Model Independent Primordial Power Spectrum from Maxima, Boomerang, and DASI Data

A model-independent determination of the primordial power spectrum of matter density fluctuations could uniquely probe physics of the very early universe, and provide powerful constraints on inflationary models. We parametrize the primordial power spectrum $A_s^2(k)$ as an arbitrary function, and deduce its binned amplitude from the cosmic microwave background radiation anisotropy (CMB) measurements of Maxima, Boomerang, and DASI. We find that for a flat universe with $A_s^2(k)=1$ (scale-invariant) for scales $k<0.001$h/Mpc, the primordial power spectrum is marginally consistent with a scale-invariant Harrison-Zeldovich spectrum. However, we deduce a rise in power compared to a scale-invariant power spectrum for 0.001 h/{Mpc} \la k \la 0.01 h/{Mpc}. Our results are consistent with large-scale structure data, and seem to suggest that the current observational data allow for the possibility of unusual physics in the very early universe.Comment: substantially revised and final version, accepted by Ap

On a Conjecture of Givental

These brief notes record our puzzles and findings surrounding Givental's recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a projective line, whose Gromov-Witten invariants are well-known and easy to compute. We make some simple checks supporting his conjecture.Comment: 13 pages, no figures; v.2: new title, minor change

Nonaxisymmetric Evolution of Magnetically Subcritical Clouds: Bar Growth, Core Elongation, and Binary Formation

We have begun a systematic numerical study of the nonlinear growth of nonaxisymmetric perturbations during the ambipolar diffusion-driven evolution of initially magnetically subcritical molecular clouds, with an eye on the formation of binaries, multiple stellar systems and small clusters. In this initial study, we focus on the $m=2$ (or bar) mode, which is shown to be unstable during the dynamic collapse phase of cloud evolution after the central region has become magnetically supercritical. We find that, despite the presence of a strong magnetic field, the bar can grow fast enough that for a modest initial perturbation (at 5% level) a large aspect ratio is obtained during the isothermal phase of cloud collapse. The highly elongated bar is expected to fragment into small pieces during the subsequent adiabatic phase. Our calculations suggest that the strong magnetic fields observed in some star-forming clouds and envisioned in the standard picture of single star formation do not necessarily suppress bar growth and fragmentation; on the contrary, they may actually promote these processes, by allowing the clouds to have more than one (thermal) Jeans mass to begin with without collapsing promptly. Nonlinear growth of the bar mode in a direction perpendicular to the magnetic field, coupled with flattening along field lines, leads to the formation of supercritical cores that are triaxial in general. It removes a longstanding objection to the standard scenario of isolated star formation involving subcritical magnetic field and ambipolar diffusion based on the likely prolate shape inferred for dense cores. Continuted growth of the bar mode in already elongated starless cores, such as L1544, may lead to future binary and multiple star formation.Comment: 5 pages, 2 figures, accepted by ApJ

Quiescent Cores and the Efficiency of Turbulence-Accelerated, Magnetically Regulated Star Formation

The efficiency of star formation, defined as the ratio of the stellar to total (gas and stellar) mass, is observed to vary from a few percent in regions of dispersed star formation to about a third in cluster-forming cores. This difference may reflect the relative importance of magnetic fields and turbulence in controlling star formation. We investigate the interplay between supersonic turbulence and magnetic fields using numerical simulations, in a sheet-like geometry. We demonstrate that star formation with an efficiency of a few percent can occur over several gravitational collapse times in moderately magnetically subcritical clouds that are supersonically turbulent. The turbulence accelerates star formation by reducing the time for dense core formation. The dense cores produced are predominantly quiescent, with subsonic internal motions. These cores tend to be moderately supercritical. They have lifetimes long compared with their local gravitational collapse time. Some of the cores collapse to form stars, while others disperse away without star formation. In turbulent clouds that are marginally magnetically supercritical, the star formation efficiency is higher, but can still be consistent with the values inferred for nearby embedded clusters. If not regulated by magnetic fields at all, star formation in a multi-Jeans mass cloud endowed with a strong initial turbulence proceeds rapidly, with the majority of cloud mass converted into stars in a gravitational collapse time. The efficiency is formally higher than the values inferred for nearby cluster-forming cores, indicating that magnetic fields are dynamically important even for cluster formation.Comment: submitted to Ap

Spectral dimensions of hierarchical scale-free networks with shortcuts

The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractals or non-fractals depending on the weight of shortcuts. Applying the renormalization group (RG) approach to the Gaussian model, we obtain the spectral dimension exactly. While the spectral dimension varies between $1$ and $2$ for the fractal case, it remains at $2$, independent of the variation of network structure for the non-fractal case. The crossover behavior between the two cases is studied through the RG flow analysis. The analytic results are confirmed by simulation results and their implications for the architecture of complex systems are discussed.Comment: 10 pages, 3 figure