8.2 Friedmann equations

The Friedmann equations are fundamental to our understanding of cosmic expansion. They describe how the universe evolves over time, relating its expansion rate to energy density and curvature. These equations form the backbone of modern cosmology, providing insights into the universe's past, present, and future.

Derived from Einstein's field equations, the Friedmann equations assume a homogeneous and isotropic universe. They incorporate various components like matter, radiation, and dark energy, allowing cosmologists to model different scenarios for the universe's evolution and ultimate fate.

Derivation of Friedmann equations

• Friedmann equations describe the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity
• Provide a set of equations that govern the evolution of the universe, relating the expansion rate, energy density, and curvature of space-time

Assumptions in derivation

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• Universe is homogeneous and isotropic on large scales, meaning it appears the same from every location and in every direction
• Matter in the universe can be described as a perfect fluid with a given density and pressure
• Universe is either expanding or contracting, but the rate of expansion or contraction is the same everywhere at a given time
• Gravitational interactions are described by Einstein's theory of general relativity

Einstein field equations

• Relate the curvature of space-time to the presence of matter and energy
• Expressed as $G_{\mu\nu} = 8\pi G T_{\mu\nu}$, where $G_{\mu\nu}$ is the Einstein tensor, $G$ is Newton's gravitational constant, and $T_{\mu\nu}$ is the stress-energy tensor
• Einstein tensor describes the curvature of space-time, while the stress-energy tensor represents the distribution of matter and energy

FLRW metric

• Friedmann-Lemaître-Robertson-Walker metric describes the geometry of a homogeneous and isotropic universe
• Expressed as $ds^2 = -dt^2 + a^2(t)[\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2)]$, where $a(t)$ is the scale factor, $k$ is the curvature parameter, and $r$, $\theta$, $\phi$ are comoving coordinates
• Scale factor $a(t)$ represents the relative size of the universe at a given time, with $a=1$ at the present time

Components of stress-energy tensor

• For a perfect fluid, the stress-energy tensor has components $T^{00} = \rho$, $T^{ij} = p\delta^{ij}$, where $\rho$ is the energy density, $p$ is the pressure, and $\delta^{ij}$ is the Kronecker delta
• Energy density includes contributions from matter, radiation, and possibly dark energy or a cosmological constant
• Pressure is related to the energy density through the equation of state, which depends on the type of fluid (matter, radiation, or dark energy)

Friedmann equations

• Two independent equations that describe the evolution of the scale factor $a(t)$ and the energy density $\rho(t)$ in a homogeneous and isotropic universe
• Derived by applying the FLRW metric and the stress-energy tensor for a perfect fluid to the Einstein field equations

First Friedmann equation

• Relates the expansion rate (Hubble parameter) to the energy density and curvature of the universe
• Expressed as $H^2 \equiv (\frac{\dot{a}}{a})^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}$, where $H$ is the Hubble parameter, $\rho$ is the total energy density, and $k$ is the curvature parameter
• Describes how the expansion rate changes with the energy density and curvature

Second Friedmann equation

• Relates the acceleration of the expansion to the energy density and pressure
• Expressed as $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3p)$, where $p$ is the total pressure
• Shows that the expansion of the universe can accelerate if the pressure is negative and sufficiently large (as in the case of dark energy)

Hubble parameter

• Measures the expansion rate of the universe at a given time
• Defined as $H \equiv \frac{\dot{a}}{a}$, where $a$ is the scale factor and the dot represents a time derivative
• Current value (Hubble constant) is approximately $H_0 \approx 70$ km/s/Mpc, meaning that the universe expands by about 70 km/s for every megaparsec of distance

Critical density

• Density required for a flat universe ($k=0$) at a given Hubble parameter
• Expressed as $\rho_c \equiv \frac{3H^2}{8\pi G}$
• Current critical density is approximately $\rho_{c,0} \approx 10^{-26}$ kg/m$^3$, or about 6 hydrogen atoms per cubic meter

Density parameters

• Dimensionless ratios of the actual density to the critical density for various components of the universe
• Defined as $\Omega_i \equiv \frac{\rho_i}{\rho_c}$, where $i$ represents the component (matter, radiation, curvature, or dark energy)
• Sum of density parameters determines the geometry of the universe: $\Omega_m + \Omega_r + \Omega_k + \Omega_\Lambda = 1$, with $\Omega_k = -\frac{k}{a^2H^2}$ and $\Omega_\Lambda = \frac{\Lambda}{3H^2}$

Solutions to Friedmann equations

• Friedmann equations can be solved for the scale factor $a(t)$ and the energy density $\rho(t)$ under different assumptions about the curvature and content of the universe
• Solutions depend on the relative contributions of matter, radiation, and dark energy, as well as the value of the curvature parameter $k$

Flat universe solution

• For a flat universe ($k=0$) dominated by matter ($\Omega_m=1$), the solution is $a(t) \propto t^{2/3}$ and $\rho(t) \propto a^{-3} \propto t^{-2}$
• Expansion decelerates due to the attractive nature of gravity, but never stops or reverses

Closed universe solution

• For a closed universe ($k>0$) dominated by matter, the solution is a cycloid function for $a(t)$
• Universe expands to a maximum size, then recollapses due to the positive curvature and the attractive nature of gravity
• Maximum size and lifetime depend on the initial density and expansion rate

Open universe solution

• For an open universe ($k<0$) dominated by matter, the solution is a hyperbolic function for $a(t)$
• Expansion decelerates but never stops, and the universe approaches a constant expansion rate at late times
• Density decreases faster than in a flat universe due to the negative curvature

