Noether Symmetry

This paper proposes a fundamentally different ontological picture than the Standard Model and General Relativity. Space is represented by four equal axes (x 1 , x 2 , x 3 , x 4), the evolution of which occurs along a single absolute time t. All "relativistic" effects (slowing down of processes, length shortening, frequency shifts) are explained not by the change of the metric tensor, but by the geometry of the projection of 4D trajectories onto our 3D slice. The two-way (average) speed of light along any closed route is fixed by the postulate ⟨c⟩ = c, while the one-way speed can be anisotropic and depend on the synchronization procedure. The global symmetry SL(4, R) generates four independent Noether currents: conservation of charge, energy, momentum and "scale" (plateau law). The integrals of the flows of these currents give universal quantizing and plateau conditions from atomic orbits to galactic disks. The work is structured in increasing complexity: 1) axiomatics and ontology of the TCS; 2) kinematics of light and absolute time; 3) the dynamics of the field Q(x, t) and the derivation of the "master equation"; 4) applications to astrophysical and cosmological phenomena (galactic curves, CMB); 5) generalizations to microphysics and the standard model.

В работе вводится и разворачивается Теория Искривлённого Пространства (ТИП) — онтология, где пространство имеет четыре равноправные координаты, время t является абсолютным параметром эволюции, а физика наблюдаемого мира возникает как проекция 4D‑динамики на 3D‑срез. Ключевой аксиомой ТИП служит сохранение 4D‑объёма (глобальная SL(4,ℝ)‑симметрия). Из неё по теореме Нётер следуют четыре фундаментальных сохраняющихся тока: заряда, энергии, осевой (момент) и гомотетический (масштаб). Центральный результат — закон плато: в устойчивых геометриях (диск, сфера) соответствующие интегралы потока становятся константами, задавая асимптотическую кинематику от атома до галактики без привлечения тёмной материи.

Закон плато формулируется в операциональном виде на 3D‑срезе с мерой J3=det Q3 и анизотропией согласования Aij:
- Сфера (радиальный перенос, изотропия): Ks=R^2 J3 ρ(R) u∗(R)=const, что при медленном изменении J3 восстанавливает ньютоновский спад v(R)∝R^−1/2 во внешних областях.
- Тонкий диск (перенос в плоскости): Ka=R J3 ARR Σ(R) uR(R)=const, и при квазипостоянных J3 ARR и u∗ даёт плоскую кривую вращения v(R)=const (закон Местела) — без тёмной материи.
- В разреженных гало из инварианта R^2ρ=const следует M(R)∝R и выход ускорения к порогу a0≈10^−10 м/с^2, связанного в ТИП с глобальным вращением вокруг четвёртой оси.

На уровне частиц и полей ТИП даёт единую кинематическую оболочку: скалярные потенциалы входят мультипликативно в «маску» энергии покоя 𝓜(x,t)=√∏i(1−2Φi/c^2), а все калибровочные взаимодействия — через минимальное сцепление в импульсе. Квантование универсальной связи энергии–импульса приводит к обобщённым уравнениям Клейна–Гордона и Дирака; нерелятивистский предел даёт уравнение Паули. В электромагнетизме магнитное поле имеет проекционную (кинематическую) природу B=(1/c^2) v×E; в неабелевых полях (слабое/сильное) «магнитные» компоненты являются динамическими и не редуцируются к проекции из‑за самодействия.

Гравитационный сектор формализован через эффективное 3D‑действие, воспроизводящее ньютоновскую нормировку ΔΦ=4πGρ в слабом поле и позволяющее решать прикладные задачи: вращательные кривые, линзирование, чёрные дыры (тень, фотосфера, ISCO, задержка Шапиро). Показано, что классические наблюдаемые величины совпадают со стандартными результатами, при этом ТИП предсказывает фазовые эффекты (моно-дромии, эхо‑сигналы при кольцевании), доступные для проверки гравитационно‑волновыми и поляриметрическими измерениями.

