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Wednesday, September 05, 2018

Superfluid dark matter passes another check: strong gravitational lensing

[image: kamere.com]
Physicists still haven’t figured out what dark matter is made of, if anything. The idea that it’s made of particles that interact so weakly we haven’t yet measured them works well to explain some of the observational evidence. Notably the motions of galaxies bound to clusters and the features of the cosmic microwave background fit with theories of particle dark matter straight-forwardly. The galaxies themselves, not so much.

Astronomers have found that galaxies have regularities that are difficult to accommodate in theories of particle dark matter, for example the Tully-Fisher relation and the Radial Acceleration Relation. These observed patterns in the measurements don’t follow all that easily from the simple models of particle dark matter. Thrifty theorists have to invoke additional effects that are assigned to various astrophysical processes, notably stellar feedback. While these processes arguably exist, it isn’t clear that they actually act in galaxies in amounts necessary to explain the observations.

In the past 20 years or so, astrophysicists have improved computer simulations for galaxy formation until everything fit with the data, sometimes adapting the models to new observations. These computer simulations now contain about a dozen or so parameters (there are various simulations and not all of them list the parameters, so it’s hard to tell exactly) and the results agree well with observation.

But I find it somewhat hard to swallow that regularities that seem to be generic in galaxies follow from the theory only after much fiddling. Indeed, the very fact that it took astrophysicists so long to get galaxies right tells me that the patters in our observations are not generic to particle dark matter. It signals that the theories are missing something important.

One of the proposals for the missing piece has long been that gravity must be modified. But I, as many theorists, have not been particularly convinced by this idea, the reason being that it’s hard to change anything about Einstein’s theory of general relativity without running into conflict with the many high precision measurements that are in excellent agreement with the theory. On the other hand, modified gravity works dramatically well for galaxies and explains the observed regularities.

For a long time I’ve been rather agnostic about the whole issue. Then, three years ago, I read a paper in which Berezhiani and Khoury proposed that dark matter is a superfluid. The reason I even paid attention to this had nothing to do with dark matter; at the time I was working on superfluid condensates that can mimic gravitational effects and I was looking for inspiration. But I have since become a big fan of superfluid dark matter – because it makes so much sense!

You see, the superfluid that Berezhiani and Khoury proposed at isn’t just any superfluid. It has an interaction with normal matter and this interaction creates a force. This force looks like modified gravity. Indeed, I think, it is justified to call it modified gravity because the pull acting on galaxies it now no longer that of general relativity alone.

However, to get the stuff to condense, you need sufficient pressure, and the pressure comes from the gravitational attraction of the matter itself. Only if you have matter sufficiently clumped together will the fluid become a superfluid and generate the additional force. If the matter isn’t sufficiently clumped, or is just too warm, it’ll not condense.

This simple idea works remarkably well to explain why the observations that we assign to dark matter seem to fall into two categories: Those that fit better to particle dark matter and those that fit better to modified gravity. It’s because the dark matter is a fluid with two phases. In galaxies it’s condensed. In galaxy clusters, most of it isn’t condensed because the average potential isn’t deep enough. And in the early universe it’s too warm for condensation. On scales of the solar system, finally, it doesn’t make sense to even speak of the superfluid’s force, it would be like talking about van der Waals forces inside a proton. The theory just isn’t applicable there.

I was pretty excited about this until it occurred to me there’s a problem with this idea. The problem is that we know at least since the 170817 gravitational wave event with an optical counterpart that gravitational waves travel to good precision at the same speed as light. This by itself is easy to explain with the superfluid idea: Light just doesn’t interact with the superfluid. There could be various reason for this, but regardless of what the reason, it’s simple to accommodate this in the model.

This has the consequence however that light which travels through the superfluid region of galaxies will not respond to the bulk of what we usually refer to as dark matter. The superfluid does have mass and therefore also has a gravitational pull. Light notices that and will bend around it. But most of the dark matter that we infer from the motion of normal matter is a “phantom matter” or an “impostor field”. It’s really due to the additional force from the superfluid. And light will not respond to this.

