🔭 astrophysics

Search for Lorentz Invariance Violation with spectral lags of GRB 190114C using profile likelihood

This paper presents a frequentist profile likelihood analysis of GRB 190114C spectral lag data from Fermi-GBM, yielding Lorentz invariance violation constraints that align with previous Bayesian estimates and establish best-fit energy scales of $2.81^{+0.50}_{-0.37}\times 10^{14}$ GeV for linear and $9.85^{+0.84}_{-0.60}\times 10^{5}$ GeV for quadratic violations.

Original authors: Vyaas Ramakrishnan, Shantanu Desai

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vyaas Ramakrishnan, Shantanu Desai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Testing the Universe's Speed Limit

Imagine the universe has a strict speed limit: the speed of light. For over a century, physicists have believed this limit is absolute and unbreakable, no matter how much energy a particle has. This rule is called Lorentz Invariance.

However, some theories about "Quantum Gravity" (a theory trying to unite the big stuff like stars with the tiny stuff like atoms) suggest that at incredibly high energies, this speed limit might get a little "wobbly." Maybe a super-energetic photon (a particle of light) travels just a tiny bit slower than a low-energy photon. If this is true, it would mean the universe's speed limit isn't actually a hard wall, but more like a speed bump that depends on how fast you're going.

This paper is a detective story trying to find out if that "wobbly speed limit" exists.

The Crime Scene: A Cosmic Explosion (GRB 190114C)

The detectives (the authors) looked at a specific event: GRB 190114C. This was a massive explosion of a star (a Gamma-Ray Burst) that happened in 2019.

Think of this explosion like a firework display in space. When a firework goes off, it sends out sparks of different colors (energies) at the same time.

Low-energy sparks are like the red embers.
High-energy sparks are like the bright white sparks.

If the universe's speed limit is perfect, all the sparks should arrive at Earth at the exact same time. But if the "wobbly speed limit" theory is true, the super-fast, high-energy sparks might get slightly delayed compared to the slower, low-energy ones. This delay is called a "spectral lag."

The Investigation: Two Ways to Solve the Puzzle

The authors re-analyzed data from this explosion that had been studied before. The previous study used a method called Bayesian analysis (let's call it the "Gut Feeling" method, which weighs probabilities based on prior knowledge).

This team wanted to try a different method called Frequentist analysis using something called Profile Likelihood.

The Analogy: Imagine you are trying to find the best setting on a radio to hear a song clearly.
- The signal is the "wobbly speed limit" (the thing you are looking for).
- The static is the natural noise of the explosion itself (astrophysical effects).
- The Profile Likelihood method is like turning the volume knob (the nuisance parameters) up and down to see if you can isolate the signal. It asks: "If I assume the signal is at this specific energy level, what is the best way to explain the static? And does that explanation make sense?"

They did this mathematically to strip away the "static" and see if a clear "signal" remained.

The Findings: A False Alarm?

Here is the twist in the story:

The "Smoking Gun": When they ran their math, they found a very strong "signal." It looked like the high-energy photons were indeed arriving late. Their calculations suggested the "wobbly speed limit" happens at a specific energy scale.
The Match: Interestingly, their "Gut Feeling" (Frequentist) results matched the previous "Gut Feeling" (Bayesian) results almost perfectly. This is good news; it means both mathematical methods agree on what the data looks like.
The Reality Check: However, the authors immediately put on their "Skeptic's Hat." They compared their "smoking gun" to other, more powerful investigations.
- Other telescopes (like MAGIC and LHAASO) have looked at even more powerful explosions and found no delay at all. Those telescopes set a much stricter speed limit.
- The "signal" found in this paper (the delay) is actually ruled out by those stricter limits.

The Verdict: It's Not the Speed Limit, It's the Firework

So, what caused the delay they saw in GRB 190114C?

The authors conclude that the delay wasn't because the universe's speed limit was broken. Instead, it was likely due to intrinsic astrophysical mechanisms.

