Cosmological constraints of Modified Gravity (MG) models are seldom carried out rigorously. First, even though general MG models evolve differently (i.e., background and perturbations) to the standard cosmological model, it is usual to assume a ΛCDM background. This treatment is not correct and in the era of precision cosmology could induce undesired biases in cosmological parameters. Second, neutrino mass is usually held fixed in the analyses which could obscure its relation to MG parameters. In a couple of previous papers we showed that by using the Effective Fluid Approach we can accurately compute observables in fairly general MG models. An appealing advantage of our approach is that it allows a pretty easy implementation of this kinds of models in Boltzmann solvers (i.e., less error-prone) while having a useful analytical description of the effective fluid to understand the underlying physics. This paper illustrates how an effective fluid approach can be used to carry out proper analyses of cosmological constraints in MG models. We investigated three MG models including the sum of neutrino masses as a varying parameter in our Markov Chain Monte Carlo analyses. Two models (i.e., Designer f(R) [DES-fR] and Designer Horndeski [HDES]) have a background matching ΛCDM, while in a third model (i.e., Hu & Sawicki f(R) model [HS]) the background differs from the standard model. In this way we estimate how relevant the background is when constraining MG parameters along with neutrinos' masses. We implement the models in the popular Boltzmann solver CLASS and use recent, available data (i.e., Planck 2018, CMB lensing, BAO, SNIa Pantheon compilation, H ₀ from SHOES, and RSD Gold-18 compilation) to compute tight cosmological constraints in the MG parameters that account for deviation from the ΛCDM model. For both the DES-fR and the HS model we obtain log₁₀ b < -8 at 68% confidence when all data are included. In the case of the HDES model we find a somewhat weaker value of log₁₀ J_c > -5 at 68% confidence. We also find that constraints on MG parameters are a bit weakened when compared to the case where neutrinos' masses are held fixed in the analysis.

修正重力（MG）模型的宇宙学约束很少被严格执行。首先，尽管一般 MG 模型与标准宇宙学模型的演化（即背景和扰动）不同，但通常假设 ΛCDM 背景。这种处理是不正确的，在精密宇宙学时代，可能会导致宇宙学参数出现不希望的偏差。其次，中微子质量通常在分析中保持固定，这可能会掩盖其与 MG 参数的关系。在之前的几篇论文中，我们展示了通过使用有效流体方法，我们可以准确地计算相当一般的 MG 模型中的可观察量。我们的方法的一个吸引人的优势是它允许在玻尔兹曼求解器中非常容易地实现这种模型（即，不易出错），同时对有效流体进行有用的分析描述，以了解基础物理。本文说明了如何使用有效的流体方法对 MG 模型中的宇宙学约束进行适当的分析。我们研究了三个 MG 模型，包括中微子质量的总和作为我们的马尔可夫链蒙特卡罗分析中的一个变化参数。两种型号（即设计师f ( R ) [DES-fR] 和设计师 Horndeski [HDES]) 具有与 ΛCDM 匹配的背景，而在第三种模型（即 Hu & Sawicki f ( R ) 模型 [HS]）中，背景与标准模型不同。通过这种方式，我们估计了在约束 MG 参数和中微子质量时背景的相关性。我们在流行的玻尔兹曼求解器 CLASS 中实现模型，并使用最近的可用数据（即 Planck 2018、CMB 透镜、BAO、SNIa Pantheon 编译、 SHOES 中的H ₀和 RSD Gold-18 编译）来计算在MG 参数说明与 ΛCDM 模型的偏差。对于 DES-fR 和 HS 模型，我们得到 log ₁₀ b< -8，当包含所有数据时，置信度为 68%。_{在 HDES 模型的情况下，我们发现 log 10} J _c > -5 的值稍弱，置信度为 68%。我们还发现，与在分析中保持中微子质量固定的情况相比，对 MG 参数的限制有所减弱。