Gravitational Primordial Perturbation Theory

Cosmology II, Course note book

Content

  1. Introduction to Cosmological Perturbation Theory
  2. Background Cosmology: The Friedmann-Robertson-Walker (FRW) Universe
  3. Einstein's Field Equations for the FRW Universe
  4. Introducing Perturbations: Scalar, Vector, and Tensor Modes
  5. Perturbed FRW Metric
  6. Gauge Freedom and Gauge Choices
  7. Einstein's Equations with Perturbations
  8. Deriving the Perturbation Equations
  9. Einstein's Equations for Scalar Perturbations
  10. Evolution of Scalar Perturbations
  11. Growth of Density Perturbations
  12. Acoustic Oscillations
  13. Tensor Perturbations and Gravitational Waves
  14. Power Spectrum of Primordial Perturbations
  15. Another way to derive these equations (Conformal system)
  16. Reference

Introduction to Cosmological Perturbation Theory

The universe, on large scales, is remarkably homogeneous and isotropic, a principle encapsulated in the Cosmological Principle. However, this perfect uniformity is perturbed by small inhomogeneities—tiny fluctuations in density and gravitational potential—that seed the formation of all cosmic structures. Cosmological perturbation theory is the framework that allows us to study these deviations from perfect homogeneity and isotropy within the context of General Relativity.
  • Importance of Perturbation Theory:
    1. Structure Formation: Explains how initial fluctuations grow under gravity to form galaxies, clusters, and larger structures.
    2. CMB Anisotropies: Describes the temperature fluctuations observed in the Cosmic Microwave Background (CMB), providing a snapshot of the early universe.
    3. Gravitational Waves: Predicts tensor perturbations that manifest as gravitational waves, offering insights into inflation and other high-energy processes in the early universe.
  • Linear vs. Non-Linear Perturbations:
    1. Linear Perturbation Theory: Assumes that perturbations are small (\( \delta \ll 1 \)) and can be treated as linear deviations from the background. This simplifies the equations and allows for analytical solutions.
    2. Non-Linear Perturbation Theory: Deals with larger perturbations where linear approximations break down, requiring numerical methods for solutions. Essential for understanding the late-time universe where structures have become non-linear.

Background Cosmology: The Friedmann-Robertson-Walker (FRW) Universe

Before introducing perturbations, it's essential to establish the background cosmological model. The universe on large scales is well-described by the Friedmann-Robertson-Walker (FRW) metric, which embodies homogeneity and isotropy.

The FRW metric encapsulates the Cosmological Principle, describing a universe that is the same in all directions (isotropic) and at all locations (homogeneous). It is given by:

\[ ds^2 = -c^2 dt^2 + a^2(t) \left[ \frac{dr^2}{1 - kr^2} + r^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right) \right] \] where:
  • \( ds \) is the spacetime interval,
  • \( c \) is the speed of light,
  • \( t \) is cosmic time,
  • \( r, \theta, \phi \) are comoving spherical coordinates,
  • \( a(t) \) is the scale factor, describing the expansion of the universe,
  • \( k \) determines the spatial curvature:
    • \( k = 0 \) for flat space,
    • \( k > 0 \) for positive curvature (closed universe),
    • \( k < 0 \) for negative curvature (open universe).

FLRW Universe in confromally flat space:

The FRW metric in conformal coordinates \((\eta, \vec{x})\) is given by: \[ ds^2 = a^2(\eta) \left[ -d\eta^2 + \gamma_{ij} dx^i dx^j \right] \] where:
  • \( a(\eta) \) is the scale factor, a function of conformal time \(\eta\).
  • \( \gamma_{ij} \) is the spatial metric, typically flat (\(\mathbb{R}^3\)), open, or closed.
For simplicity, we assume a spatially flat universe (\(k = 0\)): \[ \gamma_{ij} = \delta_{ij} \]

