Complex Inflaton Potentials with Nonminimal Coupling: Robust Inflation and Geometric Reheating

S. D. Campos (Federal University of S\~ao Carlos)

REVIEW 4 major objections 5 minor 73 references

Complex Inflaton Potentials with Nonminimal Coupling: Robust Inflation and Geometric Reheating

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A complex scalar field with a nonminimal coupling to gravity can drive inflation and, through the imaginary part of its potential, reheat the universe without extra fields.

desk verdict The reheating idea is buried under an undefined dynamical system: the complex field equations are never reduced to real dynamics, so the central claim is unsupported. read the letter →

arxiv 2602.20355 v2 pith:SPQZBNQ2 submitted 2026-02-23 gr-qc astro-ph.CO

S. D. Campos (Federal University of São Carlos) This is my paper

classification gr-qcastro-ph.CO PACS 98.80.Cq04.50.Kd

keywords complexinflatonnonminimalcouplingnon-Hermitiancosmologyreheatingalpha-attractorPTsymmetryghost-freeconditionscalar-tensortheory

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 aims to show that a single complex scalar field, with a real plateau potential and a small imaginary term that breaks Hermiticity, can account for both cosmic inflation and its exit into reheating. The real part of the field's energy-momentum tensor drives a standard slow-roll expansion whose spectral index and tensor-to-scalar ratio sit inside current CMB bounds. The imaginary part is negligible during slow roll but grows to the same size as the real part near the end of inflation, acting as an energy-transfer channel that reheats the universe without additional fields or hand-added friction. If the claim holds, inflation and reheating emerge from one geometric, non-Hermitian scalar-tensor structure.

What carries the argument

The central object is the complex potential V(Φ) = V_R(φ,χ) + i Δε φχ, with V_R built from a smoothed plateau form factor P(x) = 1/(1 + (x/μ)²), together with the nonminimal coupling F(φ,χ) R, where F = M_P² − ζ(φ² + χ²) must stay positive to avoid ghosts. The carrying mechanism is the decomposition of the energy-momentum tensor into real and imaginary parts, with spacetime geometry responding only to the real part; this defines a complex equation-of-state parameter w = p/ρ = w_R + i w_I and a relevance parameter R(N) = |ρ_I|/√(ρ_R² + ρ_I²). The complex rotation that diagonalizes the quadratic sector while keeping the kinetic terms canonical licenses interpreting complex mass eigenvalues as

What would settle it

Take the imaginary parts of Eqs. (18) for the reported background solutions: if Im[φ''] and Im[χ''] do not vanish (or are not removed by an explicit projection prescription such as 'keep the real part'), then no real trajectory satisfies the equations and the reported n_s, r, N_end are solutions of a different, unspecified system. A reader could check this by requiring Im[φ''] = Im[χ''] = 0 along the plotted trajectories.

Watch

Extended reading notes

Core claim

The paper's central claim is that a complex inflaton potential V = V_R(φ,χ) + i Δε φχ, combined with a nonminimal coupling ζΦ*Φ R, yields a viable inflationary background whose real sector behaves like a plateau-type α-attractor T-model while the imaginary sector acts as an effective PT-symmetric dissipation channel. Numerical integration of the Jordan-frame equations shows that ζ controls the duration of inflation (N_end ≈ 50–55 in the reported fits) and that Δε changes the real energy density by less than 10⁻⁵, so the CMB observables are insensitive to the non-Hermitian deformation: mapping to a single effective field in the Einstein frame gives n_s ≈ 0.961–0.969 and r < 10⁻³. The same num

Load-bearing premise

The load-bearing premise is that the real spacetime evolution is governed only by Re(T_μν) while the equations of motion for φ and χ (Eq. 18) contain complex source terms iΔεφ and iΔεχ; the paper never specifies how those complex equations are reduced to real trajectories, so the numerical background and the derived n_s, r, and N_end depend on that unstated reduction.

Editorial extensions

If this is right

Inflation models would not need a separate reheating sector: the same complex potential that sustains slow roll ends it through its imaginary part.
The predicted n_s and r are almost independent of Δε, so the model makes a narrow, testable prediction in the n_s–r plane.
The nonminimal coupling ζ becomes the main tuning dial for the duration of inflation, while Δε can be varied freely without disturbing the background.
Complex masses in this framework acquire a physical meaning as decay channels, connecting unstable-particle physics with cosmological reheating.

Reading between the lines

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

If the reduction of the complex equations of motion to real trajectories can be made explicit, the same construction may give a general prescription for open-system inflation beyond this model.
The claimed decoupling of Δε suggests a practical parameterization in which reheating efficiency can be tuned independently of the CMB spectrum; this could be probed through non-Gaussian signatures or reheating-temperature constraints.
The PT-symmetric interpretation points to laboratory analogues in optical or mechanical gain-loss systems, where the imaginary potential plays the same role as here.
Connecting the imaginary sector to Standard Model fields would determine the reheating temperature and the baryogenesis window, a step the paper leaves qualitative.

