Big Bang Nucleosynthesis Hunts Chameleon Dark Matter

Hua Chen; Shinya Matsuzaki; Taishi Katsuragawa; Taotao Qiu

REVIEW 4 major objections 4 minor 47 references

Big Bang Nucleosynthesis Hunts Chameleon Dark Matter

T0 review · 4 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 Big Bang Nucleosynthesis, not fifth-force tests, sets the tightest bound on the scalaron mass and the R² coupling in F(R) gravity.

desk verdict The headline α bound rests on a thermal criterion that doesn't apply to a condensate, but the BChPT derivation is genuine and the scenario deserves refereeing. read the letter →

arxiv 1908.04146 v3 pith:RTBF2CHZ submitted 2019-08-12 hep-ph astro-ph.COgr-qchep-th

Hua Chen , Taishi Katsuragawa , Shinya Matsuzaki , Taotao Qiu This is my paper · ORCID

classification hep-phastro-ph.COgr-qchep-th

keywords chameleondarkmatterscalaronF(R)gravitybigbangnucleosynthesisR^2correctionheliumabundancebaryonchiralperturbationtheoryfifthforceconstraint

topics Dark Energy

open problems Dark Energy

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 argues that the scalaron of F(R) gravity—a chameleon field that can serve as both dark energy and dark matter—has early-universe behaviour fixed almost entirely by the R² term added to cure singularity problems. Across the viable F(R) dark energy models, this makes the scalaron's BBN-epoch dynamics identical: a damped bouncing oscillation with mass $m(\phi_{\mathrm{min}}) \simeq \sqrt{1/(6\alpha)}$. Requiring the scalaron to be non-relativistic at BBN yields $\alpha \lesssim 2\times 10^{5}\,\mathrm{GeV}^{-2}$, a bound several orders of magnitude stronger than the fifth-force limit. The same dynamics keeps the scalaron fluctuation in the window $-0.05 < \kappa\phi < 0.03$ allowed by Planck 2018 helium data, so BBN can both constrain and potentially detect chameleon dark matter.

What carries the argument

The central object is the scalaron field $\phi$ in the Einstein frame, whose Weyl factor $e^{\sqrt{1/6}\,\kappa\phi}$ controls all matter couplings. In the early universe the model reduces to $F(R)\simeq R+\alpha R^{2}$, giving an effective potential whose second derivative yields the mass formula $m(\phi_{\mathrm{min}}) \simeq \sqrt{1/(6\alpha)}$; the chameleon mechanism balances the $R^{2}$ term against the trace of the matter energy-momentum tensor and keeps the minimum close to $\phi=0$. For the BBN observable, the load-bearing identity is the leading-order baryon chiral perturbation theory coupling, which gives $\Delta m_{N}(\phi) = \Delta m_{N}\,e^{-\kappa\phi/\sqrt{6}}$ and $\tau_{n}(\phi)=\tau_{n}\,e^{+\kappa\phi/\sqrt{6}}$, inserting the scalaron into the standard helium-yield formula.

What would settle it

A next-to-leading-order baryon chiral perturbation theory calculation of the scalaron coupling to the electromagnetic isospin-breaking operator; if that coupling is not proportional to the same $e^{-\kappa\phi/\sqrt{6}}$ factor as the quark-mass term, the predicted helium shift in Eq. (4.19) is not controlled. Observationally, a helium abundance measurement tighter than the current roughly $\pm 0.02$ error that remains centred on the standard prediction would exclude scalaron initial amplitudes of order 0.1–1 percent.

