the Universe - accelerating expansion vs. the Hubble constant

Factual Questions
1

The cover story of this month’s Discover magazine (not yet available online) brought to mind a question that has been bugging me for years. I’ve searched extensively, but been unable to find an answer. I’m hoping folks here (especially our physics types) can explain.

In a nutshell, the problem is this. We have two data sets. One is a set relating to the Hubble constant, the other relates to Type 1a supernova. The former shows a constant rate of expansion going back about 1.3 billion years (the limit of the data), presently estimated at something like 70 km/s/Mpc. The latter shows that the rate of expansion in the distant past, e.g., 5 billion years ago, was significantly less than we observe today. For reasons I’ve never understood, the latter is widely described as demonstrating that the universe’s current rate of acceleration is increasing (emphasis on present tense), as opposed to demonstrating only that it changed in the past (by definition, an acceleration, but not necessarily one that’s still occurring).

The difference is enormous. If the rate of expansion is increasing in the present tense, it should show up in the Hubble data. Conversely, the fact that it doesn’t show up there is strong evidence the rate isn’t currently increasing. Moreover, it isn’t hard to reconcile the two data sets. Suppose the Hubble expansion is basic and universal, but subject to gravity. This is scarcely surprising. It has always been recognized that Hubble flow only dominates on very large scales. Gravitationally bound systems, e.g., our solar system and the Milky Way, show no Hubble effects. Move to a middle ground, where galaxies aren’t gravitationally bound but not yet completely independent and one would expect just the result the Type 1a supernova data suggest.

To see this, consider a thought experiment. Imagine a long conveyer belt, like the people movers they use at airports. At one end of the belt, place a small robotic car driven by a photo-voltaic cell. Just off the belt, place a focused spotlight. The spotlight will power the cell, and hence the robotic car. Switch on both the belt and the spotlight. The belt takes the car away from the light; the light powers the car towards it. If the two actions are equal, the car stays in place. If the light is stronger, the car advances; if the belt is stronger, the car recedes. Our universe, based on the data, corresponds to the third case. That is, Hubble flow takes the car one way and gravity the other, with the former being stronger. Importantly, notice two things. First, the further the robotic car gets from the spotlight, the less power it draws and hence the less it is able to resist the movement of the belt. Second, as the power from the spotlight fades in significance, the car accelerates, i.e. goes faster, but the speed of the belt is a limit. Take the spotlight out of consideration entirely and the fastest the car will recede is the belt speed. Stated a little differently, observing a historical change in speed doesn’t necessarily imply acceleration in the present or future tense.

Now, I readily admit I’m not a physicist. Moreover, it won’t surprise me if there’s some simple answer to the question that I’ve overlooked. So, please, someone, explain it to me.

Notes:

  1. I give Wiki links above for the benefit of folks who might want to brush up quickly on the background of the issue. Naturally, these articles cover only the basics, but to cite all the sites I’ve read, e.g., here and here, would clutter up the thread to no real advantage. In fact, most of my understanding is based on an introductory college level astronomy text, The Universe by Freedman & Kaufmann, which I picked up used several years ago. It’s a bit dated - I have the 2005 edition - but have seen nothing in my other reading which suggests it has been rendered obsolete on the question I’m raising.

  2. One of the curious things about this topic is that almost everyone seems to regard the Type 1a supernova data as raising the question of how the universe is expanding. Except, well, no. That problem has been around since Hubble. Call it the cosmological constant. Call it dark energy. Call is quintessence. Call it the vacuum energy of space. Whatever. Why the universe expands is a question that has been with us for almost a hundred years.

  3. It’s easiest to understand the thought experiment given above if we assume the conveyer belt is moving at a constant speed. This isn’t quite analogous to Hubble flow, as that’s an acceleration, i.e., the rate of expansion increases with distance. This isn’t important for my purposes, as all I’m trying to demonstrate is how two forces operating in opposite directions can produce a changing acceleration yet terminate at a constant. If you prefer, assume the belt accelerates over time (or whatever you think best corresponds to Hubble flow). The effect of the reverse force falling off over time and/or distance is the same. In any event, let’s not get sidetracked on the conveyer-belt-keeping-a-plane-from-taking-off paradox. Yeah, it’s an obvious joke, but not relevant here.

