Universe, Volume 10, Issue 5 (May 2024) – 26 articles

With the development of large-scale sky surveys, an increasing number of stellar photometric images have been obtained. However, most stars lack spectroscopic data, which hinders stellar classification. Vision Transformer (ViT) has shown superior performance in image classification tasks compared to most convolutional neural networks (CNNs). In this study, we propose an stellar classification network based on the Transformer architecture, named stellar-ViT, aiming to efficiently and accurately classify the spectral class for stars when provided with photometric images. By utilizing RGB images synthesized from photometric data provided by the Sloan Digital Sky Survey (SDSS), our model can distinguish the seven main stellar categories: O, B, A, F, G, K, and M. Particularly, our stellar-ViT-gri model, which reaches an accuracy of 0.839, outperforms traditional CNNs and the current state-of-the-art stellar classification network SCNet when processing RGB images synthesized from the gri bands. Furthermore, with the introduction of urz band data, the overall accuracy of the stellar-ViT model reaches 0.863, further demonstrating the importance of additional band information in improving classification performance. Our approach showcases the effectiveness and feasibility of using photometric images and Transformers for stellar classification through simple data augmentation strategies and robustness analysis of training dataset sizes. The stellar-ViT model maintains good performance even in small sample scenarios, and the inclusion of urz band data reduces the likelihood of misclassifying samples as lower-temperature subtypes. Full article

Previously, a class of regular and asymptotically flat gravitating scalar solitons (scalarons) has been constructed in the Einstein–Klein–Gordon (EKG) theory by adopting a phantom field with Higgs-like potential where the kinetic term has the wrong sign and the scalaron possesses the negative Arnowitt–Deser–Misner (ADM) mass as a consequence. In this paper, we demonstrate that the use of the phantom field can be avoided by inverting the Higgs-like potential in the EKG system when the kinetic term has a proper sign, such that the corresponding gravitating scalaron can possess the positive ADM mass. We systematically study the basic properties of the gravitating scalaron, such as the ADM mass, the energy conditions, the geodesics of test particles, etc. Moreover, we find that it can be smoothly connected to the counterpart hairy black hole solutions from our recent work in the small horizon limit. Full article

We consider F-term hybrid inflation (FHI) and SUSY breaking in the context of a $B-L$ extension of the MSSM that largely respects a global $U\left(1\right)$R symmetry. The hidden sector Kaehler manifold enjoys an enhanced $SU(1,1)/U\left(1\right)$ symmetry, with the scalar curvature determined by the achievement of a SUSY-breaking de Sitter vacuum without undesirable tuning. FHI turns out to be consistent with the data, provided that the magnitude of the emergent soft tadpole term is confined to the range $(1.2$–$100)$ TeV, and it is accompanied by the production of $B-L$ cosmic strings. If these are metastable, they are consistent with the present observations from PTA experiments on the stochastic background of gravitational waves with dimensionless tension $G{\mu }_{\mathrm{cs}}\simeq (1-9.2)·{10}^{-8}$. The $\mu$ parameter of the MSSM arises by appropriately adapting the Giudice–Masiero mechanism and facilitates the out-of-equilibrium decay of the R saxion at a reheat temperature lower than about 71 GeV. Due to the prolonged matter-dominated era, the gravitational wave signal is suppressed at high frequencies. The SUSY mass scale turns out to lie in the PeV region. Full article

With new artificial intelligence (AI) technologies and application scenarios constantly emerging, AI technology has become widely used in astronomy and has promoted notable progress in related fields. A large number of papers have reviewed the application of AI technology in astronomy. However, relevant articles seldom mention telescope intelligence separately, and it is difficult to understand the current development status of and research hotspots in telescope intelligence from these papers. This paper combines the development history of AI technology and difficulties with critical telescope technologies, comprehensively introduces the development of and research hotspots in telescope intelligence, conducts a statistical analysis of various research directions in telescope intelligence, and defines the merits of these research directions. A variety of research directions are evaluated, and research trends in each type of telescope intelligence are indicated. Finally, according to the advantages of AI technology and trends in telescope development, potential future research hotspots in the field of telescope intelligence are given. Full article