Watch

Extended reading notes

Core claim

In a viable class of F(R) gravity models with an $\alpha R^{2}$ correction added to remove the curvature singularity, the scalaron's BBN-epoch dynamics is model-independent. The low-energy dark-energy modification drops out, leaving $F(R) \simeq R + \alpha R^{2}$, and the chameleon mechanism fixes the potential minimum near $\kappa\phi \sim 10^{-17}$. The resulting damped bouncing oscillation has mass $m(\phi_{\mathrm{min}}) \simeq \sqrt{1/(6\alpha)}$, so the non-relativistic condition at $T\sim 1$ MeV gives $\alpha \lesssim 2\times 10^{5}\,\mathrm{GeV}^{-2}$, which is more stringent than the fifth-force bound $\alpha \lesssim 10^{22}\,\mathrm{GeV}^{-2}$. Coupled to BBN through leading-order baryon chiral perturbation theory, the scalaron rescales the neutron-proton mass difference and neutron lifetime by opposite exponentials, shifting the helium abundance by $1 - 2.27(\kappa\phi/\sqrt{6})$. Since the oscillation amplitude naturally sits at $|\kappa\phi| \sim 10^{-3}$ to $10^{-2}$, the scalaron evades the current Planck 2018 bound while remaining within reach of more precise light-element measurements.

Load-bearing premise

The central bound rests on the assumption that the scalaron's only relevant coupling at BBN is the leading-order dilatonic scaling that multiplies both the quark-mass and electromagnetic parts of the neutron-proton mass difference and the neutron lifetime by a single exponential, which the paper acknowledges could be drastically modified by nonperturbative nucleon interactions.

Editorial extensions

If this is right

The R² coupling in viable F(R) dark energy models must obey $\alpha \lesssim 2\times 10^{5}\,\mathrm{GeV}^{-2}$, a bound far tighter than the fifth-force limit used in the paper.
At BBN the scalaron mass is essentially fixed by $\alpha$ alone, so helium-abundance data probe the high-curvature structure of F(R) gravity without knowing the late-time dark energy modification.
With natural initial conditions, scalaron fluctuations stay at $|\kappa\phi| \sim 10^{-3}$–$10^{-2}$ and satisfy the Planck 2018 helium bound, so the chameleon dark matter scenario survives current BBN data.
More precise helium-4 abundance and baryon density measurements can exclude scalaron models with initial amplitudes of order 0.1–1 percent or reveal a deviation from standard BBN.
A smaller $\alpha$ makes the scalaron oscillate faster and damp sooner, so future helium data can constrain $\alpha$ from the high-curvature side as well as from the non-relativistic condition.

Reading between the lines

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

If the universal BBN dynamics is taken at face value, the same $\alpha$ bound could constrain any $\alpha R^{2}$ inflation model in this F(R) family, a consequence the authors do not state.
A lattice-QCD check of the nucleon mass splitting under simultaneous variation of quark masses and electromagnetic coupling could test the single-exponential scaling; a different scaling would change the helium shift and the resulting window for scalar-tensor dark matter.
Because the scalaron amplitude at BBN depends on initial conditions, future helium measurements may probe the pre-BBN history of the field, including phase transitions that kick it, rather than only the high-curvature coupling.
The BBN constraint is environment-specific: it applies at BBN densities, where the chameleon mass is different from its laboratory value, so BBN and fifth-force searches are complementary probes rather than directly interchangeable limits.

Share X Bluesky LinkedIn Reddit HN

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

Bill,

You asked what I think of the scalaron-BBN paper. Short version: the headline α≲2×10^5 GeV^-2 bound in Eq. (4.2) does not hold as stated, but there is a real technical contribution in the appendix, and the paper is not a waste of time.

What's new: the baryon chiral perturbation theory derivation of the scalaron coupling to the nucleon mass splitting is genuine. At leading order the Weyl rescaling factor e^{-κϕ/√6} appears uniformly on nucleon masses and on the EM part of the mass splitting, and they are honest that higher-order terms could change this. The bouncing-oscillation dynamics of the scalaron in the early universe for generic R^2-corrected F(R) models is also a clean observation, and the figures support it.

The problem is the claim that BBN demands m(ϕ_min)≳T. That is a thermal particle criterion. This scalaron is a homogeneous condensate; it behaves non-relativistically when m≫H, not m≫T. Plugging in their own numbers, at α=10^22 you get m~4×10^-12 GeV, while H at T~1 MeV is around 10^-24 GeV, so m/H is huge. The field is cold and oscillating. The m≳T condition is not necessary, and Eq. (4.2) is unsupported. The paper's own numerics use α=10^22, which lies far above the bound they claim from Eq. (4.2); that inconsistency is not resolved anywhere.