Accelerating universe solution

• In the presence of dark energy or a cosmological constant ($\Omega_\Lambda>0$), the expansion can accelerate at late times
• For a flat universe dominated by a cosmological constant, the solution is $a(t) \propto e^{Ht}$, where $H=\sqrt{\frac{\Lambda}{3}}$ is constant
• Density of matter and radiation decreases exponentially, while the density of dark energy remains constant, leading to an exponential expansion (de Sitter universe)

Cosmological parameters

• Observationally determined quantities that describe the properties and evolution of the universe
• Include the Hubble constant, density parameters, deceleration parameter, age of the universe, and the nature of dark energy

Hubble constant

• Current value of the Hubble parameter, denoted as $H_0$
• Measured using various methods, such as Type Ia supernovae, cosmic microwave background, and baryon acoustic oscillations
• Different methods yield slightly different values, leading to the "Hubble tension" (discrepancy between early and late universe measurements)

Deceleration parameter

• Dimensionless measure of the deceleration or acceleration of the universe's expansion
• Defined as $q \equiv -\frac{\ddot{a}a}{\dot{a}^2} = -\frac{\ddot{a}}{aH^2}$, where $a$ is the scale factor and $H$ is the Hubble parameter
• Positive values indicate deceleration, while negative values indicate acceleration
• Current observations suggest $q_0 \approx -0.6$, implying that the expansion is accelerating

Age of the universe

• Time elapsed since the Big Bang, estimated using the Friedmann equations and observational data
• Depends on the values of the Hubble constant and the density parameters
• Current estimate is approximately 13.8 billion years, with an uncertainty of a few hundred million years

Cosmological constant vs dark energy

• Cosmological constant $\Lambda$ is a term in Einstein's field equations that can lead to an accelerated expansion
• Dark energy is a more general term for the unknown cause of the accelerated expansion, which may or may not be a cosmological constant
• Observational data is consistent with a cosmological constant (equation of state $w=-1$), but other forms of dark energy with different equations of state are possible

Observational evidence

• Various astronomical observations support the Friedmann equations and the current cosmological model ($\Lambda$CDM)
• Key evidence includes Hubble's law, the cosmic microwave background, Type Ia supernovae, and baryon acoustic oscillations

Hubble's law

• Empirical relationship between the distance to a galaxy and its recessional velocity due to the expansion of the universe
• Expressed as $v = H_0 d$, where $v$ is the recessional velocity, $H_0$ is the Hubble constant, and $d$ is the distance
• Discovered by Edwin Hubble in 1929 using observations of distant galaxies
• Provides evidence for the expansion of the universe and allows the measurement of the Hubble constant

Cosmic microwave background

• Relic radiation from the early universe, observed as a nearly uniform background of microwave radiation
• Discovered by Arno Penzias and Robert Wilson in 1965
• Spectrum is an almost perfect black body with a temperature of 2.725 K
• Tiny anisotropies (fluctuations) in the CMB provide information about the early universe and the values of cosmological parameters

Type Ia supernovae

• Bright stellar explosions that occur when a white dwarf star accretes matter from a companion star and reaches a critical mass (Chandrasekhar limit)
• Have a nearly uniform intrinsic brightness, making them useful as "standard candles" for measuring cosmic distances
• Observations of distant Type Ia supernovae in the late 1990s revealed that the expansion of the universe is accelerating
• Provide evidence for the existence of dark energy and constrain its properties

Baryon acoustic oscillations

• Regular pattern of density fluctuations in the distribution of galaxies, caused by sound waves in the early universe
• Sound waves propagated through the plasma of the early universe until the epoch of recombination, when neutral atoms formed and the waves were "frozen" into the matter distribution
• Characteristic scale of the BAO (sound horizon) depends on the properties of the early universe and serves as a "standard ruler" for measuring cosmic distances
• Measurements of the BAO scale at different redshifts provide information about the expansion history of the universe and the values of cosmological parameters

Implications for cosmology

• Friedmann equations and observational evidence support the standard cosmological model, known as the $\Lambda$CDM model (Cold Dark Matter with a cosmological constant)
• Key implications include the Big Bang theory, the expansion history of the universe, and its ultimate fate

Big Bang theory

• Theory that the universe originated from a singularity and has been expanding and cooling ever since
• Supported by evidence such as Hubble's law, the cosmic microwave background, and the abundance of light elements
• Friedmann equations provide the mathematical framework for describing the evolution of the universe from the Big Bang to the present

Expansion history of the universe

• Universe has undergone different stages of expansion, depending on the dominant form of energy density
• Early universe was radiation-dominated, followed by a matter-dominated era, and then a dark energy-dominated era (present)
• Transition times between the eras depend on the values of the density parameters
• Friedmann equations describe the expansion history and allow the calculation of the transition times

Ultimate fate of the universe

• Long-term evolution and end state of the universe depend on the values of the density parameters and the nature of dark energy
• If dark energy is a cosmological constant ($w=-1$), the universe will continue to expand exponentially, leading to a "Big Freeze" (all matter diluted and cooled to absolute zero)
• If dark energy has $w<-1$ (phantom energy), the expansion could lead to a "Big Rip" (all structures torn apart by the accelerating expansion)
• If dark energy decays over time or has $w>-1$, the universe could recollapse in a "Big Crunch" or approach a steady state

Inflation vs alternative theories

• Inflation is a theory proposing that the early universe underwent a brief period of exponential expansion, driven by a hypothetical scalar field (inflaton)
• Inflation addresses several problems in the standard Big Bang model, such as the horizon problem, flatness problem, and magnetic monopole problem
• Alternatives to inflation include variable-speed-of-light theories, cyclic models, and string gas cosmology
• Observational tests, such as the search for primordial gravitational waves and non-Gaussianity in the CMB, can help distinguish between inflation and alternative theories