Late time Cosmological Phase Transition and Galactic Halo as Bose-liquid

Sang-Jin Sin

REVIEW

Late time Cosmological Phase Transition and Galactic Halo as Bose-liquid

Sang-Jin Sin

High Energy Physics - Phenomenology · Submitted 8 May 1992 · 18 references

Pith's verdict: The first real treatment of a galactic halo as a single ultra-light-boson wavefunction — the structural idea holds up, the 1992 data case does not.

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arXiv IDhep-ph/9205208 (version 1)

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CitationSang-Jin Sin. "Late time Cosmological Phase Transition and Galactic Halo as Bose-liquid." arXiv:hep-ph/9205208 (1992). https://pith.science/paper/BIMHP2NY

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Editor's note

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This is the paper that treats a galactic halo as one macroscopic wavefunction of an ultra-light boson, and that move is the reason to read it. The nonlinear Schrödinger equation in Eq. 2, the scaling symmetry that removes the M0 ambiguity, and the observation that the r^-1.56 profile and the rotation-curve ripples fall out of the equation rather than being fitted — that is genuine structural content, and it is the ancestor of what people now call fuzzy or wave dark matter. The scaling argument in particular is clean and he actually proves it, so the claim that physical masses are independent of the arbitrary normalization is not hand-waving. Credit where due: the r^-1.6 slope and the ripples are outputs, not inputs, and the v_rot times lambda_ripple relation is a real falsifiable number with no free function in it. That is more than most dark-matter profile papers of the period offer. The soft spots are in the confrontation with data, and they are not small. The halo is put in a highly excited eigenstate — n=5 for the fit, n=6 for the quoted profile — justified only by the sentence that there is no dissipation mechanism. No stability analysis, no formation channel, and mergers and dynamical friction are exactly the kind of effective dissipation that could relax the state or make it a mixture. The stress-test note lands here; the paper acknowledges the choice of excited state and the Omega bound n<=7, but not its stability, and that assumption carries the distinctive predictions. Second, the fit rests on NGC2998, picked from 63 curves as the one where ripples are visible; m and M0 come from that single galaxy, and the other galaxies are called 'roughly satisfied' with no numbers and no error bars. So the v_rot-lambda prediction is never actually tested out of sample — it is the relation used to fix m. He is honest about the fine-tuning, which counts for something. The WKB justification is loose but flagged in footnote 12, so I would call that minor. Read it as a model proposal with a real equation behind it, not as evidence. Worth a serious referee and worth citing as the origin of the wave-DM halo picture; do not trust the numbers.

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Major objections (4)