Методика проверена на реальных данных без подгонок «по месту»: кривые вращения и фотометрия каталога SPARC (например, NGC 3198, NGC 2403, UGC 2885), карликовые сфероидалы Draco и Sculptor, ультрадиффузная NGC 1052‑DF2 (согласованный инвариант в сферической зоне), а также системы линзирования различного масштаба (SDSS J2141, ESO 325‑G004, Abell 1689) — расчёт углов отклонения выполняется из Φтип(r) при фиксированных входах (MGE, M/L, fgas, угловые расстояния), согласуясь с наблюдениями в пределах статистических ошибок. Космологический раздел формулирует фоновые уравнения и диагностирует совместимость с положением первого акустического пика СМВ без введения тёмной материи как компоненты.

Работа демонстрирует три свойства самодостаточной альтернативы: (1) внутренняя непротиворечивость аксиоматики и выводов; (2) количественное согласие с данными на множестве масштабов при фиксированных нормировках; (3) новые фальсифицируемые предсказания (пороговое ускорение, фазовая структура гравитационных сигналов, инварианты потоков), которые отличают ТИП от стандартных моделей.

---

## Ключевая формулировка закона плато (для карточки записи)

- Сферический режим: dR(R^2 J3 ρ u∗)=0 ⇒ R^2 J3 ρ u∗=const; следствие: M(R)∝R, v(R)∝R^−1/2, a(R)→a0 на хвосте гало.
- Дисковый режим: dR(R J3 ARR Σ uR)=0 ⇒ R J3 ARR Σ uR=const; следствие: v(R)=const при квазипостоянных ARR, J3 и переносе в плоскости.
- Операционная мера и анизотропия: J3=det Q3>0, Aij — безразмерный тензор «быстрых» направлений согласования det Q=1; u∗ — скорость фронта согласования, |u∗|<c.

---

## Новизна

- Универсальная операциональная формулировка законов сохранения через нётеровские токи ТИП с явной мерой J3 и анизотропией Aij.
- Дедуктивный вывод закона плато для дисковой и сферической геометрий без тёмной материи.
- Единая формула энергии–импульса с маской, охватывающая все четыре взаимодействия, и её квантование (КГФ/Дирак/Паули).
- Проекционная интерпретация магнитного поля (абелева) и разграничение с неабелевыми полями.
- Предсказуемые фазовые эффекты в сильнополевых режимах (чёрные дыры) при согласии классических наблюдаемых.

---

## Валидирующие расчёты и данные

- Галактики SPARC: проверка инвариантов Ka, Ks; плоские кривые вращения без тёмной материи; оценка a0≈10^−10 м/с^2.
- Карликовые сфероидалы Draco, Sculptor: плоскость R^2ρ u∗=const в сферической зоне; сравнение с MOND/ΛCDM.
- NGC 1052‑DF2: реконструкция сферического инварианта из динамической массы.
- Линзирование: SDSS J2141, ESO 325‑G004, Abell 1689 — согласие углов отклонения при фиксированных входах (MGE, M/L, fgas, D_A).

---

## Ключевые слова

Теорема Нётер; закон плато; Теория Искривлённого Пространства; SL(4,ℝ); инварианты потоков; плоские кривые вращения; критическое ускорение a0; гравитационное линзирование; обобщённые уравнения Клейна–Гордона и Дирака; маска скалярных потенциалов; проекционная природа магнитного поля; SU(3)×SU(2)×U(1); чёрные дыры; эхосигналы; СМВ.

---

## Рекомендуемое краткое описание (для превью)

Мы формулируем закон плато — универсальное следствие нётеровских симметрий в ТИП: в дисках R J3 ARR Σ uR=const, в сферах R^2 J3 ρ u∗=const. Из него следуют плоские кривые вращения и пороговое ускорение a0≈10^−10 м/с^2 без тёмной материи. Теория даёт единую формулу энергии–импульса (маска скалярных потенциалов + минимальное сцепление), обобщённые КГФ/Дирак, и согласуется с данными (SPARC, Draco/Sculptor, DF2, линзирование), предлагая новые фальсифицируемые предс

A ”reduced ” differential geometry adapted to the presence of abelian isometries is constructed. Classical T-duality diagonalizes in this setting, allowing us to get conveniently the transformation of the relevant geometrical objects such as connections, pullbacks and generalized curvatures. Moreover we can induce privileged maps from the viewpoint of covariant derivatives in the target-space and in the world-sheet generalizing previous results, at the same time that we can correct connections and curvatures covariantly in order to have a proper transformation under T-duality.

