Standard cosmology has long been plagued by a number of persistent problems. The origin of the apparent acceleration of the cosmic expansion remains enigmatic. The cosmological constant has been reintroduced as a free parameter with a value in energy density units that “happens” to be of the same order as the present matter energy density. There is an internal inconsistency with regards to the Hubble constant, the so-called H0 tension. The derived value of H0 depends on the type of data that is used. With supernovae as standard candles, one gets a H0 that is 4–5 σ larger than the value that one gets from CMB (Cosmic Microwave Background) data for the early universe. Here we show that these problems are related and can be solved if the cosmological constant represents a covariant integration constant that arises from a spatial boundary condition, instead of being a new type of hypothetical physical field, “dark energy”, as assumed by standard cosmology. The boundary condition only applies to the bounded 3D subspace that represents the observable universe, the hypersurface of the past light cone.

Angle-frequency coupling in scattering of polarized light on atoms is represented by the angle-dependent (AD) partial frequency redistribution (PRD) matrices. There are several lines in the linearly polarized solar spectrum, for which PRD combined with quantum interference between hyperfine structure states play a significant role. Here we present the solution of the polarized line transfer equation including the AD-PRD matrix for scattering on a two-level atom with hyperfine structure splitting and an unpolarized lower level. We account for the effects of arbitrary magnetic fields (including the incomplete Paschen-Back effect regime) and elastic collisions. For exploratory purposes we consider a self-emitting isothermal planar atmosphere and use atomic parameters that represent an isolated NaiD(2)line. For this case we show that the AD-PRD effects are significant for field strengths below about 30 G, but that the computationally much less demanding approximation of angle-averaged PRD may be used for stronger fields.

The evolution of the wave function in quantum mechanics is deterministic like that of classical waves. Only when we bring in observers the fundamentally different quantum reality emerges. Similarly the introduction of observers changes the nature of spacetime by causing a split between past and future, concepts that are not well defined in the observer-free world. The induced temporal boundary leads to a resonance condition for the oscillatory vacuum solutions of the metric in Euclidean time. It corresponds to an exponential de Sitter evolution in real time, which can be represented by a cosmological constant Lambda = 2 pi(2)/r(u)(2), where r(u) is the radius of the particle horizon at the epoch when the observer exists. For the present epoch we get a value of Lambda that agrees with the observed value within 2 sigma of the observational errors. This explanation resolves the cosmic coincidence problem. Our epoch in cosmic history does not herald the onset of an inflationary phase driven by some dark energy. We show that the observed accelerated expansion that is deduced from the redshifts is an 'edge effect' due to the observer-induced boundary and not representative of the intrinsic evolution. The new theory satisfies the BBN (Big Bang nucleosynthesis) and CMB (cosmic microwave background) observational constraints equally well as the concordance model of standard cosmology. There is no link between the dark energy and dark matter problems. Previous conclusions that dark matter is mainly non-baryonic are not affected.

Quantum interference between the hyperfine structure states is known to depolarize the cores of some of the lines in the linearly polarized spectrum of the Sun (the Second Solar Spectrum). The presence of external magnetic fields in the line forming regions modify these signatures through the Hanle, Zeeman, and incomplete/ complete Paschen -Back effects (PBE), depending on the strength of the magnetic field. In an earlier paper, Sowmya et al. (2014) derived the relevant collisionless partial frequency redistribution (PFR) matrix for scattering on a two-level atom with hyperfine structure splitting (HFS) and in the presence of arbitrary strength magnetic fields (including the PBE regime). In the present paper we solve the problem of polarized line transfer in a magnetized atmosphere, including this PFR matrix. For this purpose, we apply a scattering expansion method which is based on orders of scattering approach. We present the results on the combined effects of PBE and PFR on the polarized line profiles using the atomic parameters relevant to the Nai D-2 line.