2

‘Hubble’s constant’ is the relationship between recession velocity and co-moving distance and it’s a constant in space, but not in time. The Hubble constant is not the same as the rate of expansion of the Universe, indeed a Universe where the Hubble constant was a constant in both time and space would be expanding exponentially.

There mainstream cosmological model of the Universe has never had a constant rate of expansion. Prior to the observation of the Universe’s accelerated expansion, it was believed that the rate of expansion was decreasing over time, as predicted by general relativity for a Universe with a zero cosmological constant.

There was a thread quite recently on the implications of expansion on smaller scales (brief answer is that nobody really knows the effects of expansion on scales at and below roughly the level of a galactic cluster).

3

And in fact, “Hubble’s Constant” would only be truly constant in a universe dominated by dark energy or something similar, and which was constantly accelerating.

To the main point, however, there is no conflict between the different lines of evidence. I’m not sure what you mean by contrasting the Hubble’s constant data with the supernova data, since the supernova data is data of the same sort that Hubble himself was working with. But in any event, all of the various cosmological observations have been drawing error ellipses in the parameter space, and all of the ellipses seem to be intersecting rather tightly at one single point. Basically, from any two of the experiments, you can predict fairly well what all of the other experiments would be expected to show you, and these predictions turn out to be quite close.

4

Asympotically fat: Your first point is obviously correct. I didn’t mean to imply otherwise. Your second point, though, seems to me a nitpick. That is, yes, the Hubble constant doesn’t of itself address expansion. Yet, AFAICT, everyone agrees expansion is the logical inference from the observed fact that recessional velocity is proportional to distance. If I’ve misunderstood your point, please explain.

Chronos: The conflict is simply this. Is the rate of expansion of the universe constant or is it increasing? Lots of very smart minds claim the latter, but I don’t see why. Do you?

5

Expansion is described by something called the scale factor a(t), which is a function of cosmological time t. At the present time t[sub]0[/sub], a(t[sub]0[/sub])=1 by definition, at some time t[sub]1[/sub] where a distant galaxy (i.e. a galaxy which is faraway enough so that it’s distance is only really a function of cosmology expansion) is twice it’s current cosmological distance a(t[sub]1[/sub])=2, at some other time t[sub]2[/sub] where it is half its current cosmological distance a(t[sub]2[/sub])=0.5.

The rate of expansion is usually identified as the first time derivative of the scale factor, a’(t). If at a certain time a’(t) is positive the Universe is expanding at that time, if a’(t) is negative then it is contracting at that time and if a’(t) is zero, then it is neither expanding nor contracting.

The second time derivative of the scale factor a’‘(t) tells you whether the Universe’s expansion (or contraction) is accelerating or not at a certain time. If a’‘(t) is positive then it is accelerating, if a’‘(t) is negative it is decelerating and if a’'(t) is zero, then it’s rate of expansion/contraction is constant.

The Hubble constant at time t is H=a’(t)/a(t) and it is clear from it’s definition only if a(t) is an exponential function of t can H be a constant in time, which in turn means that that a’(t) and a’'(t) are also exponential functions of t. The Hubble constant actually has a limited physical value as we infer that galaxies are receding from us by their redshift, but their cosmological red shift is not a function of the recession velocity; instead it is given by 1/a(t[sub]e[/sub])-1, where the light is emitted at time t[sub]e[/sub] (assuming we are talking about light received at the present time).

It’s worth noting that the scale factor, it’s time derivatives and other functions that are compositions of them (e.g. the Hubble constant/parameter) only tell you how the Universe is expanding, what they don’t tell you is what governs that expansion. What does govern that expansion is Einstein’s field equations, the central equations of general relativity, and what big bang theory does is relate the field equations and the expansion of the Universe (of course historically big bang theory actually predicted the expansion of the Universe). The field equations depend on the stress-energy tensor which describes the configuration of mass-energy.

