Dark Matter

<h1>
              Search for scalar dark matter and variation of fundamental constants with spatially separated sensors
            </h1>

<br>

<h3>
              <it>QSNET Seminar, 5 February 2024</it>
            </h3>

<br>

<h4>

Melina Filzinger, Ashlee Caddell, Dhruv Jani, Martin Steinel, 
              Leo Giani, Nils Huntemann, BMR, [arXiv:2312.13723](https://arxiv.org/abs/2312.13723)

</h4>

### Benjamin M. Roberts

<br>

School of Mathematics and Physics,
            University of Queensland, Australia

* Slides [broberts.io/talks](https://broberts.io/talks)

<img src="img/uq-logo.svg" width="25%"> 
            
            
                             Supported by: ARC and 
                       
            <img src="img/BQI.png" width="20%">
            
            <br><br>

# Unexplained mysteries

<br>

<h2>Why is there so much more matter than anti-matter?</h2>
                <h3> • CP-violation in the Standard Model is too small</h3>
              </div>

<h2>Quantum theory for gravitation?</h2>

</div>

<h2>No explanation for dark energy</h2>
                <h3> • Explain accelerated expansion of universe
                  • Cosmologists: Why is Λ so large?
                  • Particle theorists: Why is Λ so small?
                </h3>

</div>

<h2>No explanation for dark matter...</h2>

</div>

# Galactic Rotation Curves

<div style="text-align: center; float: left; width: 70%">
              <br><br>
              <img
                src="https://upload.wikimedia.org/wikipedia/commons/c/cd/Rotation_curve_of_spiral_galaxy_Messier_33_%28Triangulum%29.png"
                width="95%"><br>
              <font size="3">[Mario De Leo, Wikimedia Commons]</font>
            </div>

<div style="text-align: left; float: right; width: 30%">
              <br><br>
              <h2>Missing Matter?</h2>
              <br>
              <p data-markdown>• Not a small correction</p>
              <br>
              <p data-markdown>• 80 - 95% of mass is "missing"</p>
              <br>
              <div data-markdown class="fragment fade-in">
                <p data-markdown>• Implies <font color="orange">dark matter</font></p><br>
                \[a = \frac{G \, [M(r) + {\color{orange}\delta M(r)}]}{r^2} \]
                <br>
                <p data-markdown>   • Simulations + observations: DM "halo"</p>
              </div>
              <br>
              <p data-markdown class="fragment fade-in">• ... or <font
                  color="green">modified
                  gravity</font><br><br>
                \[a = \frac{[G + {\color{green}\delta G(M,r)}] \, M(r)}{r^2} \]
              </p>
              <br>
            </div>

# Dark Matter: What we know

<br>
            <h3> There's lots of it</h3>

<h3> It gravitationally clusters in halos around galaxies</h3>
            <h4> (really, galaxies gravitationally cluster around DM)</h4>

<div style="text-align: left; float: left; width: 50%"
              class="fragment">
              <br><br>
              <h2>Very strong evidence:</h2>
              <br>
              <h3>• Rotation curves, gravitational lensing, CMB </h3>
              <br>
              <h3>• Also: large scale structure, BAO etc. </h3>
              <br>
              <h3>• It's not strange for particles to not interact with
                light (e.g., neutrinos)</h3>
              <br>
              <br>
              <h3 class="fragment">• ΛCDM model works extremely well
                <br>  • Ωc =
                0.2589(57) [Planck] </h3>
              <h3 class="fragment">• (Just need to work out the Λ and CDM parts)</h3>
            </div>

# Dark Matter: What we don't know

<br>
            <div class="fragment" data-fragment-index="0">
              <h2> ... everything else</h2>
            </div>

<br>

<div class="r-stack">
              <img class="fragment" data-fragment-index="1"
                src="img/DM_masses2.png" width="80%">
              <img class="fragment" data-fragment-index="3"
                src="img/DM_masses3.png" width="80%">
            </div>
            <div class="fragment" data-fragment-index="1">
              <font size="3">US "Cosmic Visions" report [arXiv:1707.04591]</font>
            </div>

<div class="fragment" data-fragment-index="1">
              <h3>• Possible mass range: 90 orders of magnitude!</h3>
            </div>
            <div class="fragment" data-fragment-index="2">
              <h3>• Fundamental particle: 50 orders (de Broglie to Planck)</h3>
            </div>

<br>

<div class="fragment">
              <h3> • Vast majority of focus on WIMPs, but field is very wide</h3>
            </div>

