The Sun as a Detector for Dark Matter

THESIS TO OBTAIN THE MASTER SCIENCE DEGREE IN ENG. PHYSICS

José Vargas Lopes

Supervised by

Prof. Ilidio P. Lopes

About me

My name is José Vargas Lopes, and I have recently obtained the Master Science Degree in Engineering Physics by Instituto Superior Técnico, Universidade de Lisboa. Currently, my research interests focus on studying the phenomenology of Dark Matter inside Stars. I also believe that outreach plays a major role in the advancement of science. Thus, I decided to create this website, where I present the work I have developed during my MSc Thesis Dissertation, supervised by Prof. Ilídio Lopes in the Multidisciplinary centre for Astrophysics.

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— OUTLINE —

INTRODUCTION

FRAMEWORK

RESULTS

INTRODUCTION

2021

Lets travel to the very first time where the term Dark Matter was used. While studying the movement of galaxies within the Coma Cluster, Fritz Zwicky, a Swiss astronomer, found that the recorded movement patterns could not be explained by the sheer amount of mass visible within the System. To solve this discrepancy, Zwicky postulated that the majority of the system's mass was in a non-visible type of matter, the so-called Dark Matter.

If this should be true, the surprising result shows that Dark Matter exists in much greater density than luminous matter.

Fritz Zwicky

in “Die Rotvershiebung von extragalaktischen Nebeln”,
Helvetica Physica Acta, vol. 6, p. 110-127

The Case for Particle Dark Matter

Since 1933, more discrepancies in the mass content of astronomical systems were found. Despite some advocating that modifications to Einstein's General Relativity would solve the problem of missing mass without postulating a new type of matter, the preferred paradigm is that Dark Matter is indeed a massive Beyond the Standard Model Particle. This picture is favoured by different observations at different astronomical scales.

Galaxy Rotation Profiles

The observation of an asymptotically flattened behavior in galactic rotation profiles hints for the presence of Dark Matter beyond the stellar disk.

The Bullet Cluster

The off-set in the mass profile of colliding galaxy clusters points to a dominant non-interactive Dark Matter contribution to the system's total mass.

Cosmology

A Universe populated by the right abundance of Cold Dark Matter particles successfully accounts for the structure of the CMB and the existence of large-scale structures.

Galactic Rotation Profiles

Keplerian dynamics state that the orbital velocity of stars beyond the visible stellar disk, i.e., outside the visible mass content of the galaxy, decreases as the distance from the its center increases (green curve). However, it was observed that, for most galaxies, this orbital velocity of stars is approximatelly constant (purple curve). Such can be explained by a Dark Matter halo which extends beyond the span of the stellar visible disk.

The Bullet Cluster

The Bullet cluster is the aftermath of the collision between two galaxy clusters. Initially, each of the clusters was composed by a stellar population (green) and intracluster hot plasma (purple). During the collision, these components decoupled due to the different forces acting on them. Today, we can use different techniques to infer the mass distribution of this system. Surprisingly, it was observed that the bulk of the system's mass was near that of the stellar population, rather than that of the hot plasma, as expected from prior considerations. It was quickly understood that this result could be explained by the presence of a particle Dark Matter halo located near the stellar population, contributing dominantly to the total system mass.

The Standard Cosmology

The ΛCDM (Lambda Cold Dark Matter) model is generally accepted as the current standard model of our Universe. Despite not being a direct proof per se, this model's success is a benchmark evidence for the existence of particle Dark Matter, since it successfully predicts: the accelerated expansion of the Universe as measured by the Hubble Space Telescope; the existence of large-scale structures in the Cosmos; and the existence and structure of the Cosmic Microwave Background. Moreover, regarding the latter, measurements from the Planck Satellite show that Dark Matter accounts for approximately 24% of the universe's energy content, which is approximatelly 5 times the contribution of ordinary matter.

