A mechanism for formation of Bose-Einstein condensation in cosmology

Kemal G\"ultekin; Recai Erdem

REVIEW 3 major objections 4 minor 2 cited by

A mechanism for formation of Bose-Einstein condensation in cosmology

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper argues that cosmic expansion itself can drive the relative momentum of the particles produced in χχ→φφ scattering to zero, producing the zero-momentum phase-space peak characteristic of Bose-Einstein condensation without any…

desk verdict A clever toy mechanism for scalar BEC in cosmology, but the central delta-function derivation has a mathematical flaw; worth a careful referee, not a desk reject. read the letter →

arxiv 1908.08784 v2 pith:MDRYJ6P2 submitted 2019-08-23 gr-qc astro-ph.COhep-phhep-th

Recai Erdem , Kemal Gültekin This is my paper · ORCID

classification gr-qcastro-ph.COhep-phhep-th PACS 04.62.+v98.80.Cq95.35.+d

keywords Bose-EinsteincondensationcosmologyscalarfieldstrilinearcouplingRobertson-Walkerspacetimekinetictheoryparticleproductiondarkmatter

topics Dark Matter

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

The paper introduces a toy cosmology with two scalar fields, a trilinear $\varphi^2\chi$ interaction, and initially only relativistic $\chi$ particles. It claims that in an expanding Robertson-Walker spacetime the production of non-relativistic $\varphi$ particles through $\chi\chi\to\varphi\varphi$ is progressively pushed toward zero relative momentum: the effective mass gap between the two species grows with the scale factor while comoving momenta stay fixed, so energy conservation leaves less and less room for the final-state momentum. The authors show that, under conditions making each collision effectively Minkowskian, the $\varphi$ phase-space distribution develops a delta-function peak at zero comoving momentum, the de Broglie-wavelength overlap condition is satisfied, and the $\varphi$ number density grows to finite values. They explicitly frame the result as a tendency toward Bose-Einstein condensation rather than a proof of full condensate formation at late times.

What carries the argument

The central object is the comoving relative momentum $\tilde{\mathbf{k}}$ of the two outgoing $\varphi$ particles, governed by energy conservation $|\tilde{\mathbf{p}}|^2-|\tilde{\mathbf{k}}|^2=\tilde m_\varphi^2-\tilde m_\chi^2$ and by the effective mass gap $\tilde m_\varphi^2-\tilde m_\chi^2=a^2(m_\varphi^2-m_\chi^2)$ in the conformally rescaled theory. Since $\tilde{\mathbf{k}}$ is comoving while the gap grows with $a$, the expansion squeezes the allowed final relative momentum toward zero; the calculation turns the sharp cutoff in the initial $\chi$ distribution into a delta-function peak at zero final momentum (Eq. 21). The supporting machinery is the piecewise-Minkowski decomposition: conditions (5) and (6) keep effective masses and the coupling nearly constant over one collision time, so the field expansion (Eq. 7), the Boltzmann equation (Eq. 9), and the tree-level matrix element (Eq. 11) can be applied interval by interval.

What would settle it

Numerically solve the full Boltzmann equation (Eq. 9) with the time-dependent effective masses of Eq. (4) and no piecewise-Minkowski approximation, starting from the $\chi$ distribution (Eq. 16); if the $\varphi$ phase-space density does not develop a growing peak at zero comoving momentum as the scale factor increases, the claimed mechanism is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a dynamical route to the pre-conditions of Bose-Einstein condensation. For a $\chi\chi\to\varphi\varphi$ event in the center-of-mass frame, energy conservation reads $|\tilde{\mathbf{p}}|^2 - |\tilde{\mathbf{k}}|^2 = \tilde m_\varphi^2 - \tilde m_\chi^2$, and the effective mass gap equals $a^2(m_\varphi^2-m_\chi^2)$ in the conformally rescaled theory. Because the comoving momenta are redshift-independent while the scale factor $a$ grows, the allowed final relative momentum $|\tilde{\mathbf{k}}|$ shrinks as the universe expands. The calculation converts the sharp cutoffs in the initial $\chi$ distribution into a $\delta(|\tilde{\mathbf{p}}_4|)$ factor in the production rate (Eq. 21), which the paper identifies as one of the main properties of Bose-Einstein condensation. It also checks the other pre-conditions, overlap of de Broglie wavelengths and a finite produced number density, and stresses that this establishes a tendency, not a proof, of condensation.

