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The profiles in Eq.\eqref{basis_pol}, Eq.\eqref{eq:lamda} and Eq.\eqref{eq:dos_torres} for $\mathcal{C}(r)$, $\delta \rho(r)/ \rho_{b}$ and $\zeta(\tilde{r})$ are plotted as function of $r$ for $\delta=\delta_{c}$. The dashed black line corresponds $q \rightarrow \infty$, and the dotted black line to $q=0$. The parameters used for $\mathcal{C}_{tt}(r)$ are $\delta_{1}=\delta_{c}(q_{1})$, $q_{2}=3$, $r_{m1}=r_{m2}=1$, $r_{j}=2r_{m1}$, $\mathcal{C}_{tt(\rm peak,2)}=0.3$, with the corresponding $\delta_{2}$ obtained from Eq.\eqref{eq:pico2} using the previous values, and $q_{1}=1$ (orange) and $q_{1}=5$ (violet).

The profiles in Eq.\eqref{basis_pol}, Eq.\eqref{eq:lamda} and Eq.\eqref{eq:dos_torres} for $\mathcal{C}(r)$, $\delta \rho(r)/ \rho_{b}$ and $\zeta(\tilde{r})$ are plotted as function of $r$ for $\delta=\delta_{c}$. The dashed black line corresponds $q \rightarrow \infty$, and the dotted black line to $q=0$. The parameters used for $\mathcal{C}_{tt}(r)$ are $\delta_{1}=\delta_{c}(q_{1})$, $q_{2}=3$, $r_{m1}=r_{m2}=1$, $r_{j}=2r_{m1}$, $\mathcal{C}_{tt(\rm peak,2)}=0.3$, with the corresponding $\delta_{2}$ obtained from Eq.\eqref{eq:pico2} using the previous values, and $q_{1}=1$ (orange) and $q_{1}=5$ (violet).

The profiles in Eq.\eqref{basis_pol}, Eq.\eqref{eq:lamda} and Eq.\eqref{eq:dos_torres} for $\mathcal{C}(r)$, $\delta \rho(r)/ \rho_{b}$ and $\zeta(\tilde{r})$ are plotted as function of $r$ for $\delta=\delta_{c}$. The dashed black line corresponds $q \rightarrow \infty$, and the dotted black line to $q=0$. The parameters used for $\mathcal{C}_{tt}(r)$ are $\delta_{1}=\delta_{c}(q_{1})$, $q_{2}=3$, $r_{m1}=r_{m2}=1$, $r_{j}=2r_{m1}$, $\mathcal{C}_{tt(\rm peak,2)}=0.3$, with the corresponding $\delta_{2}$ obtained from Eq.\eqref{eq:pico2} using the previous values, and $q_{1}=1$ (orange) and $q_{1}=5$ (violet).

The ratio $t_{AH}/t_{H}$ is plotted as a function of $\bar{\delta}_{m}$ for different values of $q$. As expected, the minimum value of $\bar{\delta}_{m}$, i.e. $\delta_c$, decreases as $q$ decreases. Circle corresponds to Eq.\eqref{basis_pol}, star to Eq.\eqref{eq:lamda} with $\lambda=0$ and square to Eq.\eqref{eq:lamda} with $\lambda=1$.

The ratio $R_{\rm BH,i}/R_{\rm H,i}$ is plotted for different values of $q$. As expected, the minimum value of $\bar{\delta}_{m}$, i.e. $\delta_c$, decreases as $q$ decreases. Circle corresponds to Eq.\eqref{basis_pol}, star to Eq.\eqref{eq:lamda} with $\lambda=0$ and square to Eq.\eqref{eq:lamda} with $\lambda=1$.

The ratio $R_{\rm BH,i}/R_{\rm H,i}$ is plotted as function of $q$ for $\bar{\delta}_{m} = \delta_{\rm max}-10^{-5}$, using the profiles in Eq.\eqref{basis_pol} (black), in Eq.\eqref{eq:lamda} with $\lambda=0$ (green) and in Eq.\eqref{eq:lamda} with $\lambda=1$ (blue). The red line corresponds to the analytical estimation of the upper bound obtained in \cite{size1}.

The ratio $M_{\rm BH,i}/M_{\rm H}$ is plotted for different values of $q$. As expected, the minimum value of $\bar{\delta}_{m}$, i.e. $\delta_c$, decreases as $q$ decreases. Circles corresponds to Eq.\eqref{basis_pol}, stars to Eq.\eqref{eq:lamda} with $\lambda=0$ and squares to Eq.\eqref{eq:lamda} with $\lambda=1$.

Top: The ratio $R_{\rm BH,i}/R_{\rm H , i}$ is plotted as a function of $\bar{\delta}_{m}$ near the maximum value $\delta_{max} = f(w)$. Bottom: The time evolution of the Hamiltonian constraint is plotted for the profiles in Eq.\eqref{basis_pol} with $q=1$, and for different values of $\bar{\delta}_{m}$.

Top: The ratio $R_{\rm BH,i}/R_{\rm H , i}$ is plotted as a function of $\bar{\delta}_{m}$ near the maximum value $\delta_{max} = f(w)$. Bottom: The time evolution of the Hamiltonian constraint is plotted for the profiles in Eq.\eqref{basis_pol} with $q=1$, and for different values of $\bar{\delta}_{m}$.

The absolute value of relative percentual difference between different profile families is plotted for the quantities $R_{\rm BH,i}/R_{H,i}$ (top), $M_{\rm BH,i}$ (middle) and $t_{AH}$ (bottom). The circle points corresponds to the comparison between Eq.\eqref{basis_pol} and Eq.\eqref{eq:lamda} with $\lambda=0$, and square points with q.\eqref{basis_pol} and Eq.\eqref{eq:lamda} with $\lambda=1$.

Top: Time evolution of the PBH mass for the profiles $q=1.5$, $q=3$ and $q=10$. Solid line corresponds to Eq.\eqref{basis_pol} and dashed line to Eq.\eqref{eq:lamda} with $\lambda=0$. Bottom: Time evolution of the corresponding Hamiltonian constraints. In all cases $\bar{\delta}_{m}-\delta_{c}=0.005$.

The constant $\cal K$ defined in Eq.\eqref{eq:scaling} is plotted as a function $q$ for the profiles in Eq.\eqref{basis_pol} (red), Eq.\eqref{eq:lamda} with $\lambda=0$ (black), Eq.\eqref{eq:lamda} with $\lambda=1$ (blue) and Eq.\eqref{eq:dos_torres} (green). The parameters used for the profile $\mathcal{C}_{tt}(r)$ are $q_{2}=3$, $r_{j}=2r_{m1}$, $\mathcal{C}_{tt(\rm peak,2)}=0.3$, and $\delta_{2}$ is obtained from Eq.\eqref{eq:pico2} using the previous values, and $q_{1}=q$.

The ratio $M_{\rm BH,f}/M_{\rm BH,i}$ is plotted as a function of $\bar{\delta}_{m}-\delta_{c}(q)$ for different profiles. Circles correspond to Eq.\eqref{basis_pol}, stars to Eq.\eqref{eq:lamda} with $\lambda=0$ and triangles to Eq.\eqref{eq:dos_torres}. The subplot shows the ratio $M_{\rm BH,f}/M_{\rm BH,i}$ for PBHs with $M_{\rm BH,f} \simeq M_{H}$.