The existence of ‘peculiar’ velocities due to the formation of cosmic structure marks a point of discord between the real universe and the usually assumed Friedmann–Lemaítre–Robertson–Walker metric, which accomodates only the smooth Hubble expansion on large scales. In the standard $\Lambda$CDM model framework, Type Ia supernovae data are routinely “corrected” for the peculiar velocities of both the observer and the supernova host galaxies relative to the cosmic rest frame, in order to infer evidence for acceleration of the expansion rate from their Hubble diagram. However, observations indicate a strong, coherent local bulk flow that continues outward without decaying out to a redshift $z\gtrsim 0.1$, contrary to the $\Lambda$CDM expectation. By querying the halo catalogue of the Dark Sky Hubble-volume N-body simulation, we find that an observer placed in an unusual environment like our local universe should see correlations between supernovae in the JLA catalogue that are 2–8 times stronger than seen by a typical or Copernican observer. This accounts for our finding that peculiar velocity corrections have a large impact on the value of the cosmological constant inferred from supernova data. We also demonstrate that local universe-like observers will infer a downward biased value of the clustering parameter $S_8$ from comparing the density and velocity fields. More realistic modelling of the peculiar local universe is thus essential for correctly interpreting cosmological data. Full article

By extending the dipole Pomeron (DP) model, successful in describing elastic nucleon–nucleon scattering, to proton single diffractive dissociation (SD), we predict a dip-bump structure in the squared four-momentum transfer (t) distribution of proton’s SD. Structures in the t distribution of single diffractive dissociation are predicted around $t=-4$${\mathrm{GeV}}^2$ at LHC energies in the range of 3 ${\mathrm{GeV}}^2$$\lesssim \left|t\right|\lesssim$ 7 ${\mathrm{GeV}}^2$. Apart from the dependence on s (total energy squared) and t (squared momentum transfer), we predict also a dependence on missing masses. We include the minimum set of Regge trajectories, namely the Pomeron and the Odderon, indispensable at the LHC. Further generalization, e.g., by the inclusion of non-leading Regge trajectories, is straightforward. The present model contains two types of Regge trajectories: those connected with $t$-channel exchanges (the Pomeron, the Odderon, and non-leading (secondary) reggeons) appearing at small and moderate $-t$, where they are real and nearly linear, as well as direct-channel trajectories $\alpha \left(M^2\right)$ related to missing masses. In this paper, we concentrate on structures in t neglecting (for the time being) resonances in $M^2$. Full article

Fast radio bursts (FRBs) have been found in great numbers, but the physical mechanism of these sources is still a mystery. The redshift evolutions of the FRB energy distribution function and the volumetric rate shed light on the origin of FRBs. However, such estimations rely on the dispersion measurement (DM)–redshift (z) relationship. A few FRBs that have been detected recently show large excess DMs beyond the expectation from the cosmological and Milky Way contributions, which indicates large spread of DMs from their host galaxies. In this work, we adopt two lognormal-distributed ${\mathrm{DM}}_{\mathrm{host}}$ models and estimate the energy function using the non-repeating FRBs selected from the Canadian Hydrogen Intensity Mapping Experiment (CHIME)/FRB Catalog 1. By comparing the lognormal-distributed ${\mathrm{DM}}_{\mathrm{host}}$ models to a constant ${\mathrm{DM}}_{\mathrm{host}}$ model, the FRB energy function results are consistent within the measurement uncertainty. We also estimate the volumetric rate of the non-repeating FRBs in three different redshift bins. The volumetric rate shows that the trend is consistent with the stellar-mass density redshift evolution. Since the lognormal-distributed ${\mathrm{DM}}_{\mathrm{host}}$ model increases the measurement errors, the inference of FRBs tracking the stellar-mass density is nonetheless undermined. Full article

