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 existence and nature of Dark Matter (DM) remains an enigma. The existence of a non-luminous matter is based on the continuation of the matter paradigm. As such it is hypothesized to be the reason for the flat rotational curves in galaxies. Modified Newtonian Dynamics (MOND) is an alternative view point on the flat rotational curves phenomenon. This work builds upon the recently introduced Reparametrization Invariant Scaling Symmetry (RISS), that introduces scale factor λ ( t ) relevant for cosmic reparametrization, which can be used to remove the Einstein cosmological constant Λ E from the relevant extended equations of General Relativity (GR) and therefore defines the Scale Invariant Vacuum (SIV) gauge for λ ( t ). Furthermore, by insisting on a reparametrization symmetry for the equation of motion, the paper demonstrates how to address the missing mass problem at galactic and extragalactic scales, where unproper (non co-moving) time parametrization gives rise to forces needed for appropriate reparametrization symmetry considerations. This approach leads to the derivation of the MOND fundamental relation g ~ a 0 g N within the RISS paradigm, where g is the gravitational acceleration, a 0 is the MOND fundamental acceleration, and g N is the Newtonian gravitational constant. Remarkably, the derived value for a 0 ≈ 10 −10 m/s 2 is found to be consistent with its observed order of magnitude for MOND applications.

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.

In a recent paper: “On the time dependency of a0” the authors claim that they have tested “one of the predictions of the Scale Invariant Vacuum (SIV) theory on MOND” by studying the dependence of the Modified Newtonian Dynamics (MOND) accelera- tion at two data sets, low-z (3.2 × 10−4 ≤ z ≤ 3.2 × 10−2) and high-z (0.5 ≤ z ≤ 2.5). They claim “both samples show a dependency of a0 from z”. Here, the work men- tioned above is revisited. The explicit analytic expression for the z-dependence of the a0 within the SIV theory is given. Furthermore, the first estimates of the Ωm within SIV theory give Ωm = 0.28 ± 0.04 using the low-z data only, while a value of Ωm = 0.055 is obtained using both data sets. This much lower Ωm leaves no room for non-baryonic matter! Unlike in the mentioned paper above, the slope in the z- dependence of A0 = log10(a0) is estimated to be consistent with zero Z-slope for the two data sets. Finally, the statistics of the data are consistent with the SIV predic- tions; in particular, the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm; however, the uncertainty in the data is too big for the clear demonstration of a z-dependence yet.

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].

Abstract The nature of Dark Energy (DE) remains an enigma. This work explores the non-zero Einstein Cosmological Constant (Λ E ) as a manifestation of DE – viewing it through the lens of Reparametrization Invariant Scaling Symmetry (RISS), where Λ E is interpreted as a “kinetic energy” term within the energy density. The scale factor λ(t) relevant for cosmic reparametrization removes Λ E from the extended Einstein GR, its defining equations, and their relations to Λ E are derived. The value for Λ E = 1.8 × 10−52 m−2 is consistent with the current observations for Λ E . The current approach avoids the puzzles associated with interpreting Λ E as vacuum zero-point energy (often called dark energy). Further applications are related to the Dark Matter problem by insisting on a reparametrization symmetry for the equations of motion, where un-proper (non-co-moving) time parametrization gives rise to fictitious forces.

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.

We discuss some of the challenges that future nuclear modeling may face in order to improve the description of the nuclear structure. One challenge is related to the need for A-body nuclear interactions justified by various contemporary nuclear physics studies. Another challenge is related to the discrepancy in the NNN contact interaction parameters for 3He and 3H that suggests the need for accurate proton and neutron masses in the future precision calculations. MSC2010 Classification: 17B81 Applications to physics, 17B80 Applications to integrable systems, 81R12 Relations with integrable systems, 81V70 Many-body theory, 81V35 Nuclear physics, 81U15 Exactly and quasi-solvable systems, 82B23 Exactly solvable models; Bethe ansatz.

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.

Recently it was found from Cassini data that the mean recession speed of Titan from Saturn is v = 11.3 ± 2.0 cm/yr which corresponds to a tidal quality factor of Saturn Q ≈ 100 while the standard estimate yields Q ≥ 6 · 104 . It was assumed that such a large speed v is due to a resonance locking mechanism of five inner mid-sized moons of Saturn. In this paper, we show that an essential part of v may come from a local Hubble expansion, where the Hubble-Lemaˆıtre constant H0 recalculated to the Saturn-Titan distance D is 8.15 cm/(yrD). Our hypothesis is based on many other observations showing a slight expansion of the Solar system and also of our Galaxy at a rate comparable with H0. We demonstrate that the large disproportion in estimating the Q factor can be just caused by the local expansion effect. [Accepted for publication in "Gravitation and Cosmology". The paper is to appear in Vol. 28, Issue 2 (2022) of the journal Gravitation and Cosmology.]