# WIMP miracle

<div style="text-align: left; float: left; width: 30%">
              <br><br>
              <img src="img/Feynman.png" width="95%">
              <br><br>
              <h3> • Early universe: hot and dense</h3>
              <h3> DM + SM in thermal equilibrium</h3>
              <br>
              <h3 class="fragment fade-in" data-fragment-index="0">
                • Universe cools: <br>
                Boltzmann suppression
              </h3>
              <br>
              <h3 class="fragment fade-in" data-fragment-index="1">
                • Universe expands: <br>
                Shuts off annihilation
              </h3>
            </div>

<div style="text-align: center; float: right; width: 65%">
              <div class="r-stack">
                <img class="fragment fade-in" data-fragment-index="0"
                  src="img/YDM/Yeq.png" width="90%">
                <img class="fragment fade-in" data-fragment-index="1"
                  src="img/YDM/YDM1.png" width="90%">
                <img class="fragment fade-in" data-fragment-index="2"
                  src="img/YDM/YDM2.png" width="90%">
              </div>
              <h3 class="fragment fade-in" data-fragment-index="3">
                • Weak interaction ($G_F \simeq 10^{-5}\,{\rm
                GeV}^{-2}$): new physics scale<br>
                • $m_{\rm weak} \simeq 100\,{\rm GeV}$ <br>
              </h3>
              <h3 class="fragment fade-in" data-fragment-index="4">
                • $\langle\sigma v\rangle_{\rm weak} \simeq
                3\times10^{-26}\,{\rm
                cm}^3/{\rm s}$ !!!
              </h3>
            </div>

# WIMP Searches: XENON

<div style="text-align: left; float: left; width: 60%">
              <div class="r-stack">
                <div class="fragment fade-out" data-fragment-index="1">
                  <br><br><br>
                  <img src="img/Feynman.png" width="65%">
                  <br><br>
                  <h3 class="fragment highlight-red" data-fragment-index="0">•
                    Direct detection
                    (scattering)</h3>
                  <h3>• Indirect detection (annihilation)</h3>
                  <h3>• Collider searches (production)</h3>
                </div>
                <div>
                  <div class="fragment fade-in" data-fragment-index="1">
                    <br>
                    <h3>Excluded cross-section for WIMPs:</h3>
                    <img
                      src="img/WIMP-constraints.png" width="90%"><br>
                      <font size="3">[GAMBIT]</font>
                  </div>
                  <h3 class="fragment fade-in" data-fragment-index="2">
                     • Mass below nuclear mass: energy threshold
                  </h3>
                  <h3 class="fragment fade-in" data-fragment-index="2">
                     • High mass: fewer particles
                  </h3>
                </div>
              </div>
            </div>

# En*light*ening the search for Dark Matter?
            ## Atomic Physics!

<br>

<h2> Mass drops below nuclear mass:</h2>
                <h3> • No nuclear recoils
                   • Instead: electron recoils + ionisation</h3>
              </div>
              <br>

<h2> Mass drops below electron mass:</h2>
                <h3> • No electron recoils
                   • Instead: absorption (dark photoelectric effect)</h3>
              </div>
              <br>

<h2> Mass drops below eV:</h2>
                <h3> • Classical DM field
                   • Quantum sensing (atomic clocks)</h3>
              </div>
              <br>

</div>

<div style="text-align: center; float: right; width: 40%">
              <br>
              <img class="fragment" data-fragment-index="0"
                src="img/atom.png" width="40%">
              <br><br>
              <div class="r-stack">
                <div class="fragment" data-fragment-index="2">
                  <img
                    src="img/laser-JILA.jpg" width="90%"><br>
                  <font size="3">[JILA]</font>
                </div>
                
              </div>
            </div>

# Light and Ultralight Dark Matter

$$ \Huge
            \rho_{\rm DM}\simeq 0.3 ~ \frac{\rm GeV}{{\rm cm}^3}
            $$

<br>

## Mass decreases $\implies$ number density increases:
            ## Classical boson field (e.g., axions, scalars)

<br>

<div style="text-align: left; float: right; width: 70%">
              <div class="fragment fade-in">
                <h3>
                  1. No interaction: oscillation: $\phi = \phi_0 \cos(m_\phi
                  t)$
                </h3>
                \[\phi_0^2\propto \rho_{\rm DM}\]
              </div>

<h3 class="fragment fade-in">
                2. Interactions: clumps
              </h3>
              <br>
              <h3 class="fragment fade-in">
                3. Also: constant build-up (local over-densities)
              </h3>
              <br><br>
            </div>

