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Scalar Field Kantowski--Sachs Solutions in Teleparallel $F(T)$ Gravity

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Pith's one-line read This paper claims that a single master formula, Equation (42), together with the coefficient function A(T), generates all new teleparallel F(T) solutions for scalar-field Kantowski–Sachs spacetimes, and applies it to produce exact and…

desk verdict A coherent solution-generating paper with new scalar-field F(T) families; the general formula is standard and the completeness claims need domain and verification checks. read the letter →

arxiv 2501.11160 v1 pith:JQTNKQOA submitted 2025-01-19 gr-qc math-phmath.MP

Alexandre Landry This is my paper · ORCID
classification gr-qcmath-phmath.MP
keywords teleparallelgravityF(T)Kantowski-Sachsspacetimescalarfieldcosmologyexactsolutionsdarkenergyquintessencephantomquintommodel
topics Dark Energy
open problems Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that for time-dependent Kantowski–Sachs spacetimes in teleparallel $F(T)$ gravity, a scalar-field source does not prevent a complete analytic treatment: the conservation law for $\phi(t)$ fixes the potential $V(\phi)$, and the full field equations collapse into one linear first-order equation for $F(T)$. The operative result is the master formula, Equation (42), supported by the coefficient function $A(T)$ of Equation (41), which the author says generates every new $F(T)$ solution for any scalar-field potential and coframe ansatz used here. Applying it to power-law, exponential, and logarithmic scalar fields yields a large catalogue of exact and approximate new $F(T)$ families. A sympathetic reader would care because those families are the raw material for teleparallel models of dark energy quintessence, phantom energy, and quintom (mixed quintessence-phantom) behaviour, and several late-time limits match previously known isotropic Robertson–Walker solutions.

What carries the argument

The load-bearing object is the master formula (42), an integrating-factor representation of the first-order ODE (40), with $A(T)$ defined by (41). $A(T)$ encodes the geometry: it is built from the torsion scalar $T$ through the characteristic equation (21), which inverts the coframe ansatz to give $t(T)$, and from the coefficient of $F_T$ in the unified field equation (25). The scalar-field sector enters through the conservation equation (19), which determines $V(\phi)$; substituting $\phi=\phi(T)$ turns the source into $V(T)$. The zero-hypermomentum condition keeps the energy-momentum tensor symmetric so the source enters only through that potential. Once $A(T)$ and $V(T)$ are known, Equation (42) produces $F(T)$ for every subcase, with the integration constant carrying the homogeneous part of the solution. The Kantowski–Sachs coframe (12) — a spherically symmetric anisotropic cosmology with a radial translation symmetry — and spin-connection (14), with $\psi=0$, $\chi=\pi/2$, and $\delta=\pm1$, are the symmetry input that makes the reduction possible.

What would settle it

Take the simplest subcase, $c=-2b$: substitute the claimed solution (45) together with $t(T)$ from (43) and the potential (44) directly into the un-reduced field equations (23)–(25). If for any allowed parameter set the equations fail to vanish beyond numerical round-off, the master formula is not generating valid solutions. A branch-check version: for $c=-1$, the inversion (59) has two signs $\delta_1=\pm1$ and a square-root branch point at $T^2+16(1-2b)c_0^{-2}=0$; finding a trajectory that crosses that point while formulas (63)–(69) remain single-valued would disprove the completeness claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the scalar-field Kantowski–Sachs system in teleparallel $F(T)$ gravity is reducible: the symmetric field equations (23)–(25), the torsion-scalar characteristic equation (21), and the scalar-field conservation law (19) combine into the single linear ODE $\Lambda_0 + 2\kappa V(T) = -F(T) + A(T) F_T(T)$, whose integrating-factor solution is Equation (42). For each choice of scalar field (power law, exponential, logarithmic), each coframe ansatz ($A_2=t^b$, $A_3=c_0 t^c$ or their exponential counterparts), and each admissible parameter set, the formula returns an explicit $F(T)$; the paper applies it across the cases $c=-2b,1,-1,2$ and the early and late cosmological limits. The author states that these are new non-trivial teleparallel $F(T)$ solutions, with several families comparable to perfect-fluid Kantowski–Sachs solutions and to scalar-field Teleparallel Robertson–Walker solutions, and with dark-energy indices spanning quintessence, phantom, and quintom regimes.

Load-bearing premise

The whole catalogue depends on the reduction to a single equation being complete: the chosen Kantowski–Sachs coframe and the spin-connection branch with $\psi=0$, $\chi=\pi/2$, and $\delta=\pm1$, together with the condition that the scalar-field source does not couple to torsion (zero hypermomentum), must be the only relevant ones, and every inversion $t(T)$ used in the subcases must be single-valued on the range considered; if any of those fails, the listed $F(T)$ families may be incomplete.

Editorial extensions

If this is right

  • For every parameter choice admitted by the ansätze, Equation (42) gives the corresponding $F(T)$ directly, so the paper's catalogue covers whole solution families rather than isolated examples.
  • The conserved potentials include linear, quadratic, logarithmic, and exponential-integral forms, with dark-energy index $\alpha_Q$ spanning quintessence, phantom, and quintom ranges, so the solutions plug directly into those cosmological scenarios.
  • The exponential coframe ansatz needs only two structural cases, general and $c=-2b$, and yields $F(T)$ expressed through new special-function classes $N_k$ and $Q_k$.
  • In the late-universe limit, several Kantowski–Sachs $F(T)$ families approach Teleparallel Robertson–Walker forms, connecting anisotropic to isotropic scalar-field cosmology.
  • Several subcases reduce to TEGR-like (Teleparallel Equivalent of General Relativity) linear $F(T)$ or constant-torsion General Relativity (Teleparallel de Sitter) limits, providing internal consistency checks.

Reading between the lines

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

  • If the master formula is as general as claimed, the same reduction should apply to any scalar-field potential whose $\phi(T)$ and $V(T)$ can be written through the characteristic equation, not just the power-law, exponential, and logarithmic sources the paper treats; that generalisation is left implicit.
  • The paper does not test stability, energy conditions, or observational constraints of the new families; checking which of the listed $F(T)$ classes could reproduce the observed expansion history would be the natural next step.
  • The late-time collapse of several families to Teleparallel Robertson–Walker forms hints that anisotropic shear decays in those models; computing the shear-to-Hubble ratio for the $c=-2b$ families would be a direct, testable consequence.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript derives, for Kantowski–Sachs teleparallel F(T) gravity with a scalar field source, the field equations (21)–(25), solves the scalar-field conservation law (19) for power-law, exponential, and logarithmic scalar fields with power-law and exponential coframe ansatze, reduces the field equations to the linear ODE (40), and proposes Eq. (42) as a general solution formula. It then presents a large number of F(T) expressions for various parameter choices, including special-function representations and late/early-time limits, and discusses their potential relevance to quintessence, phantom, and quintom dark energy models.

Significance. The central technical contribution is the reduction of the scalar-field Kantowski–Sachs field equations to a single first-order linear ODE (40) and the compact general solution (42). This reduction is transparent and checkable, and the paper applies it systematically across three scalar-field types and two coframe ansatze, producing a large set of candidate F(T) families. Several closed-form entries (e.g., Eqs. (45), (49), (54), (58), (87)–(91), (107), (115)) are plausibly new and compare naturally with the perfect-fluid results of ref. [46]. However, the catalogue is largely formal: many entries are unevaluated integrals, and real-domain and branch issues remain unspecified. Once these are resolved, the catalogue would be a useful resource for phenomenological studies of anisotropic teleparallel cosmologies; in its present form the completeness and exactness claims are not established.