A "reduced" differential geometry adapted to the presence of abelian isometries is constructed. Classical T-duality diagonalizes in this setting, allowing us to get conveniently the transformation of the relevant geometrical objects such as connections, pullbacks and generalized curvatures. Moreover we can induce privileged maps from the viewpoint of covariant derivatives in the target-space and in the world-sheet generalizing previous results, at the same time that we can correct connections and curvatures covariantly in order to have a proper transformation under T-duality.

This paper offers a formal response to the key questions raised by N. Bellomo and B. Carbonaro in their study "Toward a Mathematical Theory of Living Systems", comparing them with the theoretical model of Evolutionary Mechanics and the Law of Quarters. Evolutionary Mechanics proposes an informational and geometrical structure based on eight symmetries of the logarithmic spiral and four emergent realms, each defined by its own rhythm or "evolutionary breath." This approach overcomes the limits of the computational and reductionist paradigm, offering a fractal, synchronic, and self-coherent vision of life and evolution.

Noether’s Theorem states that every continuous symmetry of a physical system corresponds to a conserved quantity. One of the fundamental symmetries of the universe is temporal symmetry, which is directly linked to the conservation of energy. However, in Evolutionary Mechanics, energy is not just a physical quantity but a measure of coherence and information, regulated by the Temporal Matrix and the cyclical transformations of the Law of Quarters.

Noether’s Theorem states that every continuous symmetry of a physical system corresponds to a conserved quantity. One of the fundamental symmetries of the universe is translational symmetry, which is directly related to the conservation of momentum. However, in Evolutionary Mechanics, the very concept of velocity and momentum is redefined as a rhythm of informational emergence.

Noether’s Theorem states that every continuous symmetry of a physical system corresponds to a conserved quantity. One of the fundamental symmetries of the universe is rotational symmetry, which is directly linked to the conservation of angular momentum. In this work, we explore how Evolutionary Mechanics extends this symmetry, connecting it to the multi-metric and supersymmetric dynamics of the toroid.

Ostrogradsky’s, Dirac’s, and Horowitz’s techniques in terms of higher-order theories of gravity produce identical phase-space structures. The problem with these techniques is manifested in the case of Gauss–Bonnet–dilatonic coupled action in the presence of higher-order term, in which case, classical correspondence cannot be established. Here, we explore another technique developed by Buchbinder and their collaborators (BL) and show that it also suffers from the same disease. However, when expressing the action in terms of the three-space curvature, and removing ‘the total derivative terms’, if Horowitz’s formalism or even Dirac’s constraint analysis is pursued, all pathologies disappear. Here, we show that the same is true for BL formalism, which appears to be the simplest of all the techniques to handle.

In this article, the generalized gravity theory with the curvature, torsion and nonmetricity was studied. For the FRW spacetime case, in particular, the Lagrangian, Hamilatonian and gravitational equations are obtained. The particular case F (R, T) = αR + βT + µQ + νT is investigated in detail. In quantum case, the corresponding Wheeler-DeWitt equation is obtained. Finally, some gravity theories with the curvature, torsion and nonmetricity are presented.

In this paper we study the anisotropic universe using Noether symmetries in modified gravity. In particular, we choose a locally rotationally symmetric Bianchi type-I universe for the analysis in f (R, G) gravity, where R is the Ricci scalar and G is the Gauss-Bonnet invariant. Firstly, a model f (R, G) = f 0 R l + f 1 G n is proposed and the corresponding Noether symmetries are investigated. We have also recovered the Noether symmetries for f (R) and f (G) theories of gravity. Secondly, some important cosmological solutions are reconstructed. Exponential and powerlaw solutions are reported for a well-known f (R, G) model, i.e., f (R, G) = f 0 R n G 1−n. Especially, Kasner's solution is recovered and it is anticipated that the familiar de Sitter spacetime giving CDM cosmology may be reconstructed for some suitable value of n.