Dark Matter

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Atoms

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Dark Energy

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In the paradigm of Particle Dark Matter, different candidates have been suggested to address the missing mass problem in large astronomical structures. Ranging from light axions, to sterile neutrinos, to primordial black holes, each of these candidates were proposed to address not only the Dark Matter puzzle, but other missing links in different areas of Physics as well. In this work, we adopted the general class of Weakly Interactive Massive Particles (WIMP), in which Dark Matter is a Beyond the Standard Model Massive particle, usually represented by the greek letter Chi.

DARK MATTER AND THE SUN

The hypothesis that galaxies, including ours, the Milky Way, can be embedded in an halo of Dark Matter particles has an interesting consequence for the Sun: since WIMPs are massive, they can be gravitationally trapped inside our star, resulting in a whole new phenomenology which can be used to study the properties of Dark Matter.

DARK MATTER GRAVITATIONAL CAPTURE

Upon transversing the Solar plasma, WIMPs can scatter with the barionic nuclei. If their velocity after scattering is lower than the local escape velocity, WIMPs will be gravitationally captured inside the Sun.

DARK MATTER ENERGY PRODUCTION/TRANSPORT

WIMPs captured inside the Sun will seldom interact with the nuclei in the Solar plasma, serving as additional energy production and transport mechanisms to the energetic balance of Sun.

DARK MATTER ANNIHILATION

WIMPs accumulated in the Sun will annihilate and create a flux of Standard Model particles. Amongst these particles, only neutrinos will be able to escape the Sun without losing a considerable amount of energy.

Neutrinos From Dark Matter Annihilation in the Sun

Dark Matter particles accumulated in the Sun will annihilate and create a characteristic flux of neutrinos. Due to their weakly interactive nature, these neutrinos will escape the Sun and arrive at the Earth, where they can be detected.

Neutrino Detectors

Neutrinos from Dark Matter annihilation in the Sun can be detected in large water Cherenkov detectors. These detectors consist in large vessels, usually filled with water or ice, and an array of light detectors spatially distributed in order to detect Cherenkov radiation. This radiation is produced by particles cruising at velocities exceeding the local speed of light. The IceCube neutrino observatory, for instance, consists in a total of 516 light detectors arranged in a kilometer sided cube, buried 2 kilometers beneath the ice cap in the South Pole.

Cherenkov Radiation

Neutrinos arriving from the Sun will interact with the material in the experiment's vicinity, producing charged particles which will create detectable Cherenkov radiation.

Neutrino Flux

The presence of Dark Matter in the Sun can thus be inferred by a neutrino signal in Water Cherenkov Neutrino Telescopes, such has the IceCube, located in the South Pole.

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IceCube Array

South Pole

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Cristo Rei

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on each one of the

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arranged in the Ice.

Cherenkov Radiation

Neutrinos arriving from the Sun will interact with the material in the experiment's vicinity, producing charged particles which will create detectable Cherenkov radiation.

Neutrino Flux

The presence of Dark Matter in the Sun can thus be inferred by a neutrino signal in Water Cherenkov Neutrino Telescopes, such has the IceCube, located in the South Pole.

Dark Matter Signal

Due to the large number of light detectors present in neutrino telescopes, we are able to obtain the energy of the neutrino, as well as the direction of its path. The key idea behind this work is that Dark Matter models can be constrained by the absence of an excess in the neutrino signal arriving from the Sun.

OBJECTIVE

1. MODEL THE SUN

Use a consistent stellar evolution code encompassing Dark Matter phenomenology.

2. PREDICT NEUTRINO FLUX

Compute the Dark Matter signal for different particle and astrophysical parameters.

3. LIMIT DARK MATTER MODELS

Constrain Dark Matter models exploiting the limits obtained by neutrino experiments.

FRAMEWORK

Formalism

No Thesis about theoretical physics should be worthy of the name unless it has a dozen of equations, right? Anyway, if you got all the way to this section, you are probably used to deal with mathematics, so lets procceed. In this section we briefly describe the formalism used to tackle the phenomenology of Dark Matter and the Sun, which was implement in the stellar evolution code.