[§2, single-eigenstate assumption] The paper's two distinctive outputs, the rho ~ r^-1.56 profile (quoted for n = 6) and the rotation-curve ripples (fitted with n = 5), exist only if the halo occupies a single highly excited stationary solution of Eq. (4). Section 2 justifies this with the sentence 'there is no way for the system to lose its energy to relax to its ground state, since there is no dissipation mechanism.' That assertion is load-bearing and is not supported. Excited states of a self-gravitating nonlinear Schrödinger equation need not be dynamically stable even in the absence of explicit dissipation, and no linear-stability or Bogoliubov analysis is performed. Gravitational phase mixing, dynamical friction and mergers act as effective dissipation for a coarse-grained wavefunction, so the premise that there is no dissipation mechanism is not obviously true for a galactic halo. No formation mechanism is offered that would select n = 5 or n = 6 rather than a mixture of states; the only stated constraint on n is the Omega bound. The author should either supply a stability/formation argument or state explicitly that the single-eigenstate branch is an assumption whose consequences are being explored, and discuss how the profile exponent and ripple amplitude change for a mixture.
[§3, Eqs. (10)-(15) and prediction (3)] The particle mass m = 3.3 x 10^-23 eV and the mass scale M0 = 0.69 x 10^12 M_sun are fixed by matching NGC 2998's second-ripple wavelength and rotation velocity (Eqs. 10-11). Given that m is fixed, the relation v_rot times lambda_ripple approximately constant (Eq. 6) is a genuine conditional prediction for other galaxies, so the concern that prediction (3) is circular is only partly warranted. The empirical gap is that the paper does not actually test it: it states that 'for other galaxies, by similar analysis the prediction of eq. (8) is roughly satisfied' without showing a single additional fit, even though it identifies about five galaxies (NGC 2998, NGC 1357, NGC 4594, NGC 1620, NGC 801) with small enough error bars to show ripples. A table giving, for each of those galaxies, the measured ripple wavelength, the rotation velocity, and the inferred m and M from Eqs. (14)-(15) would convert prediction (3) from an assertion into a test. As written, the cross-galaxy claim is the weakest-supported of the three headline predictions.
[Abstract and §2-§3, n = 5 versus n = 6] The abstract advertises the mass profile rho ~ r^-1.6, which §2 states is the numerical result for n = 6, while the data comparison of §3 adopts n = 5 as 'the minimal choice' and fits NGC 2998 with it. The paper never states which node number describes the physical halo, nor how the profile exponent for n = 5 compares with the quoted r^-1.56. Since the exponent and the ripple spacing are the paper's two quantitative handles on the data, the reader needs the n = 5 profile exponent stated alongside the n = 6 value, and a sentence explaining why the abstract quotes the exponent of a state that is not the one used in the fit.
[Eqs. (1)-(2), potential term] The step from the classical energy in Eq. (1) to the nonlinear term in Eq. (2) is not shown, and the sign and normalization deserve an explicit derivation. As printed, the double integral in Eq. (2) evaluates for rho proportional to r^-1.56 to a positive, increasing function of r, whereas the gravitational potential of a bound spherical halo is negative; the existence of the bound states that the paper exhibits depends on the sign convention actually used in the code. Please give the one or two lines connecting Eq. (1) to Eq. (2), state the convention for the self-consistent potential, and confirm that the numerical solutions satisfy the virial relation v^2 = GM/r used later to justify the non-relativistic treatment.

Minor objections (8)

[§2, after Eq. (8)] The sentence 'The relation (5) is just a corollary of this rather interesting property' appears to refer to Eq. (8), not to the WKB expression in Eq. (5); please correct the cross-reference.
[Figures 1-3] The figures are referenced but no captions or axis labels are supplied in the text; each panel should carry a caption stating what is plotted, in what units, and for which node number n.
[Eq. (7)] The typesetting of Eq. (7) is garbled (the radical symbols are broken); please re-set it so that f(r-hat) is legible.
[§3 versus §4] The text says 'Among 63 galaxies listed in [10]' in §3 but 'Among 62 rotation curve data of Rubin et.al.' in the Conclusion; please reconcile the count.
[§3, visible matter] 'Plumber's potential' should read 'Plummer's potential', and reference [3] should read 'Lynden-Bell' rather than 'Linden-Bell'.
[§2, node-number discussion] The statement that 'for n greater than 4, the rotation curve resembles the observed ones' should be quantified, for example by giving the profile exponent and the ripple contrast for n = 5, 6 and 7, so the reader can see how weakly the results depend on n.
[§3, Omega bound] The argument that n larger than 7 gives a halo mass above 10^13 M_sun and hence violates the Omega bias is presented as a constraint on n, but the total mass is itself fixed by the fit normalization through the scaling symmetry; please clarify whether this is an independent constraint or a restatement of the normalization.
[§2, WKB discussion] The phrase 'the last wave is almost as long as the sum of all the others' is unclear; please state what is being summed and over what range.

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The paper

The paper claims

Sin argues that an ultra-light dark-matter boson turns a galactic halo into one large quantum wave, whose excited states reproduce a slightly rising rotation curve with ripples.

The reading

Sin takes seriously the ultra-light pseudo Nambu-Goldstone boson that late-time cosmological phase transition models produce, and asks what a galactic halo made of such a particle would look like. Because the particle's Compton wavelength (tens of parsecs) is vastly larger than the inter-particle spacing, the halo cannot be treated as a gas of points; it is better described as a single, highly occupied wavefunction, a Bose liquid, obeying a nonlinear Schrödinger equation whose potential is the gravity of the wavefunction itself. Solving that equation for excited stationary states yields three checkable statements: a dark-matter density profile $rho propto r^{-1.56}$, ripple-like fine structure superimposed on the rotation curve, and the product of rotation velocity and ripple wavelength being roughly the same in every galaxy, which is a restatement of the uncertainty principle. The appeal is economy: the shape of a halo and the wiggles in its rotation curve follow from wave mechanics with essentially one free parameter, the boson mass, which Sin fixes at about $3.3 imes 10^{-23}$ eV by fitting the rotation curve of NGC 2998.

