By Sean Carroll | November 22, 2007 2:55 pm

Last year we gave thanks for the Lagrangian of the Standard Model of Particle Physics. This year, we give thanks for Hubble’s Law, the linear relationship between velocity and distance of faraway galaxies:

v = H0 d.

(We could be sticklers and call it the “effective velocity as inferred from the cosmological redshift,” but it’s a holiday and we’re in an expansive mood.) Here is the original plot, from Hubble 1929:

Hubble’s Law

And here is a modern version, from Riess, Press and Kirshner 1996 (figure from Ned Wright’s cosmology tutorial):

Hubble’s Law (recent supernovae)

Note that Hubble’s distance scale goes out to about two million parsecs, whereas the modern one goes out to 500 million parsecs. Note also that Hubble mis-labeled the vertical axis, expressing velocity in units of kilometers, but he discovered the expansion of the universe so we can forgive him. And yes, the link above is to Hubble’s original paper in the Proceedings of the National Academy of Sciences. Only 146 citations! He’d never get tenure these days. (Over 1000 citations for Freedman et al., the final paper from the Hubble Key Project to measure H0.)

Hubble was helped along in his investigations by Milton Humason; together they wrote a longer follow-up paper. (Some habits don’t change.) Here is a sobering sentence from an article about Humason: “During the period from 1930 until his retirement in 1957, he measured the velocities of 620 galaxies.” These days projects measure millions of velocities. So let’s give thanks for better telescopes, CCD cameras, and software, while we’re at it.

Hubble’s Law is an empirical fact about photons we receive in our telescopes, but it’s implications are profound: the universe is expanding. This discovery marks a seismic shift in how we think about the cosmos, as profound as the Copernican displacement away from the center. It was so important, Einstein felt the need to visit Hubble on Mt. Wilson and check that he wasn’t making any mistakes.

Einstein and Hubble

  • fh

    I call a caption contest on the Einstein picture.

  • fh

    Let me amend that, I call for a contest for the intertitle following this scene.

    “Why indeed professor, it is true, Tesla has established a base on the moon!”
    “He must have found Jules Vernes lost diaries, I shall telegraph immediately to call a gathering of the League!”

  • Neil B.

    The way the universe expansion and deceleration (at least, before negative energy/cosmological constant) was very elegant when thinking of little particles on the surface of an expanding balloon, etc. However, I have always wondered about consistency problems if the universe is filled with hard balls and in a state of deceleration. What happens when the balls touch each other? They should crunch up a bit and just sit there, but then we have strange problems trying to apply the pseudoNewtonian approximation. For example, any point in the center of a ball can imagine that light propagating from itself loses energy traveling up through successive spheres of matter (the radii of which are always just wherever the photon is, since the stuff beyond that cancels out. So, the receiver should get a red shift, but the receiver could imagine himself at the bottom of the same self-defined spherical region etc. and expect a blue shift.

    Then there’s the issue of what happens to little bodies inside bigger spheres: By simple extrapolation of the incoming motion of the little bodies as seen by any given observer, they should continue moving toward that central observer until the smack into the sides of whatever sphere they’re in. But again, this does not provide a consistent picture because it produces contradictory preferred central observers in every place.

    Sure, GR is not Newtonian gravity, but then what would happen if our universe was filled with hard spheres with contents, sending light signals to each other, when the spheres scrunched together? I hope this is thought-provoking. “Problems” may deserve more thanks than anything else, since they stimulate science to grow.

  • Sid

    “You’re right, Ed. That’s one hot babe on the 87th floor.”

    OK, OK, You’re right, too crass. How ‘bout

    “Dang! I can’t see the back of my head. I guess it’s not closed after all.”

  • Neil B.

    Well, regarding problems with body surfaces stopping continued contraction: I already suspect that Doppler shift is going to continue to be based on relative scale factor, and presumably motion of contained particles would be correlated with that, etc. However, it starts getting sticky so to speak if a region of space has bodies clump together before the rest of space over a wide area does: then, to what extent and under what rules does such a local region act like an isolated Newtonian or small GR system (where the test bodies in the centers of larger spheres keep moving towards the extrapolated contraction point, grav Doppler shifts depend on local distributions of mass, etc. ) versus acting consistently with concepts properly applying in GR to “space as a whole”?