Unlike F (R) gravity, pure metric F (T) gravity in the vacuum dominated era, ends up with an imaginary action and is therefore not feasible. This eerie situation may only be circumvented by associating a scalar field, which can also drive inflation in the very early universe. We show that, despite diverse claims, F (T) theory admits Noether symmetry only in the pressure-less dust era in the form F (T) ∝ T n , n being odd integers. A suitable form of F (T) , admitting a viable Friedmann-like radiation dominated era, together with early deceleration and late-time accelerated expansion in the pressure-less dust era, has been proposed.

Unlike F(R) gravity, pure F(T) gravity in the vacuum dominated era does not produce any form, viable to study cosmological evolution. Factually, it does not even produce any dynamics. This eerie situation may be circumvented by associating a scalar field, which can also drive inflation in the very early universe. Here, we particularly show that, despite diverse claims, F(T) theory admits Noether symmetry only in the pressure-less dust era in the form $$F(T) \propto T^n, ~n$$ F ( T ) ∝ T n , n being odd integers. A suitable form of F(T), admitting inflation in the very early universe, a viable Friedmann-like radiation dominated era, together with early deceleration and late-time accelerated expansion in the pressure-less dust era, has been proposed.

Unlike F(R) gravity, pure F(T) gravity in the vacuum dominated era does not produce any form, viable to study cosmological evolution. Factually, it does not even produce any dynamics. This eerie situation may be circumvented by associating a scalar field, which can also drive inflation in the very early universe. Here, we particularly show that, despite diverse claims, F(T) theory admits Noether symmetry only in the pressure-less dust era in the form $$F(T) \propto T^n, ~n$$ F ( T ) ∝ T n , n being odd integers. A suitable form of F(T), admitting inflation in the very early universe, a viable Friedmann-like radiation dominated era, together with early deceleration and late-time accelerated expansion in the pressure-less dust era, has been proposed.

We discuss the f (R) gravity model in which the origin of dark energy is identified as a modification of gravity. The Noether symmetry with gauge term is investigated for the f (R) cosmological model. By utilization of the Noether Gauge Symmetry (NGS) approach, we obtain two exact forms f (R) for which such symmetries exist. Further it is shown that these forms of f (R) are stable.

Metric variation of higher order theory of gravity requires fixing of the Ricci scalar in addition to the metric tensor at the boundary. Fixing Ricci scalar at the boundary implies that the classical solutions are fixed once and forever to the de Sitter or anti-de Sitter (dS/AdS) solutions. Here, we justify such requirement from the standpoint of Noether symmetry.

Noether symmetry of F (R) theory of gravity in vacuum or in matter dominated era yields F (R) ∝ R 3 2. We show that this particular curvature invariant term is very special in the context of isotropic and homogeneous cosmological model as it makes the first fundamental form hij cyclic. As a result, it allows a unique power law solution, typical for this particular fourth order theory of gravity, both in the vacuum and in the matter dominated era. This power law solution has been found to be quite good to explain the early stage but not so special and useful to explain the late stage of cosmological evolution. The usefulness of Palatini variational technique in this regard has also been discussed.

In this article, we investigate the modified F (T ) gravity, which is non-minimally coupled with the Dirac (fermion) field in Friedmann-Robertson-Walker space-time. Point-like Lagrangian is derived and modified Friedmann equations and Dirac equations for the fermion field are obtained by using the Lagrange multiplier. The Noether symmetry method related to differential equations is a useful tool for studying conserved quantities. In addition, this method is very useful for determining the unknown functions that exist in the point-like Lagrangian. Using this method, the form of the coupling between gravity and matter, the self-consistent potential, the symmetry generators, the form of F (T ) gravity and the first integral (Noether charge) or a conserved quantity for this model are determined. Cosmological solutions that have a power-law form and describe the late time accelerated expansion of the Universe are obtained.

