Comprehensive Study Notes on Cosmic Expansion

Overview of Cosmic Expansion

• Definition: The expansion of the universe is the intrinsic increase in distance between gravitationally unbound parts of the observable universe over time. It is a spatial expansion within spacetime itself, meaning the universe does not expand into an existing external space, nor does an outside space exist around it.

• Apparent Recessional Motion: To any observer, all non-gravitationally bound galaxies appear to recede at speeds proportional to their distance. Local bounds, such as galaxies bound within clusters by mutual gravity, do not expand.

• Speed Limits and Local vs. Global Frames: Objects cannot travel faster than light relative to local inertial reference frames. However, this restriction does not limit the recessional rates of cosmologically distant objects across curved spacetime.

• Theoretical Framework: Cosmic expansion is modeled mathematically using General Relativity via the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and forms a cornerstone of Big Bang cosmology. The consensus Lambda-CDM ($\Lambda\text{CDM}$) model accounts for varying expansion rates across distinct cosmic epochs based on the physical energy densities of the contents of spacetime.

• Chronological Milestones:

  • Inflation: Occurred approximately $10^{-32}\,\text{seconds}$ after the Big Bang, expanding the scale factor by at least $10^{26}$ (a factor of $e^{60}$) in every direction.

  • Deceleration Epoch: Gravitational attraction slowed the expansion rate for several billion years.

  • Accelerated Expansion: Began approximately $9.8\,\text{billion years}$ after the Big Bang ($4\,\text{billion years}$ ago). This late-time acceleration is attributed to dark energy, represented as a cosmological constant ($\Lambda$) in gravitational models.

Historical Development of the Concept

• 1912–1914 (Vesto Slipher): Discovered that light from remote galaxies was redshifted, providing initial observational evidence of receding galactic motion.

• 1922 (Alexander Friedmann): Derived theoretical solutions to the Einstein field equations of General Relativity demonstrating that the universe could be expanding.

• 1924 (Knut Lundmark): Observed galaxy diameters and estimated distances to the Andromeda Galaxy to deduce expansion, arriving at reasonably accurate conclusions prior to verified distance indicators.

• 1927 (Georges Lemaître): Derived general relativity solutions and combined Slipher's radial velocity data with Edwin Hubble's distance measurements to propose a linear distance-velocity relationship.

• 1929 (Edwin Hubble): Firmly established the linear distance-velocity relationship (Hubble's law or the Hubble–Lemaître law) using Cepheid variable star distance measurements. Though Hubble's initial proportionality constant ($H_0$) was overestimated by a factor of 7 and he did not explicitly attribute the relationship to spatial expansion, his work spurred modern cosmological measurement efforts.

• 1940s (Walter Baade): Recalibrated Cepheid period-luminosity relations and doubled the estimated size of the known universe.

• Mid-to-Late 20th Century: The Hubble constant was loosely constrained between $50\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$ and $90\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

• January 13, 1994: NASA announced the successful completion of optics correction repairs on the Hubble Space Telescope main mirror, enabling precise galactic observations.

• 1994 (Wendy Freedman Key Project): Analyzed Cepheid variables in M100 within the Virgo Cluster core, yielding $H_0 = 80 \pm 17\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

• 1994 (Adam Riess et al.): Used visual-band light-curve shapes of Type Ia supernovae to reduce systematic errors, measuring $H_0 = 67 \pm 7\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$. Subsequent Cepheid variable calibrations refined this to $73 \pm 7\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

• 2003 (David Spergel et al.): Analyzed first-year Cosmic Microwave Background (CMB) data from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite, finding $H_0 = 72 \pm 5\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

Kinematics and Geometry of Expansion

• Hubble's Law Equation:   $\mathbf{v} = H_0 \mathbf{x}$   where $\mathbf{v}$ is the recessional velocity vector, $\mathbf{x}$ is the spatial position vector, and $H_0$ is the Hubble constant quantifying the expansion rate at the present day.

• Hubble Parameter: Expansion rate varies across cosmic time $t$. The current value is $H_0 = H(t = t_0)$, where $t_0$ represents present cosmic time.

• Homogeneity and Isotropy: Cosmological observations show the universe is homogeneous (identical in all locations on large scales) and isotropic (identical in all directions). Uniform expansion dictates that distances between any two unconstrained galaxies increase by the same multiplicative factor over time.

