The first, and simplest, method assembles the data into bins. This consists of dividing the data into a particular redshift interval (bin), and for each of these intervals one calculates the average value of the subset of the data contained in that bin. The corresponding redshift and error of each bin are computed as weighted averages.

Another way to reconstruct a continuous function from a dataset is by using a Gaussian process algorithm as explained in [115]. This process can be regarded as the generalization of Gaussian distributions to function space since it provides a distribution over functions instead of a distribution of a random variable. Considering a dataset $D = \{(x_i, y_i)|i = 1,...,n\}$, where $x_i$ represents deterministic variables and $y_i$ random variables, the goal is to obtain a continuous function $f(x)$ that best describes the dataset. A function $f$ evaluated at a point $x$ is a Gaussian random variable with mean $\mu(x)$ and variance $\text{Var}(x)$. The $f(x)$ values depend on the function value evaluated at other $\bar{x}$ points (particularly if they are close points). The relation between these can be given by a covariance function $\text{cov}(f(x), f(\bar{x})) = k(x, \bar{x})$. The covariance function $k(x, \bar{x})$ is in principle arbitrary. Since we are interested in reconstructing the derivatives of the data, a Gaussian covariance function expressed as

$$k(x, \bar{x}) = \sigma_f^2 \exp \left[ - \frac{(x - \bar{x})^2}{2\ell^2} \right] \qquad (145)$$

is the chosen function since it is the most common and has the least number of additional parameters. This function depends on the hyperparameters $\sigma_f$ and $\ell$, which allow us to set the strength of the covariance function. These hyperparameters can be regarded as the typical scale and change in the $x$ and $y$ direction. The full covariance function takes the data covariance matrix $C$ into account by $M(x, \bar{x}) = k(x, \bar{x}) + C$. The log likelihood is then

$$\ln \mathcal{L} = -\frac{1}{2} \sum_{i,j=1}^{N} \left\{ [y_i - \mu(x_i)] (M^{-1})_{ij} [y_j - \mu(x_j)] \right\} + \ln |M| + N \ln 2\pi \qquad (146)$$

where $|M|$ is the determinant of $M(x_i, x_j)$. The distribution Equation (146) is usually sharply peaked and so we maximize the distribution to optimize the hyperparameters, although this is an approximation to the marginalization process and it may not be the best approach for all datasets. We employ the Python publicly available GaPP code from Seikel et al. (2012) [116].

As a third method, we use a generalized linear regression. Let us assume we have $N$ data $y_i$, one for each value of the *independent* variable $x_i$ and that

$$y_i = f_i + e_i \qquad (147)$$

where $e_i$ are errors (random variables) which are assumed to be distributed as Gaussian variables. Here, $f_i$ represents theoretical functions that depend linearly on a number of parameters $A_\alpha$:

$$f_i = \sum_{\alpha=0}^{n} A_{\alpha} g_{i\alpha} \qquad (148)$$

where $g_{i\alpha}(x_i)$ are functions of the variable $x_i$, chosen to be simple powers, and $g_{i\alpha} = x_i^\alpha$, such that $f_i$ represents polynomials of order $n$.

The order of the polynomial is in principle arbitrary, up to the number $N$ of datapoints. However, it is clear that, with too many free parameters, the resulting $\chi^2$ will be very close to zero, i.e., statistically unlikely. At the same time, too many parameters also render the numerical Fisher matrix computationally unstable (producing, e.g., a non-positive definite matrix) and the polynomial wildly oscillating. On the other hand, too few parameters restrict the allowed family of functions. Therefore, we select the order of the polynomial function by choosing the degree for which the reduced chi-squared $\chi^2_{red} = \frac{\chi^2_{min}}{N-n-1}$, is closest to unity and such that the Fisher matrix is positive definite.
The first, and simplest, method assembles the data into bins. This consists of dividing the data into a particular redshift interval (bin), and for each of these intervals one calculates the average value of the subset of the data contained in that bin. The corresponding redshift and error of each bin are computed as weighted averages.

Another way to reconstruct a continuous function from a dataset is by using a Gaussian process algorithm as explained in [115]. This process can be regarded as the generalization of Gaussian distributions to function space since it provides a distribution over functions instead of a distribution of a random variable. Considering a dataset $D = \{(x_i, y_i)|i = 1,...,n\}$, where $x_i$ represents deterministic variables and $y_i$ random variables, the goal is to obtain a continuous function $f(x)$ that best describes the dataset. A function $f$ evaluated at a point $x$ is a Gaussian random variable with mean $\mu(x)$ and variance $\text{Var}(x)$. The $f(x)$ values depend on the function value evaluated at other $\bar{x}$ points (particularly if they are close points). The relation between these can be given by a covariance function $\text{cov}(f(x), f(\bar{x})) = k(x, \bar{x})$. The covariance function $k(x, \bar{x})$ is in principle arbitrary. Since we are interested in reconstructing the derivatives of the data, a Gaussian covariance function expressed as

$$k(x, \bar{x}) = \sigma_f^2 \exp \left[ - \frac{(x - \bar{x})^2}{2\ell^2} \right] \qquad (145)$$

is the chosen function since it is the most common and has the least number of additional parameters. This function depends on the hyperparameters $\sigma_f$ and $\ell$, which allow us to set the strength of the covariance function. These hyperparameters can be regarded as the typical scale and change in the $x$ and $y$ direction. The full covariance function takes the data covariance matrix $C$ into account by $M(x, \bar{x}) = k(x, \bar{x}) + C$. The log likelihood is then

$$\ln \mathcal{L} = -\frac{1}{2} \sum_{i,j=1}^{N} \left\{ [y_i - \mu(x_i)] (M^{-1})_{ij} [y_j - \mu(x_j)] \right\} + \ln |M| + N \ln 2\pi \qquad (146)$$

where $|M|$ is the determinant of $M(x_i, x_j)$. The distribution Equation (146) is usually sharply peaked and so we maximize the distribution to optimize the hyperparameters, although this is an approximation to the marginalization process and it may not be the best approach for all datasets. We employ the Python publicly available GaPP code from Seikel et al. (2012) [116].

As a third method, we use a generalized linear regression. Let us assume we have $N$ data $y_i$, one for each value of the *independent* variable $x_i$ and that

$$y_i = f_i + e_i \qquad (147)$$

where $e_i$ are errors (random variables) which are assumed to be distributed as Gaussian variables. Here, $f_i$ represents theoretical functions that depend linearly on a number of parameters $A_\alpha$:

$$f_i = \sum_{\alpha=0}^{n} A_{\alpha} g_{i\alpha} \qquad (148)$$

where $g_{i\alpha}(x_i)$ are functions of the variable $x_i$, chosen to be simple powers, and $g_{i\alpha} = x_i^\alpha$, such that $f_i$ represents polynomials of order $n$.

The order of the polynomial is in principle arbitrary, up to the number $N$ of datapoints. However, it is clear that, with too many free parameters, the resulting $\chi^2$ will be very close to zero, i.e., statistically unlikely. At the same time, too many parameters also render the numerical Fisher matrix computationally unstable (producing, e.g., a non-positive definite matrix) and the polynomial wildly oscillating. On the other hand, too few parameters restrict the allowed family of functions. Therefore, we select the order of the polynomial function by choosing the degree for which the reduced chi-squared $\chi^2_{red} = \frac{\chi^2_{min}}{N-n-1}$, is closest to unity and such that the Fisher matrix is positive definite.
## 16. Results

Let us now discuss the results of the final observable $\eta_{\text{obs}}$ for each of these methods. The binning method contains fewer assumptions compared to the polynomial regression or Gaussian process methods. It is essentially a weighted average over the data points and the error bars at each redshift bin. Since we need to take derivatives in order to calculate $P_3$ and $E'$, and we have few data points, we opt to compute finite difference derivatives. This has the caveat that it introduces correlations among the errors of the function and its derivatives, that we cannot take into account with this simple method. Moreover, for the binning method, we do not take into account possible non-diagonal covariance matrices for the data, which we do for polynomial regression and the Gaussian process reconstruction.

Figure 2 shows the reconstructed functions obtained by the binning method, the Gaussian process, and polynomial regression, alongside the theoretical prediction of the standard $\Lambda$CDM model. In all cases, the error bars or the bands represent the $1\sigma$ uncertainty.

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Comparison of the three reconstruction methods for each of the model-independent variables. The binning method is shown in blue squares with error bars, the Gaussian process is shown as a green dotted line with green bands, and polynomial regression is shown as a solid yellow line with yellow bands. All of them depict the $1\sigma$ uncertainty. **Left panel:** Plot of the reconstructed $E(z)$ function on the top and its derivative $E'(z)$ on the bottom. **Right panel:** Plot of the reconstructed $P_2(z)$ function on the top and the reconstructed $P_3(z)$ function on the bottom. For each case, we show the theoretical prediction of our reference $\Lambda$CDM model as a red dashed curve.</figcaption></figure>

With the binning method, the number of bins is limited by the maximum number of existing data redshifts from the smallest data set corresponding to one of our model-independent observables. In this case, this is the quantity $E_g$, for which we have effectively only three redshift bins. There are nine $E_g$ data points, but most of them are very close to each other in redshift, due to being measured by different collaborations or at different scales in real space for the same $z$. As explained in the data section above, we only regard this data as an average over different scales, assuming that non-linear corrections have been correctly taken into account by the respective experimental collaboration. Since we do not have to take derivatives of $E_g$, or equivalently $P_2$, this leaves us with three possible
## 16. Results

Let us now discuss the results of the final observable $\eta_{\text{obs}}$ for each of these methods. The binning method contains fewer assumptions compared to the polynomial regression or Gaussian process methods. It is essentially a weighted average over the data points and the error bars at each redshift bin. Since we need to take derivatives in order to calculate $P_3$ and $E'$, and we have few data points, we opt to compute finite difference derivatives. This has the caveat that it introduces correlations among the errors of the function and its derivatives, that we cannot take into account with this simple method. Moreover, for the binning method, we do not take into account possible non-diagonal covariance matrices for the data, which we do for polynomial regression and the Gaussian process reconstruction.

Figure 2 shows the reconstructed functions obtained by the binning method, the Gaussian process, and polynomial regression, alongside the theoretical prediction of the standard $\Lambda$CDM model. In all cases, the error bars or the bands represent the $1\sigma$ uncertainty.

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Comparison of the three reconstruction methods for each of the model-independent variables. The binning method is shown in blue squares with error bars, the Gaussian process is shown as a green dotted line with green bands, and polynomial regression is shown as a solid yellow line with yellow bands. All of them depict the $1\sigma$ uncertainty. **Left panel:** Plot of the reconstructed $E(z)$ function on the top and its derivative $E'(z)$ on the bottom. **Right panel:** Plot of the reconstructed $P_2(z)$ function on the top and the reconstructed $P_3(z)$ function on the bottom. For each case, we show the theoretical prediction of our reference $\Lambda$CDM model as a red dashed curve.</figcaption></figure>

With the binning method, the number of bins is limited by the maximum number of existing data redshifts from the smallest data set corresponding to one of our model-independent observables. In this case, this is the quantity $E_g$, for which we have effectively only three redshift bins. There are nine $E_g$ data points, but most of them are very close to each other in redshift, due to being measured by different collaborations or at different scales in real space for the same $z$. As explained in the data section above, we only regard this data as an average over different scales, assuming that non-linear corrections have been correctly taken into account by the respective experimental collaboration. Since we do not have to take derivatives of $E_g$, or equivalently $P_2$, this leaves us with three possible
redshift bins, centered at $z_1 = 0.294$, $z_2 = 0.580$, and $z_3 = 0.860$, all of them with an approximate bin width of $\Delta z \approx 0.29$. At these redshifts, we obtain $\eta_{\text{obs}}(z_1) = 0.48 \pm 0.45$, $\eta_{\text{obs}}(z_2) = -0.03 \pm 0.34$, and $\eta_{\text{obs}}(z_3) = -2.78 \pm 6.84$. These values and the estimation of the intermediate model-independent quantities can be seen in Table A2.

Regarding the Gaussian process method, we computed the normalized Hubble function and its derivative, $E(z)$ and $E'(z)$, with the *dgp* module of the GaPP code. We reconstructed $E(z)$ and $E'(z)$ for the redshift interval of the data using the Gaussian function as the covariance function and the initial values of the hyperparameters $\theta = [\sigma_f = 0.5, \ell_f = 0.5]$ that were later estimated by the code. The same procedure was done for the $P_2(z)$ data. We obtained, for the $E(z)$ and $E'(z)$ functions, the hyperparameters $\sigma_f = 2.12$ and $\ell_f = 2.06$ and, for the $P_2$ function, $\sigma_f = 0.58$ and $\ell_f = 0.67$.

For the $P_3(z)$ observable, the hyperparameters obtained by the GaPP code led to a very flat and unrealistic reconstruction, so we took another approach for obtaining the optimal hyperparameters. We sampled the logarithm of the marginal likelihood on a grid of hyperparameters $\sigma_f, \ell_f$ from 0.01 to 2, setting thus a prior with the redshift range of the dataset, and 300 points equally separated in log-space for each dimension. We remind the reader that the hyperparameter $\ell_f$ constrains the typical scale on the independent variable $z$. Thus, as an additional prior, we impose that $\ell_f$ needs to be smaller than the redshift range of the data, which was not guaranteed by the default GaPP code. We then chose a pair of hyperparameters corresponding to the maximum of the log-marginal likelihood. Therefore, for the $\ln(f\sigma_8(z))$ data, we obtained $\sigma_f = 0.549$ and $\ell_f = 1.361$. Its reconstructed derivative $P_3$ can be seen in the lower right panel of Figure 2. The function remains relatively flat, compared to the one given by other methods, but this approach has improved the determination of this observable.

