Matter density profile shape effects at DUNE

Quantum mechanical interactions between neutrinos and matter along the path of propagation, the Wolfenstein matter effect, are of particular importance for the upcoming long-baseline neutrino oscillation experiments, specifically the Deep Underground Neutrino Experiment (DUNE). Here, we explore specifically what about the matter density profile can be measured by DUNE, considering both the shape and normalization of the profile between the neutrinos’ origin and detection. Additionally, we explore the capability of a perturbative method for calculating neutrino oscillation probabilities and whether this method is suitable for DUNE. We also briefly quantitatively explore the ability of DUNE to measure the Earth’s matter density, and the impact of performing this measurement on measuring standard neutrino oscillation parameters.

Figure 1

Measurement precision of a single-bin experiment with only statistical uncertainty as a function of the oscillation probability $P_{\alpha \beta }$. Orange lines give precision in terms of fractional uncertainty $|\mathrm{\Delta }{P}_{\alpha \beta }|/P_{\alpha \beta }$ where solid lines give precision in terms of $|\mathrm{\Delta }{P}_{\alpha \beta }|$. The right axis, along with annotations, denotes the number of unoscillated events $N_1$ necessary in a bin to attain the given precision.

Figure 2

Black lines: Naïve prediction of measurement precision of oscillation probability $|\mathrm{\Delta }{P}_{\alpha \beta }|$ assuming $N_1={10}^3$ unoscillated events for all energies. Solid black lines include a 5% uncorrelated bin-to-bin systematic uncertainty, where dashed black lines are for only statistical uncertainties. We have included a bin-to-bin measurement improvement factor of $\sqrt{5}$ to this naïve estimate as discussed in the text. Green: Change to oscillation probabilities while changing oscillation parameters between their central values and $\pm 1\sigma$ extremes, as given in Table 1. In the left panel, we show the impact on appearance probability $P_{\mu e}$ for the parameters measured (predominantly) by this channel, ${\sin }^2{\theta }_{13}$ (solid) and $\delta$ (dashed). We show the maximum change $|\mathrm{\Delta }{P}_{\mu e}|$ between assuming $\delta =0\pm 0.2$ and $\delta =\pi /2\pm 0.3$. In the right panel, we show the impact on disappearance probability $P_{\mu \mu }$ and its associated parameters, $\mathrm{\Delta }{m}_{31}^2$ (dotted-dashed) and ${\sin }^2{\theta }_{23}$ (dotted). We do not display antineutrino probability precisions here, but the result is qualitatively the same.

Figure 3

The density maps considered here, given in Ref. [10] and scaled such that ${\rho }_{\mathrm{Avg}}=2.845\mathrm{g}/{\mathrm{cm}}^3$. Each density map is divided into $N=100$ segments.

Figure 4

Shaded regions (red): the range of change in oscillation probabilities obtained while changing the shape of the matter density profile, comparing the Crustal map, Shen-Ritzwoller map, and the PEMC map, as detailed in Ref. [10]. Matter density profiles have been normalized so that ${\rho }_{\mathrm{Avg}}=2.845\mathrm{g}/{\mathrm{cm}}^3$. Solid lines (blue) display the change in oscillation probabilities when changing a constant matter density by $\pm 1\%$ of ${\rho }_{\mathrm{Avg}}=2.845\mathrm{g}/{\mathrm{cm}}^3$. The left panel displays the change in appearance probability $P_{\mu e}$ and the right panel displays the change in disappearance probability $P_{\mu \mu }$, both for neutrino oscillation.

Figure 5

Change in oscillation probabilities between the DMP perturbative method at zeroth order (purple) and first order (pink) as discussed in Sec. 3d and a matrix-exponential-calculated oscillation probability assuming one layer, both with constant matter density $\rho =2.845\mathrm{g}/{\mathrm{cm}}^3$. The left panel displays differences for appearance channel neutrino oscillation probabilities $P_{\mu e}$, and the right panel displays differences for disappearance channel probabilities $P_{\mu \mu }$.

Figure 6

A summary of the scales of $|\mathrm{\Delta }{P}_{\mu e}|$ (appearance, top) and $|\mathrm{\Delta }{P}_{\mu \mu }|$ (disappearance, bottom) discussed in this paper: naïve sensitivity estimates in Sec. 3a (black), differences from parameter changes in Sec. 3b (green), changes to the matter density profile average density (blue) and shape (red) as discussed in Sec. 3c, and the precision of the zeroth-order (purple) and first-order (pink) DMP perturbative approach from Sec. 3d. We restrict the neutrino energy to be in the range $2\mathrm{GeV}<E_{\nu }<4\mathrm{GeV}$, where the number of unoscillated events is highest for each channel.

Figure 7

Measurement precision of a single-bin experiment with statistical uncertainty and 5% normalization uncertainty in the bin as a function of the oscillation probability $P_{\alpha \beta }$. Orange lines give precision in terms of fractional uncertainty $|\mathrm{\Delta }P|/P$ and black lines give precision in terms of $|\mathrm{\Delta }P|$. The right axis, along with annotations, denotes the number of unoscillated events necessary in a bin to attain the given precision.

Figure 8

Simple, two-layer matter density profile with two free parameters, either ${\rho }_0$ and ${\rho }_1$ or ${\rho }_{\mathrm{Avg}}=({\rho }_0+{\rho }_1)/2$ and $\mathrm{\Delta }\rho ={\rho }_1-{\rho }_0$. For simplicity, this model assumes that the density profile is symmetric, and that the middle, raised portion has the same distance as the two outside portions combined.

Figure 9

Change in oscillation probability $P_{\mu e}$ when changing the shape (left) or average density (right) of the matter density profile discussed in Appendix pp2 and shown in Fig. 8. In the left panel, we keep ${\rho }_{\mathrm{Avg}}=2.845\mathrm{g}/{\mathrm{cm}}^3$ fixed and vary $\mathrm{\Delta }\rho$, and in the right panel, we keep $\mathrm{\Delta }\rho =0$ fixed, and vary ${\rho }_{\text{Avg}}$. Black lines indicate contours of constant $|\mathrm{\Delta }{P}_{\mu e}|$, with labels indicating contours of interest.

Figure 10

Expected measurement sensitivity of DUNE with 300 kt-MW-yr of exposure to the parameters $\delta$ and $\rho$, where $\rho$ is assumed to be constant. Four different physical values of $\delta$ are assumed, from left to right: $-\pi /2$, 0, $\pi /2$, and $\pi$. Stars denote the assumed physical values of $\delta$ and $\rho$ in each panel. Contours correspond to 68.3% (blue), 95% (orange) and 99% (red) credibility regions, where all other parameters have been marginalized. Solid lines correspond to analysis including the Gaussian prior $\rho =(2.845\pm 0.028)\mathrm{g}/{\mathrm{cm}}^3$, and dotted lines correspond to analysis with $\rho$ free.