Gamma ray limits

Overview

hazma includes functionality for using existing gamma-ray data to constrain theories and for projecting the discovery reach of proposed gamma-ray detectors. For the first case, hazma defines a container class called FluxMeasurement for storing information about gamma-ray datasets, and Theory contains a method for using these to set limits. The second case is also handled by a method in Theory which takes arguments specifying various detector and target characteristics.

Limits from existing data

Theory.binned_limit(measurement, n_sigma=2.0)

Determines the limit on \(<sigma v>\) from gamma-ray data.

We define a signal to be in conflict with the measured flux for bin \(i\) for an experiment if

\[\Phi_{\chi}^{(i)} > n_{\sigma} \sigma^{(i)} + \Phi^{(i)},\]

where \(\Phi_\chi^{(i)}\) is the integrated flux due to DM annihilations for the bin, \(\Phi^{(i)}\) is the measured flux in the bin, \(\sigma^{(i)}\) is size of the upper error bar for the bin and \(n_{\sigma} = 2\) is the significance. The overall limit on \(\langle\sigma v\rangle\) is computed by minimizing over the limits determined for each bin.

Parameters:

• measurement (FluxMeasurement) – Information about the flux measurement and target.
• n_sigma (float) – See the notes for this function.

Returns:

<sigma v>_tot – Largest allowed thermally averaged total cross section in cm^3 / s

Return type:

float

Discovery reach for upcoming detectors

Theory.unbinned_limit(A_eff, energy_res, T_obs, target_params, bg_model, n_sigma=5.0, debug_msgs=False)

Computes smallest-detectable value of <sigma v> for given target and experiment parameters.

We define a signal to be detectable if

\[N_S / \sqrt{N_B} \geq n_{\sigma},\]

where \(N_S\) and \(N_B\) are the number of signal and background photons in the energy window of interest and \(n_\sigma\) is the significance in number of standard deviations. Note that \(N_S \propto \langle \sigma v \rangle\). While the photon count statistics are properly taken to be Poissonian and using a confidence interval would be more rigorous, this procedure provides a good estimate and is simple to compute. The energy window is chosen to maximize N_S/sqrt(N_B).

Parameters:

• A_eff (float -> float) – Effective area of experiment in cm^2 as a function of photon energy.
• energy_res (float -> float) – The detector’s energy resolution (\(\Delta E / E\)) as a function of photon energy in MeV.
• T_obs (float) – Experiment’s observation time in s
• target_params (TargetParams) – Object containing information about the observation target.
• bg_model (BackgroundModel) – Object representing a gamma ray background model.
• n_sigma (float) – Number of standard deviations the signal must be above the background to be considered detectable
• debug_msgs (bool) – If True, the energy window found by the optimizer will be printed.

Returns:

<sigma v> – Smallest-detectable thermally averaged total cross section in units of cm^3 / s.

Return type:

float

Classes, functions and constants

These data, functions and classes are relevant for setting constraints and projecting discovery reach.

class hazma.flux_measurement.FluxMeasurement(obs_rf, energy_res, target, power=2)

Container for all information about a completed gamma ray analysis.

e_lows

Lower edges of energy bins.

Type:np.array

e_highs

Upper edges of energy bins.

Type:np.array

fluxes

Flux measurements for each bin (MeV^-1 cm^-2 s^-1 sr^-1).

Type:np.array

upper_errors

Size of upper error bars on flux measurements (MeV^-1 cm^-2 s^-1 sr^-1).

Type:np.array

lower_errors

Size of lower error bars on flux measurements (MeV^-1 cm^-2 s^-1 sr^-1).

Type:np.array

energy_res

Function returning energy resolution (Delta E / E) as a function of photon energy.

Type:callable

target

Information about the target observed for this measurement.

Type:TargetParams

__init__(obs_rf, energy_res, target, power=2)

Constructor.