Regarding the choice of the kernel function, several functions were compared, each of them with a different number of parameters, to see the impact on the output. We tested the Gaussian kernel with two parameters, $(\sigma_f, \ell_f)$: the rational quadratic kernel with three parameters and the double Gaussian kernel with four parameters (see the original reference for the explicit implemented formula [116]). We performed tests using the $H(z)$ data obtained with the cosmic chronometer technique and the $f\sigma_8(z)$ data. Our tests show that the different choices shift the reconstructed function up to 6% on its central value compared to the Gaussian kernel function. This happens for $H(z)$, but the effect is negligible for $f\sigma_8(z)$. Taking into account the above choices and procedure, we report that with the Gaussian process method we obtain $\eta_{\text{obs}}(z_1) = 0.38 \pm 0.23$, $\eta_{\text{obs}}(z_2) = 0.91 \pm 0.36$, and $\eta_{\text{obs}}(z_3) = 0.58 \pm 0.93$.

For the polynomial regression method, we find $\eta_{\text{obs}}(z_1) = 0.57 \pm 1.05$, $\eta_{\text{obs}}(z_2) = 0.48 \pm 0.96$, and $\eta_{\text{obs}}(z_3) = -0.11 \pm 3.21$. Note that we applied the criteria of a $\chi^2_{\text{red}}$ closest to one and a positive definite Fisher matrix to choose the order of the polynomial for each of the datasets. These criteria led to a choice of a polynomial of order 3 for the $E(z)$ and $E_g(z)$ data and order 6 for the $\ln(f\sigma_8(z))$ data. These polynomials can be seen in Figure 2 as solid yellow lines, together with their $1\sigma$ uncertainty bands. The higher order of the polynomial of $\ln(f\sigma_8(z))$ explains the “bumpiness” of the reconstruction of $P_3$, leading to larger errors on this observable in comparison to the GP method.

In Figure 3, we show the reconstructed $\eta_{\text{obs}}$ as a function of redshift with the three different methods, again with GP in a green dashed line, polynomial regression in a yellow solid line, and the binning method in blue squares with error bars. It is possible to conclude that the methods are consistent with each other, within their $1\sigma$ uncertainties, and that in most bins the results are consistent with the standard gravity scenario. We find that the error bars of the Gaussian process reconstruction are generally smaller than the other methods, such that at the lowest redshift, GP is not compatible with $\eta_{\text{obs}} = 1$ at nearly $2\sigma$, while in the case of the binning method at the intermediate redshift, $z = 0.58$, the tension is nearly $3\sigma$.

Finally, we can combine the estimates at three redshifts of Table A2 into a single value. Assuming a constant $\eta_{\text{obs}}$ in this entire observed range and performing a simple weighted average, we find $\eta_{\text{obs}} = 0.15 \pm 0.27$ (binning), $\eta_{\text{obs}} = 0.53 \pm 0.19$ (Gaussian process), and $\eta_{\text{obs}} = 0.49 \pm 0.69$ (polynomial regression). The Gaussian process method yields the smallest error and excludes standard gravity. However, despite being sometimes advertised as “model-independent”, we believe that this method
redshift bins, centered at $z_1 = 0.294$, $z_2 = 0.580$, and $z_3 = 0.860$, all of them with an approximate bin width of $\Delta z \approx 0.29$. At these redshifts, we obtain $\eta_{\text{obs}}(z_1) = 0.48 \pm 0.45$, $\eta_{\text{obs}}(z_2) = -0.03 \pm 0.34$, and $\eta_{\text{obs}}(z_3) = -2.78 \pm 6.84$. These values and the estimation of the intermediate model-independent quantities can be seen in Table A2.

Regarding the Gaussian process method, we computed the normalized Hubble function and its derivative, $E(z)$ and $E'(z)$, with the *dgp* module of the GaPP code. We reconstructed $E(z)$ and $E'(z)$ for the redshift interval of the data using the Gaussian function as the covariance function and the initial values of the hyperparameters $\theta = [\sigma_f = 0.5, \ell_f = 0.5]$ that were later estimated by the code. The same procedure was done for the $P_2(z)$ data. We obtained, for the $E(z)$ and $E'(z)$ functions, the hyperparameters $\sigma_f = 2.12$ and $\ell_f = 2.06$ and, for the $P_2$ function, $\sigma_f = 0.58$ and $\ell_f = 0.67$.

For the $P_3(z)$ observable, the hyperparameters obtained by the GaPP code led to a very flat and unrealistic reconstruction, so we took another approach for obtaining the optimal hyperparameters. We sampled the logarithm of the marginal likelihood on a grid of hyperparameters $\sigma_f, \ell_f$ from 0.01 to 2, setting thus a prior with the redshift range of the dataset, and 300 points equally separated in log-space for each dimension. We remind the reader that the hyperparameter $\ell_f$ constrains the typical scale on the independent variable $z$. Thus, as an additional prior, we impose that $\ell_f$ needs to be smaller than the redshift range of the data, which was not guaranteed by the default GaPP code. We then chose a pair of hyperparameters corresponding to the maximum of the log-marginal likelihood. Therefore, for the $\ln(f\sigma_8(z))$ data, we obtained $\sigma_f = 0.549$ and $\ell_f = 1.361$. Its reconstructed derivative $P_3$ can be seen in the lower right panel of Figure 2. The function remains relatively flat, compared to the one given by other methods, but this approach has improved the determination of this observable.

Regarding the choice of the kernel function, several functions were compared, each of them with a different number of parameters, to see the impact on the output. We tested the Gaussian kernel with two parameters, $(\sigma_f, \ell_f)$: the rational quadratic kernel with three parameters and the double Gaussian kernel with four parameters (see the original reference for the explicit implemented formula [116]). We performed tests using the $H(z)$ data obtained with the cosmic chronometer technique and the $f\sigma_8(z)$ data. Our tests show that the different choices shift the reconstructed function up to 6% on its central value compared to the Gaussian kernel function. This happens for $H(z)$, but the effect is negligible for $f\sigma_8(z)$. Taking into account the above choices and procedure, we report that with the Gaussian process method we obtain $\eta_{\text{obs}}(z_1) = 0.38 \pm 0.23$, $\eta_{\text{obs}}(z_2) = 0.91 \pm 0.36$, and $\eta_{\text{obs}}(z_3) = 0.58 \pm 0.93$.

For the polynomial regression method, we find $\eta_{\text{obs}}(z_1) = 0.57 \pm 1.05$, $\eta_{\text{obs}}(z_2) = 0.48 \pm 0.96$, and $\eta_{\text{obs}}(z_3) = -0.11 \pm 3.21$. Note that we applied the criteria of a $\chi^2_{\text{red}}$ closest to one and a positive definite Fisher matrix to choose the order of the polynomial for each of the datasets. These criteria led to a choice of a polynomial of order 3 for the $E(z)$ and $E_g(z)$ data and order 6 for the $\ln(f\sigma_8(z))$ data. These polynomials can be seen in Figure 2 as solid yellow lines, together with their $1\sigma$ uncertainty bands. The higher order of the polynomial of $\ln(f\sigma_8(z))$ explains the “bumpiness” of the reconstruction of $P_3$, leading to larger errors on this observable in comparison to the GP method.

In Figure 3, we show the reconstructed $\eta_{\text{obs}}$ as a function of redshift with the three different methods, again with GP in a green dashed line, polynomial regression in a yellow solid line, and the binning method in blue squares with error bars. It is possible to conclude that the methods are consistent with each other, within their $1\sigma$ uncertainties, and that in most bins the results are consistent with the standard gravity scenario. We find that the error bars of the Gaussian process reconstruction are generally smaller than the other methods, such that at the lowest redshift, GP is not compatible with $\eta_{\text{obs}} = 1$ at nearly $2\sigma$, while in the case of the binning method at the intermediate redshift, $z = 0.58$, the tension is nearly $3\sigma$.

Finally, we can combine the estimates at three redshifts of Table A2 into a single value. Assuming a constant $\eta_{\text{obs}}$ in this entire observed range and performing a simple weighted average, we find $\eta_{\text{obs}} = 0.15 \pm 0.27$ (binning), $\eta_{\text{obs}} = 0.53 \pm 0.19$ (Gaussian process), and $\eta_{\text{obs}} = 0.49 \pm 0.69$ (polynomial regression). The Gaussian process method yields the smallest error and excludes standard gravity. However, despite being sometimes advertised as “model-independent”, we believe that this method
actually makes a strong assumption, since it compresses the ignorance of the reconstruction into a kernel function that depends on two or a small number of parameters, which are often not even fully marginalized over, which was done in our case. Furthermore, the binning method taken at face value would rule out standard gravity. However, as already mentioned, we did not take into account the correlation induced by the finite differences, and this might have decreased the overall error. Overall, we think the polynomial regression method is the most satisfactory one, providing the best compromise between the least number of assumptions and the best estimation of the data derivative. Therefore, we consider it as our “fiducial” result.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Plot of the reconstructed $\eta_{\text{obs}}$ as a function of redshift, using the binning method (blue squares), the Gaussian process (green dotted line), and linear regression (yellow solid line). The corresponding error bands (error bars for the binning method), represent the 1 estimated error on the reconstruction. As a reference, we show, in a dashed red line, the value in a standard gravity scenario.</figcaption></figure>

# 17. Conclusions

Measuring the properties of gravity at large scales will be one of the main tasks of cosmology for the next few years. Several large observational campaigns that are underway, or will soon be [117–121], will collect enough data on galaxy clustering and lensing to render this task possible to a high level of accuracy.

In order to test gravity, one has to provide an alternative, either a full model or at least some parametrization, that goes beyond Einstein’s gravity. Here we chose to consider the Horndeski Lagrangian because, although it is based on a single scalar field, it displays most of the properties that make modified gravity models such a rich area of research. We connected a more theoretically oriented parameterization, the $\alpha_i$ parameters of [22], with more phenomenologically oriented ones, the $h_i$ parameters. To gain a deeper physical understanding, we also discussed some interesting limiting cases: how the Newtonian potentials look in real space, and the impact of the constraints from gravitational wave speed.

The first practical goal in cosmology is to test and possibly rule out specific models of gravity and background expansion. For instance, one can rule out ΛCDM in a number of ways, the simplest of which being measuring a deviation from the predicted $H(z)$ behavior (which is not equivalent, as we have seen, to simply finding deviations from a $w = -1$ equation of state). Models in which gravity is modified can often be designed to have a perfect ΛCDM background, so it is necessary to test them at the perturbation level.
actually makes a strong assumption, since it compresses the ignorance of the reconstruction into a kernel function that depends on two or a small number of parameters, which are often not even fully marginalized over, which was done in our case. Furthermore, the binning method taken at face value would rule out standard gravity. However, as already mentioned, we did not take into account the correlation induced by the finite differences, and this might have decreased the overall error. Overall, we think the polynomial regression method is the most satisfactory one, providing the best compromise between the least number of assumptions and the best estimation of the data derivative. Therefore, we consider it as our “fiducial” result.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Plot of the reconstructed $\eta_{\text{obs}}$ as a function of redshift, using the binning method (blue squares), the Gaussian process (green dotted line), and linear regression (yellow solid line). The corresponding error bands (error bars for the binning method), represent the 1 estimated error on the reconstruction. As a reference, we show, in a dashed red line, the value in a standard gravity scenario.</figcaption></figure>

# 17. Conclusions

Measuring the properties of gravity at large scales will be one of the main tasks of cosmology for the next few years. Several large observational campaigns that are underway, or will soon be [117–121], will collect enough data on galaxy clustering and lensing to render this task possible to a high level of accuracy.

In order to test gravity, one has to provide an alternative, either a full model or at least some parametrization, that goes beyond Einstein's gravity. Here we chose to consider the Horndeski Lagrangian because, although it is based on a single scalar field, it displays most of the properties that make modified gravity models such a rich area of research. We connected a more theoretically oriented parameterization, the $\alpha_i$ parameters of [22], with more phenomenologically oriented ones, the $h_i$ parameters. To gain a deeper physical understanding, we also discussed some interesting limiting cases: how the Newtonian potentials look in real space, and the impact of the constraints from gravitational wave speed.

The first practical goal in cosmology is to test and possibly rule out specific models of gravity and background expansion. For instance, one can rule out ΛCDM in a number of ways, the simplest of which being measuring a deviation from the predicted $H(z)$ behavior (which is not equivalent, as we have seen, to simply finding deviations from a $w = -1$ equation of state). Models in which gravity is modified can often be designed to have a perfect ΛCDM background, so it is necessary to test them at the perturbation level.
Here, however, a problem arises, namely that many more assumptions generally need to be made. Some of them, listed in Section 9, are in some sense of fundamental character, and we follow them in this work. However, most cosmological analyses that test gravity assume, in addition, one or more of the following assumptions: (1) that the initial conditions are given by a simple inflationary spectrum described by one or two parameters; (2) that the cosmological evolution at a z value larger than a few is given by a pure CDM-dominated universe living in standard gravity; (3) that the linear bias depends only on time and not on scale; and (4) that the value of some parameters, such as Ωm0 obtained from CMB analyses assuming ΛCDM, can also be applied to different models.

