Quantum Cosmology with many fluids and the choice of cosmological time

C. G. M. Santos; E. V. Corr\^ea Silva; F. G. Alvarenga; G. A. Monerat; G. Oliveira-Neto; R. Fracalossi; S. V. B. Gon\c{c}alves

REVIEW 3 major objections 6 minor 35 references

Quantum Cosmology with many fluids and the choice of cosmological time

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that in a quantized closed universe filled with five barotropic fluids, the vacuum fluid is the only viable choice of physical time because it alone produces a tunneling transition from the Planck era to classical…

desk verdict A competent five-fluid minisuperspace construction whose vacuum-time conclusion rests on a tunneling probability that is not actually a tunneling probability. read the letter →

arxiv 1908.05337 v2 pith:I6S5WTPC submitted 2019-08-14 gr-qc astro-ph.CO

G. A. Monerat , C. G. M. Santos , F. G. Alvarenga , S. V. B. Gonçalves , R. Fracalossi , E. V. Corrêa Silva , G. Oliveira-Neto This is my paper · ORCID

classification gr-qcastro-ph.CO PACS 98.80.-k98.80.Cq04.30.-w

keywords quantumcosmologyWheeler-DeWittequationSchutzformalismbarotropicfluidsvacuumenergytunnelingproblemoftimescalefactor

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

Quantum cosmology inherits the problem of time: the Wheeler–DeWitt equation does not contain a time parameter, so one must choose a material fluid to play that role. This paper quantizes a closed Friedmann–Lemaître–Robertson–Walker universe filled with five non-interacting barotropic fluids—radiation, dust, cosmic strings, domain walls, and vacuum—and asks which fluid, used as the time variable, gives the most physically sensible primordial evolution. For the first four, the effective potentials are confining, producing bound states and oscillating, non-singular scale factors with no exit to a classical phase. For the vacuum, the effective potential is a small barrier, and the numerically evolved wave packet tunnels through it with probability close to one, after which the scale factor expands. The paper concludes that vacuum is the best candidate for cosmological time because it is the only one that provides a mechanism for the universe to emerge from the Planck era into classical expansion.

What carries the argument

The key machinery is the Schutz velocity-potential formalism for perfect fluids, which makes the conjugate momentum of a fluid potential appear linearly in the super-Hamiltonian, so that after canonical quantization one obtains a time-dependent Schrödinger equation with the fluid variable as time. For the vacuum ($\omega = -1$) case, the equation initially has a time-derivative term multiplied by $a^4$, but a canonical change of variable to $x$ recasts it as a standard Schrödinger equation with an effective potential $V_{\rm ef}(x)$ that has a small barrier. The numerical workhorse is the Crank–Nicolson finite-difference method, used to evolve the initial packet and to compute the tunneling probability $TP = \int_{x_2}^{x_f} |\Psi(x,t_f)|^2 dx \,/\, \int_0^{x_f} |\Psi(x,t_f)|^2 dx$, where $x_2$ is the right turning point of the barrier.

What would settle it

Evolve the same initial packet with its center shifted from $x \approx 0.189$ to a position clearly to the left of the barrier turning point $x_1 \approx 0.095$, and compute the transmitted fraction $\int_{x_2}^{65} |\Psi|^2 dx$ at $t_f = 10$; if this fraction falls to the order of the WKB values in Table 2, the near-unity tunneling probability is an artifact of launching the packet inside the forbidden region.

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Extended reading notes

Core claim

Within the Schutz formalism, each fluid choice turns the Wheeler–DeWitt equation into a Schrödinger equation with that fluid's canonical momentum as time. The paper finds that radiation, dust, cosmic strings, and domain walls all yield bound-state spectra; their wave packets are normalizable, remain well-defined at $a \to 0$, and give expectation values of the scale factor that oscillate inside the classical oscillation region. The vacuum case is qualitatively different: after a canonical transformation to a new variable $x$, the effective potential becomes a small potential barrier rather than a well. Evolving a wave packet with energy $E_m = 3.5$ just below the barrier top through the Crank–Nicolson method gives a tunneling probability $TP \approx 1$ (and $TP \geq 0.98$ for $E_m \geq 0.5$), compared with WKB estimates below 0.6. The expected scale factor contracts slightly, never vanishes, and then expands, matching the classical trajectory after tunneling. The paper's central conclusion is that vacuum is the only one of the five fluids that provides a quantum-to-classical transition, and therefore the best choice for the role of time.

Load-bearing premise

The near-unity tunneling probability rests on the assumption that the initial wave packet (43), with $E_m = 3.5$, is the correct pre-tunneling state, even though its maximum lies inside the potential barrier rather than in the classically allowed region to the left.

