Vacuum Polarization, Geodesic Equation and Sachs-Wolfe Effect

Ali Kaya

REVIEW 4 major objections 6 minor 18 references

Vacuum Polarization, Geodesic Equation and Sachs-Wolfe Effect

T0 review · 4 major / 6 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 Vacuum polarization by a charged scalar field makes photon mean paths timelike and adds a frequency-dependent Sachs-Wolfe term to the CMB.

desk verdict A coherent but physically fragile argument that vacuum polarization gives photons an effective mass and a frequency-dependent Sachs-Wolfe term; the math follows, but the key step is a gauge-invariance problem. read the letter →

arxiv 2502.08748 v3 pith:VXK7TXQO submitted 2025-02-12 astro-ph.CO gr-qchep-th

Ali Kaya This is my paper · ORCID

classification astro-ph.COgr-qchep-th

keywords vacuumpolarizationphotongeodesicequationeffectivemassSachs-WolfeeffectCMBspectraldistortionmu-distortionchargedscalarfieldcosmicmicrowavebackground

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 quantum fluctuations of a charged scalar field act as an effective spacetime-dependent mass for photons, replacing the null condition $k_\mu k^\mu=0$ with $k_\mu k^\mu=-2e^2\phi^*\phi$ and modifying the geodesic equation accordingly. In this mean-field description photons follow timelike curves rather than null geodesics, without acquiring a fundamental mass. The modified propagation adds a frequency-dependent term to the Sachs-Wolfe effect and produces a $\mu$-type distortion of the CMB blackbody spectrum. In a standard inflationary scenario the correction to the CMB power spectrum can be significant for a light scalar, while the size of the $\mu$-distortion depends strongly on how the vacuum expectation value $\langle\phi^*\phi\rangle$ is regularized, a limitation the paper itself emphasizes.

What carries the argument

The central object is the modified photon geodesic equation (10), $k_\mu k^\mu=-2e^2\phi^*\phi$ and $k^\mu\nabla_\mu k^\nu=-e^2\nabla^\nu(\phi^*\phi)$, which replaces the null condition with a spacetime-dependent effective mass and turns the mean photon path into a timelike curve. The derivation relies on the geometric-optics expansion $A_\mu=a_\mu e^{iS}$ with $k_\mu=\nabla_\mu S$, treating the scalar-field term as a small correction to a null geodesic of the unperturbed spacetime. The associated action (23) preserves Liouville's theorem in phase space, so the distribution function $f=f(\omega/T)$ imposes $\delta T/T=\delta\omega/\omega$. Combined with the frequency shift (22), this yields the Sachs-Wolfe equation (25) whose last term is the new frequency-dependent scalar contribution.

What would settle it

Measure the CMB temperature anisotropy in the same sky direction at several well-separated frequencies, e.g. channels spanning 30-857 GHz, after careful foreground removal. The standard adiabatic Sachs-Wolfe contribution is frequency-independent, so any residual frequency dependence of $\delta T/T$ that scales as $e^2/\omega_0^2$ would support the mechanism, while a null result at the predicted amplitude for a given coupling would rule out the effect.

Watch

Extended reading notes

Core claim

The central claim is that vacuum polarization by a charged scalar field changes the propagation of photons in curved spacetime at the level of the geometric-optics equations. Starting from the field equation $\nabla_\mu F^{\mu\nu}=j^\nu$ with $j^\nu=-ie(\phi\partial^\nu\phi^*-\phi^*\partial^\nu\phi)+2e^2\phi^*\phi A^\nu$, the second term acts like a mass term, and the WKB ansatz $A_\mu=a_\mu e^{iS}$ yields $k_\mu k^\mu=-2e^2\phi^*\phi$ and $k^\mu\nabla_\mu k^\nu=-e^2\nabla^\nu(\phi^*\phi)$. The paper then solves these equations in a perturbed FLRW spacetime and shows the observed frequency acquires a term proportional to $e^2 a^2\phi^*\phi/\omega_0^2$. This produces an extra Sachs-Wolfe contribution $e^2\omega_0^{-2}(a^2\phi^*\phi)'$ that is frequency-dependent, and a corresponding $\mu$-distortion of the CMB spectrum. The author presents this as an effective mean description, analogous to light propagating in a medium, and estimates the observable consequences in a standard inflationary scenario.