As a result, the amount of dark matter inferred from lensing on galaxies should not match the amount of dark matter inferred from the motion of stars. My student, Tobias Mistele, and I hence sent out to have a look at strong gravitational lensing. We just completed our paper on this and it’s now available on the arXiv.
    Strong lensing with superfluid dark matter
    Sabine Hossenfelder, Tobias Mistele
    arXiv:1809.00840 [astro-ph.GA]
It turns out that the observations from strong gravitational lenses are not hard to accommodate with superfluid dark matter. The reason is, loosely speaking, that the amount of superfluid can be adjusted or, somewhat more technically, that the additional fields require additional initial conditions and those allow us to always find solutions that fit the data.

This finding hence exemplifies why criticisms on modified gravity that insist on there only being one way to fit a galaxy are ill-founded. If you modify gravity by introducing additional fields – and that’s how almost all modifications of gravity work – the additional fields will have additional degrees of freedom and generally require additional initial conditions. There will hence usually be several solutions for galaxies. Indeed, some galaxies may by some statistical fluke not have attracted enough of the fluid for it to condense to begin with, though we have found no evidence of that.

We have been able to fit all lenses in our sample – 65 in total – except for one. The one outlier is a near-miss. It could be off for a variety of reasons, either because the measurement is imprecise, or because our model is overly simplistic. We assume, for example, that the distribution of the superfluid is spherically symmetric and time-independent, which almost certainly isn’t the case. Actually it’s remarkable it works at all.

Of course that doesn’t mean that the model is off the hook; it could still run into conflict with data that we haven’t checked so far. That observations based on the passage of light should show an apparent lack of dark matter might have other observable consequences, for example for gravitational redshift. Also, we have only looked at one particular sample of galaxies and those have no detailed data on the motion of stars. Galaxies for which there is more data will be more of a challenge to fit.

In summary: So far so good. Suggestions for what data to look at next are highly welcome.


Further reading: My Aeon essay “The Superfluid Universe”, and my recent SciAm article with Stacy McGaugh “Is dark matter real?”

44 comments:

  1. Ok, some suggestions on data that you might want to look at.. I will throw some ideas. No doubt you have long considered them all before, but who knows.

    Have you tried matching the model against the rotation speed of ultra diffuse galaxies, as well as galaxies with a low dark matter content? Could this possibly help in constraining some parameter?

    Have you tried matching observations on colliding clusters like the bullet cluster and MACS J0025.4-1222 against the model?

    Do your calculations imply any sort of correction to the rate of expansion of the universe, or to the early rate of expansion?

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  2. Okay, one final shot in the dark.

    The hubble constant has been derived from different observations with different methods, and there is a very significant disagreement between them. This reminds me of what you are saying about lack of gravitational redshift. Could it be that your model perhaps predicts a bias in at least one of the methods that have been used to calculate the hubble constant?

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  3. What about the acoustic peaks of the CMB supposedly enhanced by dark matter? The universe was quite hot back then.

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  4. Hi Sabine,
    "the superfluid that Berezhiani and Khoury proposed at isn’t just any superfluid. It has an interaction with normal dark matter and this interaction creates a force"
    I suppose that you wanted to say "normal matter" and not "normal dark matter"...?

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  5. Being a relatively ignorant person on this site, please forgive a strange question. When the proposed dark matter "condenses" into a super fluid, is it likely that its density changes under the increased pressure? Might DM be "compressible"?

    I haven't had a chance yet to read your paper; mea culpa.

    sean s.

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  6. What is the mass of the particles that make up the superfluid?

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  7. Congratulations on the work and the paper Dr. H!

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  8. https://tritonstation.wordpress.com/2018/08/29/a-precise-milky-way/

    "superfluid dark matter...has an interaction with normal dark matter" "Dark matter" is ~0.257 GeV/cm³. Superfluid DM phonon interaction is extended versus its source particle density spacing, or DM is low mass and particle-abundant. DM searches are flawed.

    "the amount of dark matter inferred from lensing on galaxies should not match the amount of dark matter inferred from the motion of stars" Stated ~1 H atom/cm³ interstellar medium. Milky Way masses ~62 billion sols and ~6.65×10^60 m³, 1.854×10^(-23) g/cm³ versus stated ~one H/cm³ = 1.674×10^(-23) g/cm³ OK!

    Intergalactic medium is stated ~5 H/m³ or 8.37×10^(-29) g/cm³, mass spaced beyond phonon interaction.

