449 research outputs found
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 denote the total
weight of partitions on with the largest singleton . In this
paper, explicit formulas for and many combinatorial
identities involving 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
is the -th derangement number or the number of permutations of
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 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 K at z\simgt 12, can explain
the observed level (nW m sr of the 3.6 and
4.5 m fluctuations on scales . The signal on smaller scales is
instead produced by undetected galaxies at low and intermediate redshifts.
Albeit Compton-thick, at scales DCBHs produce a CXB (0.5-2
keV)-NIRB () cross-correlation signal of erg
s cm nW m sr slightly dependent on the specific
value of the absorbing gas column ()
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-, 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 () 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 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
~ Mpc at to the peak value Mpc at in our fiducial model. However, the
abundance of \textit{active} (accreting) DCBHs drops after , as gas
in the potential formation sites (unpolluted halos with virial temperature
slightly above ~K) is photoevaporated. This effect almost completely
suppresses DCBH formation after . The DCBH formation era lasts only
Myr, but it might crucially provide the seeds of the supermassive
black holes (SMBHs) powering 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
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