Non-linear Schr\"{o}dinger-type formulation of scalar field cosmology: two barotropic fluids and exact solutions

Burin Gumjudpai (IF Naresuan); Chonticha Kritpetch (IF Naresuan); Jarunee Sanongkhun (IF Naresuan); Pichet Vanichchapongjaroen (IF Naresuan)

REVIEW 3 major objections 5 minor 28 references

Non-linear Schr\"{o}dinger-type formulation of scalar field cosmology: two barotropic fluids and exact solutions

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Friedmann equations for a canonical scalar field plus two barotropic fluids are exactly equivalent to a time-independent nonlinear Schrödinger equation under the variable change $u=a^{-n/2}$, $E=-\kappa^2 n^2 D_1/12$.

desk verdict Correct and honest, but narrow: the NLS–Friedmann dictionary is extended to two fluids and one new exact solution is found, yet the paper never verifies that the reconstructed scalar field is real on the full domain. read the letter →

arxiv 1908.11265 v2 pith:DJ3ZRXAL submitted 2019-08-29 gr-qc astro-ph.CO

Chonticha Kritpetch (IF Naresuan) , Jarunee Sanongkhun (IF Naresuan) , Pichet Vanichchapongjaroen (IF Naresuan) , Burin Gumjudpai (IF Naresuan) This is my paper · ORCID

classification gr-qcastro-ph.CO MSC 35Q5583F0583C15 PACS 98.80.Cq

keywords nonlinearSchrödingerformulationscalarfieldcosmologybarotropicfluidsFriedmannequationsexactsolutionsscalefactorErmakov-Pinneyequationcanonical

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that an FRW universe containing a canonical scalar field and two non-interacting barotropic fluids is exactly equivalent to a time-independent nonlinear Schrödinger (NLS) equation, not merely analogous to it. The dictionary is the change of variables $u(x)=a^{-n/2}$ with $\dot{x}=u$, $E=-\kappa^2 n^2 D_1/12$, and an effective potential $P(x)$ that absorbs the scalar kinetic term and the second fluid; in these variables the Friedmann and acceleration equations become $u''+[E-P]u=-(nk/2)u^{(4-n)/n}$. Using this dictionary, the paper converts seven previously known exact NLS wavefunctions into scale factors, Hubble rates, redshifts, and scalar-field potentials, and adds one new exact solution, $u=-e_0\sinh^2(b_0x)$. The resulting cosmologies include oscillatory, de Sitter-like, and power-law expansions, though several require negative or zero densities for the first fluid. The paper concludes that upgrading to the time-dependent NLS equation is the natural route to a more physical quantum-cosmological picture.

What carries the argument

The load-bearing object is the variable change that identifies Friedmann variables with NLS variables: $u(x)=a^{-n/2}$ together with $\dot{x}=u$, $E=-\kappa^2 n^2 D_1/12$, and $P(x)=(\kappa^2 n/4)a^n\epsilon\dot\phi^2+(mD_2/12)\kappa^2 n a^{n-m}$. Substituting these definitions into the Friedmann and acceleration equations reproduces the time-independent NLS equation (15). The role of $P(x)$ is to absorb both the scalar-field kinetic term and the second barotropic fluid while $E$ stays constant; that constancy is what allows the known solution table to be applied unchanged. The relation $x(t)=\int u\,dt$ completes the dictionary, so any NLS wavefunction $u(x)$ can be converted into a scale factor $a(t)=u^{-2/n}$ and then into a Hubble rate, redshift, and scalar potential.

What would settle it

Choose any of the eight solutions, substitute $u(x)$ and its derivatives into (16), and scan the domain for intervals where the right-hand side is negative or where the resulting $V(\phi)$ is multi-valued; the first such interval would show that the solution is not a scalar-field cosmology.

