A&A 418, 387-392 (2004)
DOI: 10.1051/0004-6361:20040049
D.-M. Chen
National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, PR China
Received 23 June 2003 / Accepted 15 January 2004
Abstract
In the standard flat cosmological constant ()
cold dark matter (CDM) cosmology, a model of two populations of
lens halos for strong gravitational lensing can reproduce the
results of the Jodrell-Bank VLA Astrometric Survey (JVAS) and the
Cosmic Lens All-Sky Survey (CLASS) radio survey. In such a model,
lensing probabilities are sensitive to three parameters: the
concentration parameter c_{1}, the cooling mass scale
and the value of the CDM power spectrum
normalization parameter .
The value ranges of these
parameters are constrained by various observations. However, we
found that predicted lensing probabilities are also quite
sensitive to the flux density (brightness) ratio
of the multiple lensing images, which has been a very important
selection criterion of a sample in any lensing survey experiments.
We re-examine the mentioned above model by considering the flux
ratio and galactic central Super Massive Black Holes (SMBHs), in
flat, low-density cosmological models with different cosmic
equations of state ,
and find that the predicted lensing
probabilities without considering
are
over-estimated. A low value of
can be compensated
for by raising the cooling mass scale
in fitting
the predicted lensing probabilities to JVAS/CLASS observations. To
determine the cosmic equation of state ,
the uncertainty
in
must be resolved. The effects of SMBHs cannot be
detected by strong gravitational lensing methods when
.
Key words: cosmology: theory - cosmology: gravitational lensing
Gravitational lensing provides us with a powerful probe of mass distribution of the universe. By comparing the lensing probabilities predicted by various cosmological models and the density profile of lenses with observations, we are able to test the mass distribution of dark matter halos and, in particular, the inner density slope because the Jodrell-Bank VLA Astrometric Survey (JVAS) and the Cosmic Lens All-Sky Survey (CLASS; Browne et al. 2000; Helbig 2000; Browne et al. 2002; Myers et al. 2002) has provided us with observed lensing probabilities at small image separations ( ).
In a standard flat CDM cosmology, it is believed that a model of two populations of lens halos for strong gravitational lensing can reproduce the results of the JVAS/CLASS radio survey (Porciani & Madau 2000; Kochanek & White 2001; Keeton 2001; Keeton & Madau 2001; Sarbu et al. 2001; Li & Ostriker 2002; Oguri 2002; Oguri 2003; Oguri et al. 2003). The two populations of lens halos are distinguished by introducing an abrupt change in the structure of the objects at the cooling mass scale dividing galaxies and clusters. Some authors used Singular Isothermal Sphere (SIS) models below and Navarro-Frenk-White (NFW) models above and could fit the observed image separation distribution of JVAS/CLASS (Li & Ostriker 2002; Sarbu et al. 2001). Based on the fact that the image separation distribution of lenses below 1'' depends sensitively on both the inner mass profile of galactic halos and the faint end slope of the mass and luminosity functions, Ma (2003) compared the traditional approach that models the lenses as SIS and the Schechter luminosity function with a dark matter based approach that models the lenses with a certain halo mass profile and the Press-Chechter mass function. Constraints on the inner total mass profiles of halos are investigated by requiring the two approaches to give consistent predictions. Li & Ostriker (2003) further proposed a model of three populations of halos as lenses distinguished by the halos mass to calculate the lensing probability of image separation and time delay.
In this paper, we revisit the model of the two populations of lens halos to calculate the lensing probabilities in a flat quintessence cold dark matter (QCDM) cosmology with different cosmic equations of state . The focus here is on the flux density ratio and cooling mass scale , and their effects on the estimate of the equation of state of dark energy . In spite of the fact that low mass lensing halos (galactic size or less) cannot all be SIS (Ma 2003; Li & Ostriker 2003; Benson et al. 2002; Somerville & Primack 1999; Kaufmann et al. 1993), we still model the galactic lens halos as SIS in our calculations based on the following considerations. On the one hand, evidence based on stellar dynamics of elliptical galaxies (e.g., Rix et al. 1997; Romanowsky & Kochanek 1999; Treu & Koopmans 2002), modeling of lensed systems (e.g., Cohn et al. 2001) and flux ratios of multiple images (Rusin & Ma 2001; Rusin 2002) all give an inner profile for lensing galaxies that is consistent with SIS. On the other hand, since the SIS lensing cross section is several orders of magnitude higher than that of the NFW, the lensing probabilities depend strongly on the cooling mass scale because a larger (and a smaller at the lower mass end of SIS lens halos, see Li & Ostriker 2003) allows more halos to be modeled as SIS. So the importance of the subdivision for low mass lens halos is lessened unless the large uncertainty of is sufficiently reduced. We thus ignore such a subdivision in this paper, and focus on the roles played by the flux density ratio and cooling mass scale in constraining the equation of state .