The observed value of the cosmological constant corresponds to a time scale that is very close to the current conformal age of the universe. Here we show that this is not a coincidence but is caused by a periodic boundary condition, which only manifests itself when the metric is represented in Euclidian spacetime. The circular property of the metric in Euclidian spacetime becomes an exponential evolution (de Sitter or $\varLambda $ term) in ordinary spacetime. The value of $\varLambda $ then gets uniquely linked to the period in Euclidian conformal time, which corresponds to the conformal age of the universe. Without the use of any free model parameters we predict the value of the dimensionless parameter $\varOmega _{\varLambda }$ to be 67.2%, which is within $2\sigma $ of the value derived from CMB observations.

The Stokes formalism of polarization physics has astounding structural parallels with the formalism used for relativity theory in Minkowski spacetime. The structure and symmetry properties of the Mueller matrices are the same as those for the matrix representations of the electromagnetic tensor and the Lorentz transformation operator. The absorption terms $\eta_k$ in the Mueller matrix correspond to the electric field components $E_k$ in the electromagnetic tensor and the Lorentz boost terms $\gamma_k$ in the Lorentz transformation matrix, while the anomalous dispersion terms $\rho_k$ correspond to the magnetic field components $B_k$ and the spatial rotation angles $\phi_k$. In a Minkowski-type space spanned by the Stokes $I,Q,U,V$ parameters, the Stokes vector for 100 % polarized light is a null vector living on the surface of null cones, like the energy-momentum vector of massless particles in ordinary Minkowski space. Stokes vectors for partially polarized light live inside the null cones like the momentum vectors for massive particles. In this description the depolarization of Stokes vectors appears as a mass'' term, which has its origin in a symmetry breaking caused by the incoherent superposition of uncorrelated fields or wave packets, without the need to refer to a ubiquitous Higgs field as is done in particle physics. The rotational symmetry of Stokes vectors and Mueller matrices is that of spin-2 objects, in contrast to the spin-1 nature of the electromagnetic field. The reason for this difference is that the Stokes objects have substructure: they are formed from bilinear tensor products between spin-1 objects, the Jones vectors and Jones matrices. The governing physics takes place at the substructure level.

Quantum interference effects, together with partial frequency redistribution (PFR) in line scattering, produce subtle signatures in the so-called Second Solar Spectrum (the linearly polarized spectrum of the Sun). These signatures are modified in the presence of arbitrary strength magnetic fields via the Hanle, Zeeman, and Paschen–Back effects. In the present paper we solve the problem of polarized line formation in a magnetized atmosphere taking into account scattering in a two-level atom with hyperfine structure splitting together with PFR. To this end we incorporate the collisionless PFR matrix derived in Sowmya et al. in the polarized transfer equation. We apply the scattering expansion method to solve this transfer equation. We study the combined effects of PFR and the Paschen–Back effect on polarized line profiles formed in an isothermal one-dimensional planar atmosphere. For this purpose, we consider the cases of D 2 lines of Li i and Na i .

The cosmological constant, which was introduced by Einstein a century ago to allow for a static universe, experienced a revival two decades ago under the label dark energy as a parameter to model the observed accelerated expansion of the universe. Its physical nature has however remained enigmatic. Here we use the Einstein equations without cosmological constant to show that the origin of the accelerated expansion is not in the equations but has to do with the boundary conditions related to the causal horizon, which exists because the age of the universe is finite. Via transformations to conformal coordinates and Euclidian spacetime we find a resonance condition that uniquely determines the dimensionless parameter Ω_Λ that governs the observed cosmic acceleration: Ω_Λ =2 3 (π t_H/t_c)^2, where t_H is the Hubble time and t_c is the conformal age of the universe. This explanation leads to a somewhat modified cosmology, in which the expansion rate of the early universe is 2.1 times faster than in the standard model. We show that Big Bang nucleosynthesis calculations with the faster expansion rate requires the mean baryon density to be raised to the level of the total matter density to agree with the observed deuterium abundance. This appears to eliminate the need to invoke the existence of some yet to be discovered exotic particles to explain dark matter, since all of it may be baryonic while still remaining consistent with the observed abundances of the light elements.

