Notes
Slide Show
Outline
1
ASTR 5460, Fri. Mar. 28, 2003
  • Catching up, updating
  • Assignment Discussion
  • Astro-ph papers
  • Ch. 7 Longair: Friedman World Models
2
Catching Up/Updates
  • WIRO issues: weather, engineering
    • No WIRO observing project this semester


  • “Mini-TAC” project results via email


  • Fixes to Bruzual & Charlot bugs via email


  • Speed issue: Limit astro-ph discussion to 10-15 minutes total after today, but retain?
3
Assignments
  • Will have a few individual assignments
  • Will have 20-25 minute talks the last week (topics to be distributed mid-April)
  • Discuss replacement of WIRO project w/
    • “Post-starburst Quasars in the SDSS EDR”
  • For next week, read through Ch. 8 plus Sandage (1988) – who will lead it?
    • http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988ARA%26A..26..561S&db_key=AST&high=3d6571051d26540

  • Finally, not to turn in, but please learn Ned Wright’s cosmology calculator
    • http://www.astro.ucla.edu/~wright/CosmoCalc.html
  • Astro-ph papers for today.
4
Friedman World Models
  • Einstein’s Field Equations
    • Under ideas discussed previously (cosmological principle, Weyl’s postulate, isotropy, homogeneity) the field equations reduce to the simple pair of independent equations:






    • R is the scale factor, ρ is total inertial mass density of matter & radiation, p the associated pressure.  Script R is the radius of curvature, and there’s lambda.
5
The Standard Dust Models (Λ=0)
  • “Dust” means pressureless fluid, p=0
    • Field equations then reduce to:




    • Have a Newtonian Analog:
6
The Standard Dust Models (Λ=0)
  • Have a Newtonian Analog:



    • Replace x by comoving value r using the scale factor R, x = Rr, and express density in terms of its value at the present epoch ρ = ρ0R-3, then:



    • Which matches eq. 7.1 for dust and lambda=0.  Multiplying by the derivative of R and integrating gives us essentially eq. 7.2
7
The Standard Dust Models (Λ=0)
  • Critical Density and the Density Parameter:



  • The ratio of the current density to the critical density is “omega-naught” Ω0:



  • Often use different subscripts on omega to denote density contributions from baryons, dark matter, etc.
8
The Standard Dust Models (Λ=0)
  • Inserting Ω0 into eq. 7.1 and 7.2:





  • Setting eq. 7-18 to present epoch, t =t0, R=0, and derivative of R is H0, then:



  • And we see that curvature and density are intimately related.
9
The Standard Dust Models (Λ=0)
  • Dynamics: using the previous equations, we can rewrite eq. 7-18:



  • And in the limit of large values of R, we get:



  • This is easy to interpret.
    • When Ω0 < 1, universe is open, hyperbolic, and expands to infinity with finite velocity.
    • When Ω0 > 1, universe is closed, spherical, and eventually collapse after reaching a maximum size after a time:
10
The Standard Dust Models (Λ=0)
  • This is easy to interpret.
    • When Ω0 < 1, universe is open, hyperbolic, and expands to infinity with finite velocity.
    • When Ω0 > 1, universe is closed, spherical, and eventually collapse after reaching a maximum size after a time:


    • Recollapses after a time t = 2 tmax.
    • When Ω0 = 1, universe is critical, flat, and expands to infinity with velocity approaching zero.  Einstein-de Sitter model:
11
The Standard Dust Models (Λ=0)
12
The Standard Dust Models (Λ=0)
13
The Standard Dust Models (Λ=0)
  • Deceleration Parameter, q0:




  • Substituting into the first of the dynamics equation (7-20) we immediately can write:




  • Keep in mind this (and all these results so far) are for universes with zero cosmological constant.
14
The Standard Dust Models (Λ=0)
  • Cosmic Time-Redshift Relation:
    • Because R = (1 + z)-1, eq. 7-20 gives us




    • Which can be integrated to give cosmic time since the big bang.  For different types of universes need different forms of the equations:
15
The Standard Dust Models (Λ=0)
  • The Flatness Problem
    • Can determine how Hubble’s “constant” changes with time, from eq. 7-20 and writing R = (1+z)-1:



    • Similarly for Omega, using the general definition Ω=8πGρ/3H2, and expressing the density ρ = ρ0(1+z)3, then



    • And rewriting...




    • Notice the behavior at high z.  This is the origin of the problem.
16
The Standard Dust Models (Λ=0)
  • Distance Measures as a function of redshift
    • Radial comoving distance coordinate r incremental is







    • Integrate from redshift 0 to z:
17
The Standard Dust Models (Λ=0)
  • Distance Measures as a function of redshift
    • Then to find the proper distance D, recall from chapter 5 that D = R sin(r/R) where script R is given by eq. 7-19.  For an exercise you could derive the general expression:




    • Notice what happens for an empty universe.  Deriving this was a closed book exam question I had in grad school (prof assumed we had the curiosity to check this for ourselves).
18
The Standard Dust Models (Λ=0)
  • Observed Properties of Standard Objects in the Friedman World Models with zero cosmological constant (cf. Hogg 2000, chapter 5, Ned Wright’s calculator).


    • Angular Diameters (need Angular Distance f(z))


    • Flux Densities (need luminosity distance f(z))


    • Comoving volume within redshift z
      • In particular covered in more detail in section 7.2.8
19
The Standard Dust Models (Λ=0)
20
The Standard Dust Models (Λ=0)
21
Models for which Λ is not 0
  • We will cover this and inhomogeneous world models on Wednesday, discuss the Sandage article on Friday, and also start chapter 8 on Friday which mirrors much of the Sandage article.


  • Cover at a basic level thermal history of universe, big bang nucleosynthesis, simple galaxy formation, cosmic background fluctuations over the next few weeks, and then spend the remainder of the time on AGNs (1-2 weeks) and individual presentations.