major comments (4)
  1. [III.A.3, Eq. (59); III.A.4, Eq. (71); III.B, Eq. (103)] The t(T) inversions used throughout contain square roots and powers whose real domains are not stated. In Eq. (59), t^2(T) = (c0^2/4)[-T + delta1 sqrt(T^2 + 16(1-2b)c0^{-2})]; for b > 1/2 the radicand is negative for small |T|, so t^2(T) is complex. Eq. (71) requires 1 - T/C2 >= 0, and Eq. (103) requires T0 - T >= 0; these restrictions appear nowhere in the text. Since phi(T), V(phi(T)), and F(T) are all built from these inversions (e.g., Eqs. (61), (73), (113)), the printed F(T) expressions are formal algebraic expressions rather than verified real solutions unless the parameter space and T-domains are specified. This is a load-bearing issue for the claim of exact new solutions.
  2. [Eqs. (63), (66), (69), (75), (79), (83), (138), (141), (145), (149), (153), (155), (170), (173), (193), (196)] Many displayed 'solutions' are unevaluated integrals. The text itself admits 'There is no general solution' after Eqs. (63), (75), (145), (153), (170), and (193). An expression of the form F(T) = ... + integral dT' ... is an implicit integral representation, not an exact closed-form solution, unless the integral is evaluated in known or well-defined special functions. The abstract's 'large number of exact and approximate new teleparallel F(T) solutions' is therefore overstated. The authors should either evaluate these integrals, or explicitly label them as integral representations and restrict the exactness claim to the cases where closed forms are actually obtained (e.g., Eqs. (45), (49), (54), (58), (87)-(91), (107), (115)).
  3. [II.C, II.E, III-VI] The completeness claim attached to Eq. (42) is conditional on the spin-connection branch. Section II.C gives delta = +/-1 in Eq. (14), and Section II.E states that the delta = -1 field equations differ by signs in Eqs. (22)-(24); however, all derivations use only the delta = +1 case through Eqs. (23)-(25). The statement that Eq. (42) is 'the general formula applicable for any subcases' and generates 'all possible new teleparallel F(T) solutions' is therefore not established for the delta = -1 branch. The authors should either carry out the analogous reduction for delta = -1 or explicitly restrict the catalogue to the delta = +1 branch.
  4. [Eqs. (39)-(42), (43), (50), (59)] Equation (42) is the general solution of the first-order linear ODE (40), but calling it a general formula 'applicable for any subcases' overstates its status. The formula presumes a chosen real, single-valued branch of the t(T) inversion from Eq. (39); the inversions are not globally injective, e.g., Eq. (43) involves a 1/(4b) power (non-integer for most b), Eq. (50) has a denominator whose sign is unconstrained, and Eq. (59) has two branches delta1 = +/-1. No branch-cut or injectivity analysis is provided, and the assumption F_T(T) != constant is not checked on the resulting domains. A concrete test of completeness would be to specify the maximal interval of T on which each inversion is real and monotone and to verify F_T != 0 there; without this, the 'all possible solutions' claim is not justified.
minor comments (6)
  1. [II.E, Eq. (22)] B' is defined twice: in Eq. (22) as a kinematic combination of the scale factors, and in the text after Eq. (22) as d(ln F_T)/dt. Please clarify which is the definition and which is a consequence; currently the two statements impose a constraint rather than a definition.
  2. [Abstract] The phrase 'spherically symmetric teleparallel F(T) gravity' is imprecise: Kantowski–Sachs spacetimes are not globally spherically symmetric, only the two-spheres are. Consider wording such as 'spatially homogeneous Kantowski–Sachs spacetimes'.
  3. [III.A.1, Eq. (45)] The solution has denominators p - 1 and p - 1 - 6b, and the special cases p = 1 and p = 1 + 6b are excluded without discussion; please state whether a logarithmic solution covers those cases.
  4. [III.A.3, Eq. (63)] The exponent -4C1 delta1 / (3(3 - 4C1)) in Eq. (63) is singular when C1 = 3/4. The appendix treats C1 -> 3/4 as a limit, but the singularity at exactly C1 = 3/4 is not addressed; please specify how the limit is taken and whether the branch of the power function is chosen consistently.
  5. [V.B.1.a, Eq. (185)] In Eq. (185), the factor (C - 1) in the denominator is not discussed for C = 1; this case typically requires a logarithmic F(T) and should be handled separately.
  6. [Tables I-II, Eqs. (108), (116)] The new functions N_k and Q_k are only tabulated for integer k with small values, while the solutions (107), (110), (115), (118), (120) require k = 2p-2, 2p-1, or 2p for arbitrary p; unless these general-k integrals can be expressed in known special functions, the corresponding entries are still unevaluated integrals, a point that should be stated where the tables are cited.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: F(T) solutions are obtained by solving a first-order ODE from the field equations; no fitted input is renamed as a prediction.

full rationale

The paper's operative claim is that Equations (21), (23)-(25) reduce, via the chosen coframe ansatz, to the linear ODE (40), whose general solution is Equation (42). The scalar-field potentials V(phi) are derived from the conservation law (19) after fixing phi(t) as power-law, exponential, or logarithmic; they are not chosen to force a particular F(T). The torsion scalar T(t) comes from the characteristic equation (21), and F(T) is then obtained by direct integration. No parameter is fitted to a subset of data and then called a prediction, and no target F(T) is inserted as an input. The coframe (12), spin-connection (14), and symmetric field equations (21)-(25) are inherited from prior work by the same author group, but these are standard derived equations and are not equivalent to the claimed F(T) outputs; they are inputs, not conclusions being rederived. Many displayed F(T) expressions are left as unevaluated integrals or depend on branch conditions for real inversions (e.g., the radicals in Equations (59), (71), and (103)), but missing domain restrictions are a correctness/completeness concern, not circularity. The paper itself acknowledges in Section V A that the solution-finding approach is not exhaustive, which further indicates that the claims are conditional rather than tautological. No circular step can be exhibited by quoting a reduction of the output to an input, so the circularity score is 0.

Assumptions & free parameters 8 free parameters · 7 assumptions · 2 invented entities

The central claim rests on the teleparallel F(T) field equations and symmetry reduction inherited from prior work (mostly by the same author), on the scalar-field conservation equation, and on a large set of free ansatz parameters. There are no data-fitted parameters, but the extensive parameter freedom and branch choices mean the solutions are a constructed family rather than a unique prediction.

free parameters (8)
  • p0 = not fitted (free constant)
    Amplitude of the scalar field in phi(t)=p0 t^p and phi(t)=p0 exp(p t); no equation fixes it.
  • p = free exponent/rate; special limits p=1 and p>>1
    Power or exponential rate of the scalar field; appears throughout potentials and F(T) formulas; no data fixes it.
  • b = free exponent/rate
    A2=t^b or A2=e^{bt} ansatz coefficient in the coframe; free.
  • c = free exponent/rate; special values -2b, 1, -1, 2
    A3=c0 t^c or A3=c0 e^{ct} ansatz coefficient; selected to make integrals tractable.
  • c0 = free constant
    Amplitude of the A3 component; free.
  • phi0 (via Lambda0=2 kappa phi0) = free integration constant
    Integration constant in the scalar potential V(phi); shifted into Lambda0 in the master equation.
  • F0 = free integration constant
    Integration constant in the solution of the linear ODE for F(T).
  • delta1 = discrete sign +1 or -1
    Branch choice in the t(T) inversion for c=-1 and c=2 cases.
assumptions (7)
  • domain assumption Teleparallel F(T) action and covariant field equations (4)-(7) are correct.
    The paper uses these as starting point and cites refs [1-7,45-47].
  • domain assumption Kantowski-Sachs coframe (12) with A1=1 and spin-connection (14) with psi=0, chi=pi/2, delta=+/-1 are the relevant symmetry-adapted solutions.
    Imported from refs [45-47]; if a scalar source changes the connection branch, the reduced equations change.
  • domain assumption Hypermomentum is zero, so energy-momentum conservation reduces to the metric covariant derivative condition (9).
    Stated in Section II B; needed for the standard scalar field stress-energy (17)-(18).
  • domain assumption The scalar field is minimally coupled and its potential is fixed by solving the Klein-Gordon equation (19).
    Equation (19) is used to derive V(phi) for each ansatz.
  • domain assumption The torsion scalar relation t = t(T) is invertible on the chosen branch, with real T range matching the ansatz.
    Used in all cases, e.g., Eqs (43), (50), (59), (71), (103); branch choices delta1 affect the resulting F(T).
  • ad hoc to paper Power-law, exponential, and logarithmic ansatze for phi, A2, A3 are chosen to obtain closed-form integrals.
    These ansatze and the restriction to c=-2b, 1, -1, 2 are not forced by the equations.
  • standard math Standard integration techniques and special functions are used to evaluate Eq (42).
    Assumed valid for the parameter ranges; approximate p>>1 limits are used in several sections.
invented entities (2)
  • N_k(B1,x) integral special-function class, Eq (108)
    purpose: To express exponential-ansatz F(T) solutions as closed forms, e.g., Eqs (107), (110), (112).
    Defined as an integral in the paper; no external data or independent benchmark anchors it.
  • Q_k(B1,B2,B3,x) integral special-function class, Eq (116)
    purpose: Generalization of N_k used in general exponential-ansatz solutions, e.g., Eqs (115), (118), (120).
    Defined as an integral in the paper; only internal consistency is claimed.

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Cite this review

Pith. "Pith review of Scalar Field Kantowski--Sachs Solutions in Teleparallel $F(T)$ Gravity." pith.science (2026). https://pith.science/paper/JQTNKQOA

@misc{pith2026250111160,
  author       = {Pith},
  title        = {Pith review of: Scalar Field Kantowski--Sachs Solutions in Teleparallel $F(T)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQTNKQOA}},
  note         = {Machine review of arXiv:2501.11160}
}
abstract

In this paper, we investigate time-dependent Kantowski--Sachs spherically symmetric teleparallel $F(T)$ gravity with a scalar field source. We begin by setting the exact field equations to be solved and solve conservation laws for possible scalar field potential, $V\left(\phi\right)$, solutions. Then, we find new non-trivial teleparallel $F(T)$ solutions by using power-law and exponential ansatz for each potential case arising from conservation laws, such as linear, quadratic, or logarithmic, to name a few. We find a general formula allowing us to compute all possible new teleparallel $F(T)$ solutions applicable for any scalar field potential and ansatz. Then, we apply this formula and find a large number of exact and approximate new teleparallel $F(T)$ solutions for several types of cases. Some new $F(T)$ solution classes may be relevant for future cosmological applications, especially concerning dark matter, dark energy quintessence, phantom energy leading to the Big Rip event, and quintom models of physical processes.

