Variation in the size of the Photon Sphere and Black Hole Shadow in the Modified Gravity

Ajay Kumar Sharma; Kanav Ganguly; Murli Manohar Verma

REVIEW 4 major objections 5 minor 44 references

Variation in the size of the Photon Sphere and Black Hole Shadow in the Modified Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Gravity's extra scalar F rescales black hole photon spheres

desk verdict The f(R) shadow formula rests on an internally inconsistent derivation, and the claimed F-dependence is just a reparameterization of the known Kottler result; I would not send this to referees as it stands. read the letter →

arxiv 2506.10414 v3 pith:CPZNEDSV submitted 2025-06-12 gr-qc astro-ph.GA

Ajay Kumar Sharma , Kanav Ganguly , Murli Manohar Verma This is my paper

classification gr-qcastro-ph.GA MSC 83C5783D0583C10 PACS 04.70.-s04.50.Kd

keywords f(R)gravityblackholeshadowphotonspheremodifiednullgeodesicsEventHorizonTelescopeKottlerspacetimescalardegreeoffreedom

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 in f(R) modified gravity, the size of a black hole's photon sphere and the angular size of its shadow are not fixed by the mass alone: a new scalar degree of freedom $F=f'(R)$ enters both formulas. Working with a static, spherically symmetric, constant-curvature $f(R)$ metric, the paper derives a photon-sphere radius $r_{\mathrm{ph}} = 3 G_N M/F$ and a critical impact parameter $b_{\mathrm{cr}} = 3\sqrt{3}\,G_N M/(F\sqrt{1-2.25\,R_{dS}G_N^2 M^2/F^2})$, which controls the shadow through $\sin^2\alpha_{\mathrm{sh}}=b_{\mathrm{cr}}^2/h^2(r_o)$. The stated motivation is that Event Horizon Telescope images of M87* and Sagittarius A* could carry a measurable signature of modified gravity in the shadow size. The formulas reduce to the standard Schwarzschild and Kottler results when $F=1$ and $R_{dS}=4\Lambda$.

What carries the argument

The load-bearing object is the ratio $h^2(r)=D(r)/A(r)$ built from the metric (here $h^2=r^2/A(r)$). Circular photon orbits sit at the stationary points of this ratio, $dh^2/dr=0$, and the angular radius of the shadow for a static observer at $r_o$ is $\sin^2\alpha_{\mathrm{sh}}=h^2(r_{\mathrm{ph}})/h^2(r_o)$. Applying this machinery to the constant-curvature $f(R)$ metric $A(r)=1-2G_{\mathrm{eff}}M/r-R_{dS}r^2/12$, with $G_{\mathrm{eff}}=G_N/F$, is precisely how the factor $F$ enters the photon sphere and shadow formulas.

What would settle it

Compute the photon sphere and shadow of any $f(R)$ black hole solution with non-constant scalar curvature; if the radius differs from $3G_N M/F$ or the critical impact parameter differs from the formula above, the constant-curvature assumption is where the argument fails. A direct observational check would use Sgr A* with $M=(4.0^{+1.1}_{-0.6})\times10^6\,M_\odot$ and shadow angular diameter $\theta_{\mathrm{sh}}=48.7\pm0.7\,\mu$as: the value of $F$ implied by the measured shadow and an independently measured mass must be consistent with 1 within the combined uncertainties, or else either modified gravity is favored or the assumed metric is wrong.

Watch

Extended reading notes

Core claim

The paper's central discovery is a generalized pair of formulas for static black holes in $f(R)$ gravity: the photon sphere radius becomes $r_{\mathrm{ph}}=3G_N M/F$ and the shadow's critical impact parameter becomes $b_{\mathrm{cr}}=3\sqrt{3}G_N M/(F\sqrt{1-2.25 R_{dS}G_N^2M^2/F^2})$, where $F=f'(R)$ is the extra scalar degree of freedom and $R_{dS}$ is the constant curvature of the spacetime. Because the shadow's angular radius satisfies $\sin^2\alpha_{\mathrm{sh}}=b_{\mathrm{cr}}^2/h^2(r_o)$ for a static observer at $r_o$, these formulas translate a measured shadow diameter into a constraint on $F$ and $R_{dS}$. The paper reads the inverse dependence on $F$ as the observational signature of modified gravity and notes that the whole scheme collapses back to the Kottler (Schwarzschild–de Sitter) result for $F=1$, $R_{dS}=4\Lambda$.