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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 short version: the paper adds an imaginary term to an inflaton potential to drive reheating, but the field equations as written are not a well-defined dynamical system for real fields, so the central claim is unsupported. What's new: the specific imaginary potential VI = Δε φχ as a geometric dissipation channel, and the claim that this gives reheating without extra fields. The paper assembles standard ingredients—nonminimal coupling, an α-attractor plateau, a radial form factor—and the observation that Δε leaves the real background almost untouched is plausible if you only take the real part of the stress tensor. The CMB numbers are the usual α-attractor numbers, and they are fitted rather than predicted: α is adopted from a Planck fit, V0 is normalized to As, and ζ, µ are tuned so N_end lands in [50,60].

The soft spot is not minor. The action (12) is complex, and the equations of motion (18) for the real fields φ and χ contain complex source terms iΔεχ and iΔεφ. If φ and χ are real, the imaginary part of those equations forces χ=φ=0 when Δε≠0, which contradicts every trajectory in the figures. The paper never says how the complex equations are reduced to real dynamics. If you take the real part, the iΔε terms vanish and the imaginary sector is a spectator—so the claimed reheating mechanism disappears. If you don't, the numerical integration is not defined. There is also a direct internal contradiction: Sec. V B and Fig. 4 say the relevance parameter R(N) becomes negligible near and after N_end, while Sec. VI and the abstract say it grows to O(1) and triggers reheating. The abstract's n_s range (0.968–0.971) also disagrees with Table I (0.961–0.969). No particle production or radiation calculation is given; reheating is asserted from the behavior of R(N).

Who gets value: someone working on non-Hermitian or PT-symmetric cosmology might find the model element worth thinking about, but they would have to rebuild the formulation from scratch. As it stands, the paper is not coherent enough for a serious referee. I would desk reject, with a note that a resubmission would be considered if the author specifies a consistent real reduction of the complex equations and actually computes a reheating rate or particle-production spectrum.

Referee Report

4 major / 5 minor

Summary. The paper proposes an inflationary model driven by a complex scalar field Φ=(φ+iχ)/√2 with nonminimal coupling ζ and a complex potential whose real part is an α-attractor-like plateau and whose imaginary part is V_I=Δε φχ. The authors derive Jordan-frame field equations, define a complex equation-of-state parameter w=w_R+i w_I and a relevance parameter R(N), and integrate the background numerically. They report that ζ controls N_end, that Δε has negligible effect on the real background, that the predicted spectral index and tensor-to-scalar ratio fall within Planck 2018 bounds, and that the imaginary sector grows to O(1) near the end of inflation and thereby triggers a geometric reheating mechanism without additional fields.

Significance. If the framework were well-posed, the idea of a non-Hermitian, PT-symmetric imaginary potential providing a dynamical exit from inflation would be a novel and potentially interesting contribution to inflationary model building. The paper includes a parameter scan, a sensitivity measure S_ρ, and explicit numerical tables, which are valuable if the underlying equations are reliable. However, the central claims are not supported: the complex equations of motion are never reduced to a well-defined real dynamical system, and the reheating assertion is internally contradicted by the paper's own Figure 4 and by the absence of any particle-production or radiation calculation. The observational 'prediction' is also largely a fitting exercise because the α-attractor shape, the potential normalization, and the parameter choices are all fixed by the same data being compared. Thus, while the idea is suggestive, the manuscript in its current form does not establish a viable model.

major comments (4)