We explore Noether gauge symmetries of FRW and Bianchi I universe models for perfect fluid in scalar-tensor gravity with extra term R −1 as curvature correction. Noether symmetry approach can be used to fix the form of coupling function ω(φ) and the field potential V (φ). It is shown that for both models, the Noether symmetries, the gauge function as well as the conserved quantity, i.e., the integral of motion exist for the respective point like Lagrangians. We determine the form of coupling function as well as the field potential in each case. Finally, we investigate solutions through scaling or dilatational symmetries for Bianchi I universe model without curvature correction and discuss its cosmological implications.

Higher-order corrections of Einstein-Hilbert action of general relativity can be recovered by imposing the existence of a Noether symmetry to a class of theories of gravity where Ricci scalar R and its d'Alembertian 2R are present. In several cases, it is possible to get exact cosmological solutions or, at least, to simplify dynamics by recovering constants of motion. The main result is that a Noether vector seems to rule the presence of higher-order corrections of gravitaty.

The role of torsion and a scalar field ϕ in gravitation in the background of a particular class of the Riemann-Cartan geometry is considered here. Some times ago, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4,1) Pontryagin density, where the scalar field ϕ causes the de Sitter connection to have the proper dimension of a gauge field. Here it has been shown that the divergence of the axial torsion gives the Newton&#39;s constant and the scalar field becomes a function of the Ricci scalar R. The starting Lagrangian then reduces to a Lagrangian representing the metric f(R) gravity theory.

The low energy physics as predicted by strings can be expressed in two (conformally related) different variables, usually called frames. The problem is raised as to whether it is physically possible in some situations to differentiate one from the other.

In this paper, we present the Noether symmetries of a class of the Bianchi type I anisotropic model in the context of f(T) gravity. By solving the system of equations obtained from the Noether symmetry condition, we obtain the form of f(T) as a teleparallel form. This analysis shows that teleparallel gravity has the maximum number of Noether symmetries. We derive the symmetry generators and show that there are five kinds of symmetries, including time and scale invariance under metric coefficients. We classify the symmetries and we obtain the corresponding invariants.

The study of particle creation phenomena at the expense of the gravitational field is of great research interest. It might solve the cosmological puzzle singlehandedly, without the need for either dark energy or modified theory of gravity. In the early universe, following graceful exit from inflationary phase, it serves the purpose of reheating the cold universe, which gave way to the hot Big-Bang model. In the late universe, it led to late time cosmic acceleration, without affecting standard Big-Bang-Nucleosynthesis (BBN), Cosmic Microwave Background Radiation (CMBR), or Structure Formation. In this chapter, we briefly review the present status of cosmic evolution, develop the thermodynamics for irreversible particle creation phenomena and study its consequences at the early as well as at the late universe.

Classical equivalence between Jordan’s and Einstein’s frame counterparts of [Formula: see text] theory of gravity has recently been questioned, since the two produce different Noether symmetries, which could not be translated back and forth using transformation relations. Here we add the Hamiltonian constraint equation, which is essentially the time–time component of Einstein’s equation, through a Lagrange multiplier to the existence condition for Noether symmetry and show that all the three different canonical structures of [Formula: see text] theory of gravity, including the one which follows from Lagrange multiplier technique, admit each and every available symmetry independently. This establishes classical equivalence.

Noether gauge symmetry for F(R) theory of gravity has been explored recently. The fallacy is that, even after setting gauge to vanish, the form of F(R) \propto R^n (where n \neq 1, is arbitrary) obtained in the process, has been claimed to be an outcome of gauge Noether symmetry. On the contrary, earlier works proved that any nonlinear form other than F(R) \propto R^3/2 is obscure. Here, we show that, setting gauge term zero, Noether equations are satisfied only for n = 2, which again does not satisfy the field equations. Thus, as noticed earlier, the only admissible form that Noether symmetry is F(R) \propto R^3/2 . Noether symmetry with non-zero gauge has also been studied explicitly here, to show that it does not produce anything new.