Editorial extensions

If this is right

If the vacuum-fluid time choice is correct, the universe emerges from the Planck era by tunneling through a small potential barrier into a classically expanding state, with no singularity at $a = 0$.
The near-unity tunneling probability ($TP \geq 0.998$ for $E_m \geq 0.6$) means the quantum-to-classical transition is essentially certain for a wide range of initial energies below the barrier top.
For radiation, dust, cosmic strings, and domain walls, the quantized universe would oscillate forever within a bounded range of scale factors, with no transition to a classical expanding phase, so those fluids are poor time variables.
The expected value of the scale factor in the vacuum case is always nonzero, providing a singularity-free description of the primordial universe.
The vacuum case yields a continuous energy spectrum, in contrast to the discrete bound states of the other fluids, which is why a different numerical method (Crank–Nicolson) is required.

Reading between the lines

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

If the vacuum-time result is taken at face value, the model predicts that the universe's classical expansion begins only after a tunneling event, so the Planck era is connected to the classical phase by a continuous quantum process with no singularity; this is a concrete realization of the tunneling proposal in a multi-fluid setting.
The four non-vacuum fluid-time choices produce bouncing, oscillating universes; a natural next test is whether any of those oscillations could be observationally distinguished from the standard expanding history, which would strengthen or weaken the case for vacuum time.
The computation's sensitivity to the initial packet placement suggests a direct extension: repeat the evolution with packets of different widths and centers, and compare the transmitted fraction with the WKB formula, to map the conditions under which $TP \approx 1$ actually holds.

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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

Here's my read. The genuinely new piece is the five-fluid Hamiltonian and the per-fluid effective potentials. That part is a straightforward but real extension of the one- and two-fluid Schutz-formalism papers, and the paper is transparent about its methods. The radiation, dust, cosmic strings, and domain walls cases are handled competently: the wave packets stay finite, the expectation values do not vanish at a=0, and the comparison with classical trajectories is reasonable. Credit where it is due: the numerical work is reproducible in principle, and the paper does not oversell the non-vacuum cases.

The trouble is the vacuum tunneling claim, which is the whole reason for the paper. The initial packet (43) with Em=3.5 has its maximum at x≈0.189, which is inside the classically forbidden region between the turning points x1≈0.095 and x2≈0.222. So the "tunneling probability" from Eq. (44) is just the fraction of the norm to the right of x2 at a fixed final time. At t=0, roughly 40% of the packet is already to the right of x2. That is not barrier transmission; it is initial probability plus spreading plus possibly reflection off the artificial boundary at xf=65. The WKB column in Table 2 tells the same story: it falls from 0.6 to 2e-9 while the numerical TP stays near one for Em≥1. The two are measuring different things.

There are smaller problems too. The fluid energies Ev, Ed, Ecs, Edw, Er are picked by hand; no sensitivity analysis is given. The paper's own Table 2 shows TP drops steeply below Em≈0.5, so the near-unity result is an artifact of choosing Em close to the barrier maximum. And the abstract promises a phenomenological selection of the time variable, but there is no observational comparison anywhere.

So: the construction is worth knowing, but the central conclusion—vacuum is the best time variable because it gives a quantum-to-classical tunneling transition—does not follow from the calculation as presented. I would not cite it as evidence for the tunneling mechanism. I would send it to peer review rather than desk reject, because the five-fluid setup deserves a serious referee, but I would expect the vacuum claim to be revised or substantially qualified.

Referee Report

3 major / 6 minor

Summary. The paper quantizes a closed FLRW minisuperspace model with five non-interacting barotropic fluids (radiation, dust, cosmic strings, domain walls, vacuum) in Schutz's formalism. For each fluid chosen as the time variable it obtains a reduced dynamics, solves the bound-state problems for four fluids with the Galerkin spectral method and the vacuum case with a Crank-Nicolson finite-difference scheme, and compares quantum expectation values with classical trajectories. Its central claim is that the vacuum fluid is the best time variable because it is the only one that gives a quantum-to-classical transition via tunneling, with probability near unity.

Significance. The extension from one- or two-fluid models to five fluids is a useful exercise, and the spectral data in Table 1 plus the numerical details (N, dt, xf) make the calculations reproducible. If the tunneling claim were correct, the paper would offer a concrete phenomenological criterion for choosing a fluid time variable in quantum cosmology. As it stands, however, the main conclusion depends on a misidentification of the quantity in Eq. (44) as a tunneling probability and on hand-picked energies; those issues are load-bearing rather than cosmetic.

major comments (3)

