<figure><img src="image_27.png" /><figcaption>**Fig. E.1.** For the photometric probe with optimistic settings, comparison of the 1D marginalised and un marginalised errors obtained under the assumption that the galaxy power spectrum is a biased tracer of either P<sub>mm</sub> (incorrect modelling) or P<sub>cc</sub> (correct modelling). For the parameter Σm<sub>v</sub> the relative difference in the un marginalised error goes outside the frame and is of the order of 43%.</figcaption></figure>

to Σm<sub>v</sub>. This discrepancy gets mitigated by the marginalisa-
tion over all other parameters, which reduces the difference on
σ(Σm<sub>v</sub>) to about 3%. However, in the marginalised case, the
error bars on the parameters that are correlated with Σm<sub>v</sub> also
increase – most noticeably for the bias parameters. As a mat-
ter of fact, the response of P<sub>cc</sub>(k; z) to a variation in the neu-
trino mass is more degenerate with a change of amplitude (and
thus of linear bias) than the response of P<sub>mm</sub>(k; z). Thus, when
using the wrong prescription, one underestimates the correlation
between Σm<sub>v</sub> and the bias parameters (see Fig. E.2), and artifi-
cially reduces the error not only on Σm<sub>v</sub> but also on each b<sub>i</sub>.

In the second test, we can check that fitting the correct fiducial model with the wrong prescription leads to artificially strong constrains on Σm<sub>v</sub> (see Fig. E.3). In this case, the theoretical model based on P<sub>mm</sub> cannot fit perfectly the fiducial model based on P<sub>cc</sub>: there is always a residual difference. This test is particularly interesting because it shows that the incorrect model fits this residual by a combination of a wrong Σm<sub>v</sub> and wrong biases. This could be expected, since the difference between P<sub>cc</sub> and P<sub>mm</sub> is close to a constant factor over a wide range of scales, as can be checked in Fig. 1. This explains the shifted means in the Σm<sub>v</sub> and b<sub>i</sub> posteriors visible in Fig. E.3. Looking at the Σm<sub>v</sub> posteriors, we see that if such a mistake occurred in the analysis of real data, one would incorrectly conclude that the data prefer a vanishing neutrino mass, while in reality Σm<sub>v</sub> = 60 meV is preferred.

Appendix F: Parameter correlations in different cosmologies

In this appendix we show additional triangle plots with the
1D marginalised posteriors and 2D confidence contours that
are not shown in Sect. 7. Figure F.1 shows the results
obtained in the baseline ΛCDM+Σm<sub>v</sub> model from each of
the Euclid primary probes and from their combination. Figu-
ures F.2 and F.3 show the results obtained in the intermediate
models (ΛCDM+Σm<sub>v</sub>+ΔN<sub>eff</sub>, and w<sub>0</sub>w<sub>a</sub>CDM+Σm<sub>v</sub>, respec-
tively) with the data combinations Euclid+Planck (orange), and
Euclid+CMB-S4+LiteBIRD (green). Note that, when compar-
ing Fig. F.1 with Fig. 13, the photometric probes (red contours)
show a different direction of degeneracy between Σm<sub>v</sub> and h,
which are positively correlated in the baseline ΛCDM+Σm<sub>v</sub>
model. On the other hand, when combining Euclid with CMB

<figure><img src="image_28.png" /><figcaption>**Fig. E.2.** Marginalised posteriors and confidence contours for the same Fisher forecasts as in Fig. E.1.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. E.3.** For the photometric probe with optimistic settings, marginalised posteriors and confidence contours from two MCMC forecasts based on the same fiducial model generated with the P<sub>cc</sub> prescription. The mock data are then fitted either with the P<sub>mm</sub> (incorrect modelling) or P<sub>cc</sub> (correct modelling) prescription.</figcaption></figure>

(Figs. F.2 and F.3), it is difficult to identify any significant change
in the direction of the degeneracies in parameter space between
one model and the other.
<figure><img src="image_30.png" /><figcaption>Fig. F.1. Same as Fig. 13, but for the baseline $\Lambda$CDM+$\Sigma m_{\nu}$ model.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. F.2. Same as Fig. 15, but for the $\Lambda$CDM+$\Sigma m_{\nu}+\Delta N_{\text{eff}}$ model.</figcaption></figure>

## Appendix G: Correlation with bias parameters

Figures G.1 and G.2 show the correlation between the bias parameters of photometric and spectroscopic observables, respectively, and the cosmological parameters in the extended $w_0w_a$CDM+$\Sigma m_{\nu}+\Delta N_{\text{eff}}$ model. We show only two redshift bins for each observable noticing that the correlation is the same in all the redshift bins. It appears that there is no correlation between the neutrino mass and the bias parameters, either in WL+GC<sub>ph</sub> + XC<sub>ph</sub> or in GC<sub>sp</sub>. On the other hand, a mild anti-correlation is found between the bias parameters and $\Delta N_{\text{eff}}$ in WL+GC<sub>ph</sub> + XC<sub>ph</sub>, thus arising from GC<sub>ph</sub>. The reason why this correlation appears in GC<sub>ph</sub> while it is not present in GC<sub>sp</sub> is that the projection of the matter power spectrum in angular harmonics and the convolution with the window function redistribute the power over a wider range of scales; the feature induced by

