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arxiv: 2606.06900 · v1 · pith:KGWCSHBWnew · submitted 2026-06-05 · 🌌 astro-ph.CO

Inverse-Scattering Reconstruction of Inflation from Scalar and Tensor Primordial Spectra

Jorge Mastache , Allan Hurtado This is my paper

Pith reviewed 2026-06-27 21:18 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords inverse scatteringinflationary potentialsprimordial power spectraJost functionstensor-to-scalar ratioMukhanov-Sasaki equationslow-roll features
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The pith

Inverse scattering reconstructs inflationary potentials directly from scalar and tensor primordial spectra.

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

The authors recast the equations for scalar and tensor perturbations during inflation as a scattering problem on the half-line. This identification lets the power spectra be expressed using Jost functions that encode the growing mode amplitudes. If successful, the approach provides a way to recover the effective potential shapes that produced observed features in the primordial spectra without relying on slow-roll assumptions at all scales.

Core claim

By treating the Mukhanov-Sasaki equation as a Schrödinger-like equation, the Bunch-Davies condition corresponds to the Jost solution, and the freeze-out amplitude is captured by the Jost function. The scalar and tensor spectra are then written in terms of these functions, yielding an inverse-scattering formula for the tensor-to-scalar ratio as their ratio. Tests on quadratic and step potentials show that the reconstructed potentials match the dominant behavior of z''/z and a''/a.

What carries the argument

The Jost function, which encodes the freeze-out amplitude of the growing mode in the scattering formulation of the perturbation equations.

If this is right

  • The tensor-to-scalar ratio equals a ratio of Jost amplitudes from the scalar and tensor channels.
  • Reconstruction via the Born approximation recovers the main features of smooth slow-roll potentials.
  • Sharp features in the spectra lead to localized discrepancies in the reconstructed scalar potential.
  • The method connects spectral features directly to scattering off the effective potential.

Where Pith is reading between the lines

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

  • This framework could be applied to observed CMB data to infer potential steps or other transient violations of slow roll.
  • Extensions might include higher-order scattering terms beyond the Born approximation for stronger features.
  • Similar scattering methods could apply to other cosmological perturbation equations involving mode freezing.

Load-bearing premise

The Bunch-Davies initial condition can be identified with the asymptotic Jost solution on the half-line.

What would settle it

Compute the reconstructed effective potential from the exact power spectra of a quadratic inflation model and check whether it matches the known z''/z to within the expected Born approximation error.

Figures

Figures reproduced from arXiv: 2606.06900 by Allan Hurtado, Jorge Mastache.

Figure 1
Figure 1. Figure 1: FIG. 1: Scalar (left) and tensor (right) primordial power spectra obtained from numerical solutions of the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Reconstructed and simulated effective potentials for scalar and tensor perturbations in inflation. The [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

We develop an inverse-scattering framework to reconstruct the effective inflationary potentials governing scalar and tensor perturbations. By recasting the Mukhanov--Sasaki equation as a Schr\"odinger-like problem on the half-line, we identify the Bunch--Davies initial condition with the asymptotic Jost solution and show that the freeze-out amplitude of the growing mode is encoded in the corresponding Jost function. This allows the scalar and tensor primordial power spectra to be written in terms of $F^{(s)}_{\nu_s-\frac12}(k)$ and $F^{(t)}_{\nu_t-\frac12}(k)$, respectively, and leads to an inverse-scattering expression for the tensor-to-scalar ratio as a ratio of Jost amplitudes. We then test the reconstruction in the large-$k$ regime using the Born approximation, where the Marchenko equation becomes linear. As benchmarks, we consider a smooth quadratic potential and a step potential that transiently violates slow roll and generates localized features in the primordial spectra. The reconstructed effective potentials reproduce the dominant behavior of $z^{\prime\prime}/z$ and $a^{\prime\prime}/a$ for smooth slow-roll evolution, while localized discrepancies arise in the scalar sector when sharp features induce stronger scattering. Our results show that inverse scattering provides a physically transparent method for connecting features in the primordial spectra to the underlying inflationary dynamics, and that the Jost function acts as a sensitive diagnostic of departures from canonical slow-roll evolution.

