Searches for primordial gravitational waves have resulted in constraints in a large frequency range from a variety of sources. The standard Cosmic Microwave Background (CMB) technique is to parameterise the tensor power spectrum in terms of the tensor-to-scalar ratio, $r$, and spectral index, $n_{\rm t}$, and constrain these using measurements of the temperature and polarization power spectra. Another method, applicable to modes well inside the cosmological horizon at recombination, uses the shortwave approximation, under which gravitational waves behave as an effective neutrino species. In this paper we give model-independent CMB constraints on the energy density of gravitational waves, $\Omega_\textrm{gw} h^2$, for the entire range of observable frequencies. On large scales, $f \lesssim 10^{-16}\, \text{Hz}$, we reconstruct the initial tensor power spectrum in logarithmic frequency bins, finding maximal sensitivity for scales close to the horizon size at recombination. On small scales, $f \gtrsim10^{-15}\,\mbox{Hz}$, we use the shortwave approximation, finding $\Omega_\textrm{gw} h^2 < 1.7 \times10^{-6}$ for adiabatic initial conditions and $\Omega_\textrm{gw} h^2 < 2.9 \times10^{-7}$ for homogeneous initial conditions (both $2\sigma$ upper limits). For scales close to the horizon size at recombination, we use second-order perturbation theory to calculate the back-reaction from gravitational waves, finding $\Omega_\textrm{gw} h^2 < 8.4 \times10^{-7}$, in the absence of neutrino anisotropic stress and $\Omega_\textrm{gw} h^2 < 8.6 \times10^{-7}$ when including neutrino anisotropic stress. These constraints are valid for $ 10^{-15}\, \text{Hz} \gtrsim f \gtrsim 3 \times 10^{-16}\, \text{Hz}$.

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来自宇宙微波背景的原始引力波的约束

对原始引力波的搜索导致了来自各种来源的大频率范围的限制。标准的宇宙微波背景 (CMB) 技术是根据张量与标量的比率 $r$ 和光谱指数 $n_{\rm t}$ 来参数化张量功率谱，并使用以下测量值来约束这些温度和极化功率谱。另一种适用于重组时宇宙视界内模式的方法，使用短波近似，在这种情况下，引力波表现为有效的中微子物种。在本文中，我们对引力波的能量密度 $\Omega_\textrm{gw} h^2$ 给出了与模型无关的 CMB 约束，适用于整个可观测频率范围。在大尺度上，$f \lesssim 10^{-16}\, \text{Hz}$, 我们在对数频率仓中重建初始张量功率谱，找到重组时接近视界尺寸的尺度的最大灵敏度。在小尺度上，$f \gtrsim10^{-15}\,\mbox{Hz}$，我们使用短波近似，找到 $\Omega_\textrm{gw} h^2 < 1.7 \times10^{-6}$对于绝热初始条件和 $\Omega_\textrm{gw} h^2 < 2.9 \times10^{-7}$ 对于均质初始条件（均为 $2\sigma$ 上限）。对于复合时接近视界大小的尺度，我们使用二阶微扰理论来计算引力波的反作用，发现 $\Omega_\textrm{gw} h^2 < 8.4 \times10^{-7}$，在没有中微子各向异性应力和 $\Omega_\textrm{gw} h^2 < 8.6 \times10^{-7}$ 的情况下，包括中微子各向异性应力。这些约束对 $10^{-15}\ 有效，