Density Fluctuations in the Early Universe

Cosmic Microwave Background (CMB) radiation, copyright - ©ESA.

Introduction

The study of density fluctuations in the early universe is fundamental to understanding the origin of large-scale structures such as galaxies and galaxy clusters. These fluctuations were imprinted on the Cosmic Microwave Background (CMB) and can be traced back to tiny quantum fluctuations during inflation. These initial perturbations evolved under gravity and other forces, and their imprint on the CMB provides one of the most important sources of information about the universe’s composition and geometry.

The Basics of Density Fluctuations

Density fluctuations are small perturbations in the matter density of the early universe. The universe, at large scales, is considered homogeneous and isotropic, but small deviations from perfect homogeneity existed as tiny density perturbations. These are expressed as the fractional density contrast \( \delta(\vec{x}, t) \) at time \( t \) and position \( \vec{x} \), defined as: \[ \delta(\vec{x}, t) = \frac{\rho(\vec{x}, t) - \bar{\rho}(t)}{\bar{\rho}(t)} \] where:
  • \( \rho(\vec{x}, t) \) is the local density at position \( \vec{x} \) and time \( t \),
  • \( \bar{\rho}(t) \) is the mean density of the universe at time \( t \).

Fourier Decomposition of Density Fluctuations

Since density fluctuations are small and are observed across a large range of scales, it is useful to express these fluctuations in terms of Fourier modes: \[ \delta(\vec{x}, t) = \int \frac{d^3k}{(2\pi)^3} \delta(\vec{k}, t) e^{i \vec{k} \cdot \vec{x}} \] where \( \delta(\vec{k}, t) \) is the Fourier transform of \( \delta(\vec{x}, t) \), and \( \vec{k} \) is the wavevector corresponding to a mode of fluctuation. The magnitude \( k = |\vec{k}| \) corresponds to a scale in space, with smaller \( k \) representing large-scale fluctuations and larger \( k \) representing small-scale fluctuations. The statistical properties of the density fluctuations are described by the power spectrum, \( P(k) \), which relates the amplitude of the fluctuations on different scales: \[ \langle \delta(\vec{k}, t) \delta^*(\vec{k}', t) \rangle = (2\pi)^3 P(k) \delta^3(\vec{k} - \vec{k}') \] where \( P(k) \) is the power spectrum, which quantifies the amplitude of fluctuations as a function of scale \( k \).

Gravitational Collapse and the Jeans Instability

For reference, you can look at: Notes on Jeans Instability and Gravitational Collapse.
Before recombination, the universe was composed of a tightly coupled fluid of photons, baryons, and dark matter. The evolution of density perturbations in this plasma was governed by the competition between gravitational collapse and pressure support from the photons. The critical scale for gravitational collapse is given by the Jeans length. To derive the Jeans length, we begin with the equation for the perturbation in density in the Newtonian approximation: \[ \frac{\partial^2 \delta}{\partial t^2} + 2H \frac{\partial \delta}{\partial t} = c_s^2 \nabla^2 \delta + 4\pi G \bar{\rho} \delta \] where:
  • \( H \) is the Hubble parameter,
  • \( c_s \) is the sound speed in the photon-baryon fluid,
  • \( G \) is Newton’s gravitational constant.
In Fourier space, the equation simplifies to: \[ \ddot{\delta}_k + 2H \dot{\delta}_k = -c_s^2 k^2 \delta_k + 4\pi G \bar{\rho} \delta_k \] The solution to this equation depends on the size of \( k \). For perturbations smaller than the Jeans scale (where pressure dominates), sound waves will oscillate. For perturbations larger than the Jeans scale, gravity dominates, and the perturbations will grow exponentially, leading to the formation of structure. The Jeans wavenumber \( k_J \) is given by: \[ k_J = \sqrt{\frac{4 \pi G \bar{\rho}}{c_s^2}} \] The corresponding Jeans length \( \lambda_J \) is: \[ \lambda_J = \frac{2\pi}{k_J} \]

Evolution of Perturbations in Different Epochs

The evolution of perturbations is different depending on the dominant form of energy in the universe:
  1. Radiation-dominated era: During this epoch, radiation pressure resists the collapse of baryonic matter. Perturbations in the photon-baryon plasma lead to acoustic oscillations, which are imprinted on the CMB as peaks in the power spectrum.
  2. Matter-dominated era: After recombination, photons decouple from baryons, and pressure no longer supports the perturbations. Perturbations in dark matter and baryons grow under gravity, eventually leading to the formation of galaxies and clusters.
Evolution in the Radiation-Dominated Era
In the radiation-dominated era, the tight coupling of photons and baryons leads to acoustic oscillations. The perturbations in density can be described by coupled equations for the density contrast in baryons \( \delta_b \) and photons \( \delta_\gamma \): \[ \ddot{\delta}_b + \mathcal{H} \dot{\delta}_b = \frac{4}{3} \nabla^2 \delta_\gamma \] \[ \ddot{\delta}_\gamma + \mathcal{H} \dot{\delta}_\gamma = \frac{1}{3} \nabla^2 \delta_\gamma \] Here, \( \mathcal{H} \) is the conformal Hubble parameter, which accounts for the expansion of the universe. The baryon and photon fluctuations are tightly coupled through Thomson scattering, leading to oscillations with a characteristic frequency: \[ \omega = c_s k \] where \( c_s \) is the sound speed in the plasma, which depends on the baryon-to-photon ratio. These oscillations lead to compressions and rarefactions, which manifest as peaks and troughs in the CMB angular power spectrum.
Evolution in the Matter-Dominated Era
After recombination, the photons decouple from baryons, and the baryons are no longer supported by radiation pressure. The growth of perturbations in dark matter and baryons during the matter-dominated era is governed by a simpler equation: \[ \ddot{\delta} + 2H \dot{\delta} = 4\pi G \bar{\rho}_m \delta \] In this era, perturbations grow approximately as \( \delta(t) \propto t^{2/3} \) in an Einstein-de Sitter universe. This linear growth of perturbations eventually leads to the formation of non-linear structures such as galaxies.

Temperature Anisotropies and the Sachs-Wolfe Effect

The Sachs-Wolfe effect describes how gravitational potentials affect the temperature of CMB photons. Photons climbing out of gravitational wells lose energy, resulting in a redshift and a decrease in temperature. Conversely, photons falling into gravitational wells are blue-shifted, appearing hotter. For large angular scales, the temperature fluctuations observed in the CMB are related to the gravitational potential \( \Phi \) by the Sachs-Wolfe relation: \[ \frac{\Delta T}{T} = \frac{1}{3} \Phi \] This relation holds on scales larger than the horizon at the time of last scattering.

Angular Power Spectrum of the CMB

The temperature anisotropies observed in the CMB can be expanded in spherical harmonics: \[ \frac{\Delta T}{T}(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n}) \] The angular power spectrum \( C_\ell \) is defined as the variance of the spherical harmonic coefficients: \[ C_\ell = \frac{1}{2\ell + 1} \sum_{m=-\ell}^{\ell} |a_{\ell m}|^2 \] The power spectrum encodes the amplitude of temperature fluctuations as a function of angular scale, with peaks corresponding to the acoustic oscillations in the photon-baryon fluid.

References

Some other interesting things to know:

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