[§III C, Eqs. (16)–(19)] The field equations for the real fields φ and χ contain explicit imaginary source terms +iΔε χ and +iΔε φ. Since φ and χ are real-valued, taking the imaginary part of (18) imposes Δε χ = 0 and Δε φ = 0, which contradicts every trajectory shown in Figs. 2–5. If instead one keeps only the real part of (18), the iΔε terms are discarded and the imaginary sector is a spectator, so the claimed O(1) growth of |w_I| and R(N) cannot arise. No complex-to-real reduction, averaging prescription, or projection rule is specified for the equations of motion; only the energy-momentum tensor is said to have its real part source the geometry. The numerical background and all derived quantities (N_end, n_s, r, R(N)) are therefore not well-defined.
[§V B, Fig. 4 vs. §VI and Abstract] Figure 4's caption and the text in §V B state that R(N) 'becomes negligible near and after N_end, where reheating takes place,' and that 'the relevance of the imaginary sector tends to vanish' above N_end. The abstract and §VI claim instead that non-Hermitian effects 'grow to O(1) near the end of inflation' and that R(N) 'rapidly grows to O(1) right after inflation.' These statements are mutually contradictory. Moreover, R(N) merely measures the relative size of an imaginary energy-density component; no calculation of particle production, radiation energy density, or reheating temperature is provided. The paper therefore does not demonstrate an efficient reheating mechanism.
[§IV B, Table I] The claimed agreement with Planck 2018 is not an independent prediction. The real potential is modeled as an α-attractor T-model and α is adopted from a fit to the same observational data (Ref. [33]); V0 is normalized to the observed scalar amplitude A_s; and ζ and μ are tuned such that N_end lies in [50,60] while χ²_CMB is minimized. The resulting n_s and r are therefore fitted attractor outputs, not robust falsifiable predictions. This does not by itself invalidate the model, but it substantially weakens the evidential weight of the comparison.
[§III C, Eqs. (22)–(23)] The paper states that 'spacetime geometry is determined solely by the real part of the energy–momentum tensor,' yet the modified Friedmann equations are written as 3F H^2 = ρ and -2F Ḣ = ... with ρ=ρ_R+iρ_I. No projection of ρ onto its real part is indicated in the Friedmann equations or in the numerical scheme. This is another instance of the missing reduction prescription: it is unclear whether the Friedmann equations are complex equations with complex solutions, or real equations operating on Re(ρ), and if the latter, how the complex scalar dynamics are simultaneously evolved.

minor comments (5)

[§V A, Fig. 2] The text refers to 'upper/lower panels' while the caption says 'Left/Right'; please align the wording with the actual figure layout.
[§II B, Eqs. (8), (16)–(17)] The plateau function is introduced as P(x) but is subsequently written as P(r) with r the radial field; use a single notation consistently.
[§II B] The statement 'the mean slope parameter is fixed by setting (ε_φ+ε_χ)/2 = 1' sets a dimensionful parameter to unity without specifying units; state that this is in Planck units or define the normalization explicitly.
[§IV B, χ²_CMB] The χ² definition contains a Heaviside term Θ(r−r_max) with an uncertainty σ_r, but no numerical value for σ_r is given, and the table reports A_ref_s values that appear to be unnormalized reference amplitudes. Clarify how the χ² values in Table I are computed.
[§III A, after Eq. (3)] The approximation for tanh²(y) is written as sinh²(y)/cosh(y), which appears to be a typo; it should be sinh²(y)/cosh²(y).

Circularity Check

3 steps flagged · score 6.0 of 10

The Planck agreement is a refit of α-attractor parameters (α adopted from Planck, ζ/µ tuned to χ²_CMB, V0 normalized to As), and the 'geometric reheating' claim is a definitional label on the imaginary-sector ratio rather than a dynamical mechanism.

fitted input called prediction [Sec. II B (α input, µ choice) + Sec. IV B (χ²_CMB, Table I); abstract claim]
"A recent comparison of α-attractor models with the latest cosmological data indicates that α=0.0962 +0.00046 −0.00047 [33]. Then, a simple comparison between potential (5) and (6) ... yields µ^2 = √(6α)/(1+√(6α)) ϕ^2+χ^2. ... The best χ^2-parameter values were obtained for the following pairs: (ζ, µ) = (0.1626,0.1100), (ζ, µ) = (0.0001,0.1011), and (ζ, µ) = (−0.0053,0.1000). The resulting n_s are, respectively, n_s≈0.961, n_s≈0.969, and n_s≈0.969..."

The potential shape is fixed by α, which is itself a fit to the same Planck CMB data later used as the target for n_s and r. The reported n_s and r are the standard slow-roll outputs of that α-attractor potential, so they are not independent predictions of the complex-field construction. In addition, the pairs (ζ,µ) are chosen by minimizing χ²_CMB, which contains n_s and A_s; the 'Planck-compatible' numbers in Table I and Figure 1 are therefore the fit objective, not a derived test.
fitted input called prediction [Sec. II B (V0 normalization) and Sec. IV B (χ²_CMB definition)]
"The parameter V0 sets the overall energy scale of the real part of the inflaton potential, fixed by the observed amplitude of the primordial scalar power spectrum, as noted by the COBE/Planck Collaboration [36]... V0=V^ref A_s/A^ref_s. ... χ^2_CMB ≡ (n_s−n_obs)^2/σ_ns^2 + (ln A_s−ln A_obs)^2/σ_lnAs^2 + Θ(r−r_max)(r−r_max)^2/σ_r^2."

The scalar amplitude A_s is put into the model by construction: V0 is rescaled to match the observed A_s. The same A_s then appears as a target term in χ²_CMB. Hence the A_s contribution to the goodness of fit is zero by construction and cannot be presented as a successful prediction of the model.