* Review: Hui _et al._, [PhysRevD.95.043541 (2017)](http://arxiv.org/abs/1610.08297)
            * Review: Antypas _et al._, Snowmass 2021 [arXiv:2203.14915](https://arxiv.org/abs/2203.14915)

# "Listen" for DM field using atomic clocks

<br>

<div style="text-align: left; float: left; width: 50%">
              <br> <br>
              <img src="img/NHanacek_NIST.png" width="90%"><br>
              <font size="3">[N Hanacek/NIST]</font>
            </div>

<div style="text-align: left; float: right; width: 50%">
              <br>
              <div>
                <h3>
                   • DM scalar field has small interaction with matter
                </h3>
                <h3>
                   • Shift atomic energy levels and frequencies
                </h3>
              </div>
              <br>
              <div class="fragment fade-in">
                <h3>Observable shift:</h3>
                
                <br>
                \[\huge
                \frac{\delta f}{f} = \kappa \, \phi_{\rm DM}(\vec{r},t)
                \]
              </div>
              <br>
              <div class="fragment fade-in">
                <h4>
                   • $\kappa$ - model + system dependent parameter
                </h4>
              </div>

<h3>
                Not just atomic clocks; not just dark matter
              </h3>

* Safronova _et al._, [RevModPhys.90.025008 (2018)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.90.025008)
              
            </div>
            </div>

# Variation of fundamental constants

<br>

<div>
          \[\large
            \omega^{A} = \underbrace{F_A(\alpha)}_{\text{Transition-specific}} 
            \times  \underbrace{m_e c^2\alpha^2}_{\text{Units}} 
          \]
          </div>

<div class="fragment" style="text-align: center; float: left; width: 50%">
            <br><br>
            <img src="img/Bohr.png" width="30%"><br>
            <font size="3">[JabberWok/Wikipedia]</font>
            <br><br><br>
            \[
            \frac{\Delta\omega}{\omega} = K\frac{\Delta\alpha}{\alpha}
            \qquad\qquad\qquad
            \left(
                {\color{red}K \equiv \frac{\partial \omega}{\partial \alpha}\frac{\alpha_0}{\omega}}
            \right)
            \]

<br>

* In general, $K$ calculated from atomic theory
            * Difference transitions: different sensitivity

<br>

* Dzuba, Flambaum, Webb [Phys Rev Lett 82, 888 (1999)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.888)
          </div>

<div class="fragment" style="text-align: left; float: right; width: 50%">
            <br> <br>
            <img src="img/AlphaSensitivity.png" width="90%"><br>
            <font size="3">[M. Murphy 2012]</font>
          </div>

# Complication + Resolution

### Example: H-like

<br>

<div style="text-align: left; float: left; width: 50%">
            \[
              \omega_{\rm SI} = \frac{m_e c^2Z^2{\color{red}\alpha^2}}{2}
              \left(\frac{1}{n^2}-\frac{1}{n'^2}\right)F(Z\alpha)
            \]
            <br>
            \[
              K = {\color{red}2} + K_{\rm rel}
            \]
          </div>

<div style="text-align: left; float: right; width: 50%" class="fragment">
            \[
                \omega_{\rm atomic} = \frac{Z^2}{2}
                \left(\frac{1}{n^2}-\frac{1}{n'^2}\right)F(Z\alpha)
            \]
            <br>
            \[
              K = K_{\rm rel}
            \]
          </div>

<div class="fragment">
          
            <br><br>
        
          * $K$ _and_ $\alpha$ both dimensionless, issue remains
          * $K$ only defined up to additive constant 
          * (constants only defined up to multiplicative constant)

</div>

### Resolution: ratios

$$
            \frac{\delta\left(\omega_A/\omega_B\right)}{\left(\omega_A/\omega_B\right)}
            = 
            K_A \frac{\delta\alpha}{\alpha}
            -K_B \frac{\delta\alpha}{\alpha}
            =
            (K_A  -K_B )\frac{\delta\alpha}{\alpha}
            $$

</div>

<div style="text-align: left; float: right; width: 50%">
             
                 •  Unit ambiguity cancels in ratios
                 •  Must have two lines at each spacetime location
            
            </div>

</div>

* Nice review: Kozlov, Budker [Ann. Phys. 1800254 (2018)](https://onlinelibrary.wiley.com/doi/10.1002/andp.201800254)

