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Starts With A Bang

Sorry, Astronomy Fans, The Hubble Constant Isn’t A Constant At All

If your Universe contains any matter at all, a constant Hubble parameter is absolutely impossible.

Our observable Universe is an enormous place, with some two trillion galaxies strewn across the abyss of space for tens of billions of light-years in all directions. Ever since the 1920s, when we first unambiguously demonstrated that those galaxies were well beyond the extent of the Milky Way by accurately measuring the distances to them, one fact leaped out at us: the farther away a galaxy is, on average, the more severely shifted towards the red, long-wavelength part of the spectrum its light will be.

This relationship, between redshift and distance, looks like a straight line when we first plot it out: the farther away you look, the greater the distant object’s redshift is, in direct proportion to one another. If you measure the slope of that line, you get a value, colloquially known as the Hubble constant. But it isn’t actually a constant at all, as it changes over time. Here’s the science behind why.

An illustration of how redshifts work in the expanding Universe. As a galaxy gets more and more distant, the emitted light from it must travel a greater distance and for a greater time through the expanding Universe. In a dark-energy dominated Universe, this means that individual galaxies will appear to speed up in their recession from us, but also that there will be distant galaxies whose light is just reaching us for the first time today. (LARRY MCNISH OF RASC CALGARY CENTER, VIA HTTP://CALGARY.RASC.CA/REDSHIFT.HTM)

In our Universe, light doesn’t simply propagate through a fixed and unchanging space, arriving at its destination with the same properties it possessed when it was emitted by the source. Instead, it must contend with an additional factor: the expansion of the Universe. This expansion of space, as you can see, above, affects the properties of the light itself. In particular, as the Universe expands, the wavelength of the light passing through that space gets stretched.

If space were expanding at a constant, unchanging rate, then this would account exactly for a constant, unchanging value of “the Hubble constant.” If you, as a photon, traveled through twice the amount of space (or, equivalently, for twice the amount of time) as a closer photon, your wavelength would experience twice the stretch — or redshift — compared to the photon that was closer.

The redshift-distance relationship for distant galaxies. The points that don’t fall exactly on the line owe the slight mismatch to the differences in peculiar velocities, which offer only slight deviations from the overall observed expansion. The original data from Edwin Hubble, first used to show the Universe was expanding, all fit in the small red box at the lower-left. (ROBERT KIRSHNER, PNAS, 101, 1, 8–13 (2004))

In the real Universe, the relationship isn’t quite as clean as this story, and for good reason: galaxies do more than just stay put in an expanding Universe. Additionally, they experience the gravitational attraction of every other object that’s causally connected to them, pulling them in a variety of different directions at a variety of different speeds.

The notion that the light from a galaxy appears more redshifted the farther away it is from us is only true on average; for any individual galaxy, there will be an additional redshift or blueshift superimposed atop it. That extra signal corresponds to that galaxy’s motion relative to the fabric of space itself, something that astronomers call peculiar velocity. In addition to the effects of the expanding Universe on the light traveling through it, the individual motions of the galaxies themselves — a Doppler shift — affects each individual data point we measure.

A two-dimensional slice of the overdense (red) and underdense (blue/black) regions of the Universe nearby us. The lines and arrows illustrate the direction of peculiar velocity flows, which are the gravitational pushes and pulls on the galaxies around us. However, all of these motions are embedded in the fabric of expanding space, so a measured/observed redshift or blueshift is the combination of the expansion of space and the motion of a distant, observed object. (COSMOGRAPHY OF THE LOCAL UNIVERSE — COURTOIS, HELENE M. ET AL. ASTRON.J. 146 (2013) 69)

But the expansion of space isn’t just an observational phenomeon; it was predicted theoretically before it was ever actually seen. Way back as early as 1922, a Soviet scientist named Alexander Friedmann found a very special solution for the equations governing spacetime in Einstein’s General theory of Relativity.

Friedmann realized that if you assumed that the Universe was, on the largest scales, both isotropic (meaning it was the same no matter which direction you looked in) and homogeneous (meaning it had the same density no matter where you were located), then one can derive two unique equations — the Friedmann equations — that govern the Universe.

A photo of me at the American Astronomical Society’s hyperwall in 2017, along with the first Friedmann equation at right. The first Friedmann equation details the Hubble expansion rate (squared) on the left hand side, which governs the evolution of spacetime. (PERIMETER INSTITUTE / HARLEY THRONSON)

In particular, the most important feature of these equations was that a static Universe is impossible: the Universe must be expanding (or contracting), and therefore, the light from distant objects must be redshifted (or blueshifted) accordingly. These equations were later derived by multiple scientists independently: Georges Lemaître, Howard Robertson, and Arthur Walker all have their names attached to various underlying components of how these equations were obtained.

But the biggest feature you should notice about this equation is simple: there are two sides to it, the left-hand side and the right-hand side. On the left is the expansion rate of the Universe — what we’ve been calling the Hubble constant — and on the right is a series of terms that correspond to the various densities of all the forms of matter and energy present within that same Universe.