Load-bearing premise

The argument assumes each $\chi\chi\to\varphi\varphi$ collision happens in a patch of spacetime that is effectively flat, meaning the effective masses and the coupling stay essentially constant during one process; if cosmic expansion changes them appreciably on the collision timescale, the field expansion, the Boltzmann equation, and the cross section used here are not justified.

Editorial extensions

If this is right

In the tachyon-free cosmological eras with $H=\xi a^{-s}$, the produced $\varphi$ number density $n_\varphi=C_\varphi/a^3$ reaches finite values, so particle production can compete with Hubble dilution rather than being washed out.
The reverse process $\varphi\varphi\to\chi\chi$ is suppressed as the final relative momentum tends to zero, so the early buildup of $\varphi$ particles is not immediately undone by number-changing back reactions.
The de Broglie-wavelength overlap condition $1/|\tilde{\mathbf{p}}_4| > \tilde n_\varphi^{-1/3}$ is automatically satisfied once $|\tilde{\mathbf{k}}|_{\max}\to 0$, giving the long-range correlation needed for condensation.
The same $\varphi^2\chi$ coupling induces $\varphi\varphi\to\varphi\varphi$ scattering, which can act as an effective $\lambda\varphi^4$ self-interaction with a redshift-dependent $\lambda$.

Reading between the lines

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

Editorial inference: this shrinking-$|\tilde{\mathbf{k}}|$ mechanism does not depend on the detailed trilinear form; any two-to-two production channel with $m_\varphi>m_\chi$ in an expanding background has the same energy-balance structure, so zero-momentum peaking may be a generic feature of massive-particle production in cosmology.
Editorial inference: the piecewise-Minkowski condition that enables the calculation also marks its own limit, because once $|\tilde{\mathbf{k}}|\to0$ the produced particles behave as a single coherent object and single-particle scattering amplitudes are no longer the right description; a complete theory of condensate formation would have to switch frameworks exactly at the condensate threshold.
Editorial inference: a numerical integration of the full kinetic equation with Bose enhancement and back reactions included is the natural next test; the paper notes this has not been done, and such a solution would show whether the delta-function tendency survives to late times or is modified by interactions.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: the paper proposes a concrete microscopic route—trilinear φ²χ coupling—for producing low-momentum scalar particles in an expanding universe, which could feed BEC dark matter/inflation models. That is genuinely new, and the authors are honest that they only show a tendency, not formation. But the load-bearing mathematical step that turns this into a 'delta function at zero momentum' does not survive scrutiny.

What's good: the kinematic observation is solid. In the effective Minkowskian approximation, energy conservation gives |p̃|²−|k̃|² = a²(m_φ²−m_χ²). As the universe expands, for fixed comoving χ momentum, the produced φ relative momentum |k̃| decreases. That is a real effect, and the paper correctly identifies it as a precursor to condensation. The Boltzmann treatment at early times (ignoring Bose enhancement) is standard, and the authors explicitly flag that late-time behavior requires solving the full kinetic equation. Conditions (5) and (6) for piecewise-Minkowski validity are at least plausible, and they check a phenomenologically relevant parameter space. The paper is clearly written and well referenced.