Load-bearing premise

The derivation assumes the photon phase oscillates so rapidly that the scalar-field term acts as a small, slowly varying effective mass of the same formal order as in the massive Klein-Gordon case; if the scalar fluctuations vary on scales comparable to the photon wavelength, or if the quantum background $2e^2\langle\phi^*\phi\rangle$ does not satisfy that ordering, the modified geodesic equation and the Sachs-Wolfe and $\mu$-distortion results do not follow.

Editorial extensions

If this is right

CMB temperature fluctuations are no longer automatically frequency-independent: the Sachs-Wolfe effect gains a term $e^2\omega_0^{-2}(a^2\phi^*\phi)'$ that varies with photon frequency.
The mechanism produces a $\mu$-distortion of the CMB blackbody spectrum, with amplitude $e^2/(\omega_0 T_0)(a_L^2\langle\phi^*\phi\rangle_L-\langle\phi^*\phi\rangle_0)$; for a TeV-scale scalar and order-one coupling this would exceed current FIRAS bounds unless $e$ is extremely small.
In a standard inflationary model the scalar contribution to the CMB power spectrum can rival the gravitational Sachs-Wolfe contribution when the scalar is light, roughly when the photon frequency at last scattering satisfies $\omega_0/a_L \sim (5/(3\pi\sqrt{2}C))^{1/2} e \epsilon^{1/4} (H_L/m)\sqrt{H_I M_p}$; heavier scalars suppress the effect.
The modified propagation is an effective mean-path description, so photons remain fundamentally massless; the timelike character of the mean path is analogous to light traveling through a medium.

Reading between the lines

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

Because the paper's $\mu$-distortion estimate changes with the regularization scheme, the quantitative prediction is not yet robust; choosing a physical renormalization condition would decide whether the effect is observable by FIRAS-class or PIXIE-class instruments.
The same geodesic-modification logic should apply to any light boson with an effective two-photon coupling, for example an axion-like field, with the coupling constant replacing $e$; the frequency-dependent Sachs-Wolfe signature would then probe such particles.
Differencing CMB anisotropy maps at widely separated frequencies would isolate the $e^2/\omega_0^2$ term from the frequency-independent adiabatic component, providing a direct observational test that does not rely on the regularization-dependent $\mu$-distortion.

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

Kaya's paper is a clean, self-contained derivation of a modified photon geodesic equation in the presence of a charged scalar, and then a speculative leap to CMB observables. The new part is real: the frequency-dependent Sachs-Wolfe term (25) and the mu-distortion formula (30) are not in the cited prior work, and the author is upfront that the mu-distortion magnitude is regularization-dependent. That honesty earns credit.

The formal derivation from the Lagrangian to Eq. (10) is internally consistent in the geometric optics limit, and the paper correctly notes the subtlety about mass ordering in the Klein-Gordon case. The problem is the step from the field equation to the effective photon mass. Eq. (8) has two terms; dropping the first term as a 'non-homogeneous source' and keeping the second as a mass term is exactly the sort of split that must be justified by gauge invariance. The stress-test note is right: for vacuum fluctuations with zero scalar expectation value, the one-loop photon polarization tensor is transverse and massless at zero momentum. Treating 2e^2 as a local photon mass is not a derived vacuum-polarization effect; it is an assumption that would need a genuine one-loop calculation in FLRW to support. The paper does not provide that, and the acknowledged UV sensitivity in Section IV is a symptom, not a resolution.

The μ-distortion estimate is also built on a coincidence: borrowing a cosmological-constant-like mismatch to set _0 ~ H0 Mp. That is a guess, and the author says so. The power-spectrum comparison is more concrete but depends on arbitrary parameters (e, m, H_I, epsilon) and on the same unverified mass term. There is also an apparent typo in Eq. (43): the exponent on C should probably be 1/2, not there as written—minor, but worth noting if this goes to referee.

Who is this for? Someone working on exotic CMB distortions or on the interface of QFT in curved spacetime and cosmological observables might find the derivation useful as a toy model. It is not yet a credible prediction. The central physical claim needs a gauge-invariant treatment; until then, Eq. (10) is an ansatz, not a result.

For peer review: it deserves a serious referee because the question is legitimate and the formal apparatus is mostly sound, but the referee should push hard on the gauge-invariance step and the regularization dependence. I would not cite it in its current form for the physical effect, but I might cite it as an example of the pitfalls in treating vacuum fluctuations as a medium.

