Context : In this paper we study the time-resolved spectral properties of energetic gamma-ray bursts ( GRBs ) with good high-energy photon statistics observed by the Gamma-Ray Burst Monitor ( GBM ) onboard the Fermi Gamma-Ray Space Telescope .

Aims : To constrain in detail the spectral properties of GRB prompt emission on a time-resolved basis and to discuss the theoretical implications of the fitting results in the context of various prompt emission models .

Methods : Our sample comprises eight GRBs observed by Fermi GBM in its first five years of mission , with 1 keV - 1 MeV fluence f > 1.0 \times 10 ^ { -4 } erg cm ^ { -2 } and signal-to-noise level \text { S / N } \geq 10.0 above 900 keV .
We perform time-resolved spectral analysis using a variable temporal binning technique according to optimal S/N criteria , resulting in a total of 299 time-resolved spectra .
We fit the Band function to all spectra and obtain the distributions for the low-energy power-law index \alpha , the high-energy power-law index \beta , the peak energy in the observed \nu F _ { \nu } spectrum E _ { \text } { p } , and the difference between the low- and high-energy power-law indices \Delta s = \alpha - \beta .
We also apply a physically motivated synchrotron model , which is a triple power-law with constrained power-law indices and a blackbody component , to test for consistency with a synchrotron origin for the prompt emission and obtain the distributions for the two break energies E _ { \text } { b, 1 } and E _ { \text } { b, 2 } , the middle segment power-law index \beta , and the Planck function temperature kT .

Results : The Band function parameter distributions are \alpha = -0.73 ^ { +0.16 } _ { -0.21 } , \beta = -2.13 ^ { +0.28 } _ { -0.56 } , E _ { \text } { p } = 374.4 ^ { +307.3 } _ { -187.7 } keV ( \log _ { 10 } E _ { \text } { p } = 2.57 ^ { +0.26 } _ { -0.30 } ) , and \Delta s = 1.38 ^ { +0.54 } _ { -0.31 } , with average errors \sigma _ { \alpha } \sim 0.1 , \sigma _ { \beta } \sim 0.2 , and \sigma _ { E _ { \text } { p } } \sim 0.1 E _ { \text } { p } .
Using the distributions of \Delta s and \beta , the electron population index p is found to be consistent with the ” moderately fast ” scenario which fast- and slow-cooling scenarios can not be distinguished .
The physically motivated synchrotron fitting function parameter distributions are E _ { \text } { b, 1 } = 129.6 ^ { +132.2 } _ { -32.4 } keV , E _ { \text } { b, 2 } = 631.4 ^ { +582.6 } _ { -309.6 } keV , \beta = -1.72 ^ { +0.48 } _ { -0.25 } , and kT = 10.4 ^ { +4.9 } _ { -3.7 } keV , with average errors \sigma _ { \beta } \sim 0.2 , \sigma _ { E _ { \text } { b, 1 } } \sim 0.1 E _ { \text } { b, 1 } , \sigma _ { E _ { \text } { b, 2 } } \sim 0.4 E _ { \text } { b, 2 } , and \sigma _ { kT } \sim 0.1 kT .
This synchrotron function requires the synchrotron injection and cooling break ( i.e. , E _ { \text } { min } and E _ { \text } { cool } ) to be close to each other within a factor of ten , often in addition to a Planck function .

Conclusions : A synchrotron model is found consistent with the majority of time-resolved spectra for eight energetic Fermi GBM bursts with good high-energy photon statistics , as long as both the cooling and injection break are included and the leftmost spectral slope is lifted either by inclusion of a thermal component or when an evolving magnetic field is accounted for .