13.4 Friedmann-Lemaître-Robertson-Walker models in cosmology
3 min read•august 9, 2024
The Friedmann-Lemaître-Robertson-Walker (FLRW) models are key to understanding the universe's evolution. These models describe how space expands over time, based on Einstein's field equations and the assumption of a homogeneous, isotropic cosmos.
FLRW models use a special metric that accounts for cosmic expansion through a . This factor relates physical distances to fixed coordinates, showing how galaxies move apart as space itself stretches.
FLRW Metric and Cosmic Expansion
Fundamentals of FLRW Metric and Scale Factor
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describes geometry of homogeneous and
Metric takes form ds2=−c2dt2+a2(t)[dr2+Sk2(r)(dθ2+sin2θdϕ2)]
Scale factor a(t) represents relative size of universe at time t
Sk(r) depends on spatial k: sinr (k = 1), r (k = 0), or sinhr (k = -1)
Scale factor relates physical distance to comoving distance: dphys=a(t)dcom
Normalized scale factor a(t0)=1 at present time t0
Hubble Parameter and Cosmic Expansion
H(t) measures expansion rate of universe
Defined as H(t)=a(t)a˙(t), where dot denotes time derivative
Current value H0 called Hubble constant, approximately 70 km/s/Mpc
Hubble's law states v=H0d for distant galaxies (redshift proportional to distance)
Cosmic expansion causes galaxies to move apart, increasing physical distances over time
Expansion affects large scales, while gravitationally bound systems (galaxies) remain intact
Friedmann Equations and Spatial Curvature
Derivation and Interpretation of Friedmann Equations
First Friedmann equation: (aa˙)2=38πGρ−a2kc2+3Λc2
Second Friedmann equation: aa¨=−34πG(ρ+c23p)+3Λc2
ρ represents energy density, p pressure, Λ cosmological constant
Equations derived from using FLRW metric
First equation relates expansion rate to energy content and curvature
Second equation describes acceleration of cosmic expansion
Spatial Curvature and Critical Density
Spatial curvature k determines geometry of universe: closed (k > 0), flat (k = 0), or open (k < 0)
Critical density ρc defined as density required for flat universe: ρc=8πG3H2
Ω=ρcρ measures ratio of actual density to critical density
Total density parameter Ωtot=Ωm+Ωr+ΩΛ
Flat universe corresponds to Ωtot=1, closed Ωtot>1, open Ωtot<1
Current observations suggest nearly flat universe with Ωtot≈1
Cosmological Principle and Big Bang Theory
Cosmological Principle and Its Implications
Cosmological principle states universe homogeneous and isotropic on large scales
Homogeneity implies same average properties at all locations
Isotropy means universe looks same in all directions from any point
Supported by observations of and large-scale structure
Simplifies cosmological models and allows use of FLRW metric
Breaks down on smaller scales (galaxies, clusters) due to gravitational clustering
Big Bang Theory and Observational Evidence
theory proposes universe began from hot, dense state and expanded over time
Supported by multiple lines of evidence:
Cosmic expansion observed through galactic redshifts
Alexander Friedmann: Alexander Friedmann was a Russian physicist and mathematician known for his groundbreaking work in cosmology, particularly for developing solutions to the Einstein field equations of general relativity. His work laid the foundation for the modern understanding of an expanding universe, which is crucial to the Friedmann-Lemaître-Robertson-Walker models that describe the dynamics of the universe over time.
Big Bang: The Big Bang is the prevailing cosmological model that describes the early development of the universe, which began as a singularity approximately 13.8 billion years ago and has been expanding ever since. This event marks the origin of space and time, leading to the formation of all matter, energy, and the cosmic structures we observe today.
Cosmic microwave background: The cosmic microwave background (CMB) is a faint glow of radiation that fills the universe, remnant from the early stages of the Big Bang, roughly 380,000 years after it occurred. This radiation provides critical evidence for the Big Bang theory, showcasing the universe's hot, dense beginnings and its subsequent expansion. It plays a significant role in understanding the structure and evolution of the universe, particularly in relation to models that describe its large-scale behavior.
Curvature: Curvature is a measure of how much a geometric object deviates from being flat or straight. In the context of Riemannian geometry, it describes how the geometry of a manifold bends and can be quantified through different types such as sectional curvature, Ricci curvature, and scalar curvature, affecting the behavior of geodesics and the manifold's overall structure.
Density Parameter: The density parameter, often denoted as \( \Omega \), quantifies the total energy density of the universe relative to a critical density needed for the universe to be flat. This parameter plays a crucial role in cosmology, as it helps determine the fate of the universe, including whether it will expand forever, eventually recollapse, or approach a steady state. Different components such as matter, radiation, and dark energy contribute to the total density, influencing the dynamics of the Friedmann-Lemaître-Robertson-Walker models.