Einstein's Field Equations for the FRW Universe

The dynamics of the FRW universe are governed by Einstein's Field Equations: \[ G_{\mu\nu} = 8\pi G T_{\mu\nu} \] where:
  • \( G_{\mu\nu} \) is the Einstein tensor.
  • \( T_{\mu\nu} \) is the stress-energy tensor.
  • \( G \) is Newton's gravitational constant.
In the context of the FRW metric, Einstein's field equations reduce to the Friedmann equations:
  1. First Friedmann Equation (Expansion Rate): \[ \left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} \]
  2. Second Friedmann Equation (Acceleration): \[ \frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3} \] where:
    • \( \dot{a} = \frac{da}{dt} \),
    • \( \ddot{a} = \frac{d^2a}{dt^2} \),
    • \( G \) is Newton's gravitational constant,
    • \( \rho \) is the energy density,
    • \( p \) is the pressure,
    • \( \Lambda \) is the cosmological constant.
Components of the Universe
The energy density \( \rho \) comprises various components:
  • Radiation (\( \rho_r \)): Includes photons and relativistic particles.
  • Baryonic Matter (\( \rho_b \)): Ordinary matter consisting of protons and neutrons.
  • Dark Matter (\( \rho_{dm} \)): Non-luminous matter that interacts gravitationally.
  • Dark Energy (\( \rho_\Lambda \)): Associated with the cosmological constant \( \Lambda \).
Each component evolves differently with the scale factor \( a(t) \): \[ \rho_r \propto a^{-4}, \quad \rho_b \propto a^{-3}, \quad \rho_{dm} \propto a^{-3}, \quad \rho_\Lambda = \text{constant} \]
In the comoving coordinate system, the Friedmann equations are: \[ \left( \frac{a'}{a} \right)^2 = \frac{8\pi G}{3} a^2 \bar{\rho} \] \[ \frac{a''}{a} = -\frac{4\pi G}{3} a^2 (\bar{\rho} + 3\bar{p}) \] where primes denote derivatives with respect to conformal time \(\eta\), and \(\bar{\rho}\) and \(\bar{p}\) are the background energy density and pressure, respectively.

Introducing Perturbations: Scalar, Vector, and Tensor Modes

In a perfectly homogeneous and isotropic universe, all spatial points are equivalent. However, real cosmological observations reveal deviations from this idealization. To study these deviations, we introduce perturbations to the FRW metric and the energy-momentum tensor.
Types of perturbations:
Perturbations can be classified based on their transformation properties under spatial rotations:
  1. Scalar Perturbations:
    • Represent density fluctuations.
    • Lead to the formation of structures like galaxies and clusters.
    • Governed by scalar potentials \( \Phi \) and \( \Psi \).
  2. Vector Perturbations:
    • Represent vortical (rotational) motions.
    • Typically decay with the expansion of the universe.
    • Not significant in the linear regime of structure formation.
  3. Tensor Perturbations:
    • Represent gravitational waves—ripples in spacetime.
    • Produced during inflation.
    • Decouple from matter and radiation, propagating freely.

Perturbed FRW Metric

In the Newtonian gauge (also known as the conformal Newtonian gauge), the perturbed FRW metric including scalar, vector, and tensor perturbations is written as: \[ ds^2 = -(1 + 2\Phi) c^2 dt^2 + a^2(t) \left[ (1 - 2\Psi) \delta_{ij} + h_{ij} \right] dx^i dx^j + 2a(t) B_i dx^i c dt + a^2(t) h_{ij} dx^i dx^j \] where:
  • \( \Phi \) and \( \Psi \) are scalar potentials.
  • \( B_i \) represents vector perturbations (transverse: \( \partial^i B_i = 0 \)).
  • \( h_{ij} \) represents tensor perturbations (transverse and traceless: \( \partial^i h_{ij} = 0 \), \( h^i_i = 0 \)).
Note: In the absence of anisotropic stress (e.g., for a universe dominated by perfect fluids), \( \Phi = \Psi \).
In the Newtonian (or conformal Newtonian) gauge, the perturbed metric for scalar perturbations is: \[ ds^2 = a^2(\eta) \left[ -(1 + 2\Phi) d\eta^2 + (1 - 2\Psi) \delta_{ij} dx^i dx^j \right] \] where:
  • \( \Phi(\eta, \vec{x}) \) is the gravitational potential perturbation.
  • \( \Psi(\eta, \vec{x}) \) is the spatial curvature perturbation.
For vector and tensor perturbations, additional terms are introduced, but for gravitational primordial perturbations related to density fluctuations, scalar perturbations are of primary interest.

