Measuring the Curvature of the Universe by Measuring the Curvature of the
Hubble Diagram

Several groups are measuring distant supernovae with the goal of determining
whether the Universe is open or closed by measuring the curvature in the Hubble
diagram. The figure below shows a binned version of the latest dataset: Riess et al. (2007).

The curves show a closed Universe (Ω = 2) in red, the critical density
Universe (Ω = 1) in black, the empty Universe (Ω = 0) in green, the steady state
model in blue, and the WMAP based concordance model with ΩM = 0.27
and ΩV = 0.73 in purple. This model gives Ho = 71
km/sec/Mpc which has been used to scale the luminosity distances in the plot.
The data show an accelerating Universe at low to moderate redshifts but a
decelerating Universe at higher redshifts, consistent with a model having both a
cosmological constant and a significant amount of dark matter. The dashed black
curve shows an Einstein-de Sitter model with a constant co-moving dust density
which can be ruled out. The dashed purple curve shows a closed ΛCDM model which
is a good fit to the data. The dashed blue curve shows an evolving supernova
model which is also a good fit.

Note that this Riess etal (2007) dataset is a
compilation of data from many sources and there are indications that there are
systematic differences between these subsets.

Both the Supernova Cosmology Project and the High-z Supernova Team
groups were the subject of news articles in Science, on 30 Jan 1998 and 27 Feb
1998. I have combined their two error ellipses along with another constraint
from the circa 1998 knowledge of the location of the acoustic peak in the
angular power spectrum of the CMB anisotropy. The two SNe groups gave very
similar error ellipses, and the combined CMB-SNe fit indicates that a flat
Universe with a cosmological
constant is preferred. But the systematic errors on the SNe data, shown as
the large grey (or pink) ellipse, could allow for a vanishing cosmological
constant lambda. The red, black, green and blue circles on the Figure to the
right are keyed to the colors of the curves on the Figure shown above. A larger GIF
file or a Postscript version of
this figure are available.

The supernova data as of late 2006 published by Riess et al. (2007)
provide the 0.3, 1, 2 and 3 standard deviation contours shown as the green,
blue, red and black ellipses in the figure at left. The CMB data using WMAP
three year results provide the cloud of dots from a Monte Carlo Markov chain
sampling of the likelihood function. The CMB degeneracy track does not follow
the flat Universe line, but crosses the flat line at a point reasonably
consistent with the supernova fit. Each CMB model has an implied Hubble constant
which provides the color code for the dots. A model that fits both the supernova
data and the CMB data has a Hubble constant that agrees reasonably well with the
Hubble Space Telescope Key Project value of the Hubble constant.

The addition of high redshift supernovae has had two effects on the supernova
error ellipse. The long axis of the ellipse has gotten shorter, and the slope of
the ellipse has gotten higher. Even though the best fit model has gotten closer
to the CMB degeneracy track in absolute terms, it has gotten further away in
terms of standard deviations because of the shortening of the error ellipses.

In the last few years distant supernovae with redshifts up to 1.755 have been
observed by the Hubble Space Telescope. These objects show that the trend toward
fainter supernovae seen at moderate redshifts has reversed. This reversal means
that one possible alternative to the accelerating Universe as the explanation of
the fainter supernovae at z near 0.5 can be rejected. This rejected
alternative proposed that dust between galaxies made the distant supernovae
fainter by absorbing some of their light. In the plot below, the brightness or
faintness of distant supernovae relative to the empty Universe model is plotted
vs redshift.

The black curve shows a critical density Ω=1
Universe, the green curve is the Ω=0 Universe, the red curve shows Ω=2, and the
blue curve shows the de Sitter or Steady State Universe. Adding dust to the Ω=1
model gives the black dashed curve, which fails to fit the new data point at
redshift z = 1.755. But the blue dashed curve is an Ω=1 model with
evolving supernovae with luminosity going like exp(alpha*t). This alternative to
the accelerating Universe is still a very good fit to all the supernova data.
The solid magenta curve shows the consensus or WMAP flat accelerating
vacuum-dominated model. The dashed magenta curve is the best closed dark energy
dominated fit to the supernova data alone.