    Baryogenesis’ Sakharov criteria, trace chiral anisotropic vacuum selective to hadrons, gives Noetherian angular momentum leakage without tuning. Observe whether quantitatively extreme chiral-divergent small molecule enantiomers have detectably non-identical deep cryogenic single line microwave rotational spectra re DOI:10.1002/anie.201704221 (222 spectral lines).

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  9. I don't see why superfluid dark matter would be different from particle dark matter. In fluid mechanics, fluids are made up of particles. The different phases of matter arise from differences in density and kinetic energy of particles. Van Der Waals forces arise from electromagnetic force. To make a superfluid theory different from particle dark matter, you have to modify gravity or introduce a new force. In that case, the difference is in the force not in the fluid.

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  10. Very interesting. Thank you. Posted to my Facebook account.

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  11. I like this line of reasoning a lot...

    I appreciate your bringing this work to our attention Dr. Hossenfelder. It ties in nicely with my own current research, Which I presented at FFP15, that treats gravitation as a condensation process. This is not new of course. Sakharov first proposed the idea, and Laughlin used it in his 'Emergent Relativity,' but it has gained ground after a pair of papers by Dvali and Gomez, and I think Susskind investigated it too. I have my own ideas.

    In the Dvali and Gomez version; a Schwarzschild black hole event horizon is seen to be well-modeled by the critical surface at formation of a Bose-Einstein Condensate. So the BH itself can be seen as a graviton condensate, and since the SBH has no charge or spin it is a good model for pure gravitation. But this leaves open a possibility that massive gravitons could somehow undergo spontaneous condensation in the cold dark reaches of space.

    As I understand it; massive gravity theory includes the existence of a mass multiplet state for gravitons, where the ground state is massless or at most minimally massive, but higher order states exist - which may have been more likely to appear in the early universe. Do you think that superfluid Dark Matter could be a condensate of massive gravitons? And if so; how would this affect the cosmological picture?

    All the Best,

    Jonathan

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  12. Yves,

    A very unfortunate typo, thanks for pointing out!

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  13. doktor Boktor,

    Well, the model reproduces the MOND regime, so it'll work at least as good as that. There's nothing to learn from the clusters because, as I said, in clusters most of the fluid isn't condensed, so it just works like normal dark matter. For what the expansion is concerned, again please note that for much of the history of the universe the stuff behaves like normal dark matter.

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  14. Daniel,

    Yes, right, it's hot, so the stuff isn't superfluid. It behaves like normal dark matter, hence I see a priori no problem with the CMB. Of course you still want to know that actually all the parameters fit and we'll have to look at this. But to be honest I am not particularly convinced by Khoury's treatment of the two phases of the fluid and so we'll first have to do some more theoretical work. But, yeah, the CMB is definitely on the list.

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  15. Jonathan,

    Giving masses to gravitons seems to me to create more problems than it solves. Also, for what I can recall Dvali doesn't use massive gravitons. In any case, I don't see a relation to Dvali's idea. Best,

    B.

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  16. Enrico,

    Well, I explained how it's different: it gives rise to a long range force that spans through galaxies. No, normal dark matter doesn't do that. Best,

    B.

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  17. Unknown,

    The parameter values are listed in the paper, page 4, top paragraph. We take m=1eV because that's the value that Khoury suggests. That may not be the best-fit parameter, but it seems to be the right order of magnitude.

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  18. "The parameter values are listed in the paper, page 4, top paragraph. We take m=1eV because that's the value that Khoury suggests. That may not be the best-fit parameter, but it seems to be the right order of magnitude."

    Is this parameter determined solely via a fit to the data, or is there some other reason for it?

    Assuming something like this exists, the scale on which its behaviour changes depends on the particle mass. The question then is, why is this...drumroll, please...fine-tuned such that it just so happens to kick in at the mass of a galaxy?

    A possible answer is that the mass of a galaxy, via the complicated process of galaxy formation, somehow depends on the mass of this particle.

    I asked Justin this question at a conference last year, but either I didn't understand his answer or he didn't understand my question. :-|

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  19. Phillip,

    As you hopefully know by now I think it's a bad idea to obsess over supposed numerical coincidences.

