Biased-estimations of the Variance and Skewness

Nonlinear combinations of direct observables are often used to estimate quantities of theoretical interest. Without sufficient caution, this could lead to biased estimations. An example of great interest is the skewness $S_3$ of the galaxy distribution, defined as the ratio of the third moment \xibar_3 and the variance squared \xibar_2^2. Suppose one is given unbiased estimators for \xibar_3 and \xibar_2^2 respectively, taking a ratio of the two does not necessarily result in an unbiased estimator of $S_3$. Exactly such an estimation-bias affects most existing measurements of $S_3$. Furthermore, common estimators for \xibar_3 and \xibar_2 suffer also from this kind of estimation-bias themselves: for \xibar_2, it is equivalent to what is commonly known as the integral constraint. We present a unifying treatment allowing all these estimation-biases to be calculated analytically. They are in general negative, and decrease in significance as the survey volume increases, for a given smoothing scale. We present a re-analysis of some existing measurements of the variance and skewness and show that most of the well-known systematic discrepancies between surveys with similar selection criteria, but different sizes, can be attributed to the volume-dependent estimation-biases. This affects the inference of the galaxy-bias(es) from these surveys. Our methodology can be adapted to measurements of analogous quantities in quasar spectra and weak-lensing maps. We suggest methods to reduce the above estimation-biases, and point out other examples in LSS studies which might suffer from the same type of a nonlinear-estimation-bias.Comment: 28 pages of text, 9 ps figures, submitted to Ap

Cosmological constraints from the CMB and Ly-alpha forest revisited

The WMAP team has recently highlighted the usefulness of combining the Ly-alpha forest constraints with those from the cosmic microwave background (CMB). This combination is particularly powerful as a probe of the primordial shape of the power spectrum. Converting between the Ly-alpha forest observations and the linear mass power spectrum requires a careful treatment of nuisance parameters and modeling with cosmological simulations. We point out several issues which lead to an expansion of the errors, the two most important being the range of cosmological parameters explored in simulations and the treatment of the mean transmitted flux constraints. We employ a likelihood calculator for the current Ly-alpha data set based on an extensive 6-dimensional grid of simulations. We show that the current uncertainties in the mean transmission and the flux power spectrum define a degeneracy line in the amplitude-slope plane. The CMB degeneracy due to the primordial power spectrum shape follows a similar relation in this plane. This weakens the statistical significance of the primordial power spectrum shape constraints based on combined CMB+Ly-alpha forest analysis. Using the current data the simplest n=1 scale invariant model with dn/dln k=0 and no tensors has a Delta chi^2=4 compared to the best fitting model in which these 3 parameters are free. Current data therefore do not require relaxing these parameters to improve the fit.Comment: 7 pages, 2 figures, expanded, matches the accepted version in MNRA

Holographic Principle bounds on Primordial Black Hole abundances

The generalized Second Law of thermodynamics and the Holographic Principle are combined to obtain the maximum mass of black holes formed inside a static spherical box of size $R$ filled with radiation at initial temperature $T_{i}$. The final temperature after the formation of black holes is evaluated, and we show that a critical threshold exists for the radiation to be fully consumed by the process. We next argue that if some form of Holographic Principle holds, upper bounds to the mass density of PBHs formed in the early universe may be obtained. The limits are worked out for inflationary and non-inflationary cosmological models. This method is independent of the known limits based on the background fluxes (from cosmic rays, radiation and other forms of energy) and applies to potentially important epochs of PBH formation, resulting in quite strong constraints to $\Omega_{pbh}$.Comment: Latex file, 2 .ps figures. To appear in Classical and Quantum Gravit

Eulerian bias and the galaxy density field

We investigate the effects on cosmological clustering statistics of empirical biasing, where the galaxy distribution is a local transformation of the present-day Eulerian density field. The effects of the suppression of galaxy numbers in voids, and their enhancement in regions of high density, are considered, independently and in combination. We compare results from numerical simulations with the predictions of simple analytic models. We find that the bias is generally scale-dependent, so that the shape of the galaxy power spectrum differs from that of the underlying mass distribution. The degree of bias is always a monotonic function of scale, tending to an asymptotic value on scales where the density fluctuations are linear. The scale dependence is often rather weak, with many reasonable prescriptions giving a bias which is nearly independent of scale. We have investigated whether such an Eulerian bias can reconcile a range of theoretical power spectra with the twin requirements of fitting the galaxy power spectrum and reproducing the observed mass-to-light ratios in clusters. It is not possible to satisfy these constraints for any member of the family of CDM-like power spectra in an Einstein - de Sitter universe when normalised to match COBE on large scales and galaxy cluster abundances on intermediate scales. We discuss what modifications of the mass power spectrum might produce agreement with the observational data.Comment: 14 pages, LaTeX (using mn.sty, epsfig), 17 Postscript figures included. Accepted for publication in MNRA

A first direct measurement of the intergalactic medium temperature around a quasar at z=6

The thermal state of the intergalactic medium (IGM) provides an indirect probe of both the HI and HeII reionisation epochs. Current constraints on the IGM temperature from the Lya forest are restricted to the redshift range 2<z<4.5, limiting the ability to probe the thermal memory of HI reionisation toward higher redshift. In this work, we present the first direct measurement of the IGM temperature around a z=6 quasar by analysing the Doppler widths of Lya absorption lines in the proximity zone of SDSS J0818+1722. We use a high resolution (R= 40000) Keck/HIRES spectrum in combination with detailed numerical modelling to obtain the temperature at mean density, T_0=23600\pm^5000_6900K (\pm^9200_9300K) at 68 (95) per cent confidence assuming a prior probability 13500K<T_0<38500 K following HI and HeII reionisation. This enables us to place an upper limit on the redshift of HI reionisation, z_H, within 33 comoving Mpc of SDSS J0818+1722. If the quasar reionises the HeII in its vicinity, then in the limit of instantaneous reionisation we infer z_H<9.0 (11.0) at 68 (95) per cent confidence assuming photoheating is the dominant heat source and that HI reionisation is driven by ionising sources with soft spectra, typical of population II stars. If the HI and HeII in the IGM around SDSS J0818+1722 are instead reionised simultaneously by a population of massive metal-free stars, characterised by very hard ionising spectra, we obtain a tighter upper limit of z_H<8.4 (9.4). Initiating reionisation at higher redshifts produces temperatures which are too low with respect to our constraint unless the HI ionising sources or the quasar itself have spectra significantly harder than typically assumed.Comment: 15 pages, 9 figures, accepted to MNRA