1 more flagged steps

self definitional [Sec. III C (geometry from Re Tμν), Sec. IV A (complex EoS), Sec. V B and Eq. (26) (R(N))]
"It is important to emphasize that, throughout this work, spacetime geometry is determined solely by the real part of the energy–momentum tensor, while the imaginary part acts as a signature of non-conservative dynamics. ... In this subsequent phase, |w_I| and R(N) grow rapidly, driving efficient reheating without perturbing the real FRW background during the observable inflationary window."

The quantity claimed to drive reheating, R(N)=|ρ_I|/√(ρ_R^2+ρ_I^2), is just the ratio of the imaginary energy density to the total. Since the imaginary sector is defined as not sourcing geometry (only Re Tμν is used) and as having negligible backreaction on the background, V_I does not enter the equations that determine the expansion or the real-field trajectory. The 'trigger' of reheating and the 'relevance' of the imaginary sector are therefore the same defined quantity; asserting that its growth causes reheating is a label, not a derived dynamical mechanism.

full rationale

There are no load-bearing self-citations here, so patterns 3–5 do not apply. The circularity is of the fitted-input and definitional kinds. The real-sector inflationary predictions are not independent: the α-attractor shape is fixed by an α taken from a fit to Planck data, ζ and µ are tuned to minimize a χ²_CMB that contains n_s and A_s, and V0 is normalized to A_s. The reported n_s, r, and A_s agreement is therefore the refit of a known α-attractor model, not a first-principles prediction of the complex-field mechanism. The novelty—the imaginary sector—is explicitly decoupled from the real background, and the reheating claim is supported only by the definition of |w_I| and R(N); no equation converts V_I into radiation or a decay width. Additionally, the equations of motion (18) are complex-valued for real fields, so the numerical trajectories are not well-defined from the stated system; this is a correctness risk independent of the circularity patterns. Because the central validation and the exit mechanism reduce to a fit and to a definitional parameter, a score of 6 is appropriate.

Assumptions & free parameters 9 free parameters · 6 assumptions · 2 invented entities

The central CMB numbers rest on a chain of fitted inputs: α from Planck-fit literature, V0 normalized to As, and ζ/μ tuned for N_end∈[50,60]. The new imaginary-sector physics rests on an ad hoc rule that only Re(Tμν) sources gravity and on an unsupported identification of imaginary EoS components with reheating.

free parameters (9)

nonminimal coupling ζ = 0.1626, 0.0001, -0.0053 (best-fit table)
Tuned so N_end ∈ [50,60] and χ² is minimized; controls the duration of inflation.
radial plateau scale μ = 0.1100, 0.1011, 0.1000
Scanned together with ζ; sets where the plateau begins and controls how many e-folds inflation lasts.
asymmetry parameter Δε = 1.0 in Table I; scanned over [-1,1]
Varied to test sensitivity of the real background; claimed to have negligible effect.
mass parameter m = 10^-6 M_P
Chosen by hand within the stated range 10^-8–10^-6 M_P; sets the quadratic mass terms.
quartic coupling λ = 10^-13 M_P
Chosen by hand to keep the potential sub-Planckian.
potential height V0 = rescaled to match As; reference A_ref_s ~103–162 in Table I
Normalized to the observed scalar amplitude, so As is an input rather than a prediction.
α-attractor parameter α = 0.0962
Adopted from a Planck fit in [33]; this largely determines the ns-r prediction.
plateau sharpness exponent s = 2
Chosen for a smooth transition; s≥6 would give a sharper plateau but is not used.
mean slope parameter A=(εφ+εχ)/2 = 1 (imposed)
Fixed 'without loss of generality' in Sec. II B; a parameter normalization choice.

assumptions (6)

ad hoc to paper Classical dynamics of real fields φ,χ can be defined from a complex action by using the complex equations of motion (18) and taking only the real part of Tμν to source gravity.
No derivation is given for how complex source terms iΔεχ produce real trajectories; this is the entry point for the non-Hermitian interpretation.
ad hoc to paper The imaginary part of the energy-momentum tensor does not source spacetime geometry; it is a bookkeeping channel for dissipation.
Used to set ρR and pR in the Friedmann equations (22)–(23); central to the reheating interpretation.
domain assumption Slow-roll single-field formulas apply to the effective radial field φ_eff in the Einstein frame despite two-field, non-Hermitian dynamics.
Invoked in Sec. IV B to compute ns, r, and As from Veff.
domain assumption The conformal Einstein-frame transformation with Ω²=F/M_P²>0 is valid and defines the physical perturbation variables.
Used to justify the ghost-free condition (11) and the effective single-field mapping.
ad hoc to paper VI=Δεφχ is a PT-symmetric deformation, relying on assigning opposite intrinsic parities to φ and χ.
Footnote 1 and Sec. V E; the parity assignment is chosen to make the model PT-symmetric and is not independently motivated.
standard math Standard FRW background and scalar-tensor equations with no higher-order derivative pathologies.
Assumes the classical action (12) and the usual cosmological dynamics used throughout the paper.