Context. The scattering polarization signal observed in the photospheric Sr I 4607 Å line is expected to vary at granular spatial scales. This variation can be due to changes in the magnetic field intensity and orientation (Hanle effect), but also to spatial and temporal variations in the plasma properties. Measuring the spatial variation of such polarization signal would allow us to study the properties of the magnetic fields at subgranular scales, but observations are challenging since both high spatial resolution and high spectropolarimetric sensitivity are required. Aims. We aim to provide observational evidence of the polarization peak spatial variations, and to analyze the correlation they might have with granulation. Methods. Observations conjugating high spatial resolution and high spectropolarimetric precision were performed with the Zurich IMaging POLarimeter, ZIMPOL, at the GREGOR solar telescope, taking advantage of the adaptive optics system and the newly installed image derotator. Results. Spatial variations of the scattering polarization in the Sr I 4607 Å line are clearly observed. The spatial scale of these variations is comparable with the granular size. Small correlations between the polarization signal amplitude and the continuum intensity indicate that the polarization is higher at the center of granules than in the intergranular lanes.

As my own work on the Sun's magnetic field started exactly 50 years ago at Crimea in the USSR, I have been a participant in the field during nearly half the time span since Hale's discovery in 1908 of magnetic fields in sunspots. The present historical account is accompanied by photos from my personal slide collection, which show a number of the leading personalities who advanced the field in different areas: measurement techniques, from photographic to photoelectric and imaging methods in spectro-polarimetry; theoretical foundations of MHD and the origin of cosmic magnetic fields (birth of dynamo theory); the quest for increased angular resolution from national projects to international consortia (for instruments both on ground and in space); introduction of the Hanle effect in astrophysics and the Second Solar Spectrum as its playground; small-scale nature of the field, the fundamental resolution limit, and transcending it by resolution-independent diagnostics.

The atlas of the Third Solar Spectrum (SS3) represents the ratio between the intensity spectrum at different distances from the solar limb and the intensity spectrum at disk center (µ = 1.0), both in units of the intensity of the local continuum level. The observed positions of the measurements cover 9 different µ values along the solar axis ranging from 0.1 to 0.9 in step of 0.1, where µ=cosθ is the cosine of the heliocentric angle θ. The current version of the atlas covers the range 4384- 6610 Å. In the PDF file, the first plot represents the spectrum at the center of the solar disk, recorded at IRSOL. The next 9 plots represent the 3rd solar spectrum for µ=0.1, µ=0.2, µ=0.3, …, µ=0.9 Columns of the CSV file: WL: Wavelength, 4384-6610 Å IC: I/Ic at disc center RMU01: limb / disk-center ratio at µ=0.1 RMU02: limb / disk-center ratio at µ=0.2 … … RMU09: limb / disk-center ratio at µ=0.9

Magnetic fields in the solar atmosphere leave their fingerprints in the polarized spectrum of the Sun via the Hanle and Zeeman effects. While the Hanle and Zeeman effects dominate, respectively, in the weak and strong field regimes, both these effects jointly operate in the intermediate field strength regime. Therefore, it is necessary to solve the polarized line transfer equation, including the combined influence of Hanle and Zeeman effects. Furthermore, it is required to take into account the effects of partial frequency redistribution (PRD) in scattering when dealing with strong chromospheric lines with broad damping wings. In this paper, we present a numerical method to solve the problem of polarized PRD line formation in magnetic fields of arbitrary strength and orientation. This numerical method is based on the concept of operator perturbation. For our studies, we consider a two-level atom model without hyperfine structure and lower-level polarization. We compare the PRD idealization of angle-averaged Hanle–Zeeman redistribution matrices with the full treatment of angle-dependent PRD, to indicate when the idealized treatment is inadequate and what kind of polarization effects are specific to angle-dependent PRD. Because the angle-dependent treatment is presently computationally prohibitive when applied to realistic model atmospheres, we present the computed emergent Stokes profiles for a range of magnetic fields, with the assumption of an isothermal one-dimensional medium.