Unlike the Noether symmetry, a metric independent general conserved current exists for non-minimally coupled scalar-tensor theory of gravity if the trace of the energymomentum tensor vanishes. Thus, in the context of cosmology, a symmetry exists both in the early vacuum and radiation dominated era. For slow roll, symmetry is sacrificed, but at the end of early inflation, such a symmetry leads to a Friedmann-like radiation era. Late-time cosmic acceleration in the matter dominated era is realized in the absence of symmetry, in view of the same decayed and redshifted scalar field. Thus, unification of early inflation with late-time cosmic acceleration with a single scalar field may be realized.

The role of torsion and a scalar field φ in gravitation in the background of a particular class of the Riemann-Cartan geometry is considered here. Some times ago, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4, 1) Pontryagin density, where the scalar field φ causes the de Sitter connection to have the proper dimension of a gauge field. Here it has been shown that the divergence of the axial torsion gives the Newton’s constant and the scalar field becomes a function of the Ricci scalar R. The starting Lagrangian then reduces to a Lagrangian representing the metric f(R) gravity theory.

Several new results regarding the quantum cosmology of higher-derivative gravity theories derived from superstring effective actions are presented. After describing techniques for solving the Wheeler-DeWitt equation with appropriate boundary conditions, results are compared with semiclassical theories of inflationary cosmology and implications for various different string cosmology models are outlined. 1. Classical Cosmological Space-times of Higher-Derivative Theories Since solutions to the general relativistic field equations contain initial curvature singularities whenever the dominant energy condition is satisfied, one of the motivations for developing quantum cosmology has been the theoretical justification of the elimination of the singularities. Curvature singularities predicted by general relativity can be avoided by introducing boundary conditions in the path integral defining the quantum theory which restrict the integral over all four-geometries and matter fields to Riema...

We revisit the T-duality transformation rules in heterotic string theory, pointing out that the chiral structure of the world-sheet leads to a modification of the standard Buscher's transformation rules. The simplest instance of such modifications arises for toroidal compactifications, which are rederived by analyzing a bosonized version of the heterotic world-sheet Lagrangian. Our study indicates that the usual heterotic toroidal T-duality rules naively extended to the curved case cannot be correct, leading in particular to an incorrect Bianchi identity for the field strength H of the Kalb-Ramond field B. We explicitly show this problem and provide a specific example of dual models where we are able to get new T-duality rules which, contrary to the standard ones, lead to a correct T-dual Bianchi identity for H to all orders in α ′ .

We derive a working model for the Tolman-Oppenheimer-Volkoff equation for quark star systems within the modified f (T, T)-gravity class of models. We consider f (T, T)-gravity for a static spherically symmetric spacetime. In this instance the metric is built from a more fundamental tetrad vierbein from which the metric tensor can be derived. We impose a linear f (T) parameter, namely taking f = αT (r) + βT (r) + ϕ and investigate the behaviour of a linear energy-momentum tensor trace, T. We also outline the restrictions which modified f (T, T)-gravity imposes upon the coupling parameters. Finally we incorporate the MIT bag model in order to derive the mass-radius and mass-central density relations of the quark star within f (T, T)-gravity.

The role of torsion and a scalar field φ in gravitation in the background of a particular class of the Riemann-Cartan geometry is considered here. Some times ago, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4, 1) Pontryagin density, where the scalar field φ causes the de Sitter connection to have the proper dimension of a gauge field. Here it has been shown that the divergence of the axial torsion gives the Newton's constant and the scalar field becomes a function of the Ricci scalar R. The starting Lagrangian then reduces to a Lagrangian representing the metric f (R) gravity theory.

We present teleparallel 3D gravity and we extract circularly symmetric solutions, showing that they coincide with the BTZ and Deser-de-Sitter solutions of standard 3D gravity. However, extending into f (T) 3D gravity, that is considering arbitrary functions of the torsion scalar in the action, we obtain BTZ-like and Deser-de-Sitter-like solutions, corresponding to an effective cosmological constant, without any requirement of the sign of the initial cosmological constant. Finally, extending our analysis incorporating the electromagnetic sector, we show that Maxwell-f (T) gravity accepts deformed charged BTZ-like solutions. Interestingly enough, the deformation in this case brings qualitatively novel terms, contrary to the pure gravitational solutions where the deformation is expressed only through changes in the coefficients. We investigate the singularities and the horizons of the new solutions, and amongst others we show that the cosmic censorship can be violated. Such novel behaviors reveal the new features that the f (T) structure brings in 3D gravity.