In addition to and , lensing probabilities estimated by the NFW halo are also sensitive to the concentration parameter c_{1}. We use the mass-dependent fitting formula given by Bullock et al. (2001). For a given halo mass and redshift, there is a scatter in c_{1} that is consistent with a log-normal distribution with standard deviation (Jing 2000; Bullock et al. 2001). Taking into account the scatter in c_{1} by averaging the lensing probability with the lognormal distribution will increase the probabilities considerably at larger image separations and only slightly at smaller separations (Chen 2003b). Since the well-defined sample of JVAS/CLASS is limited to small image separations ( ), we thus ignore the scatter in c_{1} and our conclusions will not be affected. Another major uncertainty in predicting lensing probability arises from the considerable uncertainty regarding the value of the CDM power spectrum normalization parameter (Chen 2003b); we adopt , suggested by the abundance of massive clusters at redshifts (Bahcall & Bode 2003) which gives . Lensing probability increases quickly with the source redshift ; since its distribution in the JVAS/CLASS survey is still poorly understood, we use the estimated mean value of (Marlow et al. 2000).
On the other hand, it should be pointed out that, like most authors, we use the spherical lensing profiles. Non-spherical lensing profiles do not change significantly the separation of observed multiple images, unless the profile deviates significantly from the spherical one. However, as pointed out by many authors (e.g. in a cluster environment, this has been discussed by Meneghetti et al. 2003a,b; for the importance of substructures see Mao & Schneider 1998), the elliptical lenses and their substructures do have significantly larger cross sections. A more realistic lensing model should include these effects, which will be discussed in another paper.
So in this paper we use the model of two populations of lens halos mentioned above to investigate the effect of image flux density ratio on lensing probabilities and find that the predicted lensing probabilities without considering are over-estimated. In flat, low-density cosmological models with different cosmic equations of state , we show that when the flux density ratio changes from 10 to 10^{4}, the corresponding fit value of the cosmic equation of state will change from to . Also, in our flux-limited statistics of strong lensing image separations, the contributions from galactic central SMBHs can be safely ignored.
The paper is organized as follows: Sect. 2 provides a brief description of the cosmology model used in this paper, Sect. 3 presents the predicted integral lensing probabilities, Sect. 4 gives our results, the discussion and comparisons with previous work. Appendix A is devoted to a detailed deduction of lensing equations for galactic halos.
In this section, we describe the cosmological model and dark halo
mass function. More and more evidence shows that the Universe at
present is dominated by a smooth component with negative pressure,
the so called dark energy. So far, two kinds of dark energy have
been proposed, quintessence and the so-called Chaplygin gas (e.g.,
Bento et al. 2002, 2003). We consider
only the quintessence in this paper. It is assumed that
quintessence offers an alternative to the cosmological constant as
the missing energy in a spatially flat universe with a subcritical
matter density
(Caldwell et al. 1998). In this paper, we study spatially flat
QCDM models in which the cold dark matter and quintessence-field
make up the critical density (i.e.,
). Throughout the paper, we
choose the most generally accepted values of the parameters for
flat, QCDM cosmology, in which, with the usual symbols, the matter
density parameter, dark energy density parameter and Hubble
constant are respectively:
,
,
h=0.75. Four negative values of
in equation of state
,
with (cosmological constant),
,
and
,
are chosen to see their effects on lensing
probabilities. We use the conventional form to express the
redshift z-dependent linear power spectrum for the matter
density perturbation in a QCDM cosmology established by Ma et al.
(1999)
(2) |
(3) |
= | |||
s | = | ||
(4) |
Similarly, the linear growth suppression factor of the density
field
for QCDM in Eq. (1) is
related to that for CDM
with
= | |||
t | = | ||
(5) |
We know that most of the consequences of quintessence follow from its effect on evolution of the expansion rate of the Universe, which in turn affects the growth of density perturbations (as described above) and the cosmological distances. From the first Friedmann equation d (a=1/(1+z) is the scale factor of the Universe), the expansion rate (Hubble constant) for a flat Universe and then the proper distance and the angular-diameter distance can be calculated in our QCDM models (Huterer & Turner 2001), which are needed in predicting lensing probabilities.
