Strongly Interacting Massive Particles
From January 2022-May 2024 I conducted dark matter research under the supervision of Dr. Walter Tangarife at Loyola University Chicago. My work began focusing on cosmology, before being assigned a project on particle physics. I have included some of the documentation of my work below.
Geometry
Metrics
A metric takes in two coordinates and gives the distance (or line element) between them. In 3 dimensional Cartesian and spherical coordinate systems:
$$\text{d}\ell^2 = \text{d}x^2 + \text{d}y^2 + \text{d}z^2 = \sum_{i,j=1}^{3} \text{d}x^i \text{d}x^j \delta_{ij}$$ $$\text{d}\ell^2 = \text{d}r^2 + r^2 \text{d}\theta^2 + r^2 \sin^2{\theta}\text{d}\phi^2 = \sum_{i,j=1}^{3} \text{d}x^i \text{d}x^j g_{ij}$$where $\delta_{ij}$ and $g_{\mu\nu}$ are the metrics in each coordinate system. In diagonal form:
$$\text{diag}(1,1,1) \quad \text{and} \quad \text{diag}(1, r^2, r^2\sin^2{\theta})$$Homogeneity and Isotropy
When viewed from a massive scale, the universe is homogeneous and isotropic. The four-dimensional line element can be written as:
$$\text{d}s^2 = \text{d}t^2 - a^2(t)\text{d}\ell^2$$where $a(t)$ represents the expansion of the universe.
Curvature
The metric can be adjusted to account for curvature of 3-spaces:
$$\text{d}\ell^2 = \frac{\text{d}r^2}{1-kr^2} + r^2(\text{d}\theta^2 + \sin^2{\theta}\text{d}\phi^2)$$Robertson-Walker Metric
Combining the expansion and curvature terms yields the Robertson-Walker metric:
$$\text{d}s^2 = \text{d}t^2 - a^2(t)\left(\frac{\text{d}r^2}{1-kr^2} + r^2(\text{d}\theta^2 + \sin^2{\theta}\text{d}\phi^2)\right)$$Coordinates in an Expanding Universe
$$r_{\text{phys}} = a(t)r$$ $$v_{\text{phys}} = \frac{\text{d}r_{\text{phys}}}{\text{d}t} = a(t)\dot{r} + \dot{a}r = v_{\text{peculiar}} + Hr_{\text{phys}}$$where $H \equiv \frac{\dot{a}}{a}$ is the Hubble parameter.
Christoffel Symbols
The Christoffel symbol $\Gamma_{\alpha\beta}^{\mu}$ describes how basis vectors change in curved spacetime:
$$\Gamma_{\alpha\beta}^{\mu} = \frac{1}{2}g^{\mu\gamma}(\partial_\alpha g_{\beta\gamma} + \partial_\beta g_{\alpha\gamma} - \partial_\gamma g_{\alpha\beta})$$For the Robertson-Walker metric with $g_{\mu\nu} = \text{diag}(1, -a^2, -a^2, -a^2)$, key non-zero components are:
$$\Gamma_{ij}^0 = a\dot{a}\delta_{ij}, \quad \Gamma_{0j}^i = \frac{\dot{a}}{a}\delta_j^i$$Dynamics
Conservation of the Stress-Energy Tensor
The stress-energy tensor satisfies:
$$\nabla_\mu T^{\mu}_{\nu} = 0$$For a perfect fluid, this yields the continuity equation:
$$\dot{\rho} + 3\frac{\dot{a}}{a}(\rho + P) = 0$$Ricci Scalar
The Ricci scalar for the Robertson-Walker metric is:
$$R = -6\left[\frac{\ddot{a}}{a} + \left(\frac{\dot{a}}{a}\right)^2 + \frac{k}{a^2}\right]$$Friedmann Equations
The Einstein field equations applied to the Robertson-Walker metric yield the Friedmann equations:
$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2}$$ $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3P)$$Age of the Universe (Matter-Dominated)
Assuming only matter in the universe:
$$\left(\frac{\dot{a}}{a}\right)^2 = H_0^2 a^{-3}$$Integrating gives $t_0 \approx 9.61 \times 10^9$ years, which is only 70% of the observed age (13.6 billion years).
