Computing the matter power spectrum, $P(k)$, as a function of cosmological parameters can be prohibitively slow in cosmological analyses, hence emulating this calculation is desirable. Previous analytic approximations are insufficiently accurate for modern applications, so black-box, uninterpretable emulators are often used. We utilise an efficient genetic programming based symbolic regression framework to explore the space of potential mathematical expressions which can approximate the power spectrum and $\sigma_8$. We learn the ratio between an existing low-accuracy fitting function for $P(k)$ and that obtained by solving the Boltzmann equations and thus still incorporate the physics which motivated this earlier approximation. We obtain an analytic approximation to the linear power spectrum with a root mean squared fractional error of 0.2% between $k = 9\times10^{-3} - 9 \, h{\rm \, Mpc^{-1}}$ and across a wide range of cosmological parameters, and we provide physical interpretations for various terms in the expression. We also provide a simple analytic approximation for $\sigma_8$ with a similar accuracy, with a root mean squared fractional error of just 0.4% when evaluated across the same range of cosmologies. This function is easily invertible to obtain $A_{\rm s}$ as a function of $\sigma_8$ and the other cosmological parameters, if preferred. It is possible to obtain symbolic approximations to a seemingly complex function at a precision required for current and future cosmological analyses without resorting to deep-learning techniques, thus avoiding their black-box nature and large number of parameters. Our emulator will be usable long after the codes on which numerical approximations are built become outdated.

翻译：在宇宙学分析中，计算作为宇宙学参数函数的物质功率谱$P(k)$可能因耗时过长而难以实现，因此对此计算进行仿真具有重要价值。先前的解析近似方法已无法满足现代应用所需的精度，故常采用黑箱式不可解释仿真器。我们基于高效遗传编程的符号回归框架，探索可近似功率谱与$\sigma_8$的潜在数学表达式空间。通过学习现有低精度$P(k)$拟合函数与玻尔兹曼方程求解结果之间的比值，我们保留了早期近似所依托的物理机制。最终获得线性功率谱的解析近似表达式，在波数$k = 9\times10^{-3} - 9 \, h{\rm \, Mpc^{-1}}$区间及广泛宇宙学参数范围内，均方根分数误差为0.2%，并对表达式各分量给出了物理解释。同时我们提供具有相似精度的$\sigma_8$简易解析近似，在相同宇宙学参数范围内评估时均方根分数误差仅0.4%。该函数易于求逆，可据此获得以$\sigma_8$及其他宇宙学参数表示的$A_{\rm s}$（若需）。研究表明，无需依赖深度学习技术即可获得满足当前及未来宇宙学分析精度要求的复杂函数符号近似，从而避免其黑箱特性与大量参数。我们的仿真器将在数值近似所依赖的代码过时后仍长期可用。