Gauge Freedom and Gauge Choices

When introducing perturbations, one must account for the gauge freedom—the freedom to choose coordinate systems in General Relativity. Different gauge choices can simplify the analysis of perturbations.
  1. Gauge Transformations: A gauge transformation involves shifting the coordinates by a small amount: \[ x^\mu \rightarrow x^\mu + \delta x^\mu \] where \( \delta x^\mu \) is a small perturbation. This affects the form of the metric and the perturbation variables.
  2. Common Gauge Choices:
    • Newtonian (Conformal Newtonian) Gauge: Eliminates vector and tensor perturbations in scalar perturbations. Simplifies the metric to: \[ ds^2 = -(1 + 2\Phi) c^2 dt^2 + a^2(t) (1 - 2\Psi) \delta_{ij} dx^i dx^j \] Useful for intuitive interpretations, akin to Newtonian gravity.
    • Synchronous Gauge: Sets \( \Phi = 0 \) and \( B_i = 0 \). In this case, the metric becomes: \[ ds^2 = -c^2 dt^2 + a^2(t) \left[ (\delta_{ij} + h_{ij}) dx^i dx^j \right] \] More complex for scalar perturbations but useful in certain analytical and numerical studies.
    • Comoving Gauge: Chooses coordinates such that the observer is moving with the fluid. Eliminates certain perturbation variables, simplifying the analysis of matter perturbations.
  3. Gauge-Invariant Variables: To avoid complications due to gauge freedom, it's often useful to construct gauge-invariant variables that remain unchanged under gauge transformations. Examples include:
    • Bardeen Potentials: Combinations of \( \Phi \) and \( \Psi \) that are gauge-invariant.
    • Comoving Density Contrast: Perturbation in density as measured in the comoving frame.

Einstein's Equations with Perturbations

To derive the perturbation equations, we need to apply Einstein's field equations to the perturbed metric and energy-momentum tensor.
Einstein's Field Equations
Einstein's equations relate the geometry of spacetime to its energy content: \[ G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \] where:
  • \( G_{\mu\nu} \) is the Einstein tensor,
  • \( T_{\mu\nu} \) is the energy-momentum tensor.
Perturbed Einstein Tensor
For the perturbed FRW metric, the Einstein tensor \( G_{\mu\nu} \) can be expanded to first order in perturbations: \[ G_{\mu\nu} = G_{\mu\nu}^{(0)} + G_{\mu\nu}^{(1)} + \mathcal{O}(\text{perturbations}^2) \] where:
  • \( G_{\mu\nu}^{(0)} \) corresponds to the background FRW universe,
  • \( G_{\mu\nu}^{(1)} \) contains the linear perturbations.
Perturbed Energy-Momentum Tensor
Similarly, the energy-momentum tensor is expanded as: \[ T_{\mu\nu} = T_{\mu\nu}^{(0)} + T_{\mu\nu}^{(1)} + \mathcal{O}(\text{perturbations}^2) \] where:
  • \( T_{\mu\nu}^{(0)} \) is the background energy-momentum tensor (perfect fluid),
  • \( T_{\mu\nu}^{(1)} \) contains the perturbations (density fluctuations, velocity perturbations, etc.).
Linearized Einstein Equations
Equating the perturbed Einstein tensor to the perturbed energy-momentum tensor yields the linearized Einstein equations: \[ G_{\mu\nu}^{(1)} = \frac{8\pi G}{c^4} T_{\mu\nu}^{(1)} \] These equations govern the evolution of perturbations in the early universe.