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  20. Hi Sabine,

    Would this change the critical density of the universe? Critical density=3H^2/(8piG)

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  21. I still think Physicists are bypassing the most probable and promising research due to influences inherent in all human behavior. It is true that General Relativity like Newtonian Mechanics is an extraordinary leap, yet there are many indications it is not a perfect representation of the dynamics it portrays. However, because of its accuracy, how long it has been accepted, groupthink, career considerations, and conceit we speculate ad hoc solutions to issues as we did for three centuries under Newtonian Mechanics. Einstein used what was learned from Newton and incorporated what he knew must be missing, the malleability of space-time, however he struggled with the mathematics for his idea. I believe we should take what was learned from GR and start yet again with a more fundamental approach to space-time and its empirical evidence.

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  22. Interesting, thank you. (I guess I didn't realize that SFDM was a bit of a blend of Mond and CDM.) Say, does it have anything to say concerning the remnant 21 cm line data, discussed on Stacy McGaugh's Triton station? That also seems to mostly be a galactic phenomenon.

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  23. Sabine: Khoury himslef says the following at the end in his latest paper (https://arxiv.org/abs/1712.01316):

    "At the same time, the standard[146], exponential [28] and Bekenstein [38] MOND interpolating functions, as well as the superfluid DM paradigm (which, with no renormalization of a_0, gives rise to Bekenstein’s interpolating function) [80–82] and Verlinde’s emergent gravity formula [73], all appear to be in tension with the data."

    What do you make of this?

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  24. I am pondering how there can be phonon modes. If these scalar particles are axions they have a tiny mass of 10^{-5} to 10^{-3} ev. The Compton wavelength is λ = ħ/mc or about 2×10^{-2} to 2×10^{-4}m. This means that in a cubic centimeter there would be 1 or more to 100 or more axions is their waves overlap. Given the mass density of the universe ~ 10^{-28}g/cm^3 there are about 2×10^{9} to 2×10^{11}axions per cubic centimeter. So if they exist and comprise dark matter or some appreciable percentage then we are fairly sure their wave functions overlap.

    In order to have phonons you have to have a lattice of some type and for the particles to be bound into it. That is a bit tough to see. However, maybe they are in a lattice or something analogous to a solid. The inflationary Lagrangian has a friction-like term that removes the kinetic energy of all axions. So axions might indeed be in a lattice or solid. This lattice might instead be more of an amorphous solid such as glass.

    What about an interaction? This one is really tough, but we can use the fact that U(3) = SU(3)×U(1) that decompose in a way similar to the electroweak U(2) = SU(2)×U(1). We would then have a new U(1) interaction in the world that decouples from SU(3). Yes, yes I know, there are some problems with this as I can hear now some gnashing of teeth. However let us just play with this. Now all you need is a Ginzburg-Landau mechanism for breaking the symmetry of this new U(1). There are a number of interesting routes with this, and one would be to consider the axion superfield and break the fermionic part with Bogoliubov coefficients. Hmmm,... that might work.

    So this is not all completely crazy. It even could fit somewhat within established physics.

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  25. I find it incredible that we have two gravity related things we can't explain: Dark matter and dark energy. Work on modifying gravity in a way that explains observations of galaxies and clusters--the things closest to us that we can measure more accurately--then apply that work to early universe. I suspect our current picture of the early universe is so wrong it's not even funny. Perhaps there will be another explanation for the hubble law. Relative velocity isn't that only thing that can cause redshift--gravity can as well.

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  26. With respect to what I wrote yesterday, I went looking to see if there was anything similar. I found the following

    https://arxiv.org/pdf/1712.03865.pdf

    Dark Matter as Ultralight Axion-Like particle in E6 × U(1)_X GUT with QCD Axion
    Claudio Corian and Paul H. Frampton

    For those not versed in Lie group and algebra lore. the exceptional group U(6) decomposes into O(10) is a double cover of U(5) = SU(5)×U(1).

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  27. @Q "modifying gravity in a way that explains observations of galaxies and clusters" is curve fitting. Rigorous Aristotelian derivation then failed empirical observation flags a non-empirical postulate.

    Baryogenesis requires trace chiral anisotropic vacuum toward hadrons (Sakharov criteria). Physics postulates exactly mirror-symmetric isotropic vacuum re Noether. Physics fails within that gap.

    Falsify failed accepted theory outside physics’ postulates with analytical chemistry. This is untouchable, unfundable, and unpublishable for contradicting failed accepted theory. Run Angew. Chem. Int. Ed., 56(37), 11209 (2017), DOI:10.1002/anie.201704221 with a molecule crafted to the task. One undergraduate-day versus 400,000 theorist-years and a million pages published to the contrary is offensive…and viable Look.