There's also a smaller issue: the text says the initial condition is φ(τ=1)=0.1, while Fig. 6 says 0.01. That matters because the fluctuation amplitude at BBN sets the helium shift and the resulting bound in Eq. (4.20). And the treatment of ξ(T) as constant is a simplifying assumption the authors concede; fine for a first estimate, but it isn't airtight.

The later BBN constraint on κϕ from helium abundance is more sensible because it goes through the physical couplings to Δm_N and τ_n, but its numerical reach depends on those initial conditions and on the leading-order coupling.

Who should read it: people working on scalaron dark matter or on BBN constraints for light scalars. The appendix is worth citing once the main text is cleaned up. I wouldn't cite it now for the α bound.

My recommendation: let a referee look at it. The authors have a clear scenario and the BChPT piece is original. But a referee should push hard on Eq. (4.2) and on the α=10^22 numerical analysis; if that can't be repaired, the paper needs major re-scoping.

Referee Report

4 major / 4 minor

Summary. The paper studies a chameleon scalar degree of freedom (the scalaron) in F(R) gravity with an R^2 correction, treating it as a unified dark energy/dark matter candidate. It analyzes the scalaron's evolution between T ~ 100 GeV and T ~ 1 MeV, claims a universal damped bouncing oscillation that is independent of the low-energy F(R) model, and derives a BBN bound alpha <~ 2 x 10^5 GeV^-2 from the requirement that the scalaron be non-relativistic at BBN. Using baryon chiral perturbation theory, the authors then model the scalaron's couplings to the nucleon mass difference and the neutron lifetime, and from Planck 2018 helium abundance data obtain -0.05 < kappa phi < 0.03, arguing that future BBN measurements can probe or exclude the scenario.

Significance. If the main bound were correct, the paper would provide a universal, model-independent constraint on the R^2 coupling that is stronger than fifth-force constraints, and a concrete BChPT framework for scalaron-nucleon couplings. The appendix contains a useful leading-order derivation of the Delta m_N scaling, and Eq. (4.20) is a falsifiable prediction tied to external data. However, the main alpha bound is not supported by the argument given, and the quantitative sensitivity claim rests on an uncontrolled treatment of the electromagnetic contribution; as presented, the paper's central advertised result is not established.

major comments (4)

[Sec. 4, Eq. (4.2)] The inequality m(phi_min) ~ sqrt(1/(6 alpha)) >~ T ~ 1 MeV is not a valid BBN condition for this field. The scalaron is a homogeneous, non-thermally-produced condensate, whose equation of state is that of cold matter whenever m >> H. For alpha = 10^22 GeV^-2, m ~ 4 x 10^-12 GeV, which is far below 1 MeV but far above H_BBN ~ 4 x 10^-25 GeV, so the field oscillates as a cold condensate and need not spoil BBN. The derivation of Eq. (4.2) and the accompanying claim that BBN constrains alpha more strongly than fifth-force experiments are therefore unsupported.
[Sec. 3.2 and Fig. 6] The text states that the numerical evolution uses the initial condition phi(tau=1) = 0.1, while the figure caption and the discussion in Sec. 4.3 refer to phi(tau=1) = 0.01; the amplitude at BBN and the projected detection sensitivity depend directly on this choice. In addition, the numerics are performed at alpha = 10^22 GeV^-2, which is far above the bound alpha <~ 2 x 10^5 GeV^-2 claimed in Eq. (4.2). The authors should specify which initial condition and which alpha are used, and should quantify whether the conclusions hold in the parameter region they claim is allowed.
[Sec. 4.2, Eq. (4.14)] The total scaling Delta m_N^total(phi) = Delta m_N^total e^{-kappa phi / sqrt(6)} assumes that the electromagnetic contribution to the nucleon mass splitting scales with the same exponential as the leading-order chiral contribution. The EM contribution involves hadronic matrix elements and a photon propagator; scaling the coupling e^2 by e^{-kappa phi / sqrt(6)} does not by itself guarantee that the nonperturbative EM contribution to Delta m_N scales in that way. Since Eq. (4.19) and the bound (4.20) rely on this assumption, and the paper itself concedes that higher-order nucleon interactions can "drastically" modify the coupling, the helium-shift prediction is not controlled beyond leading order.
[Secs. 3.1 and 3.2] The analysis approximates xi(T) as constant, but Fig. 5 shows xi varying between about 0.1 and 0.01 over the temperature range from 100 GeV down to 1 MeV. Because xi enters the effective potential and the oscillation amplitude, the sensitivity of the later amplitude estimates to this approximation should be quantified; the statement that the constant-x_i approximation suffices for the typical order of magnitude is an assertion rather than a demonstrated control of the error.