By the relationship between redshift and the scale factor we can see that by collecting enough data on the redshift of the distant objects in the Universe we can determine the evolution of the scale factor and hence it’s time derivatives. In around 1998 observations of the redshift of supernovae indicated that the rate of expansion was increasing/the expansion of the Universe was accelerating (i.e. a’‘(t[sub]0[/sub]) is positive ), this came as a bit of a shock as in big bang theory the kind of mass-energy content the Universe was believed to have (i.e. normal matter and dark matter) implied that a’'(t) was always negative (i.e. the expansion was always decelerating). This lead to a new kind of energy content being posited called ‘dark energy’, which in the current epoch of the Universe is the dominant content. Dark energy isn’t quite as much of a fudge factor as it seems as Einstein’s field equations always allowed for this extra factor in terms of a cosmological constant and vacuum energy shares similar properties to dark energy, though a definite link between vacuum energy and dark energy is far from established as the difference between the observed value of the cosmological constant and the value we might expect if it were due to vacuum energy are very many orders of magnitude different.

Going back to your original question, the view that the Universe’s expansion is accelerating comes purely from observation and there is no conflicting line of evidence in this regard (that I am aware of). The nearest model to what you describe would be a FLRW open Universe with zero cosmological constant, in this model the rate of the expansion of the Universe decelerates over time to approach a constant rate of expansion. However this model has been ruled out by the observation of accelerated expansion.

6

I took the OP to be asking whether we have direct evidence that the current (last billion years or so) rate of expansion is varying (and ignoring the portion of change due to the changing density of the universe). As opposed to making a parametric fit to the total history, and using that to get the current change in the rate.

7

Thank you, Asympotically fat, for your extremely useful and informative post. Frankly, I’m operating at the limit of my comprehension for what is, after all, a rather technical subject. I have several questions, but before raising them would like to take a day to digest and do further reading. If you would be so kind as to check in on the thread tomorrow, I’d be most appreciative. Meanwhile, I have two relatively simple questions which will help to advance the discussion.

First, as ZenBeam suggests, one of the things which puzzles me - perhaps the most important thing - is whether there’s good evidence that the rate of expansion of the universe is currently increasing. Everything I’ve read says that galaxies within 1.3 billion light years are receding at a constant rate, i.e., proportional to distance. The second derivative of this data set, I’m pretty sure, is zero. What am I missing?

Second, speaking of second derivatives, if we were to plot the velocity of the robotic car posited in my thought experiment in the OP, the second derivative would be positive early-to-middle, but fall off later to zero. I’m suggesting the same thing could be true of the Type 1a supernova data vs. the Hubble data. Again, what am I missing?

8

“Currently” is a tricky thing, when you’re talking about derivatives. Suppose your only way of measuring your car’s motion was looking at the mile markers alongside the road. You can easily see your current location-- Just look at the mile marker you’re next to. For your velocity, though, you need to look at the mile marker you’re next to, and also consider when you passed the last mile marker: You can’t really get your current velocity right this moment, but just your average velocity over some time interval spanning into the past. And if you want to know your current acceleration, you have to look even further into the past, since you need to know your velocity now (which takes some time span), and your velocity at some time in the past (which takes a time span that extends yet further into the past).

In the case of the Universe, there’s no reason to think the acceleration isn’t still continuing, but we have to look a few billion years into the past to be able to detect it.

9

Something I don’t understand. Okay, so the universe is expanding (essentially there’s more universe being created between stuff constantly). We see this reflected in the redshift of distant objects - the further away something is from us, the faster it’s moving away. Okay, makes sense.

But now how does that imply acceleration? If object A is twice as far as object B, then you’d expect object A to be moving away a lot faster even if the expansion was constant because there’s twice as much distance between you and A, therefore twice as much expansion.

It may just be that pop-science programs have done a bad job explaining this, but how does that indicate that the expansion is accelerating? Is it because B is moving away at more than twice the rate of A, and object C, which is twice as far as B, would be moving away more than twice as fast as B?

Not quite sure if this is the same issue the thread is talking about.

10

And, if everything is separating at an ever faster speed, how long until the speed of divergence exceeds the speed of light?

11

That’s just a matter of distance. A large fraction of the universe is receding away from us at faster than the speed of light. That’s actually the barrier that creates the observable universe. Everything that’s far enough away will have so much space created between us and it that we’re separated at a rate faster than the speed of light, and the light will never reach us.

