One of Purcell’s problems is labeled “Electromagnetic energy in your eyeball,” and it concludes with a provocative (and true) observation. The problem asks the reader to calculate the total energy in all the photons that are inside your eyes at any one moment. Roughly speaking — which is the point, since we’re doing back-of-the-envelope problems — these photons come from one of two sources: the visible light from the outside world that enters your pupil, and the infrared light that is emitted as blackbody radiation from your eye itself, since you are an object at body temperature. Purcell suggests that you compare the amount of energy from each source.

And the answer is: there is much more electromagnetic energy in your eye at any one moment from the infrared radiation you’re emitting yourself, than the pittance of visible light you get from the outside world. Between 100,000 and a million times as much. Which raises a question we may never have thought to ask: why does it get dark when we close our eyes? The amount of electromagnetic radiation hitting our retinas hardly changes!

Purcell’s last sentence gives the answer: “Only quantum mechanics can explain why that makes it dark!”

We see light when photons of an appropriate wavelength reach the photoreceptor cells in the retinas of our eyes. The energy from the photon is converted into chemical energy via phototransduction, which sets an electrochemical signal to the visual cortex. (Presumably unnecessary disclaimer: everything I know about vision I learned from Wikipedia.) In particular, the photons are absorbed by a chemical called retinal, which isomerizes from the 11-cis state to the all-trans state. (That last bit was a blatant cut and paste.)

Here’s the part I do understand: isomerization is a matter of nudging a chemical from one structural form to another, without actually changing the chemical formula. Molecules have energy levels, just like electrons in atoms, and in order to effect the change in the retinal via photoexcitation, a photon has to have enough energy to cause a transition between the isomers. That’s a matter of quantum mechanics, full stop. Molecules can’t take on just any old energy; the allowed energies are quantized. As a result, it doesn’t matter that the infrared light inside your eyeball has much more energy than the visible light from the outside world; the energy comes in the form of individual photons, none of which has enough energy to get the reaction going. It’s very analogous to the photoelectric effect in metals, for which Einstein won his Nobel prize.

We often say that quantum mechanics applies to the world of the very small, and involves mind-bending alterations of our everyday reality. Which is true as far as it goes, but the more simple truth is that quantum mechanics applies to absolutely everything. It underlies how the everyday world works, from the stability of matter to the darkness when you close your eyes.

Now we have a better analysis, from people who are experts: Jo Bovy and Scott Tremaine have a paper in which they examine the claim closely. They find it wanting. This was pointed out here in a comment by Ben; Jester and Peter Coles also have useful blog posts up about it.

Short version: the original authors made assumptions about the distribution of velocities of the stars they were looking at, and those assumptions are known to be wrong. Using a better model (i.e., one more compatible with known data), Bovy and Tremaine show that the observations are perfectly consistent with the conventionally-assumed dark matter density. The good news is that they are actually able to use this technique to get a more precise measurement of that density than was previously available. It’s a rare scientific lemon that can’t be turned into at least a little bit of lemonade.

I’m not sure why people get so emotional about dark matter. The original paper here by Bidin et al. was accompanied by a dramatic press release from the European Southern Observatory. I am known as a “dark matter supporter,” but I have no personal investment; I think it would be much cooler if something crazy were going on with gravity. But that’s not what the data indicate. It’s just some new particle we haven’t yet made in the lab, hardly the end of the world.

And it may very well turn out that the behavior of gravity on large scales does not precisely match the prediction of ordinary general relativity. Nevertheless, I think that by now we’ve accumulated enough data to conclude that the universe cannot be explained solely by modifying gravity; there is ample evidence of gravitational forces pointing in directions where there isn’t any (ordinary) “stuff” to create them, leading us to accept the existence of some form of dark matter. About a year ago I put up a post that explained this point of view, and took aim in particular at the popular framework known as MOND.

This led to some good discussion in the comments, and also to a behind-the-scenes email exchange between Rainer Plaga, Stacy McGaugh, and me. It’s a bit of old news, but I thought there would still be some interest in our discussion, so (with permission) I’m posting our emails here. Seeing how the sausage is made, as it were. It’s a bit of a long read, sorry about that.

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Rainer, March 1:

Dear Sean,

I discussed your recent vigorous defense of CDM your blog with Stacy, and he encouraged me to send you my – absolutely objective – position.

On the one hand I am with you that if Stacy uses terms like “serious fine-tuning problem for LCDM” in his newest paper’s abstract (which are then interpreted by science journalists in the way you exhibit), he had to quantitatively compare the expected properties of galaxies under the assumption of \LambdaCDM with his data set. If he wants to criticise an idea he has to deal with the idea not with alternatives to it. Alas, he does not do that in this paper.

On the other hand I strongly disagree out of principle to require statements like: ?of course we have more than sufficient evidence to conclude that dark matter exists, we?re just trying to understand how it works and what else might be going on.? from anybody. Really Sean, this sounds like a caricature of the holy inquisition to me, “philosophers can speculate as long as they accept that the final truth is already known from the holy scriptures ”.

Your statement: “Dark matter is real … there’s no reasonable doubt about the dark matter.” is misleading. Stacy and I of course know that dark matter in the form of massive neutrinos does exist beyond reasonable doubt. But that does not answer a crucial question. Crucial questions are: what flattens the rotation curves in galaxies? What creates the third CMB peak? CDM, MOND or something else?

In my opinion the final verdict on these questions is not in, yet. Allow me to argue why your top 3 arguments for the existence of CDM do not convince me, perhaps yet.

1. “MOND is ugly”: The alternative is not “theory for MOND” vs. GR but “theory for MOND” vs. GR + “theory for CDM particle”. The number of exhibited equations then becomes similar. How do you know that TeVeS is uglier than the “theory for CDM particle”?

2. “Clusters require DM anyway” If one could make a case that they require nonbaryonic cold dark matter, I would consider the case settled in favour of CDM. However, the dark matter required for MOND in clusters might be the ca. 40% fraction of baryonic matter that we anyway know is currently missing in clusters (even in LCDM). Do we agree? How can the argument be clinching then?

3. Your strongest argument is the one from the CMB. But still, replacing “MOND” with “CDM”, couldn’t your statement:

“Can some clever theorist tweak things so that there?s a MOND version that actually fits? Probably. Or we could just accept what the data are telling us.”

be used just as well to comment on the well known problems of CDM to reproduce the detailed properties of galaxies?

Wouldn’t this be a great topic for another “great debate” a la Shapley/Curtis 1920 between u and Stacy? In that case it turned out both were partly right and wrong, my personal bet: it would be the same this time .

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Sean, March 1:

Hi Rainer–

Ten years ago, it was perfectly respectable to speculate that there was no such thing as dark matter, just a modification of gravity. (It couldn’t have been MOND alone, which was ruled out by clusters, but it could have been some more elaborate modification.) That’s no longer true. The Bulltet Cluster and the CMB both provide straightforward evidence that there is gravity pointing in the direction of something other than the ordinary matter. The source for that gravity is “dark matter.” It could be simple, like an axion or a thermal relic, or it could be quite baroque, like TeVeS + sprinkles of other dark matter as required, but it’s definitely there.

If people want to contemplate that there is dark matter and also a modification of gravity, that’s fine. If people want to point to features of galaxy/cluster phenomenology and say that these features must be explained, that’s absolutely fine. But if people want to cling to the possibility that dark matter doesn’t exist, that’s not being appropriately cautious, it’s just ignoring the data, and it’s a disservice to the public to pretend otherwise.

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Rainer, March 2:

Dear Sean,

I do not fully understand your argument: do you argue that the bullet cluster proves that _nonbaryonic_ DM exists? To me Stacy’s argument – that MOND might work only with the baryonic cluster DM which is an additional problem even within LCDM – cannot be currently excluded (see 2. in my previous e-mail). Do you disagree with his argument, and if yes, why?

For your convenience let me summarize Stacy’s general argument in my own words (Stacy please protest if I misrepresent it):

a. even within LCDM generally uncontested facts are that in clusters of the size of the bullet cluster (< 10(13) M_sun):
1. ca. 50% of the cluster's _baryonic_ matter is probably in some invisible form
2. the hot gas is a minor component of the total baryonic matter (see e.g. fig.1 here: http://arxiv.org/abs/1007.1980)

b. suppose that this baryonic cluster DM is in some non-collisional form (e.g. jupiters). Then a.1. would quantitatively explain MOND’s missing cluster DM and a.2. the observational fact that the bullet’s cluster mass is concentrated on the galaxies and not the hot gas.

It is somewhat paradoxical, but seems clear: if you want to rule out MOND you have to deal with its details, if Stacy wants to rule out CDM he has to deal with its details. Neither of you guys is really doing this, and I can understand why: both of you would feel you are wasting time on a wrong concept. But you would not .

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Sean, March 2:

Hi Rainer–

We know how much baryonic matter there is from BBN. It’s not enough to explain the Bullet Cluster or the CMB, even with MOND. Not to mention that you would have to come up with some way to turn the large majority of baryonic matter into some collisionless form. (The paper you just cited says ” the baryons are not missing, they are simply located in cluster outskirts” right there in the abstract.)