Deriving the Perturbation Equations

To derive the perturbation equations, we will focus on scalar perturbations, which are most relevant for density fluctuations and the formation of large-scale structures.
Scalar Perturbations in the Newtonian Gauge
In the Newtonian gauge, scalar perturbations of the FRW metric are given by: \[ ds^2 = -(1 + 2\Phi) c^2 dt^2 + a^2(t) (1 - 2\Psi) \delta_{ij} dx^i dx^j \] Assuming no anisotropic stress (\( \Phi = \Psi \)), the metric simplifies further.
Energy-Momentum Tensor for a Perfect Fluid
For a perfect fluid, the energy-momentum tensor is: \[ T_{\mu\nu} = (\rho + \frac{p}{c^2}) u_\mu u_\nu + p g_{\mu\nu} \] where:
  • \( \rho \) is the energy density,
  • \( p \) is the pressure,
  • \( u_\mu \) is the four-velocity of the fluid.
Matter Content: Perfect Fluids and Other Components: To close the system of equations, we need to specify the matter content of the universe. Common components include:
  1. Cold Dark Matter (CDM): Pressureless, non-relativistic.
  2. Baryonic Matter: Ordinary matter, behaves similarly to CDM at large scales but interacts electromagnetically.
  3. Radiation: Includes photons and relativistic particles.
  4. Dark Energy: Causes the accelerated expansion of the universe.
For primordial perturbations relevant to structure formation and the CMB, CDM and baryonic matter are most pertinent.

Perfect Fluid Description: For a perfect fluid, the stress-energy tensor is fully described by its density \( \rho \), pressure \( p \), and four-velocity \( u^\mu \). Perturbations introduce:

  • Density Perturbations: \( \delta \rho \)
  • Pressure Perturbations: \( \delta p \)
  • Velocity Perturbations: \( v \)
  • Anisotropic Stress: \( \pi^i_j \) (usually negligible for scalar modes in the absence of free-streaming particles)
Perturbing the Energy-Momentum Tensor
Introducing perturbations, the energy density, pressure, and four-velocity become: \[ \rho = \bar{\rho}(t) + \delta\rho(\vec{x}, t) \] \[ p = \bar{p}(t) + \delta p(\vec{x}, t) \] \[ u^\mu = \left( \frac{1 - \Phi}{c}, v^i(\vec{x}, t) \right) \] where:
  • \( \delta\rho \) and \( \delta p \) are perturbations in density and pressure,
  • \( v^i \) is the peculiar velocity perturbation.

Einstein's Equations for Scalar Perturbations

Applying the linearized Einstein equations \( G_{\mu\nu}^{(1)} = \frac{8\pi G}{c^4} T_{\mu\nu}^{(1)} \), we obtain a set of coupled differential equations governing the evolution of \( \Phi \), \( \delta\rho \), \( \delta p \), and \( v^i \).
  1. \( G_{00} \) Component: \[ \nabla^2 \Phi - 3 \mathcal{H} (\dot{\Phi} + \mathcal{H} \Phi) = \frac{4\pi G}{c^2} a^2 \delta\rho \]
  2. \( G_{0i} \) Component: \[ \partial_i (\dot{\Phi} + \mathcal{H} \Phi) = \frac{4\pi G}{c^2} a^2 (\rho + p) v_i \]
  3. \( G_{ij} \) Component: \[ \ddot{\Phi} + \mathcal{H} \dot{\Phi} + \left( 2\dot{\mathcal{H}} + \mathcal{H}^2 \right) \Phi = \frac{4\pi G}{c^4} a^2 \delta p \]
Conservation Equations:
In addition to Einstein's equations, the conservation of energy and momentum \( \nabla_\mu T^{\mu\nu} = 0 \) provides further equations:
  1. Continuity Equation: \[ \dot{\delta} + (1 + w) \left( \nabla \cdot v - 3 \dot{\Phi} \right) + 3 \mathcal{H} \left( \frac{\delta p}{\delta \rho} - w \right) \delta = 0 \]
  2. Euler Equation: \[ \dot{v} + \mathcal{H} v + \frac{\nabla \delta p}{\delta \rho + p} + \nabla \Phi = 0 \] where:
    • \( \delta = \frac{\delta\rho}{\bar{\rho}} \) is the density contrast,
    • \( w = \frac{\bar{p}}{\bar{\rho} c^2} \) is the equation of state parameter.
Note: \( \mathcal{H} = \frac{\dot{a}}{a} \) is the conformal Hubble parameter, and dots denote derivatives with respect to conformal time \( \eta \), where \( d\eta = \frac{dt}{a(t)} \).

