The lambda-dimension of commutative arithmetic rings

It is shown that every commutative arithmetic ring $R$ has $lambda$-dimension $leq 3$. An example of a commutative Kaplansky ring with $lambda$-dimension 3 is given. If $R$ satisfies an additional condition then $lambda$-dim($R$

Almost clean rings and arithmetical rings

It is shown that a commutative B\'ezout ring $R$ with compact minimal prime spectrum is an elementary divisor ring if and only if so is $R/L$ for each minimal prime ideal $L$. This result is obtained by using the quotient space $\mathrm{pSpec} R$ of the prime spectrum of the ring $R$ modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if $R$ is arithmetical, $\mathrm{pSpec} R$ is Hausdorff and there is a bijection between this quotient space and the minimal prime spectrum $\mathrm{Min} R$, which is a homeomorphism if and only if $\mathrm{Min} R$ is compact. If $x$ is a closed point of $\mathrm{pSpec} R$, there is a pure ideal $A(x)$ such that $x=V(A(x))$. If $R$ is almost clean, i.e. each element is the sum of a regular element with an idempotent, it is shown that $\mathrm{pSpec} R$ is totally disconnected and, $\forall x\in\mathrm{pSpec} R$, $R/A(x)$ is almost clean; the converse holds if every principal ideal is finitely presented. Some questions posed by Facchini and Faith at the second International Fez Conference on Commutative Ring Theory in 1995, are also investigated. If $R$ is a commutative ring for which the ring $Q(R/A)$ of quotients of $R/A$ is an IF-ring for each proper ideal $A$, it is proved that $R_P$ is a strongly discrete valuation ring for each maximal ideal $P$ and $R/A$ is semicoherent for each proper ideal $A$

Analysis of CMB polarization on an incomplete sky

The full sky cosmic microwave background polarization field can be decomposed into 'electric' and 'magnetic' components. Working in harmonic space we construct window functions that allow clean separation of the electric and magnetic modes from observations over only a portion of the sky. Our construction is exact for azimuthally symmetric patches, but should continue to perform well for arbitrary patches. From the window functions we obtain variables that allow for robust estimation of the magnetic component without risk of contamination from the probably much larger electric signal. For isotropic, uncorrelated noise the variables have a very simple diagonal noise correlation, and further analysis using them should be no harder than analysing the temperature field. For an azimuthally-symmetric patch, such as that obtained from survey missions when the galactic region is removed, the exactly-separated variables are fast to compute allowing us to estimate the magnetic signal that could be detected by the Planck satellite in the absence of non-galactic foregrounds. We also discuss the sensitivity of future experiments to tensor modes in the presence of a magnetic signal generated by weak lensing, and give lossless methods for analysing the electric polarization field in the case that the magnetic component is negligible.Comment: 27 pages, 8 figures. New appendix on weak signal detection and revised plots using a better statistic. Other changes to match version accepted by PRD. Sample source code now available at http://cosmologist.info/pola

Observational Limits on Machos in the Galactic Halo

We present final results from the first phase of the EROS search for gravitational microlensing of stars in the Magellanic Clouds by unseen deflectors (machos: MAssive Compact Halo Objects). The search is sensitive to events with time scales between 15 minutes and 200 days corresponding to deflector masses in the range 1.e-7 to a few solar masses. Two events were observed that are compatible with microlensing by objects of mass of about 0.1 Mo. By comparing the results with the expected number of events for various models of the Galaxy, we conclude that machos in the mass range [1.e-7, 0.02] Mo make up less than 20% (95% C.L.) of the Halo dark matter.Comment: 4 pages, 3 Postscript figures, to be published in Astronomy & Astrophysic