[Sec. 3.1, Eqs. (43)-(44)] The quantity TP defined in Eq. (44) is not a tunneling probability. The initial packet (43) is real, so it has zero mean momentum and is not an incoming wave from the left; for the value Em=3.5 used in the main calculation its maximum lies at x about 0.189, inside the forbidden interval x1=0.0953, x2=0.2223 given in Table 2, with a substantial part of the initial normalization already to the right of x2. Equation (44) then measures the fraction of probability in the classically allowed region at the fixed time tf=10, which combines initially transmitted probability, spreading from inside the barrier, and possible boundary effects at xf=65, rather than a transmitted flux. Because the conclusion in Sec. 4 that the vacuum is the unique time variable with a tunneling transition rests entirely on TP, the central claim is unsupported.
[Table 2 and Sec. 3.1, parameter values] The near-unity result is selected by the parameter choices rather than derived. The barrier in Eq. (42) is built from hand-assigned fluid energies Ev=-0.0017, Ed=1/24, Edw=1/24, Er=1/24 and Ecs=5.75, and the initial packet energy is chosen as Em=3.5, just below Vef(xmax)=3.803. The same Table 2 shows that TP drops to 0.181 at Em=0.3 and to 4.33e-12 at Em=0.1, while the WKB column changes by eight orders of magnitude over the same range. The agreement of TP with the WKB transmission coefficient is also poor (TP=1.000 versus 0.599 at Em=3.5), reflecting that the two quantities measure different processes. A robust claim about 'tunneling probability near one' needs a physical justification for Em and for the fluid energies, not a regime chosen after the fact.
[Sec. 3.3, Eq. (50) and Fig. 13] The matching of the quantum expectation value to the classical trajectory is not established. Eq. (50) gives x(9.95)=13.03390565 and x'(9.95)=2.763117600, while the caption of Fig. 13 states x(10)=65 and x'(10)=2.763117600; these initial data are mutually inconsistent and the value 65 coincides with the numerical boundary xf. The paper does not explain how the classical initial data are obtained from the wave packet or why a matching at a single time at the boundary is sufficient to demonstrate that the universe 'emerges classically' after tunneling. This weakens the comparison with classical dynamics that supports the final conclusion.

minor comments (6)

[Sec. 3.1] The sentence 'According to the effective potential described by Eq. (24)' should refer to Eq. (42), since Eq. (24) is the cosmic-strings potential in a, not the vacuum potential in x.
[Eq. (22)] The right-hand side of Eq. (22) contains partial derivatives with respect to t_j, but the left-hand side is written with t_m; the notation should be made consistent by using t_m on both sides.
[Abstract and Sec. 1] The formalism name appears as 'Schultz' in the Abstract and Sec. 1; it should be 'Schutz', as used in the references and the rest of the text.
[Sec. 2, Galerkin calculations] The choice of L=15 or L=5 in the Galerkin calculations is not justified, and the dependence of the eigenvalues in Table 1 on L is not discussed; this matters because the energy eigenvalues enter the classical momentum assignments.
[Sec. 3.2] The statement that <x>(t) is nonzero 'for all t' is inferred from a numerical evolution to tf=10 in a finite box; the paper should state that this is a numerical result and not an analytic proof of singularity avoidance.
[Sec. 3.1, Eq. (46)] The energy E appearing in Eq. (46) is never defined explicitly; the authors should state whether it is the initial packet energy Em and explain how the WKB values in Table 2 are obtained from the expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the tunneling-probability result is parameter-sensitive and the TP measure is contestable, but no prediction reduces to its inputs by construction or via a self-citation chain.

full rationale

The derivation chain is self-contained in the sense required: the Hamiltonian (15)-(17), the Wheeler-DeWitt equation (20), and the five time-dependent Schrödinger equations (22)-(42) are obtained from the Schutz action by stated substitutions and canonical transformations, not by importing the conclusion. The canonical transformation (39) and the Crank-Nicolson implementation are taken from earlier works that include the present authors, but the transformation is stated explicitly and is algebraically checkable, so the self-citation is not load-bearing in a circular way. The tunneling probability formula (44) is also quoted from [32], yet it is written out in full; the numerical values in Table 2 follow from that formula, the potential (42), and the initial packet (43), rather than from an unexamined self-cited theorem. The main scientific weakness is that Eq. (44) is a right-of-barrier fraction at a single final time, and for Em=3.5 the packet (43) is centered inside the forbidden region, so the reported TP≈1 is not a standard barrier-transmission probability and is sensitive to the chosen Em and box size. That is a correctness and interpretation concern about what is being computed, not a circular reduction of the prediction to the input; the paper's conclusion is not forced by a definition, by fitting to data, or by a self-citation chain. Accordingly, the circularity score is 0.

