On the Canonical Formalism for a Higher-Curvature Gravity

Following the method of Buchbinder and Lyahovich, we carry out a canonical formalism for a higher-curvature gravity in which the Lagrangian density ${\cal L}$ is given in terms of a function of the salar curvature $R$ as ${\cal L}=\sqrt{-\det g_{\mu\nu}}f(R)$. The local Hamiltonian is obtained by a canonical transformation which interchanges a pair of the generalized coordinate and its canonical momentum coming from the higher derivative of the metric.Comment: 11 pages, no figures, Latex fil

High Excitation Molecular Gas in the Galactic Center Loops; 12CO(J =2-1 and J =3-2) Observations

We have carried out 12CO(J =2-1) and 12CO(J =3-2) observations at spatial resolutions of 1.0-3.8 pc toward the entirety of loops 1 and 2 and part of loop 3 in the Galactic center with NANTEN2 and ASTE. These new results revealed detailed distributions of the molecular gas and the line intensity ratio of the two transitions, R3-2/2-1. In the three loops, R3-2/2-1 is in a range from 0.1 to 2.5 with a peak at ~ 0.7 while that in the disk molecular gas is in a range from 0.1 to 1.2 with a peak at 0.4. This supports that the loops are more highly excited than the disk molecular gas. An LVG analysis of three transitions, 12CO J =3-2 and 2-1 and 13CO J =2-1, toward six positions in loops 1 and 2 shows density and temperature are in a range 102.2 - 104.7 cm-3 and 15-100 K or higher, respectively. Three regions extended by 50-100 pc in the loops tend to have higher excitation conditions as characterized by R3-2/2-1 greater than 1.2. The highest ratio of 2.5 is found in the most developed foot points between loops 1 and 2. This is interpreted that the foot points indicate strongly shocked conditions as inferred from their large linewidths of 50-100 km s-1, confirming the suggestion by Torii et al. (2010b). The other two regions outside the foot points suggest that the molecular gas is heated up by some additional heating mechanisms possibly including magnetic reconnection. A detailed analysis of four foot points have shown a U shape, an L shape or a mirrored-L shape in the b-v distribution. It is shown that a simple kinematical model which incorporates global rotation and expansion of the loops is able to explain these characteristic shapes.Comment: 59 pages, accepted to PAS

Third quantization of f(R)f(R)-type gravity

We examine the third quantization of $f(R)$-type gravity, based on its effective Lagrangian in the case of a flat Friedmann-Lemaitre-Robertson-Walker metric. Starting from the effective Lagrangian, we execute a suitable change of variable and the second quantization, and we obtain the Wheeler-DeWitt equation. The third quantization of this theory is considered. And the uncertainty relation of the universe is investigated in the example of $f(R)$-type gravity, where $f(R)=R^2$. It is shown, when the time is late namely the scale factor of the universe is large, the spacetime does not contradict to become classical, and, when the time is early namely the scale factor of the universe is small, the quantum effects are dominating.Comment: 9 pages, Arbitrary constants in (4.19) are changed to arbitrary functions of $\varphi$. Conclusions are not changed. References are added. Typos are correcte

Temperature and Density in the Foot Points of the Molecular Loops in the Galactic Center; Analysis of Multi-J Transitions of 12CO(J=1-0, 3-2, 4-3, 7-6), 13CO(J=1-0) and C18O(J=1-0)

Fukui et al. (2006) discovered two molecular loops in the Galactic center and argued that the foot points of the molecular loops, two bright spots at both loops ends, represent the gas accumulated by the falling motion along the loops, subsequent to magnetic flotation by the Parker instability. We have carried out sensitive CO observations of the foot points toward l=356 deg at a few pc resolution in the six rotational transitions of CO; 12CO(J=1-0, 3-2, 4-3, 7-6), 13CO(J=1-0) and C18O(J=1-0). The high resolution image of 12CO (J=3-2) has revealed the detailed distribution of the high excitation gas including U shapes, the outer boundary of which shows sharp intensity jumps accompanying strong velocity gradients. An analysis of the multi-J CO transitions shows that the temperature is in a range from 30-100 K and density is around 10^3-10^4 cm^-3, confirming that the foot points have high temperature and density although there is no prominent radiative heating source such as high mass stars in or around the loops. We argue that the high temperature is likely due to the shock heating under C-shock condition caused by the magnetic flotation. We made a comparison of the gas distribution with theoretical numerical simulations and note that the U shape is consistent with numerical simulations. We also find that the region of highest temperature of ~100 K or higher inside the U shape corresponds to the spur having an upward flow, additionally heated up either by magnetic reconnection or bouncing in the interaction with the narrow neck at the bottom of the U shape. We note these new findings further reinforce the magnetic floatation interpretation.Comment: 40 pages, 23 figures, accepted by PASJ on Vol.62 No.