Dark Matter in the Sun

The Number of WIMPs in the Sun is governed by

$$\frac{ \text{d} \class{wimpNumber}{N_{\chi}}}{\text{d}t}(t) = \class{captureRate}{C_{\bigodot}} - \class{annihilationCoef}{A_{\bigodot}}\class{wimpNumber}{N_{\chi}}.$$

(1)

The Capture Rate accounts for capture due to all the elements present in the Sun

$$ \class{captureRate}{C_{\bigodot}} = \sum_i \int_{\bigodot} \frac{ \text{d} \class{captureRateI}{C_{\bigodot,i}}}{\text{d}V} \text{d}^3r,$$

(2)

and depends on the WIMP's mass, scattering cross-section, velocity distribution in the Milky Way, the Solar mass, etc (Gould 1987).

The annihilation coefficient is given by,

$$ \class{annihilationCoef}{A_{\bigodot}} =\int_{\bigodot} \class{annihilationCS}{\langle \sigma v \rangle} \class{wimpDist}{n_{\chi}^2(r)} \text{d}^3r,$$

(3)

and depends on the thermally averaged annihilation cross-section and WIMP distribution.

The equation for the evolution of the Number of WIMPs has an analytical solution,

$$ \class{wimpNumber}{N_{\chi}(t)} = \sqrt{\frac{\class{captureRate}{C_{\bigodot}}}{\class{annihilationCoef}{A_{\bigodot}}}} \tanh \left( t / \class{equilibriumTime}{t_{\text{eq}}} \right),$$

(4)

where the equilibrium time-scale is defined as,

$$ \class{equilibriumTime}{t_{\text{eq}}} \equiv \left( \class{captureRate}{C_{\bigodot}} \class{annihilationCoef}{A_{\bigodot}} \right)^{-\frac{1}{2}}.$$

(5)

If the age of the Sun is larger than the equilibrium time-scale, the number of WIMPs has reached an equilibrium and is independent of the time,

$$ \class{equilibriumTime}{t_{\text{eq}}} \ll \class{ageSun}{t_{\bigodot}} \rightarrow \class{wimpNumber}{N_{\chi}(t)} \approx \sqrt{\frac{\class{captureRate}{C_{\bigodot}}}{\class{annihilationCoef}{A_{\bigodot}}}}. $$

(6)

— § —

Did you know?

The idea that WIMPs trapped inside the Sun would create an additional source of energy transport to the Solar energetic balance was first proposed in an attempt to solve the Solar Neutrino Problem, by Press and Spergel in 1985. WIMPs accumulated in the Solar core would decrease the Solar neutrino flux to the values observed in neutrino experiments. It was later found that the deficit in the solar neutrino flux was in fact due to neutrino oscillations, as proposed by Bruno pontecorvo in 1968, a discovery that awarded the 2015 Nobel award for Physics to Takaaki Kajita and Arthur McDonald.

Dark Matter Interplay with the Sun

The distribution of WIMPs within the Sun is defined by the Knudsen number (Faulkner et al. 1986),

$$ \class{knudsen}{K(t)} = \frac{\class{mfp}{l_{\chi}(0,t)}}{\class{wimpRadius}{r_{\chi}(t)}}, \qquad \class{wimpRadius}{r_{\chi}(t)} = \left( \frac{9}{4\pi}\frac{\class{boltzmann}{k}\class{centralTemperature}{T_{\text{c}}(t)}}{\class{gConstant}{G}\class{centralRho}{\rho_{\text{c}}(t)}\class{wimpMass}{m_{\chi}}}\right), $$

(7)

which relates the WIMP mean free path, \( l_{\chi}(0,t) \), and the characteristic radius of the WIMP distribution, \( r_{\chi}(t) \).

When \(\class{knudsen}{K(t)} \ll 1\), WIMPs are in local thermodynamical equilibrium with the Solar Plasma (Gould & Raffelt 1990). On the other hand, when \(\class{knudsen}{K(t)} \gg 1\), WIMPs are in non-local equilibrium following an isothermal distribution (Spergel & Press 1985).