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  28. Nice work, Bee! :)

    Brief but important critical comment: galaxies are no more spherical than cows (or, ignore their asymmetry at your theoretical peril).

    An interesting test of this superfluid dark matter idea, one also involving gravitational lenses: time delays. As Birrer+ (2018) (https://arxiv.org/abs/1809.01274) nicely demonstrate, one can get a darn good estimate of H0 from analyzing time delays in strong lenses, completely independently of the distance ladder (as a bonus, they seem to have used a blinded approach, and sound statistical analyses, though perhaps it might be wise to wait for the paper itself, the reviewers may require some important changes).

    If DM were a superfluid - of the Bee+ kind - would the strong lensing time delays be any different? Would an estimate of H0 based on analyses of time delays differ?

    Other thoughts for further exploration (not that you need any inputs from lil' ol' me!): the "satellite deficit problem" (how many dwarf galaxy satellites, of normal spirals, does the SFDM idea predict? what is the mass function of such? etc); ultra-diffuse, ultra-compact, and other non-standard galaxies (what does the SFDM predict for their mass profiles? throw in globular clusters too); and the most non-spherical of cows, galactic mergers. And just for fun: what effect do the highly relativistic jets from "quasar mode" AGNs have on your superfluid?

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  29. Lawrence,

    You don't need a lattice for the phonons. I think you are overinterpreting the word. The phonons are perturbations in the phase of the scalar field.

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  30. I have only read the introduction of the paper by Berezhiani and Khoury and half of the paper you coathored on. When I think of the term fluid I think of a collective system of particles. If this occurs with phonons then I think of there being some sort of interaction between particles. The abstract of the Berezhiani and Khoury paper there is:

    Remarkably, the superfluid phonon effective theory is strikingly similar to that of the unitary Fermi gas, which has attracted much excitement in the cold atom community in recent years.

    This suggests to me this is a collective phenomenon. The phonons are due to an interaction between ions in a lattice which may have a gauge theory interpretation as a local phase difference in the interacting field. On page 5 there is a description of the phonon as. “the DM is more aptly described as collective excitations, which at low energy are just phonons.” which is followed by equation 4 that defines these phonon according to the scalar field.

    I used the term lattice, though I agree this does not have to be a regular structure such as seen in a crystal. I am not certain of what has been done with amorphous solids with respect to phonons as quanta of vibrations. It may not necessarily have to be a solid, but just interacting. So I have pretty sure these phonons are due to a collective of fields that are interacting.

    I have this conjecture that this might be a case of Vafa's swampland. This is a set of conditions that exist on a de Sitter spacetime with positive vacuum energy. Quantum gravity appears to work well with manifolds that have negative vacuum energy in anti de Sitter spacetimes. Things are problematic on de Sitter spacetimes, and Vafa has conjectured the observable universe is a "broken world" of sorts, maybe due to spontaneous symmetry breaking, where gauge theory exists in a spacetime with positive vacuum energy, but where this is not consistent with quantum gravity. This appears to be a case where the vacuum of the low energy physics is different from the fundamental Lagrangian.

    Vafa states the condition for the swampland is for a scalar field φ with |∇φ| ≥ cφ, c = constant, and so equation 4 in Berezhiani and Khoury states

    X = φ' - Φ - (∇φ)^2/2m, φ' = dφ/dt

    puts a bound on the field φ with φ' = iωφ so that

    |X| ≥ sqrt{ω^2 + [Φ - (cφ)^2/2m]^2}

    then the BK potential V ~ const×X sqrt{X} has a bound. There is then maybe the prospect this strange potential is due to the breaking of the symmetry with negative energy.

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  31. If SFDM can't condense in the early universe, how do you get acoustic oscillations, or more to the point, galaxies to form. Baryonic matter can't condense by itself in the plasma regime, and after recombination is too late.

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  32. CIP,

    If it's not condensed it's not absent, it just doesn't have the long-range force that gives rise to the MOND-like effects. It should then pretty much behave like normal dark matter, ie a pressureless cold fluid. Whether you have it in the right amounts to get BAO right, however, I don't know.

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  33. I still don't understand why it is called modified gravity!

