Modern Cosmology -
Part III: The Friedmann Universe
Jacobs University Bremen
Alexander Friedmann lived much of his life in Saint Petersburg. He fought in World War I (on behalf of Imperial Russia) as a bomber and later lived through the Russian Revolution of 1917. Friedmann obtained his degree in St. Petersburg State University (1910), became a lecturer in St.-Petersburg State College of Mines, and a professor in Perm State University in 1918.
He discovered the expanding-universe solution to the general relativity field equations in 1922, which was corroborated by Edwin Hubble's observations in 1929. Friedmann's 1924 papers, including "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes" (On the possibility of a world with constant negative curvature of space) published by the German physics journal Zeitschrift für Physik (Vol. 21, pp. 326-332), demonstrated that he had command of all three Friedmann models describing positive, zero and negative curvature respectively, a decade before Robertson and Walker published their analysis.
|1. Einstein's Equations
The Einstein field equations (EFE) or Einstein's equations are a set of ten equations in Albert Einstein's theory of general relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime curvature (expressed by the Einstein tensor) with the energy and momentum within that spacetime (expressed by the stress-energy tensor).
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations define Einstein manifolds.
|2. Friedmann-Robertson-Walker Universe
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann-Lemaitre-Robertson-Walker metric and a fluid with a given mass density and pressure. The equations for negative spatial curvature were given by Friedmann in 1924.
|3. The Lambda CDM Universe
Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy. However, determining the time evolution of the expansion factor a(t) does require Einstein's field equations together with a way of calculating the density, also called the cosmological equation of state. The main results of the FLRW model without considering the cosmological constant were first derived by the Soviet mathematician Alexander Friedmann in 1922 and 1924. Although his work was published in the prestigious physics journal Zeitschrift für Physik, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein, who, on behalf of Zeitschrift für Physik, acted as the scientific referee of Friedmann's work. Eventually Einstein acknowledged the correctness of Friedmann's calculations, but failed to appreciate the physical significance of Friedmann's predictions.
Caption: The Universe of galaxies is expanding. According to the Big Bang theory, the Universe emerged from an extremely dense and hot state (singularity). Space itself has been expanding ever since, carrying galaxies with it, like raisins in a rising loaf of bread.These equations are the basis of the standard big bang cosmological model including the current LCDM model. Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the big bang model cannot account for the observed lumpiness of the universe. In a strictly FLRW model, there are no clusters of galaxies, stars or people, since these are objects much denser than a typical part of the universe. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models which calculate the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, i.e., a model which follows the FLRW metric apart from primordial density fluctuations. As of 2010, the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP.
|Exercises III - Solutions
|Ex_3 - Sol||Midterm Review Key Knowledge||Key Knowledge||SCP Union2 Supernova Data||SNe Ia Data||Lecture Notes: Part III
|LN: Part III|