minor comments (4)

[Sec. 2 title] The heading "Dark energy and eark matter in F(R) gravity" contains a typo: "eark" should be "dark".
[Eq. (2.11)] The notation for the second derivative of the effective potential should be V_eff,phi phi(phi_min) or d^2 V_eff / d phi^2; the current typesetting is ambiguous.
[Sec. 4.3] The text says t_BBN ~ 180 s corresponds to T ~ 1 MeV, but in the standard radiation-dominated relation t ~ 1.32 (T/MeV)^-2 s, 180 s corresponds to T ~ 0.1 MeV. This matters because g* = 3.36 is used, which is appropriate after e+e- annihilation, not at T ~ 1 MeV; please correct the temperature-time correspondence or justify the adopted value.
[Fig. 6 and Sec. 4.3] The initial-condition discrepancy between phi(tau=1) = 0.1 and 0.01 should also be resolved in the text discussing detection sensitivity, since the claimed future sensitivity to "somewhat larger initial value (0.1-1%)" depends on which value was actually simulated.

Circularity Check

1 steps flagged · score 4.0 of 10

The BBN bound derivation is largely self-contained, but the helium-shift prediction imports the EM part of the nucleon-mass scaling from the authors' own Ref. [22], making that self-citation load-bearing for Eq. (4.14).

self citation load bearing [Sec. 4.2, Eqs. (4.13)-(4.14); cf. Appendix A.3]
"Regarding the EM correction, the scalaron can actually modify it through the EM scale anomaly ∼ e2/(4π)2FµνFµν, as discussed in [22]. The scalaron thus makes the EM coupling e modified as e2→e2(ϕ) = e−√1/6κϕ · e2 [22] along with the beta function coefficient arising from the charged fermion loops. Note that the scaling factor for the EM correction part is the same as that for the quark mass difference (∆mN) part. Thus, the scalaron just gives an overall scaling for the neutron - proton mass difference in total, in such a way that ∆mtotalN (ϕ) = ∆mtotalN e−√1/6κϕ."

Equation (4.14) is the input that makes the helium shift in Eq. (4.19) linear in κϕ and hence converts observed helium data into the bound Eq. (4.20). The QCD part of ∆mN is re-derived in Appendix A, but the EM piece, quoted as roughly −0.76 MeV and contributing about 37% of the total 1.29 MeV splitting, is not re-derived. Its claimed universal scaling e²→e^{−√(1/6)κϕ}e², and the assertion that the EM correction scales with the same factor as the quark-mass part, are both justified by citation to Ref. [22], whose authors overlap with the present paper. Ref. [22] is not machine-checked, code-reproduced, or independently validated in the present text, and the paper itself warns that nonperturbative four-nucleon interactions could 'drastically' modify the coupling.

full rationale

The main α bound in Eq. (4.2) is not circular: it is the algebraic condition m(ϕmin) ≃ sqrt(1/(6α)) ≳ T, imposed as a non-relativistic criterion, not fitted to the BBN data. The helium-abundance shift in Eq. (4.19) and the fluctuation bound in Eq. (4.20) are also derived from external inputs (PDG, Planck 2018, Gasser-Leutwyler) rather than fitted back into the scalaron equations. The dynamics leading to the claimed small oscillation amplitude is obtained by solving the scalaron equation of motion, so that part is self-contained. The only load-bearing self-citation is the treatment of the electromagnetic contribution to the nucleon mass splitting in Eq. (4.14), which is imported from the authors' prior Ref. [22] and is essential for the quoted BBN sensitivity. That self-citation raises the circularity score to 4, but the paper retains substantial independent derivation content.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central claims rest on a small number of model parameters (notably α, ξ, and the initial amplitude) and on several approximations that the paper itself flags, especially the leading-order BChPT scaling and the constant-ξ radiation-dominated evolution.

