Fast Radio Burst Distance: Redshift & Cosmic Scale

Fast radio bursts (FRBs) are millisecond-long flashes of radio energy from deep space, and the key to measuring how far away they are lies in a property called redshift. By measuring the redshift of an FRB’s host galaxy, astronomers can convert that shift into a cosmic distance using the expansion of the universe. This article explains how redshift works, how it is measured for FRBs, and what the resulting distances tell us about the universe.

Formula

The comoving distance (D_C) to an object at redshift z in a flat ΛCDM universe is given by:

D_C = (c / H₀) × ∫₀^z dz′ / √(Ω_m(1+z′)³ + Ω_Λ)

• c = speed of light (299,792.458 km/s)
• H₀ = Hubble constant (km/s/Mpc), typically 70
• Ω_m = matter density parameter (0.3)
• Ω_Λ = dark energy density parameter (0.7)
• z = redshift (unitless)

The integral has no simple closed form, so the calculator evaluates it numerically. The lookback time uses a similar integral with an extra (1+z′) factor in the denominator.

Worked Example

Using the default values: z = 0.5, H₀ = 70 km/s/Mpc.

1. Compute the integral ∫₀^0.5 dz′ / √(0.3(1+z′)³ + 0.7) ≈ 0.4402
2. Multiply by c/H₀ = 299792.458 / 70 ≈ 4282.75 Mpc
3. D_C ≈ 4282.75 × 0.4402 ≈ 1885 Mpc ≈ 6.15 billion light-years
4. Lookback time: (1/70) × 0.3446 × 977.8 ≈ 4.81 billion years

The calculator shows 6.148 billion ly and 4.81 billion yr, matching these steps.

What Is Redshift?

Redshift (z) is the fractional increase in the wavelength of light as it travels through an expanding universe. If a galaxy emits light at a wavelength of 500 nm and we observe it at 600 nm, then z = (600-500)/500 = 0.2. For FRBs, the burst itself has a spectrum that can be used to estimate a crude redshift, but the reliable method is to pinpoint the host galaxy and measure its redshift spectroscopically. The first repeating FRB (FRB 121102) was localized in 2017 to a dwarf galaxy at z ≈ 0.19, about 3 billion light-years away. Since then, dozens of FRBs have been traced to galaxies with redshifts ranging from 0.03 to over 1.0.

How FRB Distances Are Measured

There are two main ways to estimate FRB distances:

• Host galaxy redshift: After an FRB is detected, radio interferometers like the Very Large Array (VLA) or the Australian Square Kilometre Array Pathfinder (ASKAP) can pinpoint its position to sub-arcsecond accuracy. Optical telescopes then identify the host galaxy and measure its redshift from spectral lines (e.g., hydrogen Lyman-alpha or calcium H&K). This gives the most accurate distance.
• Dispersion measure (DM): The DM is the integrated column density of free electrons between us and the source. By subtracting the Milky Way’s contribution, astronomers get the extragalactic DM, which correlates with distance but has large scatter due to the clumpy distribution of matter in the cosmic web. DM alone gives only a rough distance estimate.

The redshift distance is preferred because it is model-independent and ties the FRB to a specific cosmic epoch.

Common Problems and Misconceptions

One common mistake is confusing redshift with velocity. For nearby objects, redshift is proportional to recessional velocity (v = zc), but at large distances the relationship becomes nonlinear because of the expansion history. Another issue is using a single FRB’s DM to claim a precise distance—this can be off by tens of percent because the electron distribution is poorly known. Also, some FRBs may have a host-galaxy contribution to DM that is hard to separate. Astronomers mitigate this by using the host galaxy redshift as the gold standard, not the DM.

Comparing FRB Distances to Other Cosmic Objects

FRBs are extreme: they come from beyond the Milky Way, with most localized FRBs at redshifts 0.03–0.5 (roughly 0.4–6 billion light-years). That puts them far beyond supernova remnants in our Galaxy (thousands of light-years) but closer than the most distant quasars (z > 7, over 13 billion light-years). Some FRBs may originate at higher redshifts, but current localization efforts favor moderate distances. This range makes FRBs useful for probing the missing baryons in the intergalactic medium, because the DM measures the integrated electron density along the line of sight.

Why Redshift Matters for Cosmology

Measuring FRB redshifts allows astronomers to map the distribution of matter in the universe. The dispersion measure of an FRB, combined with its redshift, gives a direct measurement of the cosmic electron density. This can be used to test models of the intergalactic medium and to weigh the baryonic matter that is otherwise hard to detect. Additionally, if a large sample of FRBs with redshifts is collected, the average DM–redshift relation can constrain the Hubble constant (H₀) independently of other methods, helping resolve the ‘Hubble tension’ between local and early-universe measurements.

Limits and Uncertainties

The main uncertainty in converting redshift to distance is the value of H₀ and the assumed cosmology (flat ΛCDM with Ω_m=0.3, Ω_Λ=0.7). Changing H₀ by ±5 km/s/Mpc changes the distance by about ±7%. The integral also assumes a smooth universe; in reality, matter clumps, but the effect on the average distance is small. For individual FRBs, the host galaxy redshift may have measurement errors of ±0.01, which translates to a few percent in distance. The DM-derived distances have larger errors, often 20–50%.

Frequently Asked Questions

What is the farthest FRB ever detected?

As of 2024, the most distant localized FRB is FRB 20220610A, found in a galaxy at redshift z ≈ 1.0, about 8 billion light-years away. That means the burst occurred when the universe was roughly half its current age.

Can we measure redshift directly from the FRB signal?

Sometimes. If the FRB has a multi-frequency spectrum, the intrinsic frequency structure can be fitted to a model, giving a crude redshift. But this method is unreliable, so the standard is to identify the host galaxy and measure its redshift.

How does dispersion measure relate to redshift?

DM is the column density of free electrons along the line of sight. In a homogeneous universe, DM increases with distance, but because electrons are clumped in galaxies and filaments, the DM–redshift relation has large scatter. A typical DM of 500 pc/cm³ corresponds to a redshift of roughly 0.3–0.5, but the uncertainty is large.

Why do FRB distances matter?

They help us locate the sources (which may be magnetars or other exotic objects) and use FRBs as cosmological probes. By measuring the DM–redshift relation, we can measure the baryon density in the intergalactic medium and test models of dark energy.

Is the speed of light constant for FRBs?

Yes, all electromagnetic radiation travels at the same speed in a vacuum, but the arrival time of different frequencies is delayed by the ionized medium—this is dispersion. The speed itself is constant, but the group velocity is slightly less than c in plasma, which is why lower frequencies arrive later.

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in z | Redshift (z) | - | 0.5 | 0 | 10
in H0 | Hubble constant (km/s/Mpc) | - | 70 | 50 | 100
out dist | Comoving distance (billion ly) | - | (299792.458 / H0) * integral(1/(sqrt(0.3*(1+x)^3+0.7)), x, 0, z) / 1000 | 3
out lookback | Lookback time (billion yr) | - | (1/H0) * integral(1/((1+x)*sqrt(0.3*(1+x)^3+0.7)), x, 0, z) * 977.8 | 2