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Forward citations

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Reference graph

Works this paper leans on

167 extracted references · 64 canonical work pages · cited by 1 Pith paper

  1. [46]

    C = −2: F (T ) = − Λ0 + F0 T −1/2 + κ p2 p2 0 T2√ T Ei1 − T2√ T + exp T2√ T . (B6) 53

  2. [1]

    Then, Equation (104) is simplified to A(T ) = T 3 ; the scalar field will be ϕ(T ) = p0 (4b)p ln c2 0 2 (−T ) p

    c = −2b: T0 = 0, and Equation (103) is simplified to: e4bt(T ) = c2 0 2 (−T ) , (105) where T ≤ 0. Then, Equation (104) is simplified to A(T ) = T 3 ; the scalar field will be ϕ(T ) = p0 (4b)p ln c2 0 2 (−T ) p . The potential V (T ) and F (T ) solutions are, for the follow- ing subcases: (a) General: Equation (35) becomes: V (T ) = V1p 2 ln c2 0 2 (−T ) ...

  3. [2]

    c ̸= −2b (General case): From Equation (103), we find that the scalar field is: ϕ(T ) = p0 (−2c)p ln c2 0 2 (T0 − T ) p . (113) For potential V (T ) and F (T ) solutions, we find for the following subcases: (a) General: Equation (35) becomes: V (T ) = − V1p 2 ln − c2 0 2 (T − T0) 2p−1 − V2p 2 ln − c2 0 2 (T − T0) 2p−2 , (114) where V1p = 2p2 p2 0(b+2c) (2...

  4. [3]

    F0 − 2κ (b + |c|) |c|

    Late Cosmology t → ∞limit: (a) c > 0: Equation (103) will lead to T = T0 = constant under this limit, a GR solu- tion similar to a Teleparallel de Sitter (TdS) spacetime ( c ̸= −2b general case) [48]. For c = −2b, we will find that T = 0, a null torsion scalar spacetime [107]. In the previous situations, A(T ) = constant according to Equation (104), and w...

  5. [4]

    (1 + b) + δ1 s (1 + b)2 − T 8c2 0 # , (71) where δ1 = ±1. Equation (41) for Equation (71) is: A(T ) = 3 2 T + 4(3b − 4)(1 + b)2 c2 0 (b − 2)

    c = 2: Equation (39) becomes [46]: 0 = t−4 − 4c2 0 (1 + b) t−2 + c2 0 T 2 , ⇒ t−2(T ) = 2c2 0 " (1 + b) + δ1 s (1 + b)2 − T 8c2 0 # , (71) where δ1 = ±1. Equation (41) for Equation (71) is: A(T ) = 3 2 T + 4(3b − 4)(1 + b)2 c2 0 (b − 2) " 1 + δ1 s 1 − T 8c2 0(1 + b)2 #2 , = 1 (b − 2) " −T + C1 1 + δ1 r 1 − T C2 !# , = 1 (b − 2) [−T + C1 w±(T )], (72) wher...

  6. [5]

    (84) If t(T ) → ∞, we find from Equation (84) that T → 0

    Late Cosmology t → ∞limit: In this case, there are some subcases according to Equation (39), such as: (a) c > 1: The Equation (39) t(T ) relation will be, for the following cases: 0 ≈2c (c + 2b) t−2 − T, ⇒ t−2(T ) ≈ T 2c (c + 2b) . (84) If t(T ) → ∞, we find from Equation (84) that T → 0. Then, Equation (41) becomes: A(T ) ≈ 3 2 T + (3b − 2c) c2 0 (b − c)...

  7. [6]

    We will then recover the same F (T ) solution expressions as Equations (94)–(102), but under the T → −∞limit

    Early Cosmology t → 0 limit: For this limit, Equation (39) will lead to: (a) c > 1: t−2c(T ) and A(T ) are, respectively, Equations (92) and (93), but t−2c(T ) → ∞ and A(T ) → ∞ when T → −∞. We will then recover the same F (T ) solution expressions as Equations (94)–(102), but under the T → −∞limit. (b) 0 < c < 1 and c < 0: t−2(T ) and A(T ) are, respecti...

  8. [7]

    This is a TdS-like case model because of the constant torsion scalar [48]

    Early Cosmology t → 0 limit: For any non-zero value of c, Equation (103) leads to T → T0 − 2 c2 0 = constant and then A(T ) = constant, a GR solution. This is a TdS-like case model because of the constant torsion scalar [48]. All the teleparallel F (T ) solutions in this section are new and comparable to those found in ref. [46]. However, there are, in pr...

Show all 167 references
  1. [8]

    (127) 30 Equation (127), by the last integral term, is a different kind of potential, and this last can be considered as an additional interaction term

    Power-law: By substituting Equation (27) into Equation (126), we find that: 0 =p (b + 2c) ϕ ln (ϕ/p0) + p2 ϕ + dV dϕ , ⇒ V (ϕ) = ϕ0 − p2 2 ϕ2 + p (b + 2c) p2 0 Ei −2 ln ϕ p0 . (127) 30 Equation (127), by the last integral term, is a different kind of potential, and this last c...

  2. [9]

    Exponential: This is the most simple and naturally adapted ansatz for a pure exponen- tial scalar field. By substituting Equation (34) into Equation (126), the conservation law becomes: 0 =p (b + 2c + p) ϕ + dV dϕ , ⇒ V (ϕ) = ϕ0 − p 2 (b + 2c + p) ϕ2, (129) and then Equation (...

  3. [10]

    2p − c2 0 2 T 1 4b !6b exp −p − c2 0 2 T 1 4b ! × W hittakerM −6b, −6b + 1 2 , 2p − c2 0 2 T 1 4b ! + (1 − 12b) exp −2p − c2 0 2 T 1 4b !# − 6κ p p2 0 b − 1 12

    c = −2b: By using A(T ) = T 3 and substituting Equation (43) for thet(T ) solution, the scalar field is ϕ(T ) = p0 exp p c2 0 2 (−T ) 1/4b , and Equation (127) for V (T ) potential will be: V (T ) = − p2 p2 0 2 exp 2p c2 0 2 (−T ) 1/4b! − 3p b p2 0 Ei −2p c2 0 2 (−T ) 1/4b! . ...

  4. [11]

    c = 1: By using Equation (51) and substituting Equation (50), we find that: ϕ(T ) = p0 exp   √ 2p 1 + 2b − 1 c2 0 1/2 √ T   , (136) and the Equation (127) potential becomes: V (T ) = − p2 p2 0 2 exp T2√ T + p (b + 2) p2 0 Ei − T2√ T , (137) where T2 = 2 √ 2p 1 + 2b − 1 c...

  5. [12]

    F0 − 2κ p2 0 × Z T dT ′

    c = −1: By using Equation (60) and substituting Equation (59), we find that: ϕ(T ) = p0 exp c0 p 2 [u±(T )]1/2 , (143) and the Equation (127) potential becomes: V (T ) = − p2 p2 0 2 exp c0 p [u±(T )]1/2 + p (b − 2) p2 0 Ei −c0 p [u±(T )]1/2 . (144) By substituting Equation (14...

  6. [13]

    F0 + κ (b − 2) p2 0 Z T dT ′

    c = 2: By using Equation (72) and substituting Equation (71), we find that: ϕ(T ) = p0 exp p(1 + b)√ 2c0 [w±(T )]1/2 ! , (151) and the Equation (127) potential becomes: V (T ) = − p2 p2 0 2 exp 2p(1 + b)√ 2c0 [w±(T )]1/2 ! + p (b + 4) p2 0 Ei − 2p(1 + b)√ 2c0 [w±(T )]1/2 ! . (...

  7. [14]

    exp 2p (2c (c + 2b))1/2 T −1/2 − 2(b + 2c) p Ei −2p (2c (c + 2b))1/2 T −1/2 # , (156) and then Equation (86): F (T ) = − Λ0 + T 2 3

    Late Cosmology t → ∞limit: We have the same type of scenarios as in Section III A, and we can find some F (T ) solutions: (a) c > 1: By using Equations (84)–(86), we find for the following potentials with ϕ(T ) = p0 exp p (2c (c + 2b))1/2 T −1/2 : 36 • Equation (127) potential...

  8. [15]

    We will then recover Equations (161) and (163) under the same limit

    Early Cosmology t → 0 limit: (a) c > 1: t−2c(T ) and A(T ) are, respectively, Equations (92) and (93), but satisfying the T → −∞limit in the Section III A case. We will then recover Equations (161) and (163) under the same limit. (b) 0 < c < 1 and c < 0: t−2(T ) and A(T ) are,...

  9. [16]

    By substituting Equation (164) into Equation (42), the F (T ) solution is: F (T ) = − Λ0 + F0 T 3 − 6κ p b(p − 3b) p2 0 (p − 6b) c2 0 2 p/2b (−T )p/2b

    c = −2b: By using A(T ) = T 3 and Equation (105), we find thatϕ(T ) = p0 c2 0 2 p/4b (−T )p/4b, and the Equation (129) V (T ) potential is: V (T ) = − p (p − 3b) p2 0 2 c2 0 2 p/2b (−T )p/2b , (164) where p ̸= 3b. By substituting Equation (164) into Equation (42), the F (T ) s...