invented entities (2)

Imaginary sector of the potential (effective PT-symmetric energy-transfer channel)
purpose: Provides dissipation/reheating without explicit couplings to daughter fields; encoded in VI=Δεφχ.
No falsifiable prediction outside the model is given; it does not affect CMB observables by construction, and no reheating products are computed.
Complex equation-of-state components w_I, ρ_I, p_I as physical dissipation
purpose: Quantify non-conservative dynamics and trigger reheating.
These are renamed imaginary parts of the stress tensor; no independent observable such as a decay width or reheating temperature is predicted.

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Cite this review

Pith. "Pith review of Complex Inflaton Potentials with Nonminimal Coupling: Robust Inflation and Geometric Reheating." pith.science (2026). https://pith.science/paper/SPQZBNQ2

@misc{pith2026260220355,
  author       = {Pith},
  title        = {Pith review of: Complex Inflaton Potentials with Nonminimal Coupling: Robust Inflation and Geometric Reheating},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPQZBNQ2}},
  note         = {Machine review of arXiv:2602.20355}
}

abstract

We investigate an inflationary scenario driven by a complex scalar field nonminimally coupled to gravity and subject to a non-symmetric complex potential. The real part of the potential controls the cosmological background and realizes a plateau-type inflation compatible with $\alpha$-attractor $\mathrm{T}$-models, while the imaginary part acts as an effective non-Hermitian deformation encoding dissipative effects. Working in the Jordan frame and imposing ghost-free conditions on the effective Planck mass, we derive the background equations and define a complex equation-of-state parameter whose real part governs the expansion and whose imaginary part quantifies departures from conservative dynamics. Numerical integration shows that the duration of inflation is primarily controlled by the nonminimal coupling $\zeta$, whereas the complex asymmetry parameter $\Delta\varepsilon$ has a negligible impact on the real background: the real energy density and pressure vary by less than $10^{-5}$ as $\Delta\varepsilon$ is scanned over its allowed range. Mapping the two-field dynamics to an effective single-field description in the Einstein frame, we obtain a spectral index $n_s\simeq 0.968-0.971$ and a tensor-to-scalar ratio $r<10^{-3}$, fully consistent with Planck 2018 bounds. We introduce a relevance parameter and show that non-Hermitian effects remain strongly suppressed during slow roll but grow to $\mathcal{O}(1)$ near the end of inflation, triggering an efficient reheating phase without additional fields or {\it ad hoc} friction terms. In this sense, the imaginary sector behaves as an effective $\mathcal{PT}$-symmetric channel for energy transfer, providing a geometrical mechanism for inflation and its exit within a non-Hermitian scalar-tensor framework.

Figures

Figures reproduced from arXiv: 2602.20355 by the authors.

**Figure 2.** FIG. 2. Number of e-folds [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Relevance parameter [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 5.** Figure 5: FIG. 5. Absolute values of the real and imaginary parts of the effective equation of state, [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

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Works this paper leans on

73 extracted references · 8 linked inside Pith

[33]

Shojaeea, K

R. Shojaeea, K. Nozarib, and F. Darabia:α-attractors and reheating in a non-minimal infla- tionary model. Int. J. Mod. Phys. D 29(10), 2050077 (2020)

2020
[1]

A. G. Riesset al.:Observational evidence from supernova for an accelerating universe and cosmological constant. Astron. J. 116, 1009–1038 (1998)

1998
[2]

On the other hand, for the lower panel, we useµ= 0.1000 and ζ∈[−0.0060,0.0050]

For the upper panel, we fixµ= 0.1100 and restrict the analysis toζ∈[0.1600,0.1670] (near the conformal value). On the other hand, for the lower panel, we useµ= 0.1000 and ζ∈[−0.0060,0.0050]. These ranges forζand specific values forµwere based on the best results for the usual value estimated toN end ≈50∼60 [56, 57]. For both panels, we set 19 FIG. 2. Numb...
[3]

Near and aboveN end, the relevance of the imaginary sector tends to vanish; this means this sector can be viewed as a perturbative aspect for the real sector

It is interesting to note that as Nincreases, the real part of the energy density dominates; however, up toN≈40, the imaginary part of the energy density still represents nearly 20% of the total energy density. Near and aboveN end, the relevance of the imaginary sector tends to vanish; this means this sector can be viewed as a perturbative aspect for the ...