  • Neil B.

    Oh, I meant to write: “… if the universe is filled with hard balls and in a state of contraction.” (Our universe is not going to reverse expansion and contract, due to current combination of factors such as Hubble “constant” and dark energy, but it is logically possible and “could have” turned out that way if this or that was different. In any case, this serves as a proper “what if” problem for general relativity as a theoretical construct.)

  • Big Vlad

    interesting post Sean. Is it also true that the rate of expansion is accelerating, and if so what can we draw from this?

    Really enjoy the science stuff on this site, more of the same please whenever you have time! (i like the other stuff too)

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  • Supernova

    To borrow from the Simpsons:

    “Wow… the universe is soooo… boring!”

  • BH

    Thank you Hubble for all. Expecially for your extremly linear law!

  • Ben

    Humason’s story is itself quite interesting as his career path on Mt Wilson was mule driver, janitor, night assistant, observer, member of the scientific staff. It would be very difficult to do that today. If only because few observatories have entry-level mule-driving positions. Humason and Hubble measured most of those velocities with the 100-inch Hooker telescope on Mt. Wilson, which actually (by reputation) is/was a pretty good telescope. It is the same aperture as the SDSS 2.5-m telescope, in fact. The improvement comes mostly from vast increases in detector technology, sensitivity, and size. And a little bit from improvements in our abiliity to build spectrographs. Of course, we also need software to deal with the flood of data. And we’re also grateful for the invention of the off-axis TV guider.

  • (Required)

    Sean wrote, “we could be sticklers….”
    That is *not* being a stickler. That is making an honest effort to avoid propagating one of the most basic misunderstandings in cosmology! And worse by far: it is missing an opportunity to enjoy something which is a lot of fun, namely the fact that [peculiar velocities apart] the distant galaxies are *not* moving away from us — they are just becoming more distant. Pythagoras’s theorem has a function of time in it! Surely that is much more fun than a boring old Newtonian expansion!

  • John Farrell

    We could be even more ‘stickler’ and mention that Lemaitre actually derived Hubble’s Law two years before Hubble did.

    But I’m no less thankful!

  • Daniel Fischer

    As usual it is forgotten that Lemaitre wrote down the “Hubble” law – in the same mathematical form – two years before Edwin did. A major
    review recently explained this well – and highlighted the fact that “his papers remain desperately unquoted” …

  • Sean

    Surely things have not gotten so bad that we’ve lost the ability to distinguish between a theoretical prediction and an observational discovery.

  • Garth Barber

    Surely things have not gotten so bad that we’ve lost the ability to distinguish between a theoretical prediction and an observational discovery

    That’s because of all that ‘discovered’ DE and DM!


  • Neil B.

    BTW: Just a reminder, that even without dark energy etc. the Hubble “constant” isn’t really a constant over cosmic time scales. For example, in a traditional just barely open universe (asymptotic deceleration to “v” = 0), the scale factor varies as t^(2/3), v(relative distance expansion) varies as t^(-1/3), and so Hubble factor varies as t^(-1).

    Well, I was hoping to find at least some interest in the question about what happens if obstructive surfaces impede contraction of the universe (and the issue of things being tied together with elastic tethers is good too, in a case of expansion: what about the work done pulling on them, since outward forces pull in all directions?

  • Sam Cox

    The fact that the relationship is so obviously linear is profound…

  • Garth Barber

    What is interesting is the age of the universe, having allowed for the current best values of the density of matter (0.04), DM (0.23) and DE (0.73) is 13.81 Gyrs, and Hubble Time (the inverse of the H, with h = 0.704) is 13.89 Gyrs.

    Is this too great a coincidence?


  • Ike

    Always worth noting that Einstein initially predicted an expanding universe based on his GR thoery, but he felt this was so ridiculous that he introduced a “cosmological constant” into his equations. That issue is apparently still in play, however.

    Here’s a nice picture of the Hubble redshift from calcium lines.

  • Neil B.

    Garth, et al: isn’t the near equality of universe age and Hubble time a mark of the universe being so widely “open”? If it was instead an asymptopic just-open expansion, I think t(H) is 2/3 of the actual age, or is it the other way around, I keep forgetting. The really interesting thing, what is the time we get from taking the value of lambda, which has units a/r = 1/T^2, and thus taking lambda^(-1/2): IIRC and rough estimate it’s about the same as the age of the universe, right? But if lambda is really a constant, then a certain time after the BB is equal to that and therefore “special” – are we in it? See the discussion on dark energy etc. at Backreaction.