  10. [17]

    c ̸= −2b (General case): From Equations (103) and (104), we find that: ϕ(T ) = p0 c2 0 2 −p/2c (T0 − T )−p/2c , (168) and the Equation (129) potential becomes: V (T ) = − p (b + 2c + p) p2 0 2 2 c2 0 p/c (T0 − T )−p/c , (169) where p ̸= −b − 2c. By substituting Equation (169) ...

  11. [18]

    (b) c < 0: t(T ) and A(T ) satisfy Equations (121) and (122), and the F (T ) solutions are described by Equation (123) because this is the same situation as in Section III B

    Late Cosmology t → ∞limit: 40 (a) c > 0: Equation (103) leads to T = T0 = constant when c ̸= −2b (TdS-like spacetime), T = 0 (null torsion scalar spacetime) when c = −2b, and then A(T ) = constant under this limit, GR solutions [48], as in Section III B. (b) c < 0: t(T ) and A...

  12. [19]

    All the previous teleparallel F (T ) solutions are new, and most of those are comparable with the solutions found in refs

    Early Cosmology t → 0 limit: For any non-zero value of c, Equation (103) also leads to T → T0 − 2 c2 0 = constant and A(T ) = constant, a GR solution (TdS-like spacetime), as in Section III B [48, 107]. All the previous teleparallel F (T ) solutions are new, and most of those ...

  13. [20]

    (181) Equation (20) for αQ is: αQ = − 1 + ϕ0 p2 p2 0 exp 2ϕ p0 + (b + 2c) 2 −1

    Power-law: By using the Equations (27) ansatz, we find that: dV dϕ =p0 p2 (1 − b − 2c) exp − 2ϕ p0 , ⇒ V (ϕ) = ϕ0 + p2 0 p2 2 (b + 2c − 1) exp − 2ϕ p0 . (181) Equation (20) for αQ is: αQ = − 1 + ϕ0 p2 p2 0 exp 2ϕ p0 + (b + 2c) 2 −1 . (182) There are some examples leading to si...

  14. [21]

    F0 + 8κ p2 0 c (b − c) ×

    Exponential: By using the Equations (34) ansatz, we find that: dV dϕ =p0 p2 exp − 2ϕ p0 − p0 p (b + 2c) exp − ϕ p0 , ⇒ V (ϕ) = ϕ0 − p2 0 p2 2 exp − 2ϕ p0 + p2 0 p (b + 2c) exp − ϕ p0 . (190) Equation (20) for αQ is: αQ = − 1 + ϕ0 p2 0 p2 exp 2ϕ p0 + (b + 2c) p exp ϕ p0 −1 . (1...

  15. [22]

    Power-Law Ansatz c = −1 Solutions Equation (63) C2 ̸= 0 teleparallel F (T ) solutions:

  16. [23]

    F0 − 2κ Vp 3 C 2 3 2 ln [−u−(T )] # . (A2) • p = 1 3 and δ1 = −1: F (T ) = − Λ0 + [−u−(T )] 2 3

    C1 = 3 2 (b = − 4 3 and C2 = 176 3c2 0 ): There are three possible solutions: • p ̸= 1 3 : F (T ) = − Λ0 + F0[−u−(T )] 2δ1 3 + 2κ Vp (3p − 1) δp−1 1 [u±(T )]p−1 . (A1) • p = 1 3 and δ1 = +1: F (T ) = − Λ0 + [−u−(T )] 2 3 " F0 − 2κ Vp 3 C 2 3 2 ln [−u−(T )] # . (A2) • p = 1 3 a...

  17. [24]

    (A4) Equation (66) C2 ̸= 0 teleparallel F (T ) solutions ( p ≫ 1 case):

    C1 → 3 4 : By setting C1 = 3 4 + ϵ where ϵ ≪ 1, we find: F (T ) = − Λ0 + F0 + 4κ Vp 3C2 (p + 1)(p − 1) [u±(T )]p h T + δ1p p T 2 + C2 i . (A4) Equation (66) C2 ̸= 0 teleparallel F (T ) solutions ( p ≫ 1 case):

  18. [25]

    C1 = 3 2 (b = − 4 3 and C2 = 176 3c2 0 ): F (T ) = − Λ0 + F0[−u−(T )] 2δ1 3 + 2κ V∞ 3p [u±(T )]p . (A5)

  19. [26]

    (A6) Equation (69) C2 ̸= 0 teleparallel F (T ) solutions ( p = 1 case):

    C1 → 3 4 : By setting C1 = 3 4 + ϵ, where ϵ ≪ 1, we find: F (T ) = − Λ0 + F0 + 4κ V∞ 3C2 p [u±(T )]p (C2 − T [u±(T )]) . (A6) Equation (69) C2 ̸= 0 teleparallel F (T ) solutions ( p = 1 case):

  20. [27]

    C1 = 3 2 (b = − 4 3 and C2 = 176 3c2 0 ): F (T ) = − ˜Λ0 + F0[−u−(T )] 2δ1 3 + κ p2 0 (b − 2) ln [u±(T )] − 3 2 . (A7)

  21. [28]

    C1 → 3 4 : By setting C1 = 3 4 + ϵ, where ϵ ≪ 1, we find: F (T ) = − ˜Λ0 + F0 − κ p2 0 3 (b − 2) " δ1 ln [−u−(T )] − ln2 [u±(T )] + 2T C2 [u±(T )] ln [u±(T )] − 1 2 # . (A8) 50

  22. [29]

    (A9) Equation (A9) solutions are possible only for the following subcases:

    Power-Law Ansatz c = 2 Solutions The general Equation (75) teleparallel F (T ) solution for C1 = 2C2 is: F (T ) = − Λ0 + − T C2 + 2 (w±(T )) (2−b) " F0 − κ Vp (b − 2) C2 × Z T dT ′ w±(T ′) 1−p − T ′ C2 + 2 w±(T ′) (b−3) # . (A9) Equation (A9) solutions are possible only for th...

  23. [30]

    b = 3 and δ1 = 1: F (T ) = − Λ0 + − T C2 + 2 (w+(T )) −1 " F0 − κ Vp C2 (2)1−p T 3F2 1, p 2 , p − 1 2 ; 2, p; T C2 # . (A10)

  24. [31]

    b = 3 and δ1 = −1: F (T ) = − Λ0 + − T C2 + 2 (w−(T )) −1 " F0 + 2p−1κ Vp C2−p 2 (p − 2) T 2−p 2F1 2 − p 2 , 1 − p 2 ; 3 − p; T C2 # . (A11)

  25. [32]

    F0 − κ Vp 2p−1

    b = 4 and δ1 = 1: F (T ) = − Λ0 + − T C2 + 2 (w+(T )) −2 " F0 − κ Vp 2p−1 " − T C2 2 3F2 2, p 2 , p − 1 2 ; 3, p; T C2 + 8 T C2 3F2 1, p − 2 2 , p − 1 2 ; 2, p− 1; T C2 ## . (A12)

  26. [33]

    F0 − 8κ Vp (p − 3) T 2C2 (3−p) ×

    b = 4 and δ1 = −1: F (T ) = − Λ0 + − T C2 + 2 (w−(T )) −2 " F0 − 8κ Vp (p − 3) T 2C2 (3−p) × " 3F2 3 − p, 2 − p 2 , 1 − p 2 ; 4 − p, 2 − p; T C2 + 2F1 2 − p 2 , 3 − p 2 ; 4 − p; T C2 ## . (A13) The general Equation (79) teleparallel F (T ) solution for C1 = 2C2 is (p ≫ 1 case)...

  27. [34]

    b = 3 and δ1 = 1: F (T ) ≈ −Λ0 + − T C2 + 2 (w+(T )) −1 " F0 − κ Vp C2 2p T 3F2 1, p 2 , p 2 ; 2, p; T C2 # . (A15)

  28. [35]

    b = 3 and δ1 = −1: F (T ) ≈ −Λ0 − T C2 − 2 (w−(T )) −1 " F0 + (2C2)p κ Vp p T −p 2F1 − p 2 , − p 2 ; −p; T C2 # . (A16)

  29. [36]

    F0 − κ Vp 2p

    b = 4 and δ1 = 1: F (T ) ≈ −Λ0 + − T C2 + 2 (w+(T )) −2 " F0 − κ Vp 2p " − T C2 2 3F2 2, p 2 , p 2 ; 3, p; T C2 + 8 T C2 3F2 1, p 2 , p 2 ; 2, p; T C2 ## . (A17)

  30. [37]

    b = 4 and δ1 = −1: F (T ) ≈ −Λ0 + T C2 − 2 (w−(T )) −2 " F0 − 2κ Vp p T 2C2 −p 2F1 − p 2 , − p 2 ; −p; T C2 # . (A18) The general Equation (83) teleparallel F (T ) solution for C1 = 2C2 is (p = 1 case): F (T ) = − ˜Λ0 + [−T + 2C2 (w±(T ))](2−b) " F0 + κ p2 0 (b + 4) (b − 2) 2C...