2018
[4]

Perlmutteret al.:Measurements of omega and lambda from 42 high-redshift supernovae

S. Perlmutteret al.:Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–585 (1999)

1999
[5]

Jaman and M

N. Jaman and M. Sami:What is needed of a scalar field if it is to unify inflation and late time acceleration?Galaxies 10, 51 (2022)

2022
[6]

Berera:Warm inflation

A. Berera:Warm inflation. Phys. Rev. Lett. 75, 3218 (1995)

1995
[7]

W. H. Kinney:Horizon crossing and inflations with largeη. Phys. Rev. D 72, 023515 (2005)

2005

Show all 73 references

[8]

Cheunget al.:The effective field theory of inflation

C. Cheunget al.:The effective field theory of inflation. JHEP 03, 014 (2008). 26

2008
[9]

Kobayashi, M

T. Kobayashi, M. Yamaguchi, and J. Yokoyama:GeneralizedG-inflation. Prog. Theor. Phys. 126, 511 (2011)

2011
[10]

Martin, C

J. Martin, C. Ringeval, and V. Vennin:Encyclopaedia inflationaris. Phys. Dark Univ. 5-6, 75–235 (2014)

2014
[11]

Baumann and L

D. Baumann and L. McAllister:Inflation and string theory. Cambridge Univ. Press, (2015)

2015
[12]

Pinol and S

L. Pinol and S. Renaux-Petel:The effective field theory of inflation beyond slow-roll. Phys. Rev. D 92, 083513 (2015)

2015
[13]

Ach´ ucarroet al.:Effective theories of single inflation when heavy fields matter

A. Ach´ ucarroet al.:Effective theories of single inflation when heavy fields matter. JHEP 01, 131 (2018)

2018
[14]

Cabass, E

G. Cabass, E. Pajer, and D. van der Woude:The effective field theory of inflationary pertur- bations. JHEP 08, 050 (2021)

2021
[15]

A. V. Yurov:Complex field as inflaton and quintessence. arXiv:hep-th/0208129v1

arXiv
[16]

M. Dias, J. Frazer, and M. C. David Marsh:Simple emergent power spectra from complex inflationary physics. Phys. Rev. Lett. 117, 141303 (2016)

2016
[17]

Zhang and S

R. Zhang and S. Zheng:Complex inflaton field. arXiv:2106.00977

arXiv
[18]

Bastero-Gilet al.:Towards a reliable effective field theory of inflation

M. Bastero-Gilet al.:Towards a reliable effective field theory of inflation. Phys. Lett. B 813, 136055 (2021)

2021
[19]

Denner and S

A. Denner and S. Dittmaier:The complex-mass scheme for perturbative calculations with unstable particles. Phys. Rept. 864, 1–163 (2020)

2020
[20]

Ashida, Z

Y. Ashida, Z. Gong, and M. Ueda:Non-hermitian physics. Adv. Phys. 69, 249 (2020)

2020
[21]

Chavanis:Cosmological models based on a complex scalar field with a power-law potential associated with a polytropic equation of state

P.-H. Chavanis:Cosmological models based on a complex scalar field with a power-law potential associated with a polytropic equation of state. Phys. Rev. D 106, 043502 (2022)

2022
[22]

Ahmadet al.:Baryogenesis in the paradigm of quintessential inflation

S. Ahmadet al.:Baryogenesis in the paradigm of quintessential inflation. Phys. Rev. D 100, 103525 (2019)

2019
[23]

I. M. Khalatnikov and A. Mezhlumian:The classical and quantum cosmology with a complex scalar field. Phys. Lett. A 169, 308–312 (1992)

1992
[24]

Amendola, I

L. Amendola, I. M. Khalatnikov, M. Litterio, and F. Occhionero:Quantum cosmology with a complex field. Phys. Rev. D 49, 1881 (1994)

1994
[25]

Bertolami, F

O. Bertolami, F. S. N. Lobo, and J. P´ aramos:Nonminimal coupling of perfect fluids to curvature. Phys. Rev. D 78, 064036 (2008)

2008
[26]

Faraoni:Cosmology in scalar-tensor gravity

V. Faraoni:Cosmology in scalar-tensor gravity. Kluwer Academic Publ., (2004). 27

2004
[27]

A. Yu. Kamenshchik, I. M. Khalatnikov, and A. V. Toporensky:Complex inflaton field in quantum cosmology. Int. J. Mod. Phys. D 6, 649–672 (1997)

1997
[28]

C. P. Burgess:Introduction to effective field theory. Ann. Rev. Nucl. Part. Sci 57, 329 (2007)

2007
[29]

Kallosh, A

R. Kallosh, A. Linde, and D. Roest:Superconformal inflationaryα-attractors. JHEP 2013, 198 (2013)

2013
[30]

Kallosh and A

R. Kallosh and A. Linde:Universality class in conformal inflation. JCAP 07, 002 (2013)

2013
[31]

