Model-independent Estimation of H₀ and Ω_K from Strongly Lensed Fast Radio Bursts

The determination of the present value of the expansion rate of the universe, the Hubble constant H₀, is not only central to describing the present state of our universe and setting expectations for its fate, but also may be pointing toward a new wrinkle in the cosmological model. Measurements of the distance ladder method with improved precision and control of systematics from the Supernovae, H0, for the Equation of State of Dark energy (SH0ES) Team (Riess et al. 2016, 2019) indicate that the present expansion rate of the universe is about 9% larger than that inferred from the ΛCDM model calibrated by Planck cosmic microwave background (CMB) observations from the early universe (Planck Collaboration et al. 2020), with a significance of >4σ. The use of any one of the five independent geometric distance estimators, including masers in NGC 4258 (Humphreys et al. 2013; Riess et al. 2016), eight detached eclipsing binaries in the Large Magellanic Cloud (Pietrzyński et al. 2013, 2019), and three distinct approaches to measure the Milky Way parallaxes with the most recent Hubble Space Telescope (HST) spatial scanning and Gaia DR2 (Benedict et al. 2007; Riess et al. 2018), consistently lead to a higher local value of H₀. This "H₀ Tension" between the early and late universe, as it is widely known, may be interpreted as evidence of a new cosmological feature such as exotic dark energy (Khosravi et al. 2019; Poulin et al. 2019), a new relativistic particle (Renk et al. 2017; D'Eramo et al. 2018), dark-matter radiation or neutrino–neutrino interactions (Barenboim et al. 2019; Kreisch et al. 2020), dark-matter decay (Vattis et al. 2019; Pandey et al. 2020), or a small curvature (Aab et al. 2018). Each proposal can produce a different-sized shift with some of them covering the full discrepancy and thus improving the agreement between the model and observational data. Pinpointing the cause of this tension requires further improvement in the local distance measurements, with continued efforts on precision, accuracy, and experimental design to control systematics. Alternatively, new, independent, precise, and accurate measurements of H₀ could help clarify whether the tension arises from beyond ΛCDM physics or unrecognized systematics.

In addition to the Hubble constant H₀, cosmic curvature, usually referred as curvature density parameter Ω_K , describes the spatial property of the universe and also is at the root of cosmology. In the standard Friedmann–Lemaître–Robertson–Walker (FLRW) framework, Ω_K > 0, Ω_K = 0, and Ω_K < 0 correspond to spatially open, flat, closed universe, respectively. In the context of standard Big Bang cosmology, exponential expansion during an inflationary phase was proposed to solve several problems with Big Bang cosmology, like the flatness and horizon problem (Guth 1981; Linde 1990). And thus, a spatially flat universe is a natural prediction of inflation. Moreover, a spatially flat universe is also significantly favored by most popular observations. For instance, the recent Planck Legacy 2018 data of CMB anisotropies combined with Baryon Acoustic Oscillations (BAO measurements give Ω_K = 0.001 ± 0.002 (Planck Collaboration et al. 2020), suggesting that our universe is spatially flat to a 1σ accuracy of 0.2%. However, without CMB lensing or BAO, the Planck data alone does favor Ω_K < 0 at the 3.4σ level (Park & Ratra 2019; Di Valentino et al. 2020; Handley 2021). In these works, this high significance level deviation from the flat case is interpreted as evidence for either undetected systematics in the Planck data, new physics beyond standard ΛCDM, or an unusual statistical fluctuation (or some combination of all three). More recently, Efstathiou & Gratton (2020) revisited this problem and claimed that the Planck temperature and polarization spectra remain consistent with a spatially flat universe. At this moment, whether this crisis really exists and the interpretation for it are in debate. Similar to the "H₀ Tension," new, independent, precise, and accurate measurements of Ω_K are also of great importance for clarifying the curvature crisis.