free parameters (4)

α (R² coupling strength) = α ≲ 10^22 GeV^-2 (fifth-force); α ≲ 2×10^5 GeV^-2 (BBN bound, Eq. 4.2)
Sets the scalaron mass m ≈ 1/√(6α) at the BBN epoch; the paper uses α = 10^22 for numerical evolution and derives a new upper bound from the non-relativistic requirement.
Starobinsky parameters β, n = β = 2, n = 1
Chosen for the illustrative Starobinsky model in figures; the paper argues the early-universe result is independent of these low-energy parameters.
Matter trace parameter ξ = 0.1, 0.05, 0.01 treated as constants
ξ(T) is imported from prior work and approximated as constant in the numerical evolution (Sec. 3.2); the bouncing oscillation depends on its value.
Initial scalaron field value φ(τ=1) = 0.1 in text, 0.01 in Fig. 6 caption
The amplitude at BBN, and hence whether the scenario survives Eq. (4.20), depends on this hand-picked initial condition; the paper concedes this in Sec. 4.3.

assumptions (6)

standard math Weyl transformation to the Einstein frame maps F(R) gravity to general relativity plus a scalaron field with potential V(ϕ) = (R F_R - F)/(2κ² F_R²).
Used throughout Sec. 2.1 (Eqs. 2.2 to 2.4) to define the scalaron and its coupling to matter.
domain assumption The early universe from T~100 GeV to T~1 MeV is radiation dominated with H=1/(2t), and the scalaron does not backreact on the Hubble parameter.
Assumed in the equation of motion (Eq. 3.16) and in the BBN analysis (Sec. 4.3); if scalaron energy density were significant, the helium yield formula would change.
domain assumption The trace parameter ξ(T) is small and can be treated as constant in the numerical evolution.
Stated in Sec. 3.2 and acknowledged as an approximation in the Conclusion; the bouncing oscillation and its damping rate depend on ξ.
domain assumption For r=R/R_c >> 1, every viable F(R) model reduces to F(R) ≈ R + αR², and the effective potential can be replaced by the Heaviside-truncated quadratic form.
Used to derive the analytic potentials (Eqs. 3.7 to 3.15) and the mass formula (Eq. 4.1); the paper validates it only in the positive-φ region.
ad hoc to paper At leading order in baryon chiral perturbation theory, the scalaron rescales Δm_N and τ_n by the same exponential factors e^(∓κϕ/√6), including the electromagnetic contribution to the nucleon mass splitting.
Used in Eqs. (4.12) to (4.15) and Appendix A.3; the paper admits nonperturbative nucleon dynamics could drastically modify this coupling, and the EM scaling is an assumption.
domain assumption The scalaron starts near the effective potential minimum with a small amplitude in the early universe; larger initial values are excluded as unnatural.
The BBN survival claim depends on this initial-condition choice (Sec. 4.3); the paper notes the amplitude depends on the initial condition.

invented entities (1)

Scalaron as chameleon dark matter independent evidence
purpose: A scalar degree of freedom of F(R) gravity that acts as dark matter; its BBN-epoch oscillations alter helium production through couplings to nucleons and the neutron lifetime.
Not introduced in this paper (proposed in Refs. [21,22]), but the central claim depends on it. The BBN abundance shift and the bound on α give falsifiable handles for future measurements, though no positive detection exists yet.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Big Bang Nucleosynthesis Hunts Chameleon Dark Matter." pith.science (2026). https://pith.science/paper/RTBF2CHZ