12

Right, if there were no continuing force (gravity or cosmological constant or whatever) acting on the Universe, that’s exactly what you’d expect to see: An object twice as far away would be moving twice as fast. If we had (normal, attractive) gravity and nothing else, then you’d instead expect an object twice as far away moving a little more than twice as fast, since gravity would be causing it to slow down. But what we actually see is that an object twice as far away is moving a little less than twice as fast.

13

Asympotically fat: Reading that again this evening, especially after reading the Wiki on scale factor for context, I understand the point you’re making. Among other things, I’m talking about a second derivative of the wrong thing. Fair enough. Still don’t understand how the rate of expansion can be increasing in the present tense when the recent data show a constant. Maybe I never will.

Chronos: To my knowledge, no one has said the Hubble data is wrong. Only that it doesn’t accurately describe the distant past. What inference should be drawn from there is the question I’m asking.

SenorBeef: My impression is that Hubble expansion is regarded as an acceleration (albeit an unusual one) because it compounds. As opposed, say, to everything moving away from everything else at a constant velocity, which is what we would expect if expansion were a ballistic event arising from the Big Bang.

MikeF: If you’re curious, here’s a longish blog post by a then-Cornell graduate astronomy student explaining faster than light expansion. Short answer, yes, it’s not only possible but expected.

14

PBear42, one thing you may be missing is that galaxies have random velocities in addition to their overall expansion (it’s not strictly random, some of it is due to gravity). Andromeda, for example, is actually moving closer to the Milky Way. You need a lot of galaxies, over a large distance, to overcome that randomness.

To use Chronos’s example with the mile markers, imagine they were placed by a drunk road crew, and have a large error in their position. You measure that it takes you a minute to get from one to the next, but you don’t know if they are really a mile apart. Could be half a mile, could be a mile and a half. So you might be going 120, or you might be going 40.

Then you measure the time to the next one, and it’s 45 seconds. Are you accelerating? You have no idea. Maybe those markers were just closer together. The fourth one appears after 63 seconds, so you average, and figure you’re going somewhere around 56 MPH, but it could really be 60 or it could be 50. If you think you might be accelerating at 1 MPH every ten minutes, it would take a long time to get enough measurements to tell whether you were accelerating.

After ten mile markers, you might know your average speed reasonably well, but there would still be too much error to tell if you were going 1 MPH faster at the end of those ten minutes than at the start.

15

Thanks. Yes, I’m aware of the first point. It’s discussed extensively in Freedman & Kaufmann, the intro college astronomy text cited in the OP. (BTW, I notice a typo; the title is just Universe, with no “the.”) As I understand it, that’s already taken into account in calculating the Hubble constant. Galaxies in our cluster (e.g., Andromeda) are disregarded. Galaxies in clusters not gravitationally bound with ours are moving away under Hubble flow. Other movements are basically noise in the signal and not regarded as sufficient to undermine the overall calculation.

Bear in mind that the Hubble constant was considered well established in 1998, when the far distant Type 1a supernova observations were made showing slower expansion in the distant past (i.e., at great distances from us today). That’s why they were such a big deal, ultimately winning the 2011 Nobel Prize in Physics. Which seems to me to answer your second point (and, by extension, Chronos’). The Hubble constant isn’t based on a few highway markers. It’s based on observation of thousands of galaxies by several methods, including Cepheid variables, the Tully-Fisher relation and Type 1a supernova (which had been used for intermediate cases before the breakthrough discovery at greater ones). Harmonizing the data from these various observations was a complex task, but all of them showed a constant relation at close-to-medium distances (which was all anyone had measured accurately prior to 1998). The precise rate of expansion (within various error bands) was debated, but not whether it was constant.

Now, it may be that something like your (and Chronos’) argument is the right answer to my question. Importantly, the Hubble data are a bit messy, mainly due to observational constraints rather than random movement of galaxies. The current consensus value of about 70 km’s/Mpc is a best fit line, not a bright and clear one. On the other hand, it has been subjected to lots of scrutiny, peer review and corroboration. It’s not revealed wisdom from on high, but is considered a robust data set and, within its limits, a robust conclusion. The scientific consensus is that the 1998 observations (and later corroborations) establish the Hubble constant does not hold at great distances (i.e., when the universe was about half the size it is today). And this data, too, has been subjected to careful scrutiny.

