> > Hubble's Dark Constant

Hubble's Dark Constant

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

Dr. Eric Gaze
Director of the Quantitative Reasoning Program
Bowdoin College

Numb: The Hubble Constant is currently estimated in 2009 to be 74.2 ± 3.6 (km/s)/Mpc.

Number: The Hubble Constant is currently estimated in 2009 to be 74.2 ± 3.6 (km/s)/Mpc. This constant of proportionality represents the ratio of the velocity of a galaxy receding from us to its distance from Earth.

Num-best: Based on astronomical observations, Edwin Hubble first formulated his law in 1929 that the velocity (kilometers per second) of a galaxy receding from Earth is proportional to its distance (Megaparsecs) from us. Hubble's original estimate, 500 (km/s)/Mpc, indicates the difficulty in measuring this constant of proportionality which is currently calculated to be 74.2 ± 3.6 (km/s)/Mpc, using the latest data from the Hubble Space telescope.

The basic idea seems simple enough, astronomers Vesto Slipher (1917) and Edwin Hubble (1929) analyze light spectrum from distant celestial objects and notice the tell-tale shift of the signature spectral lines towards the red end of the spectrum:

Redshift Medium


See my column on Planck's constant for more information on the light spectrum. The reason Hubble has a constant, and not poor Vesto, is that Hubble was specifically looking for such redshift to confirm that the universe was expanding as boldly predicted by Lemaitre. This prediction was bold because it contradicted the great Einstein, who started the modern era in cosmology with his theory of general relativity, which describes gravitation as a consequence of the curvature of space-time. Einstein believed the universe was static (the prevailing view at the time) and thus was concerned that gravity would make the universe collapse according to his field equations; so he introduced the infamous "cosmological constant"into his equations to prevent such collapse in 1917:

Cosmological Constant Medium

The capital lambda represents his "anti-gravity" term. Can you find the capital lambda? This reminds me of the old math joke, where a question on a math test asks the students to find x ( an all too common question!) and a bright student simply circles the letter x. This innocuous looking equation actually contains tensors, and if there was one thing that drove me to fits in grad school it was tensor notation. So all we are going to do with this equation is admire its simplicity, be amazed that it can describe the fate of the entire universe, and circle lambda!

Of interest to us, is that following Einstein's published solutions to his equation, the Russian Alexander Alexandrovich Friedman showed that solutions exist which would indicate an expanding or contracting universe depending on how matter was distributed. This was just so much mathematical theoretical hogwash to the physicists until the Belgian priest and astronomer Georges Henri Joseph Edouard Lemaitre independently published the Friedman solutions in 1927 with the additional important caveat: an expanding universe could be empirically verified! Lemaitre noted that every galaxy in an expanding universe would be receding from every other galaxy at a rate that increases with distance separation. This is what Hubble directly verified in his 1929 paper and partly why he is so famous; also note that this expanding universe scenario is where the "big bang theory" comes from. The following is a graphic, illustrating the proportionality of distance and velocity:

Redshift v. Distance


There are several gedankenexperiments ("thought experiments") that need to be more fully explored before moving on to the darker side of this column. The first is trying to imagine how every galaxy can recede from every other galaxy. A standard visual aid is a balloon being blown up with dots on it, as the balloon expands all the dots get farther away from each other. Martin Gardner, the math puzzle guy, offers a more substantial image of a ball of dough with raisins in it expanding as it cooks. I prefer a 4 dimensional glob of dark matter with isolated singularities dispersing into a 12 dimensional space time manifold due to the mysterious quintessence of a dark energy field... but that's just me. Next gedanken, the redshift of the light from the galaxies moving apart is due to the actual stretching of the universe itself (think ball of dough stretching as it expands), not from the traditional Doppler shift of moving objects. Finally try to imagine why the velocity of raisins in the doughball is proportional to the distance between them, i.e. why is a raisin moving twice as fast from another raisin if it is twice as far? Mathematically this is a consequence of the theorem that states any two points which are moving away from the origin, each along straight lines and with speed proportional to distance from the origin, will be moving away from each other with a speed proportional to their distance apart (like that actually helps).