Load-bearing premise

The formulas hold only if the $f(R)$ black hole spacetime really has constant scalar curvature $R=R_{dS}$, so that the metric takes the form $A(r)=1-2G_N M/(Fr)-R_{dS}r^2/12$; if realistic $f(R)$ black holes have non-constant curvature, the derived photon-sphere and shadow sizes do not follow.

Editorial extensions

If this is right

The EHT angular diameters of M87* and Sgr A* become direct constraints on the product $G_N M/F$ and on the effective curvature term $R_{dS}$.
For the same mass, $F>1$ shrinks both the photon sphere and the shadow relative to the Schwarzschild prediction, while $F<1$ enlarges them.
The f(R) shadow formula reproduces the Kottler result exactly when $F=1$ and $R_{dS}=4\Lambda$, making general relativity a clean limit of the modified formulas.
The angular radius of the shadow retains an observer-position dependence through $h^2(r_o)$, so finite-distance and redshift corrections matter in precision comparisons.
The generalized photon-sphere condition $dh^2/dr=0$ can be reused for other static metrics to quickly test whether a modified theory changes the shadow size.

Reading between the lines

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

Beyond the paper's claims, the same $r_{\mathrm{ph}}=3G_N M/F$ form means a single shadow image cannot separate $F$ from the black hole mass; independent dynamical mass estimates are required to break the degeneracy.
Beyond the paper's claims, the constant-curvature metric is the fragile input; computing shadows from a non-constant-curvature $f(R)$ solution would show how much these predictions shift.
Beyond the paper's claims, the $h^2$ machinery is theory-agnostic, so the same procedure could be applied to other metric modifications of gravity to produce a common shadow-size parametrization.
Beyond the paper's claims, if EHT precision improves and still finds no deviation from GR, that would bound $F$ to within a few percent of 1 in the strong-field regime, complementing weak-field constraints.

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

The central claim—that f(R) gravity generically rescales the photon sphere and shadow by F=f'(R)—rests on a derivation that is internally inconsistent, and the result itself is a reparameterization of the known Kottler shadow. I wouldn't send this to referees as it stands.

What the paper does well: it's a self-contained walk through the Perlick–Tsupko shadow formalism, with the Schwarzschild and Kottler cases worked out in detail and the Kerr shadow summarized. The exposition is clear and the references to the standard review [21] are appropriate. If the only goal were a pedagogical note on how to apply h^2(r) to a given static spherical metric, this would be acceptable.

The soft spots are serious. In Section IV the photon sphere is derived twice. From the effective potential (eqs. 71–74) the paper gets r_ph = (3/2) G_N M/F, but that comes from a wrong derivative: the coefficient should be 3, not 3/2, so the (3/2) result is spurious. From the critical-impact-parameter condition (eqs. 75–78) it gets r_ph = 3 G_N M/F, and only the latter is used in the shadow formula. So the paper doesn't actually have a single prediction for the F-dependence; the two halves of Section IV contradict each other. The trace equation (65) is also misprinted ('32f'(R)' should be '3□f'(R)') and lacks a citation. Equation (82) has a typo in the denominator when converting G_eff to G_N. None of these are fatal to the idea of computing shadows in f(R), but they are exactly the kind of errors a careful author should catch before claiming a generalized formula.

On novelty: the metric (66) is the constant-curvature f(R) solution from Capozziello et al. [44]. Plugging that metric into the general h^2(r) formalism from [21] and recognizing G_eff = G_N/F is a substitution, not a new mechanism. The paper itself notes the similarity to the Kottler result. There is no comparison to EHT data, no new spacetime, and no parameter estimation.

Who this is for: instructors or students who want a worked example of the shadow formalism on a slightly generalized metric could use it after corrections. As a research contribution, it does not support its claims. Recommendation: desk reject or send back for major revision on the internal inconsistency; I would not spend referee time on the current version, but I'd encourage the authors to fix the derivative and the typos and resubmit a shorter, clearly pedagogical version if they don't want to add real data analysis.

