A modified friedmann equation for a system with varying. We will see that the universe is expanding and this expansion can take many different forms. We illustrate this analytic study by a numerical integration of this set of equations x 4 and nish by a discussion aboutthe initial conditions explaining the current observed state of our universe x 5. The wave function of the universe in wheelerdewitt equation is written in polar form. Friedmann equation and stability of inflationary higher derivative. This is the first derivation of friedmann equations in these gravity theories in a nonflat friedmannrobertsonwalker universe by using the novel idea proposed by padmanabhan. There is a subtlety that the apparent horizon, the hubble horizon and the cosmological event horizon cannot be distinguished clearly in some cases where one uses eqs. This is the first derivation of friedmann equations in these gravity theories in a nonflat friedmann robertsonwalker universe by using the novel idea proposed by padmanabhan.
The friedmann model is a model of the universe governed by the friedmann equations, which describes how the universe expands or contracts. In this note by use of the holographic principle together with the equipartition law of energy and the unruh temperature, we derive the friedmann equations of a friedmannrobertsonwalker universe. In 1922, alexander friedmann proposed a set of equations that can describe the expansion of the universe 21. More problems with the friedmann metric miles mathis. The second equality is because we can take the second crest given that the photon is a wave. Solutions of friedmann equations a man gazing at the stars is proverbially at the mercy of the puddles in the road.
Cosmological histories from the friedmann equation. Hence, the second friedmann equation is also named as friedmanns acceleration equation. The term friedmann model is therefore often used to indicate a matteronly cosmology, even though his equation includes contributions from all equations of state. Pdf the spatially flat friedmann equation for early. In this note by use of the holographic principle together with the equipartition law of energy and the unruh temperature, we derive the friedmann equations of a friedmann robertsonwalker universe. The time evolution of the universe from the big bang until today is described by general relativity, i. Either way, this is the equation that relates how the universe expands, quantitatively, to what makes up the.
Some types of massenergy and their values of w and n. Oct 09, 2011 we point this out simply as historical context for what follows. Since issues have repeatedly arisen concerning the origin of the equations and the correct values of the constants in them, i think it would be helpful to derive them here. We give a simple method to reduce friedmann equations to a second order. Hubbles law wasnt discovered until 1929, so of course it wasnt called this in friedmanns paper, but he did present.
An introduction to modern cosmology second edition. Petersburg russia tt friedmann laboratory for theoretical physics, stc cornputing, st. Pdf the spatially flat friedmann equation for early rainbow. Later we will add corrections due to effects of gr. An introduction to modern cosmology university of belgrade. Friedmann 18881925 the man who made the universe expand soviet mathematician and meteorologist most famous for contributions to cosmology first person to mathematically predict an expanding universe 1922 derived from einsteins general relativity einstein initially dismissed friedmanns equations. Our study indicates that the approach presented here is powerful enough and further supports the viability of padmanabhans perspective of emergence gravity.
First law of thermodynamics and friedmann equations of. Dividing both sides by 3 leads to the rst friedmann equations as given in equation 16 in. General relativity, friedmann equations, and accelerating. In this report we make a detailed derivation of friedman equations, which are the dynamical equations of a homogeneous and isotropic universe. We point this out simply as historical context for what follows. The modified friedmann equations national university of. The friedmann equations were rst derived by alexander friedmann in 19221, which gave a theoretical explanation for hubbles law even before there was a hubbles law. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. Friedman and fluid equations are energy conservation. The derivation of the redsfhit for a photon, using the frw metric we have. The modified friedmann equations the national university. Either way, this is the equation that relates how the universe expands, quantitatively, to.
So equating the two we have if we assume at is unchanging during these small intervals we can take a out of the integral. Pdf in this paper, we present mathematical evidence that the beginning of the. Indeed, the derivation of this equations is intrinsically relativistic. From now on the time dependence of the scale factor can be implicit, so at a. We summarize the value of n and w for some types of massenergy in the following table 3, 4, 5. Cosmological solutions of the einsteinfriedmann equations summary of friedmanns equations ingredients are the einstein equations of grt. Derivation of the friedmann equations from general relativity. Friedmanns equations in all dimensions and chebyshevs theorem. Finding solutions to these equations and their analysis is a big part of our lectures. Friedmann equations from entropic force internet archive. One is for the case of a flat universe containing only matter. The friedmann equations university of north carolina at. Any components of tensors or pseudotensors omitted. He used general relativity to derive these equations but for simplicity, we will use newtonian approach and we will derive the same equation that friedmann did.
Apr 17, 2018 all three of these solutions are derivable from the friedmann equations. F riedmann p etersburg received june 29, 1922 zeitschrift f ur physik 1. Page 4 to solve the friedmann equations, we can use the above results to express the density n term in terms of, suitably scaled matter and radiation parts, based upon observed densities measured in the present epoch. Article pdf available in journal of cosmology and astroparticle physics 201412. These equations are a solution to einsteins field equations, and with two very important assumptions they form the basis for our understanding of the evolution and structure of our universe. More problems with the friedmann metric by miles mathis first posted december 21, 2012 the friedmann or flrw or robertsonwalker metric is a centerpiece of applied einstein field equations, being used for many things in astrophysics, includingmost famouslycalculating the expansion of the universe. A method to solve friedmann equations for time dependent. Solution of the friedmann equation determining the time evolution, acceleration and the age of the universe franksteiner abstract. Here, we have presented models of the friedmann equations incorporating different eras of the evolution of the universe the novel solution to the big bounce time period of a positively curved universe is presented here. All structured data from the file and property namespaces is available under the creative commons cc0 license.