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Rainer, March 2:

Hi Sean,

We know how much baryonic matter there is from BBN. It’s not enough to explain the Bullet Cluster or the CMB, even with MOND.

They claim ca. a factor 2 more dark baryonic matter than seen is needed in the clusters. What problem would that pose with BBN? (Don’t forget that the baryonic matter/CDM ratios derived from LCDM in clusters are meaningless if MOND were the answer).

Not to mention that you would have to come up with some way to turn the large majority of baryonic matter into some collisionless form.

Yes, this would need some ad-hoc gastrophysics to produce enormous amounts of e.g. jupiters especially in the cluster centre. Not nice, but not impossible, cooling flows etc… But if all that were true, the bullet cluster would be OK.

(The paper you just cited says ” the baryons are not missing, they are simply located in cluster outskirts” right there in the abstract.)

But that’s exactly what is needed also for MOND: the dark baryons are really hiding somewhere… They are not claiming a detection of these baryons! But let us take a step back on this paper:

What it discusses is the fact that clusters need some dark baryonic matter even in LCDM, ca. 30% of the baryonic matter is apparently unseen. This was unexpected, some gastrophysics will be needed to explain it. (They mention “AGN feedack” and stuff…)

MOND’s problem is more severe, ca. 70% of the baryonic mater would apparently be unseen in the central parts of the clusters. This was unexpected some gastrophysics will be needed to explain it.

Sorry, Sean, this seems like an open problem to me both for LCDM and MOND, admittedly a bigger one for MOND (but then clusters are their worst problem…), but not the ultraclean evidence for CDM that you are claiming…

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Stacy, March 2:

OK, I think at least we all agree that BBN tells us the baryon density of the universe. Lets deal with one thing at a time, the dark matter in clusters. If I understand you, you are saying MOND is falsified because there is dark matter in clusters. Rainer is suggesting that a logical way out of this is if the excess mass in clusters is in some dark, baryonic, collisionless form. I agree it is tough to imagine what that would be (and have consistently said as much) but I am not willing to grant that I know it to be impossible. So the real leap to falsify MOND is to say that the dark mass in clusters is not just dark baryons, but WIMPs (or whatever non-baryonic particles compose CDM). And that follows how? Because Omega_m > Omega_b?

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Sean, March 3:

MOND without non-baryonic DM is falsified by clusters, because you can’t fit them with the baryons implied by BBN regardless of what form they take. That’s admitted by most people, e.g. Sanders’ paper.

More interesting is the question of whether you could get around the need for non-baryonic DM with some other theory of modified gravity. The Bullet Cluster and CMB, again to most people, imply not. Could you wriggle out of that conclusion by combining some new as-yet-unformulated modification of gravity with a huge population of mysterious intergalactic Jupiters? No, because you would still be completely wrong on the CMB. It’s time to accept what the data are telling us and move on.

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Stacy, March 3:

MOND without non-baryonic DM is falsified by clusters, because you can’t fit them with the baryons implied by BBN regardless of what form they take. That’s admitted by most people, e.g. Sanders’ paper.

Ah. I thought this was the conceptual error you were making. Clusters you certainly could fit just with baryons. They’re rare systems. If that is the only place we need dark baryons, then do the integrals. You can satisfy the residual mass discrepancy in clusters in MOND without making much dent in the BBN missing baryon budget.

Do I *like* such a solution? Certainly not. Neither do I like that fact that clusters are the only systems that come close to having the right baryon content in LCDM. Whay are galaxies missing more than half of their baryons? Dwarfs > 90%? I can imagine how this might happen, but the solutions are comparably contrived. The more basic point is that I am not willing to condemn a theory for needing some dark baryons if its competitor also needs dark baryons.

More interesting is the question of whether you could get around the need for non-baryonic DM with some other theory of modified gravity. The Bullet Cluster and CMB, again to most people, imply not. Could you wriggle out of that conclusion by combining some new as-yet-unformulated modification of gravity with a huge population of mysterious intergalactic Jupiters? No, because you would still be completely wrong on the CMB. It’s time to accept what the data are telling us and move on.

The CMB is really interesting. I correctly predicted the amplitude of the second peak (a prediction that is still quantitatively correct) by making the ansatz that there was whatever generally covariant theory might grow out of MOND looked just like GR in the early universe. Obviously that has to change later in order to grow structure, but at least it gives some proxy for what MOND might do with the CMB. At the time, I discussed some of the ways in which this would inevitably fail.

The response initially was that MOND itself made no prediction for the CMB, therefore we should disregard the chance success of this prediction. Now you want to treat the low third peak as an absolute prediction of MOND. You can’t have it both ways. Which is it?

A low third peak would have falsified LCDM. It survives that test. That does not automatically falisify MOND. It just means that the relativistic parent theory (whatever that might be – it is not obvious to me it has to be TeVeS) has to have a net forcing term a la CDM. Does that seem reasonable to me? No, and (as I said with the ultrafaint dwarfs) I too was ready to write off MOND on this point. But Skordis & Ferreira showed that the scalar field in TeVeS might have just such an effect. So I can not, in good conscience, say it is impossible.

You should not accuse me of ignoring data. I have written papers on these subjects. Indeed, one of the things that surprised and impressed me about MOND, when I first got over my initial revulsion and started to look into it, was what a great breadth and wealth of data it did quite in explaining. From the tone of your statements, I imagine you have no idea what I’m talking about. You really ought to check your facts before making ignorant statements to the effect that “MOND only does rotation curves.”

Indeed, you yourself appear to be ignoring facts. Why do any MOND predictions come true? Let’s suppose it is only true that all MOND does is fit rotation curves. That demands an explanation – one you nowhere attempt to provide. Your reasoning appears to boil down to “We’re sure that CDM exists, so somehow it must work out.” Well, I’ve tried – very hard – to see how it could work out. It aint easy. I won’t say it is impossible. But it is as absurd as some of the above dodges are with MOND. Dark matter in galaxies is like epicycles – you can fit anything you like, but it doesn’t explain why a simple formula does better.

You may find it hard to believe, but I started from exactly the same perspective as you. I am far more comfortable with CDM than with MOND. I will breathe a great sigh of relief if and when WIMPs are detected in the laboratory. Then we’ll know the answer, and we won’t have to have these bitter debates. However, I am not being unreasonable in holding the theory to a high standard of proof. If you want to convince me that, for sure, the universe is filled with some till-now hypothetical particle from a hypothetical dark sector outside of the Standard Model of particle physics, then show me a piece. Until then, you are over-reaching.

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Sean, March 3:

You can’t just wave your hands and say that a mysterious “forcing term” will help explain the CMB. If there is no non-baryonic dark matter, there is no way that even-numbered peaks can be different from odd-numbered peaks; the configuration of baryons is precisely analogous. You can mimic the situation in TeVeS (although the numbers don’t seem to work out) because you’ve introduced an independently propagating scalar degree of freedom whose energy density doesn’t follow the baryons. You can give that scalar whatever name you like, but it is “non-baryonic dark matter.” A particularly contrived version, but that’s what it is.

You can’t explain the third peak without a source for gravity that propagates independently of the baryons.

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Rainer, March 3:

MOND without non-baryonic DM is falsified by clusters, because you can’t fit them with the baryons implied by BBN regardless of what form they take.

Why is that? I just don’t get it, and am very open to be persuaded. 90% of all cosmic baryons are presently undetected, right? Only a fraction of the baryonic matter we see directly is in clusters (O(a few percent), let’s say 10%) So why can’t a small fraction, say O(2%), of all the cosmic dark baryons be in the form of e.g. jupiters in the central parts of clusters? They and stars would then dominate the cluster mass and be dissipationless —> no problem with the bullet cluster in MOND.

That’s admitted by most people, e.g. Sanders’ paper.

Where? In http://arxiv.org/abs/astro-ph/0703590 he states about cluster dark matter in MOND: “For example, there are more than enough undetected baryons to make up the missing dark component; they need only be present in some non-dissipative form which is difficult to observe.”

He also likes massive neutrinos, but not to the exclusion of baryonic dark matter.

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Stacy, March 4:

Hi Sean,

OK, now we are discussing science again.

I take your point about the CMB very seriously. It seems to me that you are putting a lot of weight on the third peak, which is not all THAT well constrained. WMAP really has to scrape to get there, so the result is dominated by the systematics of PSF modeling. I presume they’ve done that right, but there are double exponential corrections involved in subtracting the foreground and then getting back to the cosmic signal, so they don’t have to go far wrong to make a bad mistake with the third peak. Presumably PLANCK will clarify this soon, though a glance at their first release images does not provide a lot of confidence about the foreground MW masks that WMAP used. I also wonder, given the visceral reaction you and others have at any suggestion that LCDM might be questionsed, if the PLANCK team would let themselves admit a low third peak even if the saw it.

For now, we have an apparently clear detection of a high third peak in WMAP, and we need to explain the data we have rather than the data we hope soon to have. And honestly, I expect the most likely outcome to be a confirmation of WMAP, with only minor tweaks. So we have to understand the third peak along with clusters and rotation curves and dwarf spheroidals and everything else.

I freely admit that I don’t know how to make the third peak high. I also don’t know that a high-ish thrid peak can’t be obtained in a more general theory. I agree with your point that pure baryons shouldn’t do that – the vector is wrong, as you say. I’m not even convinced TeVeS can do it. But lots of theories (not just MOND-inspired ones) invoke scalar fields, so I can’t exclude the possibility.