  31. [38]

    F0 − 7κ p2 0 2C2

    b = 3: F (T ) = − ˜Λ0 + [−T + 2C2 (w+(T ))]−1 " F0 − 7κ p2 0 2C2 " T 2 + C2 w±(T ) − T ln [w±(T )] ## . (A20)

  32. [39]

    F0 + 8κ p2 0 C2

    b = 4: F (T ) = − ˜Λ0 + [−T + 2C2 (w+(T ))]−2 " F0 + 8κ p2 0 C2 "" − 4δ1 C2 2 3 1 − T C2 3/2 − 3 T 2 + 12C2 T − 8C2 2 # ln [w±(T )] + δ1 C2 18 (2C2 − 5 T ) r 1 − T C2 + T 2 8 − C2 3 T + C2 2 9 ## . (A21) 52 Appendix B: Additional Exponential Scalar Field F (T ) Solutions

  33. [40]

    Power-Law Ansatz c = 1 Solutions Equation (138) C1 ̸= 1 teleparallel F (T ) solutions:

  34. [41]

    C = −1: F (T ) = − Λ0 + F0 T −1 + κ p2 0 " p2 T −1 Ei1 − T2√ T T 2 2 + exp T2√ T T2 √ T + T − p (b + 2)   −T2 √ T + T exp − T2√ T T + T 2 2 − 2T Ei1 T2√ T T   # . (B1)

  35. [42]

    C = 2: F (T ) = − Λ0 + F0 T 1/2 + κ p2 0 " p2 − 2p (b + 2) √ T T2 exp T2√ T − 2p (b + 2) Ei − T2√ T # . (B2)

  36. [43]

    (B3) Equation (141) C1 ̸= 1 teleparallel F (T ) solutions ( p ≫ 1 case):

    C = −2: F (T ) = − Λ0 + F0 T −1/2 + κ p2 0 " p2 exp T2√ T + T2√ T Ei1 − T2√ T − 2p (b + 2) exp − T2√ T − T2√ T + 1 Ei1 T2√ T !# . (B3) Equation (141) C1 ̸= 1 teleparallel F (T ) solutions ( p ≫ 1 case):

  37. [44]

    C = −1: F (T ) = − Λ0 + F0 T −1 + κ p2 p2 0 T 2 2 T Ei1 − T2√ T + T2√ T + 1 . (B4)

  38. [45]

    C = 2: F (T ) = − Λ0 + F0 T 1/2 + κ p2 p2 0 T 1/2 T2 exp T2√ T . (B5)

  39. [47]

    Power-Law Ansatz c = 2 Solutions Equation (153) teleparallel F (T ) solutions are:

  40. [48]

    F0 − κ (b − 2) p2 0

    C1 = 0: F (T ) = − Λ0 + (−T )(2−b) " F0 − κ (b − 2) p2 0 " p2 Z T dT ′ (−T ′)(b−3) exp √ 2p(1 + b) c0 [w±(T ′)]1/2 ! − 2p (b + 4) Z T dT ′ (−T ′)(b−3) Ei − √ 2p(1 + b) c0 [w±(T ′)]1/2 ! ## . (B7) There are solutions for specific values of b, such as: • b = 1: F (T ) = − Λ0 + (...

  41. [49]

    F0 − κ (b − 2) p2 0 Z T dT ′

    C1 = 2C2: F (T ) = − Λ0 + [−T + 2C2 (w±(T ))](2−b) " F0 − κ (b − 2) p2 0 Z T dT ′ " p2 exp √ 2p(1 + b) c0 [w±(T ′)]1/2 ! − 2p (b + 4) Ei − √ 2p(1 + b) c0 [w±(T ′)]1/2 ! # −T ′ + 2C2 w±(T ′) (b−3) # . (B11) There are solutions for specific values of δ1 and b: 55 • b = 3: F (T )...

  42. [50]

    F0 − κ p2 p2 0 Z T dT ′ exp √ 2p(1 + b) c0 [w±(T ′)]1/2 ! (−T ′)(b−3) # . (B14) There are solutions for specific values of b: • b = 1: F (T ) ≈ −Λ0 − T

    C1 = 0: F (T ) = − Λ0 + (−T )(2−b) " F0 − κ p2 p2 0 Z T dT ′ exp √ 2p(1 + b) c0 [w±(T ′)]1/2 ! (−T ′)(b−3) # . (B14) There are solutions for specific values of b: • b = 1: F (T ) ≈ −Λ0 − T " F0 − κ p2 0 p2 4 c0C2 " p exp 2p c0 Ei1  − 2p √ 2 − p w±(T ) c0 p w±(T )   + (w±(T...

  43. [51]

    F0 + 64κ p5 p2 0 3c3 0 √ 2 C2 p w±(T ) exp 4 √ 2 p c0 p w±(T ) ! # . (B19) • b = 4: F (T ) ≈ −Λ0 + [−T + C1 (w±(T ))]−2 ×

    C1 = 2C2: F (T ) = − Λ0 + [−T + C1 (w±(T ))](2−b) [y(T )] (b−2)C1 2(C1−2C2) T + (C1 − 2C2) (w±(T )) T + (C1 − 2C2) (2 − w±(T )) δ1 (b−2)C1 2(C1−2C2) × " F0 − κ p2 p2 0 Z T dT ′ exp √ 2p(1 + b) c0 [w±(T ′)]1/2 ! −T ′ + C1 w±(T ′) (b−3) × y(T ′) − (b−2)C1 2(C1−2C2) T ′ + (C1 − 2...

  44. [52]

    Exponential Ansatz Solutions Equation (170) T1 ̸= T0 and b ≥ 3c 2 teleparallel F (T ) solutions:

  45. [53]

    b = 3c 2 : F (T ) = − Λ0 + (T − T1)−1 " F0 + κ p p2 0 c(7c + 2p) 2 (c − p) − 2 c2 0 p/c (T − T0)1−p/c # . (B21)

  46. [54]

    (B22) 58

    b = 2c: F (T ) = − Λ0 + (T − T1)−2 " F0 + 2κ p p2 0 c(4c + p) − 2 c2 0 p/c × (T − T0) −p+c c ((T + T0 − 2T1) c − p (T − T1)) 2c2 − 3cp + p2 # . (B22) 58

  47. [55]

    (B23) Equation (173) T1 ̸= T0 and b ≥ 3c 2 teleparallel F (T ) solutions ( p ≫ 1 case):

    b = 3c: F (T ) = − Λ0 + (T − T1)−4 " F0 + 24κ p p2 0 c(3c + 2c + p) (24c4 − 50c3p + 35c2p2 − 10c p3 + p4) − 2 c2 0 p/c × (T − T0) −p+c c 2T 2 1 + (−2T − 2T0) T1 + T 2 + T 2 0 (T + T0 − 2T1) c3 − 11 6 26T 2 1 11 + − 31T 11 − 21T0 11 T1 + T 2 + 9T T0 11 + 6T 2 0 11 p (T − T1) c2...

  48. [56]

    b = 3c 2 : F (T ) ≈ −Λ0 + F0 (T − T1)−1 − κ p p2 0 c − 2 c2 0 p/c (T − T0)−p/c (T − T1)−1. (B24)

  49. [57]

    b = 2c: F (T ) ≈ −Λ0 + F0 (T − T1)−2 − 2κ p p2 0 c − 2 c2 0 p/c (T − T0)− p c (T − T1)−1. (B25)

  50. [58]

    b = 3c: F (T ) ≈ −Λ0 + F0 (T − T1)−4 − 4κ p p2 0 c − 2 c2 0 p/c (T − T0)− p c (T − T1)−1 . (B26)

  51. [59]

    Teleparallel Gravity, An Introduction ; Springer: Berlin/Heidelberg, Germany, 2013

    Aldrovandi, R.; Pereira, J.G. Teleparallel Gravity, An Introduction ; Springer: Berlin/Heidelberg, Germany, 2013

    2013

  52. [60]

    Teleparallel Gravity: From Theory to Cos- mology

    Bahamonde, S.; Dialektopoulos, K.; Escamilla-Rivera, C.; Farrugia, G.; Gakis, V.; Hendry, M.; Hohmann, M.; Said, J.L.; Mifsud, J.; Di Valentino, E. Teleparallel Gravity: From Theory to Cos- mology. Rep. Prog. Phys. 2023, 86, 026901

    2023

  53. [61]

    Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach

    Krssak, M.; van den Hoogen, R.; Pereira, J.; Boehmer, C.; Coley, A. Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach. Class. Quantum Gravity 2019, 36, 183001

    2019

  54. [62]

    A frame based approach to computing symmetries with non-trivial isotropy groups

    McNutt, D.D.; Coley, A.A.; van den Hoogen, R.J. A frame based approach to computing symmetries with non-trivial isotropy groups. J. Math. Phys. 2023, 64, 032503

    2023

  55. [63]

    Symmetry and Equivalence in Teleparallel Gravity

    Coley, A.A.; van den Hoogen, R.J.; McNutt, D.D. Symmetry and Equivalence in Teleparallel Gravity. J. Math. Phys. 2020, 61, 072503

    2020

  56. [64]

    Regularizing role of teleparallelism

    Lucas, T.G.; Obukhov, Y.; Pereira, J.G. Regularizing role of teleparallelism. Phys. Rev. D 2009, 80, 064043

    2009

  57. [65]

    Spin Connection and Renormalization of Teleparallel Action

    Krssak, M.; Pereira, J.G. Spin Connection and Renormalization of Teleparallel Action. Eur. Phys. J. C 2015, 75, 519. 59