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Extended reading notes

Core claim

The central claim is an exact identity between two dynamical descriptions. The flat or curved Friedmann equations sourced by a canonical scalar field, with density $\rho_\phi=\frac12\epsilon\dot\phi^2+V(\phi)$, and two non-interacting barotropic fluids with densities $D_1/a^n$ and $D_2/a^m$, are equivalent to the stationary nonlinear Schrödinger equation $u''+[E-P]u=-(nk/2)u^{(4-n)/n}$ once one sets $u=a^{-n/2}$, $E=-\kappa^2 n^2 D_1/12$, and $P=(\kappa^2 n/4)a^n\epsilon\dot\phi^2+(mD_2/12)\kappa^2 n a^{n-m}$. Every NLS solution therefore yields a cosmological solution: the scale factor is $a=u^{-2/n}$, and the scalar kinetic term and potential are recovered from $u$ and its derivatives. The paper carries out this translation for seven exact NLS solutions taken from the literature and one new solution, $u=-e_0\sinh^2(b_0x)$, giving explicit $a(t)$, $H(t)$, $z(t)$, and $V(\phi)$ for each. It also observes that all eight NLS wavefunctions are non-normalizable, that the implied first-fluid density is often negative or zero, and that the time-independent formulation should therefore be upgraded to the time-dependent NLS case.

Load-bearing premise

The argument assumes that the scalar field recovered from each NLS solution is a genuine real canonical field: the kinetic term from (16) must be non-negative everywhere and $V(\phi)$ must be single-valued, but none of the eight solutions is checked against that condition.

Editorial extensions

If this is right

Every solution of the stationary NLS equation yields a two-fluid FRW cosmology with scale factor $a=u^{-2/n}$, so the eight listed $u(x)$ forms translate directly into explicit cosmic histories.
The first barotropic fluid is fixed by the constant $E$, while the second fluid enters only through $P(x)$; the same $u(x)$ can therefore be paired with different second-fluid equations of state without changing the wavefunction.
Because all eight NLS states are non-normalizable and have negative total $E$, the stationary wavefunction cannot support a probabilistic quantum-cosmological reading, a limitation the paper itself states.
The paper proposes the time-dependent NLS formulation as the next step, expecting more realistic solutions and deeper physical insight.

Reading between the lines

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

The paper never tests whether the right-hand side of $\epsilon\dot\phi^2=(4/\kappa^2 n)uu''+\dots$ stays non-negative on the whole domain for each listed solution; if it does not, that solution is not a valid canonical scalar-field cosmology and would need a phantom interpretation or rejection.
The second-fluid constants $D_2$ and $m$ drop out of the reconstructed scalar sector, suggesting the dictionary underdetermines the second fluid: an observed cosmology could be matched to several different $m$ values without changing $u(x)$.
The new solution 8 produces the same scale factor, redshift, and Hubble rate as solution 6; a plausible reading is that the two are related by a simple transformation of $x$, making solution 8 a re-expression rather than an independent model.
A direct test of physical relevance would be to compute the scalar-field equation-of-state $w_\phi(t)$ for each solution and compare it with the observed accelerating expansion; most of the listed models are unlikely to survive that comparison.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does what it says: it derives an NLS-type formulation of flat/non-flat FRW cosmology with a canonical scalar field plus two barotropic fluids, and it lists eight exact solutions, one of them new. The central algebra checks out — substituting the dictionary (13)–(14) into the Friedmann equations indeed recovers the NLS equation (15), and the conversion formulas (16)–(27) are internally consistent. The new solution u = −e0 sinh²(b0 x) is correctly identified as an exact solution, even though it reproduces the scale factor and expansion history of Solution 6. The authors are also honest about the limits: they note the solutions are non-normalizable, that the time-independent NLS interpretation is unlikely to give realistic physics, and that the time-dependent generalization is the next step.

