In this work we introduced
a new proposal to study the gravitational lensing theory by spherical lenses,
starting from its surface mass density ∑(x) written
in terms of a decreasing function f of
a dimensionless coordinate x on
the lens plane. The main result is the use of the function f(x) to find directly the lens
properties, at the same time that the lens problem is described by a first
order differential equation which encodes all information about the lens. SIS
and NIS profiles are used as examples to find their functions f(x). Using the Poisson equation we find
that the deflection angle is directly proportional to f(x), and therefore the lens
equation can be written in terms of the function and the parameters of the
lens. The critical and caustic curves, as well as image formation and
magnification generated by the lens are analyzed. As an example of this method,
the properties of a lens modeled by a NFW profile are determined. Although the
puntual mass is spherically symmetric, its mass density is not continuous so
that its f(x) function is discussed
in Appendix 1.
Gravitational Lensing Strong Dark Matter1. Introduction
Gravitational lensing is one of the greatest achievements of General Relativity and is one of the most useful tools of galactic astronomy, not only because the distortion of background sources carries information from the mass distribution deflecting light (called lens), but also it provides a direct test of cosmological theories [1] -[4] .
The deflection angle of the light, as well as the image multiplicities [5] and its magnifications, depends on the properties of the lens. In fact, the position and shape of the source, and the matter distribution of the lens are unknown, so you can try to resolve the problem inverting positions and shapes of the images, for expample by the Kaiser & Squires method [6] ; or you can model the lens using known mass profiles, e.g. isolated mass (PM), non-singular isothermal sphere (NIS), non-singular isothermal ellipsoid (NIE), etc., depending on parameters to be adjusted so that the model reproduces the observed data [7] [8] ; the basis of these parametric methods rely on theoretical assumptions, which encourages us to study the properties of one of the most important families of mass models: the spherical mass distribution. Due to the symmetry of these profiles, the relation between the properties of the lens-source system and its observables is reduced to a one-dimensional equation, which pro- vides some important results from a general point of view of the theory, including image position, distortion and magnification.
Of course, due to the intrinsic ellipticity of a cluster or a galaxy, it is not physically possible to model such systems using a spherical profile. However, computer simulations suggest that the dark matter halo present in these systems can be described by a spherical mass distribution1 [9] , and in this sense, we shall describe our me- thod to the NFW profile.
For a basic and comprehensive reference on gravitational lensing see [1] [10] [11] .
2. Convergence and Lens Equation
Suppose a spherically symmetric mass profile lying at a distance2, acting as a gravitational lens on the light emitted by a source at a distance from us, and assume that the distance between lens and source is. The mass projection on the lens plane, called surface mass density, is obtained through
where is a dimensionless radius vector on the lens plane and the coordinate is perpendicular to it, that is to say, it is the line of sight coordinate. In this paper we suppose that
where and are monotonically decreasing functions because the surface mass density must de- scribe a realistic and localized lens model, and that functions are depending on the mass distribution of the lens, and defined on the interval. Worth noting that, or, and this can be accom- plished by assuming and. Since may be divergent at origin, we impose the condition
with this, the convergence is defined by
where depends on both, the distances which are functions of the cosmological model, and the physical pa- rameters of the lens mass distribution
Moreover, the Poisson equation relates the convergence and the deflection potential of the lens
which, for a spherically symmetric mass distribution can be expressed as
where denotes the derivative with respect to. The Poisson equation leads to the deflection angle from
thus, from Equation (1.7) can be found that
or, by Equation (1.4)
Now, since is a decreasing function, and since we write the function from to which an amount is subtracted, that is
this assumption is made in order to use the fundamental theorem of calculus in the integral expression of the deflection angle, Equation (1.10), so that
that is
The anterior result shows that for a spherically symmetric mass profile, whose surface mass density can be written in the form of Equation (1.2), the deflection angle is proportional to the function.
The lens equation, which relates the image and source positions, and respectively, for a spherically symmetric situation, is a scalar and takes the one dimensional form,
which can be written in terms of the function as
Joining the results given above, the function satisfies the following equation
which comes from inserting Equation (1.11) in Equation (1.4), according to the initial condition (1.3). Thus, the problem is reduced to solve the first-order ordinary differential Equation (1.16) for.
<img src="htmlimages\3-4500298x\caaaf482-d199-4aba-93cc-7212cfc0c06c.png" width="70" height="43.75" />for SIS and NIS Profiles
A spherical model widely used in the gravitational lensing theory is the singular isothermal sphere (SIS) [10] , whose convergence is given by
where is the one-dimensional velocity dispersion. With Equation (1.17) plugged into Equation (1.16) and Equation (1.3), one obtains the function for a SIS
with
The function is then for a SIS
To find the deflection angle, make the product with
One generalization of the SIS model is frequently used with a finite core, that is the non-singular iso- thermal sphere (NIS), which is more realistic for modeling galaxies. In this case, the convergence is given by
Through a procces similar to the SIS, we can found, and for the NIS profile
and
where is given by Equation (1.20).