To the first post-Newtonian order, the orbital angular momentum of the fast-revolving inner binary of the triple system PSR J0337+1715, made of a millisecond pulsar and a white dwarf, induces an annular gravitomagnetic field which displaces the line of apsides of the slower orbit of the other, distant white dwarf by $-1.2$ milliarcseconds per year. The current accuracy in determining the periastron of the outer orbit is $63.9$ milliarcseconds after 1.38 years of data collection. By hypothesizing a constant rate of measurement of the pulsar’s times of arrivals over the next 10 years, assumed equal to the present one, it can be argued that the periastron will be finally known to a ≃$0.15$ milliarcseconds level, while its cumulative gravitomagnetic retrograde shift will be as large as $-12$ milliarcseconds. The competing post-Newtonian gravitolectric periastron advance due to the inner binary’s masses, nominally amounting to $74.3$ milliarcseconds per year, can be presently modelled to an accuracy level as good as ≃$0.04$ milliarcseconds per year. The mismodeling in the much larger Newtonian periastron rate due to the quadrupolar term of the multipolar expansion of the gravitational potential of a massive ring representing the inner binary, whose nominal size for PSR J0337+1715 is $0.17$ degrees per year, might be reduced down to the ≃$0.5$ milliarcseconds per year level over the next 10 years. Thus, a first measurement of such a novel form of gravitomagnetism, although undoubtedly challenging, might be, perhaps, feasible in a not too distant future. Full article

At the moment, there are two neutron star X-ray binaries with massive red supergiants as donors. Recently, De et al. (2023) proposed that the system SWIFT J0850.8-4219 contains a neutron star at the propeller stage. We study this possibility by applying various models of propeller spin-down. We demonstrate that the duration of the propeller stage is very sensitive to the regime of rotational losses. Only in the case of a relatively slow propeller model proposed by Davies and Pringle in 1981, the duration of the propeller is long enough to provide a significant probability to observe the system at this stage. Future determination of the system parameters (orbital and spin periods, magnetic field of the compact object, etc.) will allow putting strong constraints on the propeller behavior. Full article

Over the last three years, the rates of all the main nuclear reactions involving the destruction and production of ²⁶Al in stars (²⁶Al(n, p)²⁶Mg, ²⁶Al(n, α)²³Na, ²⁶Al(p, γ)²⁷Si and ²⁵Mg(p, γ)²⁶Al) have been re-evaluated thanks to new high-precision experimental measurements of their crosssections at energies of astrophysical interest, considerably reducing the uncertainties in the nuclear physics affecting their nucleosynthesis. We computed the nucleosynthetic yields ejected by the explosion of a high-mass star (20 M_⊙, Z = 0.0134) using the FRANEC stellar code, considering two explosion energies, 1.2 × 10⁵¹ erg and 3 × 10⁵¹ erg. We quantify the change in the ejected amount of ²⁶Al and other key species that is predicted when the new rate selection is adopted instead of the reaction rates from the STARLIB nuclear library. Additionally, the ratio of our ejected yields of ²⁶Al to those of 14 other short-lived radionuclides (³⁶Cl, ⁴¹Ca, ⁵³Mn, ⁶⁰Fe, ⁹²Nb, ⁹⁷Tc, ⁹⁸Tc, ¹⁰⁷Pd, ¹²⁶Sn, ¹²⁹I, ³⁶Cs, ¹⁴⁶Sm, ¹⁸²Hf, ²⁰⁵Pb) are compared to early solar system isotopic ratios, inferred from meteorite measurements. The total ejected ²⁶Al yields vary by a factor of ~3 when adopting the new rates or the STARLIB rates. Additionally, the new nuclear reaction rates also impact the predicted abundances of short-lived radionuclides in the early solar system relative to ²⁶Al. However, it is not possible to reproduce all the short-lived radionuclide isotopic ratios with our massive star model alone, unless a second stellar source could be invoked, which must have been active in polluting the pristine solar nebula at a similar time of a core-collapse supernova. Full article

In the 1960s, the remnants of supernova explosions (SNRs) were indicated as a possible source of galactic cosmic rays through the Diffusive Shock Acceleration (DSA) mechanism. Since then, the observation of gamma-ray emission from relativistic ions in these objects has been one of the main goals of high-energy astrophysics. A few dozen SNRs have been detected at GeV and TeV photon energies in the last two decades. However, these observations have shown a complex phenomenology that is not easy to reduce to the standard paradigm based on DSA acceleration. Although the understanding of these objects has greatly increased, and their nature as efficient electron and proton accelerators has been observed, it remains to be clarified whether these objects are the main contributors to galactic cosmic rays. Here, we review the observations of $\gamma$-ray emission from SNRs and the perspectives for the future. Full article

