12.2 Friedmann Equations and Cosmic Expansion
3 min read•august 9, 2024
The Friedmann equations are the backbone of modern cosmology, describing how the universe expands over time. These equations, derived from Einstein's general relativity, link the universe's expansion rate to its energy density and curvature.
Cosmic expansion, first observed by Edwin Hubble, is now known to be accelerating due to mysterious . This discovery has profound implications for our understanding of the universe's past, present, and future, challenging previous assumptions about its evolution.
Friedmann Equations and Cosmological Models
Fundamental Equations and Parameters
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- Friedmann equations describe the expansion of the universe derived from in general relativity
- First Friedmann equation relates the expansion rate to the energy density and curvature of the universe:
- Second Friedmann equation describes the acceleration of the expansion:
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represents the relative size of the universe at time t compared to its current sizea(t)- Directly related to the of distant objects
- Evolves according to the Friedmann equations
- Einstein-de Sitter model assumes a flat universe dominated by matter
- Predicts a deceleration in the expansion rate over time
- Scale factor in this model evolves as
Density Concepts and Cosmic Parameters
- Critical density defines the boundary between an open and closed universe
- Calculated using the :
- Ω compares the actual density of the universe to the critical density
- Ω < 1 indicates an open universe (negative curvature)
- Ω = 1 represents a flat universe (zero curvature)
- Ω > 1 suggests a closed universe (positive curvature)
- Current observations indicate Ω ≈ 1, suggesting a nearly flat universe
- Total density parameter includes contributions from matter, radiation, and dark energy:
Cosmic Expansion and Acceleration
Expansion Dynamics and Observational Evidence
- Cosmic expansion describes the ongoing increase in distance between all points in the universe
- First observed by Edwin Hubble through the redshift of distant galaxies
- Characterized by Hubble's Law: v = H₀d, where H₀ is the Hubble constant
- Acceleration of the universe discovered in 1998 through observations of Type Ia supernovae
- Contradicted previous assumptions of a decelerating expansion
- Implies the existence of a repulsive force counteracting gravity at large scales
- Observational techniques for studying cosmic expansion include:
- Measuring the redshift of distant galaxies
- Analyzing the
- Studying baryon acoustic oscillations in the large-scale structure of the universe
Dark Energy and Cosmological Implications
- Cosmological constant Λ introduced by Einstein to represent a constant energy density of space
- Initially proposed to create a static universe model
- Later repurposed to explain the accelerating expansion
- Dark energy serves as the leading explanation for cosmic acceleration
- Comprises about 68% of the energy content of the universe
- Exhibits negative pressure, causing the expansion to accelerate
- Potential forms of dark energy include:
- Vacuum energy from quantum field theory
- Scalar fields (quintessence models)
- Modified gravity theories
- Implications of cosmic acceleration for the fate of the universe:
- Continued acceleration could lead to a "Big Rip" scenario
- Possibility of dark energy density changing over time (dynamic dark energy models)
- Challenges for structure formation in the far future as galaxies become increasingly isolated
Key Terms to Review (17)
Accelerating universe: An accelerating universe refers to the observation that the expansion of the universe is increasing over time, rather than slowing down as previously expected. This phenomenon suggests that a mysterious force, often termed dark energy, is driving the acceleration and affecting the rate at which galaxies are moving apart. Understanding this concept is crucial for grasping the dynamics of cosmic expansion and the implications for the fate of the universe.
Alexander Friedmann: Alexander Friedmann was a Russian physicist and mathematician who is best known for developing the Friedmann equations, which describe the expansion of the universe in the context of general relativity. His work laid the groundwork for modern cosmology, providing essential insights into how the universe evolves over time and influencing our understanding of cosmic phenomena such as the Big Bang and cosmic inflation.
Baryonic matter: Baryonic matter refers to the normal matter composed of baryons, which are particles like protons and neutrons that make up atoms. This type of matter is responsible for the physical structures we observe in the universe, such as stars, planets, and galaxies, and is essential in understanding the dynamics of galaxy clusters and cosmic expansion.
Big bang theory: The big bang theory is the leading explanation for the origin of the universe, proposing that it began as an extremely hot and dense point approximately 13.8 billion years ago, and has been expanding ever since. This theory provides a framework for understanding cosmic expansion, the formation of structures in the universe, and the observed cosmic microwave background radiation.