At this juncture in cosmology, independent and complementary probes with considerable precision are very helpful for providing clues about the origin of the abovementioned crises. Strong-lensing systems are one of the most promising probes for investigating these issues. The time delay between images of strongly lensed time-variable sources was proposed to directly determine H₀ (Refsdal 1964; Treu 2010). Recently, with elaborate time-delay measurements and lens modeling for five selected strongly lensed quasar systems, the H0 Lenses in COSMOGRAIL's Wellspring (H0LiCOW) team yielded an estimation of H₀ that is also 2.3σ higher than the Planck-calibrated value (Birrer et al. 2019). It should be mentioned that this result is completely independent of all rungs of the distance ladder. Moreover, on the basis of the distance sum rule (DSR), distance ratios derived from angular separation measurements of images in strong-lensing systems were proposed to test the FLRW metric and model-independently determine Ω_K (Räsänen et al. 2015). This method has been intensively implemented with updated observations (Xia et al. 2017; Li et al. 2018a; Liu et al. 2020; Zhou & Li 2020). In order to reduce the systematics resulting from oversimplified lensing modeling in the distance ratio method, Liao et al. (2017b) reformulated the DSR in terms of the time-delay distance, which is a combination of three angular diameter distances in a strong-lensing system. With the latest strongly lensed quasar observations and type Ia supernovae (SNe Ia) luminosity distance measurements, H₀ and Ω_K were simultaneously and model-independently determined to a precision of ∼6% and ∼0.3 (Collett et al. 2019), respectively. Meanwhile, strongly lensed gravitational waves (GWs) and their electromagnetic (EM) counterparts from the binary of compact object coalescence are proposed as a powerful tool for precision cosmology since the time delay between images in these systems can be precisely measured (Liao et al. 2017a; Li et al. 2019). However, both traditional strongly lensed quasar and proposed strongly lensed GW systems face shortages. For lensed quasars, the precision of time-delay measurements is limited to ∼3% and lens modeling is difficult to improve because of the bright active-galactic nuclei (AGNs) contamination in the source host galaxy. For the expected lensed GWs, the main challenges may be the event rate and localization ability for images of GW signals, which is crucial for lens modeling.

Fast radio bursts (FRBs) are millisecond-duration radio transients and almost all of them have been observed at extragalactic distances (Lorimer et al. 2007; Thornton et al. 2013; Petroff et al. 2015, 2016; Cordes & Chatterjee 2019). Although the progenitors and radiation mechanism are still debated, ³ FRBs have been proposed to be promising tools for constraining cosmological parameters (Gao et al. 2014; Zhou et al. 2014; Yang & Zhang 2016; Walters et al. 2018; Zhao et al. 2020). Recently, Li et al. (2018b) proposed strongly lensed FRBs as more precise cosmological probes due to several prominent advantages, including short-duration, high event rates (∼10³–10⁴ per day all sky; Thornton et al. 2013; Champion et al. 2016), and precision localization for both repeating and apparent one-off FRBs (Chatterjee et al. 2017; Bannister et al. 2019). Here, we propose strongly lensed FRB systems as promising tools for simultaneous and model-independent determination of H₀ and Ω_K , and investigate the constraining power on these two fundamental parameters from upcoming observations.

In this paper, we take the standard ΛCDM model with the matter density parameter Ω_M = 0.3 and Hubble constant H₀ = 70 km s⁻¹Mpc⁻¹ as the fiducial model in our following analysis.

According to the cosmological principle, on large scales the universe is homogeneous and isotropic, which can be described by the FLRW metric

$$\begin{eqnarray}&&{{ds}}^{2}=-{c}^{2}{{dt}}^{2}+\displaystyle \frac{a{\left(t\right)}^{2}}{1-{{Kr}}^{2}}{{dr}}^{2}+a{\left(t\right)}^{2}{r}^{2}d{{\rm{\Omega }}}^{2},\end{eqnarray} \tag{ 1 }$$

where c is the speed of light, and K is related to the curvature density parameter Ω_K with the relationship between them being $-K/{H}_{0}^{2}={{\rm{\Omega }}}_{K}$. Therefore the dimensionless distance d(z_l , z_s ) =(1 + z_s )H₀ D_A (z_l , z_s )/c has the following form in the FLRW metric,

