Inflation with the standard and Randall-Sundrum model in the Two-time Physics

Vo Quoc Phong

arxiv: 2603.15172 · v5 · pith:P375XGORnew · submitted 2026-03-16 · ✦ hep-ph

Inflation with the standard and Randall-Sundrum model in the Two-time Physics

Vo Quoc Phong This is my paper

Pith reviewed 2026-05-21 10:56 UTC · model grok-4.3

classification ✦ hep-ph

keywords cosmological inflationRandall-Sundrum modeltwo-time physicstensor-to-scalar ratioextra dimensionsslow-roll parametersBICEP2Planck data

0 comments

The pith

The Randall-Sundrum II model with a two-time physics potential produces a higher tensor-to-scalar ratio that fits BICEP2 and Planck data while estimating M5 near 10^16 GeV.

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

This paper introduces a scalar field potential motivated by the Higgs-dilaton sector of two-time physics that obeys Z2 symmetry for arbitrary exponents. The slow-roll dynamics are evaluated both in ordinary four-dimensional spacetime and inside the Randall-Sundrum II braneworld. The five-dimensional geometry systematically increases the tensor-to-scalar ratio relative to the four-dimensional case. The resulting predictions lie inside the joint BICEP2–Planck allowed region and fix the fundamental scale M5 near 10^16 GeV. The same potential also permits smaller field exponents while remaining consistent with observation.

Core claim

The central claim is that the potential V(φ)=M⁴φ^{2n-2}(φ^{2n}+m^{2n})^{1/n-1} produces slow-roll inflation whose tensor-to-scalar ratio is larger in the Randall-Sundrum II model than in the standard four-dimensional model, lies in agreement with BICEP2 and Planck data, and thereby determines the five-dimensional Planck mass to be approximately 1–2 × 10^{16} GeV.

What carries the argument

The proposed inflationary potential derived from two-time physics, which is inserted into the slow-roll equations for both the four-dimensional and Randall-Sundrum II cosmologies.

If this is right

The tensor-to-scalar ratio is always higher in the RSII framework than in four dimensions for the same potential parameters.
Current data constrain the five-dimensional Planck mass M5 to the interval [1-2]×10^{16} GeV.
Lower values of the exponent n remain compatible with observations.
Inflation measurements can serve as a probe for the presence of extra dimensions.

Where Pith is reading between the lines

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

If the extra dimension modifies the effective potential, the quoted M5 bound would shift.
Future polarization data could separate the 4D and RSII predictions for the same potential.
Similar constructions may be explored in other warped geometries.

Load-bearing premise

The potential taken from the two-time physics Higgs-dilaton sector experiences no further warping corrections when placed in the Randall-Sundrum II background.

What would settle it

Observation of a tensor-to-scalar ratio lying below the lowest RSII prediction for any n and M5 consistent with the data would rule out the model.

Figures

Figures reproduced from arXiv: 2603.15172 by Vo Quoc Phong.

**Figure 2.** Figure 2: FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Compatibility between experimental and calculated [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

read the original abstract

We propose a scalar inflationary potential as $V(\phi)=M^4\phi^{2n-2}(\phi^{2n}+m^{2n})^{1/n-1}$. This potential is similar to the shaft inflation one. However, they satisfy the $Z_2$ symmetry for all $n$. The potential may come from the Higgs-dilaton potential in the two-time (2T) physics. The slow-roll scenario is recomputed in the 4-dimension (4D) and Randall-Sundrum II (RSII) frameworks. The tensor-to-scalar ratio in the RSII model is always higher than in the 4D model and is in good agreement with the experimental data of BICEP2 and Planck. Comparing this with Planck data, we estimate $M_5$ to be around $[1-2]\times 10^{16}$ GeV. Furthermore, the potential allows much lower scalar field exponents than other potentials, which results in high agreement with experimental data. Moreover, the results also reinforce the models that have the extra dimensions, should be focused. The inflation data can be used to test for the existence of the extra dimensions.

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.

