Bias in apparent dispersion measure due to de-magnification of plasma lensing on background radio sources

Propagation of light in the universe can be affected by several phenomena, e.g. gravitational field generated by massive objects. An example of such phenomena is gravitational lensing, which provides a powerful tool in studying matter distributions throughout the Universe (e.g. Schneider et al. 2006). In addition to gravitational lensing, inhomogeneous distributions of free electrons can cause deflection of light rays as well, called plasma lensing (e.g. Clegg et al. 1998; Tuntsov et al. 2016; Cordes et al. 2017). Plasma lensing shares several similar features with gravitational lensing, especially in terms of the mathematical description. However, different features introduced by plasma lensing (such as wavelength dependence) become significant only at low frequency, e.g. usually in the radio band. Moreover, the deflection caused by plasma lensing is opposite to the deflection due to gravitational lensing, which makes an over-dense clump of plasma to act as a diverging lens. Observations of plasma lensing began with the discovery of wild changes in the flux density of compact radio sources (Fiedler et al. 1987), a phenomenon known as Extreme Scattering Events (ESEs). Although a detailed physical picture of the phenomenon remains elusive, it is believed that wavelength-dependent refraction due to plasma along the line of sight in an ESE cannot be neglected. Since the introduction of the Gaussian plasma lens model, it has been widely used to describe the phenomenon (Romani et al. 1987; Clegg et al. 1998).

Millisecond duration pulses, known as fast radio bursts (Lorimer et al. 2007; Cordes & Chatterjee 2019; Petroff et al. 2019), are dispersed during propagating in the ionised gas of the interstellar and intergalactic media. Several observational aspects of FRBs are modulated by plasma lensing, such as the dispersion relation and amplitude. The cosmological origin of FRBs provide a potential probe to study the total integrated column density of free electrons along the line of sight. Studies of the baryon fraction in the intergalactic medium using FRBs with redshift have been recently introduced in the literature (e.g. Macquart et al. 2020; James et al. 2021). However, the frequency-time delay relation (i.e. the Dispersion Measure, DM), is not an exact estimate of the projected electron density (e.g. Kulkarni 2020; Er et al. 2020; Ocker et al. 2021). Moreover, it has been noted that plasma lensing is responsible for both magnification and strong de-magnification of background sources from both observations (e.g. Clegg et al. 1998) and analytical models (e.g. Tuntsov et al. 2016; Er & Rogers 2018). The diverging properties of plasma lensing deflects radio signals that would otherwise be received by us without a plasma clump along the line of sight. Therefore, the probability of detecting background sources is modified by the existence of plasma lenses. The distribution of the observed background sources will not be uniform, and will always appear biased toward the low density region of the plasma clumps. With such a biased tracer, one expects another systematic error in the estimation of the electron density in addition to all the other uncertainties already mentioned. In this study, we perform a toy simulation to evaluate the effect of such a systematic density bias, neglecting all the other uncertainties. We briefly summarise the basics of plasma lensing in Sect. 2 and discuss the lensing cross section for magnification and de-magnification in Sect. 3. A toy simulation to evaluate the bias in estimating the electron density is presented in Sect. 4. Finally we discuss our results in Sect. 5. We adopt the standard $\Lambda$CDM cosmology with parameters based on the results from the Planck 2018 (Planck Collaboration et al. 2020): $\Omega_{\Lambda}=0.6847$, $\Omega_{m}=0.315$, and Hubble constant $H_{0}=100h$ km s^-1 Mpc^-1 with $h=0.674$.

The description of plasma lensing follows from gravitational lensing, e.g. Schneider et al. (2006). One can also find details from, e.g., Tuntsov et al. (2016); Cordes et al. (2017); Er & Rogers (2018). The lens is approximated as thin and the deflection angles are assumed to be small. We consider a source at angular position $\beta$ on the source plane with respect to the line of sight. The corresponding image of this source is then formed at the angular position $\theta$ on the observer’s sky. For simplicity we only consider axisymmetric lens models. The lens equation can be thus related by the deflection angle $\alpha$

where $\psi$ is the effective lens potential, and $\nabla_{\theta}$ is the gradient on the image plane. In plasma lensing, the effective potential is determined by the projected electron density $N_{\rm e}(\theta)$, and is given by