$$\begin{eqnarray}&&d({z}_{l},{z}_{s})=\displaystyle \frac{1}{\sqrt{{{\rm{\Omega }}}_{K}}}\sinh \sqrt{{{\rm{\Omega }}}_{K}}{\int }_{{t}_{s}({z}_{s})}^{{t}_{l}({z}_{l})}\displaystyle \frac{{H}_{0}}{a(t)}{dt},\end{eqnarray} \tag{ 2 }$$

where D_A (z_l , z_s ) is the angular diameter distance of a source at redshift z_s as observed at redshift z_l . We denote d(z) ≡ d(0, z), d_ls ≡ d(z_l , z_s ). The DSR in the FLRW metric can be written as

$$\begin{eqnarray}&&{d}_{{ls}}={d}_{s}\sqrt{1+{{\rm{\Omega }}}_{K}{d}_{l}^{2}}-{d}_{l}\sqrt{1+{{\rm{\Omega }}}_{K}{d}_{s}^{2}}.\end{eqnarray} \tag{ 3 }$$

Here the function of redshift z with respect to cosmic time t is assumed to be monotonic and $d^{\prime} (z)\gt 0$. It is evident that d_l + d_ls < d_s , d_l + d_ls = d_s , and d_l + d_ls > d_s for Ω_k > 0, Ω_K = 0, and Ω_K < 0 (Bernstein 2006), respectively.

For a strong-lensing system, the relationship between time delay Δt and time-delay distance D_Δt is

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{{ij}}=(1+{z}_{l})\displaystyle \frac{{\rm{\Delta }}{\phi }_{{ij}}}{c}{D}_{{\rm{\Delta }}t},\end{eqnarray} \tag{ 4 }$$

where Δϕ_ij is the Fermat potential difference between the position of the ith image and that of the jth image, and has the form as shown below:

$$\begin{eqnarray}&&\phi ({\theta }_{i})\equiv \displaystyle \frac{1}{2}{\left({\theta }_{i}-\beta \right)}^{2}-\psi ({\theta }_{i}),\end{eqnarray} \tag{ 5 }$$

where β in the above equation is the source position and ψ(θ_i ) is the scaled gravitational potential at the image position. The $\displaystyle \frac{1}{2}{({\theta }_{i}-\beta )}^{2}-\tfrac{1}{2}{({\theta }_{j}-\beta )}^{2}$ part is called the geometric delay and the ψ(θ_i ) − ψ(θ_j ) part is called the Shapiro delay. Time-delay distance D_Δt is defined as a combination of three angular diameter distances

$$\begin{eqnarray}&&{D}_{{\rm{\Delta }}t}=\displaystyle \frac{{D}_{A}({z}_{l}){D}_{A}({z}_{s})}{{D}_{A}({z}_{l},{z}_{s})}.\end{eqnarray} \tag{ 6 }$$

In this combination, D_A (z_l ), D_A (z_s ), and D_A (z_l , z_s ) are the angular diameter distances from the observer to the lens, from the observer to the source, and from the lens to source, respectively. By combining the DSR Equation (3) and the time-delay distance Equation (6), we have

$$\begin{eqnarray}&&\displaystyle \frac{{d}_{{ls}}}{{d}_{l}{d}_{s}}=T({z}_{l})-T({z}_{s}),\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&T(z)=\displaystyle \frac{1}{d(z)}\sqrt{1+{{\rm{\Omega }}}_{k}{d}^{2}(z)}.\end{eqnarray} \tag{ 8 }$$

Finally, the DSR in terms of time-delay distance can be written as (Liao et al. 2017b)

$$\begin{eqnarray}&&{D}_{{\rm{\Delta }}t}=\displaystyle \frac{c}{{H}_{0}(1+{z}_{l})(T({z}_{l})-T({z}_{s}))}.\end{eqnarray} \tag{ 9 }$$