WIMP transport in the Sun contributes as an additional term in the Luminosity stellar equation (Bottino et al. 2002),

$$ \class{luminosity}{L_{\chi,\text{LTE}}(r)} = -\frac{16 \pi \class{radCoef}{a_{\text{rad}}}}{3}\frac{\class{solarTemperature}{T(r)^3}r^2}{\class{wimpOpacity}{\kappa_{\chi}(r)}\class{solarRho}{\rho(r)}} \frac{ \text{d} \class{solarTemperature}{T}}{\text{d}r}(r) \times \class{suppressionFactor}{f(K)}.$$

(8)

The contribution to energy production in the Sun due to WIMP annihilation is (Yoon et al. 2008),

$$ \class{wimpEnergy}{\epsilon_{\chi}(r)} = \frac{2}{3} \frac{\class{wimpMass}{m_{\chi}}\class{wimpDist}{n_{\chi}^2(r)}}{\class{solarRho}{\rho(r)}}\class{annihilationCS}{\langle \sigma v \rangle}.$$

(9)

— § —

The Code|

In order to consistently obtain the Neutrino flux from Dark Matter annihilation in the Sun, we used a modified stellar evolution code to simulate the Sun and the resulting neutrino signal from WIMP annihilation. The final code is composed by three different modules, each one in charge of dealing with different phenomena.

~/CESAM2

The code is based on CESAM2, a group of publicly available subroutines which computes a 1D Quasi-Hydrostatic evolution of the Sun, according to the standard picture of Stellar evolution. (Morel 1997)

~/DMP2K

It was modified to include the computation of Dark Matter Capture, Annihilation, Energy Production and Transport at each age-step of evolution, taking into account the different conditions found in the Solar Plasma. (Lopes et al. 2011)

~/WIMPSIM

We integrated the code with a numerical package which simulates the neutrino signal from Dark Matter annihilation including neutral/charged current interactions and three flavor neutrino Oscillation (Blennow et al. 2008)

Example Simulation

In the video below, you will find a graphical rendering of what an example simulation of the Sun looks like, including the interplay with Dark Matter particles present in the Milky Way Halo. The Sun is evolved from the Zero Age Main Sequence (ZAMS) until its present age. See legend for more details.

LEGEND: The temperature profile within the Solar radius is represented on the first color bar, while the number of WIMPs is shown in the second. In the rightmost column you can see important variables. Evolution variables: age of the simulation; the Solar effective temperature; the radius of the first convection zone; the radius of the second convection zone. Calibration variables (in Solar units): the radius of the Sun; the Solar luminosity; the metallicity fraction at the surface of the Sun. Dark Matter variables: WIMP's mass; number of WIMPs; capture rate; annihilation rate.

RESULTS

Impact of Dark Matter in the Sun

Before moving on to the actual computation of the neutrino flux from Dark Matter annihilation in the Sun - the main objective of this work - we decided to study the extent of the effect of WIMPs to the Solar structure. We carried out a large number of Solar models, with different WIMP parameters, and used two different diagnostics to assess the impact of Dark Matter in the Sun: the Sound Speed Profile (SSP), which spans the entire Solar radius; and Solar Neutrinos from nuclear reactions, which are extremely sensible to the Solar Central temperature, probing the inner Solar core.

Sound Speed

In principle, the SSP is an excellent diagnostic to assess the impact of Dark Matter particles in the Sun, since it spans the entire Solar radius and is sensible to the conditions found accross the Solar plasma, such as the temperature and pressure profiles. In fig. 2, we plotted the difference in the SSP between the Reference Model, i.e. a standard Solar model without Dark Matter, and models evolved with different WIMP models. In order to study the maximal impact of WIMPs in the Sun, we chose a benchmark mass value for WIMPs of 5 GeV.

(a) Spin Independent

(b) Spin Dependent

Fig. 2 - Squared Sound-Speed difference between the Reference Model and models evolved with different annihilation cross-section values (see legend in figure).

WIMPs cluster strongly on the Solar core. For typical values, WIMPs tend to accumulate always within the first 10% of the Solar radius.

The central Temperature profile is lowered and flattened due to WIMP energy transport. The barionic density increases in the Solar Core.

The Sound Speed Profile cannot be used to study the validity of WIMP models, due to the large experimental and theoretical uncertainty in the inner Solar core.