where $D_{ds}$, $D_{d}$, and $D_{s}$ are the angular diameter distance from lens to source, from lens to observer, and from observer to source respectively. $r_{e}$ is the classical electron radius, and the wavelength of the signal is $\lambda$. In principle, the observed wavelength is redshifted due to the expansion of the universe, i.e. $\lambda^{\rm obs}=(1+z)\lambda$. Since we only consider lenses at low redshifts, such an effect is tiny and will be neglected. In this work, we mainly focus on point-like sources, such as pulsars or FRBs. Due to the weakness of the lensing effects, image distortions cannot be observed directly. Moreover, the phases of radio signals are not included in the current study, which may affect observations of coherent sources. Thus, the magnification and frequency-dependent time-delay are the main observables. Plasma lensing usually produces behaviour analogous to a diverging lens, and causes significant de-magnification, i.e. $|\mu|\ll 1$. Magnification can be calculated from the Jacobian $A$ of the lens equation (1), such that $\mu^{-1}={\rm det(A)}$. In the axi-symmetric lens, the magnification simplifies,

Two analytical toy models are adopted to describe the number density of electrons. The first is the widely used Gaussian model (e.g. Clegg et al. 1998; Cordes et al. 2017), which can also serve as a building block for constructing more complicated density profiles. The second profile we consider is a power-law model, of which the lensing properties can be calculated analytically as well. Moreover, the maximum density gradient of the power-law lens model, which depends on the power-law index, can be much higher than that of a Gaussian profile even with a low average density. It is thus interesting to explore both profiles as they can be adopted for different scenarios. A Gaussian random field model provides a good description of the Kolmogorov spectrum of electron densities (e.g. Cordes et al. 2016), which will be studied in a future work.

Plasma lensing generates both magnification and de-magnification that affect background sources. It is interesting to compare the lensing cross section of the two effects. In plasma lensing, the weak deflection causes tiny separations between multiple images. There is only some evidence of resolved multiple images from observations with high resolution, (e.g. Backer et al. 2000; Brisken et al. 2010). We thus calculate the total magnification $\mu_{T}$ on the source plane, which is given by the sum of all the image magnifications produced by the lens

where $i$ is the index of a particular image. Such a total magnification does not contain the phase information of the images, and thus may give slightly different estimates for coherent sources. We define the area of magnification ($\mu_{T}>1.05$) or de-magnification ($\mu_{T}<0.95$) region on the source plane as, $A_{\rm mag}$ and $A_{\rm demag}$.

First, we present the magnification curves for the Gaussian and SPL models with different ratios in Fig. 3. As we see, the parameter ratio ($\theta_{0}/\sigma$ for Gaussian lens, and $\theta_{0}/\theta_{c}$ for SPL lens) fully determines the magnification properties of the lens. And it is the same to the lensing cross section. For both profiles, the de-magnifications of the sub-critical lenses are mild. Once they become critical, they can generate a clear forbidden region. The cross section of magnification increases with the ratio and is mainly between the two caustics. In Fig. 4, we show the lensing cross section as a function of the parameter ratio. For the Gaussian model, the rate of increase of the de-magnification becomes slightly slower after $\theta_{\rm crit}$, while for a SPL model, such a change does not appear. The magnification curve is somewhat complicated. A small transition appears at $\theta_{\rm crit}$ as well. The dramatic change appears after $\theta_{0}/\sigma\sim 2.1$, where the outer caustic moves outward rapidly. When the ratio becomes sufficiently large, i.e. $\theta_{0}/\sigma\sim 3.7$, the probability of magnification decreases. The reason is that in some area between the two peaks, the magnification is below 1.05 and will be excluded, e.g. see the red curve in Fig. 3. However, for a lens with high $\theta_{0}/\sigma$, the high probability of magnification does not suggest lensing strength as one can see that most of the magnification in the cross section is only slightly larger than unity.

For the SPL lens, the probability of both effects is similar as that of Gaussian lens, i.e. increases with the ratio as well. The slope of magnification curve is larger between $\theta_{0}/\theta_{c}\sim[2.6,3.1]$. After that, there is a small decrease. It is the same as that for the Gaussian model, in some area between the two peaks, the magnification is below 1.05. In order to compare the lensing efficiency of the two profiles, we show the corresponding parameter ratios of the other profile using Eq. 18 at the top x-axis in each panel of Fig. 4.

It has been proposed to use radio pulsing sources to trace the ionised baryons throughout the universe. The relation of the Dispersion Measure to electron density, however, relies on several assumptions and approximations (e.g. Kulkarni 2020). For example, the contribution from ions is not included; the temperature of the plasma is assumed to be low, and the effect of magnetic field in plasma is not taken into account. Moreover, it is necessary to assume a uniform spatial distribution of the background radio sources. In gravitational lensing, the cosmic magnification effect can be used to study the matter distribution on large scales, but the inhomogeneous distribution of background sources has to be taken into account, (e.g. Zhang & Pen 2005). A significant difference of plasma lensing is that it can cause strong de-magnification to the source (even for the brightest primary image), which therefore cannot be observed at all. Thus, the information we extract from observations of radio sources will be in principle incomplete, a distinct conclusion compared to gravitational lensing. Such a “selection effect” can lead to a systematic bias in the estimate of the electron density.