Evolution of Scalar Perturbations

With the set of equations derived, we can now analyze the evolution of scalar perturbations in the early universe.
Fourier Space Representation: To simplify the analysis, we perform a Fourier transform of the perturbation variables. For any scalar quantity \( f(\vec{x}, t) \): \[ f(\vec{x}, t) = \int \frac{d^3k}{(2\pi)^{3/2}} f(\vec{k}, t) e^{i\vec{k} \cdot \vec{x}} \] Applying this to the perturbation equations converts spatial derivatives into algebraic terms involving the wavenumber \( k \).
  1. Poisson Equation in Fourier Space The \( G_{00} \) component becomes: \[ k^2 \Phi + 3 \mathcal{H} (\dot{\Phi} + \mathcal{H} \Phi) = \frac{4\pi G}{c^2} a^2 \bar{\rho} \delta \]
  2. Relation Between Density Contrast and Potential: Assuming adiabatic perturbations (where \( \delta p = c_s^2 \delta \rho \)), and using the equation of state \( w = \frac{c_s^2}{c^2} \), we can relate \( \delta \) and \( \Phi \).
  3. Second-Order Differential Equation for \( \Phi \): Combining the equations, we derive a second-order differential equation for the gravitational potential \( \Phi \): \[ \ddot{\Phi} + (4 + 3w)\mathcal{H} \dot{\Phi} + \left( 2\dot{\mathcal{H}} + (1 + 3w)\mathcal{H}^2 \right) \Phi = 0 \] This equation describes how the gravitational potential evolves over time in different cosmological epochs.
Solutions in Different Eras
  1. Radiation-Dominated Era (\( w = \frac{1}{3} \)):
    • Scale factor: \( a(\eta) \propto \eta \)
    • Solution: \( \Phi = \text{constant} \) and \( \Phi \propto \eta^{-2} \)
  2. Matter-Dominated Era (\( w = 0 \)):
    • Scale factor:\( a(\eta) \propto \eta^2 \)
    • Solution: \( \Phi = \text{constant} \)
  3. Dark Energy-Dominated Era (\( w \approx -1 \)):
    • Gravitational potentials decay over time due to accelerated expansion.

Growth of Density Perturbations

The growth of the density contrast \( \delta \) is governed by the interplay between gravitational attraction and pressure support. In the matter-dominated era, perturbations grow as: \[ \delta \propto a(t) \propto \eta^2 \] In the radiation-dominated era, growth is suppressed due to radiation pressure.

Acoustic Oscillations

Before recombination, baryons and photons were tightly coupled, forming a photon-baryon plasma. Perturbations in this plasma undergo acoustic oscillations due to the competition between gravitational infall and radiation pressure. These oscillations leave characteristic imprints in the CMB power spectrum, known as acoustic peaks. The equation governing these oscillations in Fourier space is: \[ \ddot{\delta}_b + \mathcal{H} \dot{\delta}_b + c_s^2 k^2 \delta_b = -k^2 \Phi \] where:
  • \( \delta_b \) is the baryon density contrast,
  • \( c_s \) is the sound speed in the photon-baryon fluid.