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

2 major / 2 minor

Summary. The manuscript develops an inverse-scattering framework to reconstruct effective inflationary potentials for scalar and tensor modes. By recasting the Mukhanov-Sasaki equation as a half-line Schrödinger problem and identifying Bunch-Davies modes with asymptotic Jost solutions, the scalar and tensor power spectra are expressed in terms of the Jost functions F^{(s)}_{ν_s−1/2}(k) and F^{(t)}_{ν_t−1/2}(k). This yields an inverse-scattering formula for the tensor-to-scalar ratio as a ratio of Jost amplitudes. The approach is tested in the large-k regime via the Born approximation (linear Marchenko equation) on a smooth quadratic potential and a step potential that transiently violates slow roll, with the reconstructed z''/z and a''/a reproducing dominant slow-roll behavior while showing localized discrepancies for sharp features.

Significance. If the central mapping and reconstruction hold, the work supplies a physically transparent link between features in the primordial spectra and the underlying dynamics, with the Jost function positioned as a diagnostic for departures from canonical slow-roll. The dual treatment of scalar and tensor sectors and the use of standard quantum-mechanical scattering tools in a cosmological setting are strengths. The Born-approximation benchmarks demonstrate feasibility for smooth cases, though quantitative validation would strengthen the claim of utility for precision data analysis.

major comments (2)
  1. Abstract and benchmark section: the claim that the reconstructed potentials reproduce the dominant behavior of z''/z and a''/a is supported only by qualitative description; no quantitative error metrics (e.g., integrated squared difference or pointwise relative error between reconstructed and input potentials) are reported, which is load-bearing for assessing accuracy when sharp features induce stronger scattering.
  2. Born-approximation paragraph: the validity range of the linear Marchenko equation (Born approx) for the large-k regime is invoked without stated bounds on the potential strength or explicit comparison to the full nonlinear solution, undermining the benchmark results for the step-potential case where scattering is stronger.
minor comments (2)
  1. Notation: the precise definition of the indices ν_s and ν_t (and their relation to the slow-roll parameters) should be stated explicitly when first introducing F^{(s)}_{ν_s−1/2}(k) and F^{(t)}_{ν_t−1/2}(k).
  2. The manuscript would benefit from a short table or figure caption clarifying which quantities are reconstructed versus input for each benchmark potential.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our inverse-scattering framework. We address the major comments point-by-point below.

read point-by-point responses

  1. Referee: Abstract and benchmark section: the claim that the reconstructed potentials reproduce the dominant behavior of z''/z and a''/a is supported only by qualitative description; no quantitative error metrics (e.g., integrated squared difference or pointwise relative error between reconstructed and input potentials) are reported, which is load-bearing for assessing accuracy when sharp features induce stronger scattering.

    Authors: We agree that quantitative metrics would strengthen the assessment of reconstruction accuracy. In the revised manuscript we will add integrated squared differences and pointwise relative errors between the reconstructed and input z''/z and a''/a for both the quadratic and step-potential benchmarks. These additions will quantify the localized discrepancies noted for the step potential. revision: yes

  2. Referee: Born-approximation paragraph: the validity range of the linear Marchenko equation (Born approx) for the large-k regime is invoked without stated bounds on the potential strength or explicit comparison to the full nonlinear solution, undermining the benchmark results for the step-potential case where scattering is stronger.

    Authors: We will add an explicit discussion of the validity range of the Born approximation in the large-k regime, including standard bounds on potential strength derived from scattering theory. A direct numerical comparison to the full nonlinear Marchenko solution lies outside the scope of the present work, which focuses on the linear regime; we will note this limitation while clarifying that the approximation remains appropriate for the smooth and mildly featured cases examined. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper recasts the Mukhanov-Sasaki equation as a half-line Schrödinger problem using standard inverse scattering methods, identifying Bunch-Davies modes with asymptotic Jost solutions and expressing power spectra via Jost functions F. This mapping is a direct mathematical equivalence applied to the known equation, not a self-definition or fitted input renamed as prediction. The inverse-scattering formula for the tensor-to-scalar ratio follows from the same recasting. Benchmarks on quadratic and step potentials test reproduction of known z''/z and a''/a behavior in the Born approximation without reducing to self-referential fits. No load-bearing self-citations or uniqueness theorems from prior author work are invoked in the provided text; the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard analogy between the Mukhanov-Sasaki equation and a one-dimensional Schrödinger problem; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The Mukhanov-Sasaki equation can be recast as a Schrödinger-like problem on the half-line with Bunch-Davies conditions identified as asymptotic Jost solutions
    This mapping is the foundational step that enables the Jost-function expressions for the spectra.