We perform a toy simulation to evaluate the magnification effect of plasma lensing. For simplicity, we adopt the redshift of a repeating FRB ($z_{s}\approx 0.2$, Spitler et al. 2016) for all of our sources. Additionally, we place all the plasma clumps in the Milky Way in order to avoid the redshift effect to the estimate of electron density (Ioka 2003; Inoue 2004). The observation frequency we adopted is $\nu=1$ GHz. A realistic model of the electron density on small scales is difficult to obtain, either from observations or simulations. We thus adopt a simple model and restrict the electron density in the range consistent with observations (e.g. Cordes & Lazio 2002, 2003; Yao et al. 2017).

An axisymmetric Gaussian density model is used for all the plasma clumps in the tests in which $n_{l}$ lenses are randomly and uniformly placed in a field of $1\times 1$ arcmin². The lens width $\sigma$ follows a log-normal distribution. The central density of the lens is sampled by two constraints: i) for each $\sigma$ the ratio $\theta_{0}/\sigma$ follows a log-normal distribution, the mean value of the distribution is $0.05$ if $\sigma$ is greater than 1 arcsec, and is $0.25$ if $\sigma$ is smaller than 1 arcsec, ii) $\theta_{0}<1$ arcsec to avoid an extremely large electron density (the corresponding $N_{0}$ is about $2600$ pc cm^-3).

In Fig. 5, we present the distributions of $N_{0}$, $\sigma$ and $\theta_{0}/\sigma$ of our mock lens samples. Most of the lenses have small density and size. We calculate the magnification analytically or using finite difference. The plasma lens potential is calculated from the stacked electron density of all the lenses on a mesh with resolution of $0.24$ arcsec. The lensing properties, such as magnification can be performed by using the method in Bradač et al. (2005). We regard the region where $\mu<0.25$ as “unobservable” on the lens plane, i.e. the deflection caused by the lens is not taken into account. The threshold of being unobservable depends on the condition of observations. In reality, $\mu\sim 0.25$ is only a rough estimate of the forbidden regions. One can see from the figures of magnification curves (Fig. 3) that once the lens becomes critical, the magnification inside the caustic is extremely small around the lens centre. We also perform additional tests to calculate regions for $\mu<0.1$. The area of the unobservable regions decrease by only $10-20\%$, which is mainly caused by the sub-critical lenses. This will not change our conclusions substantially, but warrants a more detailed study with observational constraints in the future.

We mask out the unobservable regions on the lens plane, and obtain results to compare with those using all the simulated data. In Fig. 6, we present one example of the density map of $N_{\rm e}$. One is the initial simulated map of the electron density, the other is the map we mark out the exclusion regions, which is shown by red colour. Most of the exclusion regions are located at high density region. But not all the high density regions are strongly de-magnified, since the gradient of density is another important factor as well.

We compare the average electron density over the whole field (label as true) and over the field without the exclusion regions (label as estimated). 500 realisations are simulated to show the difference in Fig. 7. The distributions with or without the exclusion regions show difference, which becomes significant with the increase of the number of lenses. With a higher electron density, the cross section of the exclusion region grows, and the probability of detecting a lower electron density becomes greater. We perform comparisons for different number of lenses $n_{l}$. A probability of a background source falling in the exclusion region is defined as

where $A_{\rm ex}$ is the area of the exclusion region, and $A_{\rm tot}$ is the total field area. The probability increases from $\tau=5\%$ for $n_{l}=700$, to $11\%$ for $n_{l}=1500$ in our tests. Moreover, we calculate the mean of average electron density over $500$ realisations, and compare the true value with the estimated one by the relative ratio

