EPL 89 39002 (6pp)

Tachyonic spectral fits of γ-ray bursts

R. Tomaschitz

Department of Physics, Hiroshima University - 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan

E-mail: tom@geminga.org

Received 2 December 2009, accepted for publication 14 January 2010
Published 17 February 2010

Abstract. Evidence for superluminal radiation in γ-ray burst (GRB) spectra is pointed out. The spectral maps of GRB 941017, GRB 990123, and GRB 990104 are analyzed. The superluminal radiation modes are generated by the shock-heated ultra-relativistic source plasma. The tachyonic radiation field is a real Proca field with negative mass-square, coupled to the electron gas by a frequency-dependent fine-structure constant. At GeV energies, the coupling constant approaches a limit value, so that the radiation field is minimally coupled to the electron current. In the soft γ-ray band, the interaction with the GRB plasma becomes nonlocal, due to the varying coupling strength depending on the energy of the radiated modes. The spectral fitting with tachyonic flux densities generated by nonlocal plasma currents is explained. Estimates of the tachyonic luminosity, temperature, and internal energy of the electronic source plasma are obtained from the spectral fits.

PACS numbers: 98.70.Rz, 52.27.Ny, 95.30.Gv


γ-ray burst spectra [15] are scrutinized in search for superluminal radiation modes, by performing tachyonic spectral fits to the γ-ray bursts GRB 941017, GRB 990123, and GRB 990104. Tachyonic flux densities have shown to be efficient in reproducing GeV and TeV spectra of Galactic sources such as supernova remnants and of active galactic nuclei [6]. Here, we study lower energies, burst spectra in the 30 keV–100 MeV range. In this energy band, we have to take into account the frequency dependence of the tachyonic fine-structure constant αq(ω), by which the tachyonic radiation field couples to the shocked source plasma. At GeV energies, αq(ω) attains a limit value, so that the radiation field is coupled to the electron current by minimal substitution [7]. In the soft–γ-ray band, the energy-dependent fine-structure constant results in a nonlocal interaction of the radiation modes with the electron plasma. The goal is to quantify this interaction by way of spectral fits. This is possible since the varying coupling strength is manifested in the measured flux density of the GRBs as well as in the burst duration.

We study the long-range dispersion caused by this nonlocal coupling of tachyonic radiation modes to the electron current, and analyze the effect of the frequency-dependent fine-structure constant on the spectral functions of an ultra-relativistic electron plasma. We assemble the transversal and longitudinal tachyonic flux components radiated by the shock-heated electron gas, perform spectral fits to the above mentioned GRBs, and obtain estimates of the tachyonic luminosity and internal energy of the source plasma.

Superluminal wave modes generated by nonlocal electron currents

The tachyonic radiation field in vacuum is a real vector field with negative mass-square, satisfying the Proca equation (Δ − ∂2/∂t2 + mt2)Aμ = −jμ [7, 8]. mt is the mass of the superluminal Proca field Aμ, and jμ = (−ρ, j) the subluminal electron current. Tachyonic radiation implies superluminal signal transfer [913], the energy quanta propagating faster than light in vacuum, in contrast to rotating superluminal light sources emitting vacuum Cherenkov radiation [1416]. Tachyonic wave modes are a kind of photons with negative mass-square, and should not be confused with the apparent superluminal plasma flow of quasar jets due to relativistic beaming [17, 18] (whose intrinsic flow speed is safely subluminal), or with superluminal galactic recession velocities in expanding background geometries [19] which are themselves moving in time. The superluminal energy propagation by tachyonic vacuum modes is also to be distinguished from superluminal group velocities arising in photonic crystals, optical fibers, or metamaterials [2024]. In contrast to tachyonic quanta, the actual signal speed defined by the electromagnetic energy flow in these media is always subluminal and occasionally even opposite to the group velocity.

