Aims. We present the first systematic investigation of how rotation modifies RGB bump properties (luminosity, amplitude, and duration) across 0.8-2.2 M-circle dot, with the goal of quantifying the physical link between rotation-induced mixing, the mean molecular weight (mu) discontinuity, and the bump's observational signatures. Methods. Using the Modules for Experiments in Stellar Astrophysics (MESA) code, we computed a grid of stellar models at solar metallicity (Z = 0.014) with initial masses of 0.8-2.2 M-circle dot and initial rotational velocities ranging from 0 to 250 km/s. Our models include a comprehensive treatment of angular momentum transport and rotation-induced chemical mixing. Results. The RGB bump arises from structural readjustment when the hydrogen-burning shell (HBS) encounters the mu discontinuity left by the first dredge-up: a steeper mu gradient drives a more pronounced luminosity dip (larger bump amplitude) in the HBS, creating a distinct 'zig-zag' in the Hertzsprung-Russell (HR) diagram. Rotation-induced mixing modifies the RGB bump by smoothing the sharp mu gradient (hydrogen-helium discontinuity) and altering the depth of the first dredge-up. Rotation slightly reduces the dredge-up depth for the star with M < 1.8 M-circle dot, while it significantly shallows the convective envelope for M > 1.8 M-circle dot, flattening the mu gradient. Conclusions. Rotation-induced mixing depletes surface C-12 (lowering C-12/C-13) and enriches nitrogen (raising N-14/C-12). More critically, it affects the RGB bump by smoothing the sharpness and diminishing the amplitude of the hydrogen-helium discontinuity. This effect reduces the amplitude of the luminosity dip in the HBS. Slowly rotating stars (v(ini) similar to 100 km/s) exhibit an RGB bump at a lower luminosity and a higher effective temperature than non-rotating models, a trend most pronounced in low-mass stars; whereas faster rotation delays the bump's onset and further weakens its amplitude. Our results demonstrate that rotation is a critical, previously unquantified driver of RGB bump diversity in stellar populations.

Context. Accurate Roche-lobe radii are vital for mass transfer and binary evolution modeling. While the Eggleton (1983, ApJ, 268, 368) formula is widely used for its simplicity and 1% accuracy, observations reveal systematic underestimations in certain mass-ratio ranges–a discrepancy expected within the standard Roche framework. Retaining these standard assumptions, we refined the numerical accuracy using Eggleton’s formula as a baseline. Crucially, because Eggleton’s model only constrains the inner critical surface (L1), it cannot accommodate over-contact binaries that require outer boundary modeling. By extending numerical integrations to the outer critical surface, we derived a more precise formula for the outer Roche-lobe radii. This provides a more realistic physical representation and is a key improvement in this work. Aims. Beyond determining the inner and outer Roche-lobe radii, we also established more precise fitting formulae for the positions of L1, L2, and L3. This study aimed to construct volume-equivalent radii formulae for the equipotential surfaces passing through these three Lagrangian points, providing a more comprehensive tool for binary evolution modeling. Furthermore, observational data of contact binaries covering mass ratios q = M2/M1 q = M 2 M 1 $ q=\frac{M_{2}}{M_{1}} $ from 0 to 20 were utilized for validation. Methods. In this work, we defined a spherical coordinate system centered on the primary star (M1) and derived analytical approximations for the positions of the three collinear Lagrangian points (L1, L2, and L3). Furthermore, we proposed a new method to calculate the volume-equivalent radii for L2 and L3. By using the lowest points along the L2 and L3 equipotential surfaces as boundaries, this approach divides the peanut-shaped structure into two components, thereby yielding the corresponding volume-equivalent radii RL2 and RL3 Results. The improved Roche-lobe radii formulae in this study achieve errors ≤0.5% across all mass ratios q. The research results provide a more precise theoretical framework for simulating mass overflow and mass loss through L1, L2 and L3. Conclusions. Determining the positions of the Lagrangian points aims to calculate the Roche potential and investigate mass transfer and loss. These new formulae are applicable to any q with higher precision, and L2, and L3 facilitates the analysis of mass loss through the outer Lagrangian points.

