3.1 Introduction

The exact origin of the primordial perturbations that seeded the formation of the large-scale structure in the universe is still unknown. Our current understanding of the initial conditions is based on inflation, a phase of accelerated expansion preceding the standard evolution of the universe [416, 861, 863Jump To The Next Citation Point, 791]. In particular, inflation explains why the universe is so precisely flat, homogeneous and isotropic. During this phase, scales much smaller than the Hubble radius are inflated to super-horizon sizes, so that regions appearing today as causally disconnected were in fact very close in the past. This mechanism is also at the origin of the cosmic large-scale structure. Vacuum quantum fluctuations of any light field present during inflation are amplified by the accelerated expansion and freeze-out on super-Hubble scales acquiring a quasi-scale invariant spectrum [675, 425, 863, 417, 86Jump To The Next Citation Point].

From the early development of inflation, the simplest proposal based on a weakly-coupled single field rolling along its potential [576, 20] has gained strength and many models have been built based on this picture (see for instance [581] for a review). Although some inflationary potentials are now excluded by current data (see for instance [525Jump To The Next Citation Point]), this scenario has been extremely successful in passing many observational tests: it predicts perfectly adiabatic and almost Gaussian fluctuations with a quasi scale-invariant spectrum and a small amount of gravitational waves.

While current data have ruled out some classes of inflationary models, the next qualitative step forward is investigating the physics responsible for inflation: we still lack a complete understanding of the high energy physics describing it. In fact, most likely the physics of inflation is far out of reach of terrestrial experiments, many orders of magnitude larger than the center-of-mass energy at the Large Hadron Collider (LHC). Thus, cosmological tests of inflation offer a unique opportunity to learn about ultra-high energy physics. We can do this by targeting observations which directly probe the dynamics of inflation. One route is to accurately measure the shape of the primordial power spectrum of scalar perturbations produced during the phase of accelerated expansion, which is directly related to the shape of the inflaton potential, and to constrain the amplitude of the corresponding stochastic gravitational-wave background, which is related instead to the energy-scale of inflation.

A complementary approach is offered by constraining – or exploring – how much the distribution of primordial density perturbations departs from Gaussian statistics and purely adiabatic fluctuations. Indeed, future large-scale structure surveys like Euclid can probe these features with an unprecedented accuracy, thus providing a way to test aspects of inflationary physics that are not easily accessible otherwise. Non-Gaussianity is a very sensitive probe of self-couplings and interactions between the fields generating the primordial perturbations, whereas the presence of isocurvature modes can teach us about the number of fields present during inflation and their role in reheating and generating the matter in the universe.

Furthermore, non-minimal scenarios or proposals even radically different from single-field inflation are still compatible with the data. In order to learn something about the physics of the early universe we need to rule out or confirm the conventional slow-roll scenario and possibly discriminate between non-conventional models. Non-Gaussianities and isocurvature perturbations currently represent the best tools that we have to accomplish this task. Any deviation from the conventional Gaussian and adiabatic initial perturbations would represent important breakthroughs in our understanding of the early universe. In this section we are going to review what we can learn by constraining the initial conditions with a large-scale structure survey like Euclid.

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