The Proca equation in Fourier space reads

Equation (1)

where k(\omega) = \sqrt{\omega^{2} + m_{\rm t}^{2}} is the wave number of the tachyonic modes, subject to the Lorentz condition {\rm i}\omega {\hat A}_0 + {\rm div}\, {\bf{\hat A}} = 0. This is equivalent to a set of tachyonic Maxwell equations [7]:

Equation (2)

with {\rm rot}\, {\bf{\hat E}} - {\rm i}\omega {\bf{\hat B}} = 0 and {\rm div}\, {\bf{\hat B}} = 0. The field strengths and potentials are connected by {\bf{\hat E}} = {\rm i}\omega {\bf{\hat A}} + \nabla {\hat A}_0 and {\bf{\hat B}} = {\rm rot}\, {\bf{\hat A}}. Fourier transforms are defined by {\bf{\hat A}}({\bf x}, \omega) = \int_{ - \infty}^{ + \infty} {\bf A}({\bf x}, t){\rm e}^{{\rm i}\omega t} {\rm d}t .

The charge and current densities of a classical subluminal point particle with trajectory x0(t) read ρ(x, t) = qtδ(x − x0(t)) and {\bf j}({\bf x}, t) = {\bf{\dot x}}_0 (t)\rho ({\bf x}, t), where qt is the tachyonic charge carried by the particle. Alternatively, we may consider a Dirac current j^{\mu}= (\rho, {\bf j}) = - q_{t} {\bar \psi}\gamma^{\mu}\psi in the field equations, and replace the classical Fourier transforms (\hat \rho, {\bf{\hat j}}) by spinorial matrix elements \hat \rho_{mn} ({\bf x}) and {\bf{\hat j}}_{mn} ({\bf x}) [25]. The nonlocal coupling of the superluminal radiation field to the electron current is effected by a frequency-dependent coupling constant q(ω), which replaces qt in the Fourier amplitudes. q(ω) scales with a power of the tachyonic velocity, q(ω) = qtυtσ, where \upsilon_{\rm t} = \sqrt{1 + m_{\rm t}^{2}/\omega^2} , so that qt is recovered in the high-frequency limit q(ω → ∞) = qt. The varying tachyonic fine-structure constant thus reads

Equation (3)

The frequency dependence of αq(ω) is weak at high energy ω gg mt, but it shows in the soft γ-ray band relevant for GRB spectra [15]. In the low-frequency regime, we find αq(ω → 0) propto ω−2σ, and the constant αt = qt2/(4πhslashc) is recovered at high frequencies, αq(∞) = αt. The nonlocal charge and current densities depending on the varying coupling constant q(\omega) = q_{\rm t} {\hat \Omega}(\omega) are denoted by a subscript Ω, {\hat \rho}_{\Omega} ({\bf x}, \omega) = {\hat \Omega} (\omega){\hat \rho} ({\bf x}, \omega), and {\bf{\hat j}}_{\Omega} = {\hat \Omega} (\omega){\bf{\hat j}}, with {\hat \Omega} (\omega) in (3).

The dispersion of the charge and current densities induced by the varying coupling constant becomes apparent in real time,

Equation (4)

and analogously for the current jΩ, where

Equation (5)

so that {\hat \Omega}_{\rm reg} (\omega) = {\hat \Omega} (\omega) - 1, and δ(t) is the Dirac function. The integral (5) converges for scaling exponents σ < 1, cf. (3), which we henceforth assume. Ωreg(t) is symmetric in t, admitting power law decay Ωreg(tpropto |t|σ−1 for mt|t| → ∞. This power law tail of Ω(t) generates the long-range dispersion of the charge and current densities (4). Only at negative even integer σ, the decay is exponential propto emt|t| and the dispersion localized. The relevant interval for GRBs is 0 < σ < 1, where Ωreg(t) is positive and monotonically decaying.

Tachyonic spectral functions of GRB plasmas

The quantized tachyonic radiation densities of an inertial spinning charge read [26]

Equation (6)

where the superscripts T and L refer to the transversal/longitudinal polarization components defined by ΔT = 1 − mt2/(2m2) and ΔL = 0. m and γ denote mass and Lorentz factor of the electron, mt is the tachyon mass and \alpha_{\rm q} (\omega) = \alpha_{\rm t} {\hat \Omega}^{2}(\omega) the varying tachyonic fine-structure constant (3).

