cosmic time and distance calculations

8/9/2019 Cosmic Time and Distance Calculations

1/5


2/5


3/5

3 Friedmanns Equation

Shortly after Einstein developed his eld equations for general relativity, AlexandreFriedmann derived the R-W metric as a solution. The R-W metric is the equation of motion for the scale factor, a(t).

Dene the Hubble rate H (t) a/a ; its present value is H 0 = 71 km/s/Mpc asabove. H (t) changes in time in accordance with Friedmanns equation:

H 2 (t) =8G

3(t) +

c 0a2

[Dodelson, p 3] (2)

where (t) is the average mass-energy density of the universe at time t (gm/cc), 0 isits present value, and G is Newtons gravitational constant. The critical density, c ,

is dened below.At this point we need to recognize that the time, t, is not a useful independentvariable because we do not have expressions for the important quantities such asdensity in terms of time. A shift to using the scale factor, a, as the independentvariable is just conceptualnone of the equations needs to be changed except toreplace t with a. In the situations of interest, a(t) is monotonic and makes just asgood a time variable as t. (The relation between t and a will be derived later.)

4 Density

We need an expression for , the average density of matter and energy in the universeas a function of the expansion ratio. This is complicated by the fact that it hasthree components, each of which varies differently with time. At the present time,it consists of about 27% matter (both atomic and not), 73% vacuum energy, and0.0049% radiation. These values are accurate to a few percent.

The matter density (think atoms or particles) scales inversely as the volume of space (i.e. as a 3 .) The density of radiation particles (think photons) also scales in-versely as the volume but in addition they lose energy as their wavelength is stretched;their energy density therefore scales as a 4 . Vacuum energy has constant densityitis a property of space and more of it appears as space expands. [Dodelson, p 4 andSection 2.4]

The average mass-energy density of the universe is related to the curvature of space by Einsteins eld equations. Since at space is the boundary between positiveand negative curvature, the density for at space is called the critical density , c . If the actual density is greater than c , space is positively curved and has nite extent(but no boundary). A lower density gives negative curvature and innite extent.

3


4/5

The present value of the critical density is calculated by setting all variables in

equation (2) to their present values ( H (t) = H 0 , (t) = 0 and a = 1): c = 3H 2

0 / 8G.(It equals 9 .5 10 30 gm/cc for H 0 = 71 km/s/Mpc.) From the WMAP satellite andother sources, we now know that space is geometrically at, so its density is equal tothe critical density.

Let the density of matter, radiation and vacuum be M , R and V gm/cc. We thendene three dimensionless density ratios: M = M / c , R = R / c and V = V / c .The variation of density with the expansion is thus (a) = c (M a 3 + R a 4 + V ).As we go back in time (i.e., as a gets smaller), the radiation term grows fastest andeventually dominates.

Combining this with equation (2) gives an expression for the Hubble rate:

H (a) = H 0 M a

3+ R a

4+ V + (1 )a

2, where = M + R + V (3)

5 Times and Distances

Since the Hubble rate is dened to equal a/a = ( d a/ dt)/a , a rearrangement of equation (3) gives d t = d a/ (aH (a)), so the time it takes the expansion ratio to gofrom a 0 to a1 is

t(a0 , a 1 ) =a 1

a 0da

aH (a)(4)

The time unit chosen for H 0

determines the unit for t. Numerically integrating thisequation gives the present age of the universe as t(0, 1) = 13.7 109 years. Its agewhen the galaxy emitted our z = 6 photon was t(0, a 1 ) = 0 .95 109 years, wherea1 = 1 / (1 + z). The light travel time is the difference of these, 12 .7 109 years.

We can ask how far away that z = 6.0 galaxy is right now. The fact thatits light took 12 .7 109 years to reach us does not mean that the galaxy is now12.7 109 light-years distant. Because space is expanding, light rays are not veryuseful for measuring large distancesthe distance changes signicantly as the lightwave travels. The distance of a galaxy at a particular time is conceptually what youwould get if you could somehow line up a huge number of yardsticks between hereand there and read them all at that one time. Fortunately, the R-W metric will give

us the same answer without all that work [see Dodelson, p 34.] The proper distanceat the present time is given by:

D 0 = c1

a 1da

a 2 H (a)(5)

4


5/5

This equals 27.5 109 light years for a 1 = 1 / (1 + z) with z = 6. The galaxys proper

distance when the photon was emitted was D 1 = a1 D 0 = 3 .92

109

light years. Theuniverse expanded by a factor of z + 1 = 7 since that time.Since proper distance is not directly measurable, astronomers have devised various

ways to determine the distance of stars and galaxies. For nearby stars, the Hipparcossatellite determined accurate distances by measuring their parallax; i.e., their appar-ent shift as the Earth moves in its orbit. At the largest distances, type Ia supernovaeare used as standard candles to calculate distance from their relative brightness.Dodelsons book discusses distance measures and their relation to proper distance;formulas are given to convert between them.

The value of the Hubble constant at an earlier expansion ratio, a1 , is just theHubble rate H (a1 ). The speed of the galaxy away from us now is S 0 = H 0 D 0 = 1 .995times the speed of light; and when the light was emitted, the galaxy was moving awayat S 1 = H (a1 )D 1 = 2 .756 times the speed of light. Although the photon was travelingtoward us at the speed of light (just as it should), the space between us was stretchingfaster than that. The photons distance actually increased for about 9 billion yearsbefore it nally got enough space behind it that the remainder was not expandingfaster than light. Note that none of this violates the special relativity prohibition onmoving faster than lightthe galaxies are nearly stationary in their space, it is spaceitself that is moving them apart, and special relativity does not apply to that.

6 Bibliography

1. Scott Dodelson: Modern Cosmology, Elsevier 2003

2. John A. Peacock: Cosmological Physics, Cambridge, 1999

3. Joel Primack and Nancy Abrams: The View from the Center of the Universe,Riverhead Books, 2006

4. Tomas Weil: Another Look at Cosmic Distances, Sky and Telescope, Aug,2001, p62

5. Steven Weinberg: Gravitation and Cosmology, Wiley, 1972

6. Ned Wrights website: http://www.astro.ucla.edu/~wright/cosmolog.htm

5

cosmic time and distance calculations

Documents