The soft spot is serious. Equation (21) claims a delta function in the φ distribution at zero momentum, obtained from Eq. (17) by taking |k̃|_max→0. But Eq. (17) has the denominator squared: it's [Θ(y)−Θ(x)]/(y−x)², not [Θ(y)−Θ(x)]/(y−x). The limit is divergent, not a delta. The formal step in the paper is algebraically inconsistent. So the specific 'condensation signature' in (21) is not established. This matters because the rest of Section III.A leans on it. To be fair, the tendency toward zero momentum follows already from (15), and the paper says (21) is only a hint, not a proof. But the hint as written is wrong.

Other, smaller issues: the top-hat initial distribution is a strong assumption, tied to a particular inflationary perturbation picture; the cross-section estimate is rough; and the title overclaims 'formation' when the text says 'tendency.' These are minor relative to the delta-function problem.

In sum: the paper is not sound as a derivation, but it has a new idea and an honest framework. A serious referee could help the authors fix the delta-function claim and turn this into a useful contribution. I'd send it to review, with instructions to focus on Section III.A.

Referee Report

3 major / 4 minor

Summary. The paper proposes a toy model of two minimally coupled scalars, φ and χ, with a trilinear interaction φ²χ in a flat FLRW background. It studies the process χχ → φφ, assuming an initial population of relativistic χ particles and no φ particles. The authors impose conditions (5) and (6) so that the expanding spacetime can be treated as piecewise Minkowskian, allowing standard perturbative QFT and a Boltzmann equation (9). The central claim is that, because the effective mass difference m̃_φ²−m̃_χ² grows with the scale factor, the final-state relative momentum |k̃| in χχ → φφ decreases in time, and the φ phase-space distribution develops a delta function at zero momentum, Eq. (21), which is called one of the main properties of Bose-Einstein condensation. The paper also derives an approximate number-density evolution for φ and argues that the produced φ density can reach finite values for a range of cosmological expansion histories.

Significance. If correct, the mechanism would offer a microscopic, particle-physics description of how a scalar field could develop a condensate in an expanding universe, with possible relevance to scalar-field dark matter, dark energy, and inflation. The paper is honest about its limits, repeatedly stating that it provides a tendency or hint rather than a proof of condensation, and it does identify a phenomenologically relevant parameter region for the effective-Minkowski approximation. These are strengths. However, the mathematical step that produces the claimed delta-function signature, Eq. (21), is invalid, and the remaining kinematic argument is too weak to support the central conclusion. The significance of the paper therefore rests on a load-bearing error, and the advertised mechanism is not established by the presented derivation.

major comments (3)

[Sec. III.A, Eqs. (17) and (21)] The limit leading to the central delta-function result is mathematically incorrect. Writing x = √(k̃_max² + Δ) and y = √Δ with Δ = m̃_φ² − m̃_χ² > 0, for x > y the step-function bracket in Eq. (17) equals 1, so the right-hand side is ñ_χ² B(|p̃_4|)/(x − y)². As k̃_max → 0, x − y ≈ k̃_max²/(2√Δ), and the expression diverges as 4Δ ñ_χ² B(|p̃_4|)/k̃_max⁴. The identity lim_{x→y} (Θ(y)−Θ(x))/(y−x) = δ(x) invoked after Eq. (21) would require the denominator in Eq. (17) to be first order in (y−x), not second order. In addition, the delta argument in Eq. (21), g(p̃_4) = √(p̃_4²/4 + Δ) − √Δ, has g′(0) = 0, so δ(g(p̃_4)) is not a locally integrable distribution and cannot represent a phase-space density. Thus Eq. (21) does not follow from Eq. (17), and the claimed BEC accumulation at p̃_4 = 0 is not established.
[Sec. III.A, Eq. (15) and surrounding text] The only remaining kinematic argument for a tendency toward zero momentum is Eq. (15), which states that for a fixed comoving initial momentum |p̃| the final relative momentum |k̃| decreases as the scale factor grows. This describes a single collision and does not by itself imply that the phase-space distribution f̃_φ develops a singular peak. The singular behavior claimed in Eq. (21) is instead an artifact of taking the particular top-hat ansatz (16) to a threshold limit with ñ_χ held fixed; it is not an emergent property of the dynamics. Without a valid derivation of the distribution near p̃_4 = 0, the conclusion that the system reaches coherence about |p̃_4| = 0 is unsupported.
[Sec. III.B, Eqs. (31)–(40)] The number-density integration extends the initial-time rate (24)–(28) to arbitrarily late times without accounting for the kinematic threshold. Since the effective mass difference m̃_φ² − m̃_χ² = a²(m_φ² − m_χ²) grows with a, a χ particle with fixed comoving momentum |p̃| eventually satisfies |p̃|² < m̃_φ² − m̃_χ² and the process χχ → φφ ceases. Equations (39)–(40) therefore cannot support the claimed saturation of C_φ at C_1; the conversion shuts off before that point unless the χ distribution is replenished. The authors acknowledge neglect of statistics and backreaction, but not this threshold cutoff.

