Vanishing theorems and character formulas for the Hilbert scheme of points in the plane

Earlier we showed that the Hilbert scheme of $n$ points in the plane can be identified with the Hilbert scheme of regular $S_n$ orbits on $C^{2n}$. Using this result, together with a recent theorem of Bridgeland, King and Reid on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author. In particular we prove that the dimension of the space of diagonal harmonics is equal to $(n+1)^{n-1}$.Comment: 33 page

Luminosity Functions of Lyman Alpha Emitting Galaxies and Cosmic Reionization of Hydrogen

Recent observations imply that the observed number counts of Lya Emitters (LAEs) evolved significantly between z=5.7 and z=6.5. It has been suggested that this was due to a rapid evolution in the ionisation state, and hence transmission of the IGM which caused Lya flux from z=6.5 galaxies to be more strongly suppressed. In this paper we consider the joint evolution of the Lya and UV luminosity functions (LFs) and show that the IGM transmission evolved between z=6.5 and z=5.7 by a factor 1.1 <R < 1.8 (95% CL). This result is insensitive to the underlying model of the Lya LF (as well as cosmic variance). Using a model for IGM transmission, we find that the evolution of the mean IGM density through cosmic expansion alone may result in a value for the ratio of transmissions as high as R=1.3. Thus, the existing LFs do not provide evidence for overlap. Furthermore, the constraint R<1.8 suggests that the Universe at z=6.5 was more than half ionised by volume, i.e. x_i,V>0.5.Comment: MNRAS in press. Constraints from rest-frame UV LF added. Discussion added on cosmic variance. Lower limit on x_i,V lowered to 0.5 (from 0.8

The Growth of the Earliest Supermassive Black Holes and Their Contribution to Reionization

We discuss currently available observational constraints on the reionization history of the intergalactic medium (IGM), and the extent to which accreting black holes (BHs) can help explain these observations. We show new evidence, based on the combined statistics of Lyman alpha and beta absorption in quasar spectra, that the IGM contains a significant amount of neutral hydrogen, and is experiencing rapid ionization around redshift z=6. However, we argue that quasar BHs, even faint ones that are below the detection thresholds of existing optical surveys, are unlikely to drive the evolution of the neutral fraction around this epoch, because they would over-produce the present-day soft X-ray background. On the other hand, the seeds of the quasar BHs around z=6 likely appeared at much earlier epochs (around z=20), and produced hard ionizing radiation by accretion. These early BHs are promising candidates to account for the high redshift ionization (around z=15) implied by the recent cosmic microwave anisotropy data from WMAP. Using a model for the growth of BHs by accretion and mergers in a hierarchical cosmology, we suggest that the early growth of quasars must include a super-Eddington growth phase, and that, although not yet optically identified, the FIRST radio survey may have already detected several thousand BHs at z>6 with Mbh>10^8 Msun.Comment: 12 pages, invited contribution to Proceedings of the Conference on "Growing Black Holes" held in Garching, Germany, on June 21-25, 2004, edited by A. Merloni, S. Nayakshin and R. Sunyaev, Springer-Verlag series of "ESO Astrophysics Symposia

Fluctuations in the High-Redshift Lyman-Werner Background: Close Halo Pairs as the Origin of Supermassive Black Holes