Referee Report

4 major / 6 minor

Summary. The paper argues that, in the presence of a charged scalar field, the geometric-optics propagation of photons is governed by the modified eikonal equations k_mu k^mu = -2e^2 phi*phi and k^mu grad_mu k^nu = -e^2 grad^nu(phi*phi) (Eq. 10), so that photon mean paths become timelike. This vacuum-polarization effect is then shown to produce a frequency-dependent Sachs-Wolfe contribution e^2/omega_0^2 (a^2 phi*phi)' (Eq. 25), which distorts the CMB blackbody spectrum through a mu-distortion (Eq. 30) and modifies the CMB power spectrum, with a comparable-to-gravitational contribution at a characteristic photon frequency (Eq. 43). The estimates are made for a light charged scalar field produced in a standard inflationary scenario.

Significance. The paper addresses a timely and interesting question--whether vacuum polarization of a charged scalar field can leave observable signatures in the CMB--and it provides a systematic geometric-optics framework, explicit analytic formulas for the frequency shift (Eq. 22), the Sachs-Wolfe term (Eq. 25), the mu-distortion (Eq. 30), and the power-spectrum comparison (Eq. 43). The derivation from the stated Lagrangian is internally consistent under the stated ordering assumptions. If Eq. (10) were established, the predicted frequency-dependent Sachs-Wolfe effect would be a novel, potentially falsifiable signature, and the comparison with the gravitational contribution is instructive. However, the central derivation's neglect of the gauge-invariance-restoring current term, the unverified geometric-optics ordering for the quantum background, the regularization-dependent mu-distortion, and a numerical inconsistency in the mu estimate prevent the paper from currently supporting its main claims.

major comments (4)

[§II, Eqs. (8)–(10)] The split of the current in Eq. (8) discards the first term, -ie(phi d^nu phi* - phi* d^nu phi), treating it as a 'non-homogeneous source'. For quantum vacuum fluctuations this dismissal is not justified: the discarded term is required by gauge invariance of the current and, together with the 2e^2(phi*phi)A^nu term, determines the one-loop photon self-energy. The effective equation grad_mu F^mu nu = 2e^2<phi*phi> A^nu is not gauge invariant, and in the locally Lorentz-invariant limit the transverse vacuum-polarization tensor has Pi(0)=0, so no local photon mass of this form appears. Eq. (10) is therefore not established for vacuum fluctuations; the paper needs a gauge-invariant one-loop derivation, or an explicit argument that the derivative term is subleading in the geometric-optics limit, before the subsequent CMB predictions can be trusted.
[§II, geometric-optics expansion] The derivation of Eq. (10) requires the effective mass term 2e^2 phi*phi to be of order 1/epsilon^2 and slowly varying on the wavelength scale, as the paper states for the massive Klein-Gordon field. The paper does not verify these conditions for the quantum background <phi*phi>, which in the Bunch-Davies vacuum has a UV-divergent spectrum and, for light fields, an IR enhancement. Without such a check, the modified geodesic equation (10) is not justified even when phi is treated as a classical background, and the later use of stochastic phi*phi in Section V inherits this gap.
[§IV, Eq. (30)] The estimate 'mu ~ 10^-5 e^2' is not consistent with the preceding input <phi*phi>_0 ~ H0 Mp. With H0 ~ 10^-33 eV, Mp ~ 10^28 eV, omega0 ~ T0 ~ 10^-4 eV, the second term in Eq. (30) evaluates to mu ~ (H0 Mp)/(omega0 T0) e^2 ~ 10^3 e^2, which for e ~ 0.1 is many orders of magnitude above the COBE/FIRAS bound |mu| < 9 x 10^-5. The paper should correct this numerical statement or provide the missing calculation that yields 10^-5.
[§IV, regularization dependence] The paper explicitly states that the magnitude of the mu-distortion is strongly regularization dependent, and then fixes the coincident VEV by applying a cosmological-constant-style mismatch factor <phi*phi>_0 ~ H0 Mp. This is an external assumption, not a prediction of the scalar-QED model; different renormalization prescriptions change the result by many orders of magnitude. Consequently the mu-distortion prediction is not falsifiable as it stands, and the reach claim relative to PIXIE should be removed or conditioned on a justified renormalization scheme.