Referee Report

4 major / 5 minor

Summary. The manuscript studies photon spheres and black-hole shadows for static, spherically symmetric spacetimes, first reviewing the Schwarzschild, Kottler, and Kerr cases, and then applying the constant-curvature f(R) metric A(r)=1-2G_eff M/r - R_dS r^2/12 with G_eff=G_N/F. It claims that the photon-sphere radius and the black-hole shadow scale inversely with F=f'(R), with the central formulae being Eq. (78) for r_ph and Eqs. (85)-(86) for the shadow. It further argues that in the limit F=1 and R_dS=4Λ the result reduces to the Kottler-spacetime expression. No numerical fitting is performed; the calculation is analytic and proceeds from an assumed spacetime metric.

Significance. If the derivation were correct, the paper would provide simple analytic expressions showing how the new scalar degree of freedom F enters the photon sphere and shadow, which would be a useful pedagogical and possibly phenomenological result. The strengths are the use of the standard geodesic/shadow framework and the explicit recovery of the GR/Kottler limits. However, the central claim is currently not reliably established because the two photon-sphere derivations in Section IV disagree by a factor of two, and there is also an error in the conversion from G_eff to G_N in the impact parameter. The result is additionally conditional on the constant-curvature f(R) metric of Ref. [44], whose relevance to the EHT-observed black holes is not argued. The paper does not fit data and is not circular, but the assumed spacetime is exactly what produces the claimed modification.

major comments (4)

[Sec. IV, Eqs. (72)-(74)] The two photon-sphere derivations in Section IV are mutually inconsistent by a factor of two. Differentiating the effective potential in Eq. (72), V_eff = (J^2/(2r^2))(1 - 2G_eff M/r - R_dS r^2/12), gives dV_eff/dr = J^2(-r^{-3} + 3G_eff M r^{-4}); the condition dV_eff/dr=0 therefore yields r_ph=3G_eff M, not r_ph=(3/2)G_eff M as in Eq. (74). The coefficient 3/2 in Eq. (73) is an algebraic error. Since Eqs. (78)-(86) are built only on the h^2(r) derivation, the paper presently has no single well-defined prediction for the F-dependence, and Eq. (74) must be corrected or removed.
[Sec. IV, Eq. (82)] The substitution G_eff=G_N/F is mishandled when passing from Eq. (81) to Eq. (82). With G_eff=G_N/F, the denominator in Eq. (81) should become sqrt(1 - 2.25 R_dS G_N^2 M^2/F^2), not sqrt(1 - 2.25 R_dS G_eff^2 M^2/F^2) as printed. The printed version double-counts F and propagates into the shadow formulas in Eqs. (84)-(86).
[Sec. IV, Eq. (65)] The trace equation is misprinted: it reads f'(R)R - 2f(R) + 32f'(R) = 8πG_N T^M, with a missing citation. The displayed term '32f'(R)' is dimensionally inconsistent and should presumably be 3□f'(R) (or a properly dimensioned equivalent). Because the constant-curvature assumption R=R_dS used to obtain the metric in Eq. (66) is motivated by this trace equation, the equation must be corrected and the missing reference supplied.
[Sec. IV, Eqs. (62)-(66)] The shadow and photon-sphere calculation is performed for the specific constant-curvature f(R) metric of Ref. [44] without addressing whether such a solution describes the astrophysical black holes that EHT observes. The introductory and concluding statements about signatures in Sagittarius A* and M87* therefore go beyond what is actually derived. The authors should either justify the relevance of the constant-curvature metric or explicitly restrict the claims to this model spacetime.

minor comments (5)

[Sec. II.C, Eqs. (3) and (12)-(14)] The symbol R_s is defined as 2GM/c^2 in Eq. (3) but is then used as GM/c^2 in Eqs. (12)-(14); this notational conflict should be removed for the derivation to be unambiguous.
[Sec. II.D] The text says Synge's shadow calculation was given in 1996, but Ref. [19] is from 1966; the year should be corrected.
[Sec. III.B, Eq. (42)] Equation (42) contains garbled derivative notation, including the expression '3dr^2/dr'; the derivation should be rewritten cleanly.
[Throughout] The manuscript contains many typographical and grammatical errors, such as 'Since Lagrangian have', 'scalar degree fo freedom', and 'we can distinguished'; a careful editorial pass is needed.
[Sec. IV, after Eq. (74)] The sentence claiming that the photon sphere in f(R) gravity is larger than in GR is opposite to the behavior for F>1, which gives a smaller radius; this statement should be amended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the photon-sphere and shadow formulas are direct consequences of an assumed constant-curvature f(R) metric, not of fitted inputs or self-citation.

