05 - Fast Radio Burst Observation Applications

Introduction

Fast Radio Bursts (FRB) are among the most mysterious phenomena in the universe:

• Ultra-short: duration $\sim$ milliseconds
• Ultra-bright: energy equivalent to several days of total solar radiation
• Ultra-distant: distance $\sim$ billions of light-years (cosmological distance)

More importantly, FRB signals traverse the entire universe, carrying information about all matter and vacuum properties along their propagation path. In unified time scale theory, FRBs become natural laboratories for testing vacuum polarization, quantum gravity effects, and cosmological predictions of unified time scale.

Source theory:

• euler-gls-extend/unified-phase-frequency-metrology-frb-delta-ring-scattering.md
• euler-gls-info/16-phase-frequency-unified-metrology-experimental-testbeds.md

This chapter will show how to use FRB observations to verify the theory and provide methods for computing windowed upper bounds.

FRB as Cosmological-Scale Scattering Experiment

FRB Propagation Picture

graph LR
    A["FRB Source<br/>z ~ 1"] --> B["Intergalactic Medium<br/>IGM"]
    B --> C["Galactic Halo<br/>CGM"]
    C --> D["Milky Way<br/>ISM"]
    D --> E["Earth<br/>Telescope"]

    style A fill:#ffe8e8
    style E fill:#e8f5e8

Media traversed by signal:

1. Intergalactic Medium (IGM): low-density plasma, $n_e\sim 10^{-7}$ cm${}^{-3}$
2. Circumgalactic Medium (CGM): around host galaxy, $n_e\sim 10^{-3}$ cm${}^{-3}$
3. Galactic Interstellar Medium (ISM): our Milky Way, $n_e\sim 0.03$ cm${}^{-3}$
4. Earth’s Ionosphere: $n_e\sim 10^6$ cm${}^{-3}$ (daytime), variable

Dispersion Measure

Signals at different frequencies propagate at different speeds:

$$v_g(\omega )=c\left(1-\frac{{\omega }_p^2}{{\omega }^2}\right)$$

where ${\omega }_p=\sqrt{4\pi n_e{e}^2/m_e}$ is the plasma frequency.

Dispersion Measure:

$$\text{DM}={\int }_0^D{n}_e(z)\mathrm{d}\ell [{\text{pc cm}}^{-3}]$$

Observationally, arrival time difference between frequencies:

$$\Delta t({\omega }_1,{\omega }_2)=\frac{e^2}{2\pi m_{e}c}\text{DM}\left(\frac{1}{{\omega }_1^2}-\frac{1}{{\omega }_2^2}\right)$$

Typical FRB: $\text{DM}\sim 300-1500$ pc cm${}^{-3}$.

Phase Accumulation: From Dispersion to Unified Time Scale

Phase formula:

$$\Phi (\omega )={\int }_0^{D}k(\omega ,z)\mathrm{d}\ell ={\int }_0^D\frac{\omega n(\omega ,z)}{c}\mathrm{d}\ell$$

where $n(\omega ,z)$ is the refractive index.

Known contributions:

$$n_{\text{plasma}}(\omega )=\sqrt{1-\frac{{\omega }_p^2}{{\omega }^2}}\approx 1-\frac{{\omega }_p^2}{2{\omega }^2}$$

New physics contributions:

Unified time scale theory predicts vacuum polarization introduces correction:

$$n_{\text{total}}(\omega )=n_{\text{plasma}}+\delta n_{\text{QED}}(\omega )+\delta n_{\text{new}}(\omega )$$

where:

• $\delta n_{\text{QED}}$: QED vacuum polarization (Heisenberg-Euler effect)
• $\delta n_{\text{new}}$: new physics (e.g., axions, hidden photons, quantum gravity)

Theoretical Predictions of Vacuum Polarization

Curved Spacetime QED

In weak curvature background, effective vacuum action:

$${\mathcal{L}}_{\text{eff}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{m_e^2}\left(aR{F}_{\mu \nu }{F}^{\mu \nu }+b{R}_{\mu \nu }{F}^{\mu \alpha }{F}_{\alpha }^{\nu }+c{R}_{\mu \nu \alpha \beta }{F}^{\mu \nu }{F}^{\alpha \beta }\right)$$