The data points on the above plot are from my binning of the Riess et al.
(2007) data
table. The binned normal points are
n zmin zmax <z> d(DM) sigma
31 0.00700 0.02100 0.01484 -0.0464 0.1383
31 0.02300 0.05000 0.03352 0.0063 0.0691
16 0.05100 0.12400 0.07131 0.0725 0.0644
7 0.16000 0.24900 0.20671 0.0916 0.0878
18 0.26300 0.35900 0.32239 0.0751 0.0506
31 0.36900 0.46000 0.42323 0.1665 0.0406
31 0.46100 0.52600 0.49016 0.2700 0.0395
29 0.52600 0.62000 0.56921 0.1521 0.0375
20 0.62700 0.72100 0.67190 0.0969 0.0478
24 0.73000 0.83000 0.79029 0.0799 0.0519
17 0.83200 0.93000 0.87647 0.0464 0.0697
19 0.93500 1.02000 0.97011 0.0155 0.0696
4 1.05600 1.14000 1.11400 0.0168 0.1179
5 1.19000 1.26500 1.22280 -0.0870 0.1275
6 1.30000 1.39000 1.33533 -0.1505 0.0998
1 1.40000 1.40000 1.40000 0.0371 0.8100
1 1.55100 1.55100 1.55100 -0.4897 0.3201
1 1.75500 1.75500 1.75500 -0.5993 0.3501

where d(DM) is the difference between the distance modulus determined from
the flux and the distance modulus computed from the redshift in the empty
Universe model, and sigma is the standard deviation of the d(DM) in the bin. I
use a robust statistical technique to get the binned values and therefore
include both the Gold and Silver samples. I also include the low redshift
supernovae which of course only affect the low z bin. But I have assumed a 1500
km/sec uncertainty in the redshift when computing the d(DM) which de-weights the
low redshift bin.

I don't see much difference between the Gold+Silver data and the data
restricted to Gold, but here is a binning of the Gold data alone:
n zmin zmax <z> d(DM) sigma
22 0.01000 0.02100 0.01536 0.0186 0.1599
22 0.02300 0.04000 0.02986 0.0441 0.0892
18 0.04300 0.12400 0.06467 0.0387 0.0635
4 0.17200 0.26300 0.21600 0.1356 0.0912
12 0.27800 0.37100 0.33167 0.0720 0.0551
22 0.38000 0.47000 0.43777 0.1798 0.0446
22 0.47000 0.54000 0.50223 0.2119 0.0462
22 0.54300 0.64000 0.59268 0.1092 0.0417
11 0.64300 0.74000 0.69855 0.0930 0.0607
18 0.75600 0.85400 0.81217 0.0422 0.0607
13 0.86000 0.95400 0.91862 0.0140 0.0767
8 0.96100 1.05600 0.99863 0.1141 0.0912
4 1.12000 1.19900 1.14975 -0.0473 0.1273
4 1.23000 1.30500 1.26625 0.0566 0.1138
3 1.34000 1.39000 1.36667 -0.1848 0.1360
1 1.75500 1.75500 1.75500 -0.5993 0.3501

I have also thrown the ESSENCE dataset into the Riess et al. (2007)
dataset, getting the followed binned dataset. I needed to add 0.022 mag from the
$mu; values in Table 9 of Wood-Vasey et al.
(2007) to make the sample of objects in common consistent with the Riess
et al. scale.
n zmin zmax <z> d(DM) sigma
37 0.00700 0.02400 0.01589 -0.0518 0.1180
37 0.02450 0.05800 0.03757 0.0040 0.0573
12 0.06100 0.16000 0.09475 0.1026 0.0752
14 0.17200 0.26800 0.22071 0.1097 0.0625
36 0.27400 0.37100 0.32989 0.0963 0.0374
37 0.37400 0.45500 0.42224 0.1698 0.0373
37 0.45900 0.51100 0.48408 0.2449 0.0371
37 0.51400 0.61000 0.55297 0.1687 0.0340
31 0.61200 0.71000 0.65503 0.0999 0.0358
21 0.71900 0.81800 0.77471 0.0535 0.0526
20 0.82200 0.91000 0.85905 0.0546 0.0644
21 0.92700 1.02000 0.96614 0.0469 0.0677
4 1.05600 1.14000 1.11400 0.0168 0.1179
5 1.19000 1.26500 1.22280 -0.0870 0.1275
6 1.30000 1.39000 1.33533 -0.1505 0.0998
1 1.40000 1.40000 1.40000 0.0371 0.8100
1 1.55100 1.55100 1.55100 -0.4897 0.3201
1 1.75500 1.75500 1.75500 -0.5993 0.3501

The table above is Table 1 from Wright (2007).