We have shown that the statistics called $\eta_{\text{obs}}$ can be measured without any of these four assumptions. These statistics can be used as an estimator of the anisotropic stress parameter $\eta$, one of the two phenomenological functions of linearized, scalar, sub-horizon modified gravity. In this sense, we say that $\eta_{\text{obs}}$ is (relatively) model-independent. If $\eta_{\text{obs}}$ differs from unity, either gravity is modified, or at least one of the four “fundamental” assumptions of Section 9 are false.

We have provided a preliminary estimate of $\eta_{\text{obs}}$ based on currently available data, $\eta_{\text{obs}} = 0.49 \pm 0.69$ in the redshift range $z = (0.2 - 0.8)$. The full k- and z-dependence is still inaccessible with current data. According to [25], the Euclid mission will be able to measure $\eta$ to a few percent, which is thus almost two orders of magnitude better than current values, and will be able to begin placing interesting limits on the k- and z-dependence. As the philosopher of science Alexandre Koyré said concerning the emergence of modern science (Koyré, A. (1948). *Du monde de l’à peu près à l’univers de la précision*. In A. Koyré (Ed.), *Etudes d’histoire de la pensée philosophique* (pp. 341–362). Paris: Gallimard.), and with regard to measuring gravity at cosmological scales, we will finally move “from the world of approximation to the universe of precision.”

**Acknowledgments:** We acknowledge DFG for support through the TR33 project, “The Dark Universe.” We thank Alejandro Guarnizo Trilleras, Jiaming Zhao, Adrià Gómez, Guillem Domènech, Martin Kunz, Ippocratis Saltas, Mariele Motta, and Ignacy Sawicki for discussions and collaboration on these topics. We also wish to thank Jiaming Zhao for his valuable comments. D.B. acknowledges support from the Attraction del Talento Científico en Salamanca programme during the last stages of this work. A.M.P. gratefully acknowledges the support by the Landesgraduiertenförderung (LGF) grant of the Graduiertenakademie Universität Heidelberg. S.C. acknowledges support from CNES and CNRS grants.

**Conflicts of Interest:** The authors declare that there is no conflict of interest.

## Appendix A. Background Equations of Motion of the Horndeski Lagrangian

The equations of motion for a flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric are [17,22]

$$3M_{\star}^{2}H^{2} = \rho_{m} + \rho_{HL} \qquad (A1)$$

$$M_{\star}^{2}(2\dot{H} + 3H^{2}) = -p_{m} - p_{HL}.$$

Here, $\rho_m$, $p_m$ are the conserved background energy density and the pressure of matter, respectively; analogously $\rho_{HL}$ and $p_{HL}$ are the background energy density and pressure of the Horndeski field, defined as

$$\rho_{HL} \equiv -K + 2X (K_X - G_{3\phi}) + 6\dot{\phi}H (XG_{3X} - G_{4\phi} - 2XG_{4\phi X}) + 12H^2X (G_{4X} + 2XG_{4XX} - G_{5\phi} - XG_{5\phi X}) + 4\dot{\phi}H^3X (G_{5X} + XG_{5XX}), \qquad (A2)$$

$$p_{HL} = K - 2X (G_{3\phi} - 2G_{4\phi\phi}) + 4\dot{\phi}H (G_{4\phi} - 2XG_{4\phi X} + XG_{5\phi\phi}) - M_{\star}^2 \alpha_B H \frac{\dot{\phi}}{\dot{\phi}} \qquad (A3) \\ + 2\phi H^3 X G_{5X} - 4H^2 X^2 G_{5\phi X}$$

where $M_\star$ is defined in Equation (55), and $\alpha_B$ in Equation (58). Since in the literature there are various definitions of the energy density associated with the scalar field, we report in Table A1 the relation between the one adopted in this work and others. (We thank Jiaming Zhao for pointing out potential
Here, however, a problem arises, namely that many more assumptions generally need to be made. Some of them, listed in Section 9, are in some sense of fundamental character, and we follow them in this work. However, most cosmological analyses that test gravity assume, in addition, one or more of the following assumptions: (1) that the initial conditions are given by a simple inflationary spectrum described by one or two parameters; (2) that the cosmological evolution at a z value larger than a few is given by a pure CDM-dominated universe living in standard gravity; (3) that the linear bias depends only on time and not on scale; and (4) that the value of some parameters, such as $\Omega_{m0}$ obtained from CMB analyses assuming $\Lambda$CDM, can also be applied to different models.

We have shown that the statistics called $\eta_{\text{obs}}$ can be measured without any of these four assumptions. These statistics can be used as an estimator of the anisotropic stress parameter $\eta$, one of the two phenomenological functions of linearized, scalar, sub-horizon modified gravity. In this sense, we say that $\eta_{\text{obs}}$ is (relatively) model-independent. If $\eta_{\text{obs}}$ differs from unity, either gravity is modified, or at least one of the four “fundamental” assumptions of Section 9 are false.

We have provided a preliminary estimate of $\eta_{\text{obs}}$ based on currently available data, $\eta_{\text{obs}} = 0.49 \pm 0.69$ in the redshift range $z = (0.2 - 0.8)$. The full k- and z-dependence is still inaccessible with current data. According to [25], the Euclid mission will be able to measure $\eta$ to a few percent, which is thus almost two orders of magnitude better than current values, and will be able to begin placing interesting limits on the k- and z-dependence. As the philosopher of science Alexandre Koyré said concerning the emergence of modern science (Koyré, A. (1948). *Du monde de l’à peu près à l’univers de la précision*. In A. Koyré (Ed.), *Etudes d’histoire de la pensée philosophique* (pp. 341–362). Paris: Gallimard.), and with regard to measuring gravity at cosmological scales, we will finally move “from the world of approximation to the universe of precision.”

**Acknowledgments:** We acknowledge DFG for support through the TR33 project, “The Dark Universe.” We thank Alejandro Guarnizo Trilleras, Jiaming Zhao, Adrià Gómez, Guillem Domènech, Martin Kunz, Ippocratis Saltas, Mariele Motta, and Ignacy Sawicki for discussions and collaboration on these topics. We also wish to thank Jiaming Zhao for his valuable comments. D.B. acknowledges support from the Attraction del Talento Científico en Salamanca programme during the last stages of this work. A.M.P. gratefully acknowledges the support by the Landesgraduiertenförderung (LGF) grant of the Graduiertenakademie Universität Heidelberg. S.C. acknowledges support from CNES and CNRS grants.

**Conflicts of Interest:** The authors declare that there is no conflict of interest.

## Appendix A. Background Equations of Motion of the Horndeski Lagrangian

The equations of motion for a flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric are [17,22]

$$3M_{\star}^{2}H^{2} = \rho_{m} + \rho_{HL} \qquad (A1)$$

$$M_{\star}^{2}(2\dot{H} + 3H^{2}) = -p_{m} - p_{HL}.$$

Here, $\rho_m$, $p_m$ are the conserved background energy density and the pressure of matter, respectively; analogously $\rho_{HL}$ and $p_{HL}$ are the background energy density and pressure of the Horndeski field, defined as

$$\rho_{HL} = -K + 2X(K_{X} - G_{3\phi}) + 6\dot{\phi}H(XG_{3X} - G_{4\phi} - 2XG_{4\phi X}) + 12H^{2}X(G_{4X} + 2XG_{4XX} - G_{5\phi} - XG_{5\phi X}) + 4\dot{\phi}H^{3}X(G_{5X} + XG_{5XX}), \qquad (A2)$$

$$p_{HL} = K - 2X(G_{3\phi} - 2G_{4\phi\phi}) + 4\dot{\phi}H(G_{4\phi} - 2XG_{4\phi X} + XG_{5\phi\phi}) - M_{\star}^{2}\alpha_{B}H\frac{\dot{\phi}}{\dot{\phi}} \qquad (A3) \\ +2\phi H^{3}XG_{5X} - 4H^{2}X^{2}G_{5\phi X}$$

where $M_*$ is defined in Equation (55), and $\alpha_B$ in Equation (58). Since in the literature there are various definitions of the energy density associated with the scalar field, we report in Table A1 the relation between the one adopted in this work and others. (We thank Jiaming Zhao for pointing out potential
issues related to this.) Following [22], we write the equation of motion of the scalar field $\phi$ as a (non-)conservation equation of the “shift-charge density” $n$,

$$ \dot{n} + 3Hn = p_{HL,\phi} $$

with

$$ n \equiv \dot{\phi}(K_X - 2G_{3\phi}) + 6HX(G_{3X} - 2G_{4\phi X}) + \tag{A4} \\ +6H^2\dot{\phi}(G_{4X} + 2XG_{4XX} - G_{5\phi} - XG_{5\phi X}) + \\ +2H^3X(3G_{5X} + 2XG_{5XX}) $$

and with the non-conservation term, driven by a violation of the shift symmetry $\phi \to \phi + \text{const}$, given by the $\phi$-derivative of $p_{HL}$:

$$ p_{HL,\phi} \equiv K_{\phi} - 2XG_{3\phi\phi} + 2\ddot{\phi}(XG_{3\phi X} + 3H\dot{\phi}G_{4\phi X}) + 6\dot{H}G_{4\phi} + \tag{A5} \\ + 6H^2(2G_{4\phi} + 2XG_{4\phi X} - XG_{5\phi\phi}) + 2H^3\dot{\phi}XG_{5\phi X}. $$

<table><caption>**Table A1.** Comparison table for the different definition of energy density and pressure of the scalar field.</caption><thead><tr><th></th><th>This Work</th><th>Ref. [122]</th><th>Ref. [22]</th></tr></thead><tbody><tr><td>Energy density</td><td>ρ<sub>HL</sub></td><td>ρ<sub>DE</sub> = <sup>ρ<sub>HL</sub></sup>&frasl;<sub>3</sub> - H<sup>2</sup>(M<sub>*</sub><sup>2</sup> - 1)</td><td>ε̃ = <sup>ρ<sub>HL</sub></sup>&frasl;<sub>M<sub>*</sub><sup>2</sup></sub> ≡ ρ̃<sub>HL</sub></td></tr><tr><td>Pressure</td><td>p<sub>HL</sub></td><td>p<sub>DE</sub> = <sup>p<sub>HL</sub></sup>&frasl;<sub>3</sub> + (3H<sup>2</sup> + 2Ḣ)(M<sub>*</sub><sup>2</sup> - 1)</td><td>P̃ = <sup>p<sub>HL</sub></sup>&frasl;<sub>M<sub>*</sub><sup>2</sup></sub> ≡ P̃<sub>HL</sub></td></tr></tbody></table>

<table><caption>**Table A2.** The reconstructed or measured model-independent variables $E, E', P_2, P_3, \eta(z)$ at three different redshifts $z = (0.294, 0.58, 0.86)$, together with their $1\sigma$ errors, for each of the reconstruction methods. The polynomial regression method is compatible with the $\Lambda$CDM scenario, while the other two methods show some tension at lower redshifts.</caption><thead><tr><th rowspan="2">Method</th><th rowspan="2">Parameter</th><th colspan="3">Redshift Bins</th><th rowspan="2">Weighted Mean</th></tr><tr><th>z<sub>1</sub> = 0.294</th><th>z<sub>2</sub> = 0.58</th><th>z<sub>3</sub> = 0.86</th></tr></thead><tbody><tr><td rowspan="5">Binning</td><td>E(z)</td><td>1.12 ± 0.01</td><td>1.27 ± 0.02</td><td>1.51 ± 0.02</td><td></td></tr><tr><td>E'(z)</td><td>-0.56 ± 0.07</td><td>-0.60 ± 0.36</td><td>-1.75 ± 0.66</td><td></td></tr><tr><td>P<sub>2</sub>(z)</td><td>0.75 ± 0.10</td><td>0.54 ± 0.07</td><td>0.18 ± 0.14</td><td></td></tr><tr><td>P<sub>3</sub>(z)</td><td>-0.17 ± 0.35</td><td>0.53 ± 0.61</td><td>-1.27 ± 1.52</td><td></td></tr><tr><td>η<sub>obs</sub>(z)</td><td>0.48 ± 0.45</td><td>-0.03 ± 0.34</td><td>-2.78 ± 6.84</td><td>0.15 ± 0.27</td></tr><tr><td rowspan="5">Gaussian Process</td><td>E(z)</td><td>1.10 ± 0.01</td><td>1.30 ± 0.02</td><td>1.55 ± 0.03</td><td></td></tr><tr><td>E'(z)</td><td>-0.73 ± 0.05</td><td>-1.30 ± 0.10</td><td>-1.89 ± 0.16</td><td></td></tr><tr><td>P<sub>2</sub>(z)</td><td>0.74 ± 0.09</td><td>0.53 ± 0.06</td><td>0.23 ± 0.11</td><td></td></tr><tr><td>P<sub>3</sub>(z)</td><td>-0.10 ± 0.20</td><td>-0.03 ± 0.21</td><td>-0.21 ± 0.30</td><td></td></tr><tr><td>η<sub>obs</sub>(z)</td><td>0.38 ± 0.23</td><td>0.91 ± 0.36</td><td>0.58 ± 0.93</td><td>0.53 ± 0.19</td></tr><tr><td rowspan="5">Polynomial Regression</td><td>E(z)</td><td>1.12 ± 0.01</td><td>1.29 ± 0.02</td><td>1.50 ± 0.02</td><td></td></tr><tr><td>E'(z)</td><td>-0.73 ± 0.04</td><td>-1.06 ± 0.04</td><td>-1.45 ± 0.04</td><td></td></tr><tr><td>P<sub>2</sub>(z)</td><td>0.76 ± 0.15</td><td>0.55 ± 0.15</td><td>0.18 ± 0.14</td><td></td></tr><tr><td>P<sub>3</sub>(z)</td><td>-0.09 ± 0.80</td><td>0.14 ± 0.78</td><td>-0.17 ± 3.02</td><td></td></tr><tr><td>η<sub>obs</sub>(z)</td><td>0.57 ± 1.05</td><td>0.48 ± 0.96</td><td>-0.11 ± 3.21</td><td>0.49 ± 0.69</td></tr></tbody></table>