Coefficients (one-loop):

$$a\sim b\sim c\sim \frac{{\alpha }_{\text{em}}}{\pi }$$

Dispersion relation correction:

$$\delta n\sim \frac{{\alpha }_{\text{em}}}{\pi }{\lambda }_e^2\mathcal{R}$$

where:

• ${\lambda }_e=\hslash /(m_{e}c)\approx 3.86\times 10^{-13}$ m (Compton wavelength)
• $\mathcal{R}$: curvature scalar

Cosmological Background

Friedmann-Robertson-Walker (FRW) metric:

$$\mathrm{d}{s}^2=-\mathrm{d}{t}^2+a^2(t)(\mathrm{d}{\chi }^2+{\chi }^2\mathrm{d}{\Omega }^2)$$

Ricci scalar:

$$R=6\left(\frac{\ddot{a}}{a}+\frac{{\dot{a}}^2}{a^2}\right)$$

Current universe ($\Lambda$CDM):

$$|R|\sim H_0^2/c^2\sim (10^{-26}\text{ m}{)}^{-2}$$

Order of magnitude estimate:

$$\delta n_{\text{QED}}\sim \frac{{\alpha }_{\text{em}}}{\pi }{\lambda }_e^2\cdot H_0^2/c^2\approx 2\times 10^{-53}$$

Extremely tiny!

Phase Accumulation Upper Bound

Distance $D\sim 1$ Gpc = $3\times 10^{25}$ m, frequency $\omega /(2\pi )\sim 1$ GHz:

$$\Delta {\Phi }_{\text{QED}}={\int }_0^D\frac{\omega }{c}\delta n_{\text{QED}}\mathrm{d}\ell \sim \frac{\omega D}{c}\delta n_{\text{QED}}\approx 1.2\times 10^{-53}\text{ rad}$$

Conclusion: Unmeasurable under any realistic observation conditions!

But we can give an upper bound.

Construction of Windowed Upper Bound

Observation Equation

FRB frequency-domain complex amplitude:

$$E(\omega )=A(\omega )e^{i\Phi (\omega )}$$

Phase decomposition:

$$\Phi (\omega )={\Phi }_{\text{DM}}(\omega )+{\Phi }_{\text{scattering}}(\omega )+{\Phi }_{\text{intrinsic}}(\omega )+{\Phi }_{\text{new}}(\omega )$$

Known terms:

1. ${\Phi }_{\text{DM}}=\frac{e^2\text{DM}}{2\pi m_{e}c{\omega }^2}$ (plasma dispersion)
2. ${\Phi }_{\text{scattering}}$: multi-path scattering (modeled as $\propto {\omega }^{-4}$)
3. ${\Phi }_{\text{intrinsic}}$: source intrinsic phase (narrowband)

Unknown term:

${\Phi }_{\text{new}}$: new physics contribution

Windowed Residual

Apply PSWF window function $W_j(\omega )$:

$$R_j=⟨W_j,\Phi -{\Phi }_{\text{model}}⟩=\int W_j(\omega )[\Phi (\omega )-{\Phi }_{\text{model}}(\omega )]\mathrm{d}\omega$$

where ${\Phi }_{\text{model}}={\Phi }_{\text{DM}}+{\Phi }_{\text{scattering}}+{\Phi }_{\text{intrinsic}}$.

Generalized least squares (GLS):

$${\chi }^2=\sum _{ij}{R}_i[C^{-1}{]}_{ij}{R}_j$$

$C_{ij}$ is the covariance matrix (includes measurement noise and systematics).

Upper Bound Extraction

Profile likelihood:

For new physics parameter $\delta n$:

$$\Delta {\chi }^2(\delta n)={\chi }^2(\delta n)-{\chi }^2(\delta n_{\text{best}})$$

95% confidence upper bound:

$$\Delta {\chi }^2(\delta n_{95})=3.84$$

Gives:

$$|\delta n|<\delta n_{95}$$

Unified time scale interpretation:

$$\delta \kappa =\frac{1}{\pi }\frac{\mathrm{d}{\Phi }_{\text{new}}}{\mathrm{d}\omega }$$

Upper bound:

$$|\delta \kappa (\omega )|<\frac{\delta n_{95}c}{\omega L}$$

CHIME/FRB Data Application

CHIME Telescope

Parameters:

• Frequency range: 400-800 MHz
• Frequency resolution: $\Delta \omega /(2\pi )\approx 390$ kHz
• Number of channels: 1024
• Time resolution: $\sim 1\mu$s
• Daily FRB detections: $\sim 3$