pith-pipeline@v0.9.1-grok · 5792 in / 1375 out tokens · 21108 ms · 2026-06-27T21:18:40.842638+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

62 extracted references · 42 linked inside Pith

  1. [1]

    Scalar modes We now specialize in scalar perturbations. During slow-roll inflation, and neglecting higher-order contribu- tionsO(ϵ 2, η2), using the scalar Mukhanov variablez= a ˙ϕ H and using the relationsϵ ′ =ηHϵandH ′ =H 2(1−ϵ) the effective scalar potential becomes z′′ z = 1 τ 2 2 + 3ϵ+ 3 2 η (45) withν s, in Eq. (41), is approximately constant during...

  2. [2]

    In this sec- tor, the Mukhanov–Sasaki equation is governed by the effective potentiala ′′/a, rather than by the scalar com- binationz ′′/z

    Tensorial modes We now specialize to tensor perturbations. In this sec- tor, the Mukhanov–Sasaki equation is governed by the effective potentiala ′′/a, rather than by the scalar com- binationz ′′/z. Therefore, tensor modes probe directly the background expansion history. During slow-roll infla- tion, and neglecting higher-order contributionsO(ϵ 2, η2), ta...

  3. [3]

    Using the inverse-scattering expressions for the scalar and tensor power spectra, Eqs

    Tensor-to-scalar ratio The tensor-to-scalar ratio is defined as the ratio be- tween the tensor and scalar primordial power spectra, r(k)≡ Pt(k) Ps(k) .(51) This quantity provides a direct measure of the relative amplitude of primordial gravitational waves with respect to scalar curvature perturbations and constitutes one of the primary observational probe...

  4. [4]

    V. F. Mukhanov, Gravitational Instability of the Uni- verse Filled with a Scalar Field, JETP Lett.41, 493 (1985)

    1985

  5. [5]

    Sasaki, Large Scale Quantum Fluctuations in the In- flationary Universe, Prog

    M. Sasaki, Large Scale Quantum Fluctuations in the In- flationary Universe, Prog. Theor. Phys.76, 1036 (1986)

    1986

  6. [6]

    B. A. Bassett, S. Tsujikawa, and D. Wands, Inflation dynamics and reheating, Rev. Mod. Phys.78, 537 (2006), arXiv:astro-ph/0507632

    Pith/arXiv arXiv 2006

  7. [7]

    Martin, C

    J. Martin, C. Ringeval, and V. Vennin, Encyclopædia Inflationaris: Opiparous Edition, Phys. Dark Univ.5-6, 75 (2014), arXiv:1303.3787 [astro-ph.CO]

    arXiv 2014

  8. [8]

    J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP05, 013, arXiv:astro-ph/0210603

    Pith/arXiv arXiv

  9. [9]

    Akramiet al.(Planck), Planck 2018 results

    Y. Akramiet al.(Planck), Planck 2018 results. X. Con- straints on inflation, Astron. Astrophys.641, A10 (2020), arXiv:1807.06211 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  10. [10]

    Forconi, W

    M. Forconi, W. Giar` e, E. Di Valentino, and A. Melchiorri, Cosmological constraints on slow roll inflation: An up- date, Phys. Rev. D104, 103528 (2021), arXiv:2110.01695 [astro-ph.CO]

    arXiv 2021

  11. [11]

    Wang, Inflation 2024, arXiv preprint (2024), arXiv:2407.03577 [astro-ph.CO]

    D. Wang, Inflation 2024, arXiv preprint (2024), arXiv:2407.03577 [astro-ph.CO]

    arXiv 2024

  12. [12]

    Liu and F

    J. Liu and F. Melia, Challenges to Inflation in the Post-Planck Era, Astrophys. J.967, 109 (2024), arXiv:2404.10956 [astro-ph.CO]

    arXiv 2024

  13. [13]

    J. A. Vazquez, M. Bridges, M. P. Hobson, and A. N. Lasenby, Model selection applied to reconstruction of the Primordial Power Spectrum, JCAP06, 006, arXiv:1203.1252 [astro-ph.CO]

    Pith/arXiv arXiv

  14. [14]

    Lodha, L

    K. Lodha, L. Pinol, S. Nesseris, A. Shafieloo, W. Sohn, and M. Fasiello, Searching for local features in primordial power spectrum using genetic algorithms, Mon. Not. Roy. Astron. Soc.530, 1424 (2024), arXiv:2308.04940 [astro- ph.CO]

    arXiv 2024

  15. [15]