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

The central claim depends on hand-picked fluid momenta and initial packet choices. The only structural input from the standard formalism is Schutz's fluid time; the rest are modeling choices that steer the result.

free parameters (5)

Ev (vacuum momentum p_Tv) = -0.0017
Chosen by hand in all five-fluid sections and in the vacuum potential; controls the a^4 term and the barrier height.
Ed, Ecs, Edw, Er (other fluid energies) = 1/24 each
Assigned equal arbitrary values in the examples; no physical or observational justification is given.
Ecs = pTcs for the vacuum potential = 5.75
Taken as the mean of ten cosmic-string eigenvalues, but that spectrum itself depends on the arbitrary Section 2.1 parameters; this value strongly shapes the vacuum barrier.
Em (initial wave-packet energy) = 3.5
Parameter in Eq. (43); chosen close to the barrier maximum, which makes the computed tunneling probability nearly one.
Numerical box and truncation = L=15 (or 5), xf=65, N=4500, dt=0.05, tf=10, 10 lowest states
Approximations used for the spectral and finite-difference solutions; convergence is not demonstrated.

assumptions (5)

domain assumption Schutz's velocity-potential formalism correctly describes a barotropic perfect fluid and provides a canonical time variable.
Basis for the whole construction, taken from references [4,5].
domain assumption All five fluids are non-interacting and each obeys p = omega*rho with constant omega.
Used in the action (2) and Hamiltonian (17); a standard cosmological simplification.
ad hoc to paper For the fluid not chosen as time, its momentum is an unquantized constant with a continuum spectrum whose value can be assigned by hand.
Invoked in Sections 2.1-2.4 and 3; the values are not derived, and this is where the free parameters enter.
domain assumption The Wheeler-DeWitt equation with weight a^(1-3*omega) can be solved by the Galerkin method on a finite interval with boundary conditions Psi(0)=Psi(L)=0, and ten states suffice.
Used for all bound-state cases; no convergence proof is provided.
ad hoc to paper The initial packet (43), placed inside or at the barrier with energy Em below Vmax, represents quantum-to-classical tunneling, and TP as defined by Eq. (44) measures tunneling probability.
Load-bearing for the vacuum conclusion; the packet starts in the forbidden region, so this is not a standard scattering setup.

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Cite this review

Pith. "Pith review of Quantum Cosmology with many fluids and the choice of cosmological time." pith.science (2026). https://pith.science/paper/I6S5WTPC

@misc{pith2026190805337,
  author       = {Pith},
  title        = {Pith review of: Quantum Cosmology with many fluids and the choice of cosmological time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6S5WTPC}},
  note         = {Machine review of arXiv:1908.05337}
}

read the original abstract

In this work we propose the quantization of a cosmological model describing the primordial universe filled with five barotropic fluids, namely: radiation, dust, vacuum, cosmic strings and domain walls. We intend to identify which fluid is best suited to provide phenomenologically the temporal variable in accordance with the observable universe. Through the Galerkin spectral method and the finite difference method in the Crank-Nicolson scheme (vacuum case), the quantum cosmological solutions are obtained and compared. We, also, compare the quantum cosmological solutions with the corresponding classical ones. The vacuum case is especially interesting because it provides a tunneling transition mechanism from the quantum to the classical phase and the possibility of calculating quantum tunneling probabilities.

Figures

Figures reproduced from arXiv: 1908.05337 by the authors.

**Figure 2.** classical scale factor behavior for the corresponding model to the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Cosmological solutions for the case of a domain walls fluid. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: classical scale factor behavior for the corresponding model to the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Cosmological solutions for the case of a radiation fluid. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: classical scale factor behavior for the corresponding model to the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Cosmological solutions for the dust case. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: classical scale factor behavior for the corresponding model to the [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Effective potential Vef(a) for the vacuum case (ωv = −1). Here we consider pTv = −0.0017, pTd = 1/24, pTdw = 1/24, pTcs = 5.75, pTr = 1/24. The effective potential in the x variable will take the form of a small potential barrier as shown in [PITH_FULL_IMAGE:figures/f…

**Figure 10.** Figure 10: Effective potential Vef(x) for the vacuum case (ωv = −1). Here we consider k = 1, pTv = −0.0017, pTd = 1/24, pTdw = 1/24, pTcs = 5.75, pTr = 1/24. As initial condition have chosen the normalized wave function Ψ(x, 0) = 8192E3 m π !1/4 x e−4Em x 2 , (43) which depends …

**Figure 11.** Figure 11: |Ψ(x, tmax)| 2 = ρ for Em = 3.5 and tf = 10 when Ψ reaches the numerical infinity in xf = 65. The case of the choice of vacuum fluid for the role of time differs from other cases in certain respects. Here the energy spectrum is not discrete, providing the possibility …

**Figure 12.** Figure 12: Temporal evolution of the expected value of the universe, con [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

**Figure 13.** Figure 13: Classical evolution of the scale factor for the following initial [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

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