The real soft spot is the one the reader flagged: with ϵ = 1, eq. (16) must give a non-negative ϵφ̇² everywhere for the solution to describe a canonical scalar field, and V(φ) must be single-valued. This is never checked. The stress-test example is concrete: for Solution 6, at x = 0 the kinetic term is proportional to c0, so whenever c0 < 0 the solution fails to be a real scalar field there — and c0 < 0 is allowed by the stated conditions. This is not a fatal flaw in the equivalence claim, but it means the blanket statement that every NLS solution yields a cosmological solution outruns what has been verified. Adding the second fluid does not cure it, since the D2 term in (16) is negative. The same sign check needs to be repeated for all eight entries.

The second fluid itself is something of a phantom: its parameters m and D2 appear in P(x), but they cancel from the NLS equation, so the two-fluid extension is close to a relabeling. That is a minor point rather than an error. The citation pattern is fine: the D’Ambroise thesis is clearly the source of the solutions, and the corrections to it are flagged.

Who is this for? Someone working in the NLS–cosmology correspondence who wants the two-fluid dictionary and a compact catalogue of exact solutions. For that audience, the paper is useful and deserves a serious referee, but with a request to add the reality checks before publication. I would not cite it in my own work in the next year, but I would send it to a colleague working on exact cosmological solutions.

Referee Report

3 major / 5 minor

Summary. The paper constructs a time-independent non-linear Schrödinger (NLS) formulation of FRW cosmology with a canonical scalar field and two non-interacting barotropic fluids. It defines u = a^{-n/2}, E = -(κ² n² D1)/12, and P(x) = (κ² n/4) a^n ε φdot² + (m D2/12) κ² n a^{n-m}, and derives the exact equivalence of the Friedmann and acceleration equations to the NLS equation (15). The paper then supplies a dictionary of cosmological quantities in terms of NLS variables, examines eight exact u(x) solutions (seven attributed to D'Ambroise and one presented as new), and computes scale factors, Hubble rates, redshifts, and density parameters for each. It concludes that all listed solutions are non-normalizable and that a time-dependent NLS formulation would be needed for a more realistic description.

Significance. The formal equivalence in §III is derived transparently, and the algebraic dictionary is the main useful contribution: it is an exact map rather than a fit, and the paper does not claim to adjust parameters to observational data. The catalogue of eight solutions with their stated conditions is a compact reference, and the new solution in §IV H is correctly checked against the NLS equation. The principal limitation is physical rather than algebraic: several of the solutions will not define real canonical scalar-field cosmologies unless nontrivial sign and single-valuedness conditions are verified, so the blanket correspondence claimed near Eq. (27) is not yet established. If these checks are carried out, the dictionary would be a reliable tool for constructing two-fluid cosmologies from NLS solutions.

major comments (3)

[§III, Eq. (16) and §IV F] The paper states after Eq. (27) that only ε = 1 is considered, but it never verifies that the right-hand side of Eq. (16) is nonnegative on the whole domain. For Solution 6, u(x) = -e0 cosh²(b0 x), k = 0, E = c0 - 2b0², Eq. (16) at x = 0 gives ε φdot² = 4e0² c0/(κ² n), which is negative whenever c0 < 0, a case allowed by the stated conditions c0 < 2b0² and D1 > 0. Since the D2 term in Eq. (16) is not constrained in sign unless m is restricted, the addition of the second fluid does not automatically repair this. The same nonnegativity check is needed for the other seven entries before the claim that each NLS solution yields a cosmological solution can stand.
[§III, Eqs. (17) and (24)] Even when Eq. (16) is nonnegative, V is first computed as a function of x, while the cosmological formulation requires a potential V(φ). For the dictionary to be valid, the relation between x and φ must be invertible on the relevant domain, or V must be shown to be single-valued as a function of φ. The manuscript does not discuss monotonicity of φ(x) for any of the eight solutions, so the identification of the reconstructed object with a canonical scalar-field potential is not established.
[§III, Eqs. (14) and (16)] Substituting the expression for ε φdot² from Eq. (16) into Eq. (14) makes the terms containing D2 and m cancel identically, so P(x) and equation (15) are independent of the second-fluid parameters. The manuscript does not state this consequence or specify the admissible values of D2 and m; the only constraint on these parameters is the nonnegativity of Eq. (16), which remains unchecked. The freedom in D2 and m therefore cannot be cited as a source of new solutions unless the reality conditions are imposed.