minor comments (4)

[Eq. (16)] The first factor [Θ(|p̃|_max − |p̃|_min) − Θ(|p̃|_min − |p̃|_max)] is identically 1 under the stated assumption |p̃|_max > |p̃|_min; it can be removed for clarity.
[Eq. (21)] The absolute-value bars inside the delta function make the argument ambiguous; it should be written unambiguously as δ(√(p̃_4²/4 + Δ) − √Δ) with Δ = m̃_φ² − m̃_χ².
[Title and Abstract] The title and abstract state formation of Bose-Einstein condensation, while the text repeatedly emphasizes that only a tendency or pre-condensation is shown; the wording should be aligned with the actual strength of the result.
[Sec. III.A, definition of |α|_max] The quantity |α|_max is introduced in the text after Eq. (20) but is not defined precisely before being used in the limit; a definition such as α_max = max{|α| : |p̃_3| = |α||p̃_4|, energy conservation satisfied} would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

Delta-function BEC signal in Eq. (21) is the limiting form of the paper's own top-hat ansatz and is admitted to reiterate Eq. (15).

self definitional [Section III.A, Eqs. (16), (17), (21) and surrounding text]
"However we keep that term since it makes (16) an even function in an analytical way, this, in turn, makes the identification of the delta function in (21) easier as we shall see. ... Eq.(21) implies that the system reaches coherence about |⃗˜p4| = 0 by time which is one of the main properties of Bose-Einstein condensation. ... In fact, essentially, (15) also shows evolution towards |⃗˜p4| = 0. (21) reiterates and reinforces this result."

The claimed BEC signature is the delta-function limit in Eq. (21). That limit is not obtained by solving the late-time dynamics; it is the limiting form of the paper's own top-hat ansatz (16) after fixing |p̃|max = √(|k̃|max² + Δ) and |p̃|min = √Δ with Δ = m̃φ²−m̃χ². As a(t) grows, Δ grows, so |k̃|max→0 and the assumed initial χ distribution collapses to the χχ→φφ threshold shell; energy conservation (15) then puts the produced φs at zero momentum. The step-function bracket in Eq. (17) is exactly the bracket written into (16), and the paper explicitly says that bracket was kept to make the delta identification in (21) easier. It then concedes that (15) already shows evolution toward |p̃4|=0 and that (21) reiterates it.

