pith. sign in

arxiv: 2508.16530 · v2 · submitted 2025-08-22 · 🌀 gr-qc

ACT-Era Constraints on Single-Field Inflation in f(T) Teleparallel Gravity

Feng-Yi Zhang , Rongrong Zhai , Li-Yang Chen This is my paper

Pith reviewed 2026-05-18 21:19 UTC · model grok-4.3

classification 🌀 gr-qc
keywords single-field inflationf(T) teleparallel gravityscalar spectral indextensor-to-scalar ratioACT observationsslow-roll inflationhilltop models
0
0 comments X

The pith

Modest positive δ in f(T) teleparallel gravity suppresses the tensor-to-scalar ratio and revives sub-quadratic monomial and hilltop inflation models disfavored in GR under ACT data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines single-field slow-roll inflation in teleparallel gravity with the power-law form f(T) = C T^{2δ+1}. It uses both analytic approximations and high-precision numerics to track how the extra torsional term alters the slow-roll parameters for power-law, hilltop, and E-type plateau potentials. Positive δ reduces the tensor-to-scalar ratio while leaving the scalar spectral index close to its standard slow-roll value. This shift brings previously excluded sub-quadratic monomials and hilltop models into agreement with the higher n_s reported by ACT. The same mechanism limits the viable range of δ for plateau models and produces signatures that future B-mode measurements can test.

Core claim

For the specific f(T) = C T^{2δ+1} model, torsional corrections controlled by δ generically suppress r while keeping n_s near its slow-roll value. As a result, modest positive δ restores viability to sub-quadratic monomials and hilltop models that are disfavored in general relativity, whereas E-type plateaus remain compatible only for small δ; moderate δ drives the dynamics toward quadratic-like behavior and raises r. These predictions follow from the combined analytic and numerical treatment of the modified background and perturbation equations.

What carries the argument

The parameter δ in the torsional modification f(T) = C T^{2δ+1}, which introduces corrections that suppress the tensor-to-scalar ratio r while preserving the scalar spectral index near its slow-roll value.

If this is right

  • Sub-quadratic monomial potentials become compatible with ACT constraints once δ is modestly positive.
  • Hilltop models regain viability over a wider parameter range under the same torsional correction.
  • E-type plateau models stay viable only for small δ; larger values increase r and spoil the fit.
  • Improved B-mode data will discriminate among the potential classes and place quantitative upper bounds on δ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Teleparallel corrections offer a single-parameter route to reconcile simple single-field potentials with current data without invoking extra fields.
  • The selective suppression of r may help separate the effects of modified gravity from changes in the inflaton potential itself.
  • Similar δ-dependent behavior could appear in other teleparallel or modified-gravity cosmologies and could be checked against the same ACT and future CMB datasets.

Load-bearing premise

That analytic approximations plus high-precision numerical calculations accurately capture the full dynamics for the chosen f(T) form without significant higher-order corrections or breakdown of the slow-roll regime.

What would settle it

A future CMB B-mode measurement that finds the tensor-to-scalar ratio for sub-quadratic or hilltop potentials remains as high as in GR even when δ is positive and modest.

Figures

Figures reproduced from arXiv: 2508.16530 by Feng-Yi Zhang, Li-Yang Chen, Rongrong Zhai.

Figure 1
Figure 1. Figure 1: FIG. 1: Inflationary predictions for the power-law potential in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Inflationary predictions for the hilltop potential in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Inflationary predictions for the E-model with [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
read the original abstract

We reassess single-field slow-roll inflation in teleparallel gravity with $f(T)=C\,T^{2\delta+1}$, motivated by recent measurements from the Atacama Cosmology Telescope (ACT) that indicate a modest upward shift in the scalar spectral index $n_s$. Using analytic approximations together with high-precision numerical calculations, we compute primordial predictions for representative potentials: power-law monomials, hilltop models, and $E$-type plateaus. We find that torsional corrections controlled by $\delta$ generically suppress $r$ while keeping $n_s$ near its slow-roll value. As a result, modest positive $\delta$ can restore viability to sub-quadratic monomials and hilltop models that are disfavored in general relativity (GR), whereas $E$-type plateaus remain compatible only in a limited range of $\delta$: small $\delta$ may improve the fit but moderate $\delta$ drives the dynamics toward quadratic-like behaviour and increases $r$. These signatures are observationally testable: improved cosmic microwave background (CMB) B-mode measurements will further discriminate among potential classes and place quantitative bounds on torsional deviations from GR.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript reassesses single-field slow-roll inflation in teleparallel gravity using the specific power-law form f(T)=C T^{2δ+1}. Analytic approximations combined with high-precision numerical calculations are employed to derive primordial observables n_s and r for power-law monomials, hilltop models, and E-type plateaus. The central result is that modest positive δ suppresses r while leaving n_s near its GR slow-roll value, thereby restoring viability to sub-quadratic monomials and hilltop potentials that are disfavored under ACT constraints in general relativity; E-type plateaus remain compatible only for a limited range of small δ.