minor comments (5)

[§IV F] The sentence after Eq. (72) reads "Plot of a(t) and Ωφ(z) are in figures 1 and ." and the reference to the second figure is incomplete.
[§III, Eq. (24)] Equation (24) is difficult to parse because of unbalanced parentheses and an undefined ± sign convention; please rewrite it with explicit sign branches.
[§IV H, Eq. (80)] The notation "arcCoth" should be written as "arccoth" for consistency with standard usage elsewhere in the paper.
[§I, Eq. (1)] The introduction refers to the "Ermakov-Penny" equation; the standard spelling is Ermakov-Pinney, which is used in the rest of the paper.
[§IV H] Solution 8 is presented as new but yields the same scale factor, Hubble rate, and redshift as Solution 6; the sense in which it is new should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the NLS-Friedmann correspondence is an exact algebraic equivalence and the listed solutions are external or independently checkable.

full rationale

The derivation is self-contained: Eq. (15) is obtained from Eqs. (9)-(11) by the definitions (13)-(14), and the mapping back to epsilon phi_dot^2, V, rho_phi, etc. in Eqs. (16)-(22) is an algebraic inversion of the same relations. The eight u(x) solutions are either quoted from D'Ambroise's thesis [26], an external author rather than a self-citation, or, for Solution 8, asserted to satisfy the NLS equation; the paper does not fit any parameter to data, nor does it claim observational success, explicitly stating that the plots "do not resemble current observation." The self-citations to Gumjudpai [19,22-25] are contextual, used only to contrast earlier bottom-up ansatz approaches, and are not load-bearing for the derivation. That a solution of the NLS equation yields a Friedmann solution is a mathematical identity in both directions, not a prediction forced by a fit. The unverified positivity of epsilon phi_dot^2 for epsilon = 1 is a physical-validity concern, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the validity of the FLRW system, the NLS-Friedmann correspondence from earlier papers, D'Ambroise's exact solutions, and the unstated assumption that the reconstructed phi and V are real single-valued functions. The solution constants e0, b0, c0, d0 and fluid indices n, m are free inputs with no observational anchor.

free parameters (3)

Solution constants e0, b0, c0, d0 = not fitted
Arbitrary constants from D'Ambroise's exact solutions; signs and magnitudes determine whether the scale factor expands or contracts and whether densities are positive. No data fix them.
Barotropic index n = varies per solution (e.g., n=4 in solutions 1 and 2.2, n=1 in solution 3)
n is fixed by C = (n-4)/n for some solutions and remains arbitrary for others; it controls the equation of state of the first fluid.
Second fluid parameters m and D2 = not fitted
Enter eqs (16)-(17) and cancel from the NLS equation; effectively free model inputs.

assumptions (5)

domain assumption FLRW equations with minimally coupled canonical scalar field and two non-interacting barotropic perfect fluids accurately model the universe
Equations (9)-(11) are the starting point and are not justified in the paper.
domain assumption The NLS-Friedmann correspondence of Hawkins-Lidsey and D'Ambroise-Williams is correct
The mapping u = a^-n/2, E = -kappa^2 n^2 D1/12, and eq (15) is built on cited prior work [7,17,18].
domain assumption The seven exact solutions in Table I from D'Ambroise's thesis are correct
The paper quotes them without derivation; errors would propagate into the cosmological interpretations.
ad hoc to paper The reconstructed V(phi) and phi-dot^2 define a valid canonical scalar field
The paper assumes epsilon = 1 and never checks phi-dot^2 >= 0 or that V is single-valued in phi; this is needed for the solutions to be physical.
ad hoc to paper The scale factor a(t) = u^-2/n is real for all t
For negative u and non-integer n, u^-2/n can be complex; the paper does not discuss branch choices.

