ETD

• Backreaction
• inflation
• ultra-slow-roll
• USR

The thesis investigates the role of backreaction generated by tensor perturbations in single-field inflationary models, with particular attention to transient ultra-slow-roll phases. The interest in this type of dynamics arises from the fact that ultra-slow-roll is one of the most studied mechanisms for producing a strong enhancement of the comoving curvature perturbation and, consequently, for enabling the formation of primordial black holes. The main objective of the work is therefore to understand to what extent tensor backreaction can modify the inflationary dynamics and the phenomenology associated with these scenarios.
During an ultra-slow-roll phase, the inflaton crosses an extremely flat region of the potential. In this regime, the first slow-roll parameter decreases very rapidly, while the second slow-roll parameter takes values characteristic of a non-attractor phase. Unlike in standard slow-roll, the comoving curvature perturbation is not conserved on super-Hubble scales and can be strongly amplified. This enhancement is precisely the mechanism that can produce a scalar power spectrum large enough to favor primordial black hole formation. However, these models are often highly sensitive to the parameters of the potential, since very small variations can lead to large differences in the amplitude of the spectrum and in the resulting abundance of primordial black holes.
To study this problem, the thesis adopts a gauge-invariant and observer-dependent formalism, which allows effective cosmological quantities to be defined without introducing an artificial dependence on the choice of coordinates. In single-field inflation, the inflaton provides the natural physical clock, and the analysis is performed with respect to the comoving observer. The calculation is developed up to second perturbative order and in the long-wavelength limit. Particular attention is devoted to the tensor contribution to backreaction, which affects effective scalar quantities both through terms quadratic in first-order tensor perturbations and through scalar perturbations induced by tensor modes at second order.
In the standard slow-roll regime, the corrections produced by tensor backreaction remain small and perturbatively under control, consistently with previous results in the literature. The situation changes significantly during a non-attractor phase. In ultra-slow-roll, the rapid decrease of the first slow-roll parameter strongly enhances the corrections to the effective quantities. As a consequence, a perturbative treatment performed on a fixed background rapidly loses validity. The most relevant physical result is that tensor backreaction tends to oppose the decrease of the first slow-roll parameter, acting directly on the mechanism responsible for the amplification of the scalar spectrum.
To explore the dynamics beyond this limitation, a minimal self-consistent scheme is introduced. In this approach, the functional form of the perturbatively derived correction is retained, but it is evaluated using quantities that have already been modified by backreaction. This procedure can be interpreted as a partial resummation of tensor effects and should not be regarded as a fully non-perturbative treatment.
The application to two very close realizations of a benchmark model clearly shows the importance of this effect. Without backreaction, the non-attractor phase lasts approximately 2.2 and 2.4 e-folds in the two cases considered. Once the self-consistent backreaction is included, the duration is reduced to about 1.12 e-folds in both models. Backreaction therefore makes the dynamics significantly less sensitive to small variations in the parameters of the potential.
The strongest effect, however, concerns the scalar spectrum. Without backreaction, the two models produce peaks compatible with the amplitudes generally required for a significant production of primordial black holes. When self-consistent tensor backreaction is included, the peak of the scalar spectrum is instead suppressed by several orders of magnitude. This substantially changes the predictions for primordial black hole production and, at the same time, reduces the fine-tuning of the model. Overall, the results show that tensor backreaction, although negligible in standard slow-roll, can become a crucial effect in non-attractor regimes, shortening the ultra-slow-roll phase, suppressing the enhancement of the scalar spectrum, and making the dynamics less sensitive to the parameters of the model.