The physical number density of virialized dark halos of masses between M and is related to the comoving number density n(M,z) by , the latter originally given by Press & Schechter (1974), and the improved version is , where is the current mean mass density of the universe, and f(M,z) is the mass function for which we use the expression given by Jenkins et al. (2001).
In this section, we first give a brief description of lensing equations for galaxies and clusters of galaxies, then present the predicted integral lensing probabilities. According to the model of two populations of halos, cluster-size halos are modeled as NFW profile: , where and are constants. We can define the mass of a halo to be the mass within the virial radius of the halo : , where , and is the concentration parameter, for which we have used the fitting formula given by Bullock et al. (2001). The lensing equation for NFW lenses is as usual (Li & Ostriker 2002; Chen 2003a,b), where , is the position vector in the source plane, in which and are angular-diameter distances from the observer to the source and to the lens respectively. and , is the position vector in the lens plane. Since the surface mass density is circularly symmetric, we can extend both x and y to their opposite values in our actual calculations for convenience. The parameter is x independent, in which is the critical surface mass density, with c the speed of light, G the gravitational constant and the angular-diameter distance from the lens to the source. The function g(x) has an analytical expression originally given by Bartelmann (1996).
Since the observational evidence presented so far suggests the ubiquity of black holes in the nuclei of all bright galaxies regardless of their activity (Magorrian et al. 1998; Ferrarese & Merritt 2000; Ravindranath et al. 2001; Merritt & Ferrarese 2001a,b,c; Wandel 2002; Sarzi et al. 2002), we add a central supermassive black hole (SMBH) as a point mass to each SIS modeled galactic lens halo in its center(Keeton 2002; Chen 2003a,b). It is well known that the finite flux density ratio will definitely reduce the value of the lensing cross section (Schneider et al. 1992). In the model of one population of halos (NFW) combined with each galactic halo a central point mass, the lensing probabilities are shown to be sensitive to the flux density ratio (Chen 2003a,b). One may argue that this is the case only because we treat the galactic bulge as a point mass (some fraction of the bulge mass), since a point mass lens will always produce a larger flux density ratio of the multiple images. So it would be interesting to investigate the model of two populations of halos to see whether or not the lensing probabilities will be sensitive to . We will show later that the answer is yes, and the contribution from an individual central galactic SMBH is detectable only if . When (for the well-defined sample of JVAS/CLASS), the effect of SMBH can be safely ignored.
So, for galaxy-size lenses, we use an SIS+SMBH model. The surface
mass density is
When the quasars at the mean redshift
are
lensed by foreground CDM halos of galaxies and clusters of
galaxies, the lensing probability with image separations larger
than
and flux density ratio less than
is (Schneider et al. 1992)
The cross section for the cluster-size NFW lenses has been well studied (Li & Ostriker 2002). The lensing equation is and the multiple images can be produced only if , where is the maximum value of y when x<0, which is determined by and the cross section in the lens plane is simply . We point out that the flux density ratio would also result in a reduction of the value of the cross section for NFW lenses if we apply it in our calculations. However, since we model the cluster-size lens halos as an NFW profile, which contributes to lensing probabilities mainly at larger image separations, where no confirmed lensing events are found (i.e. the null results of the JVAS/CLAS for ), if we adopt the mentioned above usual form of an NFW lens cross section, the predicted lensing probabilities will not be severely affected at smaller image separations.
For galaxy-size SIS+SMBH modeled lenses, two images will always be
produced for all values of the source positions ,
and
larger
will produce a higher flux density ratio .
So a finite value of
will
definitely limit the source within a certain corresponding
position (which is also denoted by
)
and hence
reduce the value of the cross section. The flux density ratio
for the two images is the ratio of the
corresponding absolute values of magnifications (Schneider et al. 1992; Wu 1996),
,
where
(8) |
Figure 1: Image magnifications. Solid line: the brighter images (, x>0), dashed line: the fainter images (, x<0). The left vertical line (dot-dashed) indicates the source position at which , and the right vertical line (dotted) indicates the position for . | |
Open with DEXTER |
We plot the magnifications (each as a function of source position y) both for the brighter images and the fainter images in
Fig. 1, with the source positions at which
and
explicitly indicated,
and we have used a typical value
.
The
sensitivity of the source position to the flux density ratio is
obvious. The cross section for images with a separation greater
than
and a flux density ratio less than
in the lens plane is (Schneider et al. 1992; Chen 2003b)
The lensing probabilities predicted by Eq. (7) and calculated from the combined JVAS/CLASS data are compared in Fig. 2. Four cases with different flux density ratios or cooling scale or with and without central SMBH are investigated to show their effects on lensing probabilities. In each case (see the corresponding panel of Fig. 2), four different values of the cosmic equation of state : (cosmological constant), , and , are chosen to show the constraints from lensing probabilities. As in our previous work (Chen 2003b), the observed lensing probabilities are calculated by , where N(> is the number of lenses with separation greater than in 13 lenses.