I also agree that this is contrived. But we are WAY into contrivance with LCDM, a point I believe you’ve made yourself on occassion. We’ve just gotten familiar with the contrived parts so that they no longer bother us. That doesn’t make them any less contrived.

You make the point that the scalar field solution in TeVeS is just a contrived form of non-baryonic dark matter. But even in pure GR we could use some form of non-baryonic dark matter that gives us the MOND phenomenology. Why not consider an effect due to the physical nature of the particles? Until we detect WIMPs, surely you at least agree that we don’t really know what the dark matter is?

I know everybody invokes feedback to “fix” galaxies, but those models are just as contrived. Actually, they are considerably more contrived, as they inevitably require many more parameters, and those parameters are simply tuned to match observations. Any competent theorist can tune any model to fit a given set of data.

I must have said this to you before, but I will say it again. The MOND formula provides an apparently correct description of the effective force law in galaxies. How does the dark matter “know” to arrange itself just so as to look like MOND? If it manages this trick in galaxies, why not in the solar system? How would we know that the solar system isn’t really run by an inverse-cube force law, but there is dark matter arranged just so as to make it look like an inverse-square law?

Could anything be more contrived?

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Sean, March 6:

Hi Stacy–

I’m not sure what you are saying about the third peak in the CMB. We agree that “pure baryons shouldn’t do that.” I can only think of three possibilities.

(1) There is some sort of source for gravity other than baryons.
(2) There is a modification of gravity that doesn’t include new sources, but also doesn’t respond directly to where the sources actually are.
(3) The data aren’t good enough to say that the odd-numbered peaks are boosted relative to what we would expect from damped oscillations of baryons alone.

If it’s (1), then that’s non-baryonic dark matter and we should just admit it. I think that (2) is physically implausible, and as far as I know nobody has suggested otherwise. And I think that the time is past when anyone could credibly hang on to (3). Here’s a relatively recent figure (2 years ago) from Ned Wright’s web site.

Am I missing a possibility, or would you buy one of these three?

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Stacy, March 7:

Hi Sean,

I basically agree with the 3 possibilities you list. Indeed, I thought that was pretty much what I said.

You imply that it is hanging on to vain hope to explain the third peak of the CMB by anything other than a new source. I am saying that it is a vain hope to imagine that turning the crank on any number of CDM numerical simulations is ever going to spit out the observed MONDian phenomenology. Just because LCDM works for the CMB does not automatically guarantee that it’ll work in galaxies, any more than MOND’s success in galaxies means it must inevitably succeed as a the basis of a cosmological theory.

There is a very simple empirical result in the data for galaxies that cosmologists have, by and large, simply ignored. The stated excuse is usually something like “well, galaxies are complicated, non-linear structures” and so we should be excused from explaining them. Indeed, in LCDM galaxies probably should be complicated. But they’re not. They’re simple. So simple, the obey a single effective force law. Fitting that with dark matter is like fitting epicylces to planetary orbits. Of course you can do it – you have an infinite number of free parameters. But it don’t make no sense.

I have said for years now that they conclusion you come to depends on how you weigh the evidence. The CMB is an important piece of that evidence. So are rotation curves. It is not obvious to me that the third peak should count 100% and galaxies zero. Yet that is in effect the weighting that lots of people appear to be using.

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Sean, March 8:

Hi Stacy–

I think we’ve reached the end of what needs to be said. You agree with my three possibilities, and you agree (I think) that the CMB data are good enough to draw some conclusions. It comes down to whether you are willing to entertain the possibility that there is a mysterious new force that does not involve any new sources, yet also does not respond directly to where the actual sources are. (And in the process reproduces exactly what we would see if there were CDM.) You may think that is plausible — I, and most people in the field, do not. Therefore, we believe that there is non-baryonic DM, and the question is how it behaves.

You seem to think I am defending LCDM, when I have never mentioned it. I am defending the claim that “non-baryonic dark matter exists.” As I said in the original post, we certainly have to explain the phenomenology of galaxies and clusters, and the right explanation may very well involve a modification of gravity or interesting new physics in the dark sector — both of which I’ve written papers about. Nobody is suggesting that we ignore data from galaxies and clusters. But none of that data straightforwardly implies “non-baryonic dark matter does not exist.” It’s a complicated dynamical problem. The CMB — an enormously simpler system, where everything is in the linear regime — does straightforwardly imply “non-baryonic dark matter exists.” Admitting that will improve our chances for future progress.

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Stacy, March 8:

Yes, we’ve said what we’re going to say. But you still don’t seem to get it. The CMB is simple. It is not enormously simpler. Galaxies are also simple. One must invoke absurdly complex mechanisms to make that happen. The argument against dark matter doing this boils down to fine tuning. I don’t like fine tuning problems, especially when a theory is not otherwise falsifiable (e.g., epicycles). Note that as you claim not to be specifically defending LCDM, I am not specifically defending MOND. There is an empirical phenomenology that constitutes a fine tuning problem for ANY dark matter picture (that does not some how build it in).

Since we can’t explicitly falsify the existence of dark matter, what could be worse than this mother of all fine-tuning problems? I understand the implausibility of what you are saying in the CMB, but you seem to miss the same kind of point in galaxies. I worry that we won’t find WIMPs and keep pursuing other DM candidates indefinitely – how do we know when to stop? How would this be different from another millenium of dark epicycles?

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Rainer, March 18:

Dear disputants,

Thanks for this really informative and nearly polemic free (Stacy please stop blaming your colleagues to construct epicycles ) debate!

To me (and it seems also to Stacy) Sean’s concentration on his main argument, makes his case for some kind of “dark non-baryonic field that enters the stress-energy tensor in GR” quite convincing. It then stands to reason (but is not absolutely necessary) to identify it with a quantum field for some new massive particle.

If I may make Stacy’s main point in my own words: galaxies are observed to be simpler than they would be expected to be: at least a large fraction of them obeys a strange simple MOND rule, which is without a simple plausible motivation in known physics. In addition there are indications that galaxies sometimes behave in ways that they should not in LCDM (tidal dwarves should not contain dark matter but they seem to do).

This reminds one of atoms in classical physics, which were expected to show a very complex behaviour but obeyed strange simple rules, sometimes in contradiction to the known physical laws at the time. The old quantum condition comes to mind as somewhat analogous to MOND’s law of motion. Initially it was attempted to explain these rules within the known concepts, and that was all right and necessary.

But, as quantum mechanics showed, there is _also_ the possibility that strange simple rules for basic objects of the theory are first clues for really new concepts.

Sean, don’t you have at least a little bit of sympathy for this possibility?

I close with following proposal: CDM or MOND? is not a good question. A better question is: are the successes of the MOND rule _perhaps_ a first clue to new concepts which will modify our understanding of the “dark non-baryonic field that enters the stress-energy tensor in GR” in the sense that it is not only a new quantum field within standard QFT?

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Sean, March 18:

Hi Rainer–

Sure, I’m happy to agree with that. In fact, you will find exactly those sentiments way back in my original blog post on the topic. I just think we’re past the point where we can conclude that non-baryonic dark matter exists — what form it takes, how it interacts, and what additional things might be going on, are all crucially important questions. Of course DM faces important challenges from the phenomenology of complex structures, and that should be taken seriously; but no-DM alternatives are ruled out by the data, which should also be taken seriously.

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Stacy, March 19:

Science is dead.

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A week or so ago, I spent a couple of days back at Syracuse University, where I was a faculty member for quite a few years. I was there primarily to participate in a special event that preceded the East Coast Gravity Meeting being held there on the following weekend. The event was a celebration – GoldbergFest – of the career of Josh Goldberg, a good personal friend, and an eminent relativist at Syracuse, who has been an emeritus professor there for many years now.

Josh began as a graduate student at Syracuse in the early 1950′s, working on conservation laws in General Relativity (GR) and on canonical quantization. At the time Syracuse had one of the few well-known relativity groups in the world, led by Peter Bergmann, and an impressive group of young people were trained under him, and later under Josh, as students and postdocs; people like Ted Newman, Ray Sachs, Art Komar, Roger Penrose, and many others. I’m certainly no expert on the precise history of the Syracuse group, but fortunately, as part of a special issue of General Relativity and Gravitation dedicated to Josh, to which I was honored to also contribute, Ted Newman describes it wonderfully. The Fest was a lovely event. I enjoyed the other speakers’ talks – John Stachel, Rafael Sorkin and Peter Saulson, and Ted Newman’s hilarious and touching after dinner speech, and the reminiscences of the other people at the dinner made for what I hope Josh thought was a wonderful day.

Over the next two days quite a few of our students and postdocs from Penn gave talks at the East Coast Gravity Meeting, and I was delighted to hear that our very own Godfrey Miller won the award for the best student presentation.