With this method, a model-independent determination of H₀ and Ω_K was first obtained from strongly lensed quasars and type Ia supernova observations (Collett et al. 2019). At the same time, estimates have also been done on simulated, upcoming, strongly lensed GWs (Li et al. 2019). Strongly lensed FRB systems have precise measurements of time delay due to their short duration time, and the relative uncertainty of their Fermat potential could reach up to ∼0.8% (see Section 3.3 for details), which is more precise than the lens modeling for a lensed quasar (∼3%). Meanwhile, compared to lensed GWs, FRBs have a higher event rate and more precise localization. Therefore, here we investigate the constraining power of lensed FRBs for estimating H₀ and Ω_K model-independently.

In this work, we investigate the power of estimating H₀ and Ω_K from upcoming strongly lensed FRBs. For this purpose, three main aspects, including the redshift distribution of FRBs detected in the near future, the probability of a source at redshift z_s lensed by a foreground dark-matter halo, and the uncertainty of each factor contributing to the accuracy of the time-delay distance measurement, should be clarified. First, we briefly address these three aspects one by one in this section. In addition, we present the results of the estimation of H₀ and Ω_K from simulations with these aspects taken into consideration.

3.1. Redshift Distribution

We assume that redshifts of FRBs detected in the near future and the redshifts of currently available FRBs follow the same distribution. For the 136 already public FRBs, ⁴ 13 of them have been localized to their host galaxies and thus their redshifts are measured. Redshifts of the rest of the FRBs are roughly inferred from their observed dispersion measure (DM) using the DM-z relation proposed by Zhang (2018). The validity of this relation has been confirmed with observations of the first five localized FRBs (Li et al. 2020). The histogram of measured and approximate redshifts of these currently available 136 FRBs is shown in Figure 1. We fit this histogram with the Gamma distribution (for z > 0)

$$\begin{eqnarray}&&p(z)\propto \displaystyle \frac{{\beta }^{\alpha }}{{\rm{\Gamma }}(\alpha )}{z}^{\alpha -1}{e}^{-\beta z},\end{eqnarray} \tag{ 10 }$$

and obtain α = 2.28 and β = 3.76 (denoted as the red line in Figure 1). In the following analysis, we assume that redshifts of the upcoming FRBs or sources in strongly lensed FRB systems trace the above-fitted probability density function.

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3.2. Lensing Probability

For a distant source at redshift z_s , the probability lensed by foreground dark-matter halos is (Schneider et al. 1992)

$$\begin{eqnarray}&&P={\int }_{0}^{{z}_{s}}{{dz}}_{l}\displaystyle \frac{{{dD}}_{p}}{{{dz}}_{l}}{\int }_{0}^{\infty }{dMn}(M,{z}_{l})\sigma (M,{z}_{l}),\end{eqnarray} \tag{ 11 }$$

where D_p is the proper distance from the observer to a lens at redshift z_l , n(M, z_l )dM is the proper number density of lens objects with masses between M and M + dM, and σ(M, z_l ) is the lensing cross-section of a dark-matter halo with mass M at z_l . For generality, we consider the singular isothermal sphere (SIS) dark-matter halos in foreground galaxies and assume that the mass function of halos is given by the Press–Schechter function (Press & Schechter 1974). The cross-section for an SIS lens to produce two images with a brightness ratio < r is (Li & Ostriker 2002)

$$\begin{eqnarray}&&\sigma (\lt r)=\pi {\xi }_{0}^{2}{\left(\displaystyle \frac{r-1}{r+1}\right)}^{2},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{\xi }_{0}=4\pi {\left(\displaystyle \frac{{\sigma }_{\nu }}{c}\right)}^{2}\displaystyle \frac{{D}_{A}({z}_{l}){D}_{A}({z}_{l},{z}_{s})}{{D}_{A}({z}_{s})},\end{eqnarray} \tag{ 13 }$$

and σ_ν is the velocity dispersion. The comoving number density of dark-matter halos with mass in the range (M, M + dM) is given by

$$\begin{eqnarray}&&n(M,z){dM}=\displaystyle \frac{{\rho }_{0}}{M}f(M,z){dM},\end{eqnarray} \tag{ 14 }$$

where ρ₀ ≡ Ω_m ρ_crit,0 is the present mean mass density of the universe, and f(M, z) is the Press–Schechter function. For a source at redshift z_s , the redshift of foreground galaxy z_l , which is most likely to produce a strong-lensing system, could finally be obtained by maximizing the lensing probability (refer to Figure 2 in Li & Li 2014 for an illustration).