full rationale

The main circular element is confined to the claim that Eq. (21) demonstrates a BEC-relevant delta-function peak. Eq. (17) is not solved forward; it is the initial-time Boltzmann equation with f3=f4=0. Eq. (21) is obtained by taking |k̃|max → 0 and formally applying lim_{x→y}(Θ(y)−Θ(x))/(y−x)=δ(x) to the prefactor that came from the chosen top-hat (16). Because the lower edge of the top-hat is fixed at |p̃|min=√Δ, the approach |k̃|max→0 is exactly the collapse of the assumed initial χ distribution onto the χχ→φφ threshold; energy conservation (15) then forces φ momenta to zero. The paper itself notes that (15) already shows this and that (21) reiterates it, and it labels the result a hint rather than a proof. Thus the BEC 'prediction' is substantially built into the initial ansatz. I do not find load-bearing self-citation: ref. [6] is used only for solving the rate equation, and the top-hat ansatz is supported by external inflation-literature citations, not by the authors' own prior uniqueness claims. There is no fitted parameter called a prediction. Separately, the formal limit in (21) is mathematically questionable, since the denominator in (17) is squared while the delta identity used has a linear denominator; that is a correctness issue rather than a circularity. Overall, the central BEC-signature claim is partially circular by construction, giving score 6.

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

The central claim depends on a network of modeling choices: the piecewise-Minkowski approximation, the specific initial phase-space distribution, and the neglect of back-reaction. These are not fitted to data, so the ledger is dominated by domain assumptions rather than free parameters or invented entities.

free parameters (5)

initial χ momentum range (|p̃|min, |p̃|max) = not fitted; top-hat assumed
The top-hat distribution (16) is chosen by hand to mimic super-horizon inflaton perturbations; the delta-function limit (21) is a direct consequence of this choice.
scale factor exponent s in H = ξ a^{−s} = not fitted; set to 0, 2, 3, etc. for different eras
The time-dependence of the effective masses, and hence the rate at which |k̃| shrinks, depends on s.
ratio μ̃/m̃φ = taken ≪ 1, e.g., 0.1
Controls the cross section (30) and the validity of perturbative QFT; chosen for the order-of-magnitude estimates.
mass ratio m̃χ/m̃φ = taken ≪ 1
Required for the approximation in the cross section and for the kinematic equation (15).
initial comoving χ density C1 = Cχ(t1) = not fixed
Appears in the solution for nφ (Eqs. 34-40); the finite-density result depends on its value.

assumptions (5)

domain assumption Standard perturbative QFT with constant masses and coupling is valid in each short time interval (piecewise Minkowski approximation).
Invoked in Section II when writing the field expansion (7) and using the Minkowski-space Boltzmann equation (9); relies on conditions (5) and (6) being satisfied.
domain assumption The initial χ particles have a top-hat momentum distribution given by (16).
Used to derive Eq. (17) and the delta-function limit (21); the form is justified by analogy to inflaton perturbations, not derived from first principles.
domain assumption The Universe is spatially flat (k=0).
Used in the metric (1) and throughout; consistent with observations but still a modeling choice.
domain assumption The processes φφ→χχ and χφ→χφ are neglected at initial times, and Bose enhancement in the collision term is ignored.
Used in Eq. (17) where f3,f4≈0; the paper states this is valid only at early times.
domain assumption The effective masses are tachyonic-free for the chosen parameter ranges, and the approximation |p̃|² ≫ m̃χ², |k̃|² ≪ m̃φ² holds for the cross section (30).
Needed for the simplified cross section and the number density solution; tachyon-free ranges are discussed in Section III.B and Appendix D.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A mechanism for formation of Bose-Einstein condensation in cosmology." pith.science (2026). https://pith.science/paper/MDRYJ6P2

@misc{pith2026190808784,
  author       = {Pith},
  title        = {Pith review of: A mechanism for formation of Bose-Einstein condensation in cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDRYJ6P2}},
  note         = {Machine review of arXiv:1908.08784}
}

abstract

We introduce a toy model of scalar particles with a trilinear scalar coupling in cosmology. The trilinear coupling $\phi^2\chi$ causes production of non-relativistic $\phi$ particles through the process $\chi\chi\,\rightarrow\,\phi\phi$ where, initially, only relativistic $\chi$ particles are present. We consider the initial times of $\chi\chi\,\rightarrow\,\phi\phi$ and observe that the curved space effects promote formation of Bose-Einstein condensate of $\phi$ particles.