Canonization of F (R) theory of gravity to explore Noether symmetry is performed treating R−6(ä a +ȧ 2 a 2 + k a 2) = 0 as a constraint of the theory in Robertson-Walker space-time, which implies that R is taken as an auxiliary variable. Although it yields correct field equations, Noether symmetry does not allow linear term in the action, and as such does not produce a viable cosmological model. Here, we show that this technique of exploring Noether symmetry does not allow even a non-linear form of F (R) , if the configuration space is enlarged by including a scalar field in addition, or taking anisotropic models into account. Surprisingly enough, it does not reproduce the symmetry that already exists in the literature [15] for scalar tensor theory of gravity in the presence of R 2 term. Thus, R can not be treated as an auxiliary variable and hence Noether symmetry of arbitrary form of F (R) theory of gravity remains obscure. However, there exists in general, a conserved current for F (R) theory of gravity in the presence of a non-minimally coupled scalar-tensor theory [26, 27]. Here, we briefly expatiate the non-Noether conserved current and cite an example to reveal its importance in finding cosmological solution for such an action, taking F (R) ∝ R 3 2 .

Noether symmetry of F (R) theory of gravity reveals F (R) = R 3 2 , if the expression for the scalar curvature for R-W metric is treated as a constraint and entered into the action through a Lagrange multiplier. In the process, a cyclic coordinate is found which gives a solution that appears to explain the present cosmological evolution. The interesting issue is that out of infinite curvature invariant terms, Noether symmetry selects only R 3 2. Here, we explore the very speciality and study the attractive features and shortcomings of such term in the context of cosmological evolution.

It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so-called "Palatini formalism", i.e., treating the metric and the connection as independent variables, leads to "universal" equations. If the dimension n of space-time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians L = R n/2 √ g and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2-dimensional space-time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.

We study the matter stability in modified teleparallel gravity or f (T) theories. We show that there is no Dolgov-Kawasaki instability in these types of modified teleparallel gravity theories. This gives the f (T) theories a great advantage over their f (R) counterparts because from the stability point of view there isn't any limit on the form of functions that can be chosen.

We study the gravity in the context of a braneworld teleparallel scenario. The geometrical setup is assumed to be Randall-Sundrum II model where a single positive tension brane is embedded in an infinite AdS bulk. We derive the equivalent of Gauss-Codacci equations in teleparallel gravity and junction conditions in this setup. Using these results we derive the induced teleparallel field equations on the brane. We show that compared to general relativity, the induced field equations in teleparallel gravity contain two extra terms arising from the extra degrees of freedom in the teleparallel Lagrangian. The term carrying the effects of the bulk to the brane is also calculated and its implications are discussed.

We investigate the cosmological perturbations in f (T) gravity. Examining the pure gravitational perturbations in the scalar sector using a diagonal vierbien, we extract the corresponding dispersion relation, which provides a constraint on the f (T) ansatzes that lead to a theory free of instabilities. Additionally, upon inclusion of the matter perturbations, we derive the fully perturbed equations of motion, and we study the growth of matter overdensities. We show that f (T) gravity with f (T) constant coincides with General Relativity, both at the background as well as at the first-order perturbation level. Applying our formalism to the power-law model we find that on large subhorizon scales (O(100 Mpc) or larger), the evolution of matter overdensity will differ from ΛCDM cosmology. Finally, examining the linear perturbations of the vector and tensor sectors, we find that (for the standard choice of vierbein) f (T) gravity is free of massive gravitons.