## References

1. Lovelock, D. The Four-Dimensionality of Space and the Einstein Tensor. *J. Math. Phys.* **1972**, *13*, 874–876. [CrossRef]

2. Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified Gravity and Cosmology. *Phys. Rep.* **2012**, *513*, 1–189, [CrossRef]
issues related to this.) Following [22], we write the equation of motion of the scalar field $\phi$ as a (non-)conservation equation of the “shift-charge density” $n$,

$$ \dot{n} + 3Hn = p_{HL,\phi} $$

with

$$ \begin{aligned} n ={}& \dot{\phi}(K_X - 2G_{3\phi}) + 6HX(G_{3X} - 2G_{4\phi X}) + \\ & + 6H^2\dot{\phi}(G_{4X} + 2XG_{4XX} - G_{5\phi} - XG_{5\phi X}) + \\ & + 2H^3X(3G_{5X} + 2XG_{5XX}) \end{aligned} \quad (A4) $$

and with the non-conservation term, driven by a violation of the shift symmetry $\phi \to \phi + \text{const}$, given by the $\phi$-derivative of $p_{HL}$:

$$ \begin{aligned} p_{HL,\phi} ={}& K_{\phi} - 2XG_{3\phi\phi} + 2\ddot{\phi}(XG_{3\phi X} + 3H\dot{\phi}G_{4\phi X}) + 6\dot{H}G_{4\phi} + \\ & + 6H^2(2G_{4\phi} + 2XG_{4\phi X} - XG_{5\phi\phi}) + 2H^3\dot{\phi}XG_{5\phi X}. \end{aligned} \quad (A5) $$

<table><caption>**Table A1.** Comparison table for the different definition of energy density and pressure of the scalar field.</caption><thead><tr><th></th><th>This Work</th><th>Ref. [122]</th><th>Ref. [22]</th></tr></thead><tbody><tr><td>Energy density</td><td>ρ<sub>HL</sub></td><td>ρ<sub>DE</sub> = (ρ<sub>HL</sub>/3) - H<sup>2</sup>(M<sub>*</sub><sup>2</sup> - 1)</td><td>ε̃ = (ρ<sub>HL</sub>/M<sub>*</sub><sup>2</sup>) ≡ ρ̃<sub>HL</sub></td></tr><tr><td>Pressure</td><td>p<sub>HL</sub></td><td>p<sub>DE</sub> = (p<sub>HL</sub>/3) + (3H<sup>2</sup> + 2Ḣ)(M<sub>*</sub><sup>2</sup> - 1)</td><td>P̃ = (p<sub>HL</sub>/M<sub>*</sub><sup>2</sup>) ≡ P̃<sub>HL</sub></td></tr></tbody></table>

<table><caption>**Table A2.** The reconstructed or measured model-independent variables $E, E', P_2, P_3, \eta(z)$ at three different redshifts $z = (0.294, 0.58, 0.86)$, together with their $1\sigma$ errors, for each of the reconstruction methods. The polynomial regression method is compatible with the $\Lambda$CDM scenario, while the other two methods show some tension at lower redshifts.</caption><thead><tr><th rowspan="2">Method</th><th rowspan="2">Parameter</th><th colspan="3">Redshift Bins</th><th rowspan="2">Weighted Mean</th></tr><tr><th>z<sub>1</sub> = 0.294</th><th>z<sub>2</sub> = 0.58</th><th>z<sub>3</sub> = 0.86</th></tr></thead><tbody><tr><td rowspan="5">Binning</td><td>E(z)</td><td>1.12 ± 0.01</td><td>1.27 ± 0.02</td><td>1.51 ± 0.02</td><td></td></tr><tr><td>E'(z)</td><td>-0.56 ± 0.07</td><td>-0.60 ± 0.36</td><td>-1.75 ± 0.66</td><td></td></tr><tr><td>P<sub>2</sub>(z)</td><td>0.75 ± 0.10</td><td>0.54 ± 0.07</td><td>0.18 ± 0.14</td><td></td></tr><tr><td>P<sub>3</sub>(z)</td><td>-0.17 ± 0.35</td><td>0.53 ± 0.61</td><td>-1.27 ± 1.52</td><td></td></tr><tr><td>η<sub>obs</sub>(z)</td><td>0.48 ± 0.45</td><td>-0.03 ± 0.34</td><td>-2.78 ± 6.84</td><td>0.15 ± 0.27</td></tr><tr><td rowspan="5">Gaussian Process</td><td>E(z)</td><td>1.10 ± 0.01</td><td>1.30 ± 0.02</td><td>1.55 ± 0.03</td><td></td></tr><tr><td>E'(z)</td><td>-0.73 ± 0.05</td><td>-1.30 ± 0.10</td><td>-1.89 ± 0.16</td><td></td></tr><tr><td>P<sub>2</sub>(z)</td><td>0.74 ± 0.09</td><td>0.53 ± 0.06</td><td>0.23 ± 0.11</td><td></td></tr><tr><td>P<sub>3</sub>(z)</td><td>-0.10 ± 0.20</td><td>-0.03 ± 0.21</td><td>-0.21 ± 0.30</td><td></td></tr><tr><td>η<sub>obs</sub>(z)</td><td>0.38 ± 0.23</td><td>0.91 ± 0.36</td><td>0.58 ± 0.93</td><td>0.53 ± 0.19</td></tr><tr><td rowspan="5">Polynomial Regression</td><td>E(z)</td><td>1.12 ± 0.01</td><td>1.29 ± 0.02</td><td>1.50 ± 0.02</td><td></td></tr><tr><td>E'(z)</td><td>-0.73 ± 0.04</td><td>-1.06 ± 0.04</td><td>-1.45 ± 0.04</td><td></td></tr><tr><td>P<sub>2</sub>(z)</td><td>0.76 ± 0.15</td><td>0.55 ± 0.15</td><td>0.18 ± 0.14</td><td></td></tr><tr><td>P<sub>3</sub>(z)</td><td>-0.09 ± 0.80</td><td>0.14 ± 0.78</td><td>-0.17 ± 3.02</td><td></td></tr><tr><td>η<sub>obs</sub>(z)</td><td>0.57 ± 1.05</td><td>0.48 ± 0.96</td><td>-0.11 ± 3.21</td><td>0.49 ± 0.69</td></tr></tbody></table>

## References

1. Lovelock, D. The Four-Dimensionality of Space and the Einstein Tensor. *J. Math. Phys.* **1972**, *13*, 874–876. [CrossRef]

2. Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified Gravity and Cosmology. *Phys. Rep.* **2012**, *513*, 1–189, [CrossRef]
3. Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett. **2017**, *119*, 161101, [CrossRef] [PubMed]

4. Papantonopoulos, E. (Ed.) *Modifications of Einstein's Theory of Gravity at Large Distances*; Lecture Notes in Physics; Springer: Berlin, Germany, 2015; Volume 892. [CrossRef]

5. Horndeski, G.W. Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys. **1974**, *10*, 363-384. [CrossRef]

6. Deffayet, C.; Esposito-Farese, G.; Vikman, A. Covariant Galileon. Phys. Rev. D **2009**, *79*, 084003, [CrossRef]

7. Zumalacárregui, M.; García-Bellido, J. Transforming gravity: From derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian. Phys. Rev. D **2014**, *89*, 064046, [CrossRef]

8. Gleyzes, J.; Langlois, D.; Piazza, F.; Vernizzi, F. Exploring gravitational theories beyond Horndeski. J. Cosmol. Astropart. Phys. **2015**, *1502*, 018, [CrossRef]

9. Gleyzes, J.; Langlois, D.; Piazza, F.; Vernizzi, F. Healthy theories beyond Horndeski. Phys. Rev. Lett. **2015**, *114*, 211101, [CrossRef]

10. Crisostomi, M.; Koyama, K.; Tasinato, G. Extended Scalar-Tensor Theories of Gravity. J. Cosmol. Astropart. Phys. **2016**, *1604*, 044, [CrossRef]

11. Crisostomi, M.; Hull, M.; Koyama, K.; Tasinato, G. Horndeski: Beyond, or not beyond? J. Cosmol. Astropart. Phys. **2016**, *1603*, 038, [CrossRef]

12. Ben Achour, J.; Langlois, D.; Noui, K. Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations. Phys. Rev. D **2016**, *93*, 124005, [CrossRef]

13. Langlois, D.; Noui, K. Degenerate higher derivative theories beyond Horndeski: Evading the Ostrogradski instability. J. Cosmol. Astropart. Phys. **2016**, *1602*, 034, [CrossRef]

14. Kobayashi, T. *Horndeski Theory and Beyond: A Review*; IOP Publishing Ltd.: Bristol, UK, 2019.

15. Bettoni, D.; Liberati, S. Disformal invariance of second order scalar-tensor theories: Framing the Horndeski action. Phys. Rev. D **2013**, *88*, 084020, [CrossRef]

16. Sotiriou, T.P.; Faraoni, V. *f(R) Theories Of Gravity*. Rev. Mod. Phys. **2010**, *82*, 451-497, [CrossRef]

17. De Felice, A.; Kobayashi, T.; Tsujikawa, S. Effective gravitational couplings for cosmological perturbations in the most general scalar-tensor theories with second-order field equations. Phys. Lett. B **2011**, *706*, 123-133. [CrossRef]

18. De Felice, A.; Tsujikawa, S. Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models. J. Cosmol. Astropart. Phys. **2012**, *1202*, 007, [CrossRef]

19. Ostrogradski, M. Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Ac. St. Petersbourg **1850**, VI, 385-517.

20. Woodard, R.P. Ostrogradsky's theorem on Hamiltonian instability. *Scholarpedia* **2015**, *10*, 32243, [CrossRef]

21. Chen, T.j.; Fasiello, M.; Lim, E.A.; Tolley, A.J. Higher derivative theories with constraints: Exorcising Ostrogradski's Ghost. J. Cosmol. Astropart. Phys. **2013**, *1302*, 042, [CrossRef]

22. Bellini, E.; Sawicki, I. Maximal freedom at minimum cost: Linear large-scale structure in general modifications of gravity. J. Cosmol. Astropart. Phys. **2014**, *7*, 050, [CrossRef]

23. Amendola, L.; Tsujikawa, S. *Dark Energy: Theory and Observations*; Cambridge University Press: Cambridge, UK, 2010.

24. Weinberg, S. Damping of tensor modes in cosmology. Phys. Rev. D **2004**, *69*, 023503, [CrossRef]

25. Amendola, L.; Kunz, M.; Motta, M.; Saltas, I.D.; Sawicki, I. Observables and unobservables in dark energy cosmologies. Phys. Rev. D **2013**, *87*, 023501, [CrossRef]

26. Könnig, F.; Amendola, L. Instability in a minimal bimetric gravity model. Phys. Rev. D **2014**, *90*, 044030, [CrossRef]

27. De Felice, A.; Heisenberg, L.; Kase, R.; Mukohyama, S.; Tsujikawa, S.; Zhang, Y.L. Effective gravitational couplings for cosmological perturbations in generalized Proca theories. Phys. Rev. D **2016**, *94*, 044024, [CrossRef]

28. Vardanyan, V.; Amendola, L. How can we tell whether dark energy is composed of multiple fields? Phys. Rev. D **2015**, *92*, 024009, [CrossRef]
3. Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett. **2017**, *119*, 161101, [CrossRef] [PubMed]

4. Papantonopoulos, E. (Ed.) *Modifications of Einstein's Theory of Gravity at Large Distances*; Lecture Notes in Physics; Springer: Berlin, Germany, 2015; Volume 892. [CrossRef]

5. Horndeski, G.W. Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys. **1974**, *10*, 363-384. [CrossRef]

6. Deffayet, C.; Esposito-Farese, G.; Vikman, A. Covariant Galileon. Phys. Rev. D **2009**, *79*, 084003, [CrossRef]

7. Zumalacárregui, M.; García-Bellido, J. Transforming gravity: From derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian. Phys. Rev. D **2014**, *89*, 064046, [CrossRef]

8. Gleyzes, J.; Langlois, D.; Piazza, F.; Vernizzi, F. Exploring gravitational theories beyond Horndeski. J. Cosmol. Astropart. Phys. **2015**, *1502*, 018, [CrossRef]

9. Gleyzes, J.; Langlois, D.; Piazza, F.; Vernizzi, F. Healthy theories beyond Horndeski. Phys. Rev. Lett. **2015**, *114*, 211101, [CrossRef]