Solar Neutrinos

Solar neutrinos, produced in the thermonuclear reactions fuelling our Sun, are extremely sensible to the conditions found within the inner Solar core. Since this is the region where WIMPs will acummulate, Solar neutrinos provide an excellent probe to the impact of WIMPs in the Sun. In fig. 3, we plotted the variation of the expected Boron Solar neutrino flux due to the presence of WIMPs with different parameters.

(a) Spin Independent

(b) Spin Dependent

Fig. 3 - Impact of WIMPs in the Boron Solar neutrino flux with respect to the Reference Model.

Neutrinos from the Boron nuclear Reaction are highly sensitive to the Solar central temperature.

Lower central temperatures lead to large deficits in the Boron Solar neutrino Flux (up to 50%).

WIMP models causing a variation on the Boron Solar neutrino flux larger than 35% are in contradiction with observations.

Neutrino Flux from Dark Matter Annihilation

Having identified the parameter space where the impact of WIMPs in the Sun is not negligible, we can safely procced to the computation of the neutrino flux from Dark Matter annihilation in the Sun.

The total neutrino flux from Dark Matter annihilation in the Sun is defined by three different ingredients,

$$ \class{neutrinoFlux}{\Phi_{\nu}} = \frac{\class{annihilationRate}{\Gamma_{\text{A}}}}{4 \pi r^2} \int \class{branchingRatio}{\sum_i\text{BR}_i} \class{neutrinoSpectra}{\frac{ \text{d} N_{\nu}}{\text{d}E_{\nu}}}\text{d}E_{\nu}.$$

(10)

1
2
3

1.

The Annihilation Rate defines the flux intensity and depends on the Number of WIMPs in the Sun and on the annihilation cross-section.

2.

The Sum is carried over the possible WIMP annihilation channels and weighted with the correspondent Branching Ratio.

3.

The neutrino spectra for channel i includes neutrino interactions/oscillations in the Sun and is computed by WIMPSIM.

Dark Matter Thermal Production

The Annihilation Rate in equation 10 will depend on the Annihilation cross-section. To obtain this variable, we must take into account the standard picture of Dark Matter production, which begun in the early ages of the Universe and led to the current Dark Matter density inferred from CMB observations.

EARLY UNIVERSE

Dark Matter particles are in thermodynamical and kinetic equilibrium with the primordial thermal bath.

FREEZE-OUT

The universe’s temperature drops below the production threshold, fixing the Dark Matter Abundance

RELIC ABUNDANCE

The Dark Matter abundance inferred from the CMB is a footprint of the freeze-out epoch.

The Dark Matter abundance at the Freeze-out is defined by the thermally averaged annihilation cross-section,

$$ \class{annihilationCS}{\langle \sigma v \rangle} \approx \class{sWaveCoef}{a} + \class{pWaveCoef}{b} \langle v^2 \rangle$$

(11)

which is written as a sum of s-wave and p-wave contributions. The coefficients are obtained taking into account the Dark Matter abundance inferred from the CMB.

— § —

Annihilation Cross-section

The coefficient \( a \) and \( b \) in the annihilation cross-section expression (11) can be obtained by solving the rate equation for the number of WIMPs with the boundary condition that the abundance today is equal to the abundance measured by the Planck Satellite. For the p-wave case, we numerically solved the Boltzmann equation, which can be seen in figure 4, where we represent the Dark Matter abundance for different WIMP masses as a function of \(x \), which is a measurement of the age of the Universe.

Fig. 4 - The evolution of the number of WIMPs with p-wave annihilations. Three curves are shown for different mass values. Solid Black : WIMP density when in thermodynamical equilibrium with the photon bath. Dotted Black: Density measured today by the PLANCK satellite (Ade et al 2015).

The evolution of the Dark Matter density is obtained by solving the Boltzmann equation.

The annihilation cross-section is obtained requiring a match with the observed Dark Matter abundance by the PLANCK Satellite (Ade et al. 2015).

The coefficient for the p-wave case for which the Dark Matter density is correct is: \( b \approx 9,6 \times 10^{-25} \text{cm}^3 \text{s}^{-1}.\)

Annihilation Rate

The next step consists in the computation of the Annihilation rate, which will define the intensity of the neutrino flux from Dark Matter annihilation.