    N. Kogo, M. Sasaki, and J. Yokoyama, Reconstructing the primordial spectrum with CMB temperature and po- larization, Phys. Rev. D70, 103001 (2004), arXiv:astro- ph/0409052

    arXiv 2004

  16. [16]

    Hunt and S

    P. Hunt and S. Sarkar, Reconstruction of the primordial power spectrum of curvature perturbations using mul- tiple data sets, JCAP01, 025, arXiv:1308.2317 [astro- ph.CO]

    Pith/arXiv arXiv

  17. [17]

    D. K. Hazra, A. Shafieloo, G. F. Smoot, and A. A. Starobinsky, Primordial features and Planck polariza- tion, JCAP09, 009, arXiv:1605.02106 [astro-ph.CO]

    Pith/arXiv arXiv

  18. [18]

    M. Aich, D. K. Hazra, L. Sriramkumar, and T. Souradeep, Oscillations in the inflaton potential: 12 Complete numerical treatment and comparison with the recent and forthcoming CMB datasets, Phys. Rev. D87, 083526 (2013), arXiv:1106.2798 [astro-ph.CO]

    Pith/arXiv arXiv 2013

  19. [19]

    Z.-K. Guo, D. J. Schwarz, and Y.-Z. Zhang, Reconstruc- tion of the primordial power spectrum from CMB data, JCAP08, 031, arXiv:1105.5916 [astro-ph.CO]

    Pith/arXiv arXiv

  20. [20]

    S. L. Bridle, A. M. Lewis, J. Weller, and G. Efstathiou, Reconstructing the primordial power spectrum, Mon. Not. Roy. Astron. Soc.342, L72 (2003), arXiv:astro- ph/0302306

    arXiv 2003

  21. [21]

    Gauthier and M

    C. Gauthier and M. Bucher, Reconstructing the primor- dial power spectrum from the CMB, JCAP10, 050, arXiv:1209.2147 [astro-ph.CO]

    Pith/arXiv arXiv

  22. [22]

    P. A. R. Adeet al.(Planck), Planck 2013 results. XXII. Constraints on inflation, Astron. Astrophys.571, A22 (2014), arXiv:1303.5082 [astro-ph.CO]

    Pith/arXiv arXiv 2013

  23. [23]

    P. A. R. Adeet al.(Planck), Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys.594, A20 (2016), arXiv:1502.02114 [astro-ph.CO]

    Pith/arXiv arXiv 2015

  24. [24]

    Aslanyan, L

    G. Aslanyan, L. C. Price, K. N. Abazajian, and R. Eas- ther, The Knotted Sky I: Planck constraints on the pri- mordial power spectrum, JCAP08, 052, arXiv:1403.5849 [astro-ph.CO]

    Pith/arXiv arXiv

  25. [25]

    Finelliet al.(CORE), Exploring cosmic origins with CORE: Inflation, JCAP04, 016, arXiv:1612.08270 [astro-ph.CO]

    F. Finelliet al.(CORE), Exploring cosmic origins with CORE: Inflation, JCAP04, 016, arXiv:1612.08270 [astro-ph.CO]

    Pith/arXiv arXiv

  26. [26]

    Shafieloo and T

    A. Shafieloo and T. Souradeep, Primordial power spec- trum from WMAP, Phys. Rev. D70, 043523 (2004), arXiv:astro-ph/0312174

    Pith/arXiv arXiv 2004

  27. [27]

    D. K. Hazra, A. Shafieloo, and T. Souradeep, Pri- mordial power spectrum from Planck, JCAP11, 011, arXiv:1406.4827 [astro-ph.CO]

    Pith/arXiv arXiv

  28. [28]

    Kadota, S

    K. Kadota, S. Dodelson, W. Hu, and E. D. Stewart, Pre- cision of inflaton potential reconstruction from CMB us- ing the general slow-roll approximation, Phys. Rev. D72, 023510 (2005), arXiv:astro-ph/0505158

    Pith/arXiv arXiv 2005

  29. [29]

    Dvorkin and W

    C. Dvorkin and W. Hu, Generalized Slow Roll for Large Power Spectrum Features, Phys. Rev. D81, 023518 (2010), arXiv:0910.2237 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  30. [30]

    Hu, Generalized Slow Roll for Non-Canonical Kinetic Terms, Phys

    W. Hu, Generalized Slow Roll for Non-Canonical Kinetic Terms, Phys. Rev. D84, 027303 (2011), arXiv:1104.4500 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  31. [31]