Dark Energy and $\Omega_m$
Including a cosmological constant (dark energy):
$$H^2 = H_0^2(\Omega_m a^{-3} + \Omega_\Lambda)$$where $\Omega_m + \Omega_\Lambda = 1$. Matching the observed age of 13.6 billion years yields $\Omega_m \approx 0.36$, indicating that about 64% of the universe's energy density is dark energy.
Particle Physics: Strongly Interacting Massive Particles (SIMPs)
While I no longer have the notebooks in which the research was documented, I still have some of the background material and results from the final report. I've included them below:
The SIMP model relies on $2 \to 2$, $3 \to 2$, and $4 \to 2$ annihilation mechanisms. Each mechanism predicts a different mass range for dark matter:
$4 \to 2$ annihilations in the TeV range, $3 \to 2$ annihilations in the GeV range, and $2 \to 2$ annihilations in the keV range
This project focused on a model using the $3 \to 2$ annihilation mechanism combined with a $2 \to 2$ self-interaction process. The $2 \to 2$ self-interaction process ensures that all dark matter particles remain at equal temperatures (kinetic equilibrium), while the $3 \to 2$ annihilation mechanism is responsible for freeze-out.
The Boltzmann equation for the number density of dark matter particles is:
$$\partial_t n + 3Hn = -\big(n^3 - n^2 n_{\text{eq}}\big)\langle\sigma v^2\rangle_{3\to 2} - \big(n^2 - n_{\text{eq}}^2\big)\langle\sigma v\rangle_{\text{ann}}$$This can be transformed using the yield $Y = n/s$ (where $s$ is the entropy density) to solve for the relic abundance.
I also have some of the definitions, parameterizations, and equations used in my final model. Here are a few key ones:
$$\begin{align*} \partial_t n + 3Hn &= -\big(n^3 - n^2 n_{\text{eq}}\big)\langle\sigma v^2\rangle_{3\to 2} - \big(n^2 - n_{\text{eq}}^2\big)\langle\sigma v\rangle_{\text{ann}} \\ \langle\sigma v^2\rangle_{\text{ann}} &\equiv \frac{\epsilon^2}{m_{\text{DM}}^2} \\ \langle\sigma v^2\rangle_{3\to 2} &\equiv \frac{\alpha_{\text{eff}}^3}{m_{\text{DM}}^5} \\ \Omega h^2 &= \frac{10^9 \cdot x_F \cdot g_{\ast,F}^{-1/2}}{\text{GeV} \cdot \langle\sigma v^2\rangle_{3\to 2} \cdot M_{\text{Pl}}} \\ \alpha_{\text{eff}} &= \frac{1}{1.4} \cdot m_{\text{DM}} \cdot x_F^{4/3} \cdot g_{\ast,F}^{1/6} \cdot (\kappa \cdot T_{\text{eq}})^{-2/3} \cdot M_{\text{Pl}}^{1/3} \\ \alpha_{\text{eff}} &= \left(\frac{\Omega h^2 \cdot m_{\text{DM}}^6 \cdot x_F^{1/3} \cdot (\kappa \cdot T_{\text{eq}})^{-2/3} \cdot M_{\text{Pl}} \cdot g_{\ast,F}^{2/3} \cdot \text{GeV}}{10^9}\right)^{1/4} \\ \kappa &= \frac{2\pi^2 c g_{\ast,s}(T)}{45} \\ H &= 1.67\sqrt{g_\ast}\frac{m^2}{M_{\text{Pl}}\cdot x^2} \end{align*}$$Technologies: Mathematica, Python