When the expansion rate of the Universe at recombination is used to infer the present expansion rate H0, the value derived in the Lambda CDM model, H0=67.4 km s-1 Mpc-1, is about in 6 sigma tension with the value measured locally, H0=74 km s-1 Mpc-1. In this work, we consider instead the expansion history in the context of the symmetry of scale-invariant vacuum (SIV model). We first perform two major cosmological tests: the Hubble diagram for type-Ia supernovae and the fundamental relation between H0, the age of the Universe, and the total density of matter, Omega m. This allows us to constrain Omega m in SIV, with both tests giving the best agreement for Omega m = 0.20. We then study the physical connections of the dynamical and thermal states of the Universe at recombination with the present Hubble constant, H0, and the present temperature, T, in the Lambda CDM and SIV contexts. We find that, in SIV, the properties at recombination may be conveyed to the present ones (T=2.726 and H0 at z=0) without any tension, indicating H0=74 km s-1 Mpc-1 in spite of the anchoring on the CMB. This is due to the slightly different expansion and temperature histories of the two cosmological models. Importantly, this happens to occur for Omega m = 0.20, as constrained in SIV with supernovae and cosmic age. This suggests that the Hubble tension currently found between H0 values in the early and late Universe may simply be the result of Lambda CDM ignoring the small but still measurable effects of scale invariance.

The Scale Invariant Vacuum (SIV) paradigm is applied to the Big-Bang Nucleosynthesis using the known analytic expressions for the expansion factor $a$ and the plasma temperature $T$ as functions of the SIV time $\tau$ since the Big-Bang when $a(\tau=0)=0$. The results are compared to the known standard BBNS model as calculated with the PRIMAT code. Potential SIV-guided deviations from the local statistical equilibrium are explored. Overall, we find that smaller than usual baryon and non-zero dark matter content, by a factor of three to five times reduction, result in compatible to the standard reproduction of the light elements abundances.

The post-main-sequence evolution of massive stars is highly sensitive to key stellar model parameters, particularly initial rotational velocity and metallicity. Using the MESA code, we computed stellar models with initial masses of 16–32 M _⊙ , metallicities of Z = 0.0047–0.025, and initial rotational velocities of 0.0–400 km s ^−1 . Our results demonstrate that the evolutionary trajectory toward red or blue supergiants is primarily regulated by the parameter ϵ _grav , which is the rate of generation or absorption of gravitational energy in the energy conservation equation. Three main factors drive the evolution toward lower effective temperatures (to produce the reduced blue-to-red supergiant ratio, B/R) during central helium burning: (1) an increase in helium and CNO abundances in the hydrogen-burning shell, (2) an increase in the core mass ratio, and (3) a decrease in central helium abundance during the latter half of core helium burning. Rotation amplifies the core mass during the core He-burning phase, inducing atmospheric inflation, stronger mass loss, and redward evolution. Consequently, rapid rotation leads to a lower B/R supergiant ratio. A thinner envelope induced by strong winds favors a higher B/R supergiant ratio. The minimum envelope mass required for stars to transition from a red to a blue supergiant decreases with higher metallicity and higher initial rotational velocity. High-metallicity stars undergo an extended blue excursion after the red supergiant phase, increasing the B/R supergiant ratio. A higher B/R supergiant ratio is strongly linked to the presence of an intermediate convective region during the first crossing of the Hertzsprung gap.