The WIMP annihilation rate in the Sun is given by,

$$ \class{annihilationRate}{\Gamma_{\text{A}}} = \frac{1}{2} \class{wimpNumber}{N_{\chi}^2} \class{annihilationCoef}{A_{\bigodot}}$$

(12)

and its computations is fundamentally different for the p-wave and s-wave case.

S-WAVE

In the s-wave case, the Number of WIMPs is always in equilibrium. Thus, the Annihilation Rate is independent of the Annihilation cross-section:

$$ \class{annihilationRate}{\Gamma_{\text{A}}} = \frac{1}{2} \class{captureRate}{C_{\bigodot}}.$$

(13.1)

P-WAVE

In the p-wave case, The Number of WIMPs is not always in equilibrium. The Annihilation coefficient in the parameter region of interest is:

$$ \class{annihilationCoef}{A_{\bigodot}} = \frac{\class{centralTemperature}{T_{\text{c}}}}{\class{wimpMass}{m_{\chi}}}\frac{\class{pWaveCoef}{b}}{\sqrt{8} \pi^{\frac{3}{2}} \class{wimpRadius}{r_{\chi}^3}}.$$

(13.2)

(a) Annihilation Rate - Spin Independent

(b) Annihilation Rate - Spin Dependent

(c) Equilibrium parameter - Spin Independent

(d) Equilibrium parameter - Spin Dependent

Fig. 5 - Top: Annihilation Rate for WIMPs with s-wave (dashed) and p-wave (solid) annihilation. Bottom: The equilibrium parameter, \( K \equiv \tanh \left( t_{\odot}/t_{eq}\right)\), for WIMPs with p-wave annihilation.

The Annihilation Rate is the same for both s-wave and p-wave cases when the number of WIMPs is in equilibrium.

The Annihilation Rate is smaller for the p-wave case when the number of WIMPs is not in equilibrium.

CONSTRAINTS FROM DARK MATTER NEUTRINO

Having gathered all the necessary ingredients for the computation of the neutrino flux for a given WIMP model, we used the most recent observations of the neutrino flux arriving from the Sun, measured by the Super-Kamiokande and IceCube neutrino detectors. This allowed us to obtain the limits on the Spin Dependent and Spin Independent Scattering cross-section for WIMPs annihilating through different annihilation channels.

(a) Spin Independent

(b) Spin Dependent

Fig. 6 - Limits on the scattering cross-section for WIMPs for s-wave (dashed) and p-wave (dotted) annihilations, obtained from the IceCube (Aartsen et al. 2013) and Super-Kamiokande (Choi et al. 2015). The regions of interest obtained by direct detection experiments are also shown.

P-wave constraints are (sometimes 2 orders of magnitude) above s-wave limits, alleviating the disagreement with the regions of interest observed by other experiments.

As we improve detectors, p-wave models will be increasingly harder to constrain, since we are entering the region where the number of WIMPs is out of Equilibrium.

The uncertainty in current Solar models will result in an uncertainty up to 25% (depending on the annihilation channel) on the limits for s-wave and p-wave models.

Conclusion

We used a consistent and robust stellar evolution code to obtain the limits on WIMP models using the limits on the neutrino flux from neutrino telescopes.

We identified the parameter space for which the presence of Dark Matter in the Sun results in significant departures from the standard solar model.

To obtain the annihilation cross-section for p-wave annihilation, we numerically solved the Boltzmann equation for the number density of a relic particle satisfying the Dark Matter abundance as measured by the Planck Satellite.

We computed the p-wave limits with the most recent data-set from the Super-Kamiokande and IceCube experiments.

Using the appropriate Solar model, we can use the Sun as a unique and interdisciplinary Laboratory to study and constrain Dark Matter Models.

Further Reading

Do you want to learn more about this work? Check out the article upon which this website is based, "New Limits on Thermally annihilating Dark Matter from Neutrinos Telescopes", published in The Astrophysical Journal, Volume 827, Number 2.

See Article

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