Returning To Penn, I just about had enough time to finish putting together the take-home final exam for my graduate General Relativity course, before heading off to NYU on Wednesday with our whole group for our semesterly joint meeting. We were joined, as usual, by a nice crowd from Columbia and Case Western for a day of talks and discussion. I always find these meetings to be incredibly useful scientifically, because the group is so interactive, boisterous and interested in the material, while being such warm and friendly hosts. It makes for an enjoyable day every time. Beyond the obvious exchange of ideas, these meetings also provide an opportunity for our students to get used to giving talks on their work. This time my student – Garrett Goon – and one of my colleague Justin Khoury’s students – Austin Joyce – gave our student talks, leading to some healthy discussion Wess-Zumino terms in new field theories and conformal cosmology, respectively. Both did a terrific job, although they’re becoming old pros at this point, rather than beginning students in need of practice.

To close out last week, Greg Gabadadze from NYU came down on Friday so that we could try to finish up some details in a project that is close to completion, before we start dispersing for various summer conferences. I’ll discuss these soon, I expect.

Today my final exam will be turned in and grading starts, an old friend is delivering a seminar in our group, and Sean’s student Kim Boddy arrives for a week so that the three of us can try to finish up a paper. The end of the semester always seems to go this way. While all these things are fun, life becomes excessively hectic for two weeks, and then travel begins.

First Lawrence Krauss came out with a new book, A Universe From Nothing: Why There Is Something Rather Than Nothing (based in part on a popular YouTube lecture), which addresses this question from the point of view of a modern cosmologist. Then David Albert, speaking as a modern philosopher of science, came out with quite a negative review of the book in the New York Times. And discussion has gone back and forth since then: here’s Jerry Coyne (mostly siding with Albert), the Rutgers Philosophy of Cosmology blog (with interesting voices in the comments), a long interview with Krauss in the Atlantic, comments by Massimo Pigliucci, and another response by Krauss on the Scientific American site.

I’ve been meaning to chime in, for personal as well as scientific reasons. I do work on the origin of the universe, after all, and both Lawrence and David are friends of the blog (and of me): Lawrence was our first guest-blogger, and David and I did Bloggingheads dialogues here and here.

Executive summary

This is going to be kind of long, so here’s the upshot. Very roughly, there are two different kinds of questions lurking around the issue of “Why is there something rather than nothing?” One question is, within some framework of physical laws that is flexible enough to allow for the possible existence of either “stuff” or “no stuff” (where “stuff” might include space and time itself), why does the actual manifestation of reality seem to feature all this stuff? The other is, why do we have this particular framework of physical law, or even something called “physical law” at all? Lawrence (again, roughly) addresses the first question, and David cares about the second, and both sides expend a lot of energy insisting that their question is the “right” one rather than just admitting they are different questions. Nothing about modern physics explains why we have these laws rather than some totally different laws, although physicists sometimes talk that way — a mistake they might be able to avoid if they took philosophers more seriously. Then the discussion quickly degrades into name-calling and point-missing, which is unfortunate because these are smart people who agree about 95% of the interesting issues, and the chance for productive engagement diminishes considerably with each installment.

How the universe works

Let’s talk about the actual way physics works, as we understand it. Ever since Newton, the paradigm for fundamental physics has been the same, and includes three pieces. First, there is the “space of states”: basically, a list of all the possible configurations the universe could conceivably be in. Second, there is some particular state representing the universe at some time, typically taken to be the present. Third, there is some rule for saying how the universe evolves with time. You give me the universe now, the laws of physics say what it will become in the future. This way of thinking is just as true for quantum mechanics or general relativity or quantum field theory as it was for Newtonian mechanics or Maxwell’s electrodynamics.

Quantum mechanics, in particular, is a specific yet very versatile implementation of this scheme. (And quantum field theory is just a particular example of quantum mechanics, not an entirely new way of thinking.) The states are “wave functions,” and the collection of every possible wave function for some given system is “Hilbert space.” The nice thing about Hilbert space is that it’s a very restrictive set of possibilities (because it’s a vector space, for you experts); once you tell me how big it is (how many dimensions), you’ve specified your Hilbert space completely. This is in stark contrast with classical mechanics, where the space of states can get extraordinarily complicated. And then there is a little machine — “the Hamiltonian” — that tells you how to evolve from one state to another as time passes. Again, there aren’t really that many kinds of Hamiltonians you can have; once you write down a certain list of numbers (the energy eigenvalues, for you pesky experts) you are completely done.

We should be open-minded about what form the ultimate laws of physics will take, but almost all modern attempts to get at them take quantum mechanics for granted. That’s true for string theory and other approaches to quantum gravity — they might take very different views of what constitutes “spacetime” or “matter,” but very rarely do they muck about with the essentials of quantum mechanics. It’s certainly the case for all of the scenarios Lawrence considers in his book. Within this framework, specifying “the laws of physics” is just a matter of picking a Hilbert space (which is just a matter of specifying how big it is) and picking a Hamiltonian. One of the great things about quantum mechanics is how extremely restrictive it is; we don’t have a lot of room for creativity in choosing what kinds of laws of physics might exist. It seems like there’s a lot of creativity, because Hilbert space can be extremely big and the underlying simplicity of the Hamiltonian can be obscured by our (as subsets of the universe) complicated interactions with the rest of the world, but it’s always the same basic recipe.

So within that framework, what does it mean to talk about “a universe from nothing”? We still have to distinguish between two possibilities, but at least this two-element list exhausts all of them.

Possibility one: time is fundamental

The first possibility is that the quantum state of the universe really does evolve in time — i.e. that the Hamiltonian is not zero, it truly does push the state forward in time. This seems like the generic case (there are more ways to be not-zero than to be zero), and it’s certainly the one that we spend time considering in introductory courses when we foist quantum mechanics on fearful undergraduates for the first time. A wonderful and under-appreciated consequence of quantum mechanics is that, if this possibility is right (the universe truly evolves), time cannot truly begin or end — it goes on forever. Very unlike classical mechanics, where the universe’s trajectory through the space of states can bring it smack up against a singularity, at which point time presumably ceases. In QM, every state is just as good as every other state, and the evolution will go happily marching along.

So what does this have to do with something vs. nothing? Well, as the quantum state of the universe evolves, it can pass through phases where it looks an awful lot like “nothing,” conventionally understood — i.e. it could look like completely empty space, or like some peculiar non-geometric phase where we wouldn’t recognize it as “space” at all. And later, through the relentless influence of the Hamiltonian, it could evolve into something that looks very much like “something,” even very much like the universe we see around us today. So if your definition of “nothing” is “emptiness” or “lack of space itself,” the laws of quantum mechanics provide a nice way to understand how that nothing can evolve into the marvelous something we find ourselves inside. This is interesting, and important, and worth writing a book about, and it’s one of the possibilities Lawrence discusses.

Possibility two: time is emergent/approximate

The other possibility is that the universe doesn’t evolve at all — the Hamiltonian is zero, and there is some space of possible states, but we just sit there, without a fundamental “passage of time.” Now, you might suspect that this is a logical possibility but not a plausible one; after all, don’t we see things change around us all the time? But in fact this possibility is the one you immediately bump into if you simply take classical general relativity and try to “quantize” it (i.e., invent the quantum theory that would reduce to GR in the classical limit). We don’t know that this is the right thing to do — Tom Banks, for example, would argue that it’s not — but it’s a possibility that is on the table, so we should think about what it would mean if it turns out to be true.

We certainly think that we perceive time passing, but maybe time is just emergent rather than fundamental. (I don’t like using “illusory” in this context, but others are not so circumspect.) That is, perhaps there is an alternative description of that single, unmoving point in Hilbert space — a description that looks approximately like “a universe evolving through time,” at least for some period of duration. Think of a block of metal sitting on a hot surface, not evolving with time but with a temperature gradient from top to bottom. It might be possible to conceptually divide the block into slices of equal temperature, and then write down an equation for how the state of the block changes from slice to slice, and find that the resulting mathematical formalism looks just like “evolution through time.” In this case, unlike the previous one, time could end (or begin), because time was only a useful approximation to begin with, valid in a certain regime.

This kind of scenario is exactly what quantum cosmologists like James Hartle, Stephen Hawking, Alex Vilenkin, Andrei Linde and others have in mind when they are talking about the “creation of the universe from nothing.” In this kind of picture, there is literally a moment in the history of the universe prior to which there weren’t any other moments. There is a boundary of time (presumably at the Big Bang), prior to which there was … nothing. No stuff, not even a quantum wave function; there was no prior thing, because there is no sensible notion of “prior.” This is also interesting, and important, and worth writing a book about, and it’s another one of the possibilities Lawrence discusses.

Why is there a universe at all?

So modern physics has given us these two ideas, both of which are interesting, and both of which resonate with our informal notion of “coming into existence out of nothing” — one of which is time evolution from empty space (or not-even-space) into a universe bursting with stuff, and the other of which posits time as an approximate notion that comes to an end at some boundary in an abstract space of possibilities.

What, then, do we have to complain about? Well, a bit of contemplation should reveal that this kind of reasoning might, if we grant you a certain definition of “nothing,” explain how the universe could arise from nothing. But it doesn’t, and doesn’t even really try to, explain why there is something rather than nothing — why this particular evolution of the wave function, or why even the apparatus of “wave functions” and “Hamiltonians” is the right way to think about the universe at all. And maybe you don’t care about those questions, and nobody would question your right not to care; but if the subtitle of your book is “Why There Is Something Rather Than Nothing,” you pretty much forfeit the right to claim you don’t care.