10. Crisostomi, M.; Koyama, K.; Tasinato, G. Extended Scalar-Tensor Theories of Gravity. J. Cosmol. Astropart. Phys. **2016**, *1604*, 044, [CrossRef]

11. Crisostomi, M.; Hull, M.; Koyama, K.; Tasinato, G. Horndeski: Beyond, or not beyond? J. Cosmol. Astropart. Phys. **2016**, *1603*, 038, [CrossRef]

12. Ben Achour, J.; Langlois, D.; Noui, K. Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations. Phys. Rev. D **2016**, *93*, 124005, [CrossRef]

13. Langlois, D.; Noui, K. Degenerate higher derivative theories beyond Horndeski: Evading the Ostrogradski instability. J. Cosmol. Astropart. Phys. **2016**, *1602*, 034, [CrossRef]

14. Kobayashi, T. *Horndeski Theory and Beyond: A Review*; IOP Publishing Ltd.: Bristol, UK, 2019.

15. Bettoni, D.; Liberati, S. Disformal invariance of second order scalar-tensor theories: Framing the Horndeski action. Phys. Rev. D **2013**, *88*, 084020, [CrossRef]

16. Sotiriou, T.P.; Faraoni, V. *f(R) Theories Of Gravity*. Rev. Mod. Phys. **2010**, *82*, 451-497, [CrossRef]

17. De Felice, A.; Kobayashi, T.; Tsujikawa, S. Effective gravitational couplings for cosmological perturbations in the most general scalar-tensor theories with second-order field equations. Phys. Lett. B **2011**, *706*, 123-133. [CrossRef]

18. De Felice, A.; Tsujikawa, S. Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models. J. Cosmol. Astropart. Phys. **2012**, *1202*, 007, [CrossRef]

19. Ostrogradski, M. Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Ac. St. Petersbourg **1850**, VI, 385-517.

20. Woodard, R.P. Ostrogradsky's theorem on Hamiltonian instability. *Scholarpedia* **2015**, *10*, 32243, [CrossRef]

21. Chen, T.j.; Fasiello, M.; Lim, E.A.; Tolley, A.J. Higher derivative theories with constraints: Exorcising Ostrogradski's Ghost. J. Cosmol. Astropart. Phys. **2013**, *1302*, 042, [CrossRef]

22. Bellini, E.; Sawicki, I. Maximal freedom at minimum cost: Linear large-scale structure in general modifications of gravity. J. Cosmol. Astropart. Phys. **2014**, *7*, 050, [CrossRef]

23. Amendola, L.; Tsujikawa, S. *Dark Energy: Theory and Observations*; Cambridge University Press: Cambridge, UK, 2010.

24. Weinberg, S. Damping of tensor modes in cosmology. Phys. Rev. D **2004**, *69*, 023503, [CrossRef]

25. Amendola, L.; Kunz, M.; Motta, M.; Saltas, I.D.; Sawicki, I. Observables and unobservables in dark energy cosmologies. Phys. Rev. D **2013**, *87*, 023501, [CrossRef]

26. Könnig, F.; Amendola, L. Instability in a minimal bimetric gravity model. Phys. Rev. D **2014**, *90*, 044030, [CrossRef]

27. De Felice, A.; Heisenberg, L.; Kase, R.; Mukohyama, S.; Tsujikawa, S.; Zhang, Y.L. Effective gravitational couplings for cosmological perturbations in generalized Proca theories. Phys. Rev. D **2016**, *94*, 044024, [CrossRef]

28. Vardanyan, V.; Amendola, L. How can we tell whether dark energy is composed of multiple fields? Phys. Rev. D **2015**, *92*, 024009, [CrossRef]
29. Navarro, J.F.; Frenk, C.S.; White, S.D.M. The Structure of cold dark matter halos. Astrophys. J. **1996**, 462, 563-575, [CrossRef]

30. Pizzuti, L.; Sartoris, B.; Amendola, L.; Borgani, S.; Biviano, A.; Umetsu, K.; Mercurio, A.; Rosati, P.; Balestra, I.; Caminha, G.B.; et al. CLASH-VLT: Constraints on f(R) gravity models with galaxy clusters using lensing and kinematic analyses. J. Cosmol. Astropart. Phys. **2017**, 7, 023, [CrossRef]

31. Dodelson, S. *Modern Cosmology*; Academic Press: Amsterdam, The Netherlands, 2003.

32. Kreisch, C.D.; Komatsu, E. Cosmological Constraints on Horndeski Gravity in Light of GW170817. J. Cosmol. Astropart. Phys. **2018**, *1812*, 030, [CrossRef]

33. Saltas, I.D.; Sawicki, I.; Amendola, L.; Kunz, M. Anisotropic Stress as a Signature of Nonstandard Propagation of Gravitational Waves. Phys. Rev. Lett. **2014**, *113*, 191101, [CrossRef]

34. Sawicki, I.; Saltas, I.D.; Motta, M.; Amendola, L.; Kunz, M. Nonstandard gravitational waves imply gravitational slip: On the difficulty of partially hiding new gravitational degrees of freedom. Phys. Rev. D **2017**, *95*, 083520, [CrossRef]

35. Lombriser, L.; Lima, N.A. Challenges to self-acceleration in modified gravity from gravitational waves and large-scale structure. Phys. Lett. B **2017**, *765*, 382-385, [CrossRef]

36. Linder, E.V. No slip gravity. J. Cosmol. Astropart. Phys. **2018**, *3*, 005, [CrossRef]

37. Will, C.M. *Theory and Experiment in Gravitational Physics*; Cambridge University Press: Cambridge, UK, 1993.

38. Adelberger, E.G.; Heckel, B.R.; Nelson, A.E. Tests of the Gravitational Inverse-Square Law. Annu. Rev. Nucl. Part. Sci. **2003**, *53*, 77-121, [CrossRef]

39. Patrignani, C. Review of Particle Physics. Chin. Phys. **2016**, C40, 100001, [CrossRef]

40. Khoury, J.; Weltman, A. Chameleon cosmology. Phys. Rev. D **2004**, *69*, 044026, [CrossRef]

41. Khoury, J.; Weltman, A. Chameleon fields: Awaiting surprises for tests of gravity in space. Phys. Rev. Lett. **2004**, *93*, 171104, [CrossRef]

42. Vainshtein, A.I. To the problem of nonvanishing gravitation mass. Phys. Lett. B **1972**, *39*, 393-394, [CrossRef]

43. Babichev, E.; Deffayet, C. An introduction to the Vainshtein mechanism. Class. Quant. Grav. **2013**, *30*, 184001, [CrossRef]

44. Pietroni, M. Dark energy condensation. Phys. Rev. D **2005**, *72*, 043535, [CrossRef]

45. Olive, K.A.; Pospelov, M. Environmental dependence of masses and coupling constants. Phys. Rev. D **2008**, *77*, 043524, [CrossRef]

46. Hinterbichler, K.; Khoury, J. Symmetron Fields: Screening Long-Range Forces Through Local Symmetry Restoration. Phys. Rev. Lett. **2010**, *104*, 231301, [CrossRef] [PubMed]

47. Hinterbichler, K.; Khoury, J.; Levy, A.; Matas, A. Symmetron Cosmology. Phys. Rev. D **2011**, *84*, 103521, [CrossRef]

48. Copi, C.J.; Davis, A.N.; Krauss, L.M. New Nucleosynthesis Constraint on the Variation of G. Phys. Rev. Lett. **2004**, *92*, 171301, [CrossRef] [PubMed]

49. Dent, T.; Stern, S.; Wetterich, C. Primordial nucleosynthesis as a probe of fundamental physics parameters. Phys. Rev. D **2007**, *76*, 063513, [CrossRef]

50. Iocco, F.; Mangano, G.; Miele, G.; Pisanti, O.; Serpico, P.D. Primordial nucleosynthesis: From precision cosmology to fundamental physics. Phys. Rep. **2009**, *472*, 1-76, [CrossRef]

51. Bettoni, D.; Ezquiaga, J.M.; Hinterbichler, K.; Zumalacárregui, M. Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity. Phys. Rev. D **2017**, *95*, 084029, [CrossRef]

52. Lombriser, L.; Taylor, A. Breaking a Dark Degeneracy with Gravitational Waves. J. Cosmol. Astropart. Phys. **2016**, *1603*, 031, [CrossRef]

53. Ezquiaga, J.M.; Zumalacárregui, M. Dark Energy After GW170817: Dead Ends and the Road Ahead. Phys. Rev. Lett. **2017**, *119*, 251304, [CrossRef]

54. Creminelli, P.; Vernizzi, F. Dark Energy after GW170817 and GRB170817A. Phys. Rev. Lett. **2017**, *119*, 251302, [CrossRef]

55. Baker, T.; Bellini, E.; Ferreira, P.G.; Lagos, M.; Noller, J.; Sawicki, I. Strong constraints on cosmological gravity from GW170817 and GRB 170817A. Phys. Rev. Lett. **2017**, *119*, 251301, [CrossRef]

56. Sakstein, J.; Jain, B. Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories. Phys. Rev. Lett. **2017**, *119*, 251303, [CrossRef]

57. Amendola, L.; Bettoni, D.; Domènech, G.; Gomes, A.R. Doppelgänger dark energy: Modified gravity with non-universal couplings after GW170817. J. Cosmol. Astropart. Phys. **2018**, *1806*, 029, [CrossRef]
29. Navarro, J.F.; Frenk, C.S.; White, S.D.M. The Structure of cold dark matter halos. Astrophys. J. **1996**, 462, 563-575, [CrossRef]

30. Pizzuti, L.; Sartoris, B.; Amendola, L.; Borgani, S.; Biviano, A.; Umetsu, K.; Mercurio, A.; Rosati, P.; Balestra, I.; Caminha, G.B.; et al. CLASH-VLT: Constraints on f(R) gravity models with galaxy clusters using lensing and kinematic analyses. J. Cosmol. Astropart. Phys. **2017**, 7, 023, [CrossRef]

31. Dodelson, S. *Modern Cosmology*; Academic Press: Amsterdam, The Netherlands, 2003.

32. Kreisch, C.D.; Komatsu, E. Cosmological Constraints on Horndeski Gravity in Light of GW170817. J. Cosmol. Astropart. Phys. **2018**, *1812*, 030, [CrossRef]

33. Saltas, I.D.; Sawicki, I.; Amendola, L.; Kunz, M. Anisotropic Stress as a Signature of Nonstandard Propagation of Gravitational Waves. Phys. Rev. Lett. **2014**, *113*, 191101, [CrossRef]

34. Sawicki, I.; Saltas, I.D.; Motta, M.; Amendola, L.; Kunz, M. Nonstandard gravitational waves imply gravitational slip: On the difficulty of partially hiding new gravitational degrees of freedom. Phys. Rev. D **2017**, *95*, 083520, [CrossRef]

35. Lombriser, L.; Lima, N.A. Challenges to self-acceleration in modified gravity from gravitational waves and large-scale structure. Phys. Lett. B **2017**, *765*, 382-385, [CrossRef]

36. Linder, E.V. No slip gravity. J. Cosmol. Astropart. Phys. **2018**, *3*, 005, [CrossRef]

37. Will, C.M. *Theory and Experiment in Gravitational Physics*; Cambridge University Press: Cambridge, UK, 1993.

38. Adelberger, E.G.; Heckel, B.R.; Nelson, A.E. Tests of the Gravitational Inverse-Square Law. Annu. Rev. Nucl. Part. Sci. **2003**, *53*, 77-121, [CrossRef]

39. Patrignani, C. Review of Particle Physics. Chin. Phys. **2016**, C40, 100001, [CrossRef]

40. Khoury, J.; Weltman, A. Chameleon cosmology. Phys. Rev. D **2004**, *69*, 044026, [CrossRef]

41. Khoury, J.; Weltman, A. Chameleon fields: Awaiting surprises for tests of gravity in space. Phys. Rev. Lett. **2004**, *93*, 171104, [CrossRef]

42. Vainshtein, A.I. To the problem of nonvanishing gravitation mass. Phys. Lett. B **1972**, *39*, 393-394, [CrossRef]

43. Babichev, E.; Deffayet, C. An introduction to the Vainshtein mechanism. Class. Quant. Grav. **2013**, *30*, 184001, [CrossRef]

44. Pietroni, M. Dark energy condensation. Phys. Rev. D **2005**, *72*, 043535, [CrossRef]

45. Olive, K.A.; Pospelov, M. Environmental dependence of masses and coupling constants. Phys. Rev. D **2008**, *77*, 043524, [CrossRef]

46. Hinterbichler, K.; Khoury, J. Symmetron Fields: Screening Long-Range Forces Through Local Symmetry Restoration. Phys. Rev. Lett. **2010**, *104*, 231301, [CrossRef] [PubMed]

47. Hinterbichler, K.; Khoury, J.; Levy, A.; Matas, A. Symmetron Cosmology. Phys. Rev. D **2011**, *84*, 103521, [CrossRef]

48. Copi, C.J.; Davis, A.N.; Krauss, L.M. New Nucleosynthesis Constraint on the Variation of G. Phys. Rev. Lett. **2004**, *92*, 171301, [CrossRef] [PubMed]

49. Dent, T.; Stern, S.; Wetterich, C. Primordial nucleosynthesis as a probe of fundamental physics parameters. Phys. Rev. D **2007**, *76*, 063513, [CrossRef]

