Astroparticle Physics

8.7 Nature of Solutions to the Friedmann Equation 1838.6 The Acceleration Equation“Observations always involve theory.”Edwin Powell HubbleIn a number of cases it can be useful to combine the Friedmannand fluid equations to obtain a third equation involvingthe second derivative ¨R. Here only the relevant steps willbe outlined and the results will be given; the derivation isstraightforward (see Problem 4 in this chapter).If the Friedmann equation (8.15) is multiplied by R 2 andthen differentiated with respect to time, one will obtain anequation involving ¨R, Ṙ, R,and˙ϱ. The fluid equation (8.28)can then be solved for ˙ϱ and substituted into the derivativeof the Friedmann equation. This gives¨RR =−4πG (ϱ + 3P) , (8.29)3which is called the acceleration equation. It does not addany new information beyond the Friedmann and fluid equationsfrom which it was derived, but in a number of problemsit will provide a more convenient path to a solution.acceleration equation8.7 Nature of Solutionsto the Friedmann Equation“In every department of physical sciencethere is only so much science,properly so-called, as there is mathematics.”Immanuel KantWithout explicitly solving the Friedmann equation one canalready make some general statements about the nature ofpossible solutions. The Friedmann equation (8.15) can bewritten asH 2 = 8πG3 ϱ − k R 2 , (8.30)where, as always, H = Ṙ/R. From the observed redshiftsof galaxies, it is known that the current expansion rate H ispositive. One expects, however, that the galaxies should beslowed by their gravitational attraction. One can thereforedecelerated expansion?