What carries the argument

The carrier is the gravitational nonlinear Schrödinger equation for a condensate wavefunction, together with a scaling symmetry that makes the predictions independent of the arbitrary mass normalization $M_0$. The scaling is: if $psi(r, epsilon)$ solves the equation, so does $lambda^2 psi(lambda r, lambda^2 epsilon)$. That symmetry is what allows Sin to choose a convenient normalization for numerical work and still quote physical masses, and it is the formal content behind the claim that $v_{rot} lambda_{ripple}$ is fixed: the same scaling that rescales the wavefunction rescales the radius and the eigenvalue together, leaving $m$ and the total mass invariant. A WKB reading of the resulting linear-in-potential problem then explains qualitatively why the envelope is nearly flat but slowly rising and why a ripple must be present at all.

Extended reading notes

Core claim

The paper's claim is that if dark matter is an ultra-light boson, galactic halos are macroscopic condensates rather than particle collections, and their structure is the structure of a nonlinear Schrödinger eigenstate. Sin writes down $i hbar partial_t psi = -frac{hbar^2}{2m} Delta psi + G m int dr' r'^{-2} int dr'' 4pi r''^2 (M_0 |psi|^2 + rho_{vis}) psi$, solves it numerically for spherically symmetric stationary states, and finds that for sufficiently excited states (node number $n$ around 5 to 6) the resulting density falls as $r^{-1.56}$, giving a rotation curve that is flat but slowly rising, closer to the data than the isothermal $r^{-2}$ profile or the infall $r^{-2.25}$ profile. The wave nature also forces oscillations: the semi-classical form $u sim p_r^{-1/2} cos(int p_r dr - Cpi/4)$ shows that the amplitude and phase of the radial wave modulate the enclosed mass, so the rotation velocity ripples. Because the ripple is a quantum interference feature, its wavelength scales inversely with the rotation velocity, so $v_{rot} lambda_{ripple}$ should be nearly galaxy-independent. Fitting the second ripple of NGC 2998 gives $7 r_0 = 8$ kpc and $0.32 v_0 = 204$ km/s, hence $m = 3.3 imes 10^{-23}$ eV, a total galaxy mass of $5.9 imes 10^{12} M_odot$, and a central dark-matter density of $1.08 imes 10^{-22}$ g/cm$^3$.

Load-bearing premise

The halo is assumed to sit in a single highly excited stationary state, chosen by hand ($n=5$ for the data fit, $n=6$ for the quoted $r^{-1.56}$ profile), on the grounds that there is no dissipation to let it relax to the ground state; gravitational relaxation, dynamical friction and mergers are not modelled.

Editorial extensions

If this is right

A halo's dark-matter density profile is predicted to fall as $r^{-1.56}$, slightly shallower than the isothermal $r^{-2}$, which matches the observed tendency of rotation curves to keep rising rather than flatten exactly.
Rotation curves should contain ripple-like fine structure whose wavelength shortens in galaxies with larger rotation velocity, with the product $v_{rot} lambda_{ripple}$ roughly universal.
A single rotation curve with measurable ripples yields both the dark-matter particle mass and the total mass of that galaxy, so galaxies become weighing devices for the dark sector.
The same scaling symmetry makes the central density and the total mass independent of the arbitrary normalization used in the numerical integration, so the fit has one physical knob rather than several.
Because the total halo mass grows with the node number, requiring $Omega$ not to exceed unity bounds the excitation, which is why Sin takes $n=5$ rather than a higher state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The prediction is unusually clean for a dark-matter model because it is a shape prediction, not just an abundance: a survey that measured ripple wavelengths in a handful of galaxies could test the $v_{rot} lambda$ relation without any mass modelling.
If the picture is right, the same wave mechanics should leave a signature in the inner parts of halos where the visible bulge dominates, so the ripple phase near the centre is a second, independent handle on the boson mass.
The framework sits close to what later became known as fuzzy or wave dark matter, and the specific number quoted here, a few times $10^{-23}$ eV, is the same range that later work on dwarf-galaxy cores and large-scale structure found interesting.
A natural extension the paper does not attempt is to ask what happens when a halo is a mixture of two eigenstates rather than one; the beat between them would produce a different, possibly time-varying, ripple pattern.