The D$_1$ enigma is an anomaly, which was first observed on the Sun as a symmetric polarization peak centered in the core of the sodium D$_1$ line that is expected to be intrinsically unpolarizable. To resolve this problem the underlying physics was later explored in the laboratory for D$_1$ scattering at potassium vapor. The experiment showed that the scattering phase matrix element $P_{21}$ is positive while $P_{22}$ is negative, although standard quantum scattering theory predicts that both should be zero. This experimental contradiction is currently the main manifestation of the D$_1$ enigma. Subsequent theoretical studies showed that such polarization effects may arise if scattering theory is extended to allow for interference effects due to level splittings of the ground state, in contrast to standard scattering theory, which only allows for interferences from level splittings of the intermediate state. Previous attempts to implement this idea had to rely on heuristic arguments to allow modeling of the experimental data. In the present paper we develop a formulation of the theory that can be self-consistently applied to quantum systems with any combination of electronic and nuclear spins. No statistical equilibrium or optical pumping is needed. The atom is assumed to be unpolarized at the beginning of each scattering event. The theory is capable of explaining both the phase matrix behavior of the laboratory data and the existence of a symmetric polarization peak in the core of the solar D$_1$ line. We also use it to predict the polarization structures that we expect to see in a next-generation laboratory experiment with the rubidium isotopes $^{87}$Rb and $^{85}$Rb.

ABSTRACT In the well-established theories of polarized line formation with partial frequency redistribution (PRD) for a two-level and two-term atom, it is generally assumed that the lower level of the scattering transition is unpolarized. However, the existence of unexplained spectral features in some lines of the Second Solar Spectrum points toward a need to relax this assumption. There exists a density matrix theory that accounts for the polarization of all the atomic levels, but it is based on the flat-spectrum approximation (corresponding to complete frequency redistribution). In the present paper we propose a numerical algorithm to solve the problem of polarized line formation in magnetized media, which includes both the effects of PRD and the lower level polarization (LLP) for a two-level atom. First we derive a collisionless redistribution matrix that includes the combined effects of the PRD and the LLP. We then solve the relevant transfer equation using a two-stage approach. For illustration purposes, we consider two case studies in the non-magnetic regime, namely, the J a = 1, J b = 0 and J a = J b = 1, where J a and J b represent the total angular momentum quantum numbers of the lower and upper states, respectively. Our studies show that the effects of LLP are significant only in the line core. This leads us to propose a simplified numerical approach to solve the concerned radiative transfer problem.

The Swiss Federal Observatory, which had been founded in 1863 by Rudolf Wolf, was dissolved in connection with the retirement of Max Waldmeier in 1979. The determination of the Zurich sunpot number, which had been a cornerstone activity of the observatory, was then discontinued by ETH Zurich. A smooth transition of the responsibility for the sunspot number from Zurich to Brussels was achieved in 1980, however, through which it was possible to avoid a discontinuity in this important time series. Here we describe the circumstances that led to the termination in Zurich, how Brussels was chosen for the succession, and how the transfer was accomplished.