The theory of heavy ion double charge exchange (DCE) reactions proceeding by effective rank-2 isotensor interactions is presented. Virtual pion–nucleon charge exchange interactions are investigated as the source for induced isotensor interactions, giving rise to the Majorana DCE (MDCE) reaction mechanism. MDCE is of a generic character, proceeding through pairs of complementary (${\pi }^{\pm },{\pi }^{\mp }$) reactions in the projectile and target nucleus. The dynamics of the elementary processes is discussed, where the excitation of pion–nucleon resonances are of central importance. Investigations of initial and final state ion–ion interactions show that these effects are acting as vertex renormalizations. In closure approximation, well justified by the finite pion mass, the second-order transition matrix elements reduce to pion potentials and effective two-body isotensor DCE interactions, giving rise also to two-body correlations in either of the participating nuclei. Connections to neutrinoless Majorana double beta decay (MDBD) are elucidated at various levels of the dynamics, from the underlying fundamental electro-weak and QCD scales to the physical scales of nuclear MDBD and MDCE physics. It is pointed out that heavy ion MDCE reactions may also proceed by competing electro-weak charge exchange processes, leading to lepton MDCE by electrons, positrons, and neutrinos. Full article

We study the loop oscillations after a solar flare on 19 January 2023, in the active region N11E40 3196, which is well observed by the SDO/AIA. After tracing the loop position and fitting, we find that the loop oscillations have a period between 3 and 9 min at various locations, such as from the leg to the top or from the inner to the outer loop. Their oscillating amplitudes decrease with time. Two loops display the position oscillation simultaneously with their brightness oscillation. After the analysis of the differential emission measure (DEM), we find that two of their loop position oscillations resulted from the plasma density fluctuation. Meanwhile, it is interesting that the brightness of these two position oscillations displays a typical period of about 4 min, similar to that of the position oscillation. This is possible due to both the plasma density and temperature fluctuation there. Our findings provide the physical clues for studying and understanding the mechanism of the loop position and brightness oscillations. Full article

The $X_{CO}$ factor is defined as $X_{CO}=N\left(H_2\right)/W_{12CO}$. It is useful for estimating cloud mass. However, there is only limited research on how the $X_{CO}$ factor varies within a single cloud. Employing ¹²$CO(J=\text{1-0})$ and ¹³$CO(J=\text{1-0})$ spectral data, we computed an $X_{CO}$ factor of 3.6 $\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ for luminous gas of the N55 region. Our analysis revealed a V-shaped correlation between the $X_{CO}$ factor and $H_2$ column densities, while the relationship with excitation temperature exhibited obscurity. This suggests that the CO-to-$H_2$ conversion is not consistent on small scale (∼1 pc). Additionally, we found that star formation activity has little influence on the variability in the $X_{CO}$ factor. Full article

We review the role of the spectral condition as a characteristic of Minkowski, de Sitter, and anti-de Sitter quantum field theories. We also discuss the role of plane waves that are compatible with the relevant analyticity domains linked to the spectral condition(s) and discuss harmonic analysis in terms of them. Full article

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width. Full article

This paper reviews the dynamics of a single isotropic and homogeneous scalar field $\phi \left(t\right)$ in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution parameter for cosmic dynamics. General conclusions about the qualitative behaviour of the solutions can be drawn, and examples of how to obtain explicit solutions for some cosmological models of interest are given. For arbitrary potentials, analytical results can be obtained from the slow-roll approximation by using a series expansion for the Hubble parameter $H\left[\phi \right]$, from which a quantitative estimate for the number of e-folds of expansion is obtained. This approach is illustrated with the examples of quadratic potentials and hilltop models, with special consideration of Higgs-type potentials. The GUT-scale is shown to come out of such a model quite naturally. Finally, it is discussed how to find scalar potentials giving rise to a predetermined scalar-field behaviour and the associated evolution of the scale factor. Full article