</div>

# Scalar-SM?

* Overview: Hees _et al._, [PhysRevD.98.064051 (2018)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.98.064051)
          <br>

<div>

</div>

<div class="r-stack">
                \[\huge
                \mathcal{L}_{\rm int.} = \phi \left[ {d_e \, F_{\mu\nu}F^{\mu\nu}} + {d_{m_f} \, \bar\psi\psi}  + {d_g \, G^a_{\mu\nu}G^{a\mu\nu}}\right]
                \]
              <div class="fragment fade-in-then-out" data-fragment-index="1">
                \[\huge
                \mathcal{L}_{\rm int.} = \phi \left[ {\color{yellow}d_e \, F_{\mu\nu}F^{\mu\nu}} + {d_{m_f} \, \bar\psi\psi}  + {d_g \, G^a_{\mu\nu}G^{a\mu\nu}}\right]
                \]
              </div>
              <div class="fragment fade-in-then-out" data-fragment-index="2">
                \[\huge
                \mathcal{L}_{\rm int.} = \phi \left[ {\color{yellow}d_e \, F_{\mu\nu}F^{\mu\nu}} + {\color{orange}d_{m_f} \, \bar\psi\psi}  + {d_g \, G^a_{\mu\nu}G^{a\mu\nu}}\right]
                \]
              </div>
              <div class="fragment fade-in" data-fragment-index="3">
                \[\huge
                \mathcal{L}_{\rm int.} = \phi \left[ {\color{yellow}d_e \, F_{\mu\nu}F^{\mu\nu}} + {\color{orange}d_{m_f} \, \bar\psi\psi}  + {\color{red}d_g \, G^a_{\mu\nu}G^{a\mu\nu}}\right]
                \]
              </div>
            </div>

<div style="text-align: left; float: left; width: 50%">
            <br>
            <div class="fragment fade-in" data-fragment-index="1">
              \[\large
                {\color{yellow}\alpha \to \alpha(1 + d_e \phi(r,t))}
              \]
              <h3>
                 • nb: $d_e = d_\gamma = d_\alpha = 1/\Lambda_\gamma$
              </h3>
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="2">
              \[\large
                {\color{orange}m_e \to m_e(1 + d_{m_e} \phi(r,t))}
              \]
              <h3>
                 • electron and quark masses
              </h3>
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="3">
              \[\large
                {\color{red}m_p \to m_p(1 + d_{g} \phi(r,t))}
              \]
              <h3>
                 • Proton mass: binding energy: QCD scale $\Lambda_{\rm QCD}$<br>
                 • Nuclear moments + radius: depend on $\Lambda_{\rm QCD}$, $m_q$
              </h3>
            </div>
          </div>

<div class="fragment fade-in" data-fragment-index="4">
              <h3>
                 • Optical clock
              </h3>
              \[
              \begin{align*}\large
               \omega &\propto R_y \, F_{\rm rel}(Z\alpha) \sim \alpha^{(2+K_{\rm rel})}m_e \\
                \frac{\delta\omega}{\omega} &= (2+K_{\rm rel}){\color{yellow}d_e} +{\color{orange}d_{m_e}}
              \end{align*}
              \]
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="5">
              <h3>
                 • Microwave (hyperfine) clock
              </h3>
              \[
              \begin{align*}\large
                \omega &\propto R_y \, [\alpha^2 F_{\rm rel}(Z\alpha)] \, (\mu \, m_e/m_p) \\
                \frac{\delta\omega}{\omega} &= (4+K_{\rm rel}'){\color{yellow}d_e} + {\color{orange}d_{m_e}} + ({\color{orange}d_{m_e}} - {\color{red}d_g}) + \kappa\,  ({\color{orange}d_{m_q}} - {\color{red}d_g})
              \end{align*}
              \]
            </div>
            <br>
            <div class="fragment fade-in" data-fragment-index="6">
              <h3>
                 • Cavity-stabilised laser
              </h3>
              \[
              \begin{align*}\large
                \omega &\propto 1/a_0 \\
                \frac{\delta\omega}{\omega} &= {\color{yellow}d_e} + {\color{orange}d_{m_e}}
              \end{align*}
              \]
            </div>
            </div>

# $d_{m_e}$ alone?

* Usually, appears in ratios from $m_e/m_p$
          * True in general for any 'dimensionful' constant
          * Potential issue: universal couplings

$$\large
              \delta(m_e/m_p) ~ \propto ~ (d_{m_e} - d_g)
          $$

## Fast oscillations:

* Damped vs. undamped oscillator

<img src="img/Geraci.png" width="20%"><br>
          <font size="3">[Geraci (2019)]</font>