    S. M. Leach, Measuring the primordial power spec- trum: Principal component analysis of the cosmic mi- crowave background, Mon. Not. Roy. Astron. Soc.372, 646 (2006), arXiv:astro-ph/0506390

    Pith/arXiv arXiv 2006

  32. [32]

    Dvorkin and W

    C. Dvorkin and W. Hu, CMB Constraints on Principal Components of the Inflaton Potential, Phys. Rev. D82, 043513 (2010), arXiv:1007.0215 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  33. [33]

    Dvorkin and W

    C. Dvorkin and W. Hu, Complete WMAP Constraints on Bandlimited Inflationary Features, Phys. Rev. D84, 063515 (2011), arXiv:1106.4016 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  34. [34]

    J. M. Cline and L. Hoi, Inflationary potential reconstruc- tion for a wmap running power spectrum, JCAP06, 007, arXiv:astro-ph/0603403

    Pith/arXiv arXiv

  35. [35]

    R. Bean, D. J. H. Chung, and G. Geshnizjani, Recon- structing a general inflationary action, Phys. Rev. D78, 023517 (2008), arXiv:0801.0742 [astro-ph]

    Pith/arXiv arXiv 2008

  36. [36]

    Durakovic, P

    A. Durakovic, P. Hunt, S. P. Patil, and S. Sarkar, Recon- structing the EFT of Inflation from Cosmological Data, SciPost Phys.7, 049 (2019), arXiv:1904.00991 [astro- ph.CO]

    arXiv 2019

  37. [37]

    Kogutet al., The Primordial Inflation Explorer (PIXIE): A Nulling Polarimeter for Cosmic Mi- crowave Background Observations, JCAP07, 025, arXiv:1105.2044 [astro-ph.CO]

    A. Kogutet al., The Primordial Inflation Explorer (PIXIE): A Nulling Polarimeter for Cosmic Mi- crowave Background Observations, JCAP07, 025, arXiv:1105.2044 [astro-ph.CO]

    Pith/arXiv arXiv 2044

  38. [38]

    Andr´ eet al.(PRISM), PRISM (Polarized Radiation Imaging and Spectroscopy Mission): An Extended White Paper, JCAP02, 006, arXiv:1310.1554 [astro-ph.CO]

    P. Andr´ eet al.(PRISM), PRISM (Polarized Radiation Imaging and Spectroscopy Mission): An Extended White Paper, JCAP02, 006, arXiv:1310.1554 [astro-ph.CO]

    Pith/arXiv arXiv

  39. [39]

    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]

    Pith/arXiv arXiv 2012

  40. [40]

    Chluba, R

    J. Chluba, R. Khatri, and R. A. Sunyaev, CMB at 2x2 order: The dissipation of primordial acoustic waves and the observable part of the associated energy release, Mon. Not. Roy. Astron. Soc.425, 1129 (2012), arXiv:1202.0057 [astro-ph.CO]

    Pith/arXiv arXiv 2012

  41. [41]

    Khatri and R

    R. Khatri and R. A. Sunyaev, Forecasts for CMBµ andi-type spectral distortion constraints on the pri- mordial power spectrum on scales 8≲k≲10 4M pc−1 with the future Pixie-like experiments, JCAP06, 026, arXiv:1303.7212 [astro-ph.CO]

    Pith/arXiv arXiv

  42. [42]

    M. H. Abitbol, J. Chluba, J. C. Hill, and B. R. John- son, Prospects for Measuring Cosmic Microwave Back- ground Spectral Distortions in the Presence of Fore- grounds, Mon. Not. Roy. Astron. Soc.471, 1126 (2017), arXiv:1705.01534 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  43. [43]

    Chlubaet al., New horizons in cosmology with spectral distortions of the cosmic microwave background, Exper

    J. Chlubaet al., New horizons in cosmology with spectral distortions of the cosmic microwave background, Exper. Astron.51, 1515 (2021), arXiv:1909.01593 [astro-ph.CO]

    arXiv 2021

  44. [44]

    R. G. Newton,Inverse Schr¨ odinger scattering in three dimensions(Springer Science & Business Media, 2012)

    2012

  45. [45]

    Chadan and P

    K. Chadan and P. C. Sabatier,Inverse problems in quan- tum scattering theory(Springer Science & Business Me- dia, 2012)

    2012

  46. [46]