50. Iocco, F.; Mangano, G.; Miele, G.; Pisanti, O.; Serpico, P.D. Primordial nucleosynthesis: From precision cosmology to fundamental physics. Phys. Rep. **2009**, *472*, 1-76, [CrossRef]

51. Bettoni, D.; Ezquiaga, J.M.; Hinterbichler, K.; Zumalacárregui, M. Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity. Phys. Rev. D **2017**, *95*, 084029, [CrossRef]

52. Lombriser, L.; Taylor, A. Breaking a Dark Degeneracy with Gravitational Waves. J. Cosmol. Astropart. Phys. **2016**, *1603*, 031, [CrossRef]

53. Ezquiaga, J.M.; Zumalacárregui, M. Dark Energy After GW170817: Dead Ends and the Road Ahead. Phys. Rev. Lett. **2017**, *119*, 251304, [CrossRef]

54. Creminelli, P.; Vernizzi, F. Dark Energy after GW170817 and GRB170817A. Phys. Rev. Lett. **2017**, *119*, 251302, [CrossRef]

55. Baker, T.; Bellini, E.; Ferreira, P.G.; Lagos, M.; Noller, J.; Sawicki, I. Strong constraints on cosmological gravity from GW170817 and GRB 170817A. Phys. Rev. Lett. **2017**, *119*, 251301, [CrossRef]

56. Sakstein, J.; Jain, B. Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories. Phys. Rev. Lett. **2017**, *119*, 251303, [CrossRef]

57. Amendola, L.; Bettoni, D.; Domènech, G.; Gomes, A.R. Doppelgänger dark energy: Modified gravity with non-universal couplings after GW170817. J. Cosmol. Astropart. Phys. **2018**, *1806*, 029, [CrossRef]
58. Amendola, L.; Ballesteros, G.; Pettorino, V. Effects of modified gravity on B-mode polarization. *Phys. Rev. D* **2014**, *90*, 043009, [CrossRef]

59. Amendola, L.; Kunz, M.; Saltas, I.D.; Sawicki, I. The fate of large-scale structure in modified gravity after GW170817 and GRB170817A. *Phys. Rev. Lett.* **2018**, *120*, 131101, [CrossRef]

60. Nersisyan, H.; Lima, N.A.; Amendola, L. Gravitational wave speed: Implications for models without a mass scale. *arXiv* **2018**, arXiv:1801.06683.

61. Amendola, L.; Sawicki, I.; Kunz, M.; Saltas, I.D. Direct detection of gravitational waves can measure the time variation of the Planck mass. *J. Cosmol. Astropart. Phys.* **2018**, *2018*, 030. [CrossRef]

62. Maggiore, M. *Gravitational Waves: Volume 1: Theory and Experiments*; Gravitational Waves; OUP: Oxford, UK, 2008.

63. Nishizawa, A. Generalized framework for testing gravity with gravitational-wave propagation. *I. Formulation*. *Phys. Rev. D* **2018**, *97*, 104037, [CrossRef]

64. Belgacem, E.; Dirian, Y.; Foffa, S.; Maggiore, M. The gravitational-wave luminosity distance in modified gravity theories *Phys. Rev. D* **2018**, *97*, 104066. [CrossRef]

65. Cyburt, R.H.; Fields, B.D.; Olive, K.A.; Yeh, T.H. Big Bang Nucleosynthesis: 2015. *Rev. Mod. Phys.* **2016**, *88*, 015004, [CrossRef]

66. Riess, A.G.; Casertano, S.; Yuan, W.; Macri, L.; Anderson, J.; MacKenty, J.W.; Bowers, J.B.; Clubb, K.I.; Filippenko, A.V.; Jones, D.O.; et al. New Parallaxes of Galactic Cepheids from Spatially Scanning the Hubble Space Telescope: Implications for the Hubble Constant. *Astrophys J.* **2018**, *855*, 136, [CrossRef]

67. Planck Collaboration; Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; et al. Planck 2015 results. XIII. Cosmological parameters. *Astron. Astrophys.* **2016**, *594*, A13, [CrossRef]

68. Hildebrandt, H.; Viola, M.; Heymans, C.; Joudaki, S.; Kuijken, K.; Blake, C.; Erben, T.; Joachimi, B.; Klaes, D.; Miller, L.; et al. KiDS-450: Cosmological parameter constraints from tomographic weak gravitational lensing. *Month. Not. R. Astron. Soc.* **2017**, *465*, 1454–1498, [CrossRef]

69. Barros, B.J.; Amendola, L.; Barreiro, T.; Nunes, N.J. Coupled quintessence with a ΛCDM background: Removing the $\sigma_8$ tension. *J. Cosmol. Astropart. Phys.* **2019**, *2019*, 007. [CrossRef]

70. Weinberg, D.H.; Bullock, J.S.; Governato, F.; Kuzio de Naray, R.; Peter, A.H.G. Cold dark matter: Controversies on small scales. *Proc. Natl. Acad. Sci. USA* **2014**, *112*, 12249–12255, [CrossRef]

71. Abbott, B.P.; The LIGO Scientific Collaboration and The Virgo Collaboration. A gravitational-wave standard siren measurement of the Hubble constant. *Nature* **2017**, *551*, 85–88, [CrossRef]

72. Kunz, M. Degeneracy between the dark components resulting from the fact that gravity only measures the total energy-momentum tensor. *Phys. Rev. D* **2009**, *80*, 123001. [CrossRef]

73. Eisenstein, D.J.; Hu, W. Baryonic Features in the Matter Transfer Function. *Astrophys J.* **1998**, *496*, 605–614, [CrossRef]

74. Liske, J. On the cosmological distance and redshift between any two objects. *Month. Not. R. Astron. Soc.* **2000**, *319*, 557–561, [CrossRef]

75. Baldi, M.; Pettorino, V.; Amendola, L.; Wetterich, C. Oscillating nonlinear large scale structure in growing neutrino quintessence. *Mon. Not. R. Astron. Soc.* **2011**, *418*, 214, [CrossRef]

76. Kaiser, N. Clustering in real space and in redshift space. *Mon. Not. R. Astron. Soc.* **1987**, *227*, 1–21. [CrossRef]

77. Seo, H.J.; Eisenstein, D.J. Probing Dark Energy with Baryonic Acoustic Oscillations from Future Large Galaxy Redshift Surveys. *Astrophys. J.* **2003**, *598*, 720–740, [CrossRef]

78. Rampf, C.; Villa, E.; Amendola, L. Quasilinear observables in dark energy cosmologies. *Phys. Rev. D* **2017**, *95*, 123516, [CrossRef]

79. Motta, M.; Sawicki, I.; Saltas, I.D.; Amendola, L.; Kunz, M. Probing dark energy through scale dependence. *Phys. Rev. D* **2013**, *88*, 124035, [CrossRef]

80. Lahav, O.; Lilje, P.B.; Primack, J.R.; Rees, M.J. Dynamical effects of the cosmological constant. *Month. Not. R. Astron. Soc.* **1991**, *251*, 128–136, [CrossRef]

81. Leonard, C.D.; Ferreira, P.G.; Heymans, C. Testing gravity with EG: Mapping theory onto observations. *J. Cosmol. Astropart. Phys.* **2015**, [CrossRef]

82. Zhang, P.; Liguori, M.; Bean, R.; Dodelson, S. Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity. *Phys. Rev. Lett.* **2007**, *99*, 141302, [CrossRef] [PubMed]
58. Amendola, L.; Ballesteros, G.; Pettorino, V. Effects of modified gravity on B-mode polarization. *Phys. Rev. D* **2014**, *90*, 043009, [CrossRef]

59. Amendola, L.; Kunz, M.; Saltas, I.D.; Sawicki, I. The fate of large-scale structure in modified gravity after GW170817 and GRB170817A. *Phys. Rev. Lett.* **2018**, *120*, 131101, [CrossRef]

60. Nersisyan, H.; Lima, N.A.; Amendola, L. Gravitational wave speed: Implications for models without a mass scale. *arXiv* **2018**, arXiv:1801.06683.

61. Amendola, L.; Sawicki, I.; Kunz, M.; Saltas, I.D. Direct detection of gravitational waves can measure the time variation of the Planck mass. *J. Cosmol. Astropart. Phys.* **2018**, *2018*, 030. [CrossRef]

62. Maggiore, M. *Gravitational Waves: Volume 1: Theory and Experiments*; Gravitational Waves; OUP: Oxford, UK, 2008.

63. Nishizawa, A. Generalized framework for testing gravity with gravitational-wave propagation. *I. Formulation*. *Phys. Rev. D* **2018**, *97*, 104037, [CrossRef]

64. Belgacem, E.; Dirian, Y.; Foffa, S.; Maggiore, M. The gravitational-wave luminosity distance in modified gravity theories *Phys. Rev. D* **2018**, *97*, 104066. [CrossRef]

65. Cyburt, R.H.; Fields, B.D.; Olive, K.A.; Yeh, T.H. Big Bang Nucleosynthesis: 2015. *Rev. Mod. Phys.* **2016**, *88*, 015004, [CrossRef]

66. Riess, A.G.; Casertano, S.; Yuan, W.; Macri, L.; Anderson, J.; MacKenty, J.W.; Bowers, J.B.; Clubb, K.I.; Filippenko, A.V.; Jones, D.O.; et al. New Parallaxes of Galactic Cepheids from Spatially Scanning the Hubble Space Telescope: Implications for the Hubble Constant. *Astrophys J.* **2018**, *855*, 136, [CrossRef]

67. Planck Collaboration; Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; et al. Planck 2015 results. XIII. Cosmological parameters. *Astron. Astrophys.* **2016**, *594*, A13, [CrossRef]

68. Hildebrandt, H.; Viola, M.; Heymans, C.; Joudaki, S.; Kuijken, K.; Blake, C.; Erben, T.; Joachimi, B.; Klaes, D.; Miller, L.; et al. KiDS-450: Cosmological parameter constraints from tomographic weak gravitational lensing. *Month. Not. R. Astron. Soc.* **2017**, *465*, 1454–1498, [CrossRef]

69. Barros, B.J.; Amendola, L.; Barreiro, T.; Nunes, N.J. Coupled quintessence with a ΛCDM background: Removing the $\sigma_8$ tension. *J. Cosmol. Astropart. Phys.* **2019**, *2019*, 007. [CrossRef]

70. Weinberg, D.H.; Bullock, J.S.; Governato, F.; Kuzio de Naray, R.; Peter, A.H.G. Cold dark matter: Controversies on small scales. *Proc. Natl. Acad. Sci. USA* **2014**, *112*, 12249–12255, [CrossRef]

71. Abbott, B.P.; The LIGO Scientific Collaboration and The Virgo Collaboration. A gravitational-wave standard siren measurement of the Hubble constant. *Nature* **2017**, *551*, 85–88, [CrossRef]

72. Kunz, M. Degeneracy between the dark components resulting from the fact that gravity only measures the total energy-momentum tensor. *Phys. Rev. D* **2009**, *80*, 123001. [CrossRef]

73. Eisenstein, D.J.; Hu, W. Baryonic Features in the Matter Transfer Function. *Astrophys J.* **1998**, *496*, 605–614, [CrossRef]

74. Liske, J. On the cosmological distance and redshift between any two objects. *Month. Not. R. Astron. Soc.* **2000**, *319*, 557–561, [CrossRef]

75. Baldi, M.; Pettorino, V.; Amendola, L.; Wetterich, C. Oscillating nonlinear large scale structure in growing neutrino quintessence. *Mon. Not. R. Astron. Soc.* **2011**, *418*, 214, [CrossRef]

76. Kaiser, N. Clustering in real space and in redshift space. *Mon. Not. R. Astron. Soc.* **1987**, *227*, 1–21. [CrossRef]

77. Seo, H.J.; Eisenstein, D.J. Probing Dark Energy with Baryonic Acoustic Oscillations from Future Large Galaxy Redshift Surveys. *Astrophys. J.* **2003**, *598*, 720–740, [CrossRef]

78. Rampf, C.; Villa, E.; Amendola, L. Quasilinear observables in dark energy cosmologies. *Phys. Rev. D* **2017**, *95*, 123516, [CrossRef]

79. Motta, M.; Sawicki, I.; Saltas, I.D.; Amendola, L.; Kunz, M. Probing dark energy through scale dependence. *Phys. Rev. D* **2013**, *88*, 124035, [CrossRef]

80. Lahav, O.; Lilje, P.B.; Primack, J.R.; Rees, M.J. Dynamical effects of the cosmological constant. *Month. Not. R. Astron. Soc.* **1991**, *251*, 128–136, [CrossRef]

81. Leonard, C.D.; Ferreira, P.G.; Heymans, C. Testing gravity with EG: Mapping theory onto observations. *J. Cosmol. Astropart. Phys.* **2015**, [CrossRef]

82. Zhang, P.; Liguori, M.; Bean, R.; Dodelson, S. Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity. *Phys. Rev. Lett.* **2007**, *99*, 141302, [CrossRef] [PubMed]
83. Reyes, R.; Mandelbaum, R.; Seljak, U.; Baldauf, T.; Gunn, J.E.; Lombriser, L.; Smith, R.E. Confirmation of general relativity on large scales from weak lensing and galaxy velocities. *Nature* **2010**, *464*, 256–258, [CrossRef]

84. De La Torre, S.; Jullo, E.; Giocoli, C.; Pezzotta, A.; Bel, J.; Granett, B.R.; Guzzo, L.; Garilli, B.; Scodeggio, M.; Bolzonella, M.; et al. The VIMOS Public Extragalactic Redshift Survey (VIPERS). Gravity test from the combination of redshift-space distortions and galaxy-galaxy lensing at $0.5 < z < 1.2$. *Astron. Astrophys.* **2017**, *608*, A44, [CrossRef]

85. Amon, A.; Blake, C.; Heymans, C.; Leonard, C.D.; Asgari, M.; Bilicki, M.; Choi, A.; Erben, T.; Glazebrook, K.; Harnois-Déraps, J.; et al. KiDS+2dFLenS+GAMA: Testing the cosmological model with the Eg statistic. *Mon. Not. R. Astron. Soc.* **2018**, *479*, 3422–3437. [CrossRef]

86. Trilleras, A.G. A Model-Independent Approach to Dark Energy Cosmologies: Current and Future Constraints. Ph.D. Thesis, Rupert-Karls-Universität, Heidelberg, Germany, 2015.

