Linearity

Homogeneity

Gaussianity

Linearity, Gaussianity and homogeneity imply each other

Credit: Bjoern Malte Schaefer

Fluid equations linearised |delta| << 1

Position independent Growth

Central limit theorem

 Large Deviations statistics of the cosmic log-density field

Cora Uhlemann

 

The evolution of the density field

Initial

Gaussian density fluctuations

First two moments

Requires infinite number of moments

Gravity

Non-Gaussian

Barrier at 0!

Credit: S. Codis+16

\delta ( \vec{x},a) = D_+ (a) \delta_0(\vec{x})
\hat{\delta} ( \vec{k},a) = \int d^3 k^\prime D_+ (a, \vec{k} - \vec{k^\prime}) \delta_0(\vec{k^\prime})

Mode Coupling breaks the Central Limit theorem -> correlation produces a non-Gaussian field

Homogeneity

\delta ( \vec{x},a) = D_+ (a, \vec{x}) \delta_0(\vec{x})

Current approaches to describe the non-Gaussian field

  • Perturbation theory : Needs perturbations to be small everywhere

 

  • N-body simulations, solve the fully non-linear equations

Can we derive an analytical expression for the fully non-linear field  from first principles?

Statistics of density in concentric spheres

Only assumption : the variance of the field must be small inside the sphere

Large Deviations Principle + Spherical Collapse

Uhlemann '16

What is the most likely initial configuration the final density originated from?

Assumption: Spherical symmetry ensures the most likely path is given by spherical collapse dynamics

\int D[\tau]

Credit: S. Codis

What is the most likely one?

Valageas 2002

Large Deviations Theory

\lim_{\epsilon^2\to0} \epsilon^2 \log P_\epsilon (\tau) = - \psi (\tau)

Theory around probability families that fulfill:

P(\tau) \propto e^{-\psi(\tau)}
\lim_{\sigma^2\to0} \sigma^2 \log P(\tau) = - \psi (\tau)

Driving parameter

Rate Function

True for a Gaussian PDF

CONTRACTION PRINCIPLE If tau follows LDP:

\rho = F[\tau]

Family of (non-linear) mappings from tau to rho

\lim_{\sigma^2\to0} \sigma^2 \log P(\rho) = - \psi (\zeta ^{-1} (\rho))

Most likely mapping from tau to rho

Large deviations follow the most likely of the unlikely paths

\psi (\tau) = \frac{\tau^2}{\sigma^2}
\tau
\tau_\star = \zeta^{-1} (\rho _\star)
\psi_\star
P(\rho_\star) \propto e^{-\psi_\star}

i) Initial Gaussian random field

\sigma^2(R) = \frac{1}{(2\pi)^3} \int d^3 k P_{linear} (k) W^2(kR)

ii) Map the initial PDF into the final one assuming the most likely path  given by spherical collapse dynamics

\rho = \zeta_\nu (\delta)

iii) Final PDF depends on:

P(\rho | \nu, P_{linear}, \sigma_{NL} (R,z))

Modifications of gravity

Growth of structure

Primordial non-Gaussianity

Main result

log(P(\rho))
  • Fully analytical predictions for the density PDF in a sphere accurate at the percent level in the non-linear regime (data not used at the moment).
  • Use the cosmological dependency to test gravity in the non linear regime.
  • For density in spheres, the most likely dynamics is the one respecting the symmetry.

Summary