Mondays don't seem to agree with me!
The Jeans Equations are remarkable: They relate moments and integrals of distribution functions to the underlying gravitational potential (or really force law), for phase-mixed populations. They are true for any distribution function! But they are equations, and they are not models. As my loyal reader knows, for me a model is a likelihood function!
When people do what is called Jeans modeling, they turn the equations into some procedure for estimating the gravitational potential (or force law or mass density). And although the Equations are independent of distribution function, the performance of this heuristic procedure—that goes from velocity moments to gravitational model parameters or densities—has statistical properties that do depend strongly on the distribution function. That is, you can't make a probabilistic statement (like a measurement and an uncertainty) of anything (like a density at the Milky Way disk midplane) without assuming things about the distribution function.
And because the Jeans Equations are independent of the distribution function, it is tempting to claim or believe that the results of the inference are also independent of the DF, which they aren't. There is no procedure you can write down that isn't. I spent time this weekend writing words about this, for reasons I can't currently understand.
As my loyal reader will recall, there are AstroFest events this September at Columbia (last week), Flatiron (today), and NYU (in two weeks). Todays meeting was long but excellent. I learned many things and was pleased to see all the new faces (so many new faces)! Here are a few personal highlights:
Shy Genel (Flatiron) showed that the details of star formation and feedback affecting a simulated galaxy disk or stars is very sensitive to the initial conditions or perturbations to the conditions made mid-simulation. That caused me to wonder if it is going to be very hard to infer things about galaxies from their observed properties! But Foreman-Mackey (Flatiron) pointed out that the sensitivity might be high but also highly structured, so not necessarily a problem. Good point; but it might take a lot of simulations to find out! Whatever the case, this is an excellent line of research.
Francisco Villaescusa-Navarro (Flatiron) described a project to see if, in the non-linear regime of large-scale structure evolution, the one, two, and higher-point functions, all combined, contain as much information as the one- and two-point functions in the linear regime. That is: What is the information content in the observables? This is, in some sense, the key question of cosmology at the present day! And relates to things I have been thinking about (but doing nothing about) for years.Suvodip Mukherjee (Flatiron) delivered a beautifully simple (and yet novel) idea: He is looking at all the cosmological observables with gravitational-wave sources that we have with galaxies and the CMB. That's clever! It includes the GW LSW effect, and GW lensing. He pointed out that there might be new cosmological constraints from cross-correlating GW event properties with CMB properties, like the CMB lensing map. Clever! And possibly big, in the mid-term to long-term future.
Doyeon Avery Kim (Columbia) is building spectral-spatial models of the all-sky fields or maps that act as CMB foregrounds. She is doing this by interpolating in spatial and spectral directions the (necessarily incomplete, different sky coverage, different angular resolutions) information from many large-angular-scale surveys. This is also very much related to my (vapor-ware) latent-variable model approach here, and is looking like it is delivering exciting results.
I spoke briefly with Chris Ick (NYU) about quasi-periodic oscillations in Solar flares, Megan Bedell (Flatiron) about telluric lines in stars observed with HARPS, and Adrian Price-Whelan (Princeton) about finding overdensities in the halo in Gaia DR2 data. With Ick we discussed whether to use the Bayesian evidence or a parameter estimate to compare nested models. My loyal reader knows which side I was on! With Bedell we discussed how we might verify that our telluric model is good, using line covariances. With Price-Whelan we discussed how to estimate local overdensity in both position and proper motion that would be maximally sensitive to streams and the like.
In Stars Meeting today, visitors Greg Green (KIPAC) and Richard Teague (Michigan) both talked about mapping dust. Teague is working at protoplanetary-disk scale (using velocity maps to find planets), while Green is working at Milky Way scale (making 3-d extinction maps). Teague is working with Foreman-Mackey (Flatiron) to get better velocity maps out of ALMA data and they are getting good success with one of my favorite tricks: Fit the peak with a quadratic. We have shown, in astrometric contexts, that this saturates information-theoretic bounds. They have gorgeous maps!
Green is trying to apply more useful spatial priors to the dust maps he has made of the Milky Way, which are (currently) independently sampled in pixels. He is resampling the pixels, using neighbor information to regularize or as a prior. His method is slow, but a lot faster than using a fully general Gaussian Process prior. And it appears to be a good approximation thereto. Certainly the maps look better!
I presented my project to figure out orbits from chemistry. There was good discussion. Spergel (Flatiron) opined that I would do no better than Jeans modeling if I did the Jeans modeling conditioned on chemistry. I am sure that's wrong! But I have to demonstrate it with a good information-theoretic argument.
