Lauren Weiss ‘10
Harvard College
One of the current mysteries in galactic astronomy is the WMAP haze [1], an excess in microwave radiation from within 20 degrees of the galactic center. The WMAP haze reveals an overall “hardening” of synchrotron radiation in the range of 23-94 GHz, where the specific intensity spectrum goes as ν−0.5 compared to ν−1 elsewhere in the Galaxy. One potential mechanism for producing such hard synchrotron radiation is a model of self-annihilating cold dark matter creating ultra-relativistic e+e− pairs, which accelerate in the galactic magnetic field and produce synchrotron radiation. This research models the synchrotron radiation that results from dark matter annihilations throughout the Milky Way Galaxy with a parameter space that enables the adjustment of DM particle mass, DM particle cross-section, galactic magnetic field, electron energy injection, and electron energy diffusion. We use IDL 7.0 to integrate the volume emissivity along each line of sight to compare our model to ARCADE data, which identifies a 23 mK synchrotron excess at 3 GHz [2]. We investigate the possibility that the WMAP haze and the ARCADE excess have a common origin in dark matter annihilation. We find that (a) when normalized to the haze, vanilla WIMP annihilation model under-predicts the ARCADE data by a factor of ~ 20, and (b) this model produces a spectral index of 0.9, which is softer than the ARCADE spectral index by 0.33. Although a vanilla WIMP model cannot explain the discrepancy between the ARCADE and WMAP excesses, invoking a more exotic DM particle or considering a non-DM phenomenon such as e+e− pair production from local pulsars could explain the ARCADE data.

Introduction

Recent data from the WMAP satellite reveal an excess of synchrotron radiation at 23-94 GHz [1], which has been dubbed the WMAP “Haze”. The source of the synchrotron excess is unknown, but its hardened spectrum (index ~ 0.5) indicates that it does not originate in the usual supernova mechanism (index ~ 1.0). One possibility is that ultra-relativistic electrons and positrons produced by cold dark matter self-annihilation (e.g., from neutralinos) could create a synchrotron excess with a hardened spectrum. Although the WMAP Haze is described well with models of self-annihilating dark matter [1],[4],[5], our lack of knowledge about dark matter limits the extent to which we can constrain such models.

A new opportunity has arisen to test models of dark matter self-annihilation. The balloon mission ARCADE reports a synchrotron excess at 3-10 GHz [2]. Not only did ARCADE observe synchrotron excess in a different frequency range from WMAP, but it also observed a different part of the sky. The ARCADE footprint is roughly an annulus centered at ~ (l, b) = (70, 0) with inner radius ~ 20 degrees and outer radius ~ 40 degrees [2]. Because the region ARCADE observed is removed from the galactic center, and because it reveals a synchrotron excess in a previously un-modeled frequency range, fitting models of dark matter self-annihilation to the ARCADE data could reveal new constraints on the parameters of dark matter self-annihilation processes (see Figure 1). For these models, we consider “vanilla” WIMPs (χ), which do not experience any large-scale forces other than gravity, although more exotic DM particles exist and should be considered in the future.

Figure 1
Figure 1. Schematic of ARCADE data within the larger body of knowledge. If the ARCADE data fits the spectrum described by WMAP and Haslam, it could agree with dark matter annihilation models. Inconsistencies in the power law of the spectrum or the specific intensity at 3 GHz could rule out basic dark matter annihilation as the source of the ARCADE excess.

Methods

Constructing the model

The number of DM annihilations per unit volume per unit time is

(1)

where ρNFW is the Navarro-Frenk-White (NFW) [6] profile (see §2.2), mχ is the WIMP mass and ‹σν› is the annihilation cross section.

The number of electrons produced per dark matter particle per unit time is

(2)

where Emax and Emin are the maximum and minimum electron energies produced in annihilations, dN/dE is the number density of electrons per unit volume in a given energy bin and E is energy.