Do advances in modern physics and cosmology help us address these underlying questions, of why there is something called the universe at all, and why there are things called “the laws of physics,” and why those laws seem to take the form of quantum mechanics, and why some particular wave function and Hamiltonian? In a word: no. I don’t see how they could.

Sometimes physicists pretend that they are addressing these questions, which is too bad, because they are just being lazy and not thinking carefully about the problem. You might hear, for example, claims to the effect that our laws of physics could turn out to be the only conceivable laws, or the simplest possible laws. But that seems manifestly false. Just within the framework of quantum mechanics, there are an infinite number of possible Hilbert spaces, and an infinite number of possibile Hamiltonians, each of which defines a perfectly legitimate set of physical laws. And only one of them can be right, so it’s absurd to claim that our laws might be the only possible ones.

Invocations of “simplicity” are likewise of no help here. The universe could be just a single point, not evolving in time. Or it could be a single oscillator, rocking back and forth in perpetuity. Those would be very simple. There might turn out to be some definition of “simplicity” under which our laws are the simplest, but there will always be others in which they are not. And in any case, we would then have the question of why the laws are supposed to be simple? Likewise, appeals of the form “maybe all possible laws are real somewhere” fail to address the question. Why are all possible laws real?

And sometimes, on the other hand, modern cosmologists talk about different laws of physics in the context of a multiverse, and suggest that we see one set of laws rather than some other set for fundamentally anthropic reasons. But again, that’s just being sloppy. We’re talking here about the low-energy manifestation of the underlying laws, but those underlying laws are exactly the same everywhere throughout the multiverse. We are still left with the question of there are those deep-down laws that create a multiverse in the first place.

The end of explanations

All of these are interesting questions to ask, and none of them is addressed by modern physics or cosmology. Or at least, they are interesting questions to “raise,” but my own view is that the best answer is to promptly un-ask them. (Note that by now we’ve reached a purely philosophical issue, not a scientific one.)

“Why” questions don’t exist in a vacuum; they only make sense within some explanatory context. If we ask “why did the chicken cross the road?”, we understand that there are things called roads with certain properties, and things called chickens with various goals and motivations, and things that might be on the other side of the road, or other beneficial aspects of crossing it. It’s only within that context that a sensible answer to a “why” question can be offered. But the universe, and the laws of physics, aren’t embedded in some bigger context. They are the biggest context that there is, as far as we know. It’s okay to admit that a chain of explanations might end somewhere, and that somewhere might be with the universe and the laws it obeys, and the only further explanation might be “that’s just the way it is.”

Or not, of course. We should be good empiricists and be open to the possibility that what we think of as the universe really does exist within some larger context. But then we could presumably re-define that as the universe, and be stuck with the same questions. As long as you admit that there is more than one conceivable way for the universe to be (and I don’t see how one could not), there will always be some end of the line for explanations. I could be wrong about that, but an insistence that “the universe must explain itself” or some such thing seems like a completely unsupportable a priori assumption. (Not that anyone in this particular brouhaha seems to be taking such a stance.)

Sounds and furies

That’s all I have to say about the (fun, interesting) substantive questions, but I am not strong enough to resist a couple of remarks on the (tedious but strangely irresistible) procedural questions.

First, I think that Lawrence’s book makes a lot more sense when viewed as part of the ongoing atheism vs. theism popular debate, rather than as a careful philosophical investigation into a longstanding problem. Note that the afterword was written by Richard Dawkins, and Lawrence had originally asked Christopher Hitchens, before he became too ill — both of whom, while very smart people, are neither cosmologists nor philosophers. If your real goal is to refute claims that a Creator is a necessary (or even useful) part of a complete cosmological scheme, then the above points about “creation from nothing” are really quite on point. And that point is that the physical universe can perfectly well be self-contained; it doesn’t need anything or anyone from outside to get it started, even if it had a “beginning.” That doesn’t come close to addressing Leibniz’s classic question, but there’s little doubt that it’s a remarkable feature of modern physics with interesting implications for fundamental cosmology.

Second, after David’s review came out, Lawrence took the regrettable tack of lashing out at “moronic philosophers” and the discipline as a whole, rather than taking the high road and sticking to a substantive discussion of the issues. In the Atlantic interview especially, he takes numerous potshots that are just kind of silly. Like most scientists, Lawrence doesn’t get a lot out of the philosophy of science. That’s okay; the point of philosophy is not to be “useful” to science, any more than the point of mycology is to be “useful” to fungi. Philosophers of science aren’t trying to do science, they are trying to understand how science works, and how it should work, and to tease out the logic and standards underlying scientific argumentation, and to situate scientific knowledge within a broader epistemological context, and a bunch of other things that can be perfectly interesting without pretending to be science itself. And if you’re not interested, that’s fine. But trying to undermine the legitimacy of the field through a series of wisecracks is kind of lame, and ultimately anti-intellectual — it represents exactly the kind of unwillingness to engage respectfully with careful scholarship in another discipline that we so rightly deplore when people feel that way about science. It’s a shame when smart people who agree about most important things can’t disagree about some other things without throwing around insults. We should strive to be better than that.

Sorry for the tiny writing, there are a lot of particles! Click to embiggen and get a legible version.

The first bit of news, which has been the subject of the most internet buzz, is a new paper by Chilean astronomers C. Moni Bidin, G. Carraro, R. A. Mendez, and R. Smith, which claims that there’s no evidence for dark matter in the dynamics of stars near the Sun. If this were true, it would imply something funny going on with the distribution of nearby dark matter, which could have significant implications for direct searches here on Earth (see below). It wouldn’t really be much of a threat to the idea of dark matter itself, since there’s plenty of evidence for dark matter elsewhere. But it might mean that the distribution in the Milky Way was very different from the kinds of models we like to use, for example by being much lumpier.

We just heard a great physics colloquium here at Caltech by Katie Freese, who talked about this result very briefly. Her opinion matched those of the skeptics in Ron Cowen’s article linked above: this paper makes a lot of assumptions, some of the a bit dubious, and we would need to see something much more solid before we become convinced. The biggest issue is that they don’t actually measure the DM distribution near the Sun; they try to measure it in a region between 1500 and 4000 parsecs below the galactic plane (which is actually pretty far away), and then fit to a model and extrapolate to what we should have nearby. This kind of procedure relies on our understanding of the vertical structure of the galactic disk, which isn’t all that great. So it’s definitely an intriguing result, one that should be taken seriously and followed up by other surveys, but nothing to lose sleep over just yet.

The second bit of news is another puzzling absence: a lack of neutrinos that were predicted to be produced by gamma-ray bursts. These bursts are among the most energetic events in the post-Big-Bang universe, and for a long time were a major mystery to astrophysicists. More recently, a consensus had grown up that GRB’s (as they are called) are associated with intense beams of particles created by newly-born supernovae. That’s a model that seems to fit most of the data, anyway, and it also makes a pretty good prediction for the production of associated neutrinos. But a new paper by the marvelous IceCube experiment has thrown a spanner into the works: they should have been able to see those neutrinos, and they don’t.

IceCube consists of a series of thousands of detectors arrayed within a cubic kilometer of Antarctic ice, and looks for flashes of light associated with high-energy particles passing through. Recently they have been keeping an eye out for signs of neutrinos that should be associated with GRB’s that have been detected by the Swift and Fermi satellites — but no luck. It’s a puzzle that will send GRB theorists back to the drawing board. One of the funny aspects of the story is that particles from GRB’s are a leading candidate to serve as the origin of high-energy cosmic rays, but that seems to be out the window now. It’s still possible that the cosmic rays come from active galactic nuclei, but there’s another group of theorists who have something new to chew on.

The final bit of news is even dicier, and hasn’t received any internet buzz at all yet — I only heard about it through Katie Freese’s talk. Maybe because there was no press release and the shocking claim is hidden within the guts of a technical paper with a boring title. The paper is by our friend Juan Collar and Nicole Fields. Recall that the DAMA dark matter experiment looks for an annual modulation due to the fact that the Earth moves through the dark-matter “wind” at different velocities during different times of the year. And they see a signal — very strongly — but many people have questioned whether what they are seeing is really due to dark matter. Juan has been leading another experiment, CoGeNT, which has been trying to check DAMA’s results — and has found a very tentative signal that seems to agree with them (which wasn’t what most people were expecting).

One of the reasons for the skepticism is that there are other experiments, which aren’t tuned specifically to look for annual modulations, but nevertheless should be sensitive to dark matter at the level implied by DAMA and CoGeNT — and they see nothing. More recently, some of these experiments have started looking for annual modulations — and they see nothing. Here for example is a recent paper by the CDMS experiment that says exactly that.

But the new paper by Collar and Fields claims that CDMS have analyzed their own data incorrectly. They argue that (1) CDMS isn’t really sensitive to the kind of annual modulation purportedly seen by CoGeNT, and (2) if you look carefully there is actually a statistically significant (more than 5 sigma!) bump at low energies, consistent with the kind of low-mass dark matter particle you would need to explain the annual modulations.

