expanding universe
ExpandingUniverse
2006-07-08 02:54:42
2026-09-09 03:37:08
Topic
removed the more to come
expanding universe Hubble
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\section{Hubble's Law}
A cornerstone of cosmology is the observation of an expanding universe.
In 1929, Hubble published a relation involving the constant of
proportionality $H_0$, now known as the Hubble constant [1].
In its simplest form Hubble's law is
\[
\vec{v}=H_0\vec{r}.
\]
Hubble's original estimates were approximately 500 to
$530\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$. Modern distance measurements give
much smaller values. The Hubble Space Telescope (HST) Key Project combined
several distance-calibration methods and reported [3]
\[
H_0=72\pm8\ \mathrm{km\,s^{-1}\,Mpc^{-1}}.
\]
As an example, consider recession-velocity data from the
\PMlinkexternal{SIMBAD Astronomical Database}{http://simbad.u-strasbg.fr/Simbad}
together with distances from papers associated with the HST Key Project.
For sufficiently small redshift, the nonrelativistic approximation is
\[
z\approx\frac{v}{c}.
\]
A selection of galaxies is listed below.
\begin{center}
\begin{tabular}{rrrr}
\hline
Object (NGC) & $v$ (km/s) & Distance (Mpc) & Reference \tabularnewline
\hline
925 & 553 & 9.29 & [4] \tabularnewline
1326A & 1718 & 18.7 & [5] \tabularnewline
1365 & 1662 & 18.3 & [6] \tabularnewline
1425 & 1508.1 & 22.2 & [7] \tabularnewline
2090 & 924.6 & 12.3 & [8] \tabularnewline
3031 & -42 & 3.63 & [9] \tabularnewline
3198 & 663 & 14.5 & [10] \tabularnewline
3319 & 742.5 & 14.3 & [11] \tabularnewline
3351 & 778.5 & 10.05 & [12] \tabularnewline
3621 & 727 & 6.3 & [13] \tabularnewline
4321 & 1575 & 16.1 & [14] \tabularnewline
4414 & 718.5 & 19.1 & [15] \tabularnewline
4535 & 1964.4 & 16.0 & [16] \tabularnewline
4548 & 493.2 & 15.9 & [17] \tabularnewline
4725 & 1209.9 & 12.6 & [18] \tabularnewline
5457 & 241 & 7.4 & [19] \tabularnewline
7331 & 819.6 & 15.1 & [20] \tabularnewline
\hline
\end{tabular}
\end{center}
An unconstrained least-squares fit of velocity versus distance for these
selected data gives a slope of approximately
\[
H_0\approx78.2\ \mathrm{km\,s^{-1}\,Mpc^{-1}}.
\]
The nonzero fitted intercept reflects the scatter and local peculiar
velocities in this small sample.
\begin{center}
\includegraphics{Hubble_Data_Fit.png}
\end{center}
Hubble's law does \emph{not} imply that we are at the center of the
universe. To illustrate this, consider a square grid with the velocity
field
\[
\vec{v}=H\vec{r},
\]
using $H=0.25$ in arbitrary units.
\begin{center}
\includegraphics{Hubble_Grid_Original.png}
\end{center}
Now choose the grid point at $(1,1)$ as a new origin. Its original velocity
is
\[
\vec{v}_0=H\vec{r}_0.
\]
The velocity measured relative to this new origin is
\[
\vec{v}\,'=\vec{v}-\vec{v}_0.
\]
\begin{center}
\includegraphics{Hubble_Grid_Subtraction.png}
\end{center}
Since
\[
\vec{v}\,'
=
H\vec{r}-H\vec{r}_0
=
H(\vec{r}-\vec{r}_0)
=
H\vec{r}\,',
\]
the new observer sees exactly the same Hubble law.
\begin{center}
\includegraphics{Hubble_Grid_Shifted.png}
\end{center}
Thus every comoving point in an ideal homogeneous Hubble flow sees the
other points receding according to the same linear law. No special spatial
center is selected by the expansion.
\section{Friedmann Equation}
\emph{More to come.}
\section{References}
[1] E. Hubble,
``A Relation between Distance and Radial Velocity among Extra-Galactic
Nebulae,'' \emph{Proceedings of the National Academy of Sciences},
\textbf{15}, 168--173 (1929).
[2] A. Liddle,
\emph{An Introduction to Modern Cosmology}, 2nd ed.,
John Wiley \& Sons, West Sussex (2003).
