Cosmic Inflation, the most favoured scenario of the early Universe,
implies that all forms of matter and radiation observed today are the
outcome of quantum fluctuations occurring around the event horizon of a
exponentially fast accelerating space-time. Clearing the ground for
the incoming spatial and ground based cosmological observations,
**Pierre** and
**Christophe** have derived, at an
unprecedented level of precision, the shape of the expected power
spectra of both the quantum-generated gravitational waves and
curvature perturbations.

Cosmic Inflation is an hypothetical early phase of accelerated expansion that has occurred before the first billionth of a second of existence of our Universe. It provides a natural mechanism to explain the observed flatness of our Universe today and naturally solves the so-called horizon problems of the Big-Bang model.

In a spectacular way, the quantum fluctuations that are inherently
sourced during the inflationary era are exactly what is needed to
explain the origin of the cosmological perturbations: the **seeds of the
galaxies** of today.

These quantum fluctuations are deeply rooted in gravity and appear as both primordial gravitational waves \(h_{ij}\) and curvature perturbations \(\zeta\), with very peculiar correlation functions. In Ref. [1], we have pushed to third order the calculation of these correlation functions. They are completely determined by the Hubble parameter during inflation \(H(N)\) and its logarithmic derivatives, \(\epsilon_i(N) \equiv \mathrm{d}\ln H / \mathrm{d} \ln N\) (the so-called Hubble flow functions). Here \(N=\ln a\) is the logarithm of the scale factor \(a\).

Slow-roll inflation **predicts** the correlation functions to be
given by these spectra:

They are expanded around an observable wavenumber
\(k_*=0.05\,\mathrm{Mpc}^{-1}\) and readily testable with the
incoming cosmological observations from the
**Euclid** and
**LiteBird** space
telescopes, but also from the ground based
**CMB-S4** telescopes and **Simons
Observatory**. Are
we going to detect a non-vanishing \(\epsilon_{3*}\)?