Desk Editor's Note private letter to a colleague

Applies a 2T-inspired potential to both 4D and RSII inflation, gets higher r in RSII that fits data and backs out M5, but the direct transfer of the potential into the warped setup is the least checked step.

read the letter

This paper takes a potential inspired by the Higgs-dilaton sector in two-time physics, V(φ) = M^4 φ^{2n-2} (φ^{2n} + m^{2n})^{1/n -1}, which has Z2 symmetry for any n, and applies slow-roll inflation to it in both standard 4D and Randall-Sundrum II setups. The key result is that the tensor-to-scalar ratio comes out higher in the RSII case and aligns better with Planck and BICEP2 data, allowing them to estimate the 5D Planck scale M5 around 1-2 times 10^16 GeV. It also notes that this potential works with lower field exponents than usual. The comparison itself is straightforward and shows how the modified Friedmann equation in RSII affects the observables without altering the potential. That's the part that could interest people looking at extra dimensions in cosmology. The ability to fit data with smaller n is a practical advantage they highlight. The main concern is whether the potential really transfers directly to the RSII framework. The warped geometry and the bulk could introduce corrections to the effective 4D potential or to how the slow-roll parameters are defined, especially since the scalar field has to be embedded consistently. The M5 estimate relies on matching existing data after choosing n, M, and m, so it functions more as a fit than a sharp prediction. If the derivations in the full text handle the 5D embedding carefully, that would strengthen it, but the abstract leaves that step looking thin. This kind of work is aimed at cosmologists who study inflation in modified gravity or extra-dimensional models. Someone already thinking about RSII or 2T physics would find the direct 4D versus RSII comparison useful for seeing the impact on r. It has enough concrete claims and a link to data to merit a serious referee, even if revisions are needed on the consistency of the 5D application. I would recommend putting it through peer review, focusing the referees on whether the potential and slow-roll analysis hold up without additional 5D corrections.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes the inflationary potential V(φ)=M⁴ φ^{2n-2} (φ^{2n} + m^{2n})^{1/n-1} motivated by the Higgs-dilaton sector of two-time physics. Slow-roll parameters are recomputed in both standard 4D cosmology and the Randall-Sundrum II braneworld model. The central results are that the tensor-to-scalar ratio r is always higher in the RSII case than in 4D, agrees well with Planck and BICEP2 data, and permits an estimate M5 ≈ [1-2]×10^{16} GeV from the comparison; the potential is also said to allow lower exponents while fitting data and to support the relevance of extra dimensions.

Significance. If the effective 4D dynamics in the RSII geometry are correctly captured by the unmodified potential and standard slow-roll expressions, the comparison of r values could provide a testable signature distinguishing 4D from braneworld inflation and yield a concrete constraint on the 5D Planck scale. The numerical agreement with current data is presented as reinforcing extra-dimensional models, though the fitting of free parameters n, M, m to observations limits the predictive strength.

major comments (1)

[RSII analysis section] RSII analysis section: the modified Friedmann equation H² = (ρ/3M_p²)(1 + ρ/(2λ)) with λ tied to M5 is used to recompute slow-roll parameters, yet the identical potential V(φ) is inserted without deriving or justifying the absence of warp-factor corrections to the effective potential or to the definitions of ε and η. This assumption is load-bearing for both the reported inequality r_RSII > r_4D and the numerical extraction of M5 from Planck/BICEP2 data.

minor comments (2)

[Abstract] Abstract: the phrase 'the potential may come from the Higgs-dilaton potential in the two-time (2T) physics' should be supported by an explicit reference to the relevant 2T action or a short derivation.
[Potential definition] Potential definition: specify the allowed range of the exponent n for which Z₂ symmetry is preserved and how this form differs quantitatively from shaft inflation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the RSII analysis. We address the point raised below.

read point-by-point responses

Referee: [RSII analysis section] RSII analysis section: the modified Friedmann equation H² = (ρ/3M_p²)(1 + ρ/(2λ)) with λ tied to M5 is used to recompute slow-roll parameters, yet the identical potential V(φ) is inserted without deriving or justifying the absence of warp-factor corrections to the effective potential or to the definitions of ε and η. This assumption is load-bearing for both the reported inequality r_RSII > r_4D and the numerical extraction of M5 from Planck/BICEP2 data.