<figure><img src="image_32.png" /><figcaption>Fig. F.3. Same as Fig. 15, but for the $w_0w_a$CDM+$\Sigma m_{\nu}$ model.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Fig. G.1. 2D confidence contours showing the correlation between the bias parameters of the fifth, and of the ninth redshift bins ($b_4$ and $b_8$) and the cosmological parameters, obtained by fitting photometric observables (WL+GC<sub>ph</sub> + XC<sub>ph</sub>) to the $w_0w_a$CDM+$\Sigma m_{\nu}+\Delta N_{\text{eff}}$ model.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Fig. G.2. 2D confidence contours showing the correlation between the bias parameters of the first and of the fourth redshift bin [$\ln b_0\sigma_8(z_0)$, and $\ln b_3\sigma_8(z_3)$] and the cosmological parameters, obtained by fitting spectroscopic observables (GC<sub>sp</sub>) to the $w_0w_a$CDM+$\Sigma m_{\nu}+\Delta N_{\text{eff}}$ model.</figcaption></figure>

a variation in $\Delta N_{\text{eff}}$ on the matter power spectrum (see Fig. 2) is smoothed out and thus can be mimicked by a variation in the bias.

Besides $\Delta N_{\text{eff}}$, the bias parameters show correlations with most cosmological parameters, highlighting the importance of a careful study of bias from simulations.
# Appendix H: Impact of baryonic feedback

An accurate modelling of large scale structure observables in the nonlinear regime requires accounting for baryonic feedback effects (Semboloni et al. 2011; Parimbelli et al. 2019; Schneider et al. 2020a,b; Euclid Collaboration 2021b; Spurio Mancini & Bose 2023). Here, we test the robustness of our results on the neutrino mass sensitivity against such effects. To this end, we multiply the matter power spectrum $P_{\text{mm}}(k_\ell, z)$ of Eq. (4) in Sect. 3.1 by the boost factor obtained with the emulator BCemu (Giri & Schneider 2021). Note that we apply the boost factor only to $P_{\text{mm}}(k_\ell, z)$, thus, accounting for baryonic feedback effects only in weak lensing. We vary two BCemu parameters. The first one, $\log_{10} M_c$, is related to the critical mass $M_c'$ such that halos with a virial mass that is lower than $M_c'$ have a gas profile shallower than the Navarro-Frenk-White profile. Then, $M_c$ is the dimensionless parameter such that $M_c = M_c'/(1h^{-1}M_\odot)$. The second parameter is the index $\nu_{M_c}$ describing the power-law redshift dependence of $M_c$. This choice is motivated by the fact that $\log_{10} M_c$ is the parameter governing the suppression of $P_{\text{mm}}(k_\ell, z)$, thus, it is expected to be the most correlated with the neutrino mass. Moreover, given the cut-off at $\ell = 1500$ in our pessimistic specifications, including more than one free parameter for baryonic feedback might lead to an overfitting, leaving the additional parameters unconstrained. In Fig. H.1, we compare the constraints from the Euclid photometric probe on cosmological parameters obtained with and without baryonic feedback (red-filled and green-empty contours, respectively) in the $w_0w_a$CDM+$\sum m_\nu+\Delta N_{\text{eff}}$ model. We assume pessimistic specifications as in our final results of Sect. 7. We see that baryonic feedback does not have a dramatic impact on the constraints, and the critical mass parameter does not show a well defined correlation with either of the neutrino parameters. This outcome was expected since baryonic feedback effects are mostly relevant at $\ell > 1500$, thus, beyond the range fitted in the pessimistic scenario. Nevertheless, the variation in the $\log_{10} M_c$ parameter slightly degrades the sensitivity to the neutrino mass: more specifically, the 95% upper bound on $\sum m_\nu$ increases by about 10%. As already pointed out in Spurio Mancini & Bose (2023), if the analysis was extended to $\ell = 3000, 5000$, the degradation in sensitivity to the neutrino mass due to the inclusion of baryonic feedback effects would be more dramatic.
$$w_0 w_a \text{CDM} + \sum m_\nu + \Delta N_{\text{eff}}, \text{WL} + \text{GC}_{\text{ph}} + \text{XC}_{\text{ph}}$$

<figure><img src="image_35.png" /><figcaption>**Fig. H.1.** 1D marginalised posteriors and 2D confidence contours for cosmological parameters and for baryonic feedback parameters, obtained by fitting photometric observables (WL+GC<sub>ph</sub>+XC<sub>ph</sub>) to the $w_0 w_a \text{CDM} + \sum m_\nu$ model, with and without varying baryonic feedback parameters (red-filled and green-empty contours, respectively). The contours for baryonic feedback parameters are also shown when they are varied.</figcaption></figure>