The spectral functions of the electron plasma are calculated by averaging the tachyonic radiation densities with a thermal or nonthermal electronic power law distribution [13],

Equation (7)

The normalization factor Aα,β of the electron density {\rm d}{\hat \rho}_{\alpha, \beta} = \gamma^{ - \alpha - 1} {\rm e}^{ - \beta \gamma} \sqrt{\gamma^2 - 1} {\rm d}\gamma is related to the electron count by ne = Aα,βKα,β, where K_{\alpha, \beta} = \int_{1}^{\infty} {\rm d}{\hat \rho}_{\alpha, \beta}(\gamma) . The electron temperature can be inferred from the exponential cutoff, β = m/(kT) or kT[keV] approx 511/β. We consider ultra-relativistic multi-component plasmas in the collisionless regime [27], in stationary non-equilibrium described by power law densities [28, 29].

The averaged radiation densities read

Equation (8)


Equation (9)

The spectral functions can be reduced to incomplete gamma functions,

Equation (10)

with ΔT,L as in (6), {\hat \omega} = \omega/m_{\rm t} , and bk = βα+2−kΓ(k − α − 2, βγ1). The unpolarized density langlepT+L(ω)rangleα,β is obtained by adding the polarization components. In the following, we use keV units for the tachyon and electron mass and the radiated frequency, so that ω stands for hslashω[keV] and mt for mtc2[keV]; BT,L and langlepT,L(ω)rangleα,β are meant in keV units as well. The spectral functions (10) decay exponentially for βγ1 gg 1, since Γ propto e−βγ1, and so does B^{\rm T, L} (\omega, {\hat \gamma}(\omega)) for {\hat \omega} \to \infty . We define the cutoff frequency as {\hat \gamma} (\omega_{\rm cut}) - {\hat \gamma}(0) = 1/\beta or ωcut = ωmaxt + 1/β), where [6]

Equation (11)

The low-frequency scaling of the averaged spectral density (8) is langlepT,Lrangleα,β propto αq(ω)ω ~ ω1−2σ, valid for ω ll mt. Typical values of the electron index range in −2 ≤ α ≤ 2, and the fine-structure scaling exponent σ is usually close to 0.5 for GRBs [3, 4]. In the intermediate regime mt ll ω ll ωcut, we find langlepT,Lrangleα,β propto ω−α for α > 1, and langlepT,Lrangleα,β propto 1/ω if α ≤ 1. This power law scaling in the high-temperature regime is only approximately realized, so that the crossover is not a straight power law slope, but slightly curved in double-logarithmic plots, gradually bending into exponential decay setting in at ωcut. langlepT,Lrangleα,β is peaked at the junction ω approx mt of the two power law slopes. If ωcut < mt, which can occur in the low-temperature regime [8], the low-frequency power law is exponentially cut at ωcut without power law crossover, and langlepT,Lrangleα,β peaks in the vicinity of ωcut.

Flux densities in the soft γ-ray band: Spectral fits of GRB 941017, GRB 990123, and GRB 990104

The spectral fits in figs. 13 are based on the E2-rescaled differential flux densities

Equation (12)

where E = hslashω is the energy of the radiated tachyons, langlepT,L(ω)rangleα,β the spectral average (8), d the distance to the source, and hslash[keV s] approx 6.582 × 10−19. We substitute the radiation density (8), and replace Aα,βαt/d2 by a single parameter {\hat n} determining the flux amplitude,

Equation (13)

where d[cm] approx 3.086 × 1024d[Mpc]. The burst distance is estimated via d ~ cz/H0, with the Hubble distance c/H0 approx 4.41 Gpc (h0 approx 0.68). Thus, d[Gpc] approx 4.41 z, and we calculate the electron number of the source plasma as, cf. after (7),

Equation (14)

The asymptotic energy scaling of the flux densities (12) is E2dNT,L/dE propto E2(1−σ), applicable for E/mt ll 1, cf. after (11). In the crossover region mt ll E ll Ecut, we find in leading order E2dNT,L/dE propto E1−α if α > 1, and E2dNT,L/dE propto 1 for α ≤ 1, which terminates in exponential decay at Ecut = hslashωmax(1/β + μt).

Figure 1

Figure 1. Spectral map of γ-ray burst GRB 941017. BATSE and EGRET data points from ref. [4]. The solid line T + L depicts the unpolarized differential tachyon flux dNT+L/dE, obtained by adding the flux densities ρ1,2 radiated by an ultra-relativistic two-component plasma, cf. (12). The transversal and longitudinal densities add up to the total unpolarized flux, T + L = ρ1 + ρ2. The nonthermal low-energy flux ρ1 is fitted with the tachyon-electron mass ratio mt/m approx 0.37, cf. table 1; the exponential decay of this flux component sets in at about Ecut approx 0.75 GeV, cf. after (14). The mass ratio mt/m approx 430 and the cutoff energy Ecut approx 21 GeV of the thermal high-energy component ρ2 are tentative, owing to lack of flux data above 100 MeV.