minor comments (6)

[§II, first paragraph] The sentence 'the following discussion for does not offer a genuine geometrical optics derivation of Maxwell's equations' is grammatically broken and should be rewritten.
[§III, Eq. (24)] The use of Liouville's theorem to relate delta T/T to delta omega/omega is stated too briefly; a sentence explaining why f = f(omega/T) is preserved along the Hamiltonian flow would help the reader.
[§V, Eq. (33)] The disconnected term <0|phi^2|0>^2 in the four-point function is divergent and is dropped without specifying the subtraction; the paper should state that a renormalization scheme is implicitly assumed.
[§IV, footnote 2] The integrated scalar Sachs-Wolfe contribution is acknowledged as potentially non-negligible but is then neglected in Eq. (30); this should be quantified or the mu estimate should be labeled as missing this contribution.
[§V, Eq. (43)] The constant C in Eq. (37) is IR divergent and is set to order unity; the sensitivity of the threshold frequency to the IR cutoff or tilt should be stated.
[References] Reference [15] contains a typographical error: a stray ']' appears before the author name.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the modified photon geodesic, Sachs-Wolfe term, and power-spectrum comparison follow algebraically from the stated Lagrangian and standard vacuum assumptions, with no CMB data used to fix the predicted quantities.

full rationale

The paper's derivation chain is explicit: the charged-scalar Maxwell Lagrangian (6) gives the current (8); the geometric-optics analysis of the field equations yields the modified photon equations (10); solving those equations in the perturbed FLRW background gives the frequency shift (22); the Liouville argument converts this into the Sachs-Wolfe equation (25); and the VEV estimates produce the mu-distortion (30) and power-spectrum comparison (43). Each step is carried out in the text, and the numerical inputs are model parameters (e, m, H_I, epsilon, H_0, M_p) and standard vacuum-mode choices (Bunch-Davies, adiabatic regularization). No CMB observable is used to fit a free constant, and no predicted quantity is defined in terms of itself. The order-of-magnitude estimate <phi*phi>_0 ~ H_0 M_p borrowed from the cosmological-constant mismatch is an external assumption rather than a fit to the paper's own targets; it is not circular. The paper explicitly acknowledges that the mu-distortion magnitude is strongly regularization dependent, which is a robustness caveat rather than a circular reduction. Self-citations [6,7] are used for standard null-geodesic solution techniques that are re-derived or displayed in the equations, so they are not load-bearing in a circular sense.

Assumptions & free parameters 5 free parameters · 6 assumptions · 1 invented entities

The paper's observational claims rest on a hypothesized light charged scalar with chosen parameters (e ~ 0.1, m near 10^-4 eV for the power spectrum), an assumed Bunch-Davies vacuum and free-field evolution, and a regularization choice for the divergent coincident vev. The geometric optics ordering for the quantum background is asserted rather than verified. No new fundamental entity is derived; the light charged scalar is a model ingredient with no independent evidence.

free parameters (5)

Scalar-photon coupling e = chosen as 0.1 in estimates
Sets the size of the effective photon mass term in (10) and the mu-distortion in (30); not measured, chosen for illustrative numbers.
Scalar mass m = 10^-4 eV in the power-spectrum estimate; TeV in the mu-distortion discussion
Controls whether superhorizon mode amplitudes survive to last scattering via Z = H_L/m; the observable significance depends strongly on m.
Coincident vev <phi* phi>_0 = ~H_0 M_p
In Section IV this is set by appealing to the same mismatch factor as the cosmological constant problem; the mu-distortion estimate (30) is directly proportional to this chosen value.
Inflation scale H_I = 10^14 GeV
Input from standard inflation used to normalize the scalar and curvature power spectra (Eqs. 35, 41).
Slow-roll parameter epsilon = 10^-4
Input controlling the gravitational perturbation amplitude in Eq. (42) and the comparison threshold (43).