In this paper, we use the key parameters data set of the Neutral Gas and Ion Mass Spectrometer from the Mars Atmosphere and Volatile Evolution (MAVEN) mission. The particle density profiles of electrons, ${\mathrm{CO}}_2^{+}$/${\mathrm{N}}_2^{+}$, ${\mathrm{CO}}^{+}$, ${\mathrm{O}}_2^{+}$, ${\mathrm{O}}^{+}$, ${\mathrm{NO}}^{+}$, ${\mathrm{O}}_2$ and $\mathrm{O}$ from 90 to 500 km have been deduced by adopting the Chapman modeling methodology. The correlation of the peak density/altitude with the solar zenith angle, the changes in the profile of the Martian ionosphere during solar flares, and the effects of Martian dust storms are analyzed. The results exhibit a positive/negative correlation between the peak density/altitude of the M2 layer and the solar zenith angle. Within the MAVEN observational record available, only three C-Class flares occurred on 26 August 2016, 29 November 2020, and 26 August 2021. The analysis reveals during these solar flare events, the electron density of the M2 layer above 200 km increases obviously. The peak density of M1 increases by 33.4%, 13.2% and 7.4%, while the peak height decreases by 0.1%, 10.2% and 4.4%, respectively. The Martian dust storm causes the peak height of the M2 layer to increase by 19.5 km, and the peak density to decrease by 4.2 ×${10}^9{\mathrm{m}}^{-3}$. Our study shows that the Martian ionosphere is similar to the Earth’s, which is of great significance for understanding the planetary ionosphere. Full article

After two phases of on-site construction and testing (2010–2013 and 2013–2017), the Shanghai Tianma Radio Telescope (TMRT) can work well, with efficiencies better than 50% from 1.3 to 50.0 GHz, mainly benefiting from its low-noise cryogenic receivers and active surface system. Pulsars were chosen as important targets of research at the TMRT because of their important scientific and applied values. To meet the demands of pulsar-related observations, TMRT is equipped with some necessary backends, including a digital backend system (DIBAS) supporting normal pulsar observation modes, a real-time fast-radio-burst-monitoring backend, and baseband backends for very-long-baseline interferometry (VLBI) observations. Utilizing its high sensitivity and simultaneous dual-frequency observation capacity, a sequence of pulsar research endeavors has been undertaken, such as long-term pulsar timing, magnetar monitoring, multi-frequency (or high-frequency) observations, interstellar scintillation, pulsar VLBI, etc. In this paper, we give a short introduction about pulsar observation systems at the TMRT and briefly review the results obtained by these pulsar research projects. Full article

Solar wind is known to have different properties depending on its origin at the Sun. In addition to the differences in plasma and magnetic field parameters, these streams differ due to the properties of turbulent fluctuations involved in the flow. The present study addresses the changes in the turbulence properties in the magnetosheath—the transition region in front of the magnetosphere. This study is based on statistics from the simultaneous measurements of magnetic field fluctuations in the solar wind and in the magnetosheath. Both the dayside and flank magnetosheath regions are focused on to detect the evolution of the turbulent fluctuations during their flow around the magnetosphere. Turbulent cascade is shown to save its properties for fast solar wind streams. Conditions favorable for the preservation of the turbulence properties at the bow shock may correspond to the increased geoefficiency of large-scale solar wind structures. Full article

The accelerated expansion of the Universe is impressively well described by a cosmological constant. However, the observed value of the cosmological constant is much smaller than expected based on quantum field theories. Recent efforts to achieve consistency in these theories have proposed a relationship between Dark Energy and the most compact objects, such as black holes (BHs). However, experimental tests are very challenging to devise and perform. In this article, we present a testable model with no cosmological constant in which the accelerated expansion can be driven by black holes. The model couples the expansion of the Universe (the Friedmann equation) with the mass function of cosmological halos (using the Press–Schechter formalism). Through the observed link between halo masses and BH masses, one thus gets a coupling between the expansion rate of the Universe and the BHs. We compare the predictions of this simple BH model with SN1a data and find poor agreement with observations. Our method is sufficiently general to allow us to also test a fundamentally different model, also without a cosmological constant, where the accelerated expansion is driven by a new force proportional to the internal velocity dispersion of galaxies. Surprisingly enough, this model cannot be excluded using the SN1a data. Full article

