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Introduction

Abstract

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Analysis

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Introduction

The purpose of this paper was to analyze the probability of lensing without multiple imaging occurring. Although it treated the subject in a generalized manner, the specific purpose was to explore one of the hypothesis concerning the four z~6 quasars in the Sloan Digital Sky Survey (SDSS). These four quasars seem to be incredibly luminous, with absolute bolometric magnitudes on the order of -27. This would seem to imply a super massive black hole (BH) of around 10⁹ M_Sol in the early universe which is at odds with simple non-zero lambda, cold dark matter (LCDM) cosmology. The following table shows the hypotheses that have been proposed and have countered that attempted to explain the apparent luminosity without seemingly breaking LCDM cosmology. Of those explored, the only actual presence of a super massive BH seems to be allowed.

Drawing 1Shows the variety of explanations of the z~6 quasars

This paper will explore the effects of asymmetry, isothermal, and Navarro-Frenk-White (NFW) profiles (primarily applicable to galaxy clusters and groups).

Computation Methods

Of primary importance was computing values for the functions A_sing(m) and A_mult(m) which are the cross sections for producing a magnification greater than m for systems where there is only one image or multiple undetectable images respectively. Both functions were numerically computed using the GRAVLENS software. For computing A_sing(m), first a circle was found analytically that enclosed all image magnifications greater than a desired value m_min which was typically 1.5. Then approximately 10⁶ points were chosen inside this circle, and the cross-section computed using:

Where x is a particular image, and m(x) is a particular image magnification.

Similarly, A_mult(m) was computed by choosing the smallest circle enclosing the possible images and then choosing approximately 10⁶ points inside this circle. Those that produced a single image were thrown out immediately. Those that were not where then further reduced by only allowing that met the “single-detectable-image-criterion” (SDIC) to remain. This was set based on the limits set by a recent HST study by Richards et al (2004). One set was a flux ratio of f > 0.01 (other images 100 times dimmer) and an angular separation of Dq > 0.''3; the other was a flux ratio of f>0.1 and Dq > 0.''1. The cross-section was then computing by integrating:

Where S(u) is a function that gives 1 if the image meets the SDIC and is multiply imaged; it is zero otherwise.

Isothermal Halos

Most early type galaxies exhibit a nearly isothermal profile from multiple sources of evidence. Although a spherical model is simple, this paper will examine an elliptical model that has a more complicated surface mass density function (the mass along the line of sight):

The variable q is defined as the ratio between the axes and ellipticity is defined as e = 1-q, and the coordinate system is centered on the lensing galaxy.

Another important effect is gravitational tidal shear which is produced by objects either near the lensing galaxy's halo, or projected along the line of sight. This effect is expected to be common (Keeton et all 1997, Holder & Schehter 2003). The associated potential is:

Where g is a dimensionless constant determining shear strength and q_g denotes the shear angle. Shears for galaxies are approximately g~0.05-0.1 and for clusters g~0.2-0.3.

Some Qualitative Results

Fig. 1 Isothermal ellipsoid with ellipticity e=0.5 (see text)

For both the effects of ellipticity and shear, magnification becomes infinite near e=.606 or g=1/3 from analytic models. However since a typical shear is g~0.1 and ellipticity is e~0.3, then m_max < 3.

To better understand effects of ellipticity and shear, the cross-section is made dimensionless by dividing by the square of the Einstein radius. The results are shown in figure 2. Shear and ellipticity are not mutually exclusive, so linear combinations are allowed. Averaging over a variety of combinations, this produces a modest increase in magnification.

Fig. 2Plot showing the results of various ellipticities and shears

Fig. 3Plot showing the multiply imaged magnifications for an isothermal ellipsoid with e=0.5

Several important effects occur. Firstly, the only quads that survive lie very close to the caustic (either the ellipsoid curve of the star shaped curve in figure 1), and have very closely spaced images. When f <0.1, the curve is smooth, but when f<0.01, the distribution splits into two populations: a low magnification and a high magnification one. However, the dominant source of magnifications greater than 5 are unresolvable small separation images produced by low mass halos; this is followed by multiple images where the others are too faint. True singly imaged systems are generally not important except when e>0.6.

Overall Results

Using models developed in other papers, and averaging over 1000 random combinations, the function A(m) could be defined for observational situations. The results show that the most large magnifications correspond to multiply imaged systems with unresolved extra images. Repeating this analysis with other models gave results of the same order of magnitude, showing this model was robust to changes in distributions.

NFW Halos

NFW halos are primarily important in cases where dark mater dominates the system. The projected surface mass is described as:

Where k_s= r_s r_s/S_crit, which is a dimensionless lensing strength parameter. Using a similar method as with isothermal halos, we find that multiply imaged systems have very small cross-sections. However, there is a very large cross-section for singly imaged magnified systems.

Fig. 4Source plane of NFW halo. Singly imaged points are outside the interior ellipse.

Realistic Halo Population

The model uses two different populations of halos: NFW halos corresponding to clusters and groups, and galaxies are modeled with isothermal halos. This is in line with current research. A third population is also considered: low mass “dwarf” halos with NFW profiles. These are essentially clumps of nonluminous matter that never collapsed to form galaxies. Masses are on the order of 10¹² M_Sol. This is summarized in the function f_SIS(M) which describes the fraction of halos with a mass M.

Using data from the Cosmic Lens All-Sky Survey, there are two possible fits, one with a sharp transition to clusters and one without. The two models give indistinguishable results , so the one with the sharp transition is used for simplicity. It is also assumed that the masses M_clus and M_dwr are redshift independent.

Important Results

• For magnifications greater than 1.8, the single imaged probability is dominated by clusters.

• Large magnifications m>5, that the object is most likely lensed by a cluster, should be easy to detect.

• For m<2, the lensing objects are most likely galaxies.

• Changing the mass of M_dwr effects the net probability less than 10%, since most probability comes from clusters. The primary effect of dwarfs is to reduce the number of isothermal halos around galaxies and hence reduce the multiply imaged probability.

• Multiply imaged magnification probably comes from isothermal halos. Without dwarfs, most of it comes from low mass galaxies

• Ellipticity and shear are approximately factor of 2 effects in no dwarf models and are order of magnitude effects in dwarf models.

• These results are robust. They change less than 10% when using other data sets for the halo mass function, the galaxy cluster transition location and the scatter in correlations for the NFW halo.

SDSS Quasars

The question that needs to be answered is: what is the probability for magnifications greater than 10 with the included SDICs versus the probability of magnifications greater than 10 without the SDICs? From the analysis done in this paper, the probability of one of the quasars being magnified m>10 without multiple images being resolved is at most 29%. Thus the probability that all four are magnified without seeing multiple images is .7%. Thus there is 99.3% confidence that the quasars are not all lensed. There still could be systemic errors in the finding of quasars by the SDSS. Since the SDSS selects by colors, these are unlikely to be effected by the lensing halos in question, and the selection does not account for angular size, it is unlikely the survey is missing the 25 to 45 other quasars multiply imaged at z~6 that the SDSS should be detecting. Thus, SDSS bias can be ruled out, and we can conclude that the SDSS quasars are not magnified by lensing.