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  34. Space Time,

    Well, Khoury et al don't call it modified gravity, and since our new paper is about their model, we didn't call it modified gravity either. But the long-range force that leads to the observed motion in this case is not just gravity, so I think it makes sense to call it modified gravity. Regardless of how you want to call it though there's equations and stuff so you can just do the calculation and see if it works.

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  35. For a very long time the ratio of Dark Matter (DM) versus normal Baryonic Matter (BM) has been pegged at 1 to 5, in the purely Cold Dark Matter paradigm. With the added "force" due to the phonon field mimicking an extra gravitational attraction (assuming I'm interpreting this correctly), it seems logical that the 'actual' ratio of DM vs BM would be lower than the 1 to 5 ratio that was inferred wholly from DM's gravitational influence, as less DM would be needed for the same gravitational effect. Would this be a reasonable comclusion for the Superfluid DM paradigm?

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  36. The 1:5 ratio comes from the CMB, in which regime Khoury's stuff acts like normal dark matter. So, no.

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  37. I'd heard it said there's no such thing as a dumb question if you truly don't know the answer. (Also that dumb questions are better than dumb mistakes!)

    With that in mind, I have a dumb question (because it can't possibly be this simple):

    The problem with galaxies, AIUI, is that the outer bits rotate faster than allowed for natural orbits at that distance (given the assumed mass of the galaxy).

    I've never read anything, perhaps it's too obvious, but what about galactic gas acting to make a galaxy act somewhat like a rigid wheel? Could the outer stars be dragged along (kind of like how a torque converter works)?

    I assume this was long ago considered and discarded (plus there are effects we apparently see on larger scales than galaxies?). I'd just like to know how it was eliminated.

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  38. Chris,

    The galaxy rotates as a whole. The gas rotates with it. Also, there isn't remotely enough drag.

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  39. Just to clarify: one would expect, if mass is correlated with visible (or radio, or whatever) light, the speed of rotation to fall in a certain way with distance from the centre. What actually happens is that this velocity is approximately constant. Velocity in m/s. This does not imply a constant angular velocity; the angular velocity drops with distance even if the rotation curve is flat (referring to a plot of velocity against distance from the centre) because, at the same speed, bigger orbits take more time.

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  40. How would the superfuid idea work for dark energy as opposed to dark matter?

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  41. "How would the superfuid idea work for dark energy as opposed to dark matter?"

    Not at all.

    One can't simply combine current buzzwords into a new question and expect it to mean anything. :-|

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  42. @Bee: "The galaxy rotates as a whole. The gas rotates with it. Also, there isn't remotely enough drag."

    Thank you for your reply! So there is no chance, over millions of years, of any transfer of inertia from the inner parts to the outer parts. (Of course it wouldn't be that simple or overlooked. :)

    @Phillip: "...the angular velocity drops with distance even if the rotation curve is flat..."

    Thank you, too! I do understand that and don't mean to imply galaxies are in any way rigid. I've just been wondering about possible transfer of motion, or even some kind of frame-dragging due to the galaxy's mass.

    Mostly I think I find it bemusing that, for all we know, our theories telling us was happened mere nano-jiffies after the Big Bang, there is so much we're still just guessing at.

    (I was just reading Phil Plait's article about how we don't know exactly how far away the Pleiades are. It sometimes amazes me we're sure about anything way out there.)

    Putting this on a more cogent angle, do I understand this superfluid DM is a property of otherwise particulate DM? And that this property allows the idea of particulate DM to better fit all observations?

    If yes and yes, then we're still looking for DM particles?

    If no, how does a superfluid differ from a substance? Your post describes how its behavior differs, but what is superfluid made of? Something we can look for?

    Is the wiki article about "Superfluid vacuum theory" aligned with what you're investigating or something else entirely?

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  43. People aren't generally rude on Sabine's Blog Philip, but be a trend setter if you want.

    I understand the distinction between Dark Matter (e.g. unexplained attraction of the Coma cluster) and Dark Energy (accelerating expansion of spacetime) perfectly well.

    The notion that superfluid might be an explanation for dark matter is interesting and highly highly speculative. It is not a silly question to ask if it might also be of relevance to Dark Energy, another area where physicists have no idea what's going on.

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  44. Peter, Phillip,

    I demonstrated in this paper that you can get a cosmological constant from the superfluid & it has the right order of magnitude. It shouldn't be so surprising because you need a self-interaction for the condensation that leaves behind a potential term which (to make a long story short) can look like a cosmological constant.

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