@misc{pith2026190804146,
  author       = {Pith},
  title        = {Pith review of: Big Bang Nucleosynthesis Hunts Chameleon Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTBF2CHZ}},
  note         = {Machine review of arXiv:1908.04146}
}

abstract

We study the chameleon field dark matter, dubbed \textit{scalaron}, in $F(R)$ gravity in the Big Bang Nucleosynthesis (BBN) epoch. With an $R^{2}$-correction term required to solve the singularity problem for $F(R)$ gravity, we first find that the scalaron dynamics is governed by the $R^{2}$ term and the chameleon mechanism in the early universe, which makes the scalaron physics model-independent regarding the low-energy scale modification. In viable $F(R)$ dark energy models including the $R^{2}$ correction, our analysis suggests the scalaron universally evolves in a way with a bouncing oscillation irrespective of the low-energy modification for the late-time cosmic acceleration. Consequently, we find a universal bound on the scalaron mass in the BBN epoch, to be reflected on the constraint for the coupling strength of the $R^2$ term, which turns out to be more stringent than the one coming from the fifth force experiments. It is then shown that the scalaron naturally develops a small enough fluctuation in the BBN epoch, hence can avoid the current BBN constraint placed by the latest Planck 2018 data, and can also have a large enough sensitivity to be hunted by the BBN, with more accurate measurements for light element abundances as well as the baryon number density fraction.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 13 canonical work pages

[1]

Supernova Search Team collaboration, A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant , Astron. J. 116 (1998) 1009–1038, [astro-ph/9805201]

arXiv 1998
[2]

Planck collaboration, P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13, [ 1502.01589]

arXiv 2016
[3]

Nojiri, S

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Modiﬁed Gravity Theories on a Nutshell: Inﬂation, Bounce and Late-time Evolution , Phys. Rept. 692 (2017) 1–104, [ 1705.11098]

arXiv 2017
[4]

Khoury and A

J. Khoury and A. Weltman, Chameleon ﬁelds: Awaiting surprises for tests of gravity in space, Phys. Rev. Lett. 93 (2004) 171104, [ astro-ph/0309300]

arXiv 2004
[5]

Hu and I

W. Hu and I. Sawicki, Models of f(R) Cosmic Acceleration that Evade Solar-System Tests , Phys. Rev. D76 (2007) 064004, [ 0705.1158]

arXiv 2007
[6]

A. A. Starobinsky, Disappearing cosmological constant in f(R) gravity , JETP Lett. 86 (2007) 157–163, [0706.2041]

arXiv 2007
[7]

Tsujikawa, Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints, Phys

S. Tsujikawa, Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints, Phys. Rev. D77 (2008) 023507, [ 0709.1391]

arXiv 2008
[8]

Elizalde, S

E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, Non-singular exponential gravity: a simple theory for early- and late-time accelerated expansion , Phys. Rev. D83 (2011) 086006, [ 1012.2280]

arXiv 2011

Show all 47 references

[9]

A. V. Frolov, A Singularity Problem with f(R) Dark Energy , Phys. Rev. Lett. 101 (2008) 061103, [0803.2500]

2008 arXiv
[10]

Bamba, S

K. Bamba, S. Nojiri and S. D. Odintsov, The Universe future in modiﬁed gravity theories: Approaching the ﬁnite-time future singularity , JCAP 0810 (2008) 045, [ 0807.2575]

2008 arXiv
[11]

Kobayashi and K.-i

T. Kobayashi and K.-i. Maeda, Can higher curvature corrections cure the singularity problem in f(R) gravity? , Phys. Rev. D79 (2009) 024009, [ 0810.5664]

2009 arXiv
[12]

A. Dev, D. Jain, S. Jhingan, S. Nojiri, M. Sami and I. Thongkool, Delicate f(R) gravity models with disappearing cosmological constant and observational constraints on the model parameters, Phys. Rev. D78 (2008) 083515, [ 0807.3445]

2008 arXiv
[13]

Milgrom, A Modiﬁcation of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis , Astrophys

M. Milgrom, A Modiﬁcation of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis , Astrophys. J. 270 (1983) 365–370

1983
[14]

V. C. Rubin, N. Thonnard and W. K. Ford, Jr., Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/, Astrophys. J. 238 (1980) 471

1980
[15]

Clowe, M

D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., A direct empirical proof of the existence of dark matter , Astrophys. J. 648 (2006) L109–L113, [astro-ph/0608407]

2006 arXiv
[16]

D. J. E. Marsh, Axion Cosmology, Phys. Rept. 643 (2016) 1–79, [ 1510.07633]

2016 arXiv
[17]