The synchrotron power from a single electron is

(3)

where q is the electron charge, me is the electron mass, c is the speed of light, B(r) is the radially dependent magnetic field, α is the pitch angle between the magnetic field and the electron velocity, νc is the critical frequency as defined in [7] and K is a modified Bessel function.

With dimensional analysis and proper integration over pitch angle and energy, we can combine the above equations to create an expression for the synchrotron power produced per unit volume at a given frequency,

(4)

To find the specific intensity Iν from synchrotron radiation in a given direction, we integrate along the line of sight.

(5)

where ds is a distance element along the line of sight and we take smax = 30 kpc. A model of the synchrotron excess at 3.3 GHz and a real map of the haze are shown in Figure 2.

Figure 2
Figure 2. The Haze. Top left: Model of synchrotron excess from vanilla WIMP annihilation at 3.3 GHz, parameters 101. Top right: same model, with a mask for all sources with extinction greater than 1 mag. Bottom: A previous example of the observed haze (3), with the region of ARCADE observation overlaid in red and the haze boxed in green.

Model parameters

We used IDL 7.0 to generate a model of the synchrotron excess in the Milky Way Galaxy at 3.3 GHz (the lowest ARCADE frequency band) and 22.5 GHz (the lowest WMAP band). To model dark matter self-annihilation, we considered several parameters that could affect the synchrotron excess.

Distribution of dark matter.

We assumed an NFW profile:

(6)

where ρ(r = 8.5 kpc) = 0.3 GeV/cm3), rs = 20 kpc is the scale radius [1], ρ0 is the density of dark matter at the galactic center, and r is the radius from the galactic center.

Dark matter mass and annihilation cross section.

We assumed a particle mass of mχ = 100 GeV unless otherwise noted. To match the observed dark matter density Ωm = 0.3, the annihilation cross section should be roughly ~ 3×10−26 cm3/s for a 100 GeV neutralino, in accordance with the physics of “freeze-out” during the cooling of the universe.

Magnetic field variation

We allowed two different magnetic field models, one with constant magnetic field of B0, and one with exponential decay described by

(7)

where B0 = 10μG is the field at the galactic center and rscale = 10kpc is the scale radius. These equations assume a spherically symmetric magnetic field, whereas the true galactic magnetic field probably has a stronger disk component. Because ARCADE observes up to 40 degrees above and below the galactic plane, the spherical symmetry assumed in our model might not be a good approximation for the ARCADE data. Magnetic fields that depend on galactic latitude as well as radius should be tested in the future.

Distribution of electron energy density

Because the synchrotron spectrum of a single electron depends on the electron energy, the electron energy distribution affects the total spectrum. We consider two energy injections: a delta function at the energy of the DM particle mass, and a power law of

(8)

[4]. The normalization constant is set so that . In diffusion models, the energy distribution is modified by a diffusion code to achieve the steady state solution.

Diffusion of electron energy

Ultra-relativistic electrons lose their energy through (a) synchrotron radiation and (b) inverse Compton scattering. These losses are accounted for in a diffusion code, which solves for the steady-state system. Models without diffusion were considered as well; for these simpler models we assumed that an electron lost its energy as a step function at one million years for an electron of frequency ν = 23 GHz, and at 10 million years for ν = 3 GHz (see Figure 3).

We used several binary “switches” in the modeling process, varying the model to allow for a step function energy decay (0) versus steady-state diffusion (1), a power law electron energy injection (0) versus a delta function injection at 100 GeV (1), and a constant magnetic field (0) versus an exponentially decaying magnetic field (1).

Figure 3
Figure 3. jv as a function of electron energy. The synchrotron flux is calculated for two values of v: v1 = 3.3 GHz (ARCADE) and v2 = 22.5 GHz (WMAP). The synchrotron distributions peak at different energies: 10 GeV for ARCADE and 30 GeV for WMAP. These differing synchrotron distributions reflect the different energy loss rates for electrons at different frequencies.

Boost factors

For each set of parameters, we found the boost factor, which is the ratio of the observed flux to the modeled flux, for both ARCADE and WMAP. For WMAP, we were also able to calculate an offset.

