We present exact non-Gaussian joint likelihoods for auto- and cross-correlation functions on arbitrarily masked spherical Gaussian random fields. Our considerations apply to spin-0 as well as spin-2 fields but are demonstrated here for the spin-2 weak-lensing correlation function. We motivate that this likelihood cannot be Gaussian and show how it can nevertheless be calculated exactly for any mask geometry and on a curved sky, as well as jointly for different angular-separation bins and redshift-bin combinations. Splitting our calculation into a large- and small-scale part, we apply a computationally efficient approximation for the small scales that does not alter the overall non-Gaussian likelihood shape. To compare our exact likelihoods to correlation-function sampling distributions, we simulated a large number of weak-lensing maps, including shape noise, and find excellent agreement for one-dimensional as well as two-dimensional distributions. Furthermore, we compare the exact likelihood to the widely employed Gaussian likelihood and find significant levels of skewness at angular separations $\gtrsim 1^{\circ}$ such that the mode of the exact distributions is shifted away from the mean towards lower values of the correlation function. We find that the assumption of a Gaussian random field for the weak-lensing field is well valid at these angular separations. Considering the skewness of the non-Gaussian likelihood, we evaluate its impact on the posterior constraints on $S_8$. On a simplified weak-lensing-survey setup with an area of $10 \ 000 \ \mathrm{deg}^2$, we find that the posterior mean of $S_8$ is up to $2\%$ higher when using the non-Gaussian likelihood, a shift comparable to the precision of current stage-III surveys.

翻译：我们提出了任意掩蔽球面高斯随机场上自相关与互相关函数的精确非高斯联合似然。我们的理论适用于自旋-0及自旋-2场，但本文以自旋-2弱引力透镜相关函数为例进行演示。我们论证了该似然不可能为高斯分布，并展示了如何针对任意掩蔽几何结构、在弯曲天空上、以及联合不同角间距区间与红移区间组合的情形下精确计算该似然。通过将计算分解为大尺度与小尺度两部分，我们对小尺度部分采用了计算高效的近似方法，且不改变整体非高斯似然的形态。为将精确似然与相关函数抽样分布进行比较，我们模拟了大量包含形状噪声的弱引力透镜天图，并在一维及二维分布上均发现极佳的一致性。此外，我们将精确似然与广泛采用的高斯似然进行对比，发现在角间距 $\gtrsim 1^{\circ}$ 时存在显著的非对称性，导致精确分布的众数偏离均值向相关函数较低值方向移动。我们发现弱引力透镜场在这些角间距下满足高斯随机场的假设是成立的。基于非高斯似然的非对称性，我们评估了其对 $S_8$ 后验约束的影响。在一个面积为 $10 \ 000 \ \mathrm{deg}^2$ 的简化弱引力透镜巡天设置中，我们发现使用非高斯似然时 $S_8$ 的后验均值最高可提升 $2\%$，该偏移量与当前第三阶段巡天的测量精度相当。