There

_{ }are different distances in an expanding universe. The*comoving distance*_{ }is the distance measured in units that expand along with the Universe._{ }By following the path of a photon in the Robertson-Walker_{ }metric for flat space (k=0), we can obtain the_{ }relationfor

_{ }the comoving distance. Use the Friedmann equationto

_{ }find dt, withR(z) = R _{0}( 1 + z )^{-1}= R^{-3}where

_{ }R_{0}= 1, to obtain the formulaFor H

_{0}= 100 h km/s/Mpc, what is the comoving_{ }distance to the ``big bang'' at_{ }z = infinity?A

^{ }medium-sized cluster of galaxies like the^{ }Virgo cluster that we used in the^{ }previous problem set has a diameter of D = 1.8 h^{-1}Mpc. If this cluster were to^{ }keep the same comoving size,^{ }ie. it were to expand with the Universe^{ }D _{com}= D = 1.8 h^{-1}Mpcthen

_{ }its angular diameter would be given by_{ }the relation= D / Show

^{ }that this comoving angular size reaches a finite limiting^{ }angular diameter at z = infinty, and^{ }calculate this angular diameter for our 1.8 h^{-1}comoving Mpc cluster, in arcseconds.^{ }Consider

^{ }now a galaxy of a fixed physical angular diameter of^{ }D = 0.1 h^{-1}Mpc. This will not in general expand with^{ }the universe, and thus^{ }in comoving coordinates it will^{ }appear to shrink with the scale factor^{ }D _{com}= ( 1 + z ) Dso

_{ }its apparent size will follow= ( 1 + z ) D / = D / d _{A}where

_{ }we have used this to define the_{ }*angular diameter distance*d _{A}= / ( 1 + z )which

_{ }is true for all cosmologies.Show

_{ }that for our ``flat'' universe from the_{ }previous problemand

_{ }thus the apparent angular diameter reaches_{ }a minimum value at a finite redshift._{ }Compute these z_{min}and_{min}in arcseconds for the galaxy. (Hint: find where d_A is a maximum.)_{ }Note that this means that in a flat_{ }universe, very distant objects need not_{ }appear very small!What

_{ }is the current critical density for_{ }the universe in units_{ }of h^2 Msun/Mpc^3 for H_{0}= 100 h km/s/Mpc?Calculate

_{ }the current critical energy density (equivalent to saying_{ }E = mc^2)u _{cr}= c^{2}= 3 H_{0}^{2}c^{2}/ ( 8 G )again

_{ }for H_{0}= 100 h km/s/Mpc.Calculate

_{ }the energy density in the cosmic background radiation_{ }u _{rad}= a T^{4}= 4 T^{4}/ cfor T

_{0}= 2.73 K.What

_{ }fraction of the critical density_{rad}= u_{rad}/ u_{cr}does

_{ }the cosmic microwave background contribute? (Note: these will be_{ }parameterized by h.)If

_{ }density evolves as= ( 1 + z ) ^{3}and

_{ }temperature evolves asT = T _{0}( 1 + z ),at

_{ }what redshift z_{cr}will the energy density of the radiation_{ }equal the critical density (_{rad}= 1 )?_{ }Note that this is approximately the redshift where matter_{ }and radiation have equal ``densities''._{ }(Hint: this is an estimate, so don't_{ }worry about keeping the total energy density_{ }equal to the critical density.)_{ }The

_{ }Coma cluster of galaxies is even more_{ }impressive than the Virgo cluster,_{ }and has a radial velocity dispersion of_{r}= 977 km/s._{ }What is the true 3-D velocity dispersion_{ }of the galaxies_{v}?If

^{ }the virial radius of the Coma cluster is 1.8 h^{-1}Mpc, what is the implied total^{ }virial mass of the cluster (in Msun)?^{ }Be sure to express this^{ }in terms of the Hubble parameter h.^{ }If

_{ }the absolute visual magntitude of the Coma cluster is M_{V}= -26.92, what is the visual luminosity of Coma_{ }(in Lsun)? What is the implied_{ }mass-to-light ratio of the Coma cluster?_{ }If

_{ }an average galaxy in the Coma cluster has_{ }a mass-to-light ratio typical of spiral or_{ }the centers of normal elliptical galaxies ( ~ 10 Msun/Lsun),_{ }what is the total mass of the galaxies_{ }in the Coma cluster? Compare this to the_{ }total gravitational mass computed above._{ }What fraction of the total mass_{ }is in galaxies?We

_{ }derived the ``cosmic energy equation''_{ }using Newtonian mechanics applied_{ }to a uniform universe. In this formulation,_{ }we used the fact that only the mass_{ }interior to some scale R affects the dynamics._{ }In a sense, therefore, a cluster of_{ }galaxies can be thought of a portion of the universe_{ }that behaves like a k=1 closed model._{ }Its evolution therefore behaves similarly:_{ }it first expands with the Hubble flow, but the_{ }gravitational force from its overdensity causes_{ }the expansion to turn around,_{ }and it then recollapses into_{ }the cluster we see today.Calculate

_{ }the density of the Coma cluster_{cl}(in kg/m^3) using the previously derived mass_{ }and the given virial radius._{ }Use this and calculated above_{ }to compute the*overdensity*of_{ }the cluster_{cl}=_{cl}/ .Is

_{ }the overdensity of the Coma cluster indicative of a piece of a_{ }closed universe?What

_{ }would be the radius R_{L}of the Coma cluster if it has the same mass,_{ }but a density equal to (which we_{ }assume is the density of the universe for this problem)?_{ }This radius R_{L}is called the*Lagrangian radius*,_{ }and represents the comoving size_{ }of the patch of the Universe that collapsed to_{ }form the cluster, assuming we began_{ }long ago with a nearly uniform universe with_{ }only small density perturbations.

*smyers@nrao.edu*
*Steven T. Myers*