In this paper, we argue in favor of first-order homogeneous Lagrangians in the velocities. The relevant form of such Lagrangians is discussed and justified physically and geometrically. Such Lagrangian systems possess Reparametrization Invariance (RI) and explain the observed common Arrow of Time as related to the non-negative mass for physical particles. The extended Hamiltonian formulation, which is generally covariant and applicable to reparametrization-invariant systems, is emphasized. The connection between the explicit form of the extended Hamiltonian H and the meaning of the process parameter λ is illustrated. The corresponding extended Hamiltonian H defines the classical phase space-time of the system via the Hamiltonian constraint H=0 and guarantees that the Classical Hamiltonian H corresponds to p0—the energy of the particle when the coordinate time parametrization is chosen. The Schrödinger’s equation and the principle of superposition of quantum states emerge naturally. A connection is demonstrated between the positivity of the energy E=cp0>0 and the normalizability of the wave function by using the extended Hamiltonian that is relevant for the proper-time parametrization.

ABSTRACTMaxwell equations and the equations of general relativity are scale invariant in empty space. The presence of charge or currents in electromagnetism or the presence of matter in cosmology are preventing scale invariance. The question arises on how much matter within the horizon is necessary to kill scale invariance. The scale-invariant field equation, first written by Dirac in 1973 and then revisited by Canuto et al. in 1977, provides the starting point to address this question. The resulting cosmological models show that, as soon as matter is present, the effects of scale invariance rapidly decline from ϱ = 0 to ϱc, and are forbidden for densities above ϱc. The absence of scale invariance in this case is consistent with considerations about causal connection. Below ϱc, scale invariance appears as an open possibility, which also depends on the occurrence of inflation in the scale-invariant context. In the present approach, we identify the scalar field of the empty space in the scale-invariant vacuum context to the scalar field φ in the energy density $\varrho = \frac{1}{2} \dot{\varphi }^2 + V(\varphi)$ of the vacuum at inflation. This leads to some constraints on the potential. This identification also solves the so-called ‘cosmological constant problem’. In the framework of scale invariance, an inflation with a large number of e-foldings is also predicted. We conclude that scale invariance for models with densities below ϱc is an open possibility; the final answer may come from high redshift observations, where differences from the ΛCDM models appear.

Based on the principle of reparametrization invariance, the general structure of physically relevant classical matter systems is illuminated within the Lagrangian framework. In a straightforward way, the matter Lagrangian contains background interaction fields, such as a 1-form field analogous to the electromagnetic vector potential and symmetric tensor for gravity. The geometric justification of the interaction field Lagrangians for the electromagnetic and gravitational interactions are emphasized. The generalization to E-dimensional extended objects (p-branes) embedded in a bulk space M is also discussed within the light of some familiar examples. The concept of fictitious accelerations due to un-proper time parametrization is introduced, and its implications are discussed.The framework naturally suggests new classical interaction fields beyond electromagnetism and gravity. The simplest model with such fields is analyzed and its relevance to dark matter and dark energy phenomena on large/cosmological scales is inferred. Unusual pathological behavior in the Newtonian limit is suggested to be a precursor of quantum effects and of inflation-like processes at microscopic scales.

The Scale Invariant Vacuum (SIV) theory rests on the basic hypothesis that the macroscopic empty space is scale invariant. This hypothesis is applied in the context of the Integrable Weyl Geometry, where it leads to considerable simplifications in the scale covariant cosmological equations. After an initial explosion and a phase of braking, the cosmological models show a continuous acceleration of the expansion. Several observational tests of the SIV cosmology are performed: on the relation between H 0 and the age of the Universe, on the m − z diagram for SNIa data and its extension to z = 7 with quasars and GRBs, and on the H ( z ) vs. z relation. All comparisons show a very good agreement between SIV predictions and observations. Predictions for the future observations of the redshift drifts are also given. In the weak field approximation, the equation of motion contains, in addition to the classical Newtonian term, an acceleration term (usually very small) depending on the velocity. The two-body problem is studied, showing a slow expansion of the classical conics. The new equation has been applied to clusters of galaxies, to rotating galaxies (some proximities with Modifies Newtonian Dynamics, MOND, are noticed), to the velocity dispersion vs. the age of the stars in the Milky Way, and to the growth of the density fluctuations in the Universe. We point out the similarity of the mechanical effects of the SIV hypothesis in cosmology and in the Newtonian approximation. In both cases, it results in an additional acceleration in the direction of motions. In cosmology, these effects are currently interpreted in terms of the dark energy hypothesis, while in the Newtonian approximation they are accounted for in terms of the dark matter (DM) hypothesis. These hypotheses appear no longer necessary in the SIV context.

The Scale-Invariant Vacuum (SIV) theory is based on Weyl's Integrable Geometry, endowed with a gauge scalar field. The main difference between MOND and the SIV theory is that the first considers a global dilatation invariance of space and time, where the scale factor $\lambda$ is a constant, while the second opens the likely possibility that $\lambda$ is a function of time. The key equations of the SIV framework are used here to study the relationship between the Newtonian gravitational acceleration due to baryonic matter $g_{\mathrm{bar}}$ and the observed kinematical acceleration $g_{\mathrm{obs}}$. The relationship is applied to galactic systems of the same age where the Radial Acceleration Relation (RAR), between the $g_{\mathrm{obs}}$ and $g_{\mathrm{bar}}$ accelerations, can be compared with observational data. The SIV theory shows an excellent agreement with observations and with MOND for baryonic gravities $g_{\mathrm{bar}}>10^{-11.5}$ m s$^{-2}$. Below this value, SIV still fully agrees with the observations, as well as with the horizontal asymptote of the RAR for dwarf spheroidals, while this is not the case for MOND. These results support the view that there is no need for dark matter and that the RAR and related dynamical properties of galaxies can be interpreted by a modification of gravitation.