A method to solve friedmann equations for time dependent equation of state santripti khandai m. The simplest models of the universe jodrell bank observatory. Already in 1922, alexander friedmann presented solutions to the field equations, with a time dependent scale factor, including matter english translation. F riedmann p etersburg received june 29, 1922 zeitschrift f ur physik 1einsteins reaction remark on the w ork of a. We turn now to a derivation of the equations friedmann would have used to produce this model, and that more generally constrain the evolution of any model universe given certain initial inputs. Friedmannrobertsonwalker brane cosmological equations from.
If you are interested in how these equations were derived, i recommend reading friedmanns actual papers themselves, they are listed in reference 1 below, at the end of page in reference as well as most of the important equations. Our result shows that the spatially flat rainbowfriedmann equations of early and latetime universe. Show that among the three equations two friedman equations and the conservation equation only two are independent, i. In this note by use of the holographic principle together with the equipartition law of energy and the unruh temperature, we derive the friedmann equations of. Friedmann equations from emergence of cosmic space. Some cosmologists call the second of these two equations the friedmann acceleration equation and reserve the term friedmann equation for only the first equation. Pdf friedmanns equation and the creation of the universe. A newtonian derivation of the friedmann equation 2. Cosmological solutions of the einsteinfriedmann equations. Derive the fluid equation derive the change in density as a function of scale factor r a for a single fluid universe with given equation of state, from the fluid equation derive the acceleration equation from the friedmann equation and the fluid equation discuss the cosmological constant and dark energy. For simplicity, i will focus on the special case when k 2.
Pdf friedmanns equations in all dimensions and chebyshevs. N2 the wheelerdewitt equation is developed from the einsteinhilbert action where the gravity is coupled to the massenergy and scalar field. The cosmological constant term g nowadays is usually included as part of t. The laser interferometer gravitationalwave observatory ligo detection of gravitational waves that take away 5 per cent of the total mass of. Our present understanding of the evolution of the universe relies upon the friedmannrobertsonwalker cosmological models. Even though friedmanns work was both refuted and later unrefuted by einstein himself, the solutions were note fully acknowledged until a few years after lemaitres rediscovery. Deriving the friedmann equations from general relativity the frw metric in cartesian coordinates is ds2 g dx.
A dynamical study of the friedmann equations 2 xed points of this system and discuss their stability. As stated by wikipedia, the friedmann equations are derived from general relativity, not from newtonian mechanics. From this basis, one can study all the triumphs ofthe hot big bang cosmology the ex pansion ofthe universe, the prediction ofits age, the existence ofthe cosmic microwave. The first and second friedmann equations contain, respectively, the first and second derivatives of scale factor with respect to time. Deriving the friedmann equations from general relativity. Derivation of friedman equations joan arnau romeu points of the universe. Walker spaces based on the nonredundant friedmann equation derived from a simple effective. Examine the friedmann equation and its impact on our understanding of. The third and fourth families of solutions mathematically form two branches of the same solution of the friedmann equation. Thus encoded in friedmanns equation is previously undiscovered. K 2 is directly related to the curvature radius of the spatial hypersurface. Using mathtype with adobe acrobat to create pdf files. The new analytical solution for the friedmann equations already there exist two analytical solutions for the friedmann equations. Publication of the friedmann equations on the curvature of space a.
The modified friedmann equations 123 pw h 6 and nn an a31 w oo h, 7 where w and o n are constants. Newtonian derivation consider a sphere, which expands in a homogeneous universe. Friedmann equation expansion scale factor regularly varying functions. Solving the friedmann equation for a specific universe. Derivation of parametric equations of the expansion of a. Dividing by 2 leads to equation 17 in the lecture notes and the second friedmann equation at at 1 3 4. We illustrate this analytic study by a numerical integration of this set of equations x 4 and nish by a discussion about the initial conditions explaining the current observed state of our universe x 5. Files are available under licenses specified on their description page.
This brings us to the robertsonwalker metric, in which certain factors will remain undeter mined. Of course, the density of the universe is not the same in every position when. Is it possible to forbid a specific user from executing files on usrbin without changing all files permission to 750. All three of these solutions are derivable from the friedmann equations. The failure of the friedmann metric by miles mathis first posted december 21, 2012 the friedmann metricalso called the fl metric, the rw metric, and the flrw metricis called an exact solution of einsteins field equations, and it describes the expansion of the universe, among other things. If these fonts are not embedded in your pdf files or already present on the recipients computer, inappropriate fonts will be substituted, resulting in an incorrect appearance. F riedmann publication of the friedmann equations on the curvature of space a. For nonrelativistic particles the mass inside the sphere is constant. We need to find how the radius of the sphere changes with time. Type of massenergy w n radiation 1 3 4 pressure less matter. The spatially flat friedmann equation for early rainbow universe.