Authors: We agree that the manuscript would be strengthened by an explicit justification for employing the unmodified potential V(φ) in the RSII case. In the standard treatment of RSII braneworld models with a scalar field localized on the brane (as assumed here, consistent with the Higgs-dilaton origin in 2T physics), the effective 4D potential on the brane remains V(φ) because the warp factor enters the gravitational sector and induces the high-energy correction to the Friedmann equation, without generating additional multiplicative factors in the potential term for brane-confined fields. The slow-roll parameters are then recomputed using the modified Hubble rate. We will revise the RSII section to include a short paragraph with this reasoning and a reference to the standard effective 4D reduction in RSII inflation literature, thereby supporting the reported r_RSII > r_4D relation and the M5 estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard parameter estimation from modified Friedmann equation

full rationale

The paper proposes the potential V(φ) as possibly originating from 2T Higgs-dilaton sector, then recomputes slow-roll parameters ε and η using the standard 4D expressions and the RSII modified Friedmann equation H² = (ρ/3M_p²)(1 + ρ/(2λ)). The inequality r_RSII > r_4D follows directly from the extra ρ/(2λ) term increasing the Hubble rate for given ρ, which is an independent dynamical input. The numerical agreement with BICEP2/Planck and the M5 estimate [1-2]×10^16 GeV are obtained by matching the computed r(n, M, m, M5) to observed values; this is ordinary parameter fitting rather than a prediction that reduces to the input by construction. No self-definitional loop, no fitted quantity renamed as prediction, and no load-bearing self-citation chain appears in the derivation. The central results remain computed outputs from the two distinct background equations.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 1 invented entities

Based on abstract only; central claim rests on the proposed potential form, the validity of slow-roll in both geometries, and the assumption that RSII modifies only the tensor-to-scalar ratio without further corrections.

free parameters (3)

n
Exponent controlling potential shape and chosen to achieve data agreement.
M
Overall energy scale in the potential, adjusted to match observations.
m
Mass-scale parameter inside the potential.

axioms (2)

domain assumption Slow-roll approximation holds throughout the inflationary phase in both 4D and RSII.
Invoked to recompute slow-roll parameters without stating higher-order corrections.
ad hoc to paper The given potential originates from the Higgs-dilaton sector of two-time physics.
Stated as possible source but no derivation supplied in abstract.

invented entities (1)

RSII extra dimension no independent evidence
purpose: To raise the tensor-to-scalar ratio relative to 4D while preserving the potential form.
The extra dimension is assumed to affect only the effective gravitational dynamics without altering the scalar potential itself.

pith-pipeline@v0.9.0 · 5733 in / 1754 out tokens · 55606 ms · 2026-05-21T10:56:16.506196+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The tensor-to-scalar ratio in the RSII model is always higher than in the 4D model... estimate M5 to be around [1-2]×10^{16} GeV. ... shaft-warm inflation potential V(ϕ)=M⁴ϕ^{2n-2}(ϕ^{2n}+m^{2n})^{1/n-1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The slow-roll scenario is recomputed in the 4-dimension (4D) and Randall-Sundrum II (RSII) frameworks.

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.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 2 internal anchors

[1]

In Figure 2, it decreases withtbut asnincreases this decrease becomes less

The solution of Eq.17 has the SR form, ϕ(t) = (2n+ 2) 2M4M 2m2n(1−n)t√ 6π + ϕ2n+2 i 2(n+ 1) 1 2n+2 , (18) in whichϕ i is the initial inflaton field. In Figure 2, it decreases withtbut asnincreases this decrease becomes less. There is a similarity between Fig. 2 and 1, the SR shape is formed. This slow-roll is usually stored in the value ofm. 0.00001 0.000...

work page
[2]

and in RSII model (Eq. 33). They all have similar shapes according totas shown in Figures 2 and 3. There is a similarity between Figure 2 and 3, butϕ(t) in Figure 3 shows a significant difference betweenn= 2 andn= 3 compared to Figure 2. This is also clearly shown in Figure 8. In Figure 4, the lines with differentNvalues are quite similar, so N only affec...