Figures

Figures reproduced from arXiv: 1908.08784 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 7 citations worldwide. Full citation record

Vortex-reconnection energy bounds in Bose-Einstein-condensed and superfluid dark matter halos
astro-ph.GA 2026-07 conditional novelty 6.0 of 10

Vortex reconnections in BEC/superfluid dark matter halos produce dark-sector heating at a rate that is secular but sub-virial for relaxed non-interacting soliton cores, with the dominant uncertainty being the true vor...
Scalar model of effective field theory in curved space
hep-th 2019-08 conditional novelty 4.0 of 10

In a two-scalar model, one-loop diagrams with mixed light and heavy internal lines collapse to local tadpole contributions in the infrared, matching the effective low-energy quartic theory in flat and weakly curved spacetime.

Reference graph

Works this paper leans on

26 extracted references · 20 canonical work pages · cited by 2 Pith papers

[1]

Moreover, although, evolution toward s |⃗˜p4| = 0 is an important indication towards formation of condensation it is not suﬃcient [23]

In other words, what we have shown is not a proof but a hint towa rds formation of a condensate at later times. Moreover, although, evolution toward s |⃗˜p4| = 0 is an important indication towards formation of condensation it is not suﬃcient [23]. I n fact, essentially, (15) also shows evolution towards |⃗˜p4| = 0. (21) reiterates and reinforces this resu...

work page
[2]

Weinberg, Cosmology (Oxford Univ

S. Weinberg, Cosmology (Oxford Univ. Press, New York, 2008)

work page 2008
[3]

Weinberg, The Cosmological Constant Problem , Rev

S. Weinberg, The Cosmological Constant Problem , Rev. Mod. Phys. 61 1 (1989); S. Nobbenhuis, 2006 Categorizing Diﬀerent Approaches to the Cosmological Constan t Problem, Found. Phys. 36, 613 (2006); gr-qc/0411093

arXiv 1989
[4]

Weinberg, et al., Cold dark matter: Controversies on small scales , Proc

D.H. Weinberg, et al., Cold dark matter: Controversies on small scales , Proc. Nat. Acad. Sci. 112, 12249 (2015) P. Bull, et al., Beyond Λ CDM: Problems, solutions, and the road ahead , Physics of the Dark Universe, 12, 56 (2016)

work page 2015
[5]

Fukuyama, M

T. Fukuyama, M. Morikawa, T. Tatekawa, Cosmic structures via Bose-Einstein condensation and its collapse , JCAP 0806, 033 (2008), arXiv:0705.3091

arXiv 2008
[6]

Castellanos, C

E. Castellanos, C. Escamilla-Rivera, A. Macias, D. Nune z, Scalar ﬁeld as a Bose-Einstein condensate?, JCAP, 11, 034 (2014)

work page 2014
[7]

Erdem, Is it possible to obtain accelerated expansion of the univers e through energy transfer between diﬀerent energy densities , Phys

R. Erdem, Is it possible to obtain accelerated expansion of the univers e through energy transfer between diﬀerent energy densities , Phys. Dark Univ. 15, 57 (2017)

work page 2017
[8]

Bose-Einstein condensation of relativistic Scalar Field Dark Matter

L.A. Urena-Lopez, Bose-Einstein condensation of relativistic Scalar Field D ark Matter, JCAP 0901, 014 (2001), arXiv:0806.3093 M.P. Silverman and R.L. Mallet, Cosmic degenerate matter: a possible solution to the problem of missing mass , Gen. Rel. Grav. 18,M.P. Silverman and R.L. Mallet, Dark matter as a cosmic 16 Bose-Einstein condensate and possible sup...

work page Pith review arXiv 2001

Show all 26 references

[9]

Nishiyama, M

M. Nishiyama, M. Morita, M. Morikawa, Bose-Einstein condensation as dark energy and dark matter, preprint, astro-ph/0403571; J. Besprosvany and G. Izquierdo, Coincidence problem with dark energy as a coupled self- interacting Bose-Einstein gas , Class. Quantum Grav. 32, 055015...