87. Yu, H.; Ratra, B.; Wang, F.Y. Hubble Parameter and Baryon Acoustic Oscillation Measurement Constraints on the Hubble Constant, the Deviation from the Spatially-Flat Lambda cdm Model, The Deceleration-Acceleration Transition Redshift, and Spatial Curvature. *Astrophys. J.* **2018**, *856*, 3. [CrossRef]

88. Simon, J.; Verde, L.; Jimenez, R. Constraints on the redshift dependence of the dark energy potential. *Phys. Rev. D* **2004**, *71*, [CrossRef]

89. Stern, D.; Jimenez, R.; Verde, L.; Stanford, S.A.; Kamionkowski, M. Cosmic Chronometers: Constraining the Equation of State of Dark Energy. II. A Spectroscopic Catalog of Red Galaxies in Galaxy Clusters. *Astrophys. J. Suppl. Ser.* **2010**, *188*, 280, [CrossRef]

90. Moresco, M.; Cimatti, A.; Jimenez, R.; Pozzetti, L.; Zamorani, G.; Bolzonella, M.; Dunlop, J.; Lamareille, F.; Mignoli, M.; Pearce, H.; et al. Improved constraints on the expansion rate of the Universe up to $z \sim 1.1$ from the spectroscopic evolution of cosmic chronometers. *J. Cosmol. Astropart. Phys.* **2012**, *1208*, 006, [CrossRef]

91. Moresco, M. Raising the bar: New constraints on the Hubble parameter with cosmic chronometers at $z \sim 2$. *Mon. Not. R. Astron. Soc.* **2015**, *450*, L16–L20, [CrossRef]

92. Delubac, T.; Bautista, J.E.; Rich, J.; Kirkby, D.; Bailey, S.; Font-Ribera, A.; Slosar, A.; Lee, K.G.; Pieri, M.M.; Hamilton, J.C.; et al. Baryon acoustic oscillations in the $\Lambda\alpha$ forest of BOSS DR11 quasars. *Astron. Astrophys.* **2015**, *574*, A59, [CrossRef]

93. Font-Ribera, A.; Kirkby, D.; Miralda-Escudé, J.; Ross, N.P.; Slosar, A.; Rich, J.; Aubourg, É.; Bailey, S.; Bhardwaj, V.; Bautista, J.; et al. Quasar-Lyman $\alpha$ Forest Cross-Correlation from BOSS DR11: Baryon Acoustic Oscillations. *J. Cosmol. Astropart. Phys.* **2014**, *1405*, 027, [CrossRef]

94. Moresco, M.; Pozzetti, L.; Cimatti, A.; Jimenez, R.; Maraston, C.; Verde, L.; Thomas, D.; Citro, A.; Tojeiro, R.; Wilkinson, D. A 6% measurement of the Hubble parameter at $z \sim 0.45$: Direct evidence of the epoch of cosmic re-acceleration. *J. Cosmol. Astropart. Phys.* **2016**, *5*, 014, [CrossRef]

95. Zhang, C.; Zhang, H.; Yuan, S.; Liu, S.; Zhang, T.J.; Sun, Y.C. Four New Observational H(z) Data From Luminous Red Galaxies of Sloan Digital Sky Survey Data Release Seven. *Res. Astron. Astrophys.* **2014**, *14*, 1221. [CrossRef]

96. Alam, S.; BOSS Collaboration. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Cosmological analysis of the DR12 galaxy sample. *Mon. Not. R. Astron. Soc.* **2016**, *470*, 2617–2652, [CrossRef]

97. Blake, C.; Brough, S.; Colless, M.; Contreras, C.; Couch, W.; Croom, S.; Croton, D.; Davis, T.M.; Drinkwater, M.J.; Forster, K.; et al. The WiggleZ Dark Energy Survey: Joint measurements of the expansion and growth history at $z < 1$. *Mon. Not. R. Astron. Soc.* **2012**, *425*, 405–414, [CrossRef]

98. Riess, A.G.; Rodney, S.A.; Scolnic, D.M.; Shafer, D.L.; Strolger, L.G.; Ferguson, H.C.; Postman, M.; Graur, O.; Maoz, D.; Jha, S.W.; et al. Type Ia Supernova Distances at $z > 1.5$ from the Hubble Space Telescope Multi-Cycle Treasury Programs: The Early Expansion Rate. *Astrophys. J.* **2018**, *853*, 126. [CrossRef]

99. Blake, C.; Joudaki, S.; Heymans, C.; Choi, A.; Erben, T.; Harnois-Déraps, J.; Hildebrandt, H.; Joachimi, B.; Nakajima, R.; van Waerbeke, L.; et al. RCSLenS: Testing gravitational physics through the cross-correlation of weak lensing and large-scale structure. *Mon. Not. R. Astron. Soc.* **2016**, *456*, 2806–2828. [CrossRef]

100. Beutler, F.; Blake, C.; Colless, M.; Jones, D.H.; Staveley-Smith, L.; Poole, G.B.; Campbell, L.; Parker, Q.; Saunders, W.; Watson, F. The 6dF Galaxy Survey: $Z=0$ measurements of the growth rate and $\sigma_8$. *Mon. Not. R. Astron. Soc.* **2012**, *423*, 3430–3444, [CrossRef]
83. Reyes, R.; Mandelbaum, R.; Seljak, U.; Baldauf, T.; Gunn, J.E.; Lombriser, L.; Smith, R.E. Confirmation of general relativity on large scales from weak lensing and galaxy velocities. *Nature* **2010**, *464*, 256–258, [CrossRef]

84. De La Torre, S.; Jullo, E.; Giocoli, C.; Pezzotta, A.; Bel, J.; Granett, B.R.; Guzzo, L.; Garilli, B.; Scodeggio, M.; Bolzonella, M.; et al. The VIMOS Public Extragalactic Redshift Survey (VIPERS). Gravity test from the combination of redshift-space distortions and galaxy-galaxy lensing at $0.5 < z < 1.2$. *Astron. Astrophys.* **2017**, *608*, A44, [CrossRef]

85. Amon, A.; Blake, C.; Heymans, C.; Leonard, C.D.; Asgari, M.; Bilicki, M.; Choi, A.; Erben, T.; Glazebrook, K.; Harnois-Déraps, J.; et al. KiDS+2dFLenS+GAMA: Testing the cosmological model with the Eg statistic. *Mon. Not. R. Astron. Soc.* **2018**, *479*, 3422–3437. [CrossRef]

86. Trilleras, A.G. A Model-Independent Approach to Dark Energy Cosmologies: Current and Future Constraints. Ph.D. Thesis, Rupert-Karls-Universität, Heidelberg, Germany, 2015.

87. Yu, H.; Ratra, B.; Wang, F.Y. Hubble Parameter and Baryon Acoustic Oscillation Measurement Constraints on the Hubble Constant, the Deviation from the Spatially-Flat Lambda cdm Model, The Deceleration-Acceleration Transition Redshift, and Spatial Curvature. *Astrophys. J.* **2018**, *856*, 3. [CrossRef]

88. Simon, J.; Verde, L.; Jimenez, R. Constraints on the redshift dependence of the dark energy potential. *Phys. Rev. D* **2004**, *71*, [CrossRef]

89. Stern, D.; Jimenez, R.; Verde, L.; Stanford, S.A.; Kamionkowski, M. Cosmic Chronometers: Constraining the Equation of State of Dark Energy. II. A Spectroscopic Catalog of Red Galaxies in Galaxy Clusters. *Astrophys. J. Suppl. Ser.* **2010**, *188*, 280, [CrossRef]

90. Moresco, M.; Cimatti, A.; Jimenez, R.; Pozzetti, L.; Zamorani, G.; Bolzonella, M.; Dunlop, J.; Lamareille, F.; Mignoli, M.; Pearce, H.; et al. Improved constraints on the expansion rate of the Universe up to $z \sim 1.1$ from the spectroscopic evolution of cosmic chronometers. *J. Cosmol. Astropart. Phys.* **2012**, *1208*, 006, [CrossRef]

91. Moresco, M. Raising the bar: New constraints on the Hubble parameter with cosmic chronometers at $z \sim 2$. *Mon. Not. R. Astron. Soc.* **2015**, *450*, L16–L20, [CrossRef]

92. Delubac, T.; Bautista, J.E.; Rich, J.; Kirkby, D.; Bailey, S.; Font-Ribera, A.; Slosar, A.; Lee, K.G.; Pieri, M.M.; Hamilton, J.C.; et al. Baryon acoustic oscillations in the $\Lambda\alpha$ forest of BOSS DR11 quasars. *Astron. Astrophys.* **2015**, *574*, A59, [CrossRef]

93. Font-Ribera, A.; Kirkby, D.; Miralda-Escudé, J.; Ross, N.P.; Slosar, A.; Rich, J.; Aubourg, É.; Bailey, S.; Bhardwaj, V.; Bautista, J.; et al. Quasar-Lyman $\alpha$ Forest Cross-Correlation from BOSS DR11: Baryon Acoustic Oscillations. *J. Cosmol. Astropart. Phys.* **2014**, *1405*, 027, [CrossRef]

94. Moresco, M.; Pozzetti, L.; Cimatti, A.; Jimenez, R.; Maraston, C.; Verde, L.; Thomas, D.; Citro, A.; Tojeiro, R.; Wilkinson, D. A 6% measurement of the Hubble parameter at $z \sim 0.45$: Direct evidence of the epoch of cosmic re-acceleration. *J. Cosmol. Astropart. Phys.* **2016**, *5*, 014, [CrossRef]

95. Zhang, C.; Zhang, H.; Yuan, S.; Liu, S.; Zhang, T.J.; Sun, Y.C. Four New Observational H(z) Data From Luminous Red Galaxies of Sloan Digital Sky Survey Data Release Seven. *Res. Astron. Astrophys.* **2014**, *14*, 1221. [CrossRef]

96. Alam, S.; BOSS Collaboration. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Cosmological analysis of the DR12 galaxy sample. *Mon. Not. R. Astron. Soc.* **2016**, *470*, 2617–2652, [CrossRef]

97. Blake, C.; Brough, S.; Colless, M.; Contreras, C.; Couch, W.; Croom, S.; Croton, D.; Davis, T.M.; Drinkwater, M.J.; Forster, K.; et al. The WiggleZ Dark Energy Survey: Joint measurements of the expansion and growth history at $z < 1$. *Mon. Not. R. Astron. Soc.* **2012**, *425*, 405–414, [CrossRef]

98. Riess, A.G.; Rodney, S.A.; Scolnic, D.M.; Shafer, D.L.; Strolger, L.G.; Ferguson, H.C.; Postman, M.; Graur, O.; Maoz, D.; Jha, S.W.; et al. Type Ia Supernova Distances at $z > 1.5$ from the Hubble Space Telescope Multi-Cycle Treasury Programs: The Early Expansion Rate. *Astrophys. J.* **2018**, *853*, 126. [CrossRef]

99. Blake, C.; Joudaki, S.; Heymans, C.; Choi, A.; Erben, T.; Harnois-Déraps, J.; Hildebrandt, H.; Joachimi, B.; Nakajima, R.; van Waerbeke, L.; et al. RCSLenS: Testing gravitational physics through the cross-correlation of weak lensing and large-scale structure. *Mon. Not. R. Astron. Soc.* **2016**, *456*, 2806–2828. [CrossRef]

100. Beutler, F.; Blake, C.; Colless, M.; Jones, D.H.; Staveley-Smith, L.; Poole, G.B.; Campbell, L.; Parker, Q.; Saunders, W.; Watson, F. The 6dF Galaxy Survey: $Z = 0$ measurements of the growth rate and $\sigma_8$. *Mon. Not. R. Astron. Soc.* **2012**, *423*, 3430–3444, [CrossRef]
101. Okumura, T.; Hikage, C.; Totani, T.; Tonegawa, M.; Okada, H.; Glazebrook, K.; Blake, C.; Ferreira, P.G.; More, S.; Taruya, A.; et al. The Subaru FMOS galaxy redshift survey (FastSound). IV. New constraint on gravity theory from redshift space distortions at $z = 1.4$. *Publ. Astron. Soc. Jpn.* **2016**, *68*, 38. [CrossRef]

102. Song, Y.S.; Percival, W.J. Reconstructing the history of structure formation using redshift distortions. *J. Cosmol. Astropart. Phys.* **2009**, *2009*, 004. [CrossRef]

103. Hawken, A.J.; Granett, B.R.; Iovino, A.; Guzzo, L.; Peacock, J.A.; de la Torre, S.; Garilli, B.; Bolzonella, M.; Scodeggio, M.; Abbas, U.; et al. The VIMOS Public Extragalactic Redshift Survey: Measuring the growth rate of structure around cosmic voids. *Astron. Astrophys.* **2017**, *607*, A54. [CrossRef]

104. De la Torre, S.; Guzzo, L.; Peacock, J.A.; Branchini, E.; Iovino, A.; Granett, N.R.; Abbas, U.; Adami, C.; Arnouts, C.; Bel, J.; et al. The VIMOS Public Extragalactic Redshift Survey (VIPERS). Galaxy clustering and redshift-space distortions at $z = 0.8$ in the first data release. *Astron. Astrophys.* **2013**, *557*, A54. [CrossRef]

105. Mohammad, F.G.; Granett, B.R.; Guzzo, L.; Bel, J.; Branchini, E.; de la Torre, S.; Moscardini, L.; Peacock, J.A.; Bolzonella, M.; Garilli, B.; et al. An unbiased estimate of the growth rate of structure at $<z> = 0.85$ using the clustering of luminous blue galaxies. *Astron. Astrophys.* **2018**, *610*, A59.