Kallosh, A

R. Kallosh, A. Linde, and D. Roest:Large field inflation and doubleα-attractors. JHEP 2014, 052 (2014)

2014
[32]

Dimopoulos and C

K. Dimopoulos and C. Owen:Quintessential inflation withα-attractors. arXiv:1703.00305v3

arXiv
[34]

S. S. Mishra and V. Sahni:New models of quintessential inflation featuring plateau and hilltop potentials. Eur. Phys. J. C 85, 48 (2025)

2025
[35]

Bhattacharya, K

S. Bhattacharya, K. Dutta, M. R. Gangopadhyay, and A. Maharana:α-attractor inflation: models and predictions. Phys. Rev. D 107(10), 103530 (2023)

2023
[36]

Baumann:Tasi lectures on inflation

D. Baumann:Tasi lectures on inflation. arXiv:0907.5424v2

arXiv
[37]

N. K. Stein and W. H. Kinney:Natural inflation after Planck 2018. arXiv:2106.02089v2

2018 arXiv
[38]

Aghanimet al.(Planck Collaboration):Planck 2018 results

N. Aghanimet al.(Planck Collaboration):Planck 2018 results. VI. Cosmological parameters. A&A 641, A6 (2020)

2018
[39]

Linde:Alexei Starobinsky and modern cosmology

A. Linde:Alexei Starobinsky and modern cosmology. arXiv:2509.01675v2
[40]

Pozoet al.:Some inflationary models under the light of Planck 2018 results

D. Pozoet al.:Some inflationary models under the light of Planck 2018 results. arXiv:2311.04683

2018 arXiv
[41]

J. L. Cooket al.:Reheating predictions in single field inflation. arXiv:1502.04673

arXiv
[42]

Gorini, A

V. Gorini, A. Kamenshchik, U. Moschella, and V. Pasquier:Tachyons, scalar fields, and cosmology. Phys. Rev. D 69, 123512 (2004)

2004
[43]

Wald:General Relativity

R. Wald:General Relativity. University of Chicago Press, (1984). 17

1984
[44]

L. H. Ford:Gravitational particle creation and inflation. Phys. Rev. D 35(10), 2955 (1987)

1987
[45]

D. S. Salopek, J. R. Bond, and J. M. Bardeen:Designing density fluctuation spectra in inflation. Phys. Rev. D 40(6), 1753 (1989)

1989
[46]

Tsujikawa:Power-law inflation with a nonminimally coupled scalar field

S. Tsujikawa:Power-law inflation with a nonminimally coupled scalar field. Phys. Rev. D 62, 043512 (2000). 28

2000
[47]

H. B. Nielsen:Tachyons, monopoles and related topics(chapter Tachyons in field theory), pages 169–174. E. Recami - North Holland Publ., (1978)

1978
[48]

Sen:Field theory of tachyon matter

A. Sen:Field theory of tachyon matter. Mod. Phys. Lett. A 17, 1797–1804 (2002)

2002
[49]

Padmanabhan:Accelerated expansion of the universe driven by tachyonic matter

T. Padmanabhan:Accelerated expansion of the universe driven by tachyonic matter. Phys. Rev. D 66, 021301 (2002)

2002
[50]

Willenbrock:Mass and width of an unstable particle

S. Willenbrock:Mass and width of an unstable particle. arXiv:2203.11056

arXiv
[51]

M. N. Sergeenko:Complex masses of resonances in the potential approach. Nonlinear Phen. in Complex Syst. 17(4), 433–438 (2014)

2014
[52]

Klein and D

R. Klein and D. Roest:Exorcising the Ostrogradsky ghost in coupled systems. J. High Energ. Phys. 130 (2016)

2016
[53]

Ostrogradsky:M´ emoires sur les ´ equations diff´ erentielles, relatives au probl` eme des isop´ erim` etres

M. Ostrogradsky:M´ emoires sur les ´ equations diff´ erentielles, relatives au probl` eme des isop´ erim` etres. Mem. Ac. St. Petersbourg, VI(4), 385–517 (1850)
[54]

K. J. Ludwick:The viability of phantom dark energy: A review. Mod. Rev. Lett. A 32(28), 1730025 (2017)

2017
[55]

Vikman:Can dark energy evolve to the phantom?Phys

A. Vikman:Can dark energy evolve to the phantom?Phys. Rev. D 71, 023515 (2005)

2005
[56]

Singh, M

P. Singh, M. Sami, and N. Dadhich:Cosmological dynamics of a phantom field. Phys. Rev. D 68(2), 023522 (2003)

2003
[57]

F. S. N. Lobo, T. Harko, and M. A. S. Pinto:Modified gravity with nonminimal curva- ture–matter couplings: a framework for gravitationally induced particle creation. Universe 11(11), 356 (2025)

2025
[58]