[3] W. Freedman, B. Madore, B. Gibson et al.,
``Final Results from the Hubble Space Telescope Key Project to Measure the
Hubble Constant,'' \emph{The Astrophysical Journal},
\textbf{553}, 47--72 (2001).
[4] N. Silbermann, P. Harding, and B. Madore,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
VI. The Cepheids in NGC 925,'' \emph{The Astrophysical Journal},
\textbf{470}, 1--37 (1996).
[5] C. Prosser, R. Kennicutt, and F. Bresolin,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XXII. The Discovery of Cepheids in NGC 1326A,'' \emph{The Astrophysical
Journal}, \textbf{525}, 80--104 (1999).
[6] N. Silbermann, P. Harding, and L. Ferrarese,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XIV. The Cepheids in NGC 1365,'' \emph{The Astrophysical Journal},
\textbf{515}, 1--28 (1999).
[7] J. Mould, S. Hughes, and P. Stetson,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XXI. The Cepheid Distance to NGC 1425,'' \emph{The Astrophysical Journal},
\textbf{528}, 665--676 (2000).
[8] R. Phelps, S. Sakai, and W. Freedman,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
IX. The Discovery of Cepheids and a New Distance to NGC 3198,''
\emph{The Astrophysical Journal}, \textbf{500}, 763--788 (1998).
[9] W. Freedman, S. Hughes, and B. Madore,
``The Hubble Space Telescope Extragalactic Distance Scale Key Project I.
The Discovery of Cepheids and a New Distance to M81,''
\emph{The Astrophysical Journal}, \textbf{427}, 628--655 (1994).
[10] D. Kelson, G. Illingworth, and A. Saha,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XIX. The Discovery of Cepheids in NGC 2090,''
\emph{The Astrophysical Journal}, \textbf{514}, 614--636 (1999).
[11] S. Sakai, L. Ferrarese, and R. Kennicutt,
``The Hubble Space Telescope Extragalactic Distance Scale Key Project XXIII.
The Discovery of Cepheids in NGC 3319,'' \emph{The Astrophysical Journal},
\textbf{523}, 540--558 (1999).
[12] J. Graham, R. Phelps, and W. Freedman,
``The Hubble Space Telescope Extragalactic Distance Scale Key Project VII.
The Discovery of Cepheids in the Leo I Group Galaxy NGC 3351,''
\emph{The Astrophysical Journal}, \textbf{477}, 535--559 (1997).
[13] D. Rawson, L. Macri, and J. Mould,
``The Extragalactic Distance Scale Key Project VIII. The Discovery of
Cepheids and a New Distance to NGC 3621 Using the Hubble Space Telescope,''
\emph{The Astrophysical Journal}, \textbf{490}, 517--556 (1997).
[14] L. Ferrarese, W. Freedman, and R. Hill,
``The Extragalactic Distance Scale Key Project IV. The Discovery of Cepheids
and a New Distance to M100 Using the Hubble Space Telescope,''
\emph{The Astrophysical Journal}, \textbf{464}, 568--599 (1996).
[15] A. Turner, L. Ferrarese, and A. Saha,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XI. The Cepheids in NGC 4414,'' \emph{The Astrophysical Journal},
\textbf{505}, 207--229 (1998).
[16] L. Macri, J. Huchra, and P. Stetson,
``The Extragalactic Distance Scale Key Project XVIII. The Discovery of
Cepheids and a New Distance to NGC 4535 Using the Hubble Space Telescope,''
\emph{The Astrophysical Journal}, \textbf{521}, 155--178 (1999).
[17] J. Graham, L. Ferrarese, and W. Freedman,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XX. The Discovery of Cepheids in the Virgo Cluster Galaxy NGC 4548,''
\emph{The Astrophysical Journal}, \textbf{516}, 626--646 (1999).
[18] B. Gibson, S. Hughes, and P. Stetson,
``The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
XVII. The Cepheid Distance to NGC 4725,'' \emph{The Astrophysical Journal},
\textbf{512}, 48--64 (1999).
[19] D. Kelson, G. Illingworth, and W. Freedman,
``The Extragalactic Distance Scale Key Project III. The Discovery of Cepheids
and a New Distance to M101 Using the Hubble Space Telescope,''
\emph{The Astrophysical Journal}, \textbf{463}, 26--59 (1996).
[20] S. Hughes, H. Mingsheng, and J. Hoessel,
``The Hubble Space Telescope Extragalactic Distance Scale Key Project X.
The Cepheid Distance to NGC 7331,'' \emph{The Astrophysical Journal},
\textbf{501}, 32--53 (1998).