My impression is that the CDMS folks are unmoved by this argument. It’s certainly always very hard to analyze the data from somebody else’s experiment. This kind of controversy comes down to very particular aspects of data collection, analysis, and sources of systematic error. It’s way over my head, so I have no professional opinion about who is right. But at the very least it’s a reminder (as if we needed one) that the dark-matter-detection game is heating up, and big news might be creeping up on us. The universe loves puzzles.

In tomorrow’s New York Times, I’ve got a long story about a growing sense among scientists that science itself is getting dysfunctional. For them, the clearest sign of this dysfunction is the growing rate of retractions of scientific papers, either due to errors or due to misconduct. But retractions represent just the most obvious symptom of deep institutional problems with how science is done these days–how projects get funded, how scientists find jobs, and how they keep labs up and running.

However… essentially all the examples are from biologically-oriented fields. I’ll confess that Carl asked me if there is a similar feeling among physicists, and after some thought I decide that there really isn’t. There are certainly fumbles (faster-than-light neutrinos, anyone?) and scandals (Jan Hendrik Schön being the most obvious), but I don’t have any feeling that the problem is growing in a noticeable way. Biology and physics are fundamentally different, especially because of the tremendous pressure within medical sciences when it comes to any results that might turn out to be medically useful. Cosmologists certainly don’t have to worry about that.

But maybe this is a distorted view from within my personal bubble? Happy to hear informed opinion to the contrary. The relevant kind of informed opinion would actually involve a comparison of the situation today with the situation at some previous time, not just a litany of things you think are dysfunctional about the present day.

But that’s not what I want to talk about right now. Rather, our conversations nudged me into investigating some work that I have long known about but never really looked into: David Deutsch’s argument that probability in quantum mechanics doesn’t arise as part of a separate ad hoc assumption, but can be justified using decision theory. (Which led me to this weekend’s provocative quote.) Deutsch’s work (and subsequent refinements by another former guest blogger, David Wallace) is known to everyone who thinks about the foundations of quantum mechanics, but for some reason I had never sat down and read his paper. Now I have, and I think the basic idea is simple enough to put in a blog post — at least, a blog post aimed at people who are already familiar with the basics of quantum mechanics. (I don’t have the energy in me for a true popularization at the moment.) I’m going to try to get to the essence of the argument rather than being completely careful, so please see the original paper for the details.

The origin of probability in QM is obviously a crucial issue, but becomes even more pressing for those of us who are swayed by the Everett or Many-Worlds Interpretation. The MWI holds that we have a Hilbert space, and a wave function, and a rule (Schrödinger’s equation) for how the wave function evolves with time, and that’s it. No extra assumptions about “measurements” are allowed. Your measuring device is a quantum object that is described by the wave function, as are you, and all you ever do is obey the Schrödinger equation. If MWI is to have some chance of being right, we must be able to derive the Born Rule — the statement that the probability of obtaining a certain result from a quantum measurement is the square of the amplitude — from the underlying dynamics, not just postulate it.

Deutsch doesn’t actually spend time talking about decoherence or specific interpretations of QM. He takes for granted that when we have some observable X with some eigenstates |x_i>, and we have a system described by a state

then a measurement of X is going to return either x₁ or x₂. But we don’t know which, and at this stage of the game we certainly don’t know that the probability of x₁ is |a|² or the probability of x₂ is |b|²; that’s what we’d like to prove.

In fact let’s just focus on a simple special case, where

If we can prove that in this case, the probability of either outcome is 50%, we’ve done the hard part of the work — showing how probabilistic conclusions can arise at all from non-probabilistic assumptions. Then there’s a bit of mathematical lifting one must do to generalize to other possible amplitudes, but that part is conceptually straightforward. Deutsch refers to this crucial step as deriving “tends to from does,” in a mischievous parallel with attempts to derive ought from is. (Except I think in this case one has a chance of succeeding.)

The technique used will be decision theory, which is a way of formalizing how we make rational choices. In decision theory we think of everything we do as a “game,” and playing a game results in a “value” or “payoff” or “utility” — what we expect to gain by playing the game. If we have the choice between two different (mutually exclusive) actions, we always choose the one with higher value; if the values are equal, we are indifferent. We are also indifferent if we are given the choice between playing two games with values V₁ and V₂ or a single game with value V₃ = V₁ + V₂; that is, games can be broken into sub-games, and the values just add. Note that these properties make “value” something more subtle than “money.” To a non-wealthy person, the value of two million dollars is not equal to twice the value of one million dollars. The first million is more valuable, because the second million has a smaller marginal value than the first — the lifestyle change that it brings about is much less. But in the world of abstract “value points” this is taken into consideration, and our value is strictly linear; the value of an individual dollar will therefore depend on how many dollars we already have.

There are various axioms assumed by decision theory, but for the purposes of this blog post I’ll treat them as largely intuitive. Let’s imagine that the game we’re playing takes the form of a quantum measurement, and we have a quantum operator X whose eigenvalues are equal to the value we obtain by measuring them. That is, the value of an eigenstate |x> of X is given by

The tricky thing we would like to prove amounts to the statement that the value of a superposition is given by the Born Rule probabilities. That is, for our one simple case of interest, we want to show that

After that it would just be a matter of grinding. If we can prove this result, maximizing our value in the game of quantum mechanics is precisely the same as maximizing our expected value in a probabilistic world governed by the Born Rule.

To get there we need two simple propositions that can be justified within the framework of decision theory. The first is:

Given a game with a certain set of possible payoffs, the value of playing a game with precisely minus that set of payoffs is minus the value of the original game.

Note that payoffs need not be positive! This principle explains what it’s like to play a two-person zero-sum game. Whatever one person wins, the other loses. In that case, the value of the game to the two participants are equal in magnitude and opposite in sign. In our quantum-mechanics language, we have:

Keep that in mind. Here’s the other principle we need:

If we take a game and increase every possible payoff by a fixed amount k, the value is equivalent to playing the original game, then receiving value k.

If I want to change the value of a playing a game by k, it doesn’t matter whether I simply add k to each possible outcome, or just let you play the game and then give you k. I don’t think we can argue with that. In our quantum notation we would have

Okay, if we buy that, from now on it’s simple algebra. Let’s consider the specific choice

and plug this into (3). We get

You can probably see where this is going (if you’ve managed to make it this far). Use our other rule (2) to make this

which simplifies straightaway to

which is our sought-after result (1).

Now, notice this result by itself doesn’t contain the word “probability.” It’s simply a fairly formal manipulation, taking advantage of the additivity of values in decision theory and the linearity of quantum mechanics. But Deutsch argues — and on this I think he’s correct — that this result implies we should act as if the Born Rule is true if we are rational decision-makers. We’ve shown that the value of a game described by an equal quantum superposition of states |x₁> and |x₂> is equal to the value of a game where we have a 50% chance of gaining value x₁ and a 50% chance of gaining x₂. (In other words, if we acted as if the Born Rule were not true, someone else could make money off us by challenging us to such games, and that would be bad.) As someone who is sympathetic to pragmatism, I think that “we should always act as if A is true” is the same as “A is true.” So the Born Rule emerges from the MWI plus some seemingly-innocent axioms of decision theory.

While I certainly haven’t followed the considerable literature that has grown up around this proposal over the years, I’ll confess that it smells basically right to me. If anyone knows of any strong objections to the idea, I’d love to hear them. But reading about it has added a teensy bit to my confidence that the MWI is on the right track.

The idea of cosmological inflation is our best developed idea of how the physics of the early universe might lead to the observed universe today. This idea has been widely discussed in popular books and beyond, and in this context, many students have heard the loose description that inflation occurs when the universe is in an almost de Sitter state, and undergoes exponentially rapid expansion. There is nothing wrong with this explanation, but one consequence of accepting it before having a thorough grounding in GR is that it seems to imply that de Sitter space is a solution to GR that undergoes a rapid change over time. This leads to a few confused looks when I get to maximally symmetric spaces in my course.

You see, maximal symmetry means that you should be able to look at the space at different places and at different times and the metric should be just the same. So how are we to square that with the idea of an exponentially growing universe? Well, it all comes down to coordinate choices and the crucial existence of other matter in the universe.

Pure de Sitter space – the solution to the Einstein equations with a positive cosmological constant and no other matter sources – is, indeed, a maximally symmetric space. There exist a number of particularly useful coordinate choices for this space. In some cases, these consist of picking a useful time choice, and thus defining a family of spacelike surfaces (the spatial part of the spacetime at a constant value of this time choice). This is referred to as a slicing of the space, and it is, actually, possible to slice the space in three different ways that correspond to cosmologically expanding spaces with flat, positively-curved and negatively curved spatial parts, respectively. These are the ways of describing de Sitter space that are useful when considering inflation. However, there also exists a choice of coordinates in which the metric does not depend on time at all, and the mere existence of such a choice is enough to tell us that there is no fundamental sense in which this is an expanding cosmological spacetime. In fact, from what I just wrote, you might have a related question: even in the cosmological coordinates, what decides if the universe is flat, positively, or negatively curved?

