We present a method to introduce relativistic corrections including linear dark energy perturbations in Horndeski theory into Newtonian simulations based on the N-body gauge approach. We assume that standard matter species (cold dark matter, baryons, photons and neutrinos) are only gravitationally-coupled with the scalar field and we then use the fact that one can include modified gravity effects as an effective dark energy fluid in the total energy-momentum tensor. In order to compute the scalar field perturbations, as well as the cosmological background and metric perturbations, we use the Einstein-Boltzmann code \hiclass. As an example, we study the impact of relativistic corrections on the matter power spectrum in k-essence, a subclass of Horndeski theory, including the effects of massless and massive neutrinos. For massive neutrinos with $\sum m_{\nu} = 0.1$ eV, the corrections due to relativistic species (photons, neutrinos and dark energy) can introduce a maximum deviation of approximately $7\%$ to the power spectrum at $k \sim 10^{-3} \ \textrm{Mpc}^{-1}$ at $z=0$, for a scalar field with sound speed $c_{s}^{2}\sim 0.013$ during matter domination epoch. Our formalism makes it possible to test beyond $\Lambda$CDM models probed by upcoming large-scale structure surveys on very large scales.

中文翻译：

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修正重力结构增长的相对论修正

我们提出了一种方法，将相对论校正（包括 Horndeski 理论中的线性暗能量扰动）引入到基于 N 体规范方法的牛顿模拟中。我们假设标准物质种类（冷暗物质、重子、光子和中微子）仅与标量场引力耦合，然后我们使用这样一个事实，即可以将修正的引力效应作为有效的暗能量流体包含在总能量中——动量张量。为了计算标量场扰动以及宇宙学背景和度量扰动，我们使用爱因斯坦-玻尔兹曼代码 \hiclass。例如，我们研究了相对论修正对 k-essence 中物质功率谱的影响，k-essence 是 Horndeski 理论的一个子类，包括无质量和有质量的中微子的影响。对于 $\sum m_{\nu} = 0.1 $ eV 的大质量中微子，由于相对论物种（光子、中微子和暗能量）引起的修正可以在 $k \ 的功率谱中引入大约 $7\%$ 的最大偏差sim 10^{-3} \ \textrm{Mpc}^{-1}$ at $z=0$，对于在物质统治时期声速为 $c_{s}^{2}\sim 0.013$ 的标量场. 我们的形式主义使测试超越 $\Lambda$CDM 模型成为可能，这些模型是由即将在超大规模上进行的大规模结构调查所探索的。