Typical FRB Signal

FRB 20121102A (famous repeater):

• DM: $557$ pc cm${}^{-3}$
• Redshift: $z\approx 0.19$
• Distance: $\sim 800$ Mpc
• Signal-to-noise ratio: $>100$

Data Processing Workflow

graph TB
    A["Raw Baseband<br/>Complex Voltage Time Series"] --> B["Dedispersion<br/>Coherent dedispersion"]
    B --> C["Structure Maximization<br/>Optimize DM"]
    C --> D["Phase Extraction<br/>φ(ω)"]
    D --> E["Foreground Modeling<br/>DM + scattering"]
    E --> F["Windowed Readout<br/>PSWF Coefficients"]
    F --> G["Residual Analysis"]
    G --> H["Upper Bound Calculation"]

    style A fill:#e1f5ff
    style H fill:#e8f5e8

Covariance estimation:

Three-channel bootstrap:

1. Off-source: telescope points to empty sky, measures system noise
2. Off-band: frequency bands outside FRB signal, measures RFI
3. Sidelobe: outside telescope main lobe, measures environmental background

Synthetic covariance:

$$C_{ij}=C_{ij}^{\text{thermal}}+C_{ij}^{\text{RFI}}+C_{ij}^{\text{sys}}$$

Window Function Construction

Shannon number:

$$N_0=2\times 1024\times 0.195\approx 400$$

(Bandwidth $\approx 400$ MHz, normalized to sampling rate)

Main leakage upper bound ($\epsilon =10^{-3}$):

$$1-{\lambda }_0<10\exp \left(-\frac{(400-7{)}^2}{{\pi }^2\log (20025)}\right)\approx 10^{-1700}$$

Far exceeds requirements! Actual limitation is systematics.

Orthogonalization:

Weighted Gram-Schmidt with $C^{-1}$:

$${\tilde{W}}_j=W_j-\sum _{k<j}\frac{⟨W_j,W_k{⟩}_C}{⟨W_k,W_k{⟩}_C}{\tilde{W}}_k$$

Verify whitening: $⟨{\tilde{W}}_i,{\tilde{W}}_j{⟩}_C\approx {\delta }_{ij}$.

Systematic Basis Modeling

Basis functions:

$$\{{\Pi }_p(\omega )\}=\left\{1,\omega ,{\omega }^{-1},{\omega }^{-2},{\omega }^{-4},\log \omega \right\}$$

Physical meaning:

• $1$: overall phase shift (instrument delay)
• $\omega$: time offset
• ${\omega }^{-1}$: residual DM
• ${\omega }^{-2}$: DM correction
• ${\omega }^{-4}$: scattering tail
• $\log \omega$: nonlinear correction to dispersion relation

Robustness test:

Effect of including ${\omega }^{-1}$ basis on results:

• Model A: without ${\omega }^{-1}$
• Model B: with ${\omega }^{-1}$

Take envelope: $\delta n_{95}=\max (\delta n_{95}^A,\delta n_{95}^B)$

Expected Upper Bound

Assumptions:

• Number of events: $N_{\text{FRB}}=100$
• Typical distance: $D=1$ Gpc
• Average SNR: $\text{SNR}=50$

Error scaling:

$$\delta n_{95}\propto \frac{1}{\sqrt{N_{\text{FRB}}}}\cdot \frac{1}{\text{SNR}}$$

Order of magnitude:

$$\delta n_{95}\sim 10^{-5}\times \frac{c}{\omega D}\sim 10^{-20}$$

Comparison with QED prediction:

$$\frac{\delta n_{95}}{\delta n_{\text{QED}}}\sim \frac{10^{-20}}{10^{-53}}=10^{33}$$

QED vacuum polarization is far below detection threshold!

But for other new physics (e.g., axion dark matter), the upper bound is meaningful.

Cross-Platform Unified Scale

Joint Analysis of FRB + δ-Ring

Idea:

Two completely different scale systems, if governed by the same unified time scale ${\kappa }_{\text{univ}}(\omega )$, should satisfy consistency conditions.

Windowed residual relation:

$$R_{\text{FRB}}(W_j)={\lambda }_j{R}_{\delta }(W_j)+\mathcal{O}({\epsilon }_j)$$

where ${\lambda }_j$ is a geometric factor (can be precomputed).