S. D. Odintsov and V. K. Oikonomou, f(R) Gravity Inﬂation with String-Corrected Axion Dark Matter, Phys. Rev. D99 (2019) 064049, [ 1901.05363]

2019 arXiv
[18]

S. D. Odintsov and V. K. Oikonomou, Uniﬁcation of Inﬂation with Dark Energy in f(R) Gravity and Axion Dark Matter , Phys. Rev. D99 (2019) 104070, [ 1905.03496]. – 26 –

2019 arXiv
[19]

Nojiri and S

S. Nojiri and S. D. Odintsov, Dark energy, inﬂation and dark matter from modiﬁed F(R) gravity, TSPU Bulletin N8(110) (2011) 7–19, [ 0807.0685]

2011 arXiv
[20]

J. A. R. Cembranos, Dark Matter from R2-gravity , Phys. Rev. Lett. 102 (2009) 141301, [0809.1653]

2009 arXiv
[21]

Katsuragawa and S

T. Katsuragawa and S. Matsuzaki, Cosmic History of Chameleonic Dark Matter in F (R) Gravity, Phys. Rev. D97 (2018) 064037, [ 1708.08702]

2018 arXiv
[22]

Katsuragawa and S

T. Katsuragawa and S. Matsuzaki, Dark matter in modiﬁed gravity? , Phys. Rev. D95 (2017) 044040, [1610.01016]

2017 arXiv
[23]

Inagaki, Y

T. Inagaki, Y. Matsuo and H. Sakamoto, Dark Matter in logarithmic F (R) gravity, 1905.05503

1905 arXiv
[24]

Boehm and P

C. Boehm and P. Fayet, Scalar dark matter candidates , Nucl. Phys. B683 (2004) 219–263, [hep-ph/0305261]

2004 arXiv
[25]

Knapen, T

S. Knapen, T. Lin and K. M. Zurek, Light Dark Matter: Models and Constraints , Phys. Rev. D96 (2017) 115021, [ 1709.07882]

2017 arXiv
[26]

Lee, Brief History of Ultra-light Scalar Dark Matter Models , EPJ Web Conf

J.-W. Lee, Brief History of Ultra-light Scalar Dark Matter Models , EPJ Web Conf. 168 (2018) 06005, [ 1704.05057]

2018 arXiv
[27]

Garcia-Bellido, Dark matter with variable masses , Int

J. Garcia-Bellido, Dark matter with variable masses , Int. J. Mod. Phys. D2 (1993) 85–95, [hep-ph/9205216]

1993 arXiv
[28]

G. W. Anderson and S. M. Carroll, Dark matter with time dependent mass , in Particle physics and the early universe. Proceedings, 1st International Workshop, COSMO-97, Ambleside, UK, September 15-19, 1997 , pp. 227–229, 1997. astro-ph/9711288. DOI

1997 arXiv
[29]

P. Brax, C. van de Bruck, A.-C. Davis, J. Khoury and A. Weltman, Detecting dark energy in orbit - The Cosmological chameleon , Phys. Rev. D70 (2004) 123518, [ astro-ph/0408415]

2004 arXiv
[30]

Capozziello, S

S. Capozziello, S. Nojiri and S. D. Odintsov, The role of energy conditions in f(R) cosmology, Phys. Lett. B781 (2018) 99–106, [ 1803.08815]

2018 arXiv
[31]

Nojiri, S

S. Nojiri, S. D. Odintsov and S. Tsujikawa, Properties of singularities in (phantom) dark energy universe, Phys. Rev. D71 (2005) 063004, [ hep-th/0501025]

2005 arXiv
[32]

Capozziello, M

S. Capozziello, M. De Laurentis, S. Nojiri and S. D. Odintsov, Classifying and avoiding singularities in the alternative gravity dark energy models , Phys. Rev. D79 (2009) 124007, [0903.2753]

2009 arXiv
[33]

Nojiri and S

S. Nojiri and S. D. Odintsov, The Future evolution and ﬁnite-time singularities in F(R)-gravity unifying the inﬂation and cosmic acceleration , Phys. Rev. D78 (2008) 046006, [0804.3519]

2008 arXiv
[34]