A. R. Liddle and S. M. Leach:How long before the end of inflation were observable perturba- tions produced?Phys. Rev. D 68, 103503 (2003)

2003
[59]

Germ´ an, R

G. Germ´ an, R. G. Quaglia, and A. M. M. Colorado:Model independent bounds for the number of e-folds during the evolution of the universe. JCAP 03, 004 (2023)

2023
[60]

C. M. Bender and S. Boettcher,Real spectra in non-hermitian Hamiltonians havingPT- symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

1998
[61]

C. M. Bender:Making sense of non-hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)

2007
[62]

Mostafazadeh:Pseudo-hermitian representations of quantum mechanics

A. Mostafazadeh:Pseudo-hermitian representations of quantum mechanics. Int. J. Geom. Meth. Mod. Phys. 7, 1191 (2010)

2010
[63]

B. A. Bassett, S. Tsujikawa, and D. Wands:Inflation dynamics and reheating. Rev. Mod. Phys. 78, 537 (2006). 29

2006
[64]

Di Marco, E

A. Di Marco, E. Orazi, and G. Pradisi:Introduction to the number of e-folds in slow-roll inflation. Universe 10(7), 284 (2024)

2024
[65]

Rotter:A non-hermitian Hamiltonian operator and the physics of open quantum systems

I. Rotter:A non-hermitian Hamiltonian operator and the physics of open quantum systems. J. Phys. A 42, 153001 (2009)

2009
[66]

N. N. Bogoliubov and D. V. Shirkov:Introduction to the theory of quantized fields. Interscience, (1959)

1959
[67]

H. J. Rothe:Lattice gauge theories: an introduction. World Scientific, 2006

2006
[68]

Hatano and D

N. Hatano and D. R. Nelson:Localization transitions in non-hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996)

1996
[69]

Mostafazadeh:Pseudo-hermiticity versusPT-symmetry

A. Mostafazadeh:Pseudo-hermiticity versusPT-symmetry. J. Math. Phys. 43, 205–214 (2002)

2002
[70]

Breuer and F

H.-P. Breuer and F. Petruccione:The theory of open quantum systems. Oxford University Press, (2002)

2002
[71]

E. A. Calzetta and B. L. Hu:Nonequilibrium quantum fields. Phys. Rev. D 37, 2878–2900 (1988)

1988
[72]

Boyanovsky, H

D. Boyanovsky, H. J. de Vega, R. Holman, D.-S. Lee, and A. Singh:Dissipation via particle production in scalar field theories. Phys. Rev. D 51, 4419–4444 (1995)

1995
[73]

C. M. Bender and S. Boettcher:Real spectra in non-hermitian Hamiltonians havingPT- symmetry. Phys. Rev. Lett.80, 5243–5246 (1998)

1998

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [33]

Shojaeea, K

R. Shojaeea, K. Nozarib, and F. Darabia:α-attractors and reheating in a non-minimal infla- tionary model. Int. J. Mod. Phys. D 29(10), 2050077 (2020)

2020

[2] [1]

A. G. Riesset al.:Observational evidence from supernova for an accelerating universe and cosmological constant. Astron. J. 116, 1009–1038 (1998)

1998

[3] [2]

On the other hand, for the lower panel, we useµ= 0.1000 and ζ∈[−0.0060,0.0050]

For the upper panel, we fixµ= 0.1100 and restrict the analysis toζ∈[0.1600,0.1670] (near the conformal value). On the other hand, for the lower panel, we useµ= 0.1000 and ζ∈[−0.0060,0.0050]. These ranges forζand specific values forµwere based on the best results for the usual value estimated toN end ≈50∼60 [56, 57]. For both panels, we set 19 FIG. 2. Numb...

[4] [3]

Near and aboveN end, the relevance of the imaginary sector tends to vanish; this means this sector can be viewed as a perturbative aspect for the real sector

It is interesting to note that as Nincreases, the real part of the energy density dominates; however, up toN≈40, the imaginary part of the energy density still represents nearly 20% of the total energy density. Near and aboveN end, the relevance of the imaginary sector tends to vanish; this means this sector can be viewed as a perturbative aspect for the ...

2018

[5] [4]

Perlmutteret al.:Measurements of omega and lambda from 42 high-redshift supernovae

S. Perlmutteret al.:Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–585 (1999)

1999

[6] [5]

Jaman and M

N. Jaman and M. Sami:What is needed of a scalar field if it is to unify inflation and late time acceleration?Galaxies 10, 51 (2022)

2022

[7] [6]

Berera:Warm inflation

A. Berera:Warm inflation. Phys. Rev. Lett. 75, 3218 (1995)

1995

[8] [7]

W. H. Kinney:Horizon crossing and inflations with largeη. Phys. Rev. D 72, 023515 (2005)

2005