Big Crunch: The Big Crunch is a hypothetical scenario in cosmology where the expansion of the universe eventually reverses, leading to a catastrophic collapse of all matter and energy back into a singularity. This theory suggests that the gravitational pull of all the mass in the universe could halt the current expansion and pull everything back together, ultimately resulting in a state similar to the Big Bang.
Cosmic microwave background radiation: Cosmic microwave background radiation (CMB) is the faint glow of radiation that fills the universe, a remnant from the Big Bang, and is a critical piece of evidence for understanding cosmic evolution. This radiation is isotropic and uniform, with slight temperature fluctuations that provide insights into the density and composition of the early universe. The study of CMB has profound implications for models of cosmic expansion and the acceleration of the universe.
Dark energy: Dark energy is a mysterious form of energy that permeates all of space and is responsible for the observed accelerated expansion of the universe. It makes up about 68% of the total energy content of the universe and plays a crucial role in shaping its large-scale structure and future dynamics.
Density Parameter: The density parameter is a dimensionless quantity that describes the density of the universe relative to a critical density needed for the universe to be flat. It plays a crucial role in cosmology, particularly in understanding the dynamics of cosmic expansion and the evolution of the universe, as it helps determine the overall geometry and fate of the cosmos.
Einstein's Field Equations: Einstein's Field Equations (EFE) are a set of ten interrelated differential equations that describe how matter and energy in the universe influence the curvature of spacetime. These equations are foundational in general relativity, showing how gravity is not just a force but a result of the geometry of spacetime itself. The EFE are crucial for understanding phenomena such as black holes, cosmic expansion, and the early universe's inflationary phase, as they provide a framework to study how massive objects like supermassive black holes shape their surroundings.
Friedmann Equation for a Matter-Dominated Universe: The Friedmann equation for a matter-dominated universe describes how the expansion rate of the universe changes over time, specifically focusing on a universe where matter is the dominant form of energy. This equation is crucial in understanding cosmic dynamics as it incorporates the density of matter and the curvature of space, allowing us to predict the behavior of the universe under the influence of gravitational forces from matter.
Friedmann Equation for a Radiation-Dominated Universe: The Friedmann Equation for a radiation-dominated universe describes how the expansion of the universe evolves under the influence of radiation. It shows the relationship between the universe's expansion rate, density, and curvature, specifically when radiation is the dominant form of energy density. This equation is essential for understanding the dynamics of the early universe when it was primarily filled with high-energy photons and relativistic particles.
Georges Lemaître: Georges Lemaître was a Belgian priest and astrophysicist who is best known for proposing the idea of the expanding universe and formulating what is now known as the Big Bang theory. His work laid the foundation for understanding cosmic expansion and the evolution of the universe, connecting mathematics and observational evidence to support these groundbreaking concepts.
Hubble Parameter: The Hubble Parameter is a measure of the rate of expansion of the universe, defined as the ratio of the velocity at which a galaxy is receding from an observer to its distance from that observer. It connects the observed redshift of distant galaxies to cosmic distances, serving as a critical tool in understanding cosmic expansion, inflationary models, and the large-scale structure of the universe.
Inflationary Model: The inflationary model is a theory in cosmology that suggests the universe underwent an exponential expansion in the first moments after the Big Bang, leading to a rapid stretching of space. This model helps explain the uniformity of the cosmic microwave background radiation, the large-scale structure of the universe, and the observed flatness of space. By proposing that inflation occurred, it provides solutions to various problems such as the horizon and flatness problems, and connects closely with how we understand cosmic expansion and the age and size of the universe.
Redshift: Redshift refers to the phenomenon where light from an object in space is shifted towards longer wavelengths, making it appear more red. This effect is primarily observed in astronomical objects moving away from us, allowing scientists to measure the velocity and distance of these objects, and providing crucial insights into the expansion of the universe and the nature of cosmic phenomena.
Robertson-Walker Metric: The Robertson-Walker metric is a solution to Einstein's field equations in general relativity that describes a homogeneous and isotropic expanding or contracting universe. This metric is crucial for understanding the large-scale structure of the cosmos, providing the mathematical foundation for models of cosmic expansion, including the Friedmann equations.
Scale Factor: The scale factor is a dimensionless number that describes how much the size of the universe changes over time during cosmic expansion. It represents the ratio of the distance between two points in the universe at a given time to the distance between the same two points at a reference time, typically set to the present. The scale factor plays a crucial role in understanding the dynamics of cosmic expansion and the evolution of the universe's geometry.