Consistency test:

For multiple window functions $\{W_j\}$, test:

$${\chi }_{\text{consistency}}^2=\sum _j\frac{[R_{\text{FRB}}(W_j)-{\lambda }_j{R}_{\delta }(W_j){]}^2}{{\sigma }_j^2}$$

If ${\chi }^2<{\chi }_{\text{crit}}^2$ (e.g., 95% quantile), then the two platforms are consistent.

Geometric Factor Calculation

FRB side:

$${\lambda }_{\text{FRB}}=\frac{D_{\text{FRB}}}{c}$$

δ-ring side:

$${\lambda }_{\delta }=\frac{L}{v_F}$$

($v_F$: Fermi velocity, analogous to “light speed”)

Ratio:

$$\frac{{\lambda }_{\text{FRB}}}{{\lambda }_{\delta }}\sim \frac{10^{25}\text{ m}/c}{10^{-4}\text{ m}/v_F}\sim 10^{30}$$

If ${\kappa }_{\text{univ}}$ is truly universal, this ratio should be independent of frequency!

Statistical Analysis of FRBs

Advantages of Repeaters

FRB 20121102A: observed $>100$ bursts

Advantages:

1. Self-calibration: averaging multiple bursts eliminates random noise
2. Temporal evolution: monitor source environment changes
3. Frequency coverage: different bursts cover different frequency bands

Stacking analysis:

$${\Phi }_{\text{stack}}(\omega )=\frac{1}{N}\sum _{i=1}^N{\Phi }_i(\omega )$$

Noise reduction $\sim 1/\sqrt{N}$.

Group Delay Spectrum

Definition:

$${\tau }_g(\omega )=\frac{\mathrm{d}\Phi }{\mathrm{d}\omega }$$

Relation to unified time scale:

$${\tau }_g(\omega )=\frac{2\pi }{c}\kappa (\omega )$$

(ignoring geometric factors)

Frequency evolution:

If $\kappa (\omega )=\text{const}$ (dispersionless new physics), then ${\tau }_g$ is flat.

If dispersion exists:

$${\tau }_g(\omega )\approx {\tau }_0+{\tau }_1\omega +{\tau }_2{\omega }^2+\cdots$$

Fitting coefficients $\{{\tau }_i\}$ give Taylor expansion of $\kappa$.

Polarization Analysis

FRB signals often show high polarization ($>80\%$).

Faraday rotation:

$$\Delta \Psi (\omega )=\text{RM}\times \frac{c^2}{{\omega }^2}$$

RM: Rotation Measure

Joint constraint:

Simultaneously fit $\Phi (\omega )$ (total phase) and $\Psi (\omega )$ (polarization angle) to improve sensitivity.

Summary

This chapter demonstrates FRB applications as cosmological-scale scattering experiments:

Key Conclusions

1. QED vacuum polarization unmeasurable: $\delta n_{\text{QED}}\sim 10^{-53}$, far below observation threshold
2. Windowed upper bound feasible: use PSWF method to set constraints on new physics $\delta n<10^{-20}$
3. Cross-platform consistency: FRB and δ-ring can verify universality of unified time scale

Experimental Status

• Current: CHIME has observed $>1000$ FRBs, data publicly available
• Future: FAST, SKA, etc. will provide higher sensitivity and frequency resolution

Theoretical Significance

FRBs verify the applicability of unified time scale theory at cosmological scales, forming a multi-scale verification network with microscopic (δ-ring) and mesoscopic (optical cavity) experiments.

The next chapter will assess technical feasibility and future prospects of each experimental scheme.

References

[1] CHIME/FRB Collaboration, “First CHIME/FRB Catalog,” ApJS 257, 59 (2021).

[2] Hessels, J. W. T., et al., “FRB 121102 Bursts Show Complex Time–Frequency Structure,” ApJL 876, L23 (2019).

[3] Hollowood, T. J., Shore, G. M., “Causality, renormalizability and ultra-high energy gravitational scattering,” Nucl. Phys. B 795, 138 (2008).

[4] Drummond, I. T., Hathrell, S. J., “QED vacuum polarization in a background gravitational field,” Phys. Rev. D 22, 343 (1980).

[5] euler-gls-extend/unified-phase-frequency-metrology-frb-delta-ring-scattering.md [6] euler-gls-info/16-phase-frequency-unified-metrology-experimental-testbeds.md