Figure 2: The integral lensing probabilities with image separations larger than and flux density ratio less than , for quasars at mean redshift lensed by NFW ( ) and SIS+SMBH ( ) halos. The typically selected values of the cosmic equation of state for each curve, the flux density ratio and cooling scale are explicitly indicated. The observed lensing probabilities of JVAS/CLASS are plotted as a histogram in each panel. | |
Open with DEXTER |
When flux density (as allowed by the JVAS/CLASS well-defined sample) and , one can see from the panel (a) of Fig. 2 that none of the four values of is able to predict enough lensing probabilities to match the observations. Note that the value of the cooling mass scale used in this case is preferred by Kochanek & White (2001) and Li & Ostriker (2002), and is also close to the cutoff mass of halos below which cooling of the corresponding baryonic component will lead to concentration of the baryons to the inner parts of the mass profile (Rees & Ostriker 1977; Blumenthal et al. 1986; Porciani &Madau 2000). So a larger value of (> ) is needed for predictions to match observations. When we attribute too few predicted probabilities in panel (a) of Fig. 2 to the small value of (=10), the quite large value of (=10^{4}) also fails to match the observations, although the increment of the probabilities is also obvious, as shown in panel (c). The reason is that we have used different parameters and functions, e.g., the sensitive concentration parameter c_{1}. Although it is not explicitly displayed in the figure, our calculations indeed show that if we set and , the predicted lensing probabilities for is able to match the observations quite well, which is in agreement with the result obtained by Sarbu et al. (2001), since we have used almost the same parameters and mass (and halo) functions. Note that when is quite large (>10^{3}), the contribution from the central galactic SMBH cannot be ignored. In fact, it is just this contribution that compensates the finite value effect of , which make our results obtained in this case be the same as that of Sarbu et al. (2001), whose flux density value is infinite.
If we persist in the flux density ratio allowed by the JVAS/CLASS sample in our calculations, the cooling mass scale must have the value of for the model with to be able to match the observations, as shown in panel (b). Such a value of is close to that used by Porciani & Madau (2000), however, they used different parameters and, especially, they did not consider the finite flux density ratio effect. Note that when , the contributions from a galactic central SMBH can be ignored, no matter what the value of is, as shown in the top two panels of Fig. 2. This means that the finite small flux density ratio will reduce the cross section of SIS lenses considerably without a small central point mass. On the other hand, our calculations shows that the central SMBH can be detected only if , this value is possible for some confirmed radio-loud lens systems (Rusin & Ma 2001), but no sample suitable for analysis of the lens statistics is available with such a high flux density ratio. The sensitivity of lensing probability to the flux density ratio is obvious when we compare panel (b) with panel (d) of Fig. 2. In the latter, , the effect of which is close to that when flux density ratio is assumed to be infinite (i.e., no constraints on the flux density ratio are taken into account, as in most previous work; see Fig. 1), and other parameters, including , are the same as panel (b). Obviously, for , the "right" cosmic equation of state to match observations is . The present time deceleration parameter for a flat, dark energy dominated Universe is , and the accelerating Universe requires q_{0}<0, which immediately gives (we are using ). So the fit value of in panel (d) is still within the range required by the accelerating Universe. Clearly, when flux density ratio changes from 10 to 10^{4}, the corresponding fit value of the cosmic equation of state will change from to .
In summary, we revisit the two populations of lens halo model with mass distribution NFW ( ) and SIS+SMBH ( ), to calculate lensing probabilities in flat, low-density cosmological models with different cosmic equations of state . The finite flux density ratio effect is significant. A low value of can be compensated for by raising the cooling mass scale in fitting the predicted lensing probabilities to JVAS/CLASS observations. To determine the cosmic equation of state , the uncertainty in must be resolved. The contributions from galactic central SMBHs can be safely ignored.
Acknowledgements
I'm grateful to the anonymous referee for helpful suggestions. This work was supported by the National Natural Science Foundation of China under grant No.10233040.
In this appendix, we give a detailed deduction of the lensing
equation for SIS+SMBH model. Choose the length scales in the lens
plane and the source plane to be
(A.1) |
(A.2) |
y=x-m(x)/x, | (A.3) |
= | |||
(A.7) |
= | |||
= | (A.8) |