The largest singletons in weighted set partitions and its applications

Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let $A_{n,k}(\mathbf{t})$ denote the total weight of partitions on $[n+1]$ with the largest singleton $\{k+1\}$. In this paper, explicit formulas for $A_{n,k}(\mathbf{t})$ and many combinatorial identities involving $A_{n,k}(\mathbf{t})$ are obtained by umbral operators and combinatorial methods. As applications, we investigate three special cases such as permutations, involutions and labeled forests. Particularly in the permutation case, we derive a surprising identity analogous to the Riordan identity related to tree enumerations, namely, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, \end{eqnarray*} where $D_{k}$ is the $k$-th derangement number or the number of permutations of $\{1,2,\dots, k\}$ with no fixed points.Comment: 15page

The earliest galaxies seen in 21 cm line absorption

We investigate the 21 cm absorption lines produced by non-linear structures during the early stage of reionization, i.e. the starless minihalos and the dwarf galaxies. After a detailed modelling of their properties, with particular attention to the coupling physics, we determine their 21 cm absorption line profiles. The infalling gas velocity around minihalos/dwarf galaxies strongly affects the line shape, and with the low spin temperatures outside the virial radii of the systems, gives rise to horn-like line profiles. The optical depth of a dwarf galaxy is reduced for lines of sight penetrating through its HII region, and especially, a large HII region created by a dwarf galaxy with higher stellar mass and/or a top-heavy initial mass function results in an optical depth trough rather than an absorption line. We compute synthetic spectra of 21 cm forest for both high redshift quasars and radio afterglows of gamma ray bursts (GRBs). Even with the planned SKA, radio afterglows of most if not all GRBs would still be too dim to be the background sources for high resolution (1 kHz) observations, but absorption lines can be easily detected towards a high-z quasar. Broadband observation against GRB afterglows can also be used to reveal the evolving 21 cm signal from both minihalos and dwarf galaxies if there was no X-ray background or it was extremely weak, but it becomes difficult if an early X-ray background existed. Hence the 21 cm absorption could be a powerful probe of the presence/intensity of the X-ray background and the thermal history of the early universe.Comment: 18 pages, 17 figures. Accepted for publication in MNRA

Precise measurements of inflationary features with 21 cm observations

Future observations of 21~cm emission using HI intensity mapping will enable us to probe the large scale structure of the Universe over very large survey volumes within a reasonable observation time. We demonstrate that the three-dimensional information contained in such surveys will be an extremely powerful tool in searching for features that were imprinted in the primordial power spectrum and bispectrum during inflation. Here we focus on the "resonant" and "step" inflation models, and forecast the potential of upcoming 21~cm experiments to detect these inflationary features in the observable power- and bispectrum. We find that the full scale Tianlai experiment and the Square Kilometre Array (SKA) have the potential to improve on the sensitivity of current Cosmic Microwave Background (CMB) experiments by several orders of magnitude.Comment: 7 pages, 3 figures, replaced with revised versio

An analytical model of the large neutral regions during the late stage of reionization

In this paper we investigate the nature and distribution of large neutral regions during the late epoch of reionization. In the "bubble model" of reionization, the mass distribution of large ionized regions ("bubbles") during the early stage of reionization is obtained by using the excursion set model, where the ionization of a region corresponds to the first up-crossing of a barrier by random trajectories. We generalize this idea, and develop a method to predict the distribution of large scale neutral regions during the late stage of reionization, taking into account the ionizing background after the percolation of HII regions. The large scale neutral regions which we call "neutral islands" are not individual galaxies or minihalos, but larger regions where fewer galaxies formed and hence ionized later, and they are identified in the excursion set model with the first down-crossings of the island barrier. Assuming that the consumption rate of ionizing background photons is proportional to the surface area of the neutral islands, we obtained the size distribution of the neutral islands. We also take the "bubbles-in-island" effect into account by considering the conditional probability of up-crossing a bubble barrier after down-crossing the island barrier. We find that this effect is very important. An additional barrier is set to avoid islands being percolated through. We find that there is a characteristic scale for the neutral islands, while the small islands are rapidly swallowed up by the ionizing background, this characteristic scale does not change much as the reionization proceeds.Comment: 33 pages, 11 figures, accepted by The Astrophysical Journa