The enigmatic phenomenon of dark energy (DE) is the elusive entity driving the accelerated expansion of our Universe. A plausible candidate for DE is the non-zero Einstein Cosmological Constant ΛE manifested as a constant energy density of the vacuum, yet it seemingly defies gravitational effects. In this work, we interpret the non-zero ΛE through the lens of scale-invariant cosmology. We revisit the conformal scale factor λ and its defining equations within the Scale-Invariant Vacuum (SIV) paradigm. Furthermore, we address the profound problem of the missing mass across galactic and extragalactic scales by deriving an MOND-like relation, g∼a0gN, within the SIV context. Remarkably, the values obtained for ΛE and the MOND fundamental acceleration, a0, align with observed magnitudes, specifically, a0≈10−10ms−2 and ΛE≈1.8×10−52m−2. Moreover, we propose a novel early dark energy term, T˜μν∼κH, within the SIV paradigm, which holds potential relevance for addressing the Hubble tension.

Context. Numerous studies have established that in main-sequence binary systems, mass transfer generally proceeds from the initially more massive star to its less massive companion, thereby inducing rejuvenation in the latter. However, in certain massive close binary systems with orbital periods on the order of a few days, a reversed mass transfer scenario emerges: mass flows from the initially less massive component to the more massive one, during which the two stars simultaneously overflow their Roche lobes. This configuration potentially facilitates stellar rejuvenation in both members of the binary system. Aims. The phenomenon of reverse mass transfer in binary systems is closely linked to the efficiency of stellar rejuvenation in the mass accretor. However, the physical mechanism driving the rejuvenation of the accretor remains poorly understood. In this work, we employ the Schwarzschild criterion to define the boundaries of convective regions at the solar metallicity, as opposed to the Ledoux criterion typically used for convective conditions at low metallicities. Our study aims to investigate how mass transfer significantly influences the rejuvenation process of the mass-accreting star. Furthermore, we aim to systematically investigate how initial binary parameters (particularly the mass ratio and orbital period) regulate the initiation of reverse mass transfer from the secondary to the primary component. Methods. We constructed a new set of detailed, grid-based binary evolution models, systematically varying initial orbital periods and mass ratios, q, within a parameter space. Results. Our results show that for systems with initial primary masses of 16 M-circle dot and initial mass ratios of q(ini)= M2M1 >= 0.4, those with shorter initial orbital periods, P-orb < 1.8 days, are statistically more prone to evolving into contact binaries that exhibit reverse mass transfer. The results further demonstrate that stellar rejuvenation significantly influences the overall evolution of binary systems, notably extending the main-sequence lifetime of the mass-gaining star. Specifically, rejuvenation can induce increases in both the radius and luminosity of the accretor, potentially triggering reverse mass transfer. The transfer causes the mass of accretor's convective core to increase, facilitating the mixing of fresh fuel from outer layers into the central nuclear-burning region. Following mass transfer, the evolutionary state of the accretor is closer to the zero age main sequence. Due to the high efficiency of rejuvenation, the mass-gaining star can even outpace the mass-losing star in terms of evolution. Meanwhile, the initially more massive star can also undergo rejuvenation via reverse mass transfer, as it must rapidly adapt to its newly increased mass.

Context. About 0.2-2% of red clump stars are revealed as Lithium-rich stars and thus the surface abundance of lithium clearly increases in some red clump stars. The physical mechanism of the enrichment of lithium on the surface of these stars has not yet been explained satisfactorily by the evolutionary models of single stars. Aims. Our aim is to investigate how rotation, thermohaline mixing, and internal gravity waves have an important impact on the surface chemical abundance of lithium-rich red giants. Methods. The equations for angular momentum transport and the chemical element diffusion for rotating stars have been implemented in this paper. The diffusion coefficients of rotationally induced instabilities, thermohaline mixing, and internal gravitational waves have been included in the diffusion equation of chemical elements. Results. Rotational mixing, thermohaline mixing, and internal gravity waves have been invoked to explain this feature. Rotation impacts the evolution of the surface abundance of Lithium, but it seems an unlikely explanation for a ubiquitous mixing event occurring between the tip of the red giant star and the red clump star. Thermohaline mixing can explain the observed behaviour of C-12/C-13 and N-14/C-12 and lithium in low-mass stars that are more luminous than the red-giant branch bump, and its efficiency is decreasing with the increasing initial stellar mass. Conclusions. The internal gravity wave- (IGW-) induced mixing is located between the hydrogen-burning shell, and the outer convective envelope, and it is mainly triggered by turbulent convective motion. This physical process is beneficial to transfer the large amount of Be-7 to the cool envelope where it is converted to Li-7. Therefore, IGW-induced mixing could play a main role in explaining the red clump star with lithium enrichment. Rotation can indirectly increase the above effect by making the core-helium-burning lifetime longer. Thermohaline mixing is much smaller than the one of IGWs during the evolution of red clump stars.