Tensor Perturbations and Gravitational Waves

While scalar perturbations are responsible for density fluctuations, tensor perturbations correspond to gravitational waves—ripples in the fabric of spacetime itself.
  1. Tensor Perturbations in the FRW Metric: Tensor perturbations are introduced as transverse and traceless perturbations to the spatial part of the metric: \[ ds^2 = -c^2 dt^2 + a^2(t) \left[ \delta_{ij} + h_{ij}(\vec{x}, t) \right] dx^i dx^j \] where \( h_{ij} \) satisfies:
    • \( \partial^i h_{ij} = 0 \),
    • \( h^i_i = 0 \).
  2. Einstein's Equations for Tensor Perturbations: For tensor perturbations, Einstein's equations reduce to: \[ \ddot{h}_{ij} + 2\mathcal{H} \dot{h}_{ij} + k^2 h_{ij} = 0 \] This is a wave equation with damping due to the expansion of the universe.
  3. Solutions for Gravitational Waves: The general solution to the tensor perturbation equation depends on the epoch:
    • Radiation-Dominated Era (\( a(\eta) \propto \eta \)): \[ h_{ij}(\eta, \vec{k}) = \frac{A_{ij}}{k} \sin(k\eta) + \frac{B_{ij}}{k} \cos(k\eta) \]
    • Matter-Dominated Era (\( a(\eta) \propto \eta^2 \)): \[ h_{ij}(\eta, \vec{k}) = \frac{A_{ij}}{k} \sin(k\eta) + \frac{B_{ij}}{k} \cos(k\eta) \]
    In both eras, gravitational waves oscillate with frequency \( k \) and are damped by the expansion.
  4. Primordial Gravitational Waves: Inflation predicts the generation of a stochastic background of gravitational waves. These primordial gravitational waves can leave imprints on the CMB polarization, particularly in the form of B-modes, providing a potential window into the physics of inflation.

Power Spectrum of Primordial Perturbations

The power spectrum quantifies the distribution of perturbation amplitudes across different scales and is a crucial observable in cosmology.
  1. Definition of the Power Spectrum: For scalar perturbations, the power spectrum \( P(k) \) is defined via the Fourier transform of the two-point correlation function: \[ \langle \delta(\vec{k}) \delta^*(\vec{k}') \rangle = (2\pi)^3 \delta^3(\vec{k} - \vec{k}') P(k) \] where \( \delta(\vec{k}) \) is the Fourier transform of the density contrast \( \delta(\vec{x}) \).
  2. Primordial Power Spectrum from Inflation: Inflation generates nearly scale-invariant primordial perturbations. The primordial power spectrum is often parameterized as: \[ P_{\text{primordial}}(k) = A_s \left( \frac{k}{k_0} \right)^{n_s - 1} \] where:
    • \( A_s \) is the amplitude of the scalar perturbations,
    • \( n_s \) is the scalar spectral index,
    • \( k_0 \) is a pivot scale.
  3. Transfer Function: The transfer function \( T(k) \) encodes the evolution of perturbations from the early universe to the present day. It accounts for processes like horizon crossing, radiation pressure, and matter-radiation equality. The evolved power spectrum is given by: \[ P(k, t) = P_{\text{primordial}}(k) \, T^2(k) \, D^2(t) \] where \( D(t) \) is the growth factor.
  4. Observational Constraints: Observations of the CMB, large-scale structure, and galaxy surveys constrain the power spectrum parameters \( A_s \) and \( n_s \), providing insights into the inflationary epoch and the composition of the universe.
  5. Tensor Power Spectrum: Similarly, tensor perturbations have their own power spectrum \( P_T(k) \), related to the amplitude of primordial gravitational waves. The ratio of tensor to scalar power, \( r \), is a key observable for probing inflationary models.