assumptions (6)

domain assumption Geometric optics ordering: |grad S| is much larger than curvature and background scales, with m^2 counted as O(1/epsilon^2).
Invoked in Section II to derive Eqs. (4) and (10); not validated for the quantum background phi* phi.
domain assumption Bunch-Davies vacuum initial state for the scalar during inflation.
Assumed in Section IV to compute <phi* phi> and the scalar power spectrum (Eq. 35).
domain assumption Free-field evolution: loop corrections and backreaction from the scalar on geometry are neglected.
Stated in Section IV ('free theory') and used in the mode equation (38).
domain assumption Standard single-field inflationary curvature perturbation spectrum with Phi = 3/5 zeta.
Used in Eq. (41) to normalize the gravitational Sachs-Wolfe correlation (42).
standard math Liouville's theorem applies to the f(omega/T) distribution in the phase space of the action (23).
Basis for Eq. (24), relating temperature variation to frequency variation.
ad hoc to paper Regularization schemes (dimensional/adiabatic) or the cosmological-constant mismatch factor determine the coincident vev.
Section IV: the paper does not fix <phi* phi> from first principles; the magnitude of mu is regularization-dependent.

invented entities (1)

Light charged scalar field phi beyond the Standard Model
purpose: Provides the vacuum polarization that modifies photon propagation in Eq. (10).
No independent evidence is given; mass and coupling are chosen so the CMB corrections are noticeable. The paper notes the Higgs does not directly couple this way.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Vacuum Polarization, Geodesic Equation and Sachs-Wolfe Effect." pith.science (2026). https://pith.science/paper/VXK7TXQO

@misc{pith2026250208748,
  author       = {Pith},
  title        = {Pith review of: Vacuum Polarization, Geodesic Equation and Sachs-Wolfe Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXK7TXQO}},
  note         = {Machine review of arXiv:2502.08748}
}

abstract

We show that the null geodesic equation for photons is modified in the presence of a charged scalar field, with quantum fluctuations acting as an effective mass term that changes the null paths to timelike curves. This effect can be interpreted as a vacuum polarization phenomenon in curved spacetime. The resulting contribution to the Sachs-Wolfe effect varies with photon frequency, leading to frequency-dependent corrections to the cosmic microwave background (CMB) blackbody spectrum in the form of a $\mu$-distortion, as well as modifications to the CMB power spectrum. We estimate these within a standard inflationary scenario and find that while the correction to the CMB power spectrum is significant when the scalar field is light, the magnitude of the $\mu$-distortion depends strongly on the regularization prescription.

Figures

Figures reproduced from arXiv: 2502.08748 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 13 canonical work pages

[1]

Vacuum polarization and photon mass in inflation

T. Prokopec and R. P. Woodard, Vacuum polarization and photon mass in inflation, Am. J. Phys. 72, 60-72 (2004), [arXiv:astro-ph/0303358 [astro-ph]]

work page Pith review arXiv 2004
[3]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, Princeton University Press (October 24, 2017)

work page 2017
[4]

The Motion of Small Bodies in Space-time

R. Geroch and J. O. Weatherall, The Motion of Small Bodies in Space-time, Commun. Math. Phys. 364, 607-634 (2018), [arXiv:1707.04222 [gr-qc]]

work page Pith review arXiv 2018
[5]

J. O. Weatherall, Geometry and Motion in General Relativity, Einstein Stud. 15, 207 (2020), [arXiv:1810.09046 [physics.hist-ph]]

work page Pith review arXiv 2020
[6]

Null geodesic congruences, gravitational lensing and CMB intensity profile

A. Kaya, Null geodesic congruences, gravitational lensing and CMB intensity profile, JCAP 01, 065 (2021), [arXiv:2010.10551 [gr-qc]]

work page Pith review arXiv 2021
[7]

The imprint of primordial gravitational waves on the CMB intensity profile

A. Kaya, The imprint of primordial gravitational waves on the CMB intensity profile, Phys. Lett. B 817, 136353 (2021), [arXiv:2105.02236 [astro-ph.CO]]

work page Pith review arXiv 2021
[8]

Durrer, Gauge Invariant Cosmological Perturbation Theory With Seeds, Phys

R. Durrer, Gauge Invariant Cosmological Perturbation Theory With Seeds, Phys. Rev. D 42 (1990), 2533

work page 1990
[9]

Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, 2005

V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, 2005

work page 2005

Show all 18 references

[10]

Chluba and R

J. Chluba and R. A. Sunyaev, The evolution of CMB spectral distortions in the early Universe, Mon. Not. Roy. Astron. Soc. 419, 1294 (2012), [arXiv:1109.6552 [astro-ph.CO]]

2012 arXiv
[11]