Highly excited states in 156Gd, populated via the neutron pickup reaction 157Gd(3He,4He)156Gd, are investigated, and their spin–parity distribution P(Jπ,E) is examined. The cross section for one-neutron transfers to states above the neutron separation energy in 156Gd is calculated as coherent sum, using standard reaction codes that employ spherical basis states. Spectroscopic factors and form factors for the relevant states are obtained by expanding the deformed neutron wave functions in a spherical Sturmian basis. For the energy regime relevant to surrogate applications involving neutron absorption, 155Gd+n →156Gd⋆, the calculations show that the reaction 3He+157Gd →4He+156Gd⋆ induces a well-behaved formation probability P(Jπ,E) of approximately Gaussian shape. It is observed that the centroid and shape of the Gaussian distributions of the positive and negative parity states of the compound system can be significantly different from each other!

The oblique basis method is reviewed from engineering point of view related to vibration and control theory. Examples are used to demonstrate and relate the oblique basis in nuclear physics to the equivalent mathematical problems in vibration theory. The mathematical techniques, such as principal coordinates and root locus, used by vibration and control theory engineers are shown to be relevant to the Richardson - Gaudin pairing interaction and the A-body exactly solvable pairing models in nuclear physics.

A new perspective on the Cosmological Constant Problem (CCP) is proposed and discussed within the multiverse approach of Quantum Cosmology. It is assumed that each member of the ensemble of universes has a characteristic scale a that can be used as integration variable in the partition function. An averaged characteristic scale of the ensemble is estimated by using only members that satisfy the Einstein field equations. The averaged characteristic scale is compatible with the Planck length when considering an ensemble of solutions to the Einstein field equations with an effective cosmological constant. The multiverse ensemble is split in Planck-seed universes with vacuum energy density of order one; thus, Λ˜≈8π in Planck units and a-derivable universes. For a-derivable universe with a characteristic scale of the order of the observed Universe a≈8×1060, the cosmological constant Λ=Λ˜/a2 is in the range 10−121–10−122, which is close in magnitude to the observed value 10−123. We point out that the smallness of Λ can be viewed to be natural if its value is associated with the entropy of the Universe. This approach to the CCP reconciles the Planck-scale huge vacuum energy–density predicted by QFT considerations, as valid for Planck-seed universes, with the observed small value of the cosmological constant as relevant to an a-derivable universe as observed.

The growth of the density fluctuations is considered to be an important cosmological test. In the standard model, for a matter dominated universe, the growth of the density perturbations evolves with redshift z like (11+z)s with s=1. This is not fast enough to form galaxies and to account for the observed present-day inhomogeneities. This problem is usually resolved by assuming that at the recombination epoch the baryons settle down in the potential well of the dark matter previously assembled during the radiation era of the universe. This view is challenged in the present paper by using the recently proposed model of a scale-invariant framework for cosmology, the Scale-Invariant Vacuum Theory (Maeder 2017a,2017b,2017c), that enlarges the invariance group subtending the theory of the gravitation. From the continuity equation, the Euler and Poisson equations are written in the scale-invariant framework, the equation governing the growth of density fluctuations δ is obtained. Starting from δ=10−5 at a redshift around 1000, numerical solutions for various density background are obtained. The growth of density fluctuations is much faster than in the standard EdS model. The s values are in the range from 2.7 to 3.9 for Ωm between 0.30 and 0.02. This enables the density fluctuations to enter the nonlinear regime with δ>1 long before the present time, typically at redshifts of about 10, without requiring the presence of dark matter.

We discuss modeling of nuclear structure beyond the 2-body interaction paradigm.Our first example is related to the need of three nucleon contact interaction terms suggested by chiral perturbation theory.The relationship of the two low-energy effective coupling parameters for the relevant three nucleon contact interaction terms cD and cE that reproduce the binding energy of 3H and 3He has been emphasized and the physically relevant parameter region has been ilustrated using the binding energy of 4He.Further justification of A-body interaction terms is outlined based on the Okubo-Lee-Suzuki effective interaction method used in solving the nuclear many-body problem within a finite model space.The third example we use is an exactly solvable A-body extended paring interaction applied to heavy nuclei with a long isotopic chain;in particular using 132Sn as closed core system illustrates a remarkable relationship between the extended pairing strength G(A) and the size of the valence space dim(A) for the members of the Sn-isotope chain:G(A) =c dim(A)-β with α =259.436 andβ =0.998 5 which is actually a one parameter expression since β is practically 1.These three cases present evidence for the need of better understanding of the NNN-,NNNN-,and A-body interactions in nuclei either derived from ChPT or from a phenomenological considerations.