Another way to derive these equations:

For a perfect fluid, in the conformaly flat space (for this, see the metric given above), the stress-energy tensor \( T_{\mu\nu} \) can be perturbed as: \[ T_{\mu\nu} = (\bar{\rho} + \bar{p}) u_\mu u_\nu + \bar{p} g_{\mu\nu} + \delta T_{\mu\nu} \] where:
  • \( \delta T_{\mu\nu} \) represents the perturbations.
  • \( u^\mu \) is the four-velocity of the fluid.
In the Newtonian gauge, the perturbations are characterized by: \[ \delta T^0_0 = -\delta \rho \] \[ \delta T^0_i = (\bar{\rho} + \bar{p}) \partial_i v \] \[ \delta T^i_j = \delta p \delta^i_j + \pi^i_j \] where:
  • \( \delta \rho \) is the density perturbation.
  • \( \delta p \) is the pressure perturbation.
  • \( v \) is the velocity potential.
  • \( \pi^i_j \) represents anisotropic stress (usually negligible for scalar perturbations in the absence of free-streaming particles).
Perturbed Einstein Equations
\[ G_{\mu\nu}^{(0)} + \delta G_{\mu\nu} = 8\pi G (\bar{T}_{\mu\nu} + \delta T_{\mu\nu}) \] Subtracting the background equations \( G_{\mu\nu}^{(0)} = 8\pi G \bar{T}_{\mu\nu} \), we obtain the linearized Einstein Equations: \[ \delta G_{\mu\nu} = 8\pi G \delta T_{\mu\nu} \]
  1. 00-component: \[ \delta G^0_0 = 2 \nabla^2 \Psi - 6 \mathcal{H} (\Psi' + \mathcal{H} \Phi) \]
  2. 0i-component: \[ \delta G^0_i = 2 \partial_i (\Psi' + \mathcal{H} \Phi) \]
  3. ij-component: \[ \delta G^i_j = \left[ \nabla^2 (\Psi - \Phi) + 2 \mathcal{H} (\Psi' + \mathcal{H} \Phi) + 2 \Psi'' + 4 \mathcal{H} \Psi' + 2 (\mathcal{H}' + \mathcal{H}^2) \Phi \right] \delta^i_j - (\Psi - \Phi)_{,i,j} \]
where \( \mathcal{H} = a' / a \) is the conformal Hubble parameter, and primes denote derivatives with respect to conformal time \( \eta \).

With the perturbed Einstein equations and the matter content specified, we proceed to derive the equations governing the evolution of perturbations.

