Dispersion Measure Distribution of Unlocalized Fast Radio Bursts as a Probe of the Hubble Constant

This paper presents the first measurement of the Hubble constant ($H_0 = 73.8^{+14.0}_{-12.3}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$) derived solely from the dispersion measure distribution of 2,124 unlocalized fast radio bursts, demonstrating their potential as a new cosmological probe that can constrain $H_0$ without requiring redshift information.

The Gist

The Big Problem: The Universe is Growing, But We Can't Agree on How Fast

Imagine the universe is a giant balloon being blown up. Astronomers have been trying to measure exactly how fast that balloon is inflating. This speed is called the Hubble Constant ($H_0$).

Here's the crisis: We have two different ways of measuring it, and they give us two very different answers.

1. The "Baby Picture" Method: Looking at the Cosmic Microwave Background (the afterglow of the Big Bang), we get a speed of about 67.
2. The "Local Neighborhood" Method: Looking at nearby stars and supernovas, we get a speed of about 73.5.

The difference is small in numbers, but statistically huge. It's like if two doctors looked at the same patient and one said, "He's perfectly healthy," and the other said, "He's in critical condition." This disagreement is called the "Hubble Tension," and it suggests we might be missing something fundamental about how the universe works.

The New Detective: Fast Radio Bursts (FRBs)

Enter the Fast Radio Bursts (FRBs). These are mysterious, super-powerful flashes of radio waves from deep space that last only a millisecond. They are like cosmic "firecrackers."

For a long time, astronomers only used FRBs that were localized (we knew exactly which galaxy they came from). Think of these as labeled packages. If you know the package came from a specific city (redshift) and you know how "spread out" the signal is (Dispersion Measure), you can calculate the distance and the expansion speed.

The Problem: We only have about 100 of these "labeled packages." It's a tiny sample size, like trying to guess the average height of all humans by measuring only 100 people.

The New Idea: Counting the Unlabeled Packages

This paper proposes a brilliant new trick. Instead of waiting for the "labeled packages," let's use the thousands of "unlabeled packages" (unlocalized FRBs) we already have. We don't know exactly where they are, but we do know how "spread out" their signals are.

The Analogy: The Foggy Road Trip
Imagine you are driving down a highway in heavy fog. You can't see the exit signs (redshifts), so you don't know exactly how far you've gone. However, you know that the thicker the fog is, the longer the road must be.

• The Signal: The radio burst is your car's horn.
• The Fog: The free electrons in space.
• The Spread: As the horn sound travels through fog, it gets "smudged" or delayed. Low-pitched sounds arrive later than high-pitched ones. This "smearing" is the Dispersion Measure (DM).

The paper argues that even if we don't know where the car is, if we look at the distribution of how "smudged" all the horns are, we can figure out how much fog is on the road overall. Since the amount of fog (electrons) is tied to the expansion of the universe, we can use the smearing pattern to calculate the expansion speed ($H_0$).

What They Did

The authors took a massive list of 2,124 unlocalized FRBs from the CHIME telescope (a giant radio telescope in Canada). They didn't try to find the specific galaxy for each one. Instead, they treated the whole group like a single statistical cloud.

They built a complex computer model that simulates:

1. How the universe expands.
2. How much "fog" (electrons) exists in space.
3. How the telescope picks up signals (some signals are too weak to hear).

Then, they compared their simulation to the real data to see which expansion speed ($H_0$) fit best.

The Results

• The Measurement: They found an expansion speed of 73.8.
• The Catch: The uncertainty is still quite high (about 18%). It's like saying, "The speed is somewhere between 61 and 88." This is too wide to solve the mystery yet because it overlaps with both the "67" and "73.5" answers.
• The Breakthrough: However, they found that the main reason for this uncertainty is a "degeneracy" (a confusion) between the expansion speed and the energy of the FRBs. It's like not knowing if the horn is loud because the car is far away, or because the horn is just naturally loud.

The Good News: If they can figure out the "loudness" (energy) of the FRBs independently (perhaps by combining this data with the few "labeled" FRBs we do have), the uncertainty drops to 9%.

Why This Matters

This paper is a proof of concept. It's the first time anyone has tried to measure the Hubble Constant using only the unlocalized FRBs.

• Current Status: It's a rough sketch, but it works.
• Future Potential: As we detect more FRBs (thousands more), and as we get better at calibrating their energy, this method could become one of the most precise tools we have.

The Bottom Line:
Think of the Hubble Tension as a locked door. We have a key (localized FRBs), but it's a small key and we only have a few of them. This paper shows that we can also use a master key made of thousands of tiny pins (unlocalized FRBs). Right now, the master key is a bit wobbly, but with more data and better calibration, it might just be the key that finally unlocks the mystery of the universe's expansion.

Technical Summary

1. Problem Statement

The paper addresses the Hubble Tension, a significant crisis in modern cosmology referring to the discrepancy between the Hubble constant ($H_0$) inferred from the Cosmic Microwave Background (CMB) within the $\Lambda$CDM framework ($H_0 \approx 67.24$ km s$^{-1}$ Mpc$^{-1}$) and measurements from local distance indicators ($H_0 \approx 73.50$ km s$^{-1}$ Mpc$^{-1}$). The statistical significance of this discrepancy has risen to $7.1\sigma$.