<br>

* Free vs. fixed optical cavity -- Geraci _et al._,  [Phys.Rev.Lett.123.031304 (2019)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.123.031304)
          * Atom vs (damped) solid -- Antypas _et al._, [Phys.Rev.Lett.123.141102 (2019)](https://doi.org/10.1103/PhysRevLett.123.141102)
          * Time-delay interferometry -- Savalle _et al._, [Phys.Rev.Lett.126.051301 (2021)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.051301)
          * Overview: Antypas _et al._, [Ann. Phys. 532 (2020)](https://onlinelibrary.wiley.com/doi/10.1002/andp.201900566)

</div>

# Wait, variation of dimensionful constants?

* Almost always: appear as dimensionless ratios, so no issue
            * However, in some cases, appear alone

$$\large
            \frac{\delta\left(\omega_A/\omega_B\right)}{\left(\omega_A/\omega_B\right)}
            = 
            K_A \frac{\delta\alpha}{\alpha}(x_1)
            -K_B \frac{\delta\alpha}{\alpha}(x_2)
            ~{\color{red}\cancel{=
            (K_A  -K_B )\frac{\delta\alpha}{\alpha}}}
            $$

<br>

* Is this OK?

* Yes, so long as specific interaction is considered
              * Specific interaction Lagrangian removes ambiguity
            
            * Some care must be taken: correct $K$ factors
              * Interpretation as variation in constant may be unit dependent
              * Of course, physics/observables are not!

<br>

* Savalle, BMR, _et al._ [arXiv:1902.07192](https://arxiv.org/abs/1902.07192)
            * Savalle _et al._, [Phys.Rev.Lett.126.051301 (2021)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.051301)
            * Antypas _et al._, [Ann. Phys. 532 (2020)](https://onlinelibrary.wiley.com/doi/10.1002/andp.201900566)

</div>

# Space-time separated sensors

* Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [arXiv:2312.13723](https://arxiv.org/abs/2312.13723)

### Typical local experiment:

$$\large
              \frac{\delta(\nu_a/\nu_b)}{(\nu_a/\nu_b)} = \phi_0 \\, \Delta d_{\rm eff} \cos(\omega t)
            $$

<br>
            $$
              d_{\rm eff} = \sum_X K_X d_X
            $$

### Separated (identical) sensors:

<br>

<div>
            \[\large
            \begin{align*}
              \frac{\delta(\nu_a/\nu_b)}{(\nu_a/\nu_b)} &= \phi_0 \, d_{\rm eff} \left(\cos(\omega t) -\cos(\omega t-\delta)\right) \\
                & \approx \phi_0 d_{\rm eff} \sin(\omega t) \delta \\
                \delta &= \omega\Delta t - k_z D \\
                ~ \\
                \delta &\approx \frac{\omega L n}{c}
            \end{align*}
            \]
            </div>

# Space-time separated sensors: initial results

* Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [arXiv:2312.13723](https://arxiv.org/abs/2312.13723)
          <br>

<div style="text-align: center; float: left; width: 50%">
            <br>
            <img src="img/dme-amplitude-b.png" width="90%"><br>
            <font size="4"><a href="http://arxiv.org/abs/2312.13723">[arXiv:2312.13723]</a></font>
          </div>

* Search for oscillations in data<br><br>
            * Method exactly the same as previous local comparison search<br><br>
            * Analysis done by Melina Filzinger (PTB)<br><br>
            * Good for oscillations up to $\sim$ 1 Hz<br><br>
            * Lasers loose accuracy for longer time-scales<br>

</div>

# Space-time separated sensors: constraints on $d_{m_e}$

* Filzinger, Caddell, Jani, Steinel, Giani, Huntemann, BMR, [arXiv:2312.13723](https://arxiv.org/abs/2312.13723)

<br>

<div style="text-align: center; float: left; width: 50%">
            <img src="img/dme-limits-b.png" width="90%"><br>
            <font size="4"><a href="http://arxiv.org/abs/2312.13723">[arXiv:2312.13723]</a></font>
          </div>

* First low-mass constraints on $d_{m_e}$ alone
            * All others are |$d_{m_e} - d_g$| ($m_e/m_p$)

* Unique $L$ scaling

* Unique $m_\phi$ scaling

### Outlook

* Hope to gain significant improvement using optical clock network

* Linear distance scaling: future space-based clocks

* Linear distance scaling: method for positive signal discrimination
          </div>