We construct an updated and extended compilation of growth rate data based on recent Redshift Space Distortion (RSD) measurements. The dataset consists of 34 datapoints and includes corrections for model dependence. In order to minimize overlap and maximize the independence of the datapoints we also construct a subsample of this compilation (a 'Gold' growth dataset) which consists of 18 datapoints. We test the consistency of this dataset with the best fit Planck15/ΛCDM parameters in the context of General Relativity (GR) using the evolution equation for the growth factor δ(a) with a wCDM background. We find tension at the ∼ 3σ level between the best fit parameters w (the dark energy equation of state), Ω0m (the matter density parameter) and σ8 (the matter power spectrum normalization on scales 8h −1 Mpc) and the corresponding Planck15/ΛCDM parameters (w = −1, Ω0m = 0.315 and σ8 = 0.831). We show that the tension disappears if we allow for evolution of the effective Newton constant, parametrized as G eff (a)/GN = 1+ga(1−a) n −ga(1−a) 2n with n ≥ 2 where ga and n are parameters of the model, a is the scale factor and z = 1/a − 1 is the redshift. This parametrization satisfies three important criteria: a) positive energy of graviton (G eff > 0), b) consistency with Big Bang Nucleosynthesis constraints (G eff (a 1)/GN = 1) and c) consistency with Solar System tests (G eff (a = 1)/GN = 1 and G eff (a = 1)/GN = 0). We show that the best fit form of G eff (z) obtained from the growth data corresponds to weakening gravity at recent redshifts (decreasing function of z) and we demonstrate that this behavior is not consistent with any scalar-tensor Lagrangian with a real scalar field. Finally, we use MGCAMB to find the best fit G eff (z) obtained from the Planck CMB power spectrum on large angular scales and show that it is a mildly increasing function of z, in 3σ tension with the corresponding decreasing best fit G eff (z) obtained from the growth data.

Dark energy equation of state w(z) parametrizations with two parameters and given monotonicity are generically either convex or concave functions. This makes them suitable for fitting either freezing or thawing quintessence models but not both simultaneously. Fitting a dataset based on a freezing model with an unsuitable (concave when increasing) w(z) parametrization (like CPL) can lead to significant misleading features like crossing of the phantom divide line, incorrect w(z = 0), incorrect slope etc. that are not present in the underlying cosmological model. To demonstrate this fact we generate scattered cosmological data both at the level of w(z) and the luminosity distance DL(z) based on either thawing or freezing quintessence models and fit them using parametrizations of convex and of concave type. We then compare statistically significant features of the best fit w(z) with actual features of the underlying model. We thus verify that the use of unsuitable parametrizations can lead to misleading conclusions. In order to avoid these problems it is important to either use both convex and concave parametrizations and select the one with the best χ 2 or use principal component analysis thus splitting the redshift range into independent bins. In the latter case however, significant information about the slope of w(z) at high redshifts is lost. Finally, we propose a new family of parametrizations (nCPL) w(z) = w0 + wa( z 1+z ) n which generalizes the CPL and interpolates between thawing and freezing parametrizations as the parameter n increases to values larger than 1.

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This paper focuses on a specific class of convex multi-agent programs, prevalent in many practical applications, where agents cooperate to minimize a common cost, expressed as a function of the aggregate decision and affected by uncertainty. We model uncertainty by means of scenarios and use an epigraphic reformulation to transfer the uncertain part of the cost function to the constraints. Then, by exploiting the structure of the program under study and leveraging on existing results in the scenario approach literature, and in particular using the so called support rank notion, we provide for the optimal solution of the program distributionfree robustness certificates that are agent-independent, i.e., the constructed bound on the probability of constraint violation does not depend on the number of agents, but only on the dimension of the agents' decision. This leads to a significant improvement as it substantially reduces the number of samples required to achieve a certain level of probabilistic robustness as the number of agents increases. Our certificates can be used alongside any convex optimization algorithm centralised, decentralised or distributed, to obtain an optimal solution of the underlying problem. Our theoretical results are accompanied by a numerical example that investigates the electric vehicle charging problem and validates that the obtained robustness certificate is independent of the number of vehicles in the fleet.