Exhaustive Notes on Cosmology and the Large-Scale Structure of the Universe

Introduction to Cosmology

The fundamental characterization of the universe is its vastness, succinctly described by the concept that "Space is big."

The Hubble Constant and Scale Factors

Much of our understanding regarding the universe is contingent upon a variety of factors.
The most uncertain of these factors is the Hubble Constant ( $H$ ).
It is a common practice in the scientific community to express values in terms of a dimensionless scale factor, denoted as $h$ .
The formula for the Hubble Constant using this factor is: $H = h \times 100\,km/s/Mpc$ .
The current estimated value for this dimensionless constant is $h = 0.704$ .

The Cosmological Principle

The universe is characterized as being roughly spatially uniform when observed at scales of $100\,Mpc$ and above.
A core tenet of this principle is that the laws of physics remain constant regardless of the observer's location in space.
Perfect Cosmological Principle: This defunct theory previously proposed that the universe is uniform in both space and time; however, it is no longer considered valid.

The Friedmann-Lemaitre-Robertson-Walker (FLRW) Metric

The FLRW metric is the mathematical formalization of the Cosmological Principle.
It describes space as a medium that changes in size over time, utilizing a scale factor represented by $a$ .
The evolution and behavior of this scale factor are determined by the equation of state of the fluid that permeates the universe.
This metric provides the theoretical basis for the concept of dark energy.

Critical Density and the Density Parameter

For any given rate of expansion, there is a required energy density for the universe.
According to the provided transcript, this is represented by the relationship: $32 \quad 8$ .
This value is identifying the "critical density" of the universe.
The actual energy density of space is categorized by the density parameter, represented by the Greek letter Omega ( $\Omega$ ).
The parameter is defined as: $\Omega = \frac{\rho}{\rho_c}$ .

Curvature of Space

Cosmology explores whether space possesses a specific geometric shape. There are three primary types of space curvature: - Euclidean Space: Characterized by zero curvature. - Elliptical Space: Characterized by positive curvature. - Hyperbolic Space: Characterized by negative curvature.
The shape of space is determined by the density parameter ( $\Omega$ ): - If \Omega > 1, the space is "closed" and possesses positive curvature. - If \Omega < 1, the space is "open" and possesses negative curvature. - If $\Omega = 1$ , the space is "flat" and possesses no curvature.

Fated Geometries of the Universe

The curvature (open, flat, or closed) refers to the four-dimensional shape of space-time.
Closed Universe: This model is finite in size. It will eventually stop expanding and undergo a recollapse.
Open Universe: This model is infinite in size. It will eventually escape the gravitational bounds of its own matter.
Flat Universe: This model is infinite in size. It is balanced such that it never collapses, yet never completely escapes gravitational influence in the same manner as an open universe.

Newtonian Interpretation of Curvature

The concepts of open or closed space can be understood through the Newtonian analogy of escape velocity.
If an object travels faster than the escape velocity of a system, it will overcome the gravitational force and escape.
In this cosmological analogy, the "speed" of the universe is represented by the Hubble Constant.
A flat universe is defined as one that is traveling at exactly the escape velocity.

Quantitative Data of Our Universe

The current state of the universe is defined by the following measured quantities: - Age of the universe: $t = 13.799 \pm 0.021\,Gyr$ (Derived from many measurements). - Hubble scale: $h = 0.6774 \pm 0.0046$ (Derived from Cepheid variables and Type 1a supernovae). - Baryon (light matter) density: $\Omega_b = 0.0486 \pm 0.0010$ (Derived from the luminosity of nearby galaxies and CMB anisotropy). - Dark matter density: $\Omega_d = 0.2589 \pm 0.0057$ (Derived from the rotation curves of nearby galaxies and CMB anisotropy). - Dark energy density: $\Omega_{\Lambda} = 0.6911 \pm 0.0062$ (Derived from the redshift of Type 1a supernovae). - Density fluctuation: $\sigma_8 = 0.8159 \pm 0.0086$ (Derived from measurements of nearby galaxy clusters). - Redshift of CMB: $z = 1089.90 \pm 0.23$ (Derived from spectra of the CMB). - Total density: $\Omega = 0.9993 \pm 0.0019$ (Derived from Type 1a supernovae). - Dark energy equation of state: $w = -0.980 \pm 0.053$ (Derived from varied sources).