3. Magnification and Shear
Since gravitational lensing conserves the surface brightness, the magnification of an image is defined as the ratio between the solid angles of the image and the source. Namely
from Equation (1.11) and Equation (1.15), this is
Equation (1.28) implies that the magnification has two singularities in and and there- fore its curve has two asymptotes at these points. In the next section we will see that those points in the lens plane for and are the critical points.
Noting that the magnification Equation (1.28) can be written in terms of the convergence and shear, which measures the distortion of images,
whereby
and from Equation (1.4), the shear is
This expression allows to calculate in a straightforward way. For example, returning to the models shown in 1.2.1, through Equation (1.19) and Equation (1.21), the shear generated by a SIS profile is
and that generated by a NIS profile
where we made use of Equation (1.24) and Equation (1.25).
Now, recognizing that
and
the shear can be written in terms of the deflection potential of a mass distribution with spherical symmetry, as
Here, the definition of the function shows again its usefulness, since the shear can be found in terms of the deflection potential without have recourse to the partial derivatives of it.
4. Critical and Caustics Curves
The critical curves are those points in the lens plane where the lens equation can not be inverted, or equi- valently, those points where the magnification is infinite, which satisfy
or
but, from Equation (1.4) and Equation (1.31)
and
Thus, the critical curves are the level contours of the and functions, and are found when
Equation (1.42) are not associated forming a system, thus, given it is possible found two solutions and if, or, meaning that is increasing, and, if, so that and therefore if, with and two arbitrary constants. In this way, the only condition for forming two critital circles is that is incresing, otherwise the lens produces only a single critical curve associated to, or any critical curve if. At the same time, the caustics curves are the corresponding locations in the source plane of the critical curves through the lens equation, and if we assume that the lens produces two critical curves, that is,
with on. Thus, caustic curves will be a point and a circle concentric with the lens.
5. Image Formation
In general, the image multiplicity depends on the source position with respect to the caustic circle, changing in two as the source crosses through it. Moreover caustics depends on the critical curves and on the increase or decrease of as seen in the previous section. Figure 1 shows the two basic sketches for the function for the two lens models mentioned in Section 2.1: the SIS and the NIS profiles, where we can see that although is decreasing, the product can be increased or constant, but this depends on the lens model as follows:
• If in and is increasing, does exist and the maximum number of images are three.
• If in and is decreasing, does not exist and the total number of images are two.
• If for, then is increasing and there is only one image.
In the case where the lens produces three images the source is inside the caustic circle, that is and let us call those images, , and, which together obey
or
with; and since is decreasing in,
Now suppose the source located in the first quadrant of a cartesian coordinate system in whose center lies the lens; namely with. The lens mapping (1.15), is, by components
Function f(x) for two lens models, the singular isothermal sphere and the Non-singular isothermal sphere (NIS). In the case of SIS xf(x) is constant, while f(x) is decreasing. In the NIS model xf(x) is increased, unlike f(x)
through Equation (1.47)
This implies
therefore, the images and will be in the third quadrant. And since the angle of is,
the angle of images will be
that is, the images and lies on the same line connecting the source and lens, but are diametrically op- posed to the latter one.
Meanwhile, the third image satisfies, as
and since, it is found, with. This implies that the angle of is equal to that of the source. The third image also lies in the same line between lens and source.
If the lens produces only two images, they are diametrically opposed lying on the line lens-source. And, if the lens produces only one image, it will be located at the same angle of the source.
6. Applying Theory to the NFW Profile
The lensing effects of the NFW profile have been widely studied [12] -[14] . The NFW profile describes the dis- tribution of a dark matter halo. The dark matter halo is useful to calculate the function in this model.
Suppose a gravitational lens modeled by a NFW profile [9] with a mass density given by
where the so called scale radius and are parameters of the halo.
6.1. NFW Convergence and Lens Equation
The NFW mass density Equation (1.54) expressed in terms of, where is a radius vector on the lens plane, and Equation (1.1) leads to the convergence through, Equation (1.5), to obtain
this expression is according with the results found in [15] and [12] . Here we have defined and
Equation (1.16) and Equation (1.55) leads to the differential equation
finding that
where we use Equation (1.3). Therefore
Figure 2 shows the functions and, which, as they should be, are decreasing in for and.
The deviation angle can be calculated through Equation (1.13)
where the constant is given by Equation (1.56). Now, it is straightforward that the lens equation for a mass distribution modelled by the NFW profile, reads as
The behavior of the lens equation is shown in Figure 3. There it can be seen that the local maxima and mi- nima of the lens equation depends on the parameters of the model, these points correspond to critical curves.