Original abstract

We consider the ultra light pseudo Nambu-Goldstone boson appearing in the late time cosmological phase transition theories as a dark matter candidate. Since it is almost massless, its nature is more wave like than particle like. Hence we apply quantum mechanics to study how they form the galactic halos. Three predictions are made; (1)the mass profile $\rho\sim r^{-1.6}$, (2)there are ripple-like fine structures in rotation curve, (3) the rotation velocity times ripple's wave length is largely galaxy independent. We compare the rotation curves predicted by our theory with the data observed.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We consider the ultra light pseudo Nambu-Goldstone boson appearing in the late time cosmological phase transition theories as a dark matter candidate. Since it is almost massless, its nature is more wave like than particle like. Hence we apply quantum mechanics to study how they form the galactic halos. Three predictions are made; (1)the mass profile $\rho\sim r^{-1.6}$, (2)there are ripple-like fine structures in rotation curve, (3) the rotation velocity times ripple's wave length is largely galaxy independent. We compare the rotation curves predicted by our theory with the data observed.

background contribution method result interpretation main claim

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s0	We consider the ultra light pseudo Nambu-Goldstone boson appearing in the late time cosmological phase transition theories as a dark matter candidate.	contribution 0.69	0.22	0.06	asserted 0.57
s1	Since it is almost massless, its nature is more wave like than particle like.	background 0.84	0.13	0.03	asserted 0.95
s2	Hence we apply quantum mechanics to study how they form the galactic halos.	method 0.90	0.64	0.04	asserted 0.98
s3	Three predictions are made; (1)the mass profile $\rho\sim r^{-1.6}$, (2)there are ripple-like fine structures in rotation curve, (3) the rotation velocity times ripple's wave length is largely galaxy independent.	result 0.88	0.90	0.68	asserted 0.98
s4	We compare the rotation curves predicted by our theory with the data observed.	method 0.74	0.89	0.05	asserted 0.57

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from analysis of existing data 0.85.

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References

Works cited (18)

[1]

Introduction — Recently, motivated by the large scale structure, an ultra light Nambu-Goldstone boson was introduced in late time cosmological phase transi- tion theory.[1] The bottom line of the theory is that if a phase transition happens after decoupling, one can avoid the constraint imposed by isotropy of microwave back ground. If an event happens so ...

View on Pith
[2]

Quantum considerations of dark matter — We con- sider the spherically symmetric case for simplicity and also assume that there are no inter-particle interactions apart from gravity. The classical energy of a particle moving inside a potential generated by other particles is E = 12mv2 + ∫ r0 dr′GmM (r′)r′2 , (1) whereM (r) = ∫ r 0 dr′4πr′2ρ(r′). Quantum me...

View on Pith
[3]

Comparison with observations — We now compare the the rotation curves predicted in our theory and the actual observations. First we have to include the visible matter’s eﬀect. See ﬁgure 3(a),(b). Equation (2) has to be modiﬁed to be i∂tψ = −12m ∆ψ +Gm ∫ r0 dr′r′2 ∫ r0 dr′′4π(M0|ψ|2 +ρvis) ·ψ (9) 2 University of Florida UFIFT-HEP-92/11 We choose the eﬀecti...

View on Pith
[4]

Conclusion — In this paper we investigated the con- sequence of abundant existence of Nambu-goldstone bo- son appearing in late time phase transition. Since the particle wavelength is far larger than the inter-particle distances, galactic halos made of this species are highly correlated bose liquid. We set up a version of non-linear shr¨ odinger equation ...