    2015

  58. [66]

    Symmetries in tetrad theories

    Chinea, F. Symmetries in tetrad theories. Class. Quantum Gravity 1988, 5, 135

    1988

  59. [67]

    Moving frame formulations of 4-geometries having isometries

    Estabrook, F.; Wahlquist, H. Moving frame formulations of 4-geometries having isometries. Class. Quantum Gravity 1996, 13, 1333

    1996

  60. [68]

    Locally homogeneous spaces, induced Killing vector fields and applications to Bianchi prototypes

    Papadopoulos, G.; Grammenos, T. Locally homogeneous spaces, induced Killing vector fields and applications to Bianchi prototypes. J. Math. Phys. 2012, 53, 072502

    2012

  61. [69]

    Equivalence, Invariants and Symmetry ; Cambridge University Press: Cambridge, UK, 1995

    Olver, P. Equivalence, Invariants and Symmetry ; Cambridge University Press: Cambridge, UK, 1995

    1995

  62. [70]

    Einstein’s Other Gravity and the Acceleration of the Universe

    Linder, E. Einstein’s Other Gravity and the Acceleration of the Universe. Phys. Rev. D 2010, 81, 127301; Erratum in Phys. Rev. 2010, 82, 109902

    2010

  63. [71]

    Modified teleparallel gravity: Inflation without an inflation

    Ferraro, R.; Fiorini, F. Modified teleparallel gravity: Inflation without an inflation. Phys. Rev. D 2007, 75, 084031

    2007

  64. [72]

    On Born-Infeld Gravity in Weitzenbock spacetime

    Ferraro, R.; Fiorini, F. On Born-Infeld Gravity in Weitzenbock spacetime. Phys. Rev. D 2008, 78, 124019

    2008

  65. [73]

    New general relativity

    Hayashi, K.; Shirafuji, T. New general relativity. Phys. Rev. D 1979, 19, 3524

    1979

  66. [74]

    Non-Linear Obstructions for Consistent New General Relativity

    Jimenez, J.B.; Dialektopoulos, K.F. Non-Linear Obstructions for Consistent New General Relativity. J. Cosmol. Astropart. Phys. 2020, 2020, 018

    2020

  67. [75]

    Revisiting Stability in New General Relativity

    Bahamonde, S.; Blixt, D.; Dialektopoulos, K.F.; Hell A. Revisiting Stability in New General Relativity. arXiv 2024, arXiv:2404.02972

    2024 arXiv

  68. [76]

    Review on f (Q) Gravity

    Heisenberg, L. Review on f (Q) Gravity. arXiv 2023, arXiv:2309.15958

    2023 arXiv

  69. [77]

    Cosmological teleparallel perturbations

    Heisenberg, L.; Hohmann, M.; Kuhn, S. Cosmological teleparallel perturbations. arXiv 2023, arXiv:2311.05495

    2023 arXiv

  70. [78]

    Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity

    Flathmann, K.; Hohmann, M. Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity. Phys. Rev. D 2022, 105, 044002

    2022

  71. [79]

    General covariant symmetric teleparallel cosmology

    Hohmann, M. General covariant symmetric teleparallel cosmology. Phys. Rev. D 2021, 104, 124077

    2021

  72. [80]

    The Geometrical Trinity of Gravity

    Jimenez, J.B.; Heisenberg, L.; Koivisto, T.S. The Geometrical Trinity of Gravity. Universe 2019, 5, 173

    2019

  73. [81]

    Geometrical trinity of unimodular gravity

    Nakayama, Y. Geometrical trinity of unimodular gravity. Class. Quantum Gravity 2023, 40, 125005

    2023

  74. [82]

    f (Q, T) gravity

    Xu, Y.; Li, G.; Harko, T.; Liang, S.-D. f (Q, T) gravity. Eur. Phys. J. C 2019, 79, 708

    2019

  75. [83]

    Myrzakulov F (T, Q) gravity: Cosmological implications and constraints

    Maurya, D.C.; Yesmakhanova, K.; Myrzakulov, R.; Nugmanova, G. Myrzakulov F (T, Q) gravity: Cosmological implications and constraints. arXiv 2024, arXiv:2404.09698

    2024 arXiv

  76. [84]

    Myrzakulov, FLR W Cosmology in Myrzakulov F (R, Q) Gravity

    Maurya, D.C.; Yesmakhanova, K.; Myrzakulov, R.; Nugmanova, G. Myrzakulov, FLR W Cosmology in Myrzakulov F (R, Q) Gravity. arXiv 2024, arXiv:2403.11604

    2024 arXiv

  77. [85]

    Exact Cosmology in Myrzakulov Gravity

    Maurya, D.C.; Myrzakulov, R. Exact Cosmology in Myrzakulov Gravity. arXiv 2024, arXiv:2402.02123

    2024 arXiv

  78. [86]

    f (R, T) gravity

    Harko, T.; Lobo, F.S.N.; Nojiri, S.; Odintsov, S.D. f (R, T) gravity. Phys. Rev. D 2011, 84, 024020

    2011

  79. [87]

    Approaches to spherically symmetric solutions in f (T )-gravity

    Golovnev, A.; Guzman, M.-J. Approaches to spherically symmetric solutions in f (T )-gravity. Universe 2021, 7, 121

    2021

  80. [88]

    Issues of Lorentz-invariance in f (T )-gravity and calculations for spherically symmetric 60 solutions

    Golovnev, A. Issues of Lorentz-invariance in f (T )-gravity and calculations for spherically symmetric 60 solutions. Class. Quantum Gravity 2021, 38, 197001

    2021

  81. [89]

    Bianchi identities in f (T )-gravity: Paving the way to confrontation with astrophysics

    Golovnev, A.; Guzman, M.-J. Bianchi identities in f (T )-gravity: Paving the way to confrontation with astrophysics. Phys. Lett. B 2020, 810, 135806

    2020

  82. [90]

    On spherically symmetric vacuum solutions and horizons in covariant f (T ) gravity theory

    DeBenedictis, A.; Iliji´ c, S.; Sossich, M. On spherically symmetric vacuum solutions and horizons in covariant f (T ) gravity theory. Phys. Rev. D 2022, 105, 084020

    2022

  83. [91]

    Exact Spherically Symmetric Solutions in Modified Teleparallel gravity

    Bahamonde, S.; Camci, U. Exact Spherically Symmetric Solutions in Modified Teleparallel gravity. Symmetry 2019, 11, 1462

    2019

  84. [92]

    Revisiting diagonal tetrads: New Black Hole solutions in f (T )-gravity

    Awad, A.; Golovnev, A.; Guzman, M.-J.; El Hanafy, W. Revisiting diagonal tetrads: New Black Hole solutions in f (T )-gravity. Eur. Phys. J. C 2022, 82, 972

    2022

  85. [93]

    Black Holes in f (T, B) Gravity: Exact and Perturbed Solutions

    Bahamonde, S.; Golovnev, A.; Guzm´ an, M.-J.; Said, J.L.; Pfeifer, C. Black Holes in f (T, B) Gravity: Exact and Perturbed Solutions. J. Cosmol. Astropart. Phys. 2022, 01 037

    2022

  86. [94]

    Thick accretion disk configurations in the Born- Infeld teleparallel gravity

    Bahamonde, S.; Faraji, S.; Hackmann, E.; Pfeifer, C. Thick accretion disk configurations in the Born- Infeld teleparallel gravity. Phys. Rev. D 2022, 106, 084046

    2022

  87. [95]

    Quadratic and cubic spherically symmetric black holes in the modified teleparallel equivalent of general relativity: Energy and thermodynamics

    Nashed, G.G.L. Quadratic and cubic spherically symmetric black holes in the modified teleparallel equivalent of general relativity: Energy and thermodynamics. Class. Quantum Gravity 2021, 38, 125004

    2021

  88. [96]

    Static spherically symmetric black holes in weak f (T )-gravity

    Pfeifer, C.; Schuster, S. Static spherically symmetric black holes in weak f (T )-gravity. Universe 2021, 7, 153

    2021

  89. [97]

    Exact Teleparallel Gravity of Binary Black Holes

    El Hanafy, W.; Nashed, G.G.L. Exact Teleparallel Gravity of Binary Black Holes. Astrophys. Space Sci. 2016, 361, 68

    2016

  90. [98]

    Matter Conditions for Regular Black Holes in f (T ) Gravity

    Aftergood, J.; DeBenedictis, A. Matter Conditions for Regular Black Holes in f (T ) Gravity. Phys. Rev. D 2014, 90, 124006

    2014

  91. [99]

    Spontaneous Scalarization of Black Holes in Gauss-Bonnet Teleparallel Gravity

    Bahamonde, S.; Doneva, D.D.; Ducobu, L.; Pfeifer, C.; Yazadjiev, S.S. Spontaneous Scalarization of Black Holes in Gauss-Bonnet Teleparallel Gravity. Phys. Rev. D 2023, 107, 10, 104013

    2023

  92. [100]

    Scalarized Black Holes in Teleparallel Gravity

    Bahamonde, S.; Ducobu, L.; Pfeifer, C. Scalarized Black Holes in Teleparallel Gravity. J. Cosmol. Astropart. Phys. 2022, 04, 018