In the case of pure de Sitter space there is no answer to these questions. All the coordinate choices are equally allowed of course, and so we might as well look at the static coordinates, and there is no cosmology here. However, importantly, in cosmology we are never interested in pure de Sitter space. We know that there is other matter in the universe. This may be either in the form of particles like us, or, in the case of inflation, the background field that causes inflation in the first place – the inflaton. These types of matter mean that the behavior of the metric is at best almost de Sitter – the difference from pure de Sitter being that, crucially, there are only certain coordinate systems in which the regular matter is homogeneous and isotropic, whereas for a cosmological constant this is true in all coordinate systems. Thus an almost de Sitter space has less symmetry than pure de Sitter. One is free to transform coordinates as much as one likes, but there will no longer be any choices in which the metric is static!

Of course, we find it most convenient to discuss cosmology in the (Friedmann, Robertson-Walker) coordinates that exploit the natural homogeneity and isotropy of the relevant matter sources. This picks out a slicing of the spacetime, and in this slicing, when the universe is almost de Sitter, the universe does expand almost exponentially rapidly – inflation! This also decides among the flat, positively and negatively curved options for the spatial part of the metric.

So it matters that inflation is “quasi-de Sitter”. It is this that gives sense to statements about inflation beginning, ending, and even operating in the way we usually describe. de Sitter space is beautiful symmetric and rich, but out real universe is somewhat messier, even at its earliest times.

“Despite the unrivaled empirical success of quantum theory, the very suggestion that it may be literally true as a description of nature is still greeted with cynicism, incomprehension, and even anger.”

Fang’s research area was quantum cosmology, but he was most well-known for his political activism, fighting against repression in China. Originally a member of the Communist Party, he was expelled for protesting some of the government’s policies. The NYT obituary relates an amusing/horrifying story, according to which Fang attracted the government’s censure by co-authoring a paper entitled “A Solution of the Cosmological Equations in Scalar-Tensor Theory, with Mass and Blackbody Radiation.” Seems pretty innocuous from where we are sitting, but in Communist China the Big Bang model was considered to be a challenge to Engels’s idea that that the universe was infinite, and therefore was deemed heresy. Googling around brought me to this 1988 article in Contemporary Chinese Thought, which shows what Fang was up against. The abstract quotes Lenin, and says in all seriousness “with every new advance in science the idealists distort and take advantage of the latest results of physics to “prove” with varying sleights of hand that the universe is finite, serving the reactionary rule of the moribund exploiting classes.”

In the late 1980′s Fang helped organize resistance to China’s authoritarian regime, in the lead-up to the Tiananmen Square protests. He was fired from his job as a professor, and sought refuge in the American embassy. He was finally permitted to leave the country and emigrate to America in 1990. He finally settled down at the University of Arizona, but continued his work campaigning for human rights.

When we apply GR to cosmology, we make use of the simplifying assumptions, backed up by observations, that there exists a definition of time such that at a fixed value of time, the universe is spatially homogeneous (looks the same wherever the observer is) and isotropic (looks the same in all directions around a point). We then specialize to the most general metric compatible with these assumptions, and write down the resulting Einstein equations with appropriate sources (regular matter, dark matter, radiation, a cosmological constant, etc.). The solutions to these equations are the famous Friedmann, Robertson-Walker spacetimes, describing the expansion (or contraction) of the universe.

It is important to take a moment to emphasize what we have done here. GR is indeed a beautiful geometric theory describing curved spacetime. But practically, we are solving differential equations, subject to (in this case) the condition that the universe look the way it does today. Differential equations describe the local behavior of a system and so, in GR, they describe the local geometry in the neighborhood of a spacetime point.

Because homogeneity and isotropy are quite restrictive assumptions, there are only three possible answers for the local geometry of space at any fixed point in time – it can be spatially positively curved (locally like a 3-dimensional sphere), flat (locally like a 3-dimensional version of a flat plane) or negatively spatially curved (locally like a 3-dimensional hyperboloid). A given cosmological solution to GR tells you one of these answers around a spacetime point, and homogeneity then tells you that this is the same answer around every spacetime point. This is what we mean when we say that GR tells us about geometry – the shape of the universe – as depicted in the NASA graphic below.

This raises a very different question that is often confused with the one above. If our solution tells us that the universe is locally a 3-sphere (or flat space, or a hyperboloid) around every point, then does that mean it is a 3-sphere, or an infinite flat 3-dimensional space, or an infinite hyperboloid. This is really a question of topology – how is it connected up – which also answers the question of whether the universe is finite or infinite. To illustrate the point, suppose we have solved the cosmological equations of GR, and discovered that at every spacetime point, the universe is locally a flat 3-dimensional space. This is, by the way, what observations actually indicate our universe is like. Then, just off the top of your head, you can think of many different spaces with precisely this same property. One example is, of course, that the universe is indeed a flat, infinite 3-dimensional space. Another is that the universe is a 3-torus, in which if you were to fix time and trace out a line away from any point along the x, y or z-axis, you traverse a circle and come right back to where you started. This is a finite volume space, that is connected up in a very specific way, but which is everywhere flat, just like the infinite example. In two dimensions, one might visualize it as

Of course, I could have only made one or two directions into circles (leaving it still infinite in some directions), or made the space into a finite one with more than one hole, or any number of other possibilities.

This is the beauty of topology, but it is not something that solving the equations of GR tells us. Rather it is an extra input into our solutions. It is, however, something we can test, most precisely through measurements of the Cosmic Microwave Background radiation, as I may discuss in a later post.

Completely independent of questions of topology, the geometry of a given cosmological solution raises another issue that is often mixed up with those of geometry and topology. Suppose that the universe contains only conventional matter sources (regular matter, dark matter and radiation, say), and suppose you know (you might question whether this is truly possible) that this is all it will ever contain. Then the equations easily predict that, in the case of positive spatial curvature, an expanding universe will ultimately reach a maximum size and recollapse in a big crunch, whereas flat or negatively curved universes will expand forever. These are predictions of the destiny of the universe, and often lead to the following connection

However, as I made clear, there are some assumptions that go into the connection between geometry and destiny, and although these may have seemed reasonable ones at one time, we know today that the accelerated expansion of the universe seems to point to the existence of some kind of dark energy (a cosmological constant, for example), that behaves in a way quite different from conventional mass-energy sources. In fact, we know that for sources like this, once acceleration begins, it is easily possible for a positively curved universe, for example, to expand forever. Indeed, in the case of a cosmological constant, this is precisely what happens.

So the universe may be positively or negatively curved, or flat, and our solutions to GR tell us this. They may be finite or infinite, and connected up in interesting ways, but GR does not tell us why this is the case. And the universe may expand forever or recollapse, but this depends on detailed properties of the cosmic energy budget, and not just on geometry. Cosmological spacetimes are some of the simplest solutions to GR that we know, and even they admit all kinds of potential complexities, beyond the most obvious possibilities. Wonderful, isn’t it?

There is something rightly labeled “physics envy,” and it is a temptation justly to be resisted: the preference for reducing everything to simple and clean quantitative models whether or not they provide accurate representations of the phenomena under study. The great thing about physics is that we study systems that are so simple that it’s quite useful to invoke highly idealized models, from which fairly accurate quantitative predictions can be extracted. The messy real world of the social sciences doesn’t always give us that luxury. The envy becomes pernicious when we attack a social-science problem by picking a few simple assumptions, and then acting like those assumptions are reality just because the model is so pretty.

However, that’s not what Clarke and Primo are warning against. Their aim is at something altogether different: the idea that theories should be tested empirically! They write,

Many social scientists contend that science has a method, and if you want to be scientific, you should adopt it. The method requires you to devise a theoretical model, deduce a testable hypothesis from the model and then test the hypothesis against the world…

But we believe that this way of thinking is badly mistaken and detrimental to social research. For the sake of everyone who stands to gain from a better knowledge of politics, economics and society, the social sciences need to overcome their inferiority complex, reject hypothetico-deductivism and embrace the fact that they are mature disciplines with no need to emulate other sciences…

Unfortunately, the belief that every theory must have its empirical support (and vice versa) now constrains the kinds of social science projects that are undertaken, alters the trajectory of academic careers and drives graduate training. Rather than attempt to imitate the hard sciences, social scientists would be better off doing what they do best: thinking deeply about what prompts human beings to behave the way they do.

Sorry, but “thinking deeply” doesn’t cut it. People are not especially logical creatures, and we’re just not smart enough to gain true knowledge about the world by the power of reason alone. That’s why empiricism was invented in the first place, and why it’s been so spectacularly successful over the last few centuries.

Clarke and Primo seem to confuse “the need for empirical testing” with “the need for every model proposed to be backed up by data before it gets published.” If they had stuck to rejecting the latter narrow idea, they would have had a decent case. Certainly we physicists don’t require that every model be supported by data before it is published — otherwise my CV (and those of most of my friends) would be a lot shorter! But we all agree that the ultimate test of an idea is a confrontation with data, even if a theory might be too immature for that confrontation to take place just yet.