The Metaphor: Imagine a runner (the photon) getting tired. The delay wasn't because the track was broken (Lorentz violation); it was because the runner started running slower naturally due to the nature of the race (the physics of the explosion itself).

Why Write This Paper Then?

If they didn't find the "wobbly speed limit," why publish?

Methodology Check: The main goal was to prove that the Profile Likelihood method works. They showed that this "Frequentist" approach gives the same results as the "Bayesian" approach. It's like proving that two different types of calculators give the same answer. This is important for future scientists who might want to use this specific tool.
Open Science: They made their code public so anyone can check their math.
Honesty: They explicitly stated that while their math found a "best fit" for a broken speed limit, other evidence proves the speed limit is actually fine. They are not claiming to have found new physics; they are claiming to have successfully tested a new mathematical tool.

Summary in One Sentence

The authors used a new mathematical "magnifying glass" to look at a cosmic explosion, found a pattern that looked like a broken speed limit, but realized it was just the explosion's own noise, while proving that their new magnifying glass works just as well as the old ones.

1. Problem Statement

The paper addresses the search for Lorentz Invariance Violation (LIV), a potential signature of Quantum Gravity (QG). According to some QG theories, the speed of light in a vacuum may depend on photon energy, leading to a time delay (spectral lag) between high-energy and low-energy photons from distant astrophysical sources like Gamma-Ray Bursts (GRBs).

The Challenge: The observed spectral lag ( $\Delta t_{obs}$ ) is a combination of the LIV-induced lag ( $\Delta t_{LIV}$ ) and an intrinsic astrophysical lag ( $\Delta t_{int}$ ) caused by the GRB's emission mechanism.
The Gap: Previous analyses of GRB 190114C (specifically the work by Du et al., referred to as D20) used Bayesian inference to marginalize over the unknown astrophysical parameters. This paper aims to perform a complementary frequentist analysis using profile likelihood to handle these "nuisance" parameters, verifying if the results are consistent with the Bayesian approach.

2. Methodology

The authors re-analyzed the spectral lag data for GRB 190114C (redshift $z \approx 0.4245$ ) obtained from the Fermi-GBM instrument.

A. Physical Model

The expected time lag is modeled as:
$\Delta t_{exp} = (1 + z)\Delta t_{int} + \Delta t_{LIV}$

Intrinsic Lag ( $\Delta t_{int}$ ): Modeled as a power-law function of energy ( $E$ ):
$\Delta t_{int}(E) = \tau \left[ \left(\frac{E_0}{\text{keV}}\right)^{-\alpha} - \left(\frac{E}{\text{keV}}\right)^{-\alpha} \right]$
Where $\tau$ (scaling factor) and $\alpha$ (exponent) are free nuisance parameters, and $E_0 = 12.5$ keV.
LIV-Induced Lag ( $\Delta t_{LIV}$ ): Modeled for sub-luminal propagation in a flat $\Lambda$ CDM universe:
$\Delta t_{LIV} = - \left( \frac{1+n}{2H_0} \right) \left( \frac{E^n - E_0^n}{E_{QG,n}^n} \right) \frac{1}{(1+z)^n} \int_0^z \frac{(1+z')^n}{h(z')} dz'$
Where $n=1$ (linear) and $n=2$ (quadratic) represent the LIV models, and $E_{QG,n}$ is the energy scale of LIV (the parameter of interest).

B. Statistical Framework (Profile Likelihood)

Instead of marginalizing over nuisance parameters (as in Bayesian analysis), the authors used Profile Likelihood:

Likelihood Function: A Gaussian likelihood was constructed based on the 19 observed spectral lags and their uncertainties.
$\mathcal{L} = \prod_{i=1}^{N} \frac{1}{\sigma_{tot,i}\sqrt{2\pi}} \exp\left( -\frac{[\Delta t_i - \Delta t_{exp}]^2}{2\sigma_{tot,i}^2} \right)$
$\chi^2$ Minimization: The authors defined $\chi^2 = -2 \ln \mathcal{L}$ .
Profiling: For a grid of fixed $E_{QG}$ values (ranging from $10^6$ to $10^{19}$ GeV), they minimized $\chi^2$ with respect to the nuisance parameters ( $\tau$ and $\alpha$ ).
Confidence Intervals: Using Wilks' theorem, confidence intervals were derived from $\Delta \chi^2 = \chi^2(E_{QG}) - \chi^2_{min}$ $Δ χ^{2} = χ^{2} (E_{QG}) - χ_{min}^{2}$ .
- $68.3\%$ CL corresponds to $\Delta \chi^2 = 1.0$ .
- $95\%$ CL corresponds to $\Delta \chi^2 = 3.84$ .

3. Key Contributions

Methodological Validation: This is the first frequentist analysis of GRB 190114C spectral lags specifically using profile likelihood to handle astrophysical nuisance parameters, providing a direct comparison to the existing Bayesian results (D20).
Synthetic Data Verification: The authors generated synthetic data incorporating the best-fit parameters to verify that the profile likelihood technique could successfully recover the input $E_{QG}$ values, confirming the robustness of their statistical method.
Open Science: The analysis code has been made publicly available on GitHub.

4. Results

A. Best-Fit Parameters

The analysis found a global minimum for $\chi^2$ well below the Planck scale ( $10^{19}$ GeV) for both LIV models.

Model	Best-Fit $E_{QG}$ (68.3% CL)	Best-Fit $E_{QG}$ (95% CL)	Significance vs. No-LIV
Linear ( $n=1$ )	$2.81^{+0.50}_{-0.37} \times 10^{14}$ GeV	$2.81^{+0.99}_{-0.73} \times 10^{14}$ GeV	$\sim 19\sigma$
Quadratic ( $n=2$ )	$9.85^{+0.84}_{-0.60} \times 10^{5}$ GeV	$9.85^{+1.62}_{-1.25} \times 10^{5}$ GeV	$\sim 19\sigma$

Goodness of Fit: The reduced $\chi^2$ values were $< 1$ (5.46/16 for linear, 5.09/16 for quadratic), indicating a good fit to the data, assuming the uncertainties are not overestimated.
Agreement with Bayesian Analysis: The frequentist best-fit values and confidence intervals agree within $1\sigma$ with the marginalized credible intervals reported in D20 (the Bayesian study).

B. Synthetic Data Recovery

Using synthetic data generated with the best-fit $E_{QG}$ values:

Linear: Recovered $2.72^{+0.43}_{-0.40} \times 10^{14}$ GeV (Input: $2.81 \times 10^{14}$ ).
Quadratic: Recovered $1.05^{+0.07}_{-0.09} \times 10^{6}$ GeV (Input: $9.85 \times 10^{5}$ ).
This confirms the profile likelihood method is unbiased for this dataset.

5. Significance and Conclusion

Methodological Consistency: The primary conclusion is that frequentist analysis using profile likelihood yields results consistent with Bayesian inference for GRB 190114C. This validates the use of profile likelihood as a robust alternative for handling astrophysical nuisance parameters in LIV searches.
Physical Interpretation: Despite the high statistical significance ( $\sim 19\sigma$ $\sim 19 σ$ ) of the LIV signal within this specific dataset, the authors do not claim evidence for LIV.
- The derived $E_{QG}$ values are ruled out by stricter lower limits from other observations, specifically the LHAASO observation of GRB 221009A ( $E_{QG} \geq 10 E_{Pl}$ ) and MAGIC telescope data for GRB 190114C itself ( $E_{QG} \gtrsim 0.6 \times 10^{19}$ GeV).
- Therefore, the spectral lags observed in the keV range for GRB 190114C are most likely due to intrinsic astrophysical mechanisms, not LIV.
Future Outlook: The work serves as a "proof of principle" for applying profile likelihood in LIV searches. It highlights that while single GRB analyses can show strong statistical preferences for specific LIV scales, global constraints from higher-energy (TeV) observations and multiple GRB stacks are necessary to establish physical limits.