Detecting supernovae neutrino with Earth matter effect

We study Earth matter effect in oscillations of supernovae neutrinos. We show that detecting Earth matter effect gives an independent measurement of spectra of supernovae neutrinos, i.e., the flavor difference of the spectra of supernovae neutrinos. We study the effect of energy resolution and angular resolution of a final electron or positron on detecting the signal of Earth matter effect. We show that varying the widths of energy bins in analysis can change the signal strength of Earth matter effect and the statistical fluctuation. A reasonable choice of energy bins can both suppress the statistical fluctuation and make a good signal strength relative to the statistical fluctuation. Neutrino detectors with good energy resolution and good angular resolution are therefore preferred so that there is more freedom to vary energy bins and to optimize the signal of Earth matter effect in analyzing events of supernovae neutrinos.

Figure 1

Regeneration factor versus energy for neutrino (left) and antineutrino (right), respectively. $L=8000\mathrm{km}$ (upper) and $L=12000\mathrm{km}$ (lower), respectively. Line A (numerical): computed numerically using PREM density profile with five layers; Line B (partial analytical): computed using formulas (13) and (14) with oscillation phase computed numerically; Line C (analytical): computed using formulas (13) and (14) completely analytically including the oscillation phase.

Figure 2

Power spectrum versus k for $L=5000\mathrm{km}$ (upper left), $L=8000\mathrm{km}$ (upper right), $L=11000\mathrm{km}$ (lower left), $L=12000\mathrm{km}$ (lower right) with various assumptions of energy bin. Earth matter effect calculated numerically using PREM density profile with five layers. Line A: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.03/\sqrt{E_{\nu }/\mathrm{MeV}}+0.2\times E_{\nu }/m_p$; Line B: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.03/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Line C: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.03/\sqrt{E_{\nu }/\mathrm{MeV}}+1.0\times E_{\nu }/m_p$; Line D: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.05/\sqrt{E_{\nu }/\mathrm{MeV}}+1.0\times E_{\nu }/m_p$. No. of events: $2.\times 10^4$.

Figure 3

Power spectrum versus k for $L=5000\mathrm{km}$ (upper left), $L=8000\mathrm{km}$ (upper right), $L=11000\mathrm{km}$ (lower left), $L=12000\mathrm{km}$ (lower right) with ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.03/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$. Earth matter effect calculated numerically using PREM density profile with five layers. Line A: $N=3.\times 10^4$; Line B: $N=2.\times 10^4$; Line C: $N=1.\times 10^4$.

Figure 4

Power spectrum versus k for $L=5000\mathrm{km}$ (upper left), $L=8000\mathrm{km}$ (upper right), $L=11000\mathrm{km}$ (lower left), $L=12000\mathrm{km}$ (lower right), and for $N=2.\times 10^4$. Line A: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.03/\sqrt{E_{\nu }/\mathrm{MeV}}+0.2\times E_{\nu }/m_p$; Line B: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.10/\sqrt{E_{\nu }/\mathrm{MeV}}+0.2\times E_{\nu }/m_p$; Line C: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+0.2\times E_{\nu }/m_p$. Earth matter effect calculated numerically using PREM density profile with five layers.

Figure 5

Power spectrum versus k for $N=1.\times 10^4$ (upper left), $N=2.\times 10^4$ (upper right), $N=4.\times 10^4$ (lower left), $N=8.\times 10^4$ (lower right), with $L=12000\mathrm{km}$ and ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.03/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$. Line A: theoretical expectation; Line B: average over 20 samples; Line C: 20 samples with $1\sigma$ fluctuation. Earth matter effect calculated numerically using PREM density profile with five layers.

Figure 6

Power spectrum versus k for $L=12000\mathrm{km}$ and $N=4.\times 10^4$. Upper left: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.05/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Upper right: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.10/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Lower left: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Lower right: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.50/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$. Line A: theoretical expectation; Line B: average over 20 samples; Line C: 20 samples with $1\sigma$ fluctuation. Earth matter effect calculated numerically using PREM density profile with five layers.

Figure 7

Power spectrum versus k for $L=12000\mathrm{km}$ and $N=8.\times 10^4$. Upper left: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+0.2\times E_{\nu }/m_p$; Upper right: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Lower left: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+1.0\times E_{\nu }/m_p$; Lower right: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+1.5\times E_{\nu }/m_p$. Line A: theoretical expectation; Line B: average over 20 samples; Line C: 20 samples with $1\sigma$ fluctuation. Earth matter effect calculated numerically using PREM density profile with five layers.

Figure 8

Power spectrum versus k for $L=8000\mathrm{km}$ and $N=4.\times 10^4$. Upper left: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.05/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Upper right: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.10/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Lower left: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.20/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$; Lower right: ${\mathrm{\Delta }}_{\nu }/E_{\nu }=0.50/\sqrt{E_{\nu }/\mathrm{MeV}}+0.5\times E_{\nu }/m_p$. Line A: theoretical expectation; Line B: average over 20 samples; Line C: 20 samples with $1\sigma$ fluctuation. Earth matter effect calculated numerically using PREM density profile with five layers.