(9)

The best fit for the WMAP data was determined by finding the reduced χ2 over bfWMAP and offset. Because we did not have data from ARCADE, we could only approximate the boost factor (see Equation 12). The normalized boost factor,

(10)

describes the ratio of the synchrotron excess from ARCADE to a model that fits the WMAP data.

Although the value of bfWMAP and bfARCADE are sensitively dependent on the WIMP mass, the ratio between them is not (see Table 1). The boost factor can be interpreted as either an enhancement of the cross section above the assumed 3 × 10−26 cm3/s value, or an enhancement in the mean of the dark matter density squared ‹ρ2›/‹ρ2› (e.g., via substructure). The boost factors required to fit various models to the WMAP and ARCADE data are shown in Table 1.

Data

We used data from the ARCADE balloon survey in the 3.3 GHz band. Problems with the 8 and 10 GHz bands made these data unreliable. Because we did not have raw data from the ARCADE team, we calculated the intensity for points within an annulus of inner radius 20o and outer radius 40o centered on (l, b) = (70, 0), which approximates the region of the sky seen by ARCADE (see Figure 4). We approximated a value for the excess antenna temperature based on the relation [2].

(11)

This equation yields an antenna temperature of 23 mK in the ring at 3.3 GHz. We used the conversion factor 2.9939 mK / kJy to convert from antenna temperature to kJy/sr. We were able to compare this value to the average flux through the ARCADE footprint (see Table 1) and at four sample points (see Table 2) in our model. At 3.3 GHz,

(12)

where bfARCADE is the boost factor to the ARCADE data and Iν is the intensity at 3.3 GHz in kJy/sr.

Figure 4
Figure 4. The ARCADE footprint and calculated annulus. Left: The region of the sky observed by ARCADE, Figure from Kogut et. al. (2). Color corresponds to the total antenna temperature. Right: The calculated annulus is overlaid in red.
Table 1
Table 1. ARCADE vs. WMAP Boost Factors. For each permutation of parameters, the best χ2 was found by varying bfWMAP and offset. The boost factor and offset to the WMAP data for each fit are listed. Boost factors for the ARCADE and WMAP data were compared for each permutation of the parameters. From left to right, the parameter code indicates diffusion, energy injection, magnetic field (see §2.2 for a description of each switch). ARCADE boost factors are approximate (to a factor of 10 without diffusion, a factor of 2 with diffusion).
Table 2
Table 2. Boost Factors at different coordinates at 3 GHz. The boost factor was calculated at three points within the ARCADE ring one point (*) outside the ARCADE ring (see Figure 4).

Results

Brightness of the ARCADE excess

In our most reasonable models, we found the ARCADE excess at (l, b) = (70, 30) to be a factor of ~ 20 higher than the excess generated through the annulus our model after normalizing to the WMAP boost factor (see Table 1). The model that produces this result involve energy dissipation through diffusion, electron energy injection as a power law, and an exponentially decaying magnetic field (parameters 101, see Figure 5).

Comparative spectra

We compared the spectral indices of ARCADE and WMAP data to our model at 3 GHz and 23 GHz. A spectral index is defined by

(13)

where s is the spectral index [7]. The model shows indices of 0.90 at 3.3 GHz and 1.0 at 22.5 GHz. ARCADE quotes a spectral index of 0.57 at 1 GHz [2], which is “harder” (skewed toward higher frequencies) than the model by a spectral index of 0.33 (see Figure 6).

Figure 5
Figure 5. Preferred fit of WMAP Haze. This model allows for energy diffusion, a power law energy distribution and an exponentially decaying magnetic field (parameters 101 in Table 1). The dashed line is the model; the vertical bars are the WMAP data. bf = 2.05719, offset = 0.199734, χ2 =21.1056665.
Figure 6
Figure 6. Comparative spectra. Let the distribution of electron energies from 1-100 GeV produced by a power-law injection and subjected to energy diffusion. Right: The spectra of the model (solid) and the ARCADE data (dotted) from 1-100 GHz. Whereas the ARCADE data claims a power law of Iν α ν−0.57 (2), the vanilla WIMP model produces a power law of Iν α ν−0.90 at 3 GHz, and Iν α ν−1.0 at 23 GHz. Note that antenna temperature goes as ν2Iν , which is why the spectral index is ~ 0.5 as opposed to ~ 2.54.

