Primordial Black Hole signatures from femtolensing and spectral fringe of Gamma Ray Bursts

TL;DR

This work investigates whether asteroid-mass primordial black holes (PBHs) can constitute dark matter by searching for femtolensing fringes in gamma-ray burst (GRB) spectra observed with Swift XRT, using a wave-optics framework that includes finite-source effects. The authors implement a seven-parameter joint model combining PBH lens properties $(M_{ m PBH}, z_L, y_0)$ and GRB Band spectral parameters $(A, \alpha_1, \alpha_2, E_0)$, computing the wave-optics amplification $F(y,\Omega)$ and the resulting magnification $\mu=|F|^2$, and compare this to a null BAND spectrum. An analysis of 106 GRBs finds 22 events with oscillatory fringes that could indicate femtolensing, while 84 do not, enabling upper bounds on $f_{ m PBH}$ via an optical-depth approach; robust constraints require a small GRB emission size, $a_s \lesssim 5\times10^7$ m for $M_{ m PBH}\sim 5\times10^{-15}M_Msun$. Two candidate events (GRB091029 and GRB101219B) are highlighted as potential Milky Way PBH lenses with $M_{ m PBH}\sim (2.3-2.95)\times10^{-14}M_Msun$ at $z_L\sim 11$ kpc, illustrating the method's sensitivity. Overall, the study demonstrates the feasibility of using GRB spectral fringes to probe the asteroid-mass PBH DM window and provides a framework for tighter constraints with larger future GRB samples.

Abstract

Femtolensing of gamma ray bursts (GRBs) are vastly studied to constrain the primordial black hole (PBHs) lighter than $10^{-13}$ solar mass and may close the window for PBH dark matter. In this case, wave optics formalism is required and carefully implemented in our analysis. Incorporating the GRB observational data from Swift XRT, we perform the statistic analysis of PBH lensing, comparing with null hypothesis where BAND model is used to parametrize the GRB spectrum. We found few GRB data manifest the spectral fringe which characterize the feature of femtolensing by PBHs, and the analysis shows moderate statistical preference in terms of goodness of fit. Conversely, since most of the fitting to GRB spectral data do not improved with PBH lensing, we utilize to obtain upper bound on the PBH fractional abundance with respect to dark matter. However, the robust constraint cannot be achieved, unless the size of GRBs are smaller than $5\times10^{7}$ m for PBH mass around $5\times10^{-15}$ solar mass.

Figures (16)

• Figure 1: The orange curve represent the geometric optics approximation Eq.(\ref{['eq:3.14']}) and blue curve represent the wave optics Eq.(\ref{['eq:2.18']}) for point-like source and point-like lens. $\Omega$ is calculated by Eq.(\ref{['eq:3.8']}). The geometric optics approximation is exact with wave optics at $\Omega\gg\frac{1}{y}$. $\mu$ will approach 1 when $\Omega\rightarrow0$.
• Figure 2: The magnification $\mu$ of Eq.(\ref{['eq:2.18']}) is shown as a function of $y$ with different $\Omega$. We can see as $y$ increases, $\mu$ will approaches to 1 regardless of the value of $\Omega$.
• Figure 3: The comparison between the exact form Eq.(\ref{['eq:2.22']}) and the approximation form Eq.(\ref{['eq:2.27']}) with different $y_{0}$ and $\sigma_{y}$. The curve is the integrand of Eq.(\ref{['eq:2.22']}) and Eq.(\ref{['eq:2.27']}) that include the factor appearing before the integration. We can see that when $yy_{0}\gg\sigma_{y}^{2}$ the approximation holds. Therefore, consider our require that $\sigma_{y}\ll1$, the upper limit of $y_{0}+3\sigma_{y}$ will not lose accuracy. The peak of each curve approximately located at $y_{0}$ for different situations.
• Figure 4: The difference of the first local minimum of $\bar{\mu}$ used to determine the oscillation property of $\bar{\mu}$. The left panel is classified to oscillation because its local minimum is smaller than $1$. The local minimum of the right panel is exceeds than $1$, so it be classified to no oscillation. The value of $y_{0}$ is $1.9$ for GRB050406 and $0.2$ for GRB051016.
• Figure 5: The two of GRBs in Table \ref{['tab:label1']} that have oscillation pattern and the oscillation appear in the spectrum. Green error bar is the observed GRB data from SWIFT XRT, blue and orange curve is the fitting BAND model with and without lensing effect. GRB091029 (GRB101219B) has $T_{90}=39.2\,s$ ($T_{90}=34\,s$) and it is located $1.6\times10^{6}\,\rm{kpc}$ ($1.3\times10^{6}\,\rm{kpc}$) away from us. According to our best fit, it is lensed by PBH with mass $2.33\times10^{-14}M_{\odot}$ ($2.95\times10^{-14}M_{\odot}$) at $11\,\rm{kpc}$ ($11\,\rm{kpc}$) from us and $y_{0}=2.1$ ($y_{0}=1.6$).