Significance. If the reported shifts in the (n_s, r) plane hold under the chosen f(T) ansatz, the work supplies a concrete, observationally testable mechanism by which torsional corrections can reconcile a class of otherwise excluded inflationary models with recent ACT data on n_s. The combination of analytic insight into the modified slow-roll parameters with high-precision numerics is a methodological strength that allows both qualitative understanding and quantitative predictions for future B-mode constraints.

major comments (1)
  1. Abstract and the section describing the numerical integration: the claim that analytic approximations together with high-precision numerical calculations accurately capture the dynamics for δ ≃ 0.01–0.1 rests on the unverified assumption that higher-order corrections in δ remain negligible and that the slow-roll conditions ε, η ≪ 1 are preserved throughout the trajectory, including near the end of inflation. An explicit error budget, convergence tests against the full field equations, or direct comparison of the approximate versus exact background evolution for representative δ values would be required to substantiate the reported suppression of r.
minor comments (2)
  1. Clarify the precise range of δ values explored numerically and whether any priors or fitting procedures were applied when comparing to ACT constraints.
  2. Specify the exact definition of the slow-roll parameters used in the f(T) background (e.g., whether they are the standard GR expressions or the modified torsional versions) and provide the leading-order analytic expressions for ε and η in terms of δ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The single major comment identifies a legitimate need for stronger substantiation of the numerical and analytic methods. We address the point below and will revise the manuscript to incorporate the requested verification.

read point-by-point responses

  1. Referee: Abstract and the section describing the numerical integration: the claim that analytic approximations together with high-precision numerical calculations accurately capture the dynamics for δ ≃ 0.01–0.1 rests on the unverified assumption that higher-order corrections in δ remain negligible and that the slow-roll conditions ε, η ≪ 1 are preserved throughout the trajectory, including near the end of inflation. An explicit error budget, convergence tests against the full field equations, or direct comparison of the approximate versus exact background evolution for representative δ values would be required to substantiate the reported suppression of r.

    Authors: We agree that an explicit validation strengthens the claims. The high-precision numerical integrations solve the complete background equations obtained from the f(T) action, while the analytic expressions provide the leading-order modification to the slow-roll parameters. For the modest values δ ≃ 0.01–0.1 considered, the torsional correction remains perturbative and the slow-roll parameters ε and η stay well below unity until the end of inflation is reached (defined by ε = 1). Nevertheless, to remove any ambiguity we will add a dedicated appendix containing: (i) an error budget estimating the size of O(δ²) terms, (ii) convergence tests with respect to integrator tolerances, and (iii) direct side-by-side comparisons of the analytic slow-roll predictions against the full numerical background evolution for representative monomial, hilltop, and plateau potentials at δ = 0.05 and δ = 0.1. These additions will explicitly confirm the reported suppression of r. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper treats δ as an independent free parameter in the chosen f(T) = C T^{2δ+1} form and computes n_s and r via analytic slow-roll expansions plus separate high-precision numerical integration of the background and perturbation equations for several fixed potentials. These outputs are then compared against external ACT and Planck data. No equation reduces a reported prediction to a fitted quantity defined from the target observables, no load-bearing premise rests on a self-citation chain, and no ansatz or uniqueness result is smuggled in from prior author work. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim depends on the free parameter δ that is varied to explore viability regions and on the domain assumption that the slow-roll regime remains valid under the torsional modification.

free parameters (2)
  • δ
    Exponent controlling the strength of the torsional correction in f(T)=C T^{2δ+1}
  • C
    Overall normalization constant in the f(T) function
axioms (2)
  • domain assumption Slow-roll approximation remains valid for the modified dynamics
    Invoked when computing primordial predictions for ns and r
  • ad hoc to paper The specific power-law form f(T)=C T^{2δ+1} captures the relevant deviation from GR
    Chosen to study torsional effects in teleparallel gravity

pith-pipeline@v0.9.0 · 5740 in / 1491 out tokens · 82084 ms · 2026-05-18T21:19:39.201474+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Harrison-Zeldovich attractor: From Planck to ACT results

    astro-ph.CO 2025-10 conditional novelty 6.0

    Nonminimal derivative coupling realizes the Harrison-Zeldovich attractor for monomial, hilltop, and α-attractor E-models, pulling them to the scale-invariant spectrum suggested by ACT data.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · cited by 1 Pith paper · 5 internal anchors