• Cosmic Scale Factor ($a$): Quantifies relative spatial scale as a function of time. Conventionally defined such that $a = 1$ at the present time ($t_0$). The scale factor was smaller in the past ($a < 1$) and increases in an expanding universe.

• Age of the Universe: Extrapolating the scale factor backward to $a = 0$ yields a cosmic age of $13.787 \pm 0.020\,\text{billion years}$.

• Comoving Coordinates: Spatial coordinate system that factors out overall expansion. Objects moving purely with the uniform Hubble expansion remain fixed at constant comoving coordinates.

• Synchronous Spatial Surfaces and Shape: Surfaces where stationary comoving observers agree on the age of the universe define spatial geometry. In Special Relativity, these surfaces form hyperboloids due to time dilation. In General Relativity, gravitational curvature alters these surfaces. Astronomical data indicates the universe's spatial geometry is flat (spatial triangle angles sum to $180^\circ$).

Dynamics and Governing Equations

• Second Friedmann Equation (Acceleration Equation):   $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$   where $a$ is the scale factor, $G$ is the gravitational constant, $\rho$ is total energy density, $p$ is isotropic pressure, $c$ is the speed of light, and $\Lambda$ is the cosmological constant.

• Deceleration Conditions: A positive energy density ($\rho > 0$) and positive pressure ($p > 0$) contribute to deceleration ($\ddot{a} < 0$) due to gravitational attraction.

• Acceleration Conditions: Accelerated expansion ($\ddot{a} > 0$) requires a positive cosmological constant ($\Lambda > 0$) or a fluid possessing strong negative pressure ($p < -\frac{\rho c^2}{3}$).

• Equations of State ($p = w\rho$):

  • Nonrelativistic Matter (Dust): Possesses negligible thermal pressure ($|p| \ll \rho c^2$), giving $w = 0$.

  • Ultrarelativistic Particles (Radiation): High-velocity particle gases and photons exert positive pressure with $p = \frac{\rho c^2}{3}$, giving $w = \frac{1}{3}$.

  • Cosmological Constant / Dark Energy: Characterized by negative pressure with $w = -1$.

  • Phantom Energy: Hypothetical fluid with $w < -1$.

• Energy Density Scaling Equation:   $\rho \propto a^{-3(1+w)}$

  • Matter Density: $\rho \propto a^{-3}$ (dilutes inversely with volume).

  • Radiation Density: $\rho \propto a^{-4}$ (dilutes with volume $a^{-3}$ plus an additional factor of $a^{-1}$ from photon energy loss due to redshift).

  • Dark Energy Density: $\rho = \text{constant}$ for $w = -1$.

  • Phantom Energy Density: Increases over time as the scale factor grows ($w < -1$).

Cosmological Horizons and Observables

• Particle Horizon and Observable Universe: The comoving distance that light or particles could have traveled since the Big Bang. Defines the boundary of the observable universe.

• Cosmic Event Horizon: Boundary generated by accelerated expansion (dark energy). Photons emitted today beyond this horizon will never reach the observer in the future.

• Hubble Horizon: Spatial scale ($c H^{-1}$) separating dynamic behaviors: density perturbations larger than the Hubble horizon remain non-dynamical because gravitational signals cannot span them, whereas perturbations smaller than the horizon evolve via Newtonian gravity.

• Redshift and Doppler Origin: Cosmological redshift is mathematically equivalent to accumulated local Doppler shifts along light paths in curved spacetime.

• Peculiar Velocity and Momentum Decay: Peculiar velocity represents motion relative to the comoving coordinate grid. As the universe expands, peculiar momentum for both relativistic and nonrelativistic particles decays as $a^{-1}$. Nonrelativistic peculiar velocities decay as $a^{-1}$.

• Thermal Evolution (Cooling):

  • Radiation Temperature: Ultrarelativistic fluids cool inversely with scale factor: $T \propto a^{-1}$.

  • Nonrelativistic Matter Temperature: Matter cools faster: $T \propto a^{-2}$.

Epochs in Cosmic History

• 1. Cosmic Inflation Epoch:

  • Timing: Approximately $10^{-32}\,\text{seconds}$ after the Big Bang.

