We analyze the properties that any late-time modification of the ΛCDM expansion history must have in order to consistently solve both the H0 and the σ8 tensions. Taking a model-independent approach, we obtain a set of necessary conditions that can be applied to generic late-time extensions. Our results are fully analytical and merely based on the assumptions that the deviations from the ΛCDM background remain small. For the concrete case of a dark energy fluid with equation of state w(z), we derive the following general requirements: (i) Solving the H0 tension demands w(z) < −1 at some z (ii) Solving both the H0 and σ8 tensions requires w(z) to cross the phantom divide. Finally, we also allow for small deviations on the effective gravitational constant. In this case, our method is still able to constrain the functional form of these deviations. Contents

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With recent constraints on the propagation speed of gravitational waves, the class of scalar-tensor theories has significantly been reduced. We consider one of the surviving models still relevant for cosmology and investigate its radiative stability. The model contains operators with explicit breaking of the Galileon symmetry and we study whether they harm the re-organization of the effective field theory. Within the regime of validity we establish a non-renormalization theorem and show explicitly that the quantum corrections, to one-loop, do not detune the classical Lagrangian generating suppressed counterterms. This is striking since the non-renormalization theorem is established in the presence of a genuine Galileon symmetry breaking term.

We establish radiative stability of generalized Proca effective field theories. While standard power-counting arguments would conclude otherwise, we find non-trivial cancellations of leading order corrections by explicit computation of divergent one-loop diagrams up to four-point. These results are crosschecked against an effective action based generalized Schwinger–DeWitt method. Further, the cancellations are understood as coming from the specific structure of the theory through a decoupling limit analysis which at the same time allows for an extension of the results to higher orders.

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