  1. [1]

    To leading order, Eqs

    Small-field regime x∗ ≪ 1 (near the hilltop) In this regime we may expand (1 − xp)α ≃ 1 +α xp + · · ·. To leading order, Eqs. (31)–(33) give ϵ1 ≃ A x2p−2 ∗ , A ≡ C1 p2 2(1 + 2δ) µ−2 V 2δ 1+2δ 0 , (38) N∗ ≃ µ2 C1 p (p − 2) V 2δ 1+2δ 0 x 2−p ∗ (p > 2). (39) 9 Eliminating x∗ yields the compact relations ϵ2∗ ≃ 2(p − 1) (p − 2) 1 N∗ , (40) ϵ1∗ ≃ p xp ∗ 2(1 + 2...

    work page

  2. [2]

    Then V (ϕ) = V0[1 − (1 − ∆)p] = V0 p ∆ − p(p − 1) 2 ∆2 + O(∆3)

    Near the edge x → 1− (“linear” vicinity) When x → 1−, write x = 1 − ∆ with 0 < ∆ ≪ 1. Then V (ϕ) = V0[1 − (1 − ∆)p] = V0 p ∆ − p(p − 1) 2 ∆2 + O(∆3) . (46) To leading order, the potential is linear in the displacement from the edge: V (ϕ) ≃ λ (µ − ϕ), λ ≡ p V0 µ , ∆ = µ − ϕ µ . (47) In this linear vicinity one has V,ϕ ≃ − λ and V,ϕϕ ≃ 0, so the inflationa...

    work page

  3. [3]

    (22) and (23)

    Numerical Predictions and Observational Constraints As a representative example we take p = 4 and compute ( ns, r) using Eqs. (22) and (23). Fig. 2 displays the results for several values of the torsion parameter δ, together with the latest P–ACT–LB–BK18 constraints in the ( ns, r) plane. In the GR limit ( δ = 0), the case N∗ = 50 is excluded, while for N...

    work page

  4. [4]

    Expanding to leading order in y we obtain ϵ1∗ ≃ 2A2C1 1 + 2δ V 2δ 1+2δ 0 y2 ∗, (52) N∗ ≃ 1 2A2C1 V − 2δ 1+2δ 0 1 y∗ =⇒ y∗ ≃ 1 2A2C1 V − 2δ 1+2δ 0 1 N∗

    Plateau (large-field) limit: y∗ ≪ 1 (standard Starobinsky plateau) Horizon exit occurs on the plateau, y∗ ≡ e−Aϕ∗ ≪ 1, and the integral for N∗ is dominated by y ≳ y∗ (so the approximation V ≃ V0 is valid in the integrand). Expanding to leading order in y we obtain ϵ1∗ ≃ 2A2C1 1 + 2δ V 2δ 1+2δ 0 y2 ∗, (52) N∗ ≃ 1 2A2C1 V − 2δ 1+2δ 0 1 y∗ =⇒ y∗ ≃ 1 2A2C1 V ...

    work page

  5. [5]

    y∗ ≃ 1) Horizon exit occurs near the origin so that Aϕ∗ ≪ 1 and V (ϕ) ≃ V0(Aϕ)2 ≡eV ϕ 2, eV ≡ V0A2, i.e

    Small-field / quadratic limit: Aϕ∗ ≪ 1 (i.e. y∗ ≃ 1) Horizon exit occurs near the origin so that Aϕ∗ ≪ 1 and V (ϕ) ≃ V0(Aϕ)2 ≡eV ϕ 2, eV ≡ V0A2, i.e. the potential is well approximated by a quadratic monomial over the interval [ ϕ∗, ϕend]. For V =eV ϕ2 the slow-roll quantities become (exact-in-form) ϵ1(ϕ) = 2C1 1 + 2δ eV 2δ 1+2δ ϕ− 2 1+2δ , (64) N∗ ≃ 1 + ...

    work page

  6. [6]