Einstein Field Equations: The Einstein Field Equations (EFE) are a set of ten interrelated differential equations in the theory of general relativity that describe how matter and energy influence the curvature of spacetime. These equations form the core of Einstein's theory, linking the geometry of spacetime to the distribution of mass and energy, which is crucial for understanding phenomena like gravity, cosmology, and the behavior of various geometric structures.
FLRW Metric: The FLRW metric, named after Friedmann, Lemaître, Robertson, and Walker, is a solution to the Einstein field equations of general relativity that describes a homogeneous and isotropic expanding or contracting universe. This metric is crucial in cosmology as it provides the mathematical framework to understand the large-scale structure and dynamics of the universe, connecting geometry with cosmic expansion and density distributions.
Friedmann Equations: The Friedmann equations are a set of fundamental equations in cosmology derived from Einstein's field equations of general relativity, which describe the expansion of the universe. These equations link the dynamics of the universe's expansion to its geometry, matter content, and energy density, forming the backbone of modern cosmological models.
Friedmann-Lemaître-Robertson-Walker model: The Friedmann-Lemaître-Robertson-Walker (FLRW) model is a solution to Einstein's field equations of general relativity that describes a homogeneous and isotropic universe. This model serves as a fundamental framework in cosmology, explaining the expansion of the universe and forming the basis for modern cosmological theories, including the Big Bang theory.
Geodesics: Geodesics are the shortest paths between points in a curved space, often generalizing the concept of straight lines in Euclidean geometry. They are crucial for understanding how distances are measured on manifolds and serve as the 'straightest' possible paths that can be taken, influenced by the curvature of the space.
Georges Lemaître: Georges Lemaître was a Belgian priest, astronomer, and professor of physics, best known for proposing the Big Bang theory of the universe's origin. His work laid the foundation for modern cosmology and introduced the concept that the universe is expanding, which is a core feature of the Friedmann-Lemaître-Robertson-Walker models in cosmology.
Hubble Parameter: The Hubble parameter, often denoted as H(t), is a measure of the rate of expansion of the universe at a given time. It quantifies how fast galaxies are receding from each other due to the expansion, linking the distance of galaxies to their recessional velocity. This concept is central in cosmology, particularly in the Friedmann-Lemaître-Robertson-Walker models, which describe a homogeneous and isotropic universe.
Isotropic Universe: An isotropic universe is a cosmological model where the distribution of matter and energy is uniform in all directions, meaning there are no preferred directions in the universe. This concept is crucial for the Friedmann-Lemaître-Robertson-Walker (FLRW) models, as these models assume isotropy and homogeneity when describing the large-scale structure of the universe. Isotropy leads to simplified equations and helps cosmologists understand the expansion and evolution of the universe.
Lorentzian manifold: A Lorentzian manifold is a smooth manifold equipped with a non-degenerate, symmetric bilinear form of signature (-+++), which allows for the definition of time-like, space-like, and null intervals. This structure is essential in general relativity as it models the fabric of spacetime, where the geometry influences the motion of matter and energy. The unique properties of a Lorentzian manifold enable the exploration of fundamental concepts such as causality and the curvature of spacetime, which are critical for understanding cosmological models.
Negative curvature: Negative curvature refers to a geometric property of spaces where the sum of angles in a triangle is less than 180 degrees, leading to unique geometric behaviors and properties. This characteristic is significant as it influences the behavior of geodesics, contributes to the interpretation of sectional curvature, and plays a crucial role in certain cosmological models that describe the universe's shape and dynamics.
Positive Curvature: Positive curvature is a property of a geometric space where, intuitively, the surface bends outward, like the surface of a sphere. In such spaces, geodesics tend to converge, and triangles formed within them have angles that sum to more than 180 degrees. This concept is crucial for understanding various phenomena in Riemannian geometry, affecting properties of geodesics, curvature behavior, and geometric structures.
Scale Factor: In the context of cosmology, the scale factor is a function that describes how distances in the universe change over time, particularly in expanding models of the universe. It represents the relative size of the universe at different moments, playing a crucial role in understanding the evolution of cosmic structures and the dynamics of cosmic expansion.
Spatially homogeneous: Spatially homogeneous refers to a property of a space where the physical characteristics are uniform and do not vary with location. This concept is crucial in cosmology, particularly when analyzing models that assume the universe is the same in all directions and at every point, leading to simpler mathematical descriptions and predictions about the cosmos.