The bias field of dark matter haloes

Accepted 1998 January 26. Received 1998 January 26; in original form 1997 August 13This paper presents a stochastic approach to the clustering evolution of dark matter haloes in the Universe. Haloes, identified by a Press–Schechter-type algorithm in Lagrangian space, are described in terms of 'counting fields', acting as non-linear operators on the underlying Gaussian density fluctuations. By ensemble-averaging these counting fields, the standard Press–Schechter mass function as well as analytic expressions for the halo correlation function and corresponding bias factors of linear theory are obtained, extending the recent results by Mo & White. The non-linear evolution of our halo population is then followed by solving the continuity equation, under the sole hypothesis that haloes move by the action of gravity. This leads to an exact and general formula for the bias field of dark matter haloes, defined as the local ratio between their number density contrast and the mass density fluctuation. Besides being a function of position and 'observation' redshift, this random field depends upon the mass and formation epoch of the objects and is both non-linear and non-local. The latter features are expected to leave a detectable imprint on the spatial clustering of galaxies, as described, for instance, by statistics like the bispectrum and the skewness. Our algorithm may have several interesting applications, among which is the possibility of generating mock halo catalogues from low-resolution N -body simulations.     

Some new results on statistical properties of wind waves

For wind waves modelled by a stationary Gaussian process ζ(t) (ζ = height above m.w.l. of one point of the free surface) it is shown that, in a time interval including an instant t_m where a maximum ζ_m occurs, the ratio between ζ(t) and ζ_m tends with probability approaching one, to the ratio Ψ(t ? t_m)/Ψ(O), as $\text{ζ}{}_{\mathrm{m}}\text{/Ψ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(}\text{O}\text{)}$ tends to infinity, Ψ(t) being the autocovariance of the process.Starting from this result, it is possible to find analytically the characteristic periods of the highest waves in a sea with a given energy spectrum. These periods, calculated according to period definitions of three methods of wave record analysis, are found to be in remarkably good agreement with data from Bretschneider¹ and Svasek.²     

Temporal Variation in the Trophic Levels of Secondary Consumers in a Mediterranean Coastal Lagoon (Cabras Lagoon,Italy)

Trophic levels (TLs) of fish were estimated on three sampling dates (March, May, and August 2006) for different fish sizes in the Cabras Lagoon (Sardinia, Italy). A temporal TL variation for Atherina boyeri, Gobius niger, and Engraulis encrasicolus was observed. In March and May, the TL ranged from 3.3 to 3.4, characterizing these species as secondary consumers, while in August, this range moved to between 3.7 and 3.8, indicating a TL shift towards tertiary consumers. For Liza ramada, TL was consistently lower in small individuals (mean TL 2.5) than in larger individuals (mean TL 3.0). Increased TL of the fish species A. boyeri, G. niger, and E. encrasicolus in August was consistent with the seasonal changes in the macrobenthic assemblage in this system, with a dominance of primary consumers (benthic deposit feeders) in winter–spring and a dominance of secondary consumers (the nereidids Alitta succinea and Hediste diversicolor) in summer. These fish species took advantage of the high availability of nereidids leading to a rise in their TL values. Furthermore, the increase of TL with size of L. ramada, the most economically valuable fish species in the Cabras Lagoon, indicated an ontogenetic diet shift, the juveniles being omnivorous, while the adults being secondary consumers. We conclude that variability in the trophic levels of fish due to species traits, ontogenetic diet shifts, and variation in prey availability should be taken into account to further understand the food web structure of coastal lagoons.     

Possible stellar asterisms carved on a protohistoric stone

Chisel marks on a stone uncovered in Rupinpiccolo protohistoric hill fort from north-eastern Italy were suggested to be a representation of the night sky (Bernardini et al. 2022 Documenta Praehistorica XLIX). The patterns of the 29 marks are analyzed here to establish if they reproduce popular stellar asterisms. Nine marks are found to match the Tail of Scorpius and five the Orion's Belt, together with Rigel and Betelgeuse. Nine marks are found in the approximate position of the Pleiades showing some match with the cluster members. On the back side, 5 marks possibly reproduce Cassiopeia. One mark slightly North of Orion cannot be identified. The 28 marks show a Pearson correlation coefficient $$(28) with stellar positions higher than 0.99 with a probability $$ of a wrong correlation lower than 0.001. Departures are about one degree, or about 7 mm, as the mean diameter of the marks, which suggests a manufacturing limitation in the charting. The fort dates $$1800–400 BCE when Scorpius and Orion showed about the same orientation at the heliacal rising. The unidentified mark challenges the whole picture. We suggest it could have been the progenitor of a failed supernova, thus offering also the possibility of a verification.