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2023 ° 22(06) ° 0631-7361
https://www.wikipt.org/tphysicsletters
DOI: 10/1490/687361tpl
We thank Douglas Clowe, Jérémie Francfort, Ruth Durrer, and Han-Gil Choi for helpful discussions. The work is supported by the National Research Foundation of Korea (NRF) [NRF- 2019R1C1C1005073 and NRF-2021R1A4A2001897 (YY, JCP), NRF-2021R1A2C1094577 (HSH)] and by IBS under the project code, IBS-R018-D1 (YY, JCP).
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Gravitational lensing studies of the Bullet Cluster suggested convincingly in favour of the existence of dark matter. However, it was performed without the knowledge of the original orientation of each galaxy before gravitational lensing. A potential improvement to this issue lies in the measurement of the original orientation from the polarization direction of radio waves emitted from each galaxy. In this context, Francfort et al. derived a formula that can utilize the information about the original orientation of each galaxy to obtain what is called shear. However, we demonstrate that shear in their formula should be replaced by reduced shear when the change in sizes of images of galaxies is taken into account. As the previous gravitational lensing analysis of the Bullet Cluster used reduced shear, we suggest applying our improved formula directly for the reanalysis once we obtain the polarization direction of radio waves. In particular, we show that our new formula can yield a more accurate analysis than the previous one, if the polarization direction can be measured more precisely than 10◦.
There are many observational results that favor the existence of dark matter. One of the most convincing results is the gravitational lensing analysis of the Bullet Cluster [1]; matter present in the Bullet Cluster, be it baryonic matter or dark matter, distorts the images of galaxies behind the Bullet Cluster, by its gravitation. The authors of Ref. [1] analyzed such images to reconstruct the mass distribution at the Bullet Cluster, which did not coincide with the baryonic matter distribution obtained by X-ray image. Thus, they concluded that dark matter is responsible for the discrepancy. In order to analyze the gravitational lensing effect, certain assumptions about the original images are necessary since the observed images of galaxies alone cannot determine the distortion. In Ref. [1], it is assumed that the average orientation of the original galactic images in each small patch of sky, where variables related to gravitational lensing are determined, is zero. However, this can lead to errors if there are not enough galaxies in each patch. Although this represents the optimal approach based on the currently available observational data, the analysis requires a sufficient number of galaxies to statistically determine the gravitational lensing effect in each patch. Otherwise, accidental skewing of the original galactic orientations could lead to skewed results. However, it is now possible to determine the original orientation of galaxies from the polarization of the radio waves from each galaxy. The radio emission from each galaxy is known to have a polarization that is perpendicular to the major axis of its ellipticity [2, 3]. While the orientation of a galactic image is rotated by gravitation, polarization is not. Therefore, even if the average original galactic orientations were distorted in a certain direction by accident, possibly due to the small number of galaxies in each small patch of sky, it would not bias the data, as long as we know the original orientation and therefore are able to compensate it. Thus, we can use this information to our advantage to measure the lensing effect more accurately, as pointed out in Refs. [3–5]. Therefore, the position of dark matter at the Bullet Cluster may be corrected if we reanalyze the gravitational lensing effect with the help of the polarization data of radio waves, which would be available in the future [6–8].
Let’s estimate σg, the error of 1D g for the new method. We closely follow Ref. [3] where an error estimate for a mathematically similar but contently different case was considered. We will use the following notation:
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