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Towards a non-perturbative description of cosmological inflation

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Universitat de Barcelona. Departament de Física Quàntica i Astrofísica

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[eng] From the observation of the cosmic microwave background radiation (CMBR) we know that the universe is isotropic on scales larger than ∼ 75 Megaparsec to at least one part in 100000. The leading paradigm for explaining this observational data is a period of accelerated expansion in the earliest stages of the universe, called cosmological inflation.
Inflation also provides a causal mechanism for generating anisotropies on cosmological scales. These anisotropies result from the amplification of the unavoidable vacuum quantum excitations of the gravitational and matter fields due to the accelerated expansion. In particular, it is possible to explain the almost scale invariant power spectrum of the CMBR temperature map with a very simple inflationary regime called Slow Roll (SR) inflation.
Quantum fluctuations are usually assumed to be small enough such that they are well described with cosmological perturbation theory. However rare large quantum fluctuations can also be randomly generated during inflation. These large inhomogeneities can lead to the formation of Primordial Black Holes (PBH) and the probability of its generation is related with the amplitude of the power spectrum.
Both the amplitude of the power spectrum and the non-gaussianities measured at the CMBR are too small to generate a relevant amount of large fluctuations. To form enough PBH we need to exponentially enhance the amplitude of the power spectrum on scales which are not probed by the CMBR, for which a violation of SR is needed.
Although the growth of the power spectrum can be described at leading order in perturbation theory, the description of the tail of the Probability Distribution Function (PDF), where the large inhomogeneities are located, must be done in a non-perturbative way. We study two main approaches that aim to describe inflation in a non-perturbative way, the δN formalism and the stochastic approach.
Both approaches are based on gradient expansion, which assumes that the effect of quantum fluctuations whose characteristic wavelength is much larger than the Hubble radius is well described by an ensemble of locally homogeneous and isotropic patches, where spatial gradients are negligible, this approach is valid for any amplitude of local over-densities. In this thesis we show that spatial gradients and the momentum constraint of general relativity play an essential role when we use global coordinates, which are necessary to describe the statistical properties of the inhomogeneities by comparing different patches.
The δN formalism and the stochastic inflation represent consistent ways of providing initial conditions to gradient expansion. In the δN formalism, initial conditions on the field fluctuations are perturbatively given. The idea of stochastic inflation is fully non-perturbative. In this approach, the long-wavelength part of the field follows the evolution dictated by gradient expansion and the short-wavelength part act as a random noise continuously changing local trajectories such that the cumulative behaviour of small random kicks can induce non-perturbative effects in the local patch.
During this thesis we formulate for the first time a stochastic approach to inflation which includes spatial gradients and the momentum constraint and hence it is in principle able to describe the correct long-wavelength dynamics of inflationary inhomogeneities in a non-perturbative way. To test our results, we show that for any inflationary regime, the stochastic inflation so formulated, precisely reproduces the results of perturbative correlators in all regimes in which perturbation theory is supposed to work.
We also elucidate that, in order to take into account the whole non-perturbative dynamics of the local patches via stochastic inflation, the noises must be computed over the stochastically corrected local background rather than over the fiducial deterministic global background, as it is typically done in the literature.

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[spa] La cantidad de fluctuaciones suficientemente grandes para formar PBH depende de la amplitud del espectro de potencia (“power spectrum”) generado durante inflación. Tanto la amplitud del espectro de potencia como las no gaussianidades medidas en el fondo cósmico de microondas (CMB) son demasiado pequen˜as para generar una cantidad relevante de grandes fluctuaciones. Para formar suficientes PBH que pudieran representar, por ejemplo, una fracción significativa de materia oscura, necesitamos aumentar exponencialmente la amplitud del espectro de potencia en escalas que no son sondeadas por el CMB, para lo cual necesitamos una violación del típico régimen inflacionario de Slow-Roll (SR).
En la primera parte de esta tesis nos hemos dado cuenta de que las herramientas matemáticas existentes para estudiar la probabilidad de distribución de las inhomogeneidades escalares, aunque son bastante precisas durante inflación SR, no lo son tanto cuando abandonamos dicho régimen, lo cual es necesario para generar suficiente cantidad de PBH. Dichas herramientas matemáticas son el formalismo δN y la inflación estocástica.
Es por ello que hemos dedicado la segunda parte de esta tesis a estudiar las bases de ambos formalismos de tal forma que hemos conseguido formular por primera vez un formalismo estocástico para inflación que es capaz de describir de forma no perturbativa la evolución de las inhomogeneidades escalares durante cualquier régimen inflacionario y con una precisión arbitraria.
Aunque la aplicación práctica de este nuevo formalismo estocástico es difícil, hemos conseguido demostrar que dicho formalismo es capaz de reproducir correctamente los resultados conocidos sobre la probabilidad de distribución cuando las inhomogeneidades no son demasiado grandes, lo cual no era posible con las aproximaciones estocásticas a la inflación desarrolladas hasta la fecha.
Durante esta tesis hemos por tanto cumplido el objetivo principal de desarrollar una herramienta que permita describir de forma precisa la cola de la distribución de probabilidad de las inhomogeneidades escalares. Además, hemos dado el primer paso hacia su aplicación práctica, en lo que seguiremos trabajando los próximos años.

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Tesis Doctorals - Departament - Física Quàntica i Astrofísica

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Towards a non-perturbative description of cosmological inflation