work page
[3]

/Multiply 10/Minus 6 4

/Multiply 10/Minus 6 3. /Multiply 10/Minus 6 4. /Multiply 10/Minus 6 5. /Multiply 10/Minus 6t 1.975 /Multiply 1019 1.980 /Multiply 1019 1.985 /Multiply 1019 1.990 /Multiply 1019 1.995 /Multiply 1019 2.000 /Multiply 1019 Φ/LParen1t/RParen1 n/Equal 2 n/Equal 3 FIG. 3. Theϕ(t) field in the RSII model withn= 2,3,M≃ 1015 GeV,M 4 ≃10 19 GeV,m≃10 18 GeV, andϕ i ...

work page
[4]

We can approximate as follows:n s = 0.006N. In Figs. 6 and 5,rdecreases slightly asn s increases. Whenn >4, the values ofrare quite close to each other, whereas forn <4,rchanges significantly asnincreases. In particular, whennincreases from 2 to 3,rchanges very strongly. This is different from the shaft inflation. rin RSII model is about 100 times larger ...

work page
[5]

7, with justn= 2,3 the resultrmatches the experimental data very well

From Fig. 7, with justn= 2,3 the resultrmatches the experimental data very well. In Figures 5, 6, and 7, the largernis, the closer the lines are to each other. In other words, a clear hierarchy of values for the quantities (r, ns) only occurs with small n. 7 In most models, the largernis, the smallerrbecomes, althoughncannot be much larger than 2. In our ...

work page
[6]

Guth,Inflationary universe: A possible solution to the horizon and flatness problems, Phys

Alan H. Guth,Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D23, 347 (1981)

work page 1981
[7]

P.A.R.Adeet al.(BICEP/Keck Collaboration),Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett.127, 151301 (2021)

work page 2018
[8]

1, Cambridge Uni- versity Press (1998)

Joseph Polchinski,String theory, Vol. 1, Cambridge Uni- versity Press (1998)

work page 1998
[9]

Ngo Phuc Duc Loc,Inflation with a class of concave in- flaton potentials in Randall–Sundrum model, Eur. Phys. J. C80, 768 (2020)

work page 2020
[10]

Lisa Randall and Raman Sundrum,Large Mass Hierar- chy from a Small Extra Dimension, Phys. Rev. Lett.83, 3370 (1999)

work page 1999
[11]

Lisa Randall and Raman Sundrum,An Alternative to Compactification, Phys. Rev. Lett.83, 4690 (1999)

work page 1999
[12]

Juan Maldacena and Alexey Milekhin,Humanly traversable wormholes, Phys. Rev. D103, 066007 (2021)

work page 2021
[13]

Konstantinos Dimopoulos,Shaft inflation, Phys. Lett. B 735, 75 (2014)

work page 2014
[14]

Konstantinos Dimopoulos,Shaft Inflation and the Planck satellite observations, arXiv:1510.06593

work page internal anchor Pith review Pith/arXiv arXiv
[15]

Ngo Phuc Duc Loc,Shaft inflation in Randall-Sundrum model,JCAP02020, (2023)

work page 2023
[16]

Itzhak Bars ,”The Standard Model of Particles and Forces in the Framework of 2T-physics”, Phys.Rev.D74 085019, 2006

work page 2006
[17]

I. Bars, C. Deliduman, and D. Minic,Supersymmetric two-time physics, Phys. Rev. D59, 125004 (1999)

work page 1999
[18]

Itzhak Bars,Two-time physics in field theory, Phys. Rev. D62, 046007 (2000)

work page 2000
[19]

High En- ergy Phys.2022, 5313952 (2022)

Vo Quoc Phong and Ngo Phuc Duc Loc, Adv. High En- ergy Phys.2022, 5313952 (2022)

work page 2022
[20]

Vo Quoc Phong, Quach Ai Mi, Nguyen Xuan Vinh, Sphaleron and gravitational wave with the Higgs-Dilaton potential in the Standard Model Two-Time Physics, Nucl. Phys. B1017, 116964 (2025)

work page 2025
[21]