In sum, the rate of expansion has changed, no doubt about that. ISTM, though, in my admittedly non-expert opinion, that to conclude the rate is currently increasing, one must deal with the Hubble data. No authority I’ve seen does so, by the argument you (and Chronos) describe or otherwise. Hence my question. Especially when I can see (in theory, at least), another way to reconcile the two data sets.

16

I think you also have to realize the point that I made in my first post, generally speaking the Hubble constant isn’t a constant (in time). In fact if we assume a cosmological constant that is ‘normalized’ so it’s value is either 1,0 or -1 in an FLRW Universe a constant in time Hubble constant actually implies a Universe devoid of matter with a cosmological constant of 1 (a de Sitter Universe). A de Sitter Universe has always accelerated expansion and in fact the limit of how fast a FLRW Universe can expand.

A second point is that the Hubble constant has limited physical significance. Over short enough distances the Hubble constant can be used to approximately relate redshifts to distance, however that relationship is only approximate and the approximation breaks down over greater distances.

Prior to 1998 it was generally believed that the Universe’s expansion was decelerating, the reason for this is that the dynamics of a FLRW Universe with a cosmological constant of 0 only permit decelerating expansion. However in 1998 more accurate measurements of the Universes acceleration demonstrated that in fairly recent times its expansion has been accelerating - this is not in conflict with previous measurements.

17

This is a common misconception. That something is receding away from us with a FTL recession velocity. If we take the LCDM model currently seen as the best fit for the data, we can currently see objects with FTL velocities and secondly light emitted now by some objects with current FTL recession velocities will reach us in the future.

This again demonstrates the limited physical value of the Hubble constant and recession velocities.

18

No reconciliation is necessary. The Hubble data (assuming by that you mean the supernova 1a data) also shows the acceleration, at large distances, and is consistent with it at small distances. The further out your data set goes, the easier it is to see the acceleration, and because all data has experimental error and noise of various sorts, if you’re too close, you can’t see the acceleration at all.

19

Asympotically fat, I’ve already acknowledged the Hubble constant isn’t a constant in time, neither in theory (the formula by which it’s derived doesn’t even have a time term) nor in fact (the 1998 observations and later corroborations demonstrate it doesn’t hold at great distances). This doesn’t explain how the rate of expansion can be currently increasing yet not show up in data going out 1.3 billion light years. Your approximation argument is basically the same as Chronos’ and ZenBeam’s, no? Are you aware of any authorities making it? Cuz I’m not. For example, this paper by Riess (one of the three Nobel laureates) doesn’t as far as I can see. Yes, it could be the answer, but that would be tantamount to using a relatively small data set to throw out a much larger one. This isn’t how science is usually done.

Moreover, please understand that I’m not trying to reinstate the view of the universe prevailing prior to 1998. As I understand it, the big sea change came from cosmic background microwave radiation data, which (i) demonstrated a flat or nearly flat universe and (ii) couldn’t be accounted for by ordinary and dark matter, from which (iii) dark energy (however defined and conceived) turned out to be the dominant component of the aggregate matter-energy density of the universe. Taking this as given, I wonder whether we can’t harmonize the Hubble data and the distant Type 1a data by assuming (a) dark energy creates a constant rate of expansion but (b) in the distant past, gravity kept galaxies from sweeping away unfettered at this rate. On this view, the old gravitationally bound conception of the universe was partly right but mostly wrong. That is, gravity had an effect, but was ultimately overcome by fundamental expansion of the universe. As my thought experiment (which no one seems to want to discuss) suggests, it’s possible to have two forces operating in opposite directions which terminate at a constant expansion, notwithstanding observing a positive change in the rate of expansion over a significant portion of the curve.

Perhaps there are good theoretical reasons why this solution doesn’t work. Goodness knows I’m no physicist, as I’ve repeatedly acknowledged. But it seems to me an intuitively plausible model (much more plausible than the view that dark energy somehow increases in effect as distance increases). And, more importantly, accounts for both data sets without discarding either.

20

Can you link me to this data? I tried searching, to plot it for myself and see how noisy it really is. I found distance data, here for example, but I couldn’t find a data set with distance and with recession velocity.