Discussion and Conclusions

Although the ARCADE boost factors nicely match the WMAP boost factors for some of the non-diffusion cases, the non-diffusion model is too uncertain to be conclusive. Furthermore, the diffusion model is a more accurate representation of the physical processes in the galaxy, and so its results are more believable. The most physically accurate model is the parameter set 101, which includes energy diffusion, a power law energy injection, and an exponentially decaying magnetic field. This model requires a normalized boost factor of ~ 20 to fit the model to the ARCADE data. Is a normalized boost factor of ~ 20 reasonable? In Table 2, the boost factors at individual points within the ARCADE footprint are shown. The normalized boost factors vary from ~ 3 to ~ 40, indicating a broad range of boost factors within the ARCADE annulus. Note that at (l, b) = (30, 0) (the sampled point nearest to the galactic center), the boost factor is the smallest at ~ 3; the model approaches the WMAP portion of the haze here. However, at (l, b) = (100, 0) the boost factor is ~ 40, indicating that the model drops off too quickly at large radii to explain the synchrotron excess. This is consistent with the data from PAMELA, a satellite that observed the positron-to-electron excess and found that dark matter annihilation requires high boost factors [8]. Thus, the synchrotron excess that ARCADE and PAMELA see is simply too bright to be explained by vanilla dark matter annihilation alone. Although any number of explanations could justify this discrepancy, two come to mind easily:

1. The ARCADE synchrotron excess is dominated by a non-DM source.
Local pulsars that produce electron-positron pairs could boost the local synchrotron signal, creating the excess observed by ARCADE. A pulsar-created synchrotron excess will be modeled to determine whether there are enough local pulsars to explain the ARCADE and PAMELA excesses. However, it is important to note that finding a working model of pulsar-driven synchrotron excess does not rule out dark matter annihilation mechanisms. In particular, the WMAP haze, which is well fit by a vanilla WIMP model, cannot be explained by pulsars because the spatial distribution of pulsars in the galaxy is inconsistent with the haze morphology.

2. Dark matter particle interactions are more complex than the vanilla WIMP model permits.
Our model was built on the assumption that neutralinos only interact via gravity on large scales. Some models for dark matter allow the particles to attract or repel each other through other forces, which could drastically change the annihilation rates of the DM particles. In particular, attractive particles in over-dense regions of substructure could produce far more synchrotron radiation than the simple “vanilla” WIMPs considered in our models. More exotic dark matter particles will be considered in future simulations.

References

Finkbeiner, Douglas. “WMAP Microwave Emission Interpreted as Dark Matter Annihilation in the Inner Galaxy.” AstroPh preprint, 2005.
Kogut et. al. “ARCADE 2 Observations of Galactic Radio Emission.” AstroPhpreprint, 2008.
Dobler and Finkbeiner. “Extended Anomalous Foreground Emission in the WMAP 3-Year Data. AstroPh preprint, 10/09/06.
Blasi, Olinto,and Tyler. “Detecting WIMPs in the microwave sky.” Astroparticle Physics, 18 (2003) 649-662.
Bertone, Hooper, and Silk. “Particle Dark Matter: Evidence, Candidates and Constraints”. Fermilab, 2004.
Navarro, Frenk, White. “A Universal Density Profile from Hierarchical Clustering.” ApJ, 490:493-508,1997 December 1.
Rybicki, Lightman. Radiative Processes in Astrophysics. Wiley and Sons, Inc., 1979.
Cholis et. al. “High Energy Positrons from Annihilating Dark Matter.” arXiv:0809.1683v1 [hep-ph]

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