Recent studies of wide binary stars based on Gaia DR3 suggest that the relative orbital velocities of objects with separations s > 3'000 astronomical units are statistically larger than the standard Newtonian predictions. Obviously there is no Dark Matter halo arround binary stars that could be invoked to explain these high velocities. However, we explore the properties of two-body systems in the framework of scale invariant vacuum theory, focusing on the case of objects with extreme separations. In this regime, the additional acceleration term present in the modified Newton equation with scale invariance becomes important, and may even dominate the dynamical evolution at very low gravities. Comparisons with Gaia DR3 observations of wide binaries are performed and suggest that binaries with separations s > 3'000 astronomical units have experienced such an evolution for a few Gyr, accounting well for the observed velocity excesses.

Context. The physics of stellar rotation plays a crucial role in the evolution of stars, in their final fates, and for the properties of compact remnants. Aims. Diverse approaches have been adopted to incorporate the effects of rotation in stellar evolution models. This study seeks to explore the consequences that these various prescriptions for rotation have for the essential outputs of massive star models. Methods. We computed a grid of 15 and 60 M ⊙ stellar evolution models with the Geneva Stellar Evolution Code that accounted for both hydrodynamical and magnetic instabilities induced by rotation. Results. In the 15 and 60 M ⊙ models, the choice of the vertical and horizontal diffusion coefficients for the nonmagnetic models strongly impacts the evolution of the chemical structure, but has a weak impact on the angular momentum transport and the rotational velocity of the core. In the 15 M ⊙ models, the choice of the diffusion coefficient impacts the convective core size during the core H-burning phase, regardless of whether the model begins core He-burning as a blue or red supergiant and regardless of the core mass at the end of He-burning. In the 60 M ⊙ models, the evolution is dominated by mass loss and is less strongly affected by the choice of the diffusion coefficient. In the magnetic models, magnetic instability dominates the angular momentum transport, and these models are found to be less strongly mixed than their rotating nonmagnetic counterparts. Conclusions. Stellar models with the same initial mass, chemical composition, and rotation may exhibit diverse characteristics depending on the physics applied. By conducting thorough comparisons with observational features, we can ascertain which method(s) produce the most accurate results in different cases.

Galaxy velocities in clusters, rotation curves of galaxies, and "vertical" oscillations in the Milky Way currently show too high velocities with respect to the masses thought to be involved. While these velocity excesses are currently interpreted as the consequence of dark matter, it can also be naturally explained as a consequence of scale invariant theory, which rests on a very simple first principle: the addition of a new fundamental symmetry. In the present work, the case of scale invariance, present in General Relativity and Maxwell equations for the empty space without charge and current, is considered. Cosmological models predict a rapid decrease of these effects with increasing mean density up to the critical density, where they totally disappear. Starting from the scale invariant geodesic equation by Dirac (1973), for which a demonstration by an action principle is presented, a modified Newton equation is derived. The solutions of this equation are applied to clusters of galaxies, galactic rotation at different redshifts and "vertical" motions in the Milky Way. In this new framework, the convergence of theoretical predictions and observations, in different gravitational systems, epochs, mass range and spatial scales, opens interesting perspectives that deserve to be explored further.

