Editorial to the Special Issue “Universe: Feature Papers 2023—Cosmology”

According to recent observational data, including Supernovae Ia (SNe Ia) [1,2], the cosmic microwave background (CMB) radiation [3,4,5,6,7,8], the large-scale structure (LSS) of the universe [9,10,11], baryon acoustic oscillations (BAOs) [12,13], and the weak lensing effect [14,15,16,17,18], not only in the early universe of the inflationary stage [19,20,21,22] but also in the late-time universe, the cosmic expansion is accelerating. Moreover, the recent Planck observations [7,8] suggest that if the universe is spatially flat, the energy density of the current universe consists of dark energy (about 70%) with negative pressure and dark matter (about 25%) with only a gravitational interaction and baryon acoustic oscillations (about 5%). The identities of dark energy have not yet been understood so clearly, although a large number of studies have been carried out very actively.

Through the future observations by the Euclid satellite [23] of the European Space Agency (ESA) [24,25,26,27,28,29,30], the Roman Space Telescope [31], the Simons Observatory [32,33], and the James Webb Space Telescope [34,35], a further understanding of modern cosmology will be developed in more precise and detail.

In addition, by the direct detection of gravitational waves [36,37], the era of gravitational wave cosmology has kicked off. It is expected that not only gravitational waves originated from astrophysical objects but also that primordial gravitational waves originated from the early universe, including the inflationary stage and cosmological first-order phase transitions, such as the electroweak phase transition (EWPT) and the QCD phase transition (QCDPT) [38,39,40,41,42,43,44].

Two main methods to explain the late-time accelerated expansion of the spatially flat universe have been proposed in the literature. One approach is to explore the unknown energy component with negative pressure, the so-called dark energy, in the framework of general relativity. The other is to investigate the extension of a theory of gravitation from general relativity at cosmological scales. Such a way can be regarded as geometrical dark energy. Regarding the latter approach, many extended theories of gravitation have been proposed (for reviews in terms of extended theories of gravitation, including their applications to astrophysics and the properties of gravitational theories, as well as the dark energy problem (see, for instance, refs. [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85]).

Furthermore, there are various important subjects in modern cosmology, such as physics in the early universe [86,87,88,89,90,91,92,93,94,95,96,97], cosmological perturbation theory [98,99,100,101], the Swampland picture [102,103,104], cosmological constant [105], neutrino cosmology [106,107,108,109,110], axion cosmology [111], primordial black holes [112,113,114,115,116], dark matter [117,118,119,120,121,122,123,124,125,126,127,128], primordial magnetic fields [129,130,131,132,133], Big Bang nucleosynthesis (BBN) [134,135], the Hubble tension [136,137,138,139,140,141,142], 21 cm cosmology [143,144], gravitational lensing effects [145,146,147], and other related topics, including applications to astrophysical aspects [148,149,150,151,152,153,154,155].

In the present Special Issue, titled “Universe: Feature Papers 2023—Cosmology” of Universe, twenty-one original research manuscripts in terms of modern cosmology are collected. It was organized as follows. In the first part, the subjects of physics in the early universe [156,157] including inflationary cosmology [158] are explored. In the second part, the generic topics of observational cosmology [159,160,161] are studied. In the third part, the issues of dark energy [162,163,164,165,166,167,168] and extended theories of gravity from general relativity [169,170,171,172] are investigated. In the fourth part, the themes of gravitational waves [173] and physics of black holes [174,175] are given. As a kind of summary of this Special Issue, in the final part, a review on the cosmological recent problem of the Hubble tension [176] is presented. In the following, twenty research articles and one review are overviewed briefly.

In ref. [156], through a metric procedure, the so-called big bounce dynamics of an isotropic spacetime are studied by introducing a scalar field with its self-interaction. For the case that there is no potential, the cosmic expansion and collapse can occur along with the mode of positive and negative frequencies for the Wheeler–DeWitt equation. On the other hand, if there exists a potential, a transition from a state with a positive frequency to that with negative one can happen. It is demonstrated that the probability of a transition from the collapsing phase to the expanding one can be analyzed.

In ref. [157], a scenario to realize the de Sitter expansion of the early universe is proposed, which originated from stochastic gravitational waves based on the arguments of the Lorentzian and Euclidean spacetimes and the Wick rotation.

In ref. [158], through the Bayesian analysis, the cosmological evolution of the extrinsic energy density is numerically examined for the inflationary universe in the context of the five-dimensional spacetime. The effective potential leading to inflation is derived, originating from the extrinsic geometry. The possibility for a unified scenario of inflation with the late-time cosmic acceleration is also argued.

In ref. [159], the propagation of electromagnetic waves is studied for homogeneous and isotropic spacetimes. For plane and spherical electromagnetic waves, exact solutions are derived. Moreover, the redshift, the change in amplitude, and the dispersion of the electromagnetic waves are examined. Furthermore, the relation between the equation of electromagnetic waves and the Proca equation is argued. The importance of the relation is discussed for physics in the early universe.

In ref. [160], the methods of the number count are investigated in the context of observational cosmology, such as the number of galaxies. In particular, with the method of the so-called geodesic-light-cone gauge, the procedures to count the number are investigated.

In ref. [161], the basis of genetic algorithms and the way of parameter estimations for cosmological models are explained as a complementary technique with a conventional method of the Markov chain Monte Carlo. It is shown that the space of the parameter can be explored effectively by using models of dark energy.

