READING AND CONFERENCE
READING AND CONFERENCE PH 405
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This 2 page Class Notes was uploaded by Mrs. Demarcus Breitenberg on Monday October 19, 2015. The Class Notes belongs to PH 405 at Oregon State University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/224537/ph-405-oregon-state-university in Physics 2 at Oregon State University.
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Date Created: 10/19/15
A Summary of Useful Equations for Cosmology A W Stetz February 19 2007 The starting point for mathematical cosmology is the RobertsonWalker met ric s1n2 X closed d32 7dt2 at2R2 dX2 X2 d62 sin2 9 d452 at sin 2 X open d 2 H 1 closed d32 7dt2 at2R2 1 2 7 2 d62 sin2 9 d4 H 0 at 7 HT H 71 open These are equations for the invariant length dsl The parameter at is the cosmic scale factor By convention it is normalized to unity at the present time at0 l The time variable is cosmic time77 It is assumed that all observers can synchronize their watches to t In the language of Star Trek its the stardatel The other variables x 9 45 and 7 are angles measured in a comoving coordinates The distance R has many uses Usually it cancels out somewhere in the calculation The Friedman equations 51 2 7 87rG H62 a 7 3 p R3a2 d P 330 P 0 amp 47rG 7 77 SP a 3 P as a set predict the time dependence of a G sometimes called GN is Newtonls gravitational constant p is the density of matter and energy in the universe It not only includes matter and radiation but also in our present understanding the dark energy 7 A pA 7 87rG The equation of state gives pressure P in terms of p P 10252 Where w 0 for matter w 13 for radiation and w 71 for dark energy The critical density is 7 3H2 m 7 87rG where H da is the Hubble parameteri It is the density required to make a at universe When densities are normalized by p0 they are given the symbol 9 n pa Since most equations are linear in p it makes sense to break it up into its various constituentsi Q QTQmQAi The zero subscript always refers to the present time as in to7 H07 90 etc By de nition at0 1 These quantities scale with a in a simple way 9m QTgt0at 4 9mm Qm aars 9m QM The proper distance dz is the invariant distance d3 measured along a straight line d9 dab 0 from the observer to a distant object at one value of the cosmic timer The position of the distant object can be characterized in one of several ways the time tam at which it emitted some light we are just now observing7 the value of the cosmic scale factor at this time7 or the redshift of the light These yield three equivalent expressions cdz to Cdt 1 cda z dp em A m dpa Am W d122 0 HZ
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