Tashiro, CMB spectral distortions and energy release in the early universe, PTEP 2014, no.6, 06B107 (2014)

H. Tashiro, CMB spectral distortions and energy release in the early universe, PTEP 2014, no.6, 06B107 (2014). 12

2014
[12]

Chluba, A

J. Chluba, A. Kogut, S. P. Patil, M. H. Abitbol, N. Aghanim, Y. Ali-Ha ¨ ımoud, M. A. Amin, J. Aumont, N. Bartolo and K. Basu, et al. Spectral Distortions of the CMB as a Probe of Inflation, Recombination, Structure Formation and Particle Physics: Astro2020 Science White Paper...

2019 arXiv
[13]

Subramanian, The origin, evolution and signatures of primordial magnetic fields, Rept

K. Subramanian, The origin, evolution and signatures of primordial magnetic fields, Rept. Prog. Phys. 79, 076901 (2016), [arXiv:1504.02311 [astro-ph.CO]]

2016 arXiv
[14]

Kaya, Superhorizon Electromagnetic Field Background from Higgs Loops in Inflation, JCAP 03 (2018) 046, [arXiv:1801.02032 [gr-qc]]

A. Kaya, Superhorizon Electromagnetic Field Background from Higgs Loops in Inflation, JCAP 03 (2018) 046, [arXiv:1801.02032 [gr-qc]]

2018 arXiv
[15]

Parker and S

] L. Parker and S. A. Fulling, Adiabatic regularization of the energy momentum tensor of a quan- tized field in homogeneous spaces, Phys. Rev. D9 (1974) 341

1974
[16]

Parker, Amplitude of Perturbations from Inflation, [arXiv:hep-th/0702216 [hep-th]]

L. Parker, Amplitude of Perturbations from Inflation, [arXiv:hep-th/0702216 [hep-th]]

arXiv
[17]

Lewis and A

A. Lewis and A. Challinor, Weak gravitational lensing of the CMB, Phys. Rept. 429 (2006) 1, [arXiv:astroph/0601594 [astro-ph]]

2006
[18]

Bucciotti, A Note on the Existence of Equal Time Correlators, [arXiv:2410.01903 [hep-th]]

B. Bucciotti, A Note on the Existence of Equal Time Correlators, [arXiv:2410.01903 [hep-th]]

arXiv
[19]

J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary mod- els, JHEP 05, 013 (2003), [arXiv:astro-ph/0210603 [astro-ph]]. 13

2003 arXiv

Pith tools

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

[1] [1]

Vacuum polarization and photon mass in inflation

T. Prokopec and R. P. Woodard, Vacuum polarization and photon mass in inflation, Am. J. Phys. 72, 60-72 (2004), [arXiv:astro-ph/0303358 [astro-ph]]

work page Pith review arXiv 2004

[2] [3]

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, Princeton University Press (October 24, 2017)

work page 2017

[3] [4]

The Motion of Small Bodies in Space-time

R. Geroch and J. O. Weatherall, The Motion of Small Bodies in Space-time, Commun. Math. Phys. 364, 607-634 (2018), [arXiv:1707.04222 [gr-qc]]

work page Pith review arXiv 2018

[4] [5]

J. O. Weatherall, Geometry and Motion in General Relativity, Einstein Stud. 15, 207 (2020), [arXiv:1810.09046 [physics.hist-ph]]

work page Pith review arXiv 2020

[5] [6]

Null geodesic congruences, gravitational lensing and CMB intensity profile

A. Kaya, Null geodesic congruences, gravitational lensing and CMB intensity profile, JCAP 01, 065 (2021), [arXiv:2010.10551 [gr-qc]]

work page Pith review arXiv 2021

[6] [7]

The imprint of primordial gravitational waves on the CMB intensity profile

A. Kaya, The imprint of primordial gravitational waves on the CMB intensity profile, Phys. Lett. B 817, 136353 (2021), [arXiv:2105.02236 [astro-ph.CO]]

work page Pith review arXiv 2021

[7] [8]

Durrer, Gauge Invariant Cosmological Perturbation Theory With Seeds, Phys

R. Durrer, Gauge Invariant Cosmological Perturbation Theory With Seeds, Phys. Rev. D 42 (1990), 2533

work page 1990

[8] [9]

Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, 2005

V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, 2005

work page 2005