Benchmark Parameters for CMB Polarization Experiments

The recently detected polarization of the cosmic microwave background (CMB) holds the potential for revealing the physics of inflation and gravitationally mapping the large-scale structure of the universe, if so called B-mode signals below 10^{-7}, or tenths of a uK, can be reliably detected. We provide a language for describing systematic effects which distort the observed CMB temperature and polarization fields and so contaminate the B-modes. We identify 7 types of effects, described by 11 distortion fields, and show their association with known instrumental systematics such as common mode and differential gain fluctuations, line cross-coupling, pointing errors, and differential polarized beam effects. Because of aliasing from the small-scale structure in the CMB, even uncorrelated fluctuations in these effects can affect the large-scale B modes relevant to gravitational waves. Many of these problems are greatly reduced by having an instrumental beam that resolves the primary anisotropies (FWHM << 10'). To reach the ultimate goal of an inflationary energy scale of 3 \times 10^{15} GeV, polarization distortion fluctuations must be controlled at the 10^{-2}-10^{-3} level and temperature leakage to the 10^{-4}-10^{-3} level depending on effect. For example pointing errors must be controlled to 1.5'' rms for arcminute scale beams or a percent of the Gaussian beam width for larger beams; low spatial frequency differential gain fluctuations or line cross-coupling must be eliminated at the level of 10^{-4} rms.Comment: 11 pages, 5 figures, submitted to PR

First Detection of Polarization of the Submillimetre Diffuse Galactic Dust Emission by Archeops

We present the first determination of the Galactic polarized emission at 353 GHz by Archeops. The data were taken during the Arctic night of February 7, 2002 after the balloon--borne instrument was launched by CNES from the Swedish Esrange base near Kiruna. In addition to the 143 GHz and 217 GHz frequency bands dedicated to CMB studies, Archeops had one 545 GHz and six 353 GHz bolometers mounted in three polarization sensitive pairs that were used for Galactic foreground studies. We present maps of the I, Q, U Stokes parameters over 17% of the sky and with a 13 arcmin resolution at 353 GHz (850 microns). They show a significant Galactic large scale polarized emission coherent on the longitude ranges [100, 120] and [180, 200] deg. with a degree of polarization at the level of 4-5%, in agreement with expectations from starlight polarization measurements. Some regions in the Galactic plane (Gem OB1, Cassiopeia) show an even stronger degree of polarization in the range 10-20%. Those findings provide strong evidence for a powerful grain alignment mechanism throughout the interstellar medium and a coherent magnetic field coplanar to the Galactic plane. This magnetic field pervades even some dense clouds. Extrapolated to high Galactic latitude, these results indicate that interstellar dust polarized emission is the major foreground for PLANCK-HFI CMB polarization measurement.Comment: Submitted to Astron. & Astrophys., 14 pages, 12 Fig., 2 Table

Combined Analysis of the Binary-Lens Caustic-Crossing Event MACHO 98-SMC-1

We fit the data for the binary-lens microlensing event MACHO 98-SMC-1 from 5 different microlensing collaborations and find two distinct solutions characterized by binary separation d and mass ratio q: (d,q)=(0.54,0.50) and (d,q)=(3.65,0.36), where d is in units of the Einstein radius. However, the relative proper motion of the lens is very similar in the two solutions, 1.30 km/s/kpc and 1.48 km/s/kpc, thus confirming that the lens is in the Small Magellanic Cloud. The close binary can be either rotating or approximately static but the wide binary must be rotating at close its maximum allowed rate to be consistent with all the data. We measure limb-darkening coefficients for five bands ranging from I to V. As expected, these progressively decrease with rising wavelength. This is the first measurement of limb darkening for a metal-poor A star.Comment: 29 pages + 9 figures + 2 tables, submitted to Ap

Planck intermediate results. XXIX. All-sky dust modelling with Planck, IRAS, and WISE observations