    2022

  93. [101]

    A class of static spherically symmetric solutions in f (T )-gravity

    Calza, M.; Sebastiani, L. A class of static spherically symmetric solutions in f (T )-gravity. Eur. Phys. J. C 2024, 84, 476

    2024

  94. [102]

    Static spherically symmetric perfect fluid solutions in teleparallel F(T) gravity

    Landry, A. Static spherically symmetric perfect fluid solutions in teleparallel F(T) gravity. Axioms 2024, 13, 333

    2024

  95. [103]

    Spherically symmetric teleparallel geometries

    Coley, A.A.; Landry, A.; van den Hoogen, R.J.; McNutt, D.D. Spherically symmetric teleparallel geometries. Eur. Phys. J. C 2024, 84, 334

    2024

  96. [104]

    Kantowski-Sachs spherically symmetric solutions in teleparallel F (T ) gravity

    Landry, A. Kantowski-Sachs spherically symmetric solutions in teleparallel F (T ) gravity. Symmetry 2024 16, 953

    2024

  97. [105]

    Teleparallel Geometry with Spherical Symmetry: The diagonal and proper frames

    van den Hoogen, R.J.; Forance, H. Teleparallel Geometry with Spherical Symmetry: The diagonal and proper frames. J. Cosmol. Astrophys. 2024, 11, 033

    2024

  98. [106]

    Generalized Teleparallel de Sitter 61 geometries

    Coley, A.A.; Landry, A.; van den Hoogen, R.J.; McNutt, D.D. Generalized Teleparallel de Sitter 61 geometries. Eur. Phys. J. C 2023, 83, 977

    2023

  99. [107]

    Teleparallel Robertson-Walker Geometries and Applications

    Coley, A.A.; Landry, A.; Gholami, F. Teleparallel Robertson-Walker Geometries and Applications. Universe 2023, 9, 454

    2023

  100. [108]

    Scalar field sources Teleparallel Robertson-Walker F (T )-gravity solutions

    Landry, A. Scalar field sources Teleparallel Robertson-Walker F (T )-gravity solutions. Mathematics 2025, in press. https://doi.org/10.20944/preprints202412.1896.v1

    2025

  101. [109]

    E.; Kpadonou, A.V.; Rahaman, F.; Oliveira, P.J.; Houndjo, M.J.S

    Rodrigues, M. E.; Kpadonou, A.V.; Rahaman, F.; Oliveira, P.J.; Houndjo, M.J.S. Bianchi type-I, type-III and Kantowski-Sachs solutions in f (T ) gravity. Astrophys. Space Sci. 2015, 357, 129

    2015

  102. [110]

    Kantowski-Sachs Universe Models in f (T ) Theory of Gravity

    Amir, M.J.; Yussouf, M. Kantowski-Sachs Universe Models in f (T ) Theory of Gravity. Int. J. Theor. Phys. 2015, 54, 2798

    2015

  103. [111]

    Qualitative analysis of Kantowski-Sachs metric in a generic class of f (R) models

    Leon, G.; Roque, A.A. Qualitative analysis of Kantowski-Sachs metric in a generic class of f (R) models. J. Cosmol. Astropart. Phys. 2014, 05, 032

    2014

  104. [112]

    Curvature properties of Kantowski-Sachs metric

    Shaikh, A.A.; Chakraborty, D. Curvature properties of Kantowski-Sachs metric. J. Geom. Phys. 2021, 160, 103970

    2021

  105. [113]

    An anisotropic Kantowski-Sachs universe with radi- ation, dust and a phantom fluid

    Oliveira-Neto, G.; Canedo, D.L.; Monerat, G.A. An anisotropic Kantowski-Sachs universe with radi- ation, dust and a phantom fluid. Braz. J. Phys. 2022, 52, 130

    2022

  106. [114]

    Anisotropic spacetimes in f (T, B) theory II: Kantowski-Sachs Universe

    Leon, G.; Paliathanasis, A. Anisotropic spacetimes in f (T, B) theory II: Kantowski-Sachs Universe. Eur. Phys. J. Plus 2022, 137, 855

    2022

  107. [115]

    Anisotropic spacetimes in f (T, B) theory III: LRS Bianchi III Universe

    Leon, G.; Paliathanasis, A. Anisotropic spacetimes in f (T, B) theory III: LRS Bianchi III Universe. Eur. Phys. J. Plus 2022, 137, 927

    2022

  108. [116]

    The Study of Kantowski-Sachs Perfect Fluid Cosmological Model in Modified Gravity

    Vinutha, T.; Niharika, K.; Sri Kavya, K. The Study of Kantowski-Sachs Perfect Fluid Cosmological Model in Modified Gravity. Astrophysics 2023, 66, 64

    2023

  109. [117]

    Kantowski-Sachs Universe Filled with Perfect Fluid in f (R, T) Theory of Gravity

    Samanta, G.C. Kantowski-Sachs Universe Filled with Perfect Fluid in f (R, T) Theory of Gravity. Int. J. Theor. Phys. 2013, 52, 2647

    2013

  110. [118]

    Anisotropic Solutions in Sym- metric Teleparallel f (Q)-theory: Kantowski-Sachs and Bianchi III LRS Cosmologies

    Dimakis, N.; Roumeliotis, M.; Paliathanasis, A.; Christodoulakis, T. Anisotropic Solutions in Sym- metric Teleparallel f (Q)-theory: Kantowski-Sachs and Bianchi III LRS Cosmologies. Eur. Phys. J. C 2023, 83, 794

    2023

  111. [119]

    Kantowski-Sachs and Bianchi III dynamics in f (Q)-gravity

    Millano, A.D.; Dialektopoulos, K.; Dimakis, N.; Giacomini, A.; Shababi, H.; Halder, A.; Paliathanasis, A. Kantowski-Sachs and Bianchi III dynamics in f (Q)-gravity. Phys. Rev. D 2024, 109, 124044

    2024

  112. [120]

    Classical and Quantum Cosmological Solutions in Teleparallel Dark Energy with Anisotropic Background Geometry

    Paliathanasis, A. Classical and Quantum Cosmological Solutions in Teleparallel Dark Energy with Anisotropic Background Geometry. Symmetry 2022, 14, 1974

    2022

  113. [121]

    Kantowski-Sachs cosmology in scalar-torsion theory

    Paliathanasis, A. Kantowski-Sachs cosmology in scalar-torsion theory. Eur. Phys. J. C 2023, 83, 213

    2023

  114. [122]

    Cosmological solutions in teleparallel F (T, B) gravity

    Gholami, F.; Landry, A. Cosmological solutions in teleparallel F (T, B) gravity. Symmetry 2025, 17, 060

    2025

  115. [123]

    Quintessence, Cosmic Coincidence, and the Cosmological Constant

    Zlatev, I.; Wang, L.; Steinhardt, P. Quintessence, Cosmic Coincidence, and the Cosmological Constant. Phys. Rev. Lett. 1999, 82, 896

    1999

  116. [124]

    Cosmological tracking solutions

    Steinhardt, P.; Wang, L.; Zlatev, I. Cosmological tracking solutions. Phys. Rev. D 1999, 59, 123504. 62

    1999

  117. [125]

    Cosmological Imprint of an Energy Component with General Equation of State

    Caldwell, R.R.; Dave, R.; Steinhardt, P. Cosmological Imprint of an Energy Component with General Equation of State. Phys. Rev. Lett. 1998, 80, 1582

    1998

  118. [126]

    Quintessence and the Rest of the World

    Carroll, S.M. Quintessence and the Rest of the World. Phys. Rev. Lett. 1998, 81, 3067

    1998

  119. [127]

    Quintessence and the Separation of CMB Peaks

    Doran, M.; Lilley, M.; Schwindt, J.; Wetterich, C. Quintessence and the Separation of CMB Peaks. Astrophys. J. 2001, 559, 501

    2001

  120. [128]

    Holographic thermalization and gravitational collapse in the spacetime dominated by quintessence dark energy

    Zeng, X.-X.; Chen, D.-Y.; Li, L.-F. Holographic thermalization and gravitational collapse in the spacetime dominated by quintessence dark energy. Phys. Rev. D 2015, 91, 046005

    2015

  121. [129]

    Dynamical system analysis of quintessence dark energy model

    Chakraborty, S.; Mishra, S.; Chakraborty, S. Dynamical system analysis of quintessence dark energy model. Int. J. Geom. Methods Mod. Phys. 2025, 22, 2450250

    2025

  122. [130]

    Assessing observational constraints on dark energy

    Shlivko, D.; Steinhardt, P.J. Assessing observational constraints on dark energy. Phys. Lett. B 2024, 855, 138826

    2024

  123. [131]

    Cosmology and the Fate of Dilatation Symmetry

    Wetterich, C. Cosmology and the Fate of Dilatation Symmetry. Nucl. Phys. B 1988, 302, 668

    1988

  124. [132]

    Cosmological consequences of a rolling homogeneous scalar field

    Ratra, B.; Peebles, P.J.E. Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D 1988, 37, 3406

    1988

  125. [133]

    Attractor behavior of f (T ) modified gravity and the cosmic acceleration

    Duchaniya, L.K.; Gandhi, K.; Mishra, B. Attractor behavior of f (T ) modified gravity and the cosmic acceleration. Phys. Dark Universe 2024, 44, 101464