2015 arXiv
[10]

Fukuyama, M

T. Fukuyama, M. Morikawa, Stagﬂation: Bose-Einstein condensation in the early unive rse, Phys. Rev. D 80, 063520 (2009), arXiv:0905.0173; S. Das, Int. J. Mod. Phys. D 24, 1544001 (2015), arXiv:1509.02658

2009 arXiv
[11]

Davis, et al., Formation of Bose-Einstein condensates , arXiv:1601.06197

M.J. Davis, et al., Formation of Bose-Einstein condensates , arXiv:1601.06197

arXiv
[12]

K. Zhou, Z. Xu, P. Zhuang, and C. Greiner, Kinetic description of Bose-Einstein condensation with test particle simulations , Phys. Rev. D 96, 014020 (2017), arXiv:1703.02495

2017 arXiv
[13]

Boyanovsky, Condensates and quasiparticles in cosmology: Mass generati on and decay widths, Phys

D. Boyanovsky, Condensates and quasiparticles in cosmology: Mass generati on and decay widths, Phys. Rev. D 85, 123525 (2012), arXiv:1203.3903 J. Lankinen, I. Vilja, Decaying Massive Particle in Matter and Radiation Dominate d Eras , Phys. Rev. D 97, 065004 (2018), arXiv:1801....

2012 arXiv
[14]

Patrignani et

C. Patrignani et. al. (Particle Data Group), Review of Particle Physics (2016) , Chin. Phys. C 40, 100001 (2016)

2016
[15]

Mukhanov and S

V. Mukhanov and S. Winitzki, Introduction to Quantum Eﬀects in Gravity (Cambridge Univ. Press, UK, 2007)

2007
[16]

Itzykson, J-B

C. Itzykson, J-B. Zuber, Quantum Field Theory , (Dover Pub., New York, 1980); M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory , (Addison-Wesley Pub., New York, 1995); M. Maggiore, A Modern Introduction to Quantum Field Theory , (Oxford Univ. Press, New York, 2008)

1980
[17]

Bunch, Adiabatic Regularization for scalar ﬁelds with arbitrary c oupling to the scalar curvature, J

T.S. Bunch, Adiabatic Regularization for scalar ﬁelds with arbitrary c oupling to the scalar curvature, J. Phys. A: Math. Gen. 13, 1297 (1980); and the references therein. 17

1980
[18]

Birrell and P.C.W

N.D. Birrell and P.C.W. Davies, Quantum ﬁelds in curved space (Cambridge Univ. Press, UK, 1994)

1994
[19]

Parker, Quantized Fields and Particle Creation in Expanding Univers es

L. Parker, Quantized Fields and Particle Creation in Expanding Univers es. I , Phys. Rev. 183, 1057 (1969)

1969
[20]

Lankinen, I

J. Lankinen, I. Vilja, Gravitational particle production in a stiﬀ matter dominat ed universe , JCAP 1708, 025 (2017), arXiv:1612.02586

2017 arXiv
[21]

Albrecht, P

A. Albrecht, P. Ferreira, M. Joyce, T. Prokopec, Inﬂation and squezzed quantum states , Phys. Rev. 50, 4807 (1994), astro-ph/9303001; C. Kiefer, D. Polarski, Emergence of classicility for primordial ﬂuctuations: con cepts and analogies, Ann. Physik 7, 137 (1998), gr-qc/9805014...

1994 arXiv
[23]

G. Leon, D. Sudarsky, Origin of Structure: statistical characterization of the p rimordial density perturbations and the collapse of the wave function , JCAP 06, 020 (2015), arXiv:1503.01417; A. Perez, H. Sahlmann, D. Sudarsky, On the quantum origin of the seeds of cosmic stru...