106. Howlett, C.; Ross, A.J.; Samushia, L.; Percival, W.J.; Manera, M. The clustering of the SDSS main galaxy sample—II. Mock galaxy catalogues and a measurement of the growth of structure from redshift space distortions at $z = 0.15$. *Mon. Not. R. Astron. Soc.* **2015**, *449*, 848–866, [CrossRef]

107. Samushia, L.; Percival, W.J.; Raccanelli, A. Interpreting large-scale redshift-space distortion measurements. *Mon. Not. R. Astron. Soc.* **2012**, *420*, 2102–2119, [CrossRef]

108. Tojeiro, R.; Percival, W.J.; Brinkmann, J.; Brownstein, J.R.; Eisenstein, D.J.; Manera, M.; Maraston, C.; McBride, C.K.; Muna, D.; Reid, B.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Measuring structure growth using passive galaxies. *Mon. Not. R. Astron. Soc.* **2012**, *424*, 2339, [CrossRef]

109. Chuang, C.H.; Wang, Y. Modelling the anisotropic two-point galaxy correlation function on small scales and single-probe measurements of $H(z)$, $D_A(z)$ and $f(z)\sigma_8(z)$ from the Sloan Digital Sky Survey DR7 luminous red galaxies. *Mon. Not. R. Astron. Soc.* **2013**, *435*, 255–262, [CrossRef]

110. Gil-Marín, H.; Percival, W.J.; Brownstein, J.R.; Chuang, C.H.; Grieb, J.K.; Ho, S.; Kitaura, F.S.; Maraston, C.; Prada, F.; Rodríguez-Torres, S.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies. *Mon. Not. R. Astron. Soc.* **2016**, *460*, 4188–4209, [CrossRef]

111. Gil-Marín, H.; Percival, W.J.; Cuesta, A.J.; Brownstein, J.R.; Chuang, C.H.; Ho, S.; Kitaura, F.S.; Maraston, C.; Prada, F.; Rodríguez-Torres, S.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: BAO measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies. *Mon. Not. R. Astron. Soc.* **2016**, *460*, 4210–4219, [CrossRef]

112. Chuang, C.; Prada, F.; Pellejero-Ibanez, M.; Beutler, F.; Cuesta, A.J.; Eisenstein, D.J.; Escoffier, S.; Ho, S.; Kitaura, F.; Kneib, J.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Single-probe measurements from CMASS anisotropic galaxy clustering. *Mon. Not. R. Astron. Soc.* **2016**, *461*, 3781–3793. [CrossRef]

113. Cabré, A.; Gaztañaga, E. Clustering of luminous red galaxies—I. Large-scale redshift-space distortions. *Mon. Not. R. Astron. Soc.* **2009**, *393*, 1183–1208, [CrossRef]

114. Guzzo, L.; Pierleoni, M.; Meneux, B.; Branchini, E.; Le Fèvre, O.; Marinoni, C.; Garilli, B.; Blaizot, J.; De Lucia, G.; Pollo, A.; et al. A test of the nature of cosmic acceleration using galaxy redshift distortions. *Nature* **2008**, *451*, 541–544, [CrossRef]

115. Rasmussen, C.; Williams, C. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006.

116. Seikel, M.; Clarkson, C.; Smith, M. Reconstruction of dark energy and expansion dynamics using Gaussian processes. *J. Cosmol. Astropart. Phys.* **2012**, *2012*, 036. [CrossRef]

117. Laureijs, R.; Amiaux, J.; Arduini, S.; Auguères, J.-L.; Brinchmann, J.; Cole, R.; Cropper, M.; Dabin, C.; Duvet, L.; Ealet, A.; et al. Euclid Definition Study Report. arXiv **2011**, arXiv:1110.3193.

118. Amendola, L.; Appleby, S.; Avgoustidis, A.; Bacon, D.; Baker, T.; Baldi, M.; Bartolo, N.; Blanchard, A.; Bonvin, C.; Borgani, S.; et al. Cosmology and fundamental physics with the Euclid satellite. *Living Rev. Rel.* **2013**, *16*, 6, [CrossRef]

119. Rawlings, S. Cosmology with the Square Kilometre Array. arXiv **2011**, arXiv:1105.6333.
101. Okumura, T.; Hikage, C.; Totani, T.; Tonegawa, M.; Okada, H.; Glazebrook, K.; Blake, C.; Ferreira, P.G.; More, S.; Taruya, A.; et al. The Subaru FMOS galaxy redshift survey (FastSound). IV. New constraint on gravity theory from redshift space distortions at $z = 1.4$. *Publ. Astron. Soc. Jpn.* **2016**, *68*, 38. [CrossRef]

102. Song, Y.S.; Percival, W.J. Reconstructing the history of structure formation using redshift distortions. *J. Cosmol. Astropart. Phys.* **2009**, *2009*, 004. [CrossRef]

103. Hawken, A.J.; Granett, B.R.; Iovino, A.; Guzzo, L.; Peacock, J.A.; de la Torre, S.; Garilli, B.; Bolzonella, M.; Scodeggio, M.; Abbas, U.; et al. The VIMOS Public Extragalactic Redshift Survey: Measuring the growth rate of structure around cosmic voids. *Astron. Astrophys.* **2017**, *607*, A54. [CrossRef]

104. De la Torre, S.; Guzzo, L.; Peacock, J.A.; Branchini, E.; Iovino, A.; Granett, N.R.; Abbas, U.; Adami, C.; Arnouts, C.; Bel, J.; et al. The VIMOS Public Extragalactic Redshift Survey (VIPERS). Galaxy clustering and redshift-space distortions at $z = 0.8$ in the first data release. *Astron. Astrophys.* **2013**, *557*, A54. [CrossRef]

105. Mohammad, F.G.; Granett, B.R.; Guzzo, L.; Bel, J.; Branchini, E.; de la Torre, S.; Moscardini, L.; Peacock, J.A.; Bolzonella, M.; Garilli, B.; et al. An unbiased estimate of the growth rate of structure at $<z> = 0.85$ using the clustering of luminous blue galaxies. *Astron. Astrophys.* **2018**, *610*, A59.

106. Howlett, C.; Ross, A.J.; Samushia, L.; Percival, W.J.; Manera, M. The clustering of the SDSS main galaxy sample—II. Mock galaxy catalogues and a measurement of the growth of structure from redshift space distortions at $z = 0.15$. *Mon. Not. R. Astron. Soc.* **2015**, *449*, 848–866, [CrossRef]

107. Samushia, L.; Percival, W.J.; Raccanelli, A. Interpreting large-scale redshift-space distortion measurements. *Mon. Not. R. Astron. Soc.* **2012**, *420*, 2102–2119, [CrossRef]

108. Tojeiro, R.; Percival, W.J.; Brinkmann, J.; Brownstein, J.R.; Eisenstein, D.J.; Manera, M.; Maraston, C.; McBride, C.K.; Muna, D.; Reid, B.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Measuring structure growth using passive galaxies. *Mon. Not. R. Astron. Soc.* **2012**, *424*, 2339, [CrossRef]

109. Chuang, C.H.; Wang, Y. Modelling the anisotropic two-point galaxy correlation function on small scales and single-probe measurements of $H(z)$, $D_A(z)$ and $f(z)\sigma_8(z)$ from the Sloan Digital Sky Survey DR7 luminous red galaxies. *Mon. Not. R. Astron. Soc.* **2013**, *435*, 255–262, [CrossRef]

110. Gil-Marín, H.; Percival, W.J.; Brownstein, J.R.; Chuang, C.H.; Grieb, J.K.; Ho, S.; Kitaura, F.S.; Maraston, C.; Prada, F.; Rodríguez-Torres, S.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies. *Mon. Not. R. Astron. Soc.* **2016**, *460*, 4188–4209, [CrossRef]

111. Gil-Marín, H.; Percival, W.J.; Cuesta, A.J.; Brownstein, J.R.; Chuang, C.H.; Ho, S.; Kitaura, F.S.; Maraston, C.; Prada, F.; Rodríguez-Torres, S.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: BAO measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies. *Mon. Not. R. Astron. Soc.* **2016**, *460*, 4210–4219, [CrossRef]

112. Chuang, C.; Prada, F.; Pellejero-Ibanez, M.; Beutler, F.; Cuesta, A.J.; Eisenstein, D.J.; Escoffier, S.; Ho, S.; Kitaura, F.; Kneib, J.; et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Single-probe measurements from CMASS anisotropic galaxy clustering. *Mon. Not. R. Astron. Soc.* **2016**, *461*, 3781–3793. [CrossRef]

113. Cabré, A.; Gaztañaga, E. Clustering of luminous red galaxies—I. Large-scale redshift-space distortions. *Mon. Not. R. Astron. Soc.* **2009**, *393*, 1183–1208, [CrossRef]

114. Guzzo, L.; Pierleoni, M.; Meneux, B.; Branchini, E.; Le Fèvre, O.; Marinoni, C.; Garilli, B.; Blaizot, J.; De Lucia, G.; Pollo, A.; et al. A test of the nature of cosmic acceleration using galaxy redshift distortions. *Nature* **2008**, *451*, 541–544, [CrossRef]

115. Rasmussen, C.; Williams, C. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006.

116. Seikel, M.; Clarkson, C.; Smith, M. Reconstruction of dark energy and expansion dynamics using Gaussian processes. *J. Cosmol. Astropart. Phys.* **2012**, *2012*, 036. [CrossRef]

117. Laureijs, R.; Amiaux, J.; Arduini, S.; Auguères, J.-L.; Brinchmann, J.; Cole, R.; Cropper, M.; Dabin, C.; Duvet, L.; Ealet, A.; et al. Euclid Definition Study Report. arXiv **2011**, arXiv:1110.3193.

118. Amendola, L.; Appleby, S.; Avgoustidis, A.; Bacon, D.; Baker, T.; Baldi, M.; Bartolo, N.; Blanchard, A.; Bonvin, C.; Borgani, S.; et al. Cosmology and fundamental physics with the Euclid satellite. *Living Rev. Rel.* **2013**, *16*, 6, [CrossRef]

119. Rawlings, S. Cosmology with the Square Kilometre Array. arXiv **2011**, arXiv:1105.6333.
120. LSST Science Collaboration; Abell, P.A.; Allison, J.; Anderson, S.F.; Andrew, J.R.; Angel, J.R.P.; Armus, L.; Arnett, D.; Asztalos, S.J.; Axelrod, T.S.; et al. LSST Science Book, Version 2.0. *arXiv* 2009, arXiv:0912.0201.

121. Levi, M.; Bebek, C.; Beers, T.; Blum, R.; Cahn, R.; Eisenstein, D.; Flaugher, B.; Honscheid, K.; Kron, R.; Lahav, O.; et al. The DESI Experiment, a whitepaper for Snowmass 2013. *arXiv* 2013, arXiv:1308.0847.

122. Zumalacarregui, M.; Bellini, E.; Sawicki, I.; Lesgourgues, J.; Ferreira, P.G. hiclass: Horndeski in the Cosmic Linear Anisotropy Solving System. *J. Cosmol. Astropart. Phys.* **2017**, *1708*, 019, [CrossRef]

<figure><img src="image_6.png" /></figure>

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
120. LSST Science Collaboration; Abell, P.A.; Allison, J.; Anderson, S.F.; Andrew, J.R.; Angel, J.R.P.; Armus, L.; Arnett, D.; Asztalos, S.J.; Axelrod, T.S.; et al. LSST Science Book, Version 2.0. *arXiv* 2009, arXiv:0912.0201.

121. Levi, M.; Bebek, C.; Beers, T.; Blum, R.; Cahn, R.; Eisenstein, D.; Flaugher, B.; Honscheid, K.; Kron, R.; Lahav, O.; et al. The DESI Experiment, a whitepaper for Snowmass 2013. *arXiv* 2013, arXiv:1308.0847.

122. Zumalacarregui, M.; Bellini, E.; Sawicki, I.; Lesgourgues, J.; Ferreira, P.G. hiclass: Horndeski in the Cosmic Linear Anisotropy Solving System. *J. Cosmol. Astropart. Phys.* **2017**, *1708*, 019, [CrossRef]

<figure><img src="image_6.png" /></figure>

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).