    Wang,Seismic inversion: Theory and applications (John Wiley & Sons, 2016)

    Y. Wang,Seismic inversion: Theory and applications (John Wiley & Sons, 2016)

    2016

  47. [47]

    Habib, K

    S. Habib, K. Heitmann, and G. Jungman, Inverse- scattering theory and the density perturbations from in- flation, Phys. Rev. Lett.94, 061303 (2005), arXiv:astro- ph/0409599

    arXiv 2005

  48. [48]

    Mastache, F

    J. Mastache, F. Zago, and A. Kosowsky, Inflationary Dynamics Reconstruction via Inverse-Scattering Theory, Phys. Rev. D95, 063511 (2017), arXiv:1611.03957 [astro- ph.CO]

    Pith/arXiv arXiv 2017

  49. [49]

    A. A. Starobinsky, Spectrum of adiabatic perturbations in the universe when there are singularities in the infla- tion potential, JETP Lett.55, 489 (1992)

    1992

  50. [50]

    J. A. Adams, B. Cresswell, and R. Easther, Inflationary perturbations from a potential with a step, Phys. Rev. D 64, 123514 (2001), arXiv:astro-ph/0102236

    Pith/arXiv arXiv 2001

  51. [51]

    G. A. Palma, D. Sapone, and S. Sypsas, Constraints on inflation with LSS surveys: features in the primordial power spectrum, JCAP06, 004, arXiv:1710.02570 [astro- ph.CO]

    Pith/arXiv arXiv

  52. [52]

    Silverstein and A

    E. Silverstein and A. Westphal, Monodromy in the CMB: Gravity Waves and String Inflation, Phys. Rev. D78, 106003 (2008), arXiv:0803.3085 [hep-th]

    Pith/arXiv arXiv 2008

  53. [53]

    McAllister, E

    L. McAllister, E. Silverstein, and A. Westphal, Grav- ity Waves and Linear Inflation from Axion Monodromy, Phys. Rev. D82, 046003 (2010), arXiv:0808.0706 [hep- th]

    Pith/arXiv arXiv 2010

  54. [54]

    Flauger, L

    R. Flauger, L. McAllister, E. Pajer, A. Westphal, and G. Xu, Oscillations in the CMB from Axion Monodromy Inflation, JCAP06, 009, arXiv:0907.2916 [hep-th]

    Pith/arXiv arXiv

  55. [55]

    Flauger, L

    R. Flauger, L. McAllister, E. Silverstein, and A. West- phal, Drifting Oscillations in Axion Monodromy, JCAP 13 10, 055, arXiv:1412.1814 [hep-th]

    Pith/arXiv arXiv

  56. [56]

    Silverstein and D

    E. Silverstein and D. Tong, Scalar speed limits and cos- mology: Acceleration from D-cceleration, Phys. Rev. D 70, 103505 (2004), arXiv:hep-th/0310221

    Pith/arXiv arXiv 2004

  57. [57]

    R. P. Woodard, The Case for Nonlocal Modifications of Gravity, Universe4, 88 (2018), arXiv:1807.01791 [gr-qc]

    Pith/arXiv arXiv 2018

  58. [58]

    D. J. Brooker, N. C. Tsamis, and R. P. Woodard, Ana- lytic approximation for the primordial spectra of single scalar potential models and its use in their reconstruc- tion, Phys. Rev. D96, 103531 (2017), arXiv:1708.03253 [gr-qc]

    Pith/arXiv arXiv 2017

  59. [59]

    D. J. Brooker, N. C. Tsamis, and R. P. Woodard, From Non-trivial Geometries to Power Spectra and Vice Versa, JCAP04, 003, arXiv:1712.03462 [gr-qc]

    Pith/arXiv arXiv

  60. [60]

    Z.-G. Liu, J. Zhang, and Y.-S. Piao, Phantom Inflation with A Steplike Potential, Phys. Lett. B697, 407 (2011), arXiv:1012.0673 [gr-qc]

    Pith/arXiv arXiv 2011

  61. [61]

    Liu and Y.-S

    Z.-G. Liu and Y.-S. Piao, Phantom Inflation in Little Rip, Phys. Lett. B713, 53 (2012), arXiv:1203.4901 [gr-qc]

    Pith/arXiv arXiv 2012

  62. [62]

    T. S. Bunch and P. C. W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A360, 117 (1978)

    1978