Katsuragawa, S

T. Katsuragawa, S. Matsuzaki and E. Senaha, F (R) gravity in the early Universe: Electroweak phase transition and chameleon mechanism , 1812.00640

arXiv
[35]

A. L. Erickcek, N. Barnaby, C. Burrage and Z. Huang, Catastrophic Consequences of Kicking the Chameleon, Phys. Rev. Lett. 110 (2013) 171101, [ 1304.0009]

2013 arXiv
[36]

Belokon and A

A. Belokon and A. Tokareva, Light scalar dark matter coupled to a trace of energy-momentum tensor, 1812.04065

arXiv
[37]

Tanabashi et al., Review of Particle Physics , Phys

Particle Data Group collaboration, M. Tanabashi et al., Review of Particle Physics , Phys. Rev. D98 (2018) 030001. – 27 –

2018
[38]

Gasser and H

J. Gasser and H. Leutwyler, Quark Masses, Phys. Rept. 87 (1982) 77–169

1982
[39]

Sarkar, Big bang nucleosynthesis and physics beyond the standard model , Rept

S. Sarkar, Big bang nucleosynthesis and physics beyond the standard model , Rept. Prog. Phys. 59 (1996) 1493–1610, [ hep-ph/9602260]

1996 arXiv
[40]

Aghanim et al., Planck 2018 results

Planck collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters , 1807.06209

2018 arXiv
[41]

A. L. Erickcek, N. Barnaby, C. Burrage and Z. Huang, Chameleons in the Early Universe: Kicks, Rebounds, and Particle Production , Phys. Rev. D89 (2014) 084074, [ 1310.5149]

2014 arXiv
[42]

Gasser and H

J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop , Annals Phys. 158 (1984) 142

1984
[43]

Gasser and H

J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B250 (1985) 465–516

1985
[44]

X. L. Ren, L. S. Geng, J. Martin Camalich, J. Meng and H. Toki, Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory , JHEP 12 (2012) 073, [ 1209.3641]

2012 arXiv
[45]

G. E. Brown and M. Rho, Scaling eﬀective Lagrangians in a dense medium , Phys. Rev. Lett. 66 (1991) 2720–2723

1991
[46]

Li, Y.-L

Y.-L. Li, Y.-L. Ma and M. Rho, Chiral-scale eﬀective theory including a dilatonic meson , Phys. Rev. D95 (2017) 114011, [ 1609.07014]

2017 arXiv
[47]

Li and Y.-L

Y.-L. Li and Y.-L. Ma, Derivation of Brown-Rho scaling from scale-chiral perturbation theory, 1710.02839. – 28 –

arXiv

Pith tools

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

[1] [1]

Supernova Search Team collaboration, A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant , Astron. J. 116 (1998) 1009–1038, [astro-ph/9805201]

arXiv 1998

[2] [2]

Planck collaboration, P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13, [ 1502.01589]

arXiv 2016

[3] [3]

Nojiri, S

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Modiﬁed Gravity Theories on a Nutshell: Inﬂation, Bounce and Late-time Evolution , Phys. Rept. 692 (2017) 1–104, [ 1705.11098]

arXiv 2017

[4] [4]

Khoury and A

J. Khoury and A. Weltman, Chameleon ﬁelds: Awaiting surprises for tests of gravity in space, Phys. Rev. Lett. 93 (2004) 171104, [ astro-ph/0309300]

arXiv 2004

[5] [5]

Hu and I

W. Hu and I. Sawicki, Models of f(R) Cosmic Acceleration that Evade Solar-System Tests , Phys. Rev. D76 (2007) 064004, [ 0705.1158]

arXiv 2007

[6] [6]

A. A. Starobinsky, Disappearing cosmological constant in f(R) gravity , JETP Lett. 86 (2007) 157–163, [0706.2041]

arXiv 2007

[7] [7]

Tsujikawa, Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints, Phys

S. Tsujikawa, Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints, Phys. Rev. D77 (2008) 023507, [ 0709.1391]

arXiv 2008

[8] [8]

Elizalde, S

E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, Non-singular exponential gravity: a simple theory for early- and late-time accelerated expansion , Phys. Rev. D83 (2011) 086006, [ 1012.2280]

arXiv 2011