1. WEIGHING THE UNIVERSE
Neta A. Bahcall
Princeton University

2. Why Weigh Universe? How much matter in Universe?
Is there Dark-Matter? Where is it located?
Is there Non-baryonic (‘exotic’) dark-matter? What is it?
[Baryon limit is ~4-5% of critical-density.]
Most fundamental cosmological parameter
 Cosmology; Evolution of Universe;
Age of Universe; Galaxy Formation; Gravity

3. Mass Density of Universe
How much? How distributed?
Mass-to-Light Function
Baryon Fraction
Cluster Abundance and Evolution
Other Large-Scale Structure Obs.
 All yield m ~ 0.25
 Mass ~ Light (on large scales)

4. Mass-Density (Units) m = 1 is the critical density  ‘Flat’ Universe
Critical mass-density (= density needed to halt the Universe expansion):
critical = 3Ho2/8G
~10-29g/cm3 ~ 6 p/m3
m = m/crit
m = 1 is the critical density
 ‘Flat’ Universe
 b(baryons)(observed) ~ 0.04
[Mpc = 106pc; 1pc ~ 3 ly; Mo=2E33g]

7. Flat Rotation Curves M ~ v2R ~ R M/L ~ R [GMm/R2~mv2/R]
Kaptyen (Local) 1920’s
Zwicky (Clusters) 1930s
Rubin (Galaxies) 1970s ( M/L ~ R )
M ~ v2R ~ R
M/L ~ R
[GMm/R2~mv2/R]

8. Mass-to-Light Method m = m/critical
 <M/L>cl Luniv(Lo/Vol) = m(Mo/Vol)
 Weigh cluster mass, Mcl (<R~1Mpc)
<M/L>cl = 300h
m = m/critical
 m ~

10. Weighing Clusters Motion of galaxies [MR ~ v2R]
3 Basic Methods
Motion of galaxies [MR ~ v2R]
 Temperature of hot gas [MR~TR]
 Gravitational lensing [MR]

13. Mass-to-Light Function (Bahcall, Lubin & Dorman ‘95; Bahcall and Fan ‘98)
SDSS
Ωm=0.2

14. Theory vs. Observations
(Bahcall, Yu, et al ‘01)

15. Cluster M/Li(R) Profile (SDSS, weak lensing 2x104 clusters N= 3 to 220 (Sheldon etal 2008)
Flat >~ 1Mpc
M ~ L
X=R(vir)

16. M/Li(r=22Mpc) vs. Mcl (SDSS; Sheldon etal ‘08)
Flat M/L on large scales; SAME for ALL clusters!

17. M/L Function: Conclusions
 M/L Function Flattens on Large Scales:
 M ~ L (on large scales)
reaching the end of Dark-Matter
 Total Mass-Density of Universe:
 m =

18. Baryons in Clusters [Stars and Gas]
 Ωb/Ωm(cl)  Mb/Mtot(cl) =
0.13 (gas) (stars) = 0.16 (h=0.7)
Ωb(BBN; CMB) = (h=0.7)
Ωm = Ωb/(Ωb/Ωm) =
 corrected for gas outflow

19. Baryon Fraction vs. Scale ( 0.18) (Bahcall & Martin ‘07)

20. m from Baryon-Fraction
b/m = h=0.7
(Clusters; CMB)
b = (BBN; CMB)
 m =

21. Weighing the Universe  M/L Function m= 0.2 +- 0.05
 Baryon Fraction
 Cluster Abundance
and Evolution [8 = ]
Supernovae Ia + Flat
CMB + LSS + h + Flat
 m ≈
 4% Baryons + ~20% Dark Matter
Mass ~ Light (R >~ 1Mpc)

22. Cosmic Acceleration: Supernovae

23. Cosmic Acceleraion: Supernovae (‘07)

25. Cosmic Microwave Background (WMAP)

26. CMB Spectrum

27. Space Curvature

28. The Cosmic Triangle m +  + k = 1 Mass Density: m = 0.25
(Friedmann’s eq.)
Mass Density: m = 0.25
Dark Energy:  = 0.75
Space Curvature: k = 0

29. Mass-density, Curvature, Expansion
H2(t) = 8G(m + )/3 - k/a2(t)
k = 0 Flat geometry (no curvature)
1 Closed (positivly curved space)
-1 Open (negatively curved space)
/H2  m + + k = 1 Friedmann Eq.

 m ~ a-3
 ~ constant (IF Cosmological Constant)

31. Cosmic Triangle  Mass Density of Universe: 25% Critical
 Universe will expand forever
Dark Energy in Universe: 75%
 Universe expansion accelerates
Universe Space Curvature: 0
 Universe ‘Flat’

32. Fate of Universe 
Universe Will Become:
 Larger
 Sparser
 Darker
 Colder

34. The Cosmic Triangle

36. Hot Gas in Clusters (X-Rays; S-Z)
(Carlstrom etal)

37. Mass Density of Universe
How much? How distributed?
Mass-to-Light Function
Baryon Fraction
Cluster Abundance and Evolution
Other Large-Scale Structure Obs.
 All yield m ~ 0.25
 Mass ~ Light (on large scales)

38. Mass-to-Light Function M/L(R)
How does M/L depend on scale?
How and where is the mass distributed?
How use it to weigh Universe?
<M/L>rep Luniv(Lo/Vol) = m(Mo/Vol)
Determine M, <M/L> of clusters, SCs, LSS
 <M/L> rep [≈ 300h ]
 m ~

39. Cluster (M/L)200 versus M200
M/L~M
M/L ~ M

40. M/L Function: Conclusions
M/L Function Flattens on Large Scales
 M ~ L (reaching end of Dark-Matter)
Dark Matter located mostly in large galactic halos 100s Kpc) Group/Clusters: made up of Sp+E mix (+their DM halos); no significant additional DM
Cluster M/L increases slightly with M (mergers?)
Rich clusters M/LB is ‘Anti-biased’ (M/LB>mean)
Asymptotic Cluster M/Li(22Mpc) is same for ALL
Groups and Clusters, h !
Mass-Density of Univers: m =

41. III. Cluster Abundance and Evolution
 Powerful method to determine m and 8
8 = Amplitude of mass fluctuations
(initial ‘seeds’)
ncl (z~0)  8 m0.6 ~ 0.35
ncl (hi z)  Breaks degeneracy
 m= and 8=

8 (galaxies)(obs) ~ 0.9
If Mass ~ Light (on large scale)  8(m)~ 0.9

42. Cluster Mass-Function (SDSS) (Bahcall, Dong, et al ‘02) Best-fit MF: m=0.2 and 8=0.9

43. m - 8 constraints from MF: m = 0.2 and 8 = 0.9

44. m - 8 constraints from SDSS cluster MF [Bahcall etal ‘03 Rozo etal ’09]

45. Cluster Abundance Evolution  8 (Bahcall & Bode)

46. Cosmological Constraints (Bahcall & Bode) (from Low and Hi redshift cluster abundance)
Low z
Hi z

47. Cosmic Acceleration: Supernovae (ESSENCE ‘08)

49. Cosmological Constraints Supernovae, CMB, Clusters

50. CMB Spectrum (Seivers etal ’09)

51. SDSS Clusters (Rozo etal ‘09)

52. Mass-to-Light Function (Bahcall, Lubin & Dorman ‘95; Bahcall and Fan ‘98)
1. M/L flattens on large-scales: M ~ L. End of Dark Matter.
2. Sp + E produce M/L of groups, clusters; Clusters have no excess DM !
3. Most of the DM is in huge halos around galaxies (few-100 Kpc)