  • Garth Barber

    Neil, s grea

    If the universe is flat with a total density parameter Omega = 1 then:

    if there is no DE then the Age = 2/3 Hubble Time

    if there is DE that reduces the rate of then
    the Age > 2/3 Hubble Time, from that value up to infinity, depending on the amount and the exact equation of state of the DE.

    As Omega = 1 it seems a suspicious coincidence that Age = Hubble Time to within +/- 0.6%, which is equal to within observational error.


  • Sean

    The Moon and the Sun, as seen from the Earth, subtend the same angle in the sky. You are welcome to discern deep cosmic conspiracies underlying this fact if you like.

  • Garth Barber

    Agreed it could be just pure coincidence, but the Universe Age/Hubble Time coincidence is linked with the near equality to an OOM of the densities of baryonic matter (4%), non-baryonic Dark Matter (23%) and Dark Energy (73%).

    Why should these be approximately equal?

    The two coincidences are linked because it is the mix of DE/Matter that determines the age of the universe in terms of Hubble’s constant.


  • rgb

    This has been discussed elsewhere:

    One might want to do to read the stuff before the abstract though, … the paper did not quite manage to achieve what they had set out to do.

  • John Ramsden

    According to an article in this week’s New Scientist, the cosmological horizon is currently at a distance of 43 billion light years which, as it predates the Big Bang, is presumably fictitious or virtual in some sense.

    But if cosmic expansion keeps accelerating, so that the cosmological horizon shrinks, is there likely to be any change in the character of cosmic expansion if/when the cosmological horizon falls below 13 billion years so that the Big Bang is no longer “visible”?

  • SH

    Observational astronomy has a rich history that must be respected. These astronomers documented reality and today, we are trying to learn how to understand what they actually observed.

    Today’s cosmology, is often a form of mental masturbation.

  • Sam Cox

    Relative to us, parts of the universe with a huge doppler redshift far back in time near the BB are not only receding at velocities near the speed of light; their mass is, by relativity, much greater.

    I’m not at all convinced, especially since we now know that the universal expansion is accelerating, that this Hubble relationship really is linear, considering our limits of accuracy in measurement. Time dilation function is a pretty straight line until we are pretty close to the speed of light.

    We are dealing with relativistic relationships when are talking this distant aspect of astronomy. The simple linear relationship which Hubble proposed may not be quite so simple- or linear 13BLY out.

  • Garth Barber

    This has been discussed elsewhere:

    One might want to do to read the stuff before the abstract though, … the paper did not quite manage to achieve what they had set out to do.

    Yes that paper was a hoot! Notice the date and the co-author Ali Frolop = April Fool!


  • Sam Cox

    Sean mentioned the coincidence of the almost exact apparent size of the sun and moon in the sky, as viewed from the surface of the Earth. The shadow of the moon on the Earth at totality is no more than about 200 miles wide. Sometimes no umbra strikes the Earth.

    I did a few quick calculations using the measured angular diameter of the Moon and Sun in the sky at the apogee and perogee of appropriate orbits, and a guestimate of a rate of withdrawl of the moon from the Earth of 4 CM per year, and came up with numbers indicating it would take a full billion years for total solar eclipses to be eliminated. It seemed to me like an unbelievably long period of time for this trivial cosmic coincidence to be eliminated.

    I checked with NASA and they told me that taking the increase in the size of the Suns disc into consideration and using a correct 3 CM annual withdrawl rate for the Moon, it would take a bit less than 600,000.000 years for total solar eclipses to be eliminated….a pretty stable coincidence indeed.

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  • Ned Wright

    Neil B: the “hard” spheres would lead to a large positive pressure
    which would cause the contraction to go faster. This seems
    paradoxical but recall that objects are accelerated by
    pressure gradients, not by a uniform pressure, and that both
    pressure and density contribute with negative signs to the
    second derivative of the scale factor. So to slow down a contraction,
    which would be a positive second derivative for the scale factor,
    one needs a large negative pressure.