The velocities of the receding galaxies are easier to calculate, using the observed redshift in spectral lines, than the distances of the galaxies, which require knowing how bright the celestial object is supposed to be and comparing that with the observed brightness. Astronomers have refined their distance estimations using improved data collection, as well as finding more easily measurable bright objects in the sky. Type Ia supernovae, exploding stars that briefly shine as brightly as 10 billion suns, are rare examples. In 1998 two teams of astronomers studying just such supernovae determined that they were dimmer than expected, leading them to conclude that they were farther than estimated using Hubble's constant... meaning that the expansion of the universe must be speeding up! This completely unexpected result has been further supported with more observations, leading to cosmologists looking for some kind of anti-gravity term that could account for this acceleration... Amazingly they have turned back to Einstein's cosmological constant, which he had called "his biggest blunder" after seeing Hubble's results.

Physicists now refer to the needed "stuff" to cause the accelerating expansion: "dark energy." To be precise (if possible when speculating), there are two competing forms of this hypothetical energy: one a constant energy density field versus a variable energy density field called "quintessence." For those of you who still remember the hypothesized "aether" from the turn of the last century (actually the idea dates back to Aristotle, with aether being the fifth element with air, earth, fire and water, hence the "quint" in quintessence), the following statement should be eerily familiar: "The exact nature of dark energy is a matter of speculation. It is known to be very homogeneous, uniformly filling otherwise empty space, not very dense and it is not known to interact through any of the fundamental forces other than gravity." This is incredibly exciting news!!! For those of you who thought you missed the boat when Einstein replaced the Newtonian paradigm with relativity theory, and quantum theory blew the socks off of just about everyone, this is a chance to make a real contribution to our knowledge of the universe. Just as the "aether" and problems related to it led to a deeper understanding of light and quantum phenomena, dark energy will certainly push young scientists to come up with creative explanations. A word of caution is warranted here, note that I say young "scientists" in the previous sentence. Lots of people like to "speculate" but to really contribute to the field of physics requires years of hard work, persistence and study in said field. Getting back to dark energy, scientists have calculated how much of this stuff is out there, even though they have no idea what it actually might be:

Dark Energy Pie Chart (Medium Size)

Wow, three-quarters of the pie is made up of something that we didn't know about 10 years ago. Some of you may have noticed that almost all of the rest of the pie is "dark matter" which has at least been around since 1934. Fritz Zwicky coined this term to help explain observed galaxy rotation and galaxy cluster formation. So only 4% of the universe consists of stuff we can actually "explain", which is in quotes because quantum theory still has many conundrums that leave our understanding of ordinary matter with much to be desired. Jermey Bernstein's nice little book, Quantum Leaps, provides an excellent insider look at the development of the quantum theory with particular focus on John Bell and his theorem related to the EPR paradox and the measurement problem. On page 172 he asks: "But what is the explanation for the correlations of the widely separated entangled photons or electrons?", and then answers: "If you accept the usual interpretation of the quantum theory, there isn't one." In order to avoid taking one of his "quantum leaps" I will say no more on this subject here, except that someone now needs to write a book called, Cosmic Leaps.

Finishing up our story on Hubble and his erstwhile "constant", which now is accepted as varying according to the cosmological model being used, we find ourselves in a universe comprised largely of energy and matter that we cannot explain. Given such uncertainties in the science of cosmology it might come as no surprise that a Scientific American article was published in March 2008 titled, The End of Cosmology. The authors do not propose that all cosmologists should pack up their bags since we cannot account for most of what is being observed, but rather posit that this is the golden age of cosmology. The accelerating universe actually implies that all galaxies will eventually get so far away from us, that in roughly 100 billion years their light will not be able to overcome the ever-increasing distance and reach us! This will leave us with only light from the Milky Way and nearby galaxies which will have since formed a ball-like super- galaxy, turning our night sky and constellations into one big white smudge. Our sun is expected to burn out in 5 billion years so humans will probably be someplace else by then. What is interesting is that there will be no traces of any other galaxies, thus no evidence of the expanding universe, leading future astronomers to believe in the static universe that Einstein had originally tried to derive with his cosmological constant!

Sapere Aude!