We present all-sky modelling of the high resolution Planck, IRAS, and WISE infrared (IR) observations using the physical dust model presented by Draine and Li in 2007 (DL). We study the performance and results of this model, and discuss implications for future dust modelling. The present work extends the DL dust modelling carried out on nearby galaxies using Herschel and Spitzer data to Galactic dust emission. We employ the DL dust model to generate maps of the dust mass surface density, the optical extinction Av, and the starlight intensity parametrized by Umin. The DL model reproduces the observed spectral energy distribution (SED) satisfactorily over most of the sky, with small deviations in the inner Galactic disk and in low ecliptic latitude areas. We compare the DL optical extinction Av for the diffuse interstellar medium with optical estimates for 2 10^5 quasi-stellar objects (QSOs) observed in the Sloan digital sky survey. The DL Av estimates are larger than those determined towards QSOs by a factor of about 2, which depends on Umin. The DL fitting parameter Umin, effectively determined by the wavelength where the SED peaks, appears to trace variations in the far-IR opacity of the dust grains per unit Av, and not only in the starlight intensity. To circumvent the model deficiency, we propose an empirical renormalization of the DL Av estimate, dependent of Umin, which compensates for the systematic differences found with QSO observations. This renormalization also brings into agreement the DL Av estimates with those derived for molecular clouds from the near-IR colours of stars in the 2 micron all sky survey. The DL model and the QSOs data are used to compress the spectral information in the Planck and IRAS observations for the diffuse ISM to a family of 20 SEDs normalized per Av, parameterized by Umin, which may be used to test and empirically calibrate dust models.Comment: Final version that has appeared in A&

Planck 2015 results. XXIII. The thermal Sunyaev-Zeldovich effect--cosmic infrared background correlation

We use Planck data to detect the cross-correlation between the thermal Sunyaev-Zeldovich (tSZ) effect and the infrared emission from the galaxies that make up the the cosmic infrared background (CIB). We first perform a stacking analysis towards Planck-confirmed galaxy clusters. We detect infrared emission produced by dusty galaxies inside these clusters and demonstrate that the infrared emission is about 50% more extended than the tSZ effect. Modelling the emission with a Navarro--Frenk--White profile, we find that the radial profile concentration parameter is $c_{500} = 1.00^{+0.18}_{-0.15}$. This indicates that infrared galaxies in the outskirts of clusters have higher infrared flux than cluster-core galaxies. We also study the cross-correlation between tSZ and CIB anisotropies, following three alternative approaches based on power spectrum analyses: (i) using a catalogue of confirmed clusters detected in Planck data; (ii) using an all-sky tSZ map built from Planck frequency maps; and (iii) using cross-spectra between Planck frequency maps. With the three different methods, we detect the tSZ-CIB cross-power spectrum at significance levels of (i) 6 $\sigma$, (ii) 3 $\sigma$, and (iii) 4 $\sigma$. We model the tSZ-CIB cross-correlation signature and compare predictions with the measurements. The amplitude of the cross-correlation relative to the fiducial model is $A_{\rm tSZ-CIB}= 1.2\pm0.3$. This result is consistent with predictions for the tSZ-CIB cross-correlation assuming the best-fit cosmological model from Planck 2015 results along with the tSZ and CIB scaling relations.Comment: 18 pages, 16 figure

Planck 2015 results. XXVII. The Second Planck Catalogue of Sunyaev-Zeldovich Sources

We present the all-sky Planck catalogue of Sunyaev-Zeldovich (SZ) sources detected from the 29 month full-mission data. The catalogue (PSZ2) is the largest SZ-selected sample of galaxy clusters yet produced and the deepest all-sky catalogue of galaxy clusters. It contains 1653 detections, of which 1203 are confirmed clusters with identified counterparts in external data-sets, and is the first SZ-selected cluster survey containing > $10^3$ confirmed clusters. We present a detailed analysis of the survey selection function in terms of its completeness and statistical reliability, placing a lower limit of 83% on the purity. Using simulations, we find that the Y5R500 estimates are robust to pressure-profile variation and beam systematics, but accurate conversion to Y500 requires. the use of prior information on the cluster extent. We describe the multi-wavelength search for counterparts in ancillary data, which makes use of radio, microwave, infra-red, optical and X-ray data-sets, and which places emphasis on the robustness of the counterpart match. We discuss the physical properties of the new sample and identify a population of low-redshift X-ray under- luminous clusters revealed by SZ selection. These objects appear in optical and SZ surveys with consistent properties for their mass, but are almost absent from ROSAT X-ray selected samples