The table below is the Riess et al Gold plus the ESSENCE supernovae, from
Table 2 in Wright (2007):
n zmin zmax <z> d(DM) sigma
29 0.01000 0.02500 0.01694 -0.0351 0.1250
29 0.02500 0.05300 0.03612 0.0240 0.0667
10 0.05600 0.12400 0.07760 0.0742 0.0798
10 0.15900 0.24900 0.20320 0.1229 0.0735
27 0.26300 0.36300 0.31963 0.1054 0.0425
29 0.36800 0.45000 0.41490 0.1437 0.0390
29 0.45500 0.50800 0.48083 0.2032 0.0401
29 0.51000 0.60400 0.55145 0.1386 0.0389
25 0.61000 0.70700 0.64748 0.1190 0.0381
18 0.73000 0.83000 0.78883 0.0581 0.0587
10 0.83200 0.90500 0.86660 0.0073 0.0811
14 0.93500 1.02000 0.96957 0.0347 0.0760
4 1.05600 1.14000 1.11400 0.0168 0.1179
3 1.19900 1.23000 1.21967 0.0806 0.1434
5 1.30000 1.39000 1.34100 -0.1629 0.1054
1 1.75500 1.75500 1.75500 -0.5993 0.3501

Observationally
d(DM) = 5 log (Ho sqrt[L/(4πF)]/[cz(1+z/2)])

while theoretically
d(DM) = 5 log[Z(z) J([1-Ωtot]Z(z)2) (1+z)/(z(1+z/2))]

with Z(z) and J(x) defined here. The
Hubble constant used in computing the empty Universe Milne model which is
subtracted off is 63.8 km/sec/Mpc, to be consistent with Riess et al. (2007).
Note that any fit to this dataset should include as a free parameter an
adjustment to this Hubble constant, which gives a constant term in d(DM).

I found the following chi2 values for fits to both the unbinned
and the binned Riess et
al. (2007) Gold+Silver data:
unbinned binned
Name Omega_m Omega_vac chi^2/df chi^2/df
Best fit 0.55 1.15 290.4/289 12.1/15
Best flat 0.36 0.64 297.7/290 20.1/16
WMAP model 0.27 0.73 302.6/291 25.3/17
Milne 0.0 0.0 321.2/291 44.5/17
EdS 1.0 0.0 386.3/291 108.6/17
Evolving 1.0 0.0 295.8/290 18.2/16

The evolving model is the model with supernova luminosity evolving as
a exponential function of cosmic time, which I discussed in astro-ph/0201196. This model is
still a better fit than the flat vacuum-dominated model, but not to a
significant degree.

I have also binned the data from Schaefer (2006):
n zmin zmax <z> d(DM) sigma
1 0.17000 0.17000 0.17000 0.4532 0.3813
1 0.25000 0.25000 0.25000 0.4471 1.1402
2 0.43000 0.45000 0.44000 0.2069 0.3973
6 0.61000 0.71000 0.68000 0.4517 0.2367
7 0.78000 0.86000 0.82857 0.3608 0.2116
6 0.96000 1.10000 1.02000 -0.1046 0.2142
8 1.24000 1.51000 1.37625 -0.0509 0.1866
5 1.52000 1.71000 1.60200 -0.2954 0.2299
8 1.98000 2.35000 2.17375 -0.0616 0.2020
7 2.44000 2.90000 2.65857 -0.5738 0.2612
8 3.08000 3.53000 3.30000 -0.4595 0.2219
7 3.79000 4.50000 4.10429 -0.8771 0.2301
1 4.90000 4.90000 4.90000 -0.5275 0.9500
2 6.29000 6.60000 6.44500 -1.1004 0.4628

Note that Schaefer uses a different Ho than Riess but I have
used the appropriate Ho (72 km/sec/Mpc) when computing the Milne
model for this dataset.

The plot above shows the difference in distance modulus between the
empty model and the supernova and the GRB binned data. It looks a bit
inconsistent at redshifts near 0.5 but the errors agree with the residuals from
the fits. When fitting to both the SNe and GRB datasets, there should be two
free parameters for Hubble constant changes, one for each dataset. These free
parameters can be thought of as adjustments to the overall luminosity
calibration of SNe and GRBs respectively.

With multiple datasets it is now possible to say something about the equation of state
parameter w even without assuming the Universe is flat. The figure below
shows the constraints from the Hubble constant (vertical lines), the baryon
acoustic oscillations (nearly vertical lines), the CMB (tilted fan of
lines), and the supernovae (ellipses). In each case green is right on (or 0.3
sigma for the supernovae), blue is 1 sigma, red is 2 sigma, and black is 3
sigma. Ho taken to be 71 +/- 5 km/sec/Mpc based on an average of the
HST Key project, the SZ effect, the Cepheids in the nuclear maser
ring galaxy NGC 4258, and the double-lined eclipsing binary in
M33.

The three panels above show three different values of the
equation of state parameter, w = -0.7, -1, and -1.3. Clearly if one assumes the
Universe is flat the supernovae favor w = -1.3 which leads to a "Big Rip". But
if one looks only at the concordance between the four datasets, the standard
flat ΛCDM model with w = -1 is preferred.

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