Gravitational lensing of fast radio bursts: prospects for probing microlens populations in lensing galaxies

Abstract

Gravitational lensing by a stellar microlens of mass $M$ forms two images separated by micro-arcseconds on the sky and has a time delay of $2\times10^{-5}(M/{\rm M_\odot})$ seconds. Although we cannot resolve such micro-images in the sky, they could be resolved in time if the source is a fast radio burst (FRB). In this work, we study the magnification ($|μ|$) and time delay~($t_d$) distributions of micro-images led by different microlens populations. We find that, in microlensing of typical strongly lensed (macro-)images in galaxy lenses, micro-images stemmed from a population of stellar mass microlenses in the $[0.08, 1.5]\:{\rm M_\odot}$ range and a second (dark) microlens population in $[10^{-3} - 10^{-2}]\:{\rm M_\odot}$ range reside in different parts of $|μ|-t_d$ plane. For the global minimum macro-image, due to low stellar mass density, we find that the stellar population leads to peaks in autocorrelation at ${>}10^{-6}$ seconds, whereas the secondary population leads to peaks at ${<}10^{-6}$ seconds, allowing us to differentiate different microlens populations. However, an increase in stellar density introduces new peaks at ${<}10^{-6}$ seconds, which can pollute the inference about the presence of multiple microlens populations. In addition, we also show that the number of micro-images, hence the number of peaks in the autocorrelation, is also sensitive to the underlying stellar mass function, allowing us to constrain the stellar initial mass function (IMF) with FRB microlesning in the future. This work is a first step towards using FRB lensing to probe the microlens population within strong lenses, and more detailed studies are required to assess the effect of various uncertainties that we only discussed qualitatively.

Figures (6)

• Figure 1: Time delay ($t_d$) vs. absolute magnification ($|\mu|$) for the saddle point image formed by an isolated point mass lens. The different curves correspond to different mass values, which are shown below each curve. Each curve is color-coded according to the source position ($y$). The horizontal dashed black line corresponds to $|\mu|=1$.
• Figure 2: Left panel. Time delay ($t_d$) vs. absolute magnification ($|\mu|$) distribution of micro-images corresponding to mock populations of microlenses in the absence of external effects. The black points (which are covered up by other colored points) correspond to a stellar microlens population drawn using the Salpeter mass function assuming $\tau=10^{-2}$. The gray shaded region covers a range of $|\mu|-t_d$ values for an isolated microlens mass range of $[0.08, 1.5]~{\rm M_\odot}$. To generate green (yellow) points, we have added another microlens population with a constant mass of $10^{-2}~{\rm M_\odot}$ ( $10^{-3}~{\rm M_\odot})$ with $\tau=10^{-2}$. The green and yellow solid curves represent the $|\mu|-t_d$ values for an isolated microlens with $M=10^{-2}~{\rm M_\odot}$ and $M=10^{-3}~{\rm M_\odot}$, respectively. The horizontal black dashed line corresponds to $|\mu|=1$. Right panel. Average histogram of time delays ($\Delta t_{d,jk}$) between pairs of micro-images shown in the left panel, weighted by the corresponding geometric mean of magnifications, i.e., $\sqrt{|\mu_j \mu_k|}$. The solid and dashed histograms are made using micro-images with $|\mu|\geq10^{-3}$ and $|\mu|\geq10^{-1}$, respectively.
• Figure 3: Mock galaxy scale strong lens system assuming SIE mass density profile with $(\sigma, \epsilon)=(250\:{\rm km\:s^{-1}}, 0.3)$. Black and blue curves represent the critical curve and caustic, respectively. The red star marks the source position, and the black stars show the corresponding image positions. The green dashed curves represent the arrival time delay contours corresponding to saddle point images.
• Figure 4: Left column. Time delay ($t_d$) vs. absolute magnification ($|\mu|$) distribution of micro-images for the first three macro-images in our mock strong lens system. The macro-images are labelled according to arrival time (see Table \ref{['tab:sl_system']} for more details). In each panel, the black points (which are covered up by other colored points) correspond to a stellar microlens population drawn using the Salpeter mass function with the surface density given in Table \ref{['tab:sl_system']}. To generate green (yellow) points, for each image, we have added a second microlens population equivalent to 1% of total convergence of constant mass, $10^{-2}{\rm M_\odot}$ ($10^{-3}~{\rm M_\odot}$). The horizontal black dashed line corresponds to $|\mu|=1$. Right column. Average histogram of time delays ($\Delta t_{d,jk}$) between pairs of micro-images shown in the corresponding left panels, weighted by the corresponding geometric mean of magnifications, i.e., $\sqrt{|\mu_j \mu_k|}$. The solid and dashed histograms only include micro-images with $|\mu|\geq10^{-3}$ and $|\mu|\geq10^{-1}$, respectively.
• Figure 5: Left column. Time delay ($t_d$) vs. absolute magnification ($|\mu|$) distribution of micro-images corresponding to different stellar mass functions for Image-1/2 in Table \ref{['tab:sl_system']}. For each macro-image, in the bottom scatter plot, the black points correspond to the Salpeter mass function, whereas the blue points are for the Kroupa mass function. The gray shaded region covers the range of $|\mu|-t_d$ values for an isolated microlens mass in $[0.08, 1.5]~{\rm M_\odot}$ range. The horizontal black dashed line corresponds to $|\mu|=1$. Right column. Average histogram of time delays ($\Delta t_{d,jk}$) between pairs of micro-images shown in the corresponding left panels, weighted by the corresponding geometric mean of magnifications, i.e., $\sqrt{|\mu_j \mu_k|}$. The solid and dashed histograms include micro-images with $|\mu|\geq10^{-3}$ and $|\mu|\geq10^{-1}$, respectively.