Atomic Clocks

# Precision timing: fundamental physics

<div style="text-align: left; float: left; width: 70%">
            <br>
            <div class="fragment" data-fragment-index="0">
              <h3> 1657: Huygens designs pendulum clock</h3>
              <h4> • Accurate to $\sim15$ s / day</h4>
              <h4> • Works on solid physics principal: $T\propto\sqrt{L/g}$</h4>
            </div>

<br>

<div class="fragment">
              <h3> 1672: Jean Richer expedition to French Guiana </h3>
              <h4> • Observe Mars near equator -- measure scale of solar system</h4>
              <h4> • Failed: pendulum clocks ran slow by 2 min / day</h4>
            </div>

<br>

<div class="fragment">
              <h3> 1687: Newton's Principia (15 years post Richer)</h3>
              <h4> • Explained Richer's measurement: law of gravitation</h4>
            </div>

<br>

<div class="fragment">
              <h3> 1676: Rømer calculates speed of light</h3>
              <h4> • Observe eclipse of Jupiter's moons</h4>
              <h4> • Required accuracy of $\sim30$ s / day</h4>
            </div>

<br>

<div class="fragment">
              <h3> Modern day</h