My research tidbit for the day was a long conversation with Kate Storey-Fisher (NYU), in which we discussed our new estimator for the correlation function that can estimate vector (or tensor or higher order) quantities. That is, it doesn't have to estimate the correlation function in bins, it can estimate any aribtrary parameterized representation sensibly. This includes, say, a parameterization that is derivatives with respect to cosmological parameters. That would estimate the cosmological parameters directly from the positions of galaxies! It also includes, say, a fourier representation. That would estimate the correlation function and the power spectrum simultaneously! It also includes, say, dependencies of the correlation function on redshift or position, which would test cosmological growth of structure and cosmological homogeneity. Etc! I'm stoked.
In the course of the discussion we came up with a strong test of the estimator: An affine-invariance test: If we make an affine transformation of the model regressors, do we get the same results at the correlation-function level? That's a great test, and something we can do easily and now. If we don't pass, our estimator is just plain wrong!
For many years, Columbia Astronomy has had a tradition of having everyone in the Department give a short talk in a monster, full-day event called AstroFest. This year, we extended it to three Fridays, and covering all parts of NYC Astronomy. The first of these days was today, at Columbia, and it was great! I learned many things. Here is a smattering:
There is interesting laboratory astrophysics going on at Columbia, including experiments to measure deuterium molecular formation and dissociation rates (reported by Bowen) and experiments to measure aspects of Alfven wave propagation that might be relevant to Solar Coronal heating (reported by Bose).
Spinning black holes in a magnetic field charge up, and this might lead to pulsar-like activity in the late stages of a BH-NS inspiral (reported by Levin). After that I asked if any of the electromagnetic effects might affect the gravitational-wave signal itself, and the answer is unlikely, or only at a very low level.
You can't tell the shape of a transiting object from the shape of the transit (reported by Sandford). There are strict degeneracies! That led the audience to ask about regularization. You can break these degeneracies with regularization, but the answers will depend on the form of that regularization. I was wondering if star spots or limb darkening could break the degeneracies interestingly?
If you slowed down the rotation of the Earth, it would get colder, and more uniform in temperature between equator and pole (reported by Jansen)! That was a great use of Earth climate models to inform the study of exoplanets. And it maybe violates my simplest intuitions. New cure for global warming: Slow down Earth rotation!
And I was only there for the morning.
My only real research today was conversations with Bonaca (Harvard) about her stream–dark-matter substructure collision problem, explaining features in the GD-1 stream. We discussed ways to analyze how important the stream thickness is, without actually building a realistic model of the stream thickness. That is, our simplest simulations treat the stream as arbitrarily thin, but the best impact models may not be clearly in the thin-stream limit.
I'm back in the city and back for the Stars Meeting at Flatiron. It did not disappoint! We went around the room and did long, post-summer introductions. In that process, many good ideas came up! I learned that John Brewer (Yale) has a great result on the metallicity-dependence of the occurrence of various different kinds of planetary systems (currently under review). I learned that Spergel (Flatiron) is pursuing halo binaries at wide separations to look at halo dynamics. And I learned that Kathryn Johnston (Columbia) is thinking about how chaos might affect not just streams but also wide binaries or unbound comoving pairs. Maybe the comoving pairs will highlight regular (non-chaotic) orbits! That would be a super-cool constraint on Milky Way dynamics.
Late in the day I sat down with Bedell (Flatiron) who showed me the current state of our wobble project to model stars and the atmosphere in extreme-precision radial-velocity projects. It looks great! The data are very well described by the model, our statistical regularizations seem to be working, and there is every evidence that we are getting great telluric spectra. Now, are we doing well on the radial-velocity determination? Damn I hope so!
Eilers (MPIA) and I went on the APOGEE science telecon to describe our results. I talked about how we calibrated a (purely linear) spectro-photometric distance estimate for luminous red giants that manages to correct for dust and luminosity, and Eilers talked about how we used those tracers to measure the circular velocity of the Milky Way disk (that is, the potential). We use the Jeans equation in cylindrical symmetry. We got great feedback from the APOGEE team, which we will use to improve our discussion in our papers.
My flight home got seriously delayed and I had an extra day in Aspen. I spent it talking about (and working on) my project to infer dynamical invariants in the well-mixed parts of the Milky Way from chemical (element-abundance) invariants. I had various epiphanies and useful discussions:
Rix (MPIA) and I worked on how you explain the project to the world. One explanation is this: In addition to dynamical invariants, there are chemical abundances, which depend on the dynamical invariants (and not on the conjugate angles). Therefore inference of the dynamical invariants must be better or improved if you model the abundance invariants as well or in tandem. Another explanation is this: Imagine you do a dynamical inference (like Jeans modeling) and you (effectively) determine orbit structure. If you are slightly off, the element abundances you have measured will reveal the issues, and can be used to adjust or update or improve the orbit-structure inference, because stars don't change their abundances as they orbit!