2015 arXiv
[24]

Semikoz, I.I

D.V. Semikoz, I.I. Tkachev, Kinetics of Bose Condensation , Phys. Rev. Lett 74, 3093 (1995), hep-ph/9409202; D.V. Semikoz, I.I. Tkachev, Condensation of bosons in the kinetic regime , Phys. Rev. D 55, 489 (1997), hep-ph/9507306

1995 arXiv
[25]

Weinberg The Quantum Theory of Fields, Vol.I , (Cambridge Univ

S. Weinberg The Quantum Theory of Fields, Vol.I , (Cambridge Univ. Press, New York, 1995)

1995
[26]

Bjorken and S.D

J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill Book Company, New York, 1965)

1965
[27]

Hui, J.P

L. Hui, J.P. Ostriker, S. Tremaine, and E. Witten, Ultralight scalars as cosmological dark matter, Phys. Rev. D 95, 043541 (2017), arXiv:1610.08297. 18 Appendix A: Condition (5) in the case of radiation, matter, cosmological constant, and stiﬀ matter dominated eras If we let H...

2017 arXiv

Pith tools

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

[1] [1]

Moreover, although, evolution toward s |⃗˜p4| = 0 is an important indication towards formation of condensation it is not suﬃcient [23]

In other words, what we have shown is not a proof but a hint towa rds formation of a condensate at later times. Moreover, although, evolution toward s |⃗˜p4| = 0 is an important indication towards formation of condensation it is not suﬃcient [23]. I n fact, essentially, (15) also shows evolution towards |⃗˜p4| = 0. (21) reiterates and reinforces this resu...

work page

[2] [2]

Weinberg, Cosmology (Oxford Univ

S. Weinberg, Cosmology (Oxford Univ. Press, New York, 2008)

work page 2008

[3] [3]

Weinberg, The Cosmological Constant Problem , Rev

S. Weinberg, The Cosmological Constant Problem , Rev. Mod. Phys. 61 1 (1989); S. Nobbenhuis, 2006 Categorizing Diﬀerent Approaches to the Cosmological Constan t Problem, Found. Phys. 36, 613 (2006); gr-qc/0411093

arXiv 1989

[4] [4]

Weinberg, et al., Cold dark matter: Controversies on small scales , Proc

D.H. Weinberg, et al., Cold dark matter: Controversies on small scales , Proc. Nat. Acad. Sci. 112, 12249 (2015) P. Bull, et al., Beyond Λ CDM: Problems, solutions, and the road ahead , Physics of the Dark Universe, 12, 56 (2016)

work page 2015

[5] [5]

Fukuyama, M

T. Fukuyama, M. Morikawa, T. Tatekawa, Cosmic structures via Bose-Einstein condensation and its collapse , JCAP 0806, 033 (2008), arXiv:0705.3091

arXiv 2008

[6] [6]

Castellanos, C

E. Castellanos, C. Escamilla-Rivera, A. Macias, D. Nune z, Scalar ﬁeld as a Bose-Einstein condensate?, JCAP, 11, 034 (2014)

work page 2014

[7] [7]

Erdem, Is it possible to obtain accelerated expansion of the univers e through energy transfer between diﬀerent energy densities , Phys

R. Erdem, Is it possible to obtain accelerated expansion of the univers e through energy transfer between diﬀerent energy densities , Phys. Dark Univ. 15, 57 (2017)

work page 2017

[8] [8]

Bose-Einstein condensation of relativistic Scalar Field Dark Matter

L.A. Urena-Lopez, Bose-Einstein condensation of relativistic Scalar Field D ark Matter, JCAP 0901, 014 (2001), arXiv:0806.3093 M.P. Silverman and R.L. Mallet, Cosmic degenerate matter: a possible solution to the problem of missing mass , Gen. Rel. Grav. 18,M.P. Silverman and R.L. Mallet, Dark matter as a cosmic 16 Bose-Einstein condensate and possible sup...

work page Pith review arXiv 2001