The earliest generation of stars and black holes must have established an early 'Lyman-Werner' background (LWB) at high redshift, prior to the epoch of reionization. Because of the long mean free path of photons with energies E<13.6 eV, the LWB was nearly uniform. However, some variation in the LWB is expected due to the discrete nature of the sources, and their highly clustered spatial distribution. In this paper, we compute the probability distribution function (PDF) of the LW flux that irradiates dark matter (DM) halos collapsing at high-redshift (z~10). Our model accounts for (i) the clustering of DM halos, (ii) Poisson fluctuations in the number of corresponding star forming galaxies, and (iii) scatter in the LW luminosity produced by halos of a given mass (calibrated using local observations). We find that > 99% of the DM halos are illuminated by a LW flux within a factor of 2 of the global mean value. However, a small fraction, ~1e-8 to 1e-6, of DM halos with virial temperatures above 1e4 K have a close luminous neighbor within < 10 kpc, and are exposed to a LW flux exceeding the global mean by a factor of > 20, or to J_(21,LW)> 1e3 (in units of 1e-21 erg/s/Hz/sr/cm^2). This large LW flux can photo--dissociate H_2 molecules in the gas collapsing due to atomic cooling in these halos, and prevent its further cooling and fragmentation. Such close halo pairs therefore provide possible sites in which primordial gas clouds collapse directly into massive black holes (M_BH~ 1e4 - 1e6 M_sun), and subsequently grow into supermassive (M_BH > 1e9 M_sun) black holes by z~6.Comment: 13 pages, 8 figures, Accepted to MNRA

Evolution in the Escape Fraction of Ionizing Photons and the Decline in Strong Lya Emission from z>6 Galaxies

The rapid decline in the number of strong Lyman Alpha (Lya) emitting galaxies at z > 6 provides evidence for neutral hydrogen in the IGM, but is difficult to explain with plausible models for reionization. We demonstrate that the observed reduction in Lya flux from galaxies at z > 6 can be explained by evolution in the escape fraction of ionizing photons, f_esc. We find that the median observed drop in the fraction of galaxies showing strong Lya emission, as well as the observed evolution of the Lya luminosity function both follow from a small increase in f_esc of Delta f_esc ~ 0.1 from f_esc ~ 0.6 at z ~ 6. This high escape fraction may be at odds with current constraints on the ionising photon escape fraction, which favor smaller values of f_esc < 20%. However, models that invoke a redshift evolution of f_ esc that is consistent with these constraints can suppress the z~7 Lya flux to the observed level, if they also include a small evolution in global neutral fraction of Delta x_HI ~ 0.2. Thus, an evolving escape fraction of ionising photons can be a plausible part of the explanation for evolution in the Lya emission of high redshift galaxies. More generally, our analysis also shows that the drop in the Lya fraction is quantitatively consistent with the observed evolution in the Lya luminosity functions of Lya Emitters.Comment: 9 pages, 3 figures. Accepted for publication in MNRAS. Added references & brief discussion on line shape evolutio

Lyman Alpha Radiation From Collapsing Protogalaxies II: Observational Evidence for Gas Infall

We model the spectra and surface brightness distributions for the Lyman alpha (Lya) radiation expected from protogalaxies that are caught in the early stages of their assembly. We use the results of a companion paper to characterize the radiation emerging from spherically collapsing gas clouds. We then modify the intrinsic spectra to incorporate the effect of subsequent resonant scattering in the intergalactic medium (IGM). Using these models, we interpret a number of recent observations of extended Lya blobs (LABs) at high redshift. We suggest, based on the angular size, energetics, as well as the relatively shallow surface brightness profiles, and double-peaked spectra, that several of these LABs may be associated with collapsing protogalaxies. We suggest two follow-up observations to diagnose the presence of gas infall. High S/N spectra of LABs should reveal a preferential flattening of the surface brightness profile at the red side of the line. Complementary imaging of the blobs at redshifted Balmer alpha wavelengths should reveal the intrinsic Lya emissivity and allow its separation from radiative transfer effects. We show that Lya scattering by infalling gas can reproduce the observed spectrum of Steidel et al's LAB2 as accurately as a recently proposed outflow model. Finally, we find similar evidence for infall in the spectra of point-like Lyman alpha emitters. The presence of scattering by the infalling gas implies that the intrinsic Lya luminosities, and derived quantities, such as the star-formation rate, in these objects may have been underestimated by about an order of magnitude.Comment: Accepted for Publication in ApJ, 11 emulateapj pages with 6 figures, together with a companion pape

A raising operator formula for Macdonald polynomials

We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}_{\mu }(X;q,t)$, which follows from our recent formula for $\nabla$ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde{H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde{H}_{\mu}(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to $\mathbb{N}[q,t]$, generalizing Macdonald positivity.Comment: 20 page