Price-Whelan (Princeton) and I worked on how to compare the project with Jeans modeling, Schwarzschild modeling, or fully marginalized forward modeling of the kinematics (which has almost never been done). I have a scaling argument that my new method must be better than any of these: Each of these methods gets some amount of information out of the positions and velocities of the stars. My chemical-tangents method gets more information from every new element abundance you measure (even if each new element is fully covariant physically with the ones you have measured before; it is the measurements that are near-independent). So in some limit (and I think that limit arrives very early), it will have more information than any of these methods. But of course I need to demonstrate this quantitatively in the very near future.
Another point of comparison is related to the conditional or generative or causal structure of the model: I am modeling the abundance distribution conditioned on the phase-space positions. This means that I don't need to know the selection function of the survey, which Jeans modeling does (to some extent) and the more serious methods do (to great precision). On the other hand, because I am conditioning on the positions and not generating them, I can't (gracefully) account for measurement uncertainties in position. (Of course that's true for Jeans too.)
Anyway, the reason I am writing all this is because: The best practice for writing (the paper) is writing (this blog and emails and etc).
Today was a whirlwind of meetings and sessions. I started with a great conversation with Vasiliev (Cambridge) and Valluri (Michigan) about statistical inference problems related to Schwarzschild modeling a galaxy using an orbit library. I'm not sure I helped much! Right now no-one knows how to realistically marginalize out the effective distribution function, although I think there might be good ideas somewhere in the probabilistic machine-learning world.
We had a plenary discussion of non-steady-state and non-equilibrium aspects of the Milky Way and how we will model or understand them. Fundamentally, we only know how to infer the dynamics of the Milky Way by making strong assumptions: Either (in the case of Jeans or Schwarzschild modeling, say) that there is time symmetry and also cylindrical or spherical symmetries, or (in the case of stream modeling, say) that the stars were put onto orbits in some collectively informative way. Since the real Milky Way violates these assumptions for most stars at some level, we need qualitatively new kinds of assumptions to make. My proposal: That the Milky Way grew from the cosmological initial conditions! That's the right thing, I think, but we don't yet have any tractable way to think about (say) the Gaia data in that context. At least not precisely.
In the afternoon, there was a cross-meeting dark-matter session in which a large set of particle physicists and a large set of astrophysicists interacted over testing dark-matter models. I learned that there is a huge literature we don't know enough about. I am very interested in going down this path, because it connects Gaia and SDSS-V to fundamental properties of the Universe. That's what I would really love to do (and that's in part why my original idea for SDSS-V was called "Disco": Cosmology with the disk).
At some point in the day, I realized that we can test chemical-tangents method (my baby) in fake data! I discussed this with Loebman (Davis) and Price-Whelan (Princeton). I also realized that we can compare it to Jeans modeling, and show (I hope) that it always wins.
At the very end of the day, I had a conversation with Anderson (Flatiron) about advising and mentoring of postdocs. I feel very lost, I have to say: I want to give my attention to all my projects, my attention and time are limited, I don't spend my attention in the right places, I disappoint many of my people, and I impede their progress. I feel like I am doing it wrong! I don't feel like I understand how to be a mentor, and I am starting to feel stressed about it. One of the strange things is that the postdocs with whom I work are both the best and most fun collaborators I have, and also independent, capable scientists. That would seem to make it all easy and fun, but instead it somehow makes it confusing and existential. I went home unhappy from an absolutely great week in Aspen.
Adrian Price-Whelan (Princeton) resolved some of our code differences today as unit or dimensions differences. That was good! But we still have the problem that different elements (in comparison with kinematics) lead to different inferences about orbits in the Milky Way disk. Don't know what to do about that! Either the data are wrong, or there is a big discovery here.
Ana Bonaca (Harvard), Price-Whelan, and I discussed how to build a pseudo-likelihood for comparing the models that Bonaca has for a stream perturbation to the real data. This is a bit of a hard problem, because we want objectives that improve as the agreement improves, but we don't want to build a fully generative model of the data. Why not? because we don't have a good generative model, and perturbations away from a bad generative model could lead to very wrong inferences. All we want, after all, is a rough sense of what kinds of events are consistent with the data.
In the afternoon, I gave the Aspen Center for Physics Colloquium. I spoke about Gaia and dark matter, but I also threw in my thinking about the inference of Solar System dynamics in the 17th Century: We would do it very differently now! I have much more to say but I am too tired to write it here.
Side by side, Price-Whelan (Princeton) and I worked through and discussed code inconsistencies between my code and his code to compute the likelihood function for my chemical-tangents (working title) project, that uses chemical invariants to find dynamical invariants. It was a frustrating discussion, because we couldn't figure out either the issues or how to test. That's our project for tomorrow!
Bonaca (Harvard) reported in: She can show that the gaps and associated loops we find in the GD-1 stream cannot be caused by interaction with a molecular cloud on a disk orbit! That means that the only explanation remaining is dark-matter substructure. Awesome! I'm stoked!