  • Mechanism: Driven by the inflaton field in a positive-energy false vacuum state, producing exponential spatial expansion ($a \propto e^{H t}$) by a factor of at least $e^{60}$ ($\approx 10^{26}$).

  • Cosmological Solutions: Solves the horizon problem, flatness problem, and eliminates magnetic monopoles predicted by grand unified theories. Quantum fluctuations during inflation seeded matter density variations.

• 2. Radiation-Dominated Epoch:

  • Timing: Established by neutrino decoupling at approximately $1\,\text{second}$ after the Big Bang, lasting until approximately $50,000\,\text{years}$ post-Big Bang.

  • Expansion Kinematics: Scale factor growth followed $a \propto t^{1/2}$, causing expansion deceleration.

• 3. Matter-Dominated Epoch:

  • Timing: From approximately $50,000\,\text{years}$ to $11\,\text{billion years}$ after the Big Bang ($3\,\text{billion years}$ ago).

  • Expansion Kinematics: Scale factor expanded according to $a \propto t^{2/3}$, continuing deceleration.

  • Structure Formation: Gravitational instability was most efficient during this phase, creating galaxies and large-scale structures.

• 4. Dark Energy-Dominated Epoch:

  • Timing: Began approximately $11\,\text{billion years}$ after the Big Bang ($3\,\text{billion years}$ ago) and represents the current phase.

  • Expansion Kinematics: Constant energy density causes exponential scale factor growth ($a \propto e^{H t}$) and late-time accelerated expansion.

Measurement Techniques and Empirical Values

• 1. Standard Candles & Redshift (Local Universe):

  • Uses intrinsic brightness of Cepheid variable stars and Type Ia Supernovae.

  • Riess et al. (2016): Measured $H_0 = 73.24 \pm 1.74\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

  • Discovery of cosmic acceleration via supernovae awarded the 2011 Nobel Prize in Physics.

• 2. Cosmic Microwave Background (Early Universe):

  • Infers current expansion from CMB fluctuation angular scale fitting inside the $\Lambda\text{CDM}$ framework.

  • Planck Collaboration: Measured $H_0 = 67.4 \pm 0.5\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

  • Hubble Tension: Persistent statutory discrepancy between local distance-ladder measurements ($\approx 73\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$) and early-universe CMB predictions ($\approx 67.4\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$).

• 3. Strong Gravitational Lensing Time-Delay Cosmography:

  • Measures time delays between light arrival of multiple lensed images of distant background sources.

  • H0LiCOW Collaboration (Quasars): $H_0 = 73.3^{+1.7}_{-1.8}\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

  • TDCOSMO Collaboration (2025): $H_0 = 71.6^{+3.9}_{-3.3}\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

  • SN Refsdal: $H_0 = 64.8^{+4.4}_{-4.3}\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

  • SN H0pe: $H_0 = 75.7^{+8.1}_{-5.5}\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$.

  • MACS J0138-2155 Cluster (SN Encore & SN Requiem): Preliminary measurement $H_0 = 66.9^{+11.2}_{-8.1}\,\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$. Reappearance of SN Requiem expected April–December 2026 (if $H_0 = 73$) or March–November 2027 (if $H_0 = 67$).

• 4. Gravitational Wave Standard Sirens:

  • Evaluates distance directly from gravitational wave signals of compact binary mergers (e.g., neutron star merger GW170817).

• 5. Redshift Drift (Future Observations):

  • Direct real-time measurement of drift in redshift, angular size, and flux over long observation periods. Targeted for mid-2030s detection via the Square Kilometre Array (SKA) and Extremely Large Telescope (ELT).

Common Misconceptions and Clarifications

• Expansion as "Stretching of Space": Describing expansion as space stretching while galaxies remain stationary introduces a preferred rest frame, violating the principle of relativity. The actual physical phenomenon is galaxies moving apart across spacetime.

• Superluminal Speeds and Relativity: Objects beyond the Hubble horizon recede at speeds exceeding $c$. Special relativity applies strictly to local inertial frames in flat spacetime. In curved spacetime, relative velocities over vast cosmological distances are not uniquely defined and do not violate local light-speed limits.

• Expansion as a Physical Force: Cosmic expansion is not a repulsive physical force pushing objects apart; it represents inertial motion established by early cosmological conditions. In contrast, the cosmological constant ($\Lambda$) produces a repulsive accelerated effect, but expansion occurs independently of it.