    Garth: H_o*t_o is not equal to 1 with an accuracy of 0.6%.
    t_o is known (but model dependent) to 1% and H_o (also
    model dependent) to 4%. So while the product is very close to 1, the propagated uncertainy is 4%.

    Sean: I read Luminet’s review (pointed out by Daniel Fischer)
    and it makes the point that Le Maitre’s little cited 1927 paper
    in French to the Belgian Academy actually did the data analysis
    and derived the Hubble constant two years before Hubble. Then
    Luminet didn’t list this paper in his reference list! I wish
    this 1927 paper were in the ADS, but last time I checked only
    the translation in the 1931 MNRAS was there.

  • Paul Stankus

    Hi Sean –

    The discovery of the Hubble expansion was indeed a profound advance. But there’s even more to it than that, as you well know, so let me point out for your readers:

    1. The pattern of Hubble’s law, that the recessional velocity of distant galaxies is proportional to their distance away, directly implies that contemporary observers on any galaxy will see the same pattern, and with the same proportionality. Thus the Hubble law contains the idea that the expansion has no preferred center.

    2. If we extrapolate the Hubble-law motions of the galaxies backward, assuming no acceleration (a not entirely unreasonable thing to do), we discover that there was a singular moment at a finite time into the past when they were all on top of each other. So Hubble’s law also raises directly the idea of a Universe with finite age.

    Hubble could/would have declared an expanding Universe based on any observation of velocity going up monotonically with distance. It is the special linear result v=Hd that also leads to the also profound picture of a uniform expansion with a finite history. It all came in the same package! as long ago as 1930, for which we can indeed be thankful.



  • Garth Barber

    Garth: H_o*t_o is not equal to 1 with an accuracy of 0.6%.
    t_o is known (but model dependent) to 1% and H_o (also
    model dependent) to 4%. So while the product is very close to 1, the propagated uncertainy is 4%.

    Yes Ned, although my post #22 was submitted before it was properly prepared (I have you Preview button on this Forum), I was simply using your Cosmology Calculator,.

    The present best accepted values of cosmological parameters
    (using the table in WMAP Cosmological Parameters
    Ho = 70.4 km/sec/Mpsc
    Omega_Lambda = 0.732
    Omega_matter = 0.268

    Feeding these values into your calculator I obtain:
    The age of the universe is = 13.81 Gyrs.
    But with h_100 = 0.704,
    Hubble Time = 13.89 Gyrs. i.e. 0.6% difference!

    In other words the calculated age turns out to be equal to Hubble Time to within the limits of the observational error of 4%.

    Is this just a coincidence or is the universe trying to tell us something?


  • Neil B.

    Hey folks, what about that “axis of evil” stuff etc? You can google for it … It looks like something preferred, but too early to tell what’s up.

    As for hard spheres in the contracting universe: the point is, not how they affect the forces, but they are an obstruction – I don’t think they could be just pulverized as if nothing there. There are contradiction problems if you try to imagine what happens to all the bodies in expanding/contracting universes if some things are impeded by material barriers/obstructions and other things just move like dust in free fall.

  • Sam Cox

    I first saw that famous picture of Einstein at the telescope with Hubble looking on in high school, more years ago than I would care to discuss. Suffice it to say I saw that picture for the first time while Albert Einstein was still very much alive!

    If anybody is into non-verbals, look at the expressions on Albert and Edwins faces!

    A picture is truly worth 1,000 words!

    I realize the picture was posed…for all we know it was taken in the daytime.

  • Sam Cox

    ABSTRACT: Don N. Page; December 13, 2007…last sentence…

    “SQM also suggests the possibility that past steps along our evolutionary ancestry may be so rare, that they have occurred no where else in the past history of the universe that we can observe”.

    It is almost certain that life in the universe is a common phenomenon. It is also almost impossible that beings identical to ourselves exist anywhere else. One does not have to be a quantum theorist to fathom that! However it is interesting that certain quantum theories point to such an obvious conclusion about our uniqueness in the universe.

    For similar reasons, related to genetic inheritance and environmental upbringing, each human individual who has ever lived is also unique. Cloning people doesn’t produce robots, just individuals with identical genetic inheritance- as identical twins. However, no pair of identical twins live exactly identical lives.

    This comment in Don’s abstract has profound implications…

  • Sam Cox

    Errata: See: “A Dark, Misleading Force” thread….

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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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