full rationale

I found no circular step. Section IV takes the constant-curvature f(R) metric of [44] (eq. 66) with G_eff = G_N/F and R_dS constant, then applies the standard h^2(r) = D/A critical-impact-parameter method (eqs. 33, 75-82) to obtain r_ph = 3 G_eff M and b_cr = 3√3 G_eff M / √(1 - 2.25 R_dS G_eff^2 M^2). This is a direct calculation from an assumed spacetime, not a fit or a prediction that reproduces its own input. The F-dependence is inherited from the input metric's G_eff, but that is an assumption about the spacetime, not circular reasoning. Self-citations ([12], [13], [39]-[41]) appear only as motivational background for f(R) cosmology and are not used to derive eqs. (78)-(86). The trace equation (65) has a misprint and missing citation, and Section IV contains an internal inconsistency between eq. (74) and eq. (78); these are correctness and consistency defects, not circularity. The shadow formula is not benchmarked against EHT data, so no fitted-input circularity exists.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No data are fitted and no new particle or field is introduced; F=f'(R) is the standard scalar degree of freedom of f(R) gravity. The central formulas depend on the two model inputs F and R_dS and on three assumptions listed above.

free parameters (2)

F = f'(R_dS) = not fitted (model parameter)
F rescales the photon sphere and shadow through G_eff=G_N/F (eqs 74, 78); no f(R) model is specified, so F is an input.
R_dS = not fitted (model parameter)
The constant curvature acts like a cosmological constant in the shadow formula (eq 86); it is undetermined by the paper.

assumptions (4)

domain assumption The f(R) black hole spacetime is described by the constant-curvature metric (66): A(r)=1-2G_effM/r - R_dS r^2/12.
Taken from ref [44]; if real f(R) black holes have nonconstant R, the formulas fail.
domain assumption The vacuum trace condition f(R_dS)/F = R_dS/2 holds.
Used to reduce eq (62) to eq (66); the paper's eq (65) is misprinted and uncited.
standard math The photon sphere is located by dh^2/dr=0 with h^2=D/A, and the shadow angular radius by sin^2 α_sh = h^2(r_ph)/h^2(r_o).
Standard result from the cited review [21].
domain assumption F>0 so that G_eff=G_N/F is a positive effective Newton constant.
Required for stable gravity; not discussed in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Variation in the size of the Photon Sphere and Black Hole Shadow in the Modified Gravity." pith.science (2026). https://pith.science/paper/CPZNEDSV

@misc{pith2026250610414,
  author       = {Pith},
  title        = {Pith review of: Variation in the size of the Photon Sphere and Black Hole Shadow in the Modified Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPZNEDSV}},
  note         = {Machine review of arXiv:2506.10414}
}

abstract

Black hole shadows are a widespread topic in astrophysics. This paper searches for an optical view of the black hole and the relationship between black hole shadow and photon sphere with curvature. We were inspired by the observations of Sagittarius $A^*$ and supermassive black hole M87$*$ through the Event Horizon Telescope. We have found the signature of the modified theory of gravity on the photon sphere and shadow. We considered $f(R)$ modified theory of gravity and new scalar degree of freedom $F$ appears in the expressions of photon sphere and shadow.

Figures

Figures reproduced from arXiv: 2506.10414 by the authors.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

44 extracted references · 22 canonical work pages

[44]

Nojiri and S

S. Nojiri and S. Odintsov, Physics of the Dark Universe 47, 101785 (2025)

work page 2025
[1]

Akiyama et al

K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875, L6 (2019), arXiv:1906.11243 [astro-ph.GA]

arXiv 2019
[2]

A. G. Riess et al. (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201

arXiv 1998
[3]