We first briefly review the adventure of scale invariance in physics, from Galileo Galilei, Weyl, Einstein, and Feynman to the revival by Dirac (1973) and Canuto et al. (1977). In the way that the geometry of space–time can be described by the coefficients gμν, a gauging condition given by a scale factor λ(xμ) is needed to express the scaling. In general relativity (GR), λ=1. The “Large Number Hypothesis” was taken by Dirac and by Canuto et al. to fix λ. The condition that the macroscopic empty space is scale-invariant was further preferred (Maeder 2017a), the resulting gauge is also supported by an action principle. Cosmological equations and a modified Newton equation were then derived. In short, except in extremely low density regions, the scale-invariant effects are largely dominated by Newtonian effects. However, their cumulative effects may still play a significant role in cosmic evolution. The theory contains no “adjustment parameter”. In this work, we gather concrete observational evidence that scale-invariant effects are present and measurable in astronomical objects spanning a vast range of masses (0.5 M⊙< M <1014M⊙) and an equally impressive range of spatial scales (0.01 pc < r < 1 Gpc). Scale invariance accounts for the observed excess in velocity in galaxy clusters with respect to the visible mass, the relatively flat/small slope of rotation curves in local galaxies, the observed steep rotation curves of high-redshift galaxies, and the excess of velocity in wide binary stars with separations above 3000 kau found in Gaia DR3. Last but not least, we investigate the effect of scale invariance on gravitational lensing. We show that scale invariance does not affect the geodesics of light rays as they pass in the vicinity of a massive galaxy. However, scale-invariant effects do change the inferred mass-to-light ratio of lens galaxies as compared to GR. As a result, the discrepancies seen in GR between the total lensing mass of galaxies and their stellar mass from photometry may be accounted for. This holds true both for lenses at high redshift like JWST-ER1 and at low redshift like in the SLACS sample. Of note is that none of the above observational tests require dark matter or any adjustable parameter to tweak the theory at any given mass or spatial scale.

We present a summary of the main results within the Scale Invariant Vacuum (SIV) paradigm as related to the Weyl Integrable Geometry. After a brief review of the mathematical framework, we will highlight the main results related to inflation within the SIV [9], the growth of the density fluctuations [8], and the application of the SIV to scale-invariant dynamics of Galaxies, MOND, Dark Matter, and the Dwarf Spheroidals [7]. The connection of the weak-field SIV results to the un-proper time parametrization within the reparametrization paradigm is also discussed [14].

We study the path of light rays passing near a massive object, in the context of the scale invariant equation of the geodesics first obtained by Dirac (1973). Using the exterior Schwarzschild solution for the metric, we derive the complete equations of the geodesics in the scale invariant context. We find that scale invariance introduces two additional terms to the Einstein term producing the deflection angle and that can potentially act over cosmological distances. Numerical integration of the scale-invariant geodesics, for the specific case of the z_L=1.94 lens galaxy in the extreme system JWST-ER1 (van Dokkum et al. 2023) shows that the two additional terms introduce only negligible effects, typically 1E-06 of the Einstein term. We conclude that the lensing deflection angle derived in Einstein's General Relativity is essentially independent of the scale invariant effects and that the photon's geodesics remain unchanged. We also explore the possible origin of the differences in the mass estimates from lensing and photometry in JWST-ER1 and in the SLACS galaxies, differences which appear larger at higher redshifts.

On the basis of a general action principle, we revisit the scale invariant field equation using the co-tensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor λ, which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler's third law is a remarkable scale-invariant property.