While Fast Radio Bursts (FRBs) have been proposed as an independent probe to resolve this tension, current methods rely on localized FRBs (those with measured redshifts $z$). The sample of localized FRBs is currently limited to just over 100 sources, which restricts the precision of $H_0$ constraints. The authors aim to utilize the vastly larger population of unlocalized FRBs (thousands of sources without redshift measurements) to constrain $H_0$ by analyzing their observed Dispersion Measure (DM) distribution, which encodes cosmological information via cosmic expansion.

2. Methodology

The authors developed a statistical framework to infer $H_0$ from the distribution of observed DMs ($DM_{obs}$) without requiring individual redshift measurements.

A. Decomposition of Dispersion Measure

The observed DM is modeled as the sum of Galactic and extragalactic components:
$$DM_{obs} = DM_{MW} + DM_{ext}$$
Where $DM_{MW}$ includes the Interstellar Medium (ISM) and Halo contributions, and $DM_{ext}$ comprises the host galaxy contribution ($DM_{host}$) and the Intergalactic Medium (IGM) contribution ($DM_{cos}$).

B. Probability Density Functions (PDFs)

The core of the method involves modeling the theoretical PDF of $DM_{obs}$, denoted as $P(DM_{obs}|\Theta)$, which depends on a set of parameters $\Theta$.

1. Host Galaxy Contribution ($DM_{host}$): Modeled as a log-normal distribution with free parameters $\mu_{host}$ and $\sigma_{host}$.
2. IGM Contribution ($DM_{cos}$): Modeled based on the mean value $\langle DM_{cos} \rangle$, which is directly dependent on $H_0$ and cosmological parameters (Eq. 6). The distribution of the ratio $\Delta = DM_{cos}/\langle DM_{cos} \rangle$ follows a specific functional form involving baryon feedback parameters.
3. Redshift Distribution: The intrinsic FRB event rate $R(z)$ is parameterized relative to the cosmic Star Formation Rate (SFR) with a free parameter $n$.
4. Selection Effects: The model incorporates the telescope's detection probability ($P_{det}$) and selection functions for both fluence ($S_F$) and DM ($S_{DM}$), derived empirically from CHIME data.

C. Likelihood Analysis

The authors constructed a likelihood function $\mathcal{L}$ for a sample of $N$ unlocalized FRBs:
$$\mathcal{L}(DM_{obs}|\Theta) = \prod_{i=1}^{N} P(DM_{obs,i}|\Theta)$$
Using Bayesian inference (via the emcee sampler), they marginalized over nuisance parameters (host galaxy properties, energy distribution parameters, halo DM) to constrain $H_0$.

Data Source: The analysis utilized a selected subsample of 2,124 unlocalized FRBs from the CHIME Catalog II (2018–2023). Selection criteria included non-repeating bursts, high signal-to-noise ratio (SNR > 10), and specific DM thresholds to ensure extragalactic origin.

3. Key Contributions

• First Unlocalized $H_0$ Measurement: This is the first independent measurement of the Hubble constant derived solely from the DM distribution of unlocalized FRBs, bypassing the need for host-galaxy identification or redshift follow-up.
• Scalability: The method demonstrates that the abundant population of unlocalized FRBs (orders of magnitude larger than localized samples) can serve as a viable cosmological probe.
• Degeneracy Analysis: The study explicitly identifies and quantifies the degeneracy between $H_0$ and the characteristic cutoff energy ($E^*$) of the FRB isotropic energy distribution, showing how breaking this degeneracy significantly improves precision.

4. Results

• Primary Constraint: Using the sample of 2,124 unlocalized FRBs with free energy distribution parameters, the authors obtained:
  $$H_0 = 73.8^{+14.0}_{-12.3} \text{ km s}^{-1} \text{ Mpc}^{-1} \quad (1\sigma)$$
  This corresponds to a relative uncertainty of approximately 18%. The result is consistent with both the CMB and local distance indicator measurements within $1\sigma$.
• Impact of Energy Distribution Calibration: The analysis revealed a strong degeneracy between $H_0$ and $\log E^*$. When $\log E^*$ was fixed to its mean inferred value ($40.9$), the uncertainty on $H_0$ dropped to 9%:
  $$H_0 = 74.4^{+7.7}_{-5.4} \text{ km s}^{-1} \text{ Mpc}^{-1}$$
• Future Projections (Mock Catalogs):
  • For a simulated sample of 5,000 FRBs, the uncertainty reduces to 8%.
  • If the energy distribution ($E^*$) is independently calibrated for this larger sample, the uncertainty drops to 3%, making it competitive with current localized FRB constraints.

5. Significance

This work establishes unlocalized FRBs as a promising and complementary probe for the Hubble tension.

• Independence: It offers a route to constrain $H_0$ that is independent of the distance ladder and CMB physics, relying instead on the statistical properties of the FRB population and the expansion history of the universe imprinted on the DM.
• Path to Precision: While current precision (18%) is lower than that of localized FRBs (~3%), the method scales efficiently with sample size. As the CHIME and future telescopes (e.g., DSA-2000, SKA) detect thousands of FRBs, and as the FRB energy distribution is better calibrated, this method could achieve sub-5% precision.
• Synergy: The authors propose that future joint analyses combining localized (redshift-known) and unlocalized FRBs will provide the tightest constraints, potentially resolving the Hubble tension or revealing new physics beyond the $\Lambda$CDM model.