Emulation Strategy

This page describes the strategy used by Aletheia to emulate the non-linear power spectrum and the halo mass function, which is based on the evolution mapping framework.

Non-Linear Matter Power Spectrum

AletheiaEmu predicts the non-linear matter power spectrum, \(P(k|\mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}},z)\), for a target cosmology defined by its shape parameters, \(\mathbf{\Theta}_{\mathrm{s}} = (\omega_{\mathrm{b}}, \omega_{\mathrm{c}}, n_{\mathrm{s}})\), and evolution parameters, \(\mathbf{\Theta}_{\mathrm{e}}\), using a two-stage process.

The first stage focuses on modelling the overall change in the shape of the non-linear power spectrum for fixed evolution parameters, \(\mathbf{\Theta}_{\mathrm{e0}}\). We define the boost factor, \(B(k)\), as the ratio of the non-linear power spectrum to its de-wiggled linear counterpart, \(P_{\mathrm{dw}}(k)\), as:

\[B(k | \mathbf{\Theta}_{\mathrm{s}}, \sigma_{12}) = \frac{P(k | \mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e0}}, \sigma_{12})}{P_{\mathrm{dw}}(k | \mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e0}}, \sigma_{12})}.\]

Here, \(P_{\mathrm{dw}}(k)\) is a smoothed version of P_{mathrm{L}}(k) but has the BAO signal damped. We emulate the boost factor, \(\mathcal{E}_B(k)\), only as a function of the shape parameters \(\mathbf{\Theta}_{\mathrm{s}}\) and the clustering amplitude \(\sigma_{12}\).

A second emulator accounts for the deviations that arise from any difference between the evolution parameters of the target cosmology, \(\mathbf{\Theta}_{\mathrm{e}}\) and the reference set. To account for these deviations, we usea parameter that captures the relevant differences in the growth history. In SPT, the primary cosmology dependence of the solutions is encoded in

\[x(z) = \frac{\Omega_{\mathrm{m}}(z)}{f^2(z)},\]

where \(\Omega_{\mathrm{m}}(z)\) is the fractional matter density parameter and \(f(z) = \mathrm{d}\ln D(z) / \mathrm{d}\ln a\) is the linear growth rate parameter. More than the instantaneous value of \(x(z)\), we find that the small deviations caused bby different evolution parameters are best described by the integrated groth history parameter, \(\tilde{x}\), which is an average of \(x\) over the past history, using \(\tau = \ln(\sigma_{12})\) as the time variable, that is

\[\tilde{x}(\tau) = \int_{-\infty}^\tau x(\tau\prime) K(\tau-\tau\prime|\eta) \,d\tau\prime,\]

where \(K(\tau|\eta)\) is a Gaussian kernel with a width \(\eta\) that represents a characteristic memory of the non-linear evolution. We find that a fixed value of \(\eta=0.12\) provides an excellent description of the deviations for all relevant cosmologies and scales we consider.

We define the ratio

\[R(k | \mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}}, \sigma_{12}) = \frac{P(k | \mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}}, \sigma_{12})}{P(k | \mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e0}}, \sigma_{12})},\]

where both power spectra are evaluated at the same \(\mathbf{\Theta}_{\mathrm{s}}\) and \(\sigma_{12}\), but with different evolution parameters, leading to different integrated growth histories, \(\tilde{x}\) and \(\tilde{x}_0\). We estimate this ratio as

\[R(k | \mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}}, \sigma_{12}) \approx 1 + \left. \frac{\partial R(k)}{\partial\tilde{x}} \right|_{\tilde{x}_0} \left(\tilde{x} - \tilde{x}_0\right).\]

We build a second emulator, \(\mathcal{E}_{\partial R/\partial\tilde{x}}(k)\), to predict the derivative term \(\partial R(k)/\partial\tilde{x}\) as a function of \((\mathbf{\Theta}_{\mathrm{s}},\sigma_{12})\).

The final prediction for the non-linear power spectrum for an arbitrary cosmology \((\mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}})\) at a redshift \(z\), characterised by the clustering amplitude \(\sigma_{12}\) and the integrated growth history parameter \(\tilde{x}\), is given by

\[\mathcal{E}_P(k|\mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}}, z) = P_{\mathrm{dw}}(k|\mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}}, z) \times \mathcal{E}_B(k|\mathbf{\Theta}_{\mathrm{s}}, \sigma_{12}) \times \left[1 + \mathcal{E}_{\partial R/\partial\tilde{x}}(k|\mathbf{\Theta}_{\mathrm{s}}, \sigma_{12}) \left(\tilde{x} - \tilde{x}_0\right) \right].\]

Here, \(P_{\mathbf{dw}}(k|\mathbf{\Theta}_{\mathrm{s}}, \mathbf{\Theta}_{\mathrm{e}}, z)\) is the de-wiggled linear power spectrum of the target cosmology and \(\tilde{x}_0\) is the value of the integrated growth history parameter for the reference evolution parameters \(\mathbf{\Theta}_{\mathrm{e0}}\) evaluated at the redshift \(z_0\) that corresponds to the same value of \(\sigma_{12}\).

This design ensures that the emulator is broadly applicable to various evolution histories, including dynamic dark energy models, without explicitly sampling their parameters or redshift during the primary training of the emulators \(\mathcal{E}_B\) and \(\mathcal{E}_{dR/dx}\), which only depend on \((\omega_{\mathrm{b}}, \omega_{\mathrm{c}}, n_{\mathrm{s}}, \sigma_{12}, \ln k)\). The value of \(\ln k\) is treated as an input parameter for both emulators, allowing their evaluation at any desired \(k\) value.

Halo Multiplicity Function

The emulator of the halo mass function follows an analogous two-stage strategy. We emulate the multiplicity function, \(f(\nu)\), relative to a fixed reference prediction, \(f_{\mathrm{ref}}(\nu)\), defined using the fitting function of Fiorilli et al. (2026). Using this ratio reduces the dynamic range of the function to emulate.

This baseline ratio, \(\mathcal{E}_C(\nu\vert{}\mathbf{\Theta}_{\mathrm{s}}, \sigma_{12})\), is trained on a fixed set of evolution parameters. A second emulator is then trained on the response of that ratio to changes in the evolution parameters. With the output of the two emulators, we construct the full prediction of the multiplicity function as

\[\mathcal{E}_f(\nu|\mathbf{\Theta}_\mathrm{s}, \mathbf{\Theta}_{\mathrm{e}}, z) = f_\mathrm{ref}(\nu) \, \mathcal{E}_C(\nu\vert{}\mathbf{\Theta}_\mathrm{s}, \sigma_{12}) \cdot \left[ 1 + \mathcal{E}_{\partial R / \partial \tilde{x}}(\nu\vert{}\mathbf{\Theta}_\mathrm{s}, \sigma_{12}) \, (\tilde{x} - \tilde{x}_0) \right].\]

To accurately capture the longer characteristic memory of halo formation, the integrated growth history parameters for the mass function (\(\tilde{x}' and :math:\)tilde{x}_0`) are evaluated using a wider Gaussian kernel of \(\eta = 0.2884\).