Figure 4 shows the lens equation in the case. Depending on the source position there are four po- sibilities of image formation, if:
• , there are infinite images (Einstein’s ring of radius), if
• , there are three images, the first within the circle of radius and second one outside of it, but within of that of radius, and third outside the circle of radius, but within that of radius, if
• , there are two images in and, and if
• , there is only one image outside of the circle of radius.
6.2. NFW Critical and Caustics Curves
In Figure 5 are displayed the local maxima of the lens equation as a function of. This values are the inverse functions and and therefore, they correspond to the radii of the critical circles, in fact, their maximum values are taken when, where, and. The radius of the caustic circle, also as a function of, is shown in the same plot.
6.3. NFW Shear, Image Positions and Magnification
From Equation (1.31), we find
Shear Equation (1.62) is a continuous and decreasing function over the range, as it must be since shear is a lensing effect that should be diminish as the distance to the lens increases. In fact
An horizontal line, , determines the critical curves when crossed with the functions, Equation (1.58), and, Equation (1.59). Note that in
Lens equation by a NFW model, Equation (1.61). As shown, the image positions will depend on the magnitud of the source and the parameters of the profile, rs, ρc and δk, represented by C, Equation (1.56). The local maxi- ma and minima (red and blue lines), corresponds to the radius of the critical circles
and
Figure 6 shows position of the images for different values of the source position. The change in position of the images is smaller as increases and decreases. The greater, and the lower source position, the position of the image tends to zero, the image tends to the inner critical circle and the image tends
Lens equation by a NFW model for C = 0.1. In xc1 ≠ 0, the function intercepts the horizontal axis and takes its minimum value. In xc2 the function takes its local maximum in. If the images will be located outside the circle of radius xr
Behavior of the points xc1 and xc2 where the lens equation takes its maximum values. Radius of the caustic circle associated to xc2, as a function of C. This curves were found numerically
to the outer critical circle.
At the same time, the magnification, given by Equation (1.28) though Equation (1.58) and Equation (59) is plotted in Figure 7 for two values and. There, we can see that the image is highly demag- nified when it approaches to zero, and the same occurs to the image when increases.
7. Conclusions
In this paper we introduce a new proposal to study the gravitational lens effect by a spherically symmetric mass distribution. The main result is the use of a new function which depends on the lens properties and the lens problem is described by the first order differential Equation (1.16) which encodes all information about lensing observables. If the surface mass density of the lens is continuous, this method leads to the deflection an- gle in a direct way by multiplying the function for the dimmensionless coordinate. We describe the critical and caustic curves through an equation that relates the function and the parameter, Equation (1.56), of the
Image positions of a point source as a function of C. The critical curves xc1 (blue) and xc2 (red) divides the source plane in three regions of image formation, that is, depending of the source position we can found up to 3 images. In agreement to Section (1.5), solid black lines represents the position of the first image (α), dotted lines, the second one (β), and the dashed lines, the third (γ), for each case of. The three images are associated as follows: each of the curves from left to right and under xc1 is associated with one curve from top to bottom above xc1. If the image position approaches to zero, i.e., then the Einstein Ring of radius xc1 is formed, and the images and go to zero, as we can see from the plot
Magnification of images in the lens plane for two values of C. The asymptotes will form in xc1 and xc2 (in each curve from right to left, respec- tively). Simulation of the image formation for a circular lens, magnification, critical and caustic curves generated by a lens modeled through a NFW profile is available online
lens which contains all the physical information of the lens and also is a function of the cosmological model.
The importance of the method described in this paper is that if you resolve Equation (1.16) for, then you can find the lens observables directly in terms of that function. This implies that you do not need to solve the Poisson equation to find the deflection potential, and this is an advantage.
In the case where the convergence is not a continuous function of the space, the differential Equation (1.16) can still be used to find the function, however, the deflection angle must be calculated through Equation (1.10). In Appendix 1 we explore this approach by the point mass lens.
We apply the method to a lens modelled by the NFW profile and found explicitly the function in this case. The critical and caustic curves, shear, magnification and the image formation are found for this model us- ing the formalism proposed in the first part of this paper.
Acknowledgements
R. Hurtado is grateful with Y. Villota for some helpful suggestions that improved the presentation of the paper and the Universidad Nacional de Colombia for financial support.
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, determines the critical curves when crossed with the functions
, Equation (1.58), and
, Equation (1.59). Note that in 
. If 
the images will be located outside the circle of radius xr
. The three images are associated as follows: each of the curves from left to right and under xc1 is associated with one curve from top to bottom above xc1. If the image position approaches to zero, i.e.
, then the Einstein Ring of radius xc1 is formed, and the images 
and 
go to zero, as we can see from the plot