We show the relative difference in Table 1. From both Fig. 7 and Table 1, one can see that the “estimated” density from mock observations will be lower than the “real” one. With an increase in the electron density (i.e., the number of lenses in front of the field), the underestimate becomes larger. As one expects that the density gradient is another important aspect, we perform additional simulations with higher gradient (the mean value of $\theta_{0}/\sigma$ changes to $0.05$, $0.5$ of the two log-normal distributions in the simulation, and the distribution of $\theta_{0}/\sigma$ is shown by the red curve in Fig. 5). The comparison is presented by the bottom three rows in Table. 1. The underestimate becomes slightly stronger with the similar electron density. In Fig. 8, we present $r$ of $\bar{N_{\rm e}}$, $r$ is estimated by the average and the error bars present the standard deviation over 500 realisations. As a comparison, we show a linear relation between $r$ and $\bar{N_{\rm e}}$ by the red curve

where $a=6\times 10^{-5}$ cm³/pc, and $b=0.045$. Our test (dots) follows the trend of Eq. 22. Given the high dispersion measure of distant FRBs (a few thousands), we expect the underestimate of electron density due to de-magnification can be $9\%$ for $n_{l}=700$ (for $\tau\approx 5\%$), and up to $\sim 15\%$ for $n_{l}=1500$ (for $\tau\approx 10\%$). Such a relation (Eq. 22) only provides a rough estimate of the bias, since it depends on several factors of the density model, e.g. the distribution of $\theta_{0}/\sigma$ of the Gaussian lenses. It is clear that the relation will not be strictly valid for other density models. More constraints from observations and further studies are necessary in order to obtain a more realistic estimate of the bias.

Inhomogeneous distributions of plasma can deflect the propagation of low-frequency radio signals, similar to the gravitational lensing phenomena. This plasma lensing shares several similarities with gravitational lensing, especially in terms of the mathematical description. However, for an over-dense clump of plasma, it behaves analogously to a concave lens. The plasma lens thus diverges the light of background sources and produces significant de-magnification. We study the de-magnifying properties, and the cross section for two density profiles: the Gaussian model and the softened power-law model.

The properties of the magnification and de-magnification of Gaussian plasma lenses can be determined by the ratio of lens parameters $\theta_{0}/\sigma$. The cross sections of magnifications increase with this ratio. Softened power-law lenses can be described by $\theta_{0}/\theta_{c}$, and its cross sections increase with ratio as well. Since plasma lensing causes a strong de-magnification, i.e. $\mu\rightarrow 0$, there will be two observational consequences: 1) the luminosity function of the background sources is modified by plasma lensing. 2) in the study of using distant radio sources to trace the distribution of electron density in the universe, there is a bias: when the electron density is high and has a large gradient, we cannot receive any radio signal. Therefore, the average electron density that we estimate will always be lower than their true mean value. We use a toy model in a small field of view ($1\times 1$ arcmin²), a reasonable electron density range from observations (Yao et al. 2017), and compare the simulated and the estimated electron density. Underestimates are obtained in all our tests. The magnitude of the bias is correlated with several properties: the number of lenses, the density model, and the density parameter, e.g. $\theta_{0}/\sigma$ etc. In general, the higher the electron density, the larger the bias. From the observed electron density within the Milky Way, e.g. up to $\sim 2000$ pc cm³ along the disk(Yao et al. 2017), the underestimate can be up to $15\%$ in our test using the Gaussian density model when the lensing probability is $\tau\approx 10\%$ (see Eq. 21). We also calculate the luminosity function after plasma lensing, and find that the distribution of the flux of background sources can be broadened to both higher and lower values. One interesting point is that most of the high flux samples are magnified by lensing.

Our study is simplified in several aspects. First of all, the plasma lensing is frequency dependent. We adopt $\nu=1$ GHz in our study. In reality, for observations with lower frequency, plasma lensing effects will become even stronger. By comparing observations, e.g. luminosity function at different frequencies, one can obtain tighter constraints on both the plasma density and the background sources. Another two of simplifications are the model of electron density and the luminosity function of the radio sources. The constraint on the electron density is weak from observations so far, especially on small scales. One will be able to rely on high resolution numerical simulations, which have been undergoing rapid improvement in recent years. Interests in the luminosity function have experienced explosive growths thanks to the observations of FRBs. We expect that thousands of FRBs will be detected in the coming years. The collective study of many sources affected by foreground lenses (e.g. by variations in the local electron density or by cosmic re-ionisation) will be dramatically improved to high precision. Under such conditions the magnification effect due to plasma lensing will present a systematic bias and will need to be accounted for.

We thank the referee for a constructive and valuable report which increases the scope of our work significantly. We also thank Jenny Wagner for interesting discussions. We acknowledge use of the CHIME/FRB Public Database, provided at https://www.chime-frb.ca/ by the CHIME/FRB Collaboration. XE is supported by the NSFC Grant No. 11873006. SM is supported by the National Key Research and Development Program of China No. 2018YFA0404501, and NSFC Grant No. 11821303, 11761131004 and 11761141012.

The data underlying this article will be shared on reasonable request to the corresponding author.