Angular radius of the black hole shadow for large distance ro >> mis αsh = 3 √ 3m ro

in Schwarzschild spacetime. Angular radius of the black hole shadow for large distance ro >> mis αsh = 3 √ 3m ro . (39) The equation (39) is useful to calculate angular diameter of the black hole shadow. B. Shadow in the Kottler space-time for a static observer Here we discussed the effect of the cosmological con- stant on BHS and photon sphere. We consid...

work page
[4]

S. M. Carroll, Living Rev. Rel. 4, 1 (2001), arXiv:astro- ph/0004075

arXiv 2001
[5]

Sahni, Class

V. Sahni, Class. Quant. Grav. 19, 3435 (2002), arXiv:astro-ph/0202076

arXiv 2002
[6]

Sahni and A

V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), arXiv:astro-ph/9904398

arXiv 2000
[7]

A new recipe for $\Lambda$CDM

V. Sahni and A. A. Sen, Eur. Phys. J. C 77, 225 (2017), arXiv:1510.09010 [astro-ph.CO]

work page Pith review arXiv 2017

Show all 44 references

[8]

Bertone, D

G. Bertone, D. Hooper, and J. Silk, Phys. Rept. 405, 279 (2005), arXiv:hep-ph/0404175

2005 arXiv
[9]

Armendariz-Picon and J

C. Armendariz-Picon and J. T. Neelakanta, JCAP 03, 049 (2014), arXiv:1309.6971 [astro-ph.CO]

2014 arXiv
[10]

Ili´ c, M

S. Ili´ c, M. Kopp, C. Skordis, and D. B. Thomas, Phys. Rev. D 104, 043520 (2021), arXiv:2004.09572 [astro- ph.CO]

2021 arXiv
[11]

Perivolaropoulos and F

L. Perivolaropoulos and F. Skara, New Astron. Rev. 95, 101659 (2022), arXiv:2105.05208 [astro-ph.CO]

2022 arXiv
[12]

W. Yang, S. Pan, E. Di Valentino, O. Mena, D. F. Mota, 8 and S. Chakraborty, Phys. Rev. D 111, 103509 (2025), arXiv:2504.11973 [astro-ph.CO]

2025 arXiv
[13]

A. K. Sharma and M. M. Verma, Astrophys. J. 934, 13 (2022), arXiv:2203.06741 [astro-ph.CO]

2022 arXiv
[15]

B. K. Yadav and M. M. Verma, JCAP 10, 052 (2019), arXiv:1811.03964 [gr-qc]

2019 arXiv
[17]

V. K. Sharma, B. K. Yadav, and M. M. Verma, Eur. Phys. J. C 81, 109 (2021), arXiv:2011.02878 [astro- ph.CO]

2021 arXiv
[18]

Snepppen, Scientific reports 11, 14247 (2021)

A. Snepppen, Scientific reports 11, 14247 (2021)

2021
[19]

Akiyama, A

K. Akiyama, A. Alberdi, W. Alef, J. C. Algaba, R. Anan- tua, K. Asada, R. Azulay, U. Bach, A.-K. Baczko, D. Ball, et al. , The Astrophysical Journal Letters 930, L12 (2022)

2022
[20]

Synge, Monthly Notices of the Royal Astronomical So- ciety 131, 463 (1966)

J. Synge, Monthly Notices of the Royal Astronomical So- ciety 131, 463 (1966)

1966
[22]

Perlick and O

V. Perlick and O. Y. Tsupko, Physics Reports (2021)

2021
[23]

Takahashi, The Astrophysical Journal 611, 996 (2004)

R. Takahashi, The Astrophysical Journal 611, 996 (2004)

2004
[24]

P.-C. Li, M. Guo, and B. Chen, Physical Review D 101, 084041 (2020)

2020
[25]

S. E. Gralla, D. E. Holz, and R. M. Wald, Physical Review D 100, 024018 (2019)

2019
[26]

Event Horizon Telescope Collaboration and K. e. a. Akiyama, Astronomy & Astrophysics 681, A79 (2024)

2024
[27]

Akiyama et al

K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

2019 arXiv
[28]

Schwarzschild, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften , 189 (1916)

K. Schwarzschild, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften , 189 (1916)

1916
[29]

J. V. Narlikar, Introduction to Cosmology (3rd ed.)(Cam- bridge University Press, Cambridge, 2002)

2002
[30]

Schutz, A first course in general relativity (Cambridge university press, 2022)

