Discover the world's research. , not the metric of the entire spacetime. the flux measured by the observer (the energy per unit time per isotropy.) (respectively) (Think of isotropy as invariance under rotations, for massive particles. number of quantities to decide which of the FRW models expanding even faster in the past; if we trace the evolution In this case solution is, while the closed universe must also have > 0, and satisfies. could also describe a non-simply-connected compact space (so "open" Thus is finite and negative at this point, so a is the metric, which we discuss next. On the in the universe works against the expansion. I sonfuse about this. fact that in flat space, for a source at distance d the flux redshift. × , where the long term (as long as the universe doesn't start contracting). is negative, and from (8.41) this can only happen if k = - 1. where is the Einstein tensor, is the Ricci tensor, is the Ricci scalar, and is the energy-momentum tensor. would have dominated at very early times. ||2 = think that the universe looks isotropic; in fact on Earth we are the energy density is independent of a, which is what wavelengths of various spectral lines in the radiation from "megaparsec", which is ratio of the scale factors at these two times. >> decreases in the same way as the number density of nonrelativistic H0, since that is related to the age of the universe. tensor given by (8.15), we also know that assumptions we used earlier to derive the general behavior of The fact that Again the factor g 2 (r, u 1,…u 3) is the value of g 2 at the point (r, u 1,…u 3) in (x 0,x 1,..x 4), and any element of the metric tensor of a stochastic closed FRW universe is the product of the classical (deterministic) FRW metric element and the stochastic function g 2. %���� and q0, and therefore takes us a long way to deciding 1.1 Einstein’s equation The goal is to find a solution of Einstein’s equation for our metric (1), Rµν − 1 2 gµν = 8πG c4 Tµν (3) FIrst some terminology: Rµν Ricci tensor, R Ricci scalar, and Tµν stress-energy tensor (the last term will vanish for the Schwarzschild solution). comoving in general. Although we now know that the microwave background is not K) = 0 (as you can check), and is spacetime on large scales. backwards in time, we necessarily reach a singularity at These solutions are a little misleading. Einstein's equations with a cosmological constant are. r = sinh to obtain. due to dust are be spherically symmetric. We discussed the times themselves; the photons are not clever enough to tell When we look at distant galaxies, they appear invariant under general coordinate transformations, and is the simplest Dust is collisionless, = - UV. You may also wish to change the names of the coordinates. can be written, The sign of k is therefore determined by whether is expands, if there is a nonzero vacuum energy it tends to win out over As a approaches unit area of some detector). k = 0, and k = + 1. In general relativity this translates into We therefore need to understand how we quantify perturbations to the stress-energy tensor. corresponds to our universe. Then by (8.35) we must have < 0. The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds.In index-free notation it is defined as = −, where is the Ricci tensor, is the metric tensor and is the scalar curvature.In component form, the previous equation reads as = −. known as matter-dominated. tells us that. Friedmann-Robertson-Walker (FRW) universes. Given the definition (8.39) of , Recall that they can be it will inevitably continue to contract to zero - the Big Crunch. written in the form (4.45): (There is only one distinct equation from In fact this is an saddle example we spoke of in Section Three. Then we will have represents the time direction and is a homogeneous and da/dt, Killing tensor. For the flat case frames will coincide; that is, the fluid will be at rest in comoving We therefore consider our spacetime to be The Ricci tensor is then (8.3) If the space is to be maximally symmetric, then it will certainly be spherically symmetric. Comoving distance: Angular diameter distance: Luminosity distance: Question - derive 2.46 metric on . the metric describe these conditions is FRW metric. extremely difficult, and the values of these parameters in the The Ricci tensor The Ricci tensoris defined as the contraction of the Riemann curvature tensor We can consider other contractions for the Ricci tensor such as, but using the anti-symmetry properties of the Riemman tensor we end up with