Context. The physical mechanism leading to the formation of the blue loop in the Hertzsprung–Russell (HR) diagram is not satisfactorily explained by the evolutionary track of single stars. Rapid rotation and low metallicity drastically modify the internal structures and surface compositions of stars. Therefore, they provide a very significant pattern to investigate the evolutionary properties of the blue loop. Aims. In this paper, we mainly explore how rapid rotation and low metallicity have an important impact on the occurrence and extension of the blue loop. Methods. To this end, we implemented the rotating stellar evolution model, including the angular momentum transportation and chemical element mixing. We incorporated several initial rotational velocities and two characteristic metallicities in various models to explore the blue loop extension. Results. The blue loop can occur when the hydrogen burning shell merges with the hydrogen–helium abundance discontinuity. We find that the blue loop extension strongly depends on the amplitude and gradient of the hydrogen–helium discontinuity. The hydrogen–helium discontinuity is created by the intermediate convective region or the convective dredge-up. A steeper hydrogen gradient in association with a greater amplitude of the hydrogen abundance discontinuity may favour a hotter star. Conclusions. Both the low metallicity and rapid rotation tend to restrain the development of the outer convective envelope and thus disfavour the occurrence and extension of the blue loop. There are three main reasons for this occurrence. Firstly, the helium core and its core potential can be enlarged by rotational mixing or low metallicity. Secondly, rapid rotation reduces the convective dredge-up depth in the star with Z = 0.014 and the mass extension of the intermediate convective region in the star with Z = 0.0008. Both of these phenomena lead to a reduction of the amplitude of the hydrogen abundance gradient. Thirdly, strong rotational mixing in the model (i.e. vini = 350 km s−1) with Z = 0.0008 reduces the energy generation rate from the hydrogen burning shell. Without bending towards higher effective temperature in the HR diagram, the additional helium brought near the H-burning shell associated with the larger He core can cause the star to expand towards becoming a red giant star directly after the core hydrogen burning. Rapid rotation and low metallicity tend to produce surface enrichment of the ratio of nitrogen to carbon and reduce the 12C left in the core; this has an important influence on the stellar compactness of the supernovae progenitor.

The scale invariant theory is preserving the fundamental physical properties of General Relativity, while enlarging the group of invariances subtending gravitation theory (Dirac1973; Canuto et al.1977). The Scale Invariant Vacuum (SIV) theory assumes, as gauging condition, that:"The macroscopic empty space is scale invariant, homogeneous and isotropic". Some basic properties in Weyl's Integrable Geometry and cotensor calculus are examined in relation with scalar-tensor theories. Possible scale invariant effects are strongly reduced by matter density, both at the cosmological and local levels. The weak feld limit of SIV tends to MOND, when the scale factor is taken as constant, an approximation valid (<1%) over the last 400 Myr. A better understanding of the a0-parameter is obtained: it corresponds to the equilibrium point of the Newtonian and SIV dynamical acceleration. Parameter a0 is not a universal constant, it depends on the density and age of the Universe. As MOND is doing, SIV theory avoids the call to dark matter, moreover the cosmological models predict accelerated expansion.

The observed late-type WC Wolf-Rayet stars (WC7-9) with low luminosity below $\rm \log L/L_{\odot}<5.4$ in the HR diagram cannot be reproduced satisfactorily by the evolutionary track of single stars. The mass transfer due to Roche lobe overflow drastically modifies the internal structure and surface compositions of two components. Therefore, binaries provide a very promising evolutionary channel to produce these WC stars.

Scale invariance is expected in empty Universe models, while the presence of matter tends to suppress it. As shown recently, scale invariance is certainly absent in cosmological models with densities equal to or above the critical value ρc = 3(H0)2/(8πG). For models with densities below ρc, the possibility of limited effects remains open. If present, scale invariance would be a global cosmological property. Some traces could be observable locally. For the Earth-Moon two-body system, the predicted additional lunar recession would be increased by 0.92 cm/yr, while the tidal interaction would also be slightly increased. The Earth-Moon distance is the most systematically measured distance in the Solar System, thanks to the Lunar Laser Ranging (LLR) experiment active since 1970. The observed lunar recession from LLR amounts to 3.83 (±0.009) cm/yr; implying a tidal change of the length-of-the-day (LOD) by 2.395 ms/cy. However, the observed change of the LOD since the Babylonian Antiquity is only 1.78 ms/cy, a result supported by paleontological data, and implying a lunar recession of 2.85 cm/yr. The significant difference of (3.83-2.85) cm/yr = 0.98 cm/yr, already pointed out by several authors over the last two decades, corresponds well to the predictions of the scale-invariant theory, which is also supported by several other astrophysical tests.