Infrared background signatures of the first black holes

Angular fluctuations of the Near InfraRed Background (NIRB) intensity are observed up to scales \simlt 1^{\ensuremath{^{\circ}}}. Their interpretation is challenging as even after removing the contribution from detected sources, the residual signal is $>10$ times higher than expected from distant galaxies below the detection limit and first stars. We propose here a novel interpretation in which early, intermediate mass, accreting direct collapse black holes (DCBH), which are too faint to be detected individually in current surveys, could explain the observed fluctuations. We find that a population of highly obscured (N_{\rm H}\simgt 10^{25} \rm cm^{-2}) DCBHs formed in metal-free halos with virial temperature $10^4$ K at z\simgt 12, can explain the observed level $\approx 10^{-3}$ (nW m$^{-2}$ sr$^{-1})^2$ of the 3.6 and 4.5 $\mu$m fluctuations on scales $>100''$. The signal on smaller scales is instead produced by undetected galaxies at low and intermediate redshifts. Albeit Compton-thick, at scales $\theta> 100''$ DCBHs produce a CXB (0.5-2 keV)-NIRB ($4.5 \rm \mu m$) cross-correlation signal of $\simeq 10^{-11}$ erg s$^{-1}$ cm$^{-2}$ nW m$^{-2}$ sr$^{-1}$ slightly dependent on the specific value of the absorbing gas column ($N_{\rm H} \approx 10^{25} \rm cm^{-2}$) adopted and in agreement with the recent measurements by \cite{2012arXiv1210.5302C}. At smaller scales the cross-correlation is dominated by the emission of high-mass X-ray binaries (HMXB) hosted by the same low-$z$, undetected galaxies accounting for small scale NIRB fluctuations. These results outline the great potential of the NIRB as a tool to investigate the nature of the first galaxies and black holes.Comment: 27 pages, 8 figures, accepted for publication in MNRA

The brief era of direct collapse black hole formation

It has been proposed that the first, intermediate-mass ($\approx 10^{5-6}~M_\odot$) black holes might form through direct collapse of unpolluted gas in atomic-cooling halos exposed to a strong Lyman-Werner (LW) or near-infrared (NIR) radiation. As these systems are expected to be Compton-thick, photons above 13.6 eV are largely absorbed and re-processed into lower energy bands. It follows that direct collapse black holes (DCBHs) are very bright in the LW/NIR bands, typically outshining small high-redshift galaxies by more than 10 times. Once the first DCBHs form, they then trigger a runaway process of further DCBH formation, producing a sudden rise in their cosmic mass density. The universe enters the "DCBH era" at $z \approx 20$ when a large fraction of atomic-cooling halos are experiencing DCBH formation. By combining the clustering properties of the radiation sources with Monte Carlo simulations we show that in this scenario the DCBH mass density rises from $\sim 5$~$M_\odot$ Mpc$^{-3}$ at $z\sim 30$ to the peak value $\sim5\times10^5 M_\odot$ Mpc$^{-3}$ at $z \sim 14$ in our fiducial model. However, the abundance of \textit{active} (accreting) DCBHs drops after $z \sim 14$, as gas in the potential formation sites (unpolluted halos with virial temperature slightly above $10^4$~K) is photoevaporated. This effect almost completely suppresses DCBH formation after $z\sim 13$. The DCBH formation era lasts only $\approx 150$ Myr, but it might crucially provide the seeds of the supermassive black holes (SMBHs) powering $z\sim6$ quasars.Comment: 26 pages, 4 figures, 2 tables. Accepted for publication in MNRA

21cm Forest with the SKA

An alternative to both the tomography technique and the power spectrum approach is to search for the 21cm forest, that is the 21cm absorption features against high-z radio loud sources caused by the intervening cold neutral intergalactic medium (IGM) and collapsed structures. Although the existence of high-z radio loud sources has not been confirmed yet, SKA-low would be the instrument of choice to find such sources as they are expected to have spectra steeper than their lower-z counterparts. Since the strongest absorption features arise from small scale structures (few tens of physical kpc, or even lower), the 21cm forest can probe the HI density power spectrum on small scales not amenable to measurements by any other means. Also, it can be a unique probe of the heating process and the thermal history of the early universe, as the signal is strongly dependent on the IGM temperature. Here we show what SKA1-low could do in terms of detecting the 21cm forest in the redshift range z = 7.5-15.Comment: Accepted for publication in the SKA Science Book 'Advancing Astrophysics with the Square Kilometre Array', to appear in 2015; 10 pages, 5 figures; the manuscript is based on Ciardi et al., 2013, MNRAS, 428, 175