    Numerical Predictions and Observational Constraints In Fig. 3 we present the numerical predictions of the E-model withA = p 2/3 (Starobinsky potential) within the f(T ) gravity framework, confronted with the latest CMB measurements (P-ACT-LB-BK18). For δ = 0, corresponding to the standard GR case, the Starobinsky model is found to lie almost entirely outs...

    work page

  7. [7]

    Starobinsky, Physics Letters B 91, 99 (1980)

    A. Starobinsky, Physics Letters B 91, 99 (1980)

    work page 1980

  8. [8]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982)

    work page 1982

  9. [9]

    A. H. Guth, Phys. Rev. D 23, 347 (1981)

    work page 1981

  10. [10]

    Linde, Physics Letters B 108, 389 (1982)

    A. Linde, Physics Letters B 108, 389 (1982)

    work page 1982

  11. [11]

    Akrami et al

    Y. Akrami et al. (Planck Collaboration), Astronomy & Astrophysics 641, A10 (2020)

    work page 2020

  12. [12]

    P. A. R. Ade et al. (BICEP/Keck Collaboration), Phys. Rev. Lett. 127, 151301 (2021)

    work page 2021

  13. [13]

    The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and $\Lambda$CDM Parameters

    T. Louis et al., (2025), arXiv:2503.14452 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  14. [14]

    The Atacama Cosmology Telescope: DR6 Constraints on Extended Cosmological Models

    E. Calabrese et al., (2025), arXiv:2503.14454 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  15. [15]

    Adame et al., Journal of Cosmology and Astroparticle Physics 2025 (04), 012

    A. Adame et al., Journal of Cosmology and Astroparticle Physics 2025 (04), 012

    work page 2025

  16. [16]

    Adame et al., Journal of Cosmology and Astroparticle Physics 2025 (02), 021

    A. Adame et al., Journal of Cosmology and Astroparticle Physics 2025 (02), 021

    work page 2025

  17. [17]

    Translation of Einstein's Attempt of a Unified Field Theory with Teleparallelism

    A. Unzicker and T. Case, Translation of einstein’s attempt of a unified field theory with teleparallelism (2005), arXiv:physics/0503046 [physics.hist-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  18. [18]

    Ferraro and F

    R. Ferraro and F. Fiorini, Physical Review D 75, 10.1103/physrevd.75.084031 (2007)

    work page doi:10.1103/physrevd.75.084031 2007

  19. [19]

    Ferraro and F

    R. Ferraro and F. Fiorini, Physical Review D 78, 10.1103/physrevd.78.124019 (2008)

    work page doi:10.1103/physrevd.78.124019 2008

  20. [20]

    G. R. Bengochea and R. Ferraro, Physical Review D 79, 10.1103/physrevd.79.124019 (2009)

    work page doi:10.1103/physrevd.79.124019 2009

  21. [21]

    E. V. Linder, Phys. Rev. D 81, 127301 (2010)

    work page 2010

  22. [22]

    Wu and H

    P. Wu and H. Yu, Physics Letters B 693, 415–420 (2010)

    work page 2010

  23. [23]

    X. FU, P. WU, and H. YU, International Journal of Modern Physics D 20, 1301–1311 (2011)

    work page 2011

  24. [24]

    Matter Bounce Cosmology with the f(T) Gravity

    Y.-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta, and E. N. Saridakis, Class. Quant. Grav. 28, 215011 (2011), arXiv:1104.4349 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  25. [25]

    Wu and H

    P. Wu and H. Yu, Physics Letters B 703, 223–227 (2011)

    work page 2011

  26. [27]

    Geng, C.-C

    C.-Q. Geng, C.-C. Lee, and E. N. Saridakis, Journal of Cosmology and Astroparticle Physics 2012 (01), 002

    work page 2012

  27. [28]

    D. LIU, P. WU, and H. YU, International Journal of Modern Physics D 21, 1250074 (2012)

    work page 2012

  28. [30]

    Basilakos, S

    S. Basilakos, S. Capozziello, M. De Laurentis, A. Paliathanasis, and M. Tsamparlis, Phys. Rev. D 88, 103526 (2013)

    work page 2013

  29. [31]

    Nesseris, S

    S. Nesseris, S. Basilakos, E. N. Saridakis, and L. Perivolaropoulos, Physical Review D 88, 10.1103/physrevd.88.103010 (2013)

    work page doi:10.1103/physrevd.88.103010 2013

  30. [32]

    Iorio and E

    L. Iorio and E. N. Saridakis, Monthly Notices of the Royal Astronomical Society 427, 1555–1561 (2012)

    work page 2012

  31. [33]