The first Friedmann equation, as conventionally written today (in modern notation), where the left side details the Hubble expansion rate and the evolution of spacetime, and the right side includes all the different forms of matter and energy, along with spatial curvature. This has been called the most important equation in all of cosmology, and was derived by Friedmann in essentially its modern form back in 1922. (LATEX / PUBLIC DOMAIN)

Now, here’s the important thing you have to think about: when the Universe expands, what happens to a quantity like matter density or energy density? The correct answer is, “it depends on what type of matter or energy you have.” For example, as the Universe expands, its volume increases, but the total number of particles within it remains the same. Radiation, like photons, also gets stretched to longer wavelengths (and lower energies), while dark energy, which is a form of energy inherent to the fabric of space itself, has a constant energy density even as the Universe expands.

As time goes on, the volume of an expanding Universe increases, which means, at a base level, that the energy densities of all the individual components combined are not required to remain constant. In fact, in almost all cases, they will not.

How matter (top), radiation (middle), and a cosmological constant (bottom) all evolve with time in an expanding Universe. As the Universe expands, the matter density dilutes, but the radiation also becomes cooler as its wavelengths get stretched to longer, less energetic states. Dark energy’s density, on the other hand, will truly remain constant if it behaves as is currently thought: as a form of energy intrinsic to space itself. (E. SIEGEL / BEYOND THE GALAXY)

Because of what the Friedmann equations tell us, we know that a Universe with a greater energy density will expand at a faster rate, while one with a smaller energy density must expand at a slower rate. So long as the energy density does not remain the same at all times, the expansion rate must change as well. The big question, of how the expansion rate evolves with time, is entirely dependent on what exists within our Universe.

There are many possible ingredients that can exist in an expanding Universe, and each one will evolve according to the unique properties inherent to that particular form of energy. Radiation and neutrinos were the most important ingredients, energy-wise, a very long time ago, later replaced by normal matter and dark matter as the dominant ingredients. As we move far into the future, dark energy will dominate, eventually causing the Hubble rate to asymptote to a finite, non-zero value.

Various components of and contributors to the Universe’s energy density, and when they might dominate. Note that radiation is dominant over matter for roughly the first 9,000 years, but remains an important component, relative to matter, until the Universe is many hundreds of millions of years old, thus suppressing the gravitational growth of structure. (E. SIEGEL / BEYOND THE GALAXY)

In fact, the most useful part of the relationship between the expansion rate and the Universe’s contents is that it gives us a method to go out and physically measure two things simultaneously:

  1. how quickly the Universe is expanding at the present,
  2. and what the relative values of the different significant components of the energy density are, both today and in the past.

Think about it this way: the light that arrives at our eyes, today, had to travel through the expanding Universe to get there. Light that arrives from a nearby galaxy was only emitted a short time ago, and the Universe’s expansion rate has only changed by a small amount in that time. Therefore, the nearby Universe gives us a handle on the present expansion rate. However, light that requires a journey of many billions of years to reach us will see the expansion rate change over time.

A plot of the apparent expansion rate (y-axis) vs. distance (x-axis) is consistent with a Universe that expanded faster in the past, but where distant galaxies are accelerating in their recession today. This is a modern version of, extending thousands of times farther than, Hubble’s original work. Note the fact that the points do not form a straight line, indicating the expansion rate’s change over time. The fact that the Universe follows the curve it does is indicative of the presence, and late-time dominance, of dark energy. (NED WRIGHT, BASED ON THE LATEST DATA FROM BETOULE ET AL. (2014))

By making measurements of galaxies at a wide variety of distances, we can determine what the expansion rate was (and how it changed) over many billions of years. Those changes in the expansion rate of the Universe teach us what the different components that make up the Universe are, as all the light traveling through the Universe will experience the expansion of space.

This also motivates us to measure light from progressively farther, more distant objects. If we want to understand how the Universe came to be the way it is today, and how the expansion rate has evolved, the best thing we can do it to measure how light redshifts as it travels to us throughout our entire cosmic history. With everything we’ve measured today, we can not only reconstruct what our Universe is made of now, but what it was made of at every point throughout our past.

The relative importance of different energy components in the Universe at various times in the past. Note that when dark energy reaches a number near 100% in the future, the energy density of the Universe (and, therefore, the expansion rate) will asymptote to a constant, but will continue to drop so long as matter remains in the Universe. (E. SIEGEL)

The fact that the Hubble expansion rate of the Universe changes over time teaches us that the expanding Universe isn’t a constant phenomenon. In fact, by measuring how that rate changes over time, we can learn what our Universe is made from: this was exactly how dark energy was first discovered.

Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard!

But the “Hubble constant” itself is a misnomer. It has a value today that’s the same everywhere in the Universe, making it a constant in space, but it’s not a constant in time. In fact, so long as matter remains in our Universe, it will never become a constant, as increasing the volume will always make the density (and, a la Friedmann, the expansion rate) decrease. Perhaps it’s time to call it by its more accurate but rarely used name: the Hubble parameter. Its present value is not a constant either, and should perhaps be called the Hubble parameter today. As it changes with time, it continues to reveal the very nature of our expanding Universe.

Ethan Siegel is the author of Beyond the Galaxy and Treknology. You can pre-order his third book, currently in development: the Encyclopaedia Cosmologica.


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