The Sun's spectrum varies with center-to-limb distance. This variation is governed by the underlying temperature-density structure of the solar atmosphere. To explore the spectrally resolved center-to-limb variation (CLV) we make use of two spectral atlases recorded with the Fourier transform spectrometer (FTS) at the McMath-Pierce facility at Kitt Peak. One spectral atlas obtained 10 arcsec inside the solar limb was recorded in 1978-79 as part of the first survey of the Second Solar Spectrum, while the other atlas is the well used reference NSO/Kitt Peak FTS atlas for the disk center. Both atlases represent fully resolved spectra without any spectral stray light. We then construct an atlas of the limb/disk-center ratio between the two spectra over the wavelength range 4084-9950 \AA. This ratio spectrum, which expresses the CLV amplitude relative to the continuum, is as richly structured as the intensity spectrum itself, but the line profiles differ greatly in both shape and amplitude. It is as if we are dealing with a new, unfamiliar spectrum of the Sun, distinctly different from both the intensity spectrum (which we here refer to with the acronym SS1) and the linear polarization of the Second Solar Spectrum (for which we use acronym SS2). In analogy we refer to the new ratio spectrum as SS3. While there is hardly any resemblance between SS3 and SS2, we are able to identify a non-linear mapping that can translate SS1 to SS3 in the case of weak to medium-strong spectral lines that are mainly formed in LTE (being directly coupled to the local temperature-density structure). This non-linear mapping is successfully modeled in terms of two free parameters that are found to vary approximately linearly over the entire wavelength range covered. These parameters and the various SS3 line profiles provide a novel, rich set of observational constraints on model atmospheres.

The symmetric peak observed in linear polarization in the core of the solar sodium D-1 line at 5896 angstrom has remained enigmatic since its discovery nearly two decades ago. One reason is that the theory of polarized scattering has not been experimentally tested for multi-level atomic systems in the relevant parameter domains, although the theory is continually being used for the interpretation of astrophysical observations. A laboratory experiment that was set up a decade ago to find out whether the D-1 enigma is a problem of solar physics or quantum physics revealed that the D-1 system has a rich polarization structure in situations where standard scattering theory predicts zero polarization, even when optical pumping of the m state populations of the hyperfine-split ground state is accounted for. Here we show that the laboratory results can be modeled in great quantitative detail if the theory is extended to include the coherences in both the initial and final states of the scattering process. Radiative couplings between the allowed dipole transitions generate coherences in the initial state. Corresponding coherences in the final state are then demanded by a phase closure selection rule. The experimental results for the well understood D-2 line are used to constrain the two free parameters of the experiment, collision rate and optical depth, to suppress the need for free parameters when fitting the D-1 results.

Interference between magnetic substates of the hyperfine structure states belonging to different fine structure states of the same term influences the polarization for some of the diagnostically important lines of the Sun's spectrum, like the sodium and lithium doublets. The polarization signatures of this combined interference contain information on the properties of the solar magnetic fields. Motivated by this, in the present paper, we study the problem of polarized scattering on a two-term atom with hyperfine structure by accounting for the partial redistribution in the photon frequencies arising due to the Doppler motions of the atoms. We consider the scattering atoms to be under the influence of a magnetic field of arbitrary strength and develop a formalism based on the Kramers-Heisenberg approach to calculate the scattering cross section for this process. We explore the rich polarization effects that arise from various level-crossings in the Paschen-Back regime in a single scattering case using the lithium atomic system as a concrete example that is relevant to the Sun.

The extensive literature on the physics of polarized scattering may give the impression that we have a solid theoretical foundation for the interpretation of spectro-polarimetric data. This theoretical framework has however not been sufficiently tested by experiments under controlled conditions. While the solar atmosphere may be viewed as a physics laboratory, the observed solar polarization depends on too many environmental factors that are beyond our control. The existence of a symmetric polarization peak at the center of the solar Na D 1 line has remained an enigma for two decades, in spite of persistent efforts to explain it with available quantum theory. A decade ago a laboratory experiment was set up to determine whether this was a problem for solar physics or quantum physics. The experiment revealed a rich polarization structure of D 1 scattering, although available quantum theory predicted null results. It has now finally been possible to formulate a well-defined and self-consistent extension of the theory of quantum scattering that can reproduce in great quantitative detail the main polarization structures that were found in the laboratory experiment. Here we give a brief overview of the new physical ingredients that were missing before. The extended theory reveals that multi-level atomic systems have a far richer coherence structure than previously believed.