We investigate equations of motion and future singularities of [Formula: see text] gravity where [Formula: see text] is the Ricci scalar and [Formula: see text] is the trace of stress-energy tensor. Future singularities for two kinds of equation of state (barotropic perfect fluid and generalized form of equation of state) are studied. While no future singularity is found for the first case, some kind of singularity is found to be possible for the second. We also investigate [Formula: see text] gravity by the method of dynamical systems and obtain some fixed points. Finally, the effect of the Noether symmetry on [Formula: see text] is studied and the consistent form of [Formula: see text] function is found using the symmetry and the conserved charge.

The low energy physics as predicted by strings can be expressed in two (conformally related) different variables, usually called frames. The problem is raised as to whether it is physically possible in some situations to tell one from the other.

Classical equivalence between Jordan’s and Einstein’s frame counterparts of [Formula: see text] theory of gravity has recently been questioned, since the two produce different Noether symmetries, which could not be translated back and forth using transformation relations. Here we add the Hamiltonian constraint equation, which is essentially the time–time component of Einstein’s equation, through a Lagrange multiplier to the existence condition for Noether symmetry and show that all the three different canonical structures of [Formula: see text] theory of gravity, including the one which follows from Lagrange multiplier technique, admit each and every available symmetry independently. This establishes classical equivalence.

Metric variation of higher order theory of gravity requires fixing of the Ricci scalar in addition to the metric tensor at the boundary. Fixing Ricci scalar at the boundary implies that the classical solutions are fixed once and forever to the de Sitter or anti-de Sitter (dS/AdS) solutions. Here, we justify such requirement from the standpoint of Noether symmetry.

Noether symmetry of F (R) theory of gravity in vacuum or in matter dominated era yields We show that this particular curvature invariant term is very special in the context of isotropic and homogeneous cosmological model as it makes the first fundamental form hij cyclic. As a result, it allows a unique power law solution, typical for this particular fourth order theory of gravity, both in the vacuum and in the matter dominated era. This power law solution has been found to be quite good to explain the early stage but not so special and useful to explain the late stage of cosmological evolution. The usefulness of Palatini variational technique in this regard has also been discussed.

Noether symmetry of F (R) theory of gravity in vacuum and in the presence of pressureless dust yields F (R) ∝ R 3 2 along with the conserved current d dt (a √ R) in Robertson-Walker metric and nothing else. Still some authors recently claimed to have obtained four conserved currents setting F (R) ∝ R 3 2 a-priori, taking time translation along with a gauge term. We show that the first one of these does not satisfy the field equations and the second one is the Hamiltonian which is constrained to vanish in gravity and thus a part and parcel of the field equations. We also show that the other two conserved currents, which do not contain time translation are the same in disguise and identical to the one mentioned above. Thus the claim is wrong. PACS 04.50.+h F (R) theory of gravity has drawn lot of attention to the cosmologists in recent years, since it elegantly solves the cosmic puzzle in connection with dark energy, unifies early inflation with late time cosmic acceleration, admits Newtonian limit and passes the solar test singlehandedly [see for recent and thorough reviews]. However, in order to select a particular form of F (R) out of indefinitely large number of curvature invariant terms, Noether symmetry was invoked as a selection rule by several authors [2] - . Results in a nutshell are (i) Noether symmetry does not admit anything other than F (R) ∝ R 3 2 , along with a conserved current d dt (a √ R) in Robertson-Walker metric; (ii) Explicit solution of the field equations shows that it is well behaved in the early Universe, nevertheless it suffers from the same disease as R -n , n > 0 term in the late Universe in the presence of radiation and a pressureless dust. In the radiation era the scale factor tracks as a ∝ t 3

Noether gauge symmetry for F (R) theory of gravity has been explored recently. The fallacy is that, even after setting gauge to vanish, the form of F (R) ∝ R n (where n = 1 is arbitrary) obtained in the process, has been claimed to be an outcome of gauge Noether symmetry. On the contrary, earlier works proved that any nonlinear form other than F (R) ∝ R 3 2 is obscure. Here, we show that, setting gauge term zero, Noether equations are satisfied only for n = 2 , which again does not satisfy the field equations. Thus, as noticed earlier, the only form that Noether symmetry admits is F (R) ∝ R 3 2. Noether symmetry with non-zero gauge has also been studied explicitly here, to show that it does not produce anything new.