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We test the generalized scalar-tensor theory in static systems, namely galaxy clusters. The Degenerate higher-order scalar-tensor (DHOST) theory modifies the Newtonian potential through effective Newtonian constant and $\Xi_1$ parameter in the small scale, which modifies the hydrostatic equilibrium. We utilize the well-compiled X-COP catalog consisting of 12 clusters with Intra Cluster Medium (ICM) pressure profile by Sunyaev-Zeldovich effect data and temperature profile by X-ray data for each cluster. We perform a fully Bayesian analysis modeling Navarro-Frenk-White (NFW) for the mass profile, and the simplified Vikhlinin model for the electron density. Carefully selecting suitable clusters to present our results, we find a mild to moderate, i.e, $\sim 2\sigma$ significance for a deviation from the standard scenario in 4 of the clusters. However, in terms of Bayesian evidence, we find either equivalent or mild preference for GR. We estimate a joint constraint of $\Xi_1 = -0.030 \pm 0.043$ using 8 clusters, for a modification from a $\Lambda$CDM scenario. This limit is in very good agreement with theoretical ones and an order of magnitude more stringent than the previous constraint obtained using clusters. We also quote a more conservative limit of $\Xi_1 = -0.061 \pm 0.074$. Finally, we comment on the tentative redshift dependence ($\Xi_1(z)$), finding a mild preference ($ \lesssim 2\sigma$) for the same., Comment: Comments are Welcome

The origin of the late-time accelerated expansion of the universe is still a great mystery. Numerous cosmological models have been proposed to explain this phenomenon. Modern days' technology and equipment have allowed scientists to successfully execute many observations in cosmology and astrophysics: space missions, large ground-based telescopes and gravitational-wave antennas have led to important discoveries and ruled out many models. The Lambda-Cold Dark Matter, model provides a coherent and satisfactory framework to accommodate all fundamental observations. Therefore it is called the "standard model of cosmology''. Despite its many successes, Lambda CDM requires the introduction of dark energy in the form of an unnaturally small cosmological constant and is plagued by fine-tuning problems ("why do dark energy, dark matter and baryons have comparable energy densities today?''). The elementary particle candidates which are assumed to form the cold dark matter component have never been directly detected. These facts can be taken as possible indications of a potential crisis. This has motivated the introduction of various alternative models, among which a novel class of modified gravity theories, called "mimetic gravity'' or "mimetic dark matter-theory'', which aims at explaining both the dark energy and (at least part of) the dark matter components as consequences of a suitable modification of the gravitational theory w.r.t. Einstein General Relativity. (Chapter 1 and 2) In this PhD thesis, we propose the "generalized mimetic gravity theory", which arises in full generality by means of a non-invertible disformal transformation of the most general single scalar field scalar-tensor theory of gravity and implemented our idea for Horndeski and beyond-Horndeski models. This novel class of models is a generalization of the so-called mimetic dark matter theory recently introduced by Chamseddine and Mukhanov, as discussed in Chapters 2 and 3. It can source the background evolution of the universe by mimicking any perfect fluid, including radiation, dark matter, and dark energy. In this chapter, we also show that very general single-scalar-field scalar-tensor theories of gravity are generically invariant under invertible disformal transformations. In Chapter 4 we analyze linear scalar perturbations around a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background in mimetic Horndeski gravity and show that the sound speed is zero on all backgrounds and therefore the system does not have any wave-like scalar degrees of freedom. Further, we present mimetic vector-tensor theories. In particular, we establish that the non-invertible disformal transformation at the origin of the normalization constraint term in the Einstein-Aether theory, i.e., that the Einstein-Aether theory is also in the class of mimetic theories. We shall also show that an Einstein-Maxwell system sourced by dust can be recovered in the weak limit of a minimal Einstein-Aether theory and that vector field becomes rotation and acceleration free in such a limit (Chapter 5). Finally, in the concluding Chapter 6, we wind up the thesis by discussing some applications and future research directions in mimetic theories of gravity. % So far, it is not ruled out by any observations. The Chapters 3 and 4 are based on our published papers and Chapter 5 is based on the material which will appear in a forthcoming paper (P. Karmakar, T. Koivisto, D. Mota and S. Mukohyama)