Figure 2

Figure 2. Spectral map of GRB 990123. Flux data from ref. [4]. T and L stand for the transversal and longitudinal flux components, and T + L = ρ1 labels the unpolarized flux generated by a one-component plasma. The tachyon-electron mass ratio is mt/m approx 0.92, and the tachyonic flux density T + L is exponentially cut at Ecut approx 5.4 MeV. The least-squares fit is performed with the unpolarized flux, and subsequently split into transversal and longitudinal components. The parameters of the nonthermal electronic source plasma are recorded in tables 1 and 2.

Figure 3

Figure 3. Spectral map of GRB 990104. Data points from ref. [4], notation as in figs. 1 and 2. The superluminal flux is radiated by a thermal single-component electron plasma, cf. table 1. The cutoff energy of the tachyonic flux density ρ1 = T + L is Ecut approx 3.8 MeV, and the tachyon-electron mass ratio is mt/m approx 0.47.

The spectral fits are performed with the unpolarized flux density dNT+L = dNT + dNL (denoted by T + L in figs. 13) of the plasma components ρi specified by electronic power law distributions in table 1. The corresponding tachyonic flux components are likewise denoted by ρi, cf. fig. 1, and add up to the total unpolarized flux T + L = ∑ρi. Different plasma components ρi radiate tachyons with different mass-square. The tachyon mass extracted from the spectral fits of GRB 990123 in fig. 2 and GRB 990104 in fig. 3, as well as from the low-energy component ρ1 of GRB 941017 in fig. 1 is comparable to the electron mass, cf. table 1. A notably larger tachyon-electron mass ratio is inferred from the high-energy flux component ρ2 of GRB 941017, cf. the caption to fig. 1.

Table 1. Electron distributions ρi generating the tachyonic flux densities of the γ-ray bursts in figs. 13. The components ρ1,2 of the source plasma are specified by electronic power law densities with electron index α and cutoff parameter β in the Boltzmann factor, cf. after (7). mt is the mass parameter of the superluminal modes, and σ the scaling exponent of the frequency-dependent tachyonic fine-structure constant (3). The scale factor {\hat n} determining the amplitude of the superluminal flux and the electron number is defined in (13). kT is the temperature of the thermal (α = −2) or nonthermal plasma components.
GRB mt (keV) σ α β {\hat n} (keV cm−2 s−1) kT (MeV)
ρ1 190 0.42 1.2 2.11 × 10−4 116 2.42 × 103
ρ2 2.2 × 105 0.4  − 2 2.44 × 10−5 1.36 × 103 2.09 × 104
ρ1 470 0.5 0.6 5.88 × 10−2 689 8.69
ρ1 240 0.6  − 2 5.33 × 10−2 190 9.59

The transversal/longitudinal tachyonic luminosity langlePT,Lrangleα,β of the source plasma is obtained by a frequency integration of the averaged spectral densities langlepT,L(ω)rangleα,β,

Equation (15)

The integrand decays exponentially at high frequencies; convergence at the lower integration boundary requires σ < 1, cf. after (11). On substituting (8) into (15), and making use of (13), we obtain

Equation (16)

Conversion into erg/s units means to multiply by 1.602 × 10−9. {\hat n} is the tachyonic flux amplitude in units of keV cm−2 s−1, and 4πd2[cm] approx 2.33 × 1057 z2, cf. after (13). The redshift estimate of GRB 990123 in fig. 2 is z approx 1.6 [30], which amounts to 7 Gpc. The high-energy component ρ2 of GRB 941017 in fig. 1 does not imply a low redshift, as there is no intergalactic absorption of the tachyon flux. Tachyonic γ-rays are not attenuated by cosmic background photons, as tachyons and photons can only indirectly interact via matter fields [28, 29, 31].