Abdul Jawad, Amara Ilyas, Shamaila Rani,Warm mod- ified Chaplygin gas shaft inflation, Eur. Phys. J. C77, 131 (2017)

work page 2017
[22]

Bassett, and Imo- gen P

Roy Maartens, David Wands, Bruce A. Bassett, and Imo- gen P. C. Heard,Chaotic inflation on the brane, Phys. Rev. D62, 041301(R) (2000)

work page 2000
[23]

Planck 2015 results. XX. Constraints on inflation

P.A.R. Ade et al. [Planck Collaboration], arXiv:1502.02114

work page internal anchor Pith review Pith/arXiv arXiv
[24]

Lidsey, Reza Tavakol,Running of the Scalar Spectral Index and Observational Signatures of Inflation, Phys

James E. Lidsey, Reza Tavakol,Running of the Scalar Spectral Index and Observational Signatures of Inflation, Phys. Lett. B575, 157 (2003)

work page 2003
[25]

Gia Dvali, Gregory Gabadadze, Massimo Porrati,4D Gravity on a Brane in 5D Minkowski Space, Phys. Lett. B485, 208 (2000)

work page 2000
[26]

Itzhak Bars,Gravity in 2T-Physics, Phys.Rev.D77, 125027 (2008)

work page 2008
[27]

D62, 085015 (2000)

Itzhak Bars,Two-time physics with gravitational and gauge field backgrounds,Phys.Rev. D62, 085015 (2000)

work page 2000

[1] [1]

In Figure 2, it decreases withtbut asnincreases this decrease becomes less

The solution of Eq.17 has the SR form, ϕ(t) = (2n+ 2) 2M4M 2m2n(1−n)t√ 6π + ϕ2n+2 i 2(n+ 1) 1 2n+2 , (18) in whichϕ i is the initial inflaton field. In Figure 2, it decreases withtbut asnincreases this decrease becomes less. There is a similarity between Fig. 2 and 1, the SR shape is formed. This slow-roll is usually stored in the value ofm. 0.00001 0.000...

work page

[2] [2]

and in RSII model (Eq. 33). They all have similar shapes according totas shown in Figures 2 and 3. There is a similarity between Figure 2 and 3, butϕ(t) in Figure 3 shows a significant difference betweenn= 2 andn= 3 compared to Figure 2. This is also clearly shown in Figure 8. In Figure 4, the lines with differentNvalues are quite similar, so N only affec...

work page

[3] [3]

/Multiply 10/Minus 6 4

/Multiply 10/Minus 6 3. /Multiply 10/Minus 6 4. /Multiply 10/Minus 6 5. /Multiply 10/Minus 6t 1.975 /Multiply 1019 1.980 /Multiply 1019 1.985 /Multiply 1019 1.990 /Multiply 1019 1.995 /Multiply 1019 2.000 /Multiply 1019 Φ/LParen1t/RParen1 n/Equal 2 n/Equal 3 FIG. 3. Theϕ(t) field in the RSII model withn= 2,3,M≃ 1015 GeV,M 4 ≃10 19 GeV,m≃10 18 GeV, andϕ i ...

work page

[4] [4]

We can approximate as follows:n s = 0.006N. In Figs. 6 and 5,rdecreases slightly asn s increases. Whenn >4, the values ofrare quite close to each other, whereas forn <4,rchanges significantly asnincreases. In particular, whennincreases from 2 to 3,rchanges very strongly. This is different from the shaft inflation. rin RSII model is about 100 times larger ...

work page

[5] [5]

7, with justn= 2,3 the resultrmatches the experimental data very well

From Fig. 7, with justn= 2,3 the resultrmatches the experimental data very well. In Figures 5, 6, and 7, the largernis, the closer the lines are to each other. In other words, a clear hierarchy of values for the quantities (r, ns) only occurs with small n. 7 In most models, the largernis, the smallerrbecomes, althoughncannot be much larger than 2. In our ...

work page

[6] [6]

Guth,Inflationary universe: A possible solution to the horizon and flatness problems, Phys