The Cosmic Microwave Background (CMB)

The CMB has been mapped by several key space missions: COBE, WMAP, and Planck.
Anisotropy: Fluctuations in the CMB are on the order of 1 part in $100,000$ .
COBE (Cosmic Background Explorer): Found that the spectral distribution of the CMB perfectly matched that of a blackbody.
WMAP (Wilkinson Microwave Anisotropy Probe): Discovered large-scale harmonic structures within the CMB that could not be explained by traditional cosmological models.
Planck: Identified a "preferred direction" in the universe, a phenomenon often referred to as the "axis of evil."
Temperature Fluctuations: Planck 2 data indicates variations between $-300\,\mu K$ and $+300\,\mu K$ .

Density Fluctuations and Large-Scale Structure

While the universe is uniform at large scales, it is non-uniform on scales below $100\,Mpc$ .
Regions characterized by overdensity eventually collapse.
The nature of the collapse depends on the level of perturbation: - For small perturbations, the collapse is roughly linear. - When the local density becomes approximately $28\%$ higher than the surrounding density, the collapse becomes exponential.
Overdense regions tend to collapse along their shortest axis first, leading to specific structures: - Zeldovich pancakes: Flat structures. - Spindles: Thin structures. - Clumps: Dense structures.
Baryon Acoustic Oscillations (BAO): These oscillations form structures in the universe and serve as a "standard ruler" for cosmological measurements.

CMB Anisotropy and the Power Spectrum

Anisotropy is often analyzed using the multipole moment $l$ , with angular scales ranging from $90^{\circ}$ to $0.2^{\circ}$ .
Experiments charting these include WMAP, ACBAR, Boomerang, CBI, and VSA.
The locations of the peaks in the anisotropy power spectrum are derived from BAOs in the decoupled universe, which represent the cosmic horizon at that specific point in time.
First Peak: Suggests the overall shape of the universe.
Even/Odd Peak Relationship: Suggests the density of baryons ( $\Omega_b$ ).
First/Third Peak Relationship: Suggests the density of dark matter ( $\Omega_d$ ).
Adiabatic vs. Isocurvature Perturbations: - Adiabatic: All flavors of matter/energy have the same perturbation. They originate from primordial perturbations and result in a $1:2:3…$ peak structure. These are the primary types observed. - Isocurvature: Different flavors have different perturbations that result in no net perturbation. They originate from topological defects in the early universe and result in a $1:3:5…$ peak structure.

Late Time Anisotropy

This occurs in the universe post-CMB due to intervening matter.
Effects include: - Erasing small-scale anisotropies (making the CMB image appear "fuzzier"). - Inducing polarization anisotropies.
The presence of these effects implies that the universe underwent ionization following the creation of the CMB.

Dark Matter Candidates

There are three commonly proposed candidates for dark matter: - Hot Neutrinos: Tiny particles generated via weak decay. While originally thought massless, flavor oscillation proved they have non-zero mass. They are uncharged and do not interact with electromagnetic forces. It is unlikely they constitute even $1\%$ of the total mass of the universe. - MACHOs (MAssive Compact Halo Objects): These are Brown Dwarfs located in galactic halos. They are attractive as candidates because they do not require new physics (Standard Model expansion), though their existence would require a different H/He mixture in the Intergalactic Medium (IGM). However, expected microlensing events have not been observed. - WIMPs (Weakly Interacting Massive Particles): Currently the most likely candidate, though they remain undiscovered. The current lower limit for their mass is on the order of $12\,TeV$ . Detection efforts include "hot" detectors like the Large Hadron Collider (LHC) and "cold" detectors like the Cryogenic Dark Matter Search (CDMS).

Theoretical Frameworks for Dark Energy

Dark energy was originally proposed by Albert Einstein as an extra term in the cosmological equations.
Direct interpretation suggests it represents the energy associated with the existence of space itself.
Speculative origins include vacuum fluctuations or a dynamic field known as quintessence.