power spectra. On the top panel, difference between the linear power-spectra of the two cosmologies, $\DeltaP_{\rm lin.}^{[1]}$ (blue), and fitted approximations $\DeltaP_{\rm lin., fit, 1}^{[1]}$ (red) and $\DeltaP_{\rm lin., fit, 2}^{[1]}$ (green). Below, relative residual between the interpolated $\DeltaP_{\rm lin.}^{[1]}$, at redshift $z=0$, and the fitted function $\DeltaP_{\rm lin., fit, 1}^{[1]}$ (blue) and $\DeltaP_{\rm lin., fit, 2}^{[1]}$ (red). While the residual is always small, we see that $\Delta \mathcal P_{\rm lin., fit, 1}$ is more accurate than $\Delta \mathcal P_{\rm lin., fit, 2}$.

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: \small Top panels: One-loop (\ref{subfig:1-loop_results}) and two-loop (\ref{subfig:2-loop_results}) corrections at redshift $z=0$ computed using the exact linear power spectrum $\Plin^{[1]}$ (red) and the fitted $\DeltaP^{[1]}_{\rm lin. fit}$ (green). The exact and fitted predictions are almost visually indistinguishable over the full $k$-range shown. Bottom panels: Relative residuals between the exact and fitted results for each case. We stress that $\Pcal^{[1]}_{2\rm -loop, fit}$ has been computed up to second order in $\DeltaP_{\rm lin.}$. Error bars in the bottom panels are from Monte Carlo integration. : Caption not extracted

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: \small Above: Full power spectrum for cosmology $j = 1$ at redshift $z=0$, at LO (blue), exact numeric computation of NLO (orange), exact numeric computation of NNLO (red), and NNLO computed with cosmology independent fitting functions (green). Below: Relative residual between the next-to-next leading order power spectrum computed with the exact $\Plin^{[j]}$ and the fitting procedure at redshift $z = 0$ (blue) and at $z=1.23$ (red). {In all these plots we set the counterterms of the EFTofLSS to zero.} Error bars in the bottom panels are from Monte Carlo integration. : Caption not extracted

\small Comparison of the theoretical uncertainty {$|\Pnnlo^{[1]} - \mathcal{P}^{[1]}_{\rm NNLO, fit}|$} at $z = 1.23$ and $\sigma_{\rm DESI}$ (see Footnote~\ref{covfn}) for the $j = 1$ cosmology. We see that this error is negligible with respect to the DESI error bars, where we typically allow a $\sigma_{\rm DESI} /3$ theoretical error. Error bars are from Monte Carlo integration.

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: : \small { Relative residuals between the fitted and exact values of the one-loop integrals $I^{i, [1]}_{15}$ (\ref{subfig:I15_plots}), $I^{i, [1]}_{42}$ (\ref{subfig:I42_plots}) and $I^{[1]}_{33}$ (\ref{subfig:I33_plots}) for the cosmology $j=1$. The blue bands show the min-max envelope of the relative residuals over all components $i$ (7 for $I^{i, [1]}_{15}$, 4 for $I^{i, [1]}_{42}$), while the black data points show the residual for~$i = 1$. Error bars are from Monte Carlo integration. All these ratios are redshift independent.}

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