Parameters:

• obs_rf (str) –

  Name of file containing observation information. The columns of this file must be:

  1. Lower bin edge (MeV)
  2. Upper bin edge (MeV)
  3. \(E^2 d^2 \Phi/dE d\Omega\) (MeV cm^-2 s^-1 sr^-1)
  4. Upper error bar (MeV cm^-2 s^-1 sr^-1)
  5. Lower error bar (MeV cm^-2 s^-1 sr^-1)

  Note that the error bar values are their y-coordinates, not their relative distances from the central flux.

• energy_res (callable) – Energy resolution function.
• target (TargetParams) – The target of the analysis

class hazma.background_model.BackgroundModel(e_range, dPhi_dEdOmega)

Represents a gamma ray background model, which is required for computing projected limits for planned gamma-ray detectors.

Parameters:

• e_range ([float, float]) – Minimum and maximum photon energies for which this model is valid, in MeV.
• dPhi_dEdOmega (np.array) – Background gamma ray flux (MeV^-1 sr^-1 m^-2 s^-1) as a function of photon energy (MeV). This function must be vectorized.

dPhi_dEdOmega(es)

Computes this background model’s gamma ray flux.

Parameters:es (float or np.array) – Photon energy/energies at which to compute Returns:dPhi_dEdOmega – Background gamma ray flux, in MeV^-1 sr^-1 m^-2 s^-1. For any energies outside of self.e_range, np.nan is returned. Return type:np.array

hazma.gamma_ray_parameters.energy_res_comptel(e)

COMPTEL energy resolution \(\Delta E / E\).

Taken from ch. II, page 11.

hazma.gamma_ray_parameters.A_eff_comptel = <scipy.interpolate.interpolate.interp1d object>

COMPTEL effective area function

hazma.gamma_ray_parameters.comptel_diffuse = <hazma.flux_measurement.FluxMeasurement object>

COMPTEL diffuse gamma-ray flux measurements

hazma.gamma_ray_parameters.energy_res_egret(e)

EGRET’s energy resolution \(\Delta E / E\).

This is the most optimistic value, taken from sec. 4.3.3.

hazma.gamma_ray_parameters.A_eff_egret = <scipy.interpolate.interpolate.interp1d object>

EGRET effective area function

hazma.gamma_ray_parameters.egret_diffuse = <hazma.flux_measurement.FluxMeasurement object>

EGRET diffuse gamma-ray flux measurements

hazma.gamma_ray_parameters.energy_res_fermi(e)

Fermi-LAT’s energy resolution \(\Delta E / E\).

This is the average of the most optimistic normal and 60deg off-axis values from fig. 18.

hazma.gamma_ray_parameters.A_eff_fermi = <scipy.interpolate.interpolate.interp1d object>

Fermi-LAT effective area function

hazma.gamma_ray_parameters.fermi_diffuse = <hazma.flux_measurement.FluxMeasurement object>

Fermi diffuse gamma-ray flux measurements

hazma.gamma_ray_parameters.energy_res_e_astrogam = <scipy.interpolate.interpolate.interp1d object>

e-ASTROGAM energy resolution function. From table 1 of the e-ASTROGAM whitebook.

hazma.gamma_ray_parameters.A_eff_e_astrogam = <scipy.interpolate.interpolate.interp1d object>

e-ASTROGAM effective area function

class hazma.gamma_ray_parameters.TargetParams(J, dOmega)

Container for information about a target region.

Currently implemented for the Draco dwarf galaxy and \(10^\circ imes 10^\circ\) region around the galactic center, which can be imported using:

from hazma.gamma_ray_parameters import draco_params, gc_target

Parameters:

• J (float) – J-factor in MeV^2 cm^-5
• dOmega (float) – Angular size in sr

hazma.gamma_ray_parameters.solid_angle(l_max, b_min, b_max)

Returns solid angle subtended for a rectangular target region centered on the galactic center.

Parameters:

• l_max (float) – Maximum value of galactic longitude in deg. Note that \(l\) must lie in the interval \([-180, 180]\).
• b_max (b_min,) – Minimum and maximum values for \(b\) in deg. Note that \(b\) must lie in the interval \([-90, 90]\), with the equator at \(b = 0\).

Returns:

Omega – Solid angle subtended by the region in sr.

Return type:

float