B. Schutz, A first course in general relativity (Cambridge university press, 2022)

2022
[31]

Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley and Sons, New York, 1972)

S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley and Sons, New York, 1972)

1972
[32]

Sitzungs- berichte der k ˜A¶niglich preussischen akademie der wissenschaften zu berlin,

D. A. der Wissenschaften zu Berlin., “Sitzungs- berichte der k ˜A¶niglich preussischen akademie der wissenschaften zu berlin,” (Berlin, Deutsche Akademie der Wissenschaften zu Berlin, 1882-1918, 1915) p. 0, https://www.biodiversitylibrary.org/bibliography/42231

1918
[33]

Zeng, G.-P

X.-X. Zeng, G.-P. Li, and K.-J. He, Nucl. Phys. B 974, 115639 (2022), arXiv:2106.14478 [hep-th]

2022 arXiv
[34]

M. Wang, S. Chen, and J. Jing, Commun. Theor. Phys. 74, 097401 (2022), arXiv:2205.05855 [gr-qc]

2022 arXiv
[35]

Kottler, Annalen der Physik 361, 401 (1918)

F. Kottler, Annalen der Physik 361, 401 (1918)

1918
[36]

R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963)

1963
[37]

R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967)

1967
[38]

J. M. Bardeen, in Black Holes (Les Astres Occlus), edited by C. Dewitt and B. S. Dewitt (1973) pp. 215–239

1973
[39]

Johannsen, Astrophys

T. Johannsen, Astrophys. J. 777, 170 (2013), arXiv:1501.02814 [astro-ph.HE]

2013 arXiv
[40]

A. K. Sharma and M. M. Verma, Astrophys. J. 926, 29 (2022)

2022
[41]

V. K. Sharma, B. K. Yadav, and M. M. Verma, Eur. Phys. J. C 80, 619 (2020), arXiv:1912.12206 [gr-qc]

2020 arXiv
[42]

Kumar Sharma and M

V. Kumar Sharma and M. M. Verma, Eur. Phys. J. C 82, 400 (2022), arXiv:2201.01058 [astro-ph.CO]

2022 arXiv
[43]

Nojiri and S

S. Nojiri and S. D. Odintsov, Physics of the Dark Uni- verse 46, 101702 (2024)

2024
[45]

Capozziello, K

S. Capozziello, K. F. Dialektopoulos, and O. Luongo, International Journal of Modern Physics D 28, 1950058 (2019)

2019
[46]

De Felice and S

A. De Felice and S. Tsujikawa, Living Reviews in Rela- tivity 13, 1 (2010)

2010
[47]

Nojiri, S

S. Nojiri, S. Odintsov, and V. Oikonomou, Physics Re- ports 692, 1 (2017)

2017

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [44]

Nojiri and S

S. Nojiri and S. Odintsov, Physics of the Dark Universe 47, 101785 (2025)

work page 2025

[2] [1]

Akiyama et al

K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875, L6 (2019), arXiv:1906.11243 [astro-ph.GA]

arXiv 2019

[3] [2]

A. G. Riess et al. (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201

arXiv 1998

[4] [3]

Angular radius of the black hole shadow for large distance ro >> mis αsh = 3 √ 3m ro

in Schwarzschild spacetime. Angular radius of the black hole shadow for large distance ro >> mis αsh = 3 √ 3m ro . (39) The equation (39) is useful to calculate angular diameter of the black hole shadow. B. Shadow in the Kottler space-time for a static observer Here we discussed the effect of the cosmological con- stant on BHS and photon sphere. We consid...

work page

[5] [4]

S. M. Carroll, Living Rev. Rel. 4, 1 (2001), arXiv:astro- ph/0004075

arXiv 2001

[6] [5]

Sahni, Class

V. Sahni, Class. Quant. Grav. 19, 3435 (2002), arXiv:astro-ph/0202076

arXiv 2002

[7] [6]

Sahni and A

V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), arXiv:astro-ph/9904398

arXiv 2000

[8] [7]

A new recipe for $\Lambda$CDM

V. Sahni and A. A. Sen, Eur. Phys. J. C 77, 225 (2017), arXiv:1510.09010 [astro-ph.CO]

work page Pith review arXiv 2017