In ref. [162], a possible origin of the cosmological constant is proposed from a boundary condition. The relation between the issue of the cosmological constant and the Hubble tension is investigated. It is argued that the cosmological constant represents a covariant integration constant arising from a spatial boundary condition, which is only applicable to a three-dimensional bounded subspace representing our universe.

In ref. [163], a fractional differential equation in a quantum K-essence scalar field is studied in homogeneous and isotropic spacetime. For the Wheeler–DeWitt equation of the scalar field in quantum cosmology, a fractional differential equation is found, and the solutions of the equation are examined.

In ref. [164], cosmology is studied in the Einstein–Newcomb–de Sitter space, where antipodal points are topologically identified. In particular, with the type Ia supernovae data of Pantheon+ and the sample data of gamma-ray bursts, the fitting analysis to this cosmological model is performed. As a result, it is reported that the minimum value of the ${\chi }^2$ analysis for this model could be smaller than that for the $\mathsf{\Lambda }$ CDM model.

In ref. [165], with the autonomous system analysis, the phase-space diagram of the cosmological evolutions for a scalar field model is investigated. For two types of potentials, the stationary points of the phase-space are examined. It is demonstrated that for the case in which the potential is close to asymptotically constant, the de Sitter expansion of the universe can be realized, while for that with the exponential form of the potential, the de Sitter cosmic expansion can occur only in the infinity limit.

In ref. [166], a cosmological transition from the anti-de Sitter phase to the de Sitter one is analyzed based on a motivation to explain the so-called Hubble tension. It is assumed that there exist two kinds of fluids playing a role of the dark energy component. One of these two fluids can lead to the anti-de Sitter phase in the early universe. The self-interaction of these fluids may influence the energy density of the effective dark energy to realize late-time cosmic acceleration.

In ref. [167], a relation between a scalar field theory of K-essence and the Vaidya metric is discussed. Especially, a non-canonical action classified into a type of Dirac–Born–Infeld one is considered. The geodesics within a comoving plane are examined for certain forms of the mass function. In addition, for a kind of Vaidya spacetime, a solution of wormholes, the tunneling effect, the existence of (quasi-)circular orbits of the event horizon, and the central singularity of the spacetime are analyzed.

In ref. [168], the (im)possibility of the discrimination of the de Sitter phase of the cosmic expansion from the following stationary phase is examined. In relation with this issue, a signal (a kind of non-standard photons) related to dark energy is argued.

In ref. [169], as a possibility to avoid the initial singularity in the early universe, a context of Loop Quantum Cosmology is considered. In particular, the Friedmann equations for $f\left(R\right)$ gravity in metric formalism are analyzed. As a result, the effective actions with covariance, in which a bounce can be realized, are derived through a reduction method.

In ref. [170], cosmological fluctuations for cosmic microwave background radiations are studied in delta gravity, which is proposed as an alternative theory of gravity to general relativity to explain the type Ia SN data. With a semi-analytic method, the temperature–temperature power spectrum of the scalar mode of the cosmological fluctuations is analyzed by using the gauge transformations of the spacetime metric as well as perfect fluid, so that observational constraints on delta gravity can be obtained.

In ref. [171], a homogeneous and anisotropic Bianchi type I universe is explored for the local limit in nonlocal gravity. In such a theory of gravity, there exists a function representing susceptibility, which vanishes in the limit of general relativity. The influence of the function of susceptibility is investigated on the accelerated anisotropic expansion of the universe.

In ref. [172], late-time cosmology is studied in dilaton gravity, which consists of the scalar curvature and the trace of the energy–momentum tensor of the dilatonic field. In particular, the issue of the Hubble tension is discussed by using cosmological observational data.

In ref. [173], through the Bayesian statistical analysis with the pulsar timing of the NANOGrav 12.5-year data, the curvature perturbations from inflation and the mass of primordial black holes are investigated. The possibility that primordial black holes play a role of a fraction of dark matter is also studied. Moreover, detectability with future experiments of gravitational waves is discussed.

In ref. [174], the existence of white holes is argued. In general relativity, a white hole is recognized as a reversed process of the solution of a collapsing black hole. This phenomenon leads to the crossing of the event horizon of the black hole, at which causality is broken. It is indicated that the Big Bang of our universe is a kind of black hole, and not a white hole. It is explained why the crossing of the event horizon can be realized from the inside to the outside if an event decelerates when the cosmic expansion accelerates by discussing that an apparent solution of a white hole has an exterior of a regular solution of a Schwarzschild black hole.

In ref. [175], for the Schwarzschild–Finsler–Randers spacetime, the equation of the geodesic deviation and the Raychaudhuri equation are explored. In this spacetime, there exist multiple curvature tensors, enabling the Raychaudhuri equation to have extra degrees of freedom. The astrophysical results of the limit of the weak field and the limit of Newtonian mechanics are also presented.

In ref. [176], the issue of the “Hubble tension” is reviewed in detail, including the history of the Hubble constant. The significance of the discovery of our universe’s expansion and the pioneering measurement of the cosmic expansion rate are explained. For the measurement, high-level technology and small uncertainties in terms of parameters are important. A review of various method to measure the Hubble constant is also necessary at the present time.

In conclusion, we believe that the twenty-one articles introduced above in the present Special Issue are useful for relevant future studies for various aspects of modern physics and cosmology.