The Gravitational Lensing Effect on the CMB Polarisation Anisotropy in the {\Lambda}-LTB Model

A local void modifies the sky distribution pattern of the cosmic microwave background (CMB) polarisation by gravitational lensing and produces B-modes from E-modes for an off-center observer. In order to see whether this effect can be used to observationally test the validity of the local void model, we calculate this lensing effect by solving the propagation of CMB polarisation along null geodesics close to the central light cone in the general Lema\^itre-Tolman-Bondi (LTB) model perturbatively. In particular, we give general formulas for the correlations of E and B observed by an off-center observer and show that E_l^m and B_l^m are correlated for the same value of l, i.e., is proportional to {\delta}_{l, l'}, while <E_l^m E_l'^m'> is proportional to {\delta}_{|l'-l|, 1}. This feature can be used to distinguish the gravitational lensing effect by a local void from those by normal shear field of galaxies.Comment: 22 pages in the PTPTeX style, 2 figures, 1 table; revised argument in Section 4.4 and added Section 5.4 in response to referee's comment

Alkenylation and Allylation of Aldehydes by Using an Ni/Cr/TDAE Redox System

In 360 females, Bone mineral density (BMD) in lumbar vertebrae (L2-L4) was assessed by quantitative computed tomography (QCT), and the values obtained were compared with the frequency of vertebral transformation or fracture as assessed by lateral scan image (scanogram) by X-ray CT. A correlation was observed between the frequency of vertebral transformation (or fracture) and lumbar BMD values : BMD under 125 mg/cm(3) was observed over 90% of women with vertebral transformation, and BMD under 70mg/cm(3) was found about 50% of them. These results suggest that decrease in BMD in lumbar vertebrae leads to vertebral transformation or fracture. Thus, measurement of BMD by QCT would be very useful in predicting vertebral transformation or fractures