The high-temperature limit (β → 0) of the internal energy U = nemc2uα(β) of the ultra-relativistic electron gas defined by density {\rm d}{\hat \rho}_{\alpha, \beta} , cf. after (7), reads [32]

Equation (17)

The electron energy of each plasma component ρi is recorded in table 2, with mc2 approx 8.187 × 10−7 erg. We identify the time scale τ0 = U/langlePT+Lrangleα,β with the burst duration, typically of the order of 100 s, in which the internal energy of the electron gas is radiated off. We use U[keV] = nem[keV]uα(β) and langlePT+Lrangleα,β in (16) with 4\pi d^{2}{\hat n} = n_{\rm e} \alpha_{\rm t} m_{\rm t}^{2}/\hbar , cf. (13), to estimate the asymptotic fine-structure constant from the burst duration,

Equation (18)

Once αt is known, we can calculate the electron number and internal energy of the plasma, as well as the tachyonic power radiated, cf. table 2.

Table 2. Tachyonic luminosity, electron count, and internal energy of the source plasma. langlePT,Lrangleα,β denotes the power transversally and longitudinally radiated, cf. (16). αt is the tachyonic fine-structure constant in the high-frequency limit, cf. (18), estimated from the burst duration τ0 [4]. ne is the electron number (14), and U the internal energy (17) of the respective plasma component ρi, cf. table 1. The tachyonic power, electron number, and the energy stored in the electron gas scale propto z2; we list these quantities at z = 1, since a redshift estimate is only available for GRB 990123, z approx 1.6 [30].
GRB τ0 (s) langlePTrangleα,β/z2 (erg/s) langlePLrangleα,β/z2 (erg/s) αt ne/z2 U/z2 (erg)
941017 262.1
ρ1 1.26 × 1051 1.41 × 1051 1.24 × 10−21 3.96 × 1057 6.98 × 1053
ρ2 2.64 × 1052 2.51 × 1052 3.19 × 10−25 1.35 × 1056 1.35 × 1055
990123 98.4
ρ1 4.19 × 1051 4.55 × 1051 3.10 × 10−23 1.54 × 1059 8.60 × 1053
990104 262.2
ρ1 2.82 × 1051 3.02 × 1051 1.52 × 10−22 3.33 × 1058 1.53 × 1054


We discussed the coupling of superluminal radiation fields to a plasma current by a frequency-dependent fine-structure constant (3), resulting in a nonlocal interaction of the tachyonic modes with the GRB plasma. We studied tachyonic radiation densities subject to an energy-dependent coupling constant, and explained the spectral averaging over the plasma components constituting the GRBs. The low-frequency scaling of the averaged spectral densities (8) is determined by the scaling exponent of the tachyonic fine-structure constant. The high-frequency crossover between the spectral peak and exponential cutoff depends on the electronic power law index of the plasma, cf. after (11).

The averaged radiation densities (8) were put to test by performing tachyonic spectral fits to burst spectra in the soft γ-ray band, cf. figs. 13. We showed that the low-energy components as well as the occasionally observed high-energy slopes of GRB spectra can be fitted with tachyonic flux densities. The fits are based on the tachyonic radiation densities (6) averaged over electronic power law distributions, cf. after (7); no additional radiation mechanism is invoked for the high-energy component of GRB 941017 in fig. 1. The spectral fitting of GRBs described here does not require any detailed modeling of the burst evolution, no specific assumptions on the progenitor and heating mechanism, not even a distance estimate; only the inferred tachyonic power, the electron number, and internal energy of the source plasma scale with redshift.

The mass of the radiated quanta and the scaling exponent of the tachyonic fine-structure constant αq(ω) = αtυt depend on the respective plasma component ρi generating the tachyon flux, and can be extracted from the spectral fit, cf. table 1; the scaling amplitude αt = αq(ω → ∞) is estimated from the burst duration. The superluminal velocity υt coincides with the vacuum refractive index k/\omega = \sqrt{1 + m_{\rm t}^{2}/\omega^2}  [31]. It is possible that αt scales with the tachyon-electron mass ratio, αt = α0(mt/m)−2σ, with a universal amplitude α0 of the order of 10−22, but the spectral fits in figs. 13 do not yet allow a definite conclusion on that. The constituents of a multi-component source plasma can be disentangled by identifying the corresponding flux components ρi in the spectral maps, as done in the case of GRB 941017, cf. fig. 1. The tachyonic luminosity of each plasma component is listed in table 2.


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