Astroparticle Physics

178 8 CosmologyNewtonian gravityBirkhoff’s theoremforce is the same as what one would obtain if all of the massinside the sphere were placed at the center.Another non-trivial consequence of the 1/r 2 force is thatthe galaxies outside the sphere do not matter. Their totalgravitational force on the test galaxy is zero. In Newtoniangravity these properties of isotropically distributed matterinside and outside a sphere follow from Gauss’s law for a1/r 2 force. The corresponding law holds in general relativityas well, where it is known as Birkhoff’s theorem.If one assumes that the mass of the galaxies is distributedin space with an average density ϱ, then the mass inside thesphere isM = 4 3 πR3 ϱ. (8.11)The gravitational potential energy V of the test galaxy isthereforeV =− GmMR =−4π 3 GmR2 ϱ. (8.12)The sum of the kinetic energy T and potential energy V ofthe test galaxy gives its total energy E,E = 1 2 mṘ2 − 4π 3 GmR2 ϱ = 1 2 mR2 (Ṙ2R 2 − 8π 3 Gϱ ).(8.13)curvature parameterThe curvature parameter k is now defined by()k = −2Em = 8πR2 Ṙ2Gϱ −3 R 2 . (8.14)If one were still dragging along the factors of c, k wouldhave been defined as −2E/mc 2 ; in either case k is dimensionless.Equation (8.14) can be written asFriedmann equationṘ 2R 2 + k R 2 = 8π Gϱ , (8.15)3which is called the Friedmann equation. The terms in thisequation can be identified as representingT − E =−V, (8.16)i.e., the Friedmann equation is simply an expression of con-servation of energy applied to our test galaxy. Since theenergy conservation