    2024

  126. [134]

    Teleparallel gravity

    Hohmann, M. Teleparallel gravity. In Modified and Quantum Gravity ; Springer: Berlin/Heidelberg, Germany, 2022; p. 145

    2022

  127. [135]

    Scalar-torsion theories of gravity III: Analogue of scalar-tensor gravity and conformal invariants

    Hohmann, M. Scalar-torsion theories of gravity III: Analogue of scalar-tensor gravity and conformal invariants. Phys. Rev. D 2018, 98, 064004

    2018

  128. [136]

    Energy Constraints and Phenomenon of Cosmic Evolution in f (T, B) Framework

    Zubair, M.; Waheed, S.; Fayyaz, M.A.; Ahmad, I. Energy Constraints and Phenomenon of Cosmic Evolution in f (T, B) Framework. Eur. Phys. J. Plus 2018, 133, 452

    2018

  129. [137]

    Constraining f (T, B) teleparallel gravity from energy conditions

    Bhattacharjee, S. Constraining f (T, B) teleparallel gravity from energy conditions. New Astron. 2021, 83, 101495

    2021

  130. [138]

    Cosmological singularities in f (T, ϕ) gravity

    Trivedi, O.; Khlopov, M.; Said, J.L.; Nunes, R. Cosmological singularities in f (T, ϕ) gravity. Eur. Phys. J. C 2023, 83, 1017

    2023

  131. [139]

    f (T, B) gravity in a Friedmann–Lema ˆ ıtre–Robertson–Walker universe with nonzero spatial curvature

    Paliathanasis, A.; Leon, G. f (T, B) gravity in a Friedmann–Lema ˆ ıtre–Robertson–Walker universe with nonzero spatial curvature. Math. Methods Appl. Sci. 2023, 46, 3905

    2023

  132. [140]

    Cosmological evolution in f (T, B) gravity

    Paliathanasis, A.; Leon, G. Cosmological evolution in f (T, B) gravity. Eur. Phys. J. Plus 2021, 136, 1092

    2021

  133. [141]

    Dynamical behavior in f (T, TG) cosmology

    Kofinas, G.; Leon, G.; Saridakis, E.N. Dynamical behavior in f (T, TG) cosmology. Class. Quantum Gravity 2014, 31, 175011

    2014

  134. [142]

    Gravitoelectromagnetism, Solar System Test and Weak-Field So- lutions in f (T, B) Gravity with Observational Constraints

    Farrugia, G.; Said, J.L.; Finch, A. Gravitoelectromagnetism, Solar System Test and Weak-Field So- lutions in f (T, B) Gravity with Observational Constraints. Universe 2020, 6, 34

    2020

  135. [143]

    Kinetically Driven Quintessence

    Chiba, T.; Okabe, T.; Yamaguchi, M. Kinetically Driven Quintessence. Phys. Rev. D 2000, 62, 023511

    2000

  136. [144]

    Can the dark energy equation-of-state parameter w be less than −1? Phys

    Carroll, S.M.; Hoffman, M.; Trodden, M. Can the dark energy equation-of-state parameter w be less than −1? Phys. Rev. D 2003, 68, 023509. 63

    2003

  137. [145]

    A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state

    Caldwell, R.R. A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state. Phys. Lett. B 2002, 545, 23

    2002

  138. [146]

    A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation within a Modified ΛCDM Framework

    Farnes, J.S. A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation within a Modified ΛCDM Framework. Astron. Astrophys. 2018, 620, A92

    2018

  139. [147]

    Turnaround in Cyclic Cosmology

    Baum, L.; Frampton, P.H. Turnaround in Cyclic Cosmology. Phys. Rev. Lett. 2007, 98, 071301

    2007

  140. [148]

    Crossing the Phantom Divide: Dark Energy Internal Degrees of Freedom

    Hu, W. Crossing the Phantom Divide: Dark Energy Internal Degrees of Freedom. Phys. Rev. D 2005, 71, 047301

    2005

  141. [149]

    Phantom-Like Behavior in Modified Teleparallel Gravity

    Karimzadeh, S.; Shojaee, R. Phantom-Like Behavior in Modified Teleparallel Gravity. Adv. High Energy Phys. 2019, 2019, 4026856

    2019

  142. [150]

    Rip cosmological models in extended symmetric teleparallel gravity

    Pati, L.; Kadam, S.A.; Tripathy, S.K.; Mishra, B. Rip cosmological models in extended symmetric teleparallel gravity. Phys. Dark Universe 2022, 35, 100925

    2022

  143. [151]

    Exact solutions in teleparallel dark energy model

    Kucukakca, Y.; Akbarieh, A.R.; Ashrafi, S. Exact solutions in teleparallel dark energy model. Chin. J. Phys. 2023, 82, 47

    2023

  144. [152]

    Quintom Cosmology: Theoretical implications and observations

    Cai, Y.-F.; Saridakis, E.N.; Setare, M.R.; Xia, J.-Q. Quintom Cosmology: Theoretical implications and observations. Phys. Rep. 2010, 493, 1

    2010

  145. [153]

    Cosmological evolution of a quintom model of dark energy

    Guo, Z.-K.; Piao, Y.-S.; Zhang, X.; Zhang, Y.-Z. Cosmological evolution of a quintom model of dark energy. Phys. Lett. B 2005, 608, 177

    2005

  146. [154]

    Oscillating quintom and the recurrent universe

    Feng, B.; Li, M.; Piao, Y.-S.; Zhang, X. Oscillating quintom and the recurrent universe. Phys. Lett. B 2006, 634, 101

    2006

  147. [155]

    Dynamical system analysis of quintom dark energy model

    Mishra, S.; Chakraborty, S. Dynamical system analysis of quintom dark energy model. Eur. Phys. J. C 2018, 78, 917

    2018

  148. [156]

    The dynamics of scalar-field Quintom cosmological models

    Tot, J.; Coley, A.A.; Yildrim, B.; Leon, G. The dynamics of scalar-field Quintom cosmological models. Phys. Dark Universe 2023, 39, 101155

    2023

  149. [157]

    Generalised teleparallel quintom dark energy non-minimally coupled with the scalar torsion and a boundary term

    Bahamonde, S.; Marciu, M.; Rudra, P. Generalised teleparallel quintom dark energy non-minimally coupled with the scalar torsion and a boundary term. J. Cosmol. Astropart. Phys. 2018, 04, 056

    2018

  150. [158]

    Exact solutions of an anisotropic universe in a modified teleparallel gravity model via the Noether and B.N.S

    Tajahmad, B. Exact solutions of an anisotropic universe in a modified teleparallel gravity model via the Noether and B.N.S. approaches. Int. J. Geom. Methods Mod. Phys. 2024, 21, 2450230

    2024

  151. [159]

    Cosmological Hyperfluids, Torsion and Non-metricity

    Iosifidis, D. Cosmological Hyperfluids, Torsion and Non-metricity. Eur. Phys. J. C 2020, 80, 1042

    2020

  152. [160]

    Homogeneous and isotropic cosmology in general teleparallel gravity

    Heisenberg, L.; Hohmann, M.; Kuhn, S. Homogeneous and isotropic cosmology in general teleparallel gravity. Eur. Phys. J. C 2023, 83, 315

    2023

  153. [161]

    Gauge-invariant cosmological perturbations in general teleparallel grav- ity

    Heisenberg, L.; Hohmann, M. Gauge-invariant cosmological perturbations in general teleparallel grav- ity. Eur. Phys. J. C 2024, 84, 462

    2024

  154. [162]

    Cosmological fluids with boundary term couplings

    Bohmer, C.G.; d’Alfonso del Sordo, A. Cosmological fluids with boundary term couplings. Gen. Rel- ativ. Gravit. 2024, 56, 75

    2024

  155. [163]

    The Large Scale Structure of Space-Time ; Cambridge University Press: Cambridge, UK, 2010

    Hawking, S.W.; Ellis, G.F.R. The Large Scale Structure of Space-Time ; Cambridge University Press: Cambridge, UK, 2010

    2010

  156. [164]

    Dynamical Systems and Cosmology ; Kluwer Academic: Dordrecht, The Netherlands, 64 2003

    Coley, A.A. Dynamical Systems and Cosmology ; Kluwer Academic: Dordrecht, The Netherlands, 64 2003

    2003

  157. [165]

    Teleparallel Minkowski Spacetime with Perturbative Approach for Teleparallel Gravity on a Proper Frame

    Landry, A.; van den Hoogen, R.J. Teleparallel Minkowski Spacetime with Perturbative Approach for Teleparallel Gravity on a Proper Frame. Universe 2023 9, 232

    2023

  158. [166]

    Landau levels in a gravitational field: The Schwarzschild spacetime case

    Landry, A.; Hammad, F. Landau levels in a gravitational field: The Schwarzschild spacetime case. Universe 2021, 7, 144

    2021

  159. [167]

    Landau levels in a gravitational field: The Levi-Civita and Kerr spacetimes case

    Hammad, F.; Landry, A. Landau levels in a gravitational field: The Levi-Civita and Kerr spacetimes case. Eur. Phys. J. Plus 2020, 135, 90. 65

    2020

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