1 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 H₀, now known as the Hubble constant [1]. In its simplest form Hubble’s law is

Hubble’s original estimates were approximately 500 to 530 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]

As an example, consider recession-velocity data from the SIMBAD Astronomical Database together with distances from papers associated with the HST Key Project. For sufficiently small redshift, the nonrelativistic approximation is

A selection of galaxies is listed below.

An unconstrained least-squares fit of velocity versus distance for these selected data gives a slope of approximately

The nonzero fitted intercept reflects the scatter and local peculiar velocities in this small sample.

Hubble’s law does not imply that we are at the center of the universe. To illustrate this, consider a square grid with the velocity field

using H = 0.25 in arbitrary units.

Now choose the grid point at (1, 1) as a new origin. Its original velocity is

The velocity measured relative to this new origin is

Since

the new observer sees exactly the same Hubble law.

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.

2 References

[1] E. Hubble, “A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae,” Proceedings of the National Academy of Sciences, 15, 168–173 (1929).

[2] A. Liddle, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 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,” The Astrophysical Journal, 501, 32–53 (1998).