In this work, we will explore the effects of non-commutativity in fractional classical and quantum schemes using the flat Friedmann–Robertson–Walker (FRW) cosmological model coupled to a scalar field in the K-essence formalism. In previous work, we have obtained the commutative solutions in both regimes in the fractional framework. Here, we introduce non-commutative variables, considering that all minisuperspace variables $q_{nc}^i$ do not commute, so the symplectic structure was modified. In the quantum regime, the probability density presents a new structure in the scalar field corresponding to the value of the non-commutative parameter, in the sense that this probability density undergoes a shift back to the direction of the scale factor, causing classical evolution to arise earlier than in the commutative world. Full article

Solar wind modeling is classified into two main types: empirical models and physics-based models, each designed to forecast solar wind properties in various regions of the heliosphere. Empirical models, which are cost-effective, have demonstrated significant accuracy in predicting solar wind at the L1 Lagrange point. On the other hand, physics-based models rely on magnetohydrodynamics (MHD) principles and demand more computational resources. In this research paper, we build upon our recent novel approach that merges empirical and physics-based models. Our recent proposal involves the creation of a new physics-informed neural network that leverages time series data from solar wind predictors to enhance solar wind prediction. This innovative method aims to combine the strengths of both modeling approaches to achieve more accurate and efficient solar wind predictions. In this work, we show the variability of the proposed physics-informed loss across multiple deep learning models. We also study the effect of training the models on different solar cycles on the model’s performance. This work represents the first effort to predict solar wind by integrating deep learning approaches with physics constraints and analyzing the results across three solar cycles. Our findings demonstrate the superiority of our physics-constrained model over other unconstrained deep learning predictive models. Full article

The question of the global topology of the Universe (cosmic topology) is still open. In the $\Lambda$CDM concordance model, it is assumed that the space of the Universe possesses the trivial topology of ${\mathbb{R}}^3$, and thus that the Universe has an infinite volume. As an alternative, in this paper, we study one of the simplest non-trivial topologies given by a cubic 3-torus describing a universe with a finite volume. To probe cosmic topology, we analyze certain structure properties in the cosmic microwave background (CMB) using Betti functionals and the Euler characteristic evaluated on excursions sets, which possess a simple geometrical interpretation. Since the CMB temperature fluctuations $\delta T$ are observed on the sphere ${\mathbb{S}}^2$ surrounding the observer, there are only three Betti functionals ${\beta }_k\left(\nu \right)$, $k=0,1,2$. Here, $\nu =\delta T/{\sigma }_0$ denotes the temperature threshold normalized by the standard deviation ${\sigma }_0$ of $\delta T$. The analytic approximations of the Gaussian expectations for the Betti functionals and an exact formula for the Euler characteristic are given. It is shown that the amplitudes of ${\beta }_0\left(\nu \right)$ and ${\beta }_1\left(\nu \right)$ decrease with an increasing volume $V=L^3$ of the cubic 3-torus universe. Since the computation of the ${\beta }_k$’s from observational sky maps is hindered due to the presence of masks, we suggest a method that yields lower and upper bounds for them and apply it to four Planck 2018 sky maps. It is found that the ${\beta }_k$’s of the Planck maps lie between those of the torus universes with side-lengths $L=2.0$ and $L=3.0$ in units of the Hubble length and above the infinite $\Lambda$CDM case. These results give a further hint that the Universe has a non-trivial topology. Full article

We discuss matters related to the point that topological quantization in the strong interaction is a consequence of an infinite spacetime volume. Because of the ensuing order of limits, i.e., infinite volume prior to summing over topological sectors, $CP$ is conserved. Here, we show that this reasoning is consistent with the construction of the path integral from steepest-descent contours. We reply to some objections that aim to support the case for $CP$ violation in strong interactions that are based on the role of the $CP$-odd theta-parameter in three-form effective theories, the correct sampling of all configurations in the dilute instanton gas approximation and the volume dependence of the partition function. We also show that the chiral effective field theory derived from taking the volume to infinity first is in no contradiction with analyses based on partially conserved axial currents. Full article