Alan H. Guth,Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D23, 347 (1981)

work page 1981

[7] [7]

P.A.R.Adeet al.(BICEP/Keck Collaboration),Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett.127, 151301 (2021)

work page 2018

[8] [8]

1, Cambridge Uni- versity Press (1998)

Joseph Polchinski,String theory, Vol. 1, Cambridge Uni- versity Press (1998)

work page 1998

[9] [9]

Ngo Phuc Duc Loc,Inflation with a class of concave in- flaton potentials in Randall–Sundrum model, Eur. Phys. J. C80, 768 (2020)

work page 2020

[10] [10]

Lisa Randall and Raman Sundrum,Large Mass Hierar- chy from a Small Extra Dimension, Phys. Rev. Lett.83, 3370 (1999)

work page 1999

[11] [11]

Lisa Randall and Raman Sundrum,An Alternative to Compactification, Phys. Rev. Lett.83, 4690 (1999)

work page 1999

[12] [12]

Juan Maldacena and Alexey Milekhin,Humanly traversable wormholes, Phys. Rev. D103, 066007 (2021)

work page 2021

[13] [13]

Konstantinos Dimopoulos,Shaft inflation, Phys. Lett. B 735, 75 (2014)

work page 2014

[14] [14]

Konstantinos Dimopoulos,Shaft Inflation and the Planck satellite observations, arXiv:1510.06593

work page internal anchor Pith review Pith/arXiv arXiv

[15] [15]

Ngo Phuc Duc Loc,Shaft inflation in Randall-Sundrum model,JCAP02020, (2023)

work page 2023

[16] [16]

Itzhak Bars ,”The Standard Model of Particles and Forces in the Framework of 2T-physics”, Phys.Rev.D74 085019, 2006

work page 2006

[17] [17]

I. Bars, C. Deliduman, and D. Minic,Supersymmetric two-time physics, Phys. Rev. D59, 125004 (1999)

work page 1999

[18] [18]

Itzhak Bars,Two-time physics in field theory, Phys. Rev. D62, 046007 (2000)

work page 2000

[19] [19]

High En- ergy Phys.2022, 5313952 (2022)

Vo Quoc Phong and Ngo Phuc Duc Loc, Adv. High En- ergy Phys.2022, 5313952 (2022)

work page 2022

[20] [20]

Vo Quoc Phong, Quach Ai Mi, Nguyen Xuan Vinh, Sphaleron and gravitational wave with the Higgs-Dilaton potential in the Standard Model Two-Time Physics, Nucl. Phys. B1017, 116964 (2025)

work page 2025

[21] [21]

Abdul Jawad, Amara Ilyas, Shamaila Rani,Warm mod- ified Chaplygin gas shaft inflation, Eur. Phys. J. C77, 131 (2017)

work page 2017

[22] [22]

Bassett, and Imo- gen P

Roy Maartens, David Wands, Bruce A. Bassett, and Imo- gen P. C. Heard,Chaotic inflation on the brane, Phys. Rev. D62, 041301(R) (2000)

work page 2000

[23] [23]

Planck 2015 results. XX. Constraints on inflation

P.A.R. Ade et al. [Planck Collaboration], arXiv:1502.02114

work page internal anchor Pith review Pith/arXiv arXiv

[24] [24]

Lidsey, Reza Tavakol,Running of the Scalar Spectral Index and Observational Signatures of Inflation, Phys

James E. Lidsey, Reza Tavakol,Running of the Scalar Spectral Index and Observational Signatures of Inflation, Phys. Lett. B575, 157 (2003)

work page 2003

[25] [25]

Gia Dvali, Gregory Gabadadze, Massimo Porrati,4D Gravity on a Brane in 5D Minkowski Space, Phys. Lett. B485, 208 (2000)

work page 2000

[26] [26]

Itzhak Bars,Gravity in 2T-Physics, Phys.Rev.D77, 125027 (2008)

work page 2008

[27] [27]

D62, 085015 (2000)

Itzhak Bars,Two-time physics with gravitational and gauge field backgrounds,Phys.Rev. D62, 085015 (2000)

work page 2000