  1. 00-component: Poisson Equation: From \( \delta G^0_0 = 8\pi G \delta T^0_0 \): \[ 2 \nabla^2 \Psi - 6 \mathcal{H} (\Psi' + \mathcal{H} \Phi) = -8\pi G a^2 \delta \rho \] Rearranging: \[ \nabla^2 \Psi = 4\pi G a^2 \delta \rho + 3 \mathcal{H} (\Psi' + \mathcal{H} \Phi) \] For sub-horizon scales (\( k \gg \mathcal{H} \)), the second term on the right is negligible, yielding the Poisson equation: \[ k^2 \Psi = 4\pi G a^2 \delta \rho \]
  2. 0i-component: Momentum Constraint: From \( \delta G^0_i = 8\pi G \delta T^0_i \): \[ 2 \partial_i (\Psi' + \mathcal{H} \Phi) = 8\pi G (\bar{\rho} + \bar{p}) \partial_i v \] Dividing both sides by \( 2 \partial_i \) (assuming \( \partial_i \neq 0 \)): \[ \Psi' + \mathcal{H} \Phi = 4\pi G (\bar{\rho} + \bar{p}) v \]
  3. ij-component: Evolution of Metric Perturbations: From \( \delta G^i_j = 8\pi G \delta T^i_j \): \[ \left[ \nabla^2 (\Psi - \Phi) + 2 \mathcal{H} (\Psi' + \mathcal{H} \Phi) + 2 \Psi'' + 4 \mathcal{H} \Psi' + 2 (\mathcal{H}' + \mathcal{H}^2) \Phi \right] \delta^i_j - (\Psi - \Phi)_{,i,j} = 8\pi G a^2 \left( \delta p \delta^i_j + \pi^i_j \right) \] For scalar perturbations without anisotropic stress (\( \pi^i_j = 0 \)): \[ \Psi = \Phi \] Thus, the equation simplifies to: \[ 2 \Psi'' + 4 \mathcal{H} \Psi' + 2 (\mathcal{H}' + \mathcal{H}^2) \Psi = 8\pi G a^2 \delta p \] Dividing by \( 2 \): \[ \Psi'' + 2 \mathcal{H} \Psi' + (\mathcal{H}' + \mathcal{H}^2) \Psi = 4\pi G a^2 \delta p \]
  4. Continuity and Euler Equations: To fully describe the dynamics, we also need the continuity and Euler equations from the conservation of the stress-energy tensor \( \nabla_\mu T^{\mu\nu} = 0 \).
    • Continuity Equation: From \( \nabla_\mu T^{\mu 0} = 0 \): \[ \delta \rho' + 3 \mathcal{H} (\delta \rho + \delta p) = - (\bar{\rho} + \bar{p}) \nabla^2 v + 3 \bar{p}' \Phi \]
    • Euler Equation: From \( \nabla_\mu T^{\mu i} = 0 \): \[ (\bar{\rho} + \bar{p}) v' + 4 \mathcal{H} (\bar{\rho} + \bar{p}) v = \delta p + (\bar{\rho} + \bar{p}) \Phi \]
These equations, combined with the perturbed Einstein equations, form a closed system governing the evolution of cosmological perturbations.

Evolution of Perturbations:

While the full treatment requires General Relativity, for certain regimes, a Newtonian approximation suffices. However, the GR approach is necessary for accurately describing perturbations on scales comparable to the horizon or during radiation domination.
  1. Newtonian Approximation: In the Newtonian limit (\( c \rightarrow \infty \)), and for sub-horizon scales (\( k \gg \mathcal{H} \)), General Relativity reduces to Newtonian gravity. The key equations simplify to:
    • Poisson Equation: \[ k^2 \Psi = 4\pi G a^2 \delta \rho \]
    • Continuity Equation: \[ \delta' + \nabla^2 v = 0 \]
    • Euler Equation: \[ v' + \mathcal{H} v = -\Phi - \frac{\delta p}{\bar{\rho} + \bar{p}} \]
    2. These equations describe the growth of density perturbations under gravity and pressure forces.

Conclusion

Gravitational primordial perturbation theory serves as a cornerstone of modern cosmology, bridging the gap between the early universe's quantum fluctuations and the vast cosmic structures we observe today. By meticulously deriving and understanding the perturbation equations within the framework of General Relativity and the FRW universe, we gain profound insights into the universe's origin, composition, and ultimate fate.

As observational technologies advance, particularly in the realm of CMB measurements and gravitational wave detection, gravitational primordial perturbation theory remains an active and evolving field, continually refining our understanding of the cosmos.

References

    • Cosmology II, Notes and code
    • Baryon-Dark matter interaction in presence of magnetic fields in light of EDGES signal, Jitesh R Bhatt, Pravin Kumar Natwariya, Aleka C. Nayak, Arun Kumar Pandey [arXiv:1905.13486 [astro-ph.CO] (2019)], Eur. Phys. J. C 80, 334 (2020)
    • Implications of baryon-dark matter interaction on IGM temperature and tSZ effect with magnetic field, Arun Kumar Pandey, Sunil Malik, T. R. Seshadri, arXiv:2006.07901 [astro-ph.CO] (2020), Mon.Not.Roy.Astron.Soc. 500 (2020)
    • Thermal SZ effect in a magnetized IGM dominated by interacting DM decay/annihilation during dark ages, Arun Kumar Pandey, Sunil Malik (2022) [arXiv:2204.08088]

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