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Cite this review

Pith. "Pith review of Non-linear Schr\"{o}dinger-type formulation of scalar field cosmology: two barotropic fluids and exact solutions." pith.science (2026). https://pith.science/paper/DJ3ZRXAL

@misc{pith2026190811265,
  author       = {Pith},
  title        = {Pith review of: Non-linear Schr\"odinger-type formulation of scalar field cosmology: two barotropic fluids and exact solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJ3ZRXAL}},
  note         = {Machine review of arXiv:1908.11265}
}

read the original abstract

Time-independent non-linear Schr\"{o}dinger-type (NLS) formulation of FRW cosmology with canonical scalar field are considered in case of two barotropic fluids. We derived Friedmann formulation variables in terms of NLS variables. Seven exact solutions found by D'Ambroise \cite{DAmbroise:2010dgl} and one new found solution are explored and tested in cosmology. The result suggests that time-independent NLS formulation of cosmology case should be upgraded to the time-dependent case.

Figures

Figures reproduced from arXiv: 1908.11265 by the authors.

**Figure 2.** FIG. 2: Scalar field density parameter Ω [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Taylor expansion of the solution (64) is u(x) =−e0 [ 1 +b2 0(x−x0)2 + b4 0 3 (x−x0)4 +

The results are x(t) = 1 b0 arctanh [−e0b0(t−t0)] and u(t) = −e0 1−e2 0b2 0(t−t0)2 , (65) and the scale factor, redshift and Hubble rate are a(t) = [1−e2 0b2 0(t−t0)2 −e0 ]2/n , a (z) = 1 e2/n 0 (z + 1) (66) z(t) = 1 [1−e2 0b2 0(t−t0)2]2/n− 1, (67) H(t) = −4 n [ e2 0b2 0(t−t0) 1−e2 0b2 0(t−t0)2 ] , (68) H(z) = −4 n|e0||b0| √ z(z + 1), (69) where n⁄= 0. Ta...

work page

[2] [2]

Kazanas, Astrophys

D. Kazanas, Astrophys. J. 241, L59 (1980); A. A. Starobinsky, Phys. Lett. B91, 99 (1980); A. H. Guth, Phys. Rev. D 23, 347 (1981); K. Sato, Mon. Not. Roy. Astro. Soc. 195, 467 (1981); A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982); A. D. Linde, Phys. Lett. B108, 389 (1982)

work page 1980

[3] [3]

Masi et al

S. Masi et al. , Prog. Part. Nucl. Phys. 48, 243 (2002); C. L. Bennett et al. , Astrophys. J. Suppl. 148, 1 (2003); WMAP Collab. (D. N. Spergel et al. ), Astrophys. J. Suppl. 148, 175 (2003); SDSS Collab. (R. Scranton et al. ), arXiv: astro- ph/0307335; Supernova Search Team Collab. (A. G. Riess et al. ), Astron. J. 116, 1009 (1998); SNLS Collab. (P. Asti...

work page arXiv 2002

[4] [4]

Padmanabhan, Curr

T. Padmanabhan, Curr. Sci. 88, 1057 (2005); E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006); T. Padmanabhan, Dark energy: mystery of the millennium, in AIP Conf. Proc. Albert Einstein’s Century Int. Conf., eds. J.-M. Alimi and A. F¨ uzfa (AIP, New York, 2006),861 p. 179

work page 2005

[5] [5]

Nojiri and S

S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)

2007

[6] [6]

H. R. Lewis, J. Math. Phys. 9, 1976 (1968); J. R. Ray and J. L. Reid, Phys. Lett. A71, 317 (1979)

work page 1968

[7] [7]

V. P. Ermakov, Univ. Izv. Kiev 20, 1 (1880)

work page

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Pinney, Proc

E. Pinney, Proc. Amer. Math. Soc. 1, 681 (1950)

work page 1950