    Transition redshift in $f(T)$ cosmology and observational constraints

    S. Capozziello, O. Luongo, and E. N. Saridakis, Transition redshift in f(t) cosmology and observational constraints (2015), arXiv:1503.02832 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  32. [34]

    R. C. Nunes, A. Bonilla, S. Pan, and E. N. Saridakis, The European Physical Journal C 77, 10.1140/epjc/s10052-017-4798-5 (2017)

    work page doi:10.1140/epjc/s10052-017-4798-5 2017

  33. [35]

    Capozziello, G

    S. Capozziello, G. Lambiase, and E. N. Saridakis, The European Physical Journal C 77, 10.1140/epjc/s10052-017-5143-8 (2017)

    work page doi:10.1140/epjc/s10052-017-5143-8 2017

  34. [36]

    SpecBit, DecayBit and PrecisionBit: GAMBIT modules for com- puting mass spectra, particle decay rates and precision observables

    G. Otalora and M. J. Rebou¸ cas, The European Physical Journal C 77, 10.1140/epjc/s10052- 017-5367-7 (2017)

    work page doi:10.1140/epjc/s10052- 2017

  35. [37]

    B. Xu, H. Yu, and P. Wu, Astrophys. J. 855, 89 (2018)

    work page 2018

  36. [38]

    Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, Reports on Progress in Physics 79, 106901 (2016)

    work page 2016

  37. [39]

    Bahamonde, K

    S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud, and E. Di Valentino, Reports on Progress in Physics 86, 026901 (2023)

    work page 2023

  38. [40]

    H. A. Buchdahl, Mon. Not. Roy. Astron. Soc. 150, 1 (1970)

    work page 1970

  39. [41]

    Wu and H

    P. Wu and H. Yu, Physics Letters B 692, 176–179 (2010)

    work page 2010

  40. [42]

    Wu and H

    P. Wu and H. Yu, The European Physical Journal C 71, 10.1140/epjc/s10052-011-1552-2 (2011)

    work page doi:10.1140/epjc/s10052-011-1552-2 2011

  41. [43]

    Wei, X.-P

    H. Wei, X.-P. Ma, and H.-Y. Qi, Physics Letters B 703, 74–80 (2011). 18

    work page 2011

  42. [44]

    and, Research in Astronomy and Astrophysics 13, 757 (2013)

    work page 2013

  43. [45]

    Karami, A

    K. Karami, A. Abdolmaleki, S. Asadzadeh, and Z. Safari, The European Physical Journal C 73, 10.1140/epjc/s10052-013-2565-9 (2013)

    work page doi:10.1140/epjc/s10052-013-2565-9 2013

  44. [46]

    Rezazadeh, A

    K. Rezazadeh, A. Abdolmaleki, and K. Karami, Journal of High Energy Physics 2016, 10.1007/jhep01(2016)131 (2016)

    work page doi:10.1007/jhep01(2016)131 2016

  45. [47]

    Rezazadeh, A

    K. Rezazadeh, A. Abdolmaleki, and K. Karami, The Astrophysical Journal 836, 228 (2017)

    work page 2017

  46. [48]

    El Bourakadi, B

    K. El Bourakadi, B. Asfour, Z. Sakhi, M. Bennai, and T. Ouali, The European Physical Journal C 82, 10.1140/epjc/s10052-022-10762-7 (2022)

    work page doi:10.1140/epjc/s10052-022-10762-7 2022

  47. [49]

    Zhang, H

    F.-Y. Zhang, H. Yu, and W. Lin, Phys. Dark Univ. 44, 101482 (2024)

    work page 2024

  48. [50]

    Jawad, A

    A. Jawad, A. M. Sultan, and N. Azhar, Astrophys. Space Sci. 367, 48 (2022)

    work page 2022

  49. [51]

    A. D. Linde, Phys. Lett. B 129, 177 (1983)

    work page 1983

  50. [52]

    Boubekeur and D

    L. Boubekeur and D. H. Lyth, Journal of Cosmology and Astroparticle Physics 2005 (07), 010–010

    work page 2005

  51. [53]

    Kallosh and A

    R. Kallosh and A. Linde, Journal of Cosmology and Astroparticle Physics 2013 (10), 033–033

    work page 2013

  52. [54]

    J. J. M. Carrasco, R. Kallosh, and A. Linde, Physical Review D 92, 10.1103/phys- revd.92.063519 (2015). 19

    work page doi:10.1103/phys- 2015