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3.3. Uncertainty Contribution

3.4. Statistical Analysis and Results

With three main concerning aspects addressed, we perform 20,000 realizations based on 10 simulated lensed FRB systems, which are likely to be accumulated in several years, to study model-independent constraints on H₀ and Ω_K . First, we propagate the relative uncertainties of time delay (δΔt = 0), Fermat potential difference (δΔϕ = 0.8%), and LOS contamination (δ κ_ext = 2%) to the relative uncertainty of D_Δt . At the same time, we model the function d(z) with a polynomial from SNe Ia observations in the near future. The relationship between luminosity distances D_L of SNe Ia and the dimensionless comoving distance is

$$\begin{eqnarray}&&{D}_{L}(z)=\displaystyle \frac{1+z}{{H}_{0}}d(z).\end{eqnarray} \tag{ 15 }$$

Here, we mock SNe Ia distances on the basis of the upcoming Wide Field InfraRed Survey Telescope (WFIRST), which was the highest-ranked large space-based mission with the primary objective being able to precisely constrain the nature of dark energy with multiple probes, including SNe Ia. Optimistically, the Imaging: Allz strategy of WFIRST could collect ∼13500 SNe Ia in the redshift range 0 < z < 3 (Hounsell et al. 2018). The number of SNe Ia that can be discovered by WFIRST is expected to follow the volumetric rates (in units of 10⁻⁵ yr⁻¹ Mpc⁻³; Rodney et al. 2014; Graur et al. 2014),

$$\begin{eqnarray}R(z)=\left\{\begin{array}{cc}2.5\times {\left(1+z\right)}^{1.5}, & z\lt 1,\\ 9.7\times {\left(1+z\right)}^{0.5}, & \,\,1\lt z\lt 3.\end{array}\right.\end{eqnarray} \tag{ 16 }$$

As the expected detection rate is low for z > 3 SNe Ia, we do not simulate events at those redshifts. For a sample of SNe Ia, the total uncertainty of their distances σ_μ (μ represents distance modulus) consists of two main components, statistical uncertainty σ_stat and systematic uncertainty σ_sys. With the fractional statistical uncertainty (panel h of Figure 7 in Hounsell et al. 2018) and all ingredients of systematic uncertainty (Figures 9 and 10 in Hounsell et al. 2018) taken into consideration, we generate a sample of 13,570 mocked SNe Ia in the range 0 < z < 3 based on the fiducial ΛCDM model with the model parameters being H₀ = 70 km s⁻¹ Mpc⁻¹ and Ω_m0 = 0.3. Finally, we obtain model-independent constraints on H₀ and Ω_K simultaneously from mocked SNe Ia and the time-delay distance data by fitting the following χ² function,

$$\begin{eqnarray}&&{\chi }^{2}=\sum _{i=1}^{10}\displaystyle \frac{{\left[{D}_{{\rm{\Delta }}t(\mathrm{th})}-{D}_{{\rm{\Delta }}t(\mathrm{ob})}\right]}^{2}}{{\sigma }_{{D}_{{\rm{\Delta }}t(\mathrm{ob})}}^{2}}+\sum _{i=1}^{n}\displaystyle \frac{{\left[d{\left(z\right)}_{\mathrm{ob}}-d{\left(z\right)}_{\mathrm{th}}\right]}^{2}}{{\sigma }_{d{\left(z\right)}_{\mathrm{ob}}}^{2}},\end{eqnarray} \tag{ 17 }$$

where D_Δt(th)(z_l,i , z_s,i , H₀, Ω_K ) is the theoretical time-delay distance, while D_Δt(ob) is the corresponding simulated distance with Fermat potential difference uncertainty and an extra LOS uncertainty considered, and its uncertainty is ${\sigma }_{{D}_{{\rm{\Delta }}t,i}}=\delta {D}_{{\rm{\Delta }}t,i}{D}_{{\rm{\Delta }}t,i}$.

