From: Robert A. Knop Jr. (robert.a.knop@vanderbilt.edu)
Date: Sat May 17 2003 - 11:07:06 PDT
On Sat, May 17, 2003 at 11:05:05AM -0700, Greg Aldering wrote:
>
> Hi Rob,
>
> Please add the following to the bibliography:
>
> E.V. Linder & A. Jenkins, submitted to MNRAS, astro-ph/0305286
What is this? Where should it be cited?
> Also, perhaps you want to send me the TeX for the paragraph starting
> with "Other methods provide ..." on page 27 so that I can put in the
> proper description of what we did with the 2dFGRS measurements.
See below.
Other methods provide measurements of $\om$ and $w$ which are
complementary to the supernova results. Two of these measurements are
plotted in the middle row of Figure~\ref{fig:omw}, compared with the
supernova measurements (in solid contours). In filled contours are
results
from the redshift distortion parameter and bias factor measurement of
the 2dF Galaxy Redshift Survey (2dFGRS) \citep{haw02}. This provides a
measurement of $\om(z_s)=0.32\pm0.12$ at $z_s=0.15$. We can convert
this
into a measurement of $\om$ at $z=0$ for a given value of $w$ using the
expression
\begin{equation}
\om = \frac{\om(z_s)}{\om(z_s) + (1-\om(z_s))(1+z_s)^{-3w}}
\end{equation}
to yield joint confidence regions for $\om$ and $w$.
In solid lines on the middle row of Figure~\ref{fig:omw} are contours
representing confidence regions based on the distance to the surface of
last scattering at $z=1089$ from the Wilkinson Microwave Anisotropy
Probe (WMAP) \citep{ben03,spe03}. For a given \om\ and $w$, this
distance reduced distance to the surface of last scattering, $I$, is
given by:
\begin{eqnarray}
I=\int_0^{1089} [((1-\om)/\om) (1+z)^{3(1+w)} + \nonumber \\
(1+z)^3]^{-1/2}\ dz
\end{eqnarray}
The plotted CMB constraints come from the ``WMAPext'' sample, which
includes other CMB experiments in addition to WMAP; for $w=-1$, they
yield a measurement of $I_0=1.76\pm0.058$, corresponding to $\om=0.29$.
Confidence intervals are generated by calculating a $\chi^2 =
\left[(I-I_0)/\sigma_{I_0}\right]^2$, where $I$ is calculated for each
$\om,w$.
As both of these measurements show correlations between $\om$ and $w$ in
a different sense from that of the supernova measurement, the combined
measurements provide much tighter overall constraints on both
parameters. The confidence regions which combine these three
measurements are shown on the bottom row of Figure~\ref{fig:omw}. When
the resulting probability distribution is marginalized over \om, we
obtain a measurement of $w=-1.06^{+0.14}_{-0.18}$ (for the
low-extinction subset), or $w=-0.98^{+0.19}_{-0.22}$\checkit\ (for the
full primary subset with host-galaxy extinction corrections applied).
The 95\% confidence limits on $w$ when our data is combined with WMAP
and 2dFGRS are \mbox{$-1.53<w<-0.79$} for the low-extinction primary
subset, or \mbox{$-1.51<w<-0.67$}\checkit\ for the full
extinction-corrected primary subset. If we add an additional prior that
$w\geq-1$, we obtain a 95\% upper confidence limit of \mbox{$w<-0.80$}
for the low-extinction primary subset, or \mbox{$w<-0.67$}\checkit\ for
the extinction-corrected full primary subset. This confidence may be
compared with the limit in \citet{spe03} which combines the CMB, 2dFGRS
power spectrum, and HST key project $H_0$ measurements to yield a $95\%$
upper limit of \mbox{$w<-0.78$}. Although both our measurement and that
of \citet{spe03} include CMB data, they are complementary in that our
limit does not include the $H_0$ prior, nor does it include any of the
same external constraints, such as those from large scale structure.
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