This year’s run is all about luminosity, i.e. getting as many collisions in the can as they can. Last year they reached about 5 inverse femtobarns, while this year they’re shooting for 15 inverse femtobarns. Yes, those are the goofiest units in all of physics. Think of it this way: imagine the protons entering a detector are shooting at a tiny target with some fixed size, measured in units of area. Then we can measure the luminosity by counting the number of protons passing through that area in a fixed moment of time: i.e., the number of protons per square centimeter per second. That’s at any one moment; if we integrate up over the course of a year, the “per second” disappears and leaves us with the total number of protons that have passed through the target area, i.e. a certain number of protons per square centimeter. But that number would be enormously huge, so rather than using square centimeters, particle physicists like to use “barns,” defined as 10^-24 cm². (Broad side of a barn, get it?) But even measuring the luminosity in inverse barns would be really big, so they go for inverse femtobarns (1 fb = 10^-39 cm²). Long story short: 10 inverse femtobarns is equivalent to 10⁴⁰ protons passing through a 1 cm target area. (That’s much larger than the number of collisions — to get the number of collisions for any particular process, you need to multiply by the cross-section for that process, which is often quite tiny. That’s why particle physics is hard! Still, there will be a buttload of collisions.)

Anyway, I’m pretty sure the LHC is back to colliding protons after this year’s winter shutdown, and they’re smashing together at 8 TeV. But to be honest my only hard evidence is from Twitter, where the ATLAS collaboration has tweeted this image.

Meanwhile, results are still coming out from last year’s run. Sadly, they’re doing a great job at constraining possible new physics, but no convincing discoveries as yet. Here’s a recent result from LHCb, the experiment that looks at decays of mesons containing b quarks. This plot is from David Straub, from a talk at Moriond, based on this paper.

Horizontal axis is the fraction of time (the branching ratio) bottom/strange mesons decay into two muons, while the vertical axis is the fraction of time bottom/down mesons do the same thing. These numbers have specific predictions within the good old Standard Model, but it’s very easy for new physics such as supersymmetry to enhance the numbers quite a bit. LHCb has put an upper limit on both quantities, which rules out all the gray area of the plot, leaving only the colorful part at the bottom left. The colors correspond to possible predictions in different versions of supersymmetry. As you see, it would have been very easy to have detected a substantial deviation from the Standard Model by now, but no such luck. This doesn’t mean some other version of supersymmetry isn’t right, just that we’ll have to try harder. No question that a proper update of our likelihood functions will have to decrease the chance that we expect fo find SUSY at the LHC compared to what we would have thought a few years ago, however. This is why the march to higher energies will be so important.

If you want to ask some detailed questions about the accelerator and the experiment, the CMS and ATLAS collaborations are having a Google+ hangout this Wednesday that you are welcome to join. It starts at 7 am Los Angeles time, so I’m unlikely to make it, but let us know if anyone here participates.

Here on Earth, the Global Positioning System (GPS) gives us a highly accurate way of determining position, and many of us now use hand-held devices every day to help with directions. These work because GPS satellites provide a set of clocks, the relative timings of the signals from which can be translated into positions. This is, by the way, another place where both special and general relativity are crucial to how the system works. Out in deep space, of course, our clocks are unfortunately useless for this purpose, and the best we currently can do is by comparing the timing of signals as they are measured back on Earth by different detectors. But the accuracy of this method is limited, since the Earth is a finite size, and our terrestrial detectors can therefore only be separated by a relatively small amount. The further away a spacecraft is, the worse this method is.

What Werner Becker of the Max-Planck Institute for Extraterrestrial Physics in Garching has realized (and announced yesterday at the UK-Germany National Astronomy Meeting in Manchester), is that the universe comes equipped with its own set of exquisite clocks – pulsars – the timing of which can, in principle, be used to guide spacecraft in a similar way to how GPS is used here on Earth. Of course, it isn’t quite as simple as all that.

A significant obstacle to making this work today is that detecting signals from the pulsars requires X-ray detectors that are compact enough to be easily carried on spacecraft. However, it turns out the relevant technology is also needed by the next generation of X-ray telescopes, and should be ready in twenty years or so. Perhaps one day our spacecraft will map their routes through the cosmos thanks to yet another spinoff from basic research.

For a terrific example of this, see the Carbon Map. The idea of Cartograms has been around for a while, of course, and in particular, I recall them being used extensively here in the US to represent the voting tendencies of regions of the country in the lead up to the 2008 presidential election. For example, here’s one in which the sizes of counties have been rescaled according to their population.

What is nice about the Carbon Map is that it is an animated cartogram, in which one can choose different datasets and have the world map dynamically rescale to represent the new data. The screen shot below is of the scaling of areas by population, but as you can see there are other possibilities.

Obviously, this example (the first I’ve seen) is meant to get across a particular point, but that isn’t what I’m discussing here. What impressed me is the clarity and power of representing the data in this way. I’m sure there are many other ways in which this technique would be useful, some of them in physics and astronomy.

I recently got to hear a talk by Zimbardo, in which among other things he discussed the Stanford Marshmallow Experiment — a rather more adorable experience than the prison experiment, from what I understand. The Marshmallow experiment, originally conducted by Walter Mischel in 1972, was aimed at understanding how we think about different times — the future vs. the present. Children were asked to do some easy tasks, and then were rewarded by being given a marshmallow. But! They were told that the experimenter had to step outside for a few minutes, and if they could just sit tight and not eat their marshmallow until he came back, they could have that and also an additional marshmallow.

It’s a matter of future vs. present rewards. It’s natural (and totally rational) to discount rewards that are promised in the future — after all, the future is hard to predict, and anything can happen. If I offered you a choice between $4 today and $5 ten years from now, you’d be sensible to take the lower amount today — depending on how much you trusted me, of course. But if there is a good reason to trust, and the future isn’t that far off, it makes sense to delay gratification a bit. So what happens when some four-year-olds are put to the test?

As you see, the results were mixed. But most importantly, Mischel followed up years later, looking into how the kids who participated in the study ultimately turned out. There was a remarkable amount of correlation with this simple test and success later in life — kids who were able to hold off at age 4 for the second marshmallow turned out years later to have higher SAT scores and generally seem more competent. The hypothetical explanation is that our personalities are strongly influenced by our attitude toward time — whether we are focused primarily on the past, the present, or the future.

Zimbardo has learned a lot about how people treat different times differently, depending on personality and culture and numerous other factors. Here’s a great animated version of a talk he gave on the subject.

They also have a blog, because blogs are awesome. It has a humble title: What There Is and Why There Is Anything. They have a new post up, by Eric Winsberg, that brings up the issue of whether the multiverse can help explain the arrow of time. The post is basically a pointer to this paper by Eric, which is a close analysis of the kind of scenario I’ve been pursuing since my 2004 paper with Jennifer Chen. If this kind of thing is your bag, consider going over there and commenting on Eric’s paper.

I am working on a real science paper about some of these issues myself, but going has been admittedly slow. Let me just lay out a couple of the major issues here. One is, naturally, the question of whether the Farhi-Guth process of baby-universe creation really happens. In 2004 I was pretty confident that it did, but now I’m less sure. Aguirre and Johnson have looked closely at these kinds of tunneling events and come back pessimistic; others have looked at similar processes from the perspective of the AdS/CFT correspondence with similarly unpromising results. I don’t think the issue is settled, however; for the moment I’m willing to take the possibility of spontaneous baby-universe creation as an allowed hypothesis, while continuing to search for more well-grounded alternatives.

The other issue, which I think it should be more possible to make progress on, is a problem of counting — comparing the likelihood of different occurrences. In fact there are two sub-problems here. One is what’s now called the measure problem in cosmology. Assuming that many things (like the appearance of people exactly like you) happen an infinite number of times, how can we compare appearances to calculate probabilities? In this context the question is how we can compare the number of observers who appear in a nice warm post-Big-Bang environment to the number who pop randomly out of the nothingness as thermal fluctuations. In a scenario like ours, you need thermal fluctuations to create new universes, so there is always some possibility of making observers as well. I think that our picture is much better than most versions of eternal inflation from this perspective, as it seems easier to make a baby universe than to make an observer — the magic of inflation is that a bubble ready to inflate can be almost arbitrarily tiny, while an observer needs space for its thinking apparatus. But it’s harder to actually calculate things in a well-defined measure once your spacetime becomes disconnected by the appearance of new universes, so it’s certainly a legitimate question.

The other sub-problem, more subtle, might be called the “genericity problem.” The most important point of my paper with Jennie was to argue that a dynamical origin of the arrow of time is possible if and only if the space of states is infinitely big — the universe can keep evolving forever without reaching an equilibrium or entering a recurrent cycle. Baby universes were just the means to that end. But if there are an infinite number of possible states, how do you pick a “generic” one?

We tried to argue that “almost any” initial state would robustly evolve to a condition where baby universes were produced and became the dominant channel for creating observers. Our strategy for doing so was to say that a low-energy de Sitter vacuum was the highest-entropy state you could be in where space was still connected, and that most conditions evolve toward such a state. (At least if we discount Minkowski space with exactly vanishing vacuum energy, maybe for anthropic reasons.) Of course we then immediately evolve to a state with more than one connected component via the nucleation of baby universes. So then you could ask why we didn’t start there. Our idea is that there is no maximum number of components (separate universes), so any finite number can still grow. So why don’t we have an infinite number?

It’s a legitimate question, but not a show-stopper. In toy models it’s certainly easy to construct examples where there is no equilibrium state (as we mentioned in the paper). It could be difficult in those cases to fix what counts as a generic initial condition, but it might not be impossible. That’s something worth further investigation.

Nobody ever said explaining what there is and why there is anything would be easy.