View on Pith
[5]

J. Gott, Ap. J. 201,296

View on Pith
[6]

J. Gunn, Ap. J. 218 592

View on Pith
[7]

[8] [9] , where the mass proﬁle is given by ρ ∼r−2.25, giving a slowly decreasing rotation curve. However, the observed rotation curves are actually slightly increasing

View on Pith
[8]

C.T.Hill, D.N. Schramm, J. Fry, Comments of Nucl.& Part. Phys. 19,25. 3 University of Florida UFIFT-HEP-92/11

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[9]

J. Frieman, C. Hill and R. Watkins, Fermilab-PUB- 91/324-A

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[10]

though they are said to be ‘ﬂat’. So neither the thermal distribution nor the infall model is completely satisfactory. If dark matter consists of these particles whose mass is, say, m ∼ 10−24eV and dark matter density around us is ∼ 10−25g/cm3, then the inter-particle distance is order of 10 −13cm, while the compton wave length is of 10pc order. Hence the...

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[11]

D. Linden-Bell, MNRAS,(1967) 136,101

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[12]

I. R. King, Astron. J. 7164(1966)

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[13]

J. Filmore and P. Goldreich 281 1

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E. Bertschinger, Ap. J. Suppl. 58 39

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P. Sikivie and J. Ipser, UFIFT-HEP-91-17

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V. Rubin et.al, Ap. J. 289 81 (1985); Ap. J. 261 439(1982); Ap. J. 238 471 (1980)

View on Pith 1985
[17]

Landau and Lifshitz, Course of theoretical physics Vol.9 section 26

View on Pith
[18]

A cautious remark on terminology; There is no WKB method for non-linear equations in general. Here the non- linearity is contained only in potential that is obtained nu - merically. Hence we are applying WKB to eﬀectively linear equations. Since the solution’s behaviour is such that the resulting potential varies slowly, WKB is a good approxima- tion. 4

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How the literature uses it

80 arXiv papers cite this in 95 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 38 sentences without a confident label

Contests it 1 paper

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Stability and physical properties of spherical excited scalar boson stars · Discussion and conclusion · 0.93

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This type of DM has been dubbed “fuzzy DM (FDM)" [ref] and can be naturally identified with an almost massless axion, see for instance [ref] .

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Introduction — Recently, motivated by the large scale structure, an ultra light Nambu-Goldstone boson was introduced in late time cosmological phase transi- tion theory.[1] The bottom line of the theory is that if a phase transition happens after decoupling, one can avoid the constraint imposed by isotropy of microwave back ground. If an event happens so ...

View on Pith

[2] [2]

Quantum considerations of dark matter — We con- sider the spherically symmetric case for simplicity and also assume that there are no inter-particle interactions apart from gravity. The classical energy of a particle moving inside a potential generated by other particles is E = 12mv2 + ∫ r0 dr′GmM (r′)r′2 , (1) whereM (r) = ∫ r 0 dr′4πr′2ρ(r′). Quantum me...

View on Pith

[3] [3]

Comparison with observations — We now compare the the rotation curves predicted in our theory and the actual observations. First we have to include the visible matter’s eﬀect. See ﬁgure 3(a),(b). Equation (2) has to be modiﬁed to be i∂tψ = −12m ∆ψ +Gm ∫ r0 dr′r′2 ∫ r0 dr′′4π(M0|ψ|2 +ρvis) ·ψ (9) 2 University of Florida UFIFT-HEP-92/11 We choose the eﬀecti...

View on Pith

[4] [4]

Conclusion — In this paper we investigated the con- sequence of abundant existence of Nambu-goldstone bo- son appearing in late time phase transition. Since the particle wavelength is far larger than the inter-particle distances, galactic halos made of this species are highly correlated bose liquid. We set up a version of non-linear shr¨ odinger equation ...

View on Pith

[5] [5]

J. Gott, Ap. J. 201,296

View on Pith

[6] [6]

J. Gunn, Ap. J. 218 592

View on Pith

[7] [7]

[8] [9] , where the mass proﬁle is given by ρ ∼r−2.25, giving a slowly decreasing rotation curve. However, the observed rotation curves are actually slightly increasing

View on Pith

[8] [8]

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A cautious remark on terminology; There is no WKB method for non-linear equations in general. Here the non- linearity is contained only in potential that is obtained nu - merically. Hence we are applying WKB to eﬀectively linear equations. Since the solution’s behaviour is such that the resulting potential varies slowly, WKB is a good approxima- tion. 4

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