Figure 2 shows the result after 20,000 realizations for (H₀, Ω_K ) from 10 lensed FRBs and WFIRST SNe Ia. First, considering the 0.8% relative uncertainty on the Fermat potential, we achieve the model-independent estimation of H₀ to a precision of 1%, ${H}_{0}={69.88}_{-0.73}^{+0.76}$. Next, we also take 1.5% uncertainty on the Fermat potential into account for comparison, and the precision of H₀ also achieves ∼1.2%, ${H}_{0}={69.87}_{-0.85}^{+0.90}$. These constraints are comparable to the estimation from the latest Planck CMB observations (Planck Collaboration et al. 2020) and that from the proposed 10 lensed GW+EM systems (Li et al. 2019). Meanwhile, these determinations of H₀ are a little more precise than the one from the latest local distance ladder measurements (a precision of ∼2%; Riess et al. 2019) and the one determined from the latest but traditional strongly lensed quasar systems (a precision of ∼6%; Collett et al. 2019). Furthermore, it is shown that, along with model-independent estimation of H₀, Ω_K could be simultaneously determined to ∼0.1 from 10 lensed FRBs (∼0.11 for the 0.8% Fermat potential uncertainty and ∼0.13 for the 1.5% Fermat potential uncertainty, respectively). Although these estimations are weaker than the one constrained from the latest Planck CMB observations in the context of ΛCDM (Di Valentino et al. 2020; Planck Collaboration et al. 2020), the model-independent estimation of Ω_K from the combination of traditional lensed quasars and new lensed FRBs will be helpful for judging the curvature crisis. In addition to these improvements in precision, good agreement between constraints on H₀ and Ω_K from our simulations and their fiducial values used to mock data suggest that upcoming strongly lensed FRBs together with DSR can achieve unbiased estimation of these two concerning fundamental parameters.

In this work, we first propose to model-independently estimate H₀ and Ω_K with time-delay distance measurements of strongly lensed FRBs. Combining forthcoming WFIRST SNe Ia observations and 10 lensed FRBs systems, which are likely to come true in several years, we obtain a precision of ∼1% on H₀ and constrain Ω_K to ∼0.1 in a model-independent way in the context of DSR. In light of the recent Hubble constant tension and curvature crisis, these determinations, which are independent on any assumptions for the composition of the universe and gravity theory, will be fairly helpful for clarifying these current challengeable issues in cosmology.

In the next few years, a considerable population of repeating FRBs will be detected by wide-field sensitive radio telescopes, such as the Canadian Hydrogen Intensity Mapping Experiment (CHIME) FRB project (CHIME/FRB Collaboration et al. 2019), and thus their redshifts could be measured by localizing them with a very long baseline array. Moreover, facilities including radio interferometers, world-leading radio survey telescopes, and multimessenger discovery engines for the next decade, such as the Australian Square Kilometre Array Pathfinder (ASKAP) and DSA-2000, could simultaneously detect and localize ∼10⁴ apparent one-off FRBs each year (Hallinan et al. 2019). It is foreseen that the prospect of determining of H₀ and Ω_K from strongly lensed FRBs will be optimistic with the rapid progress in the FRB community and complementary to existing strongly lensed quasars.

We would like to thank Xuheng Ding for helpful discussions. This work was supported by the National Natural Science Foundation of China under grants Nos. 11920101003, 11722324, 11603003, 11633001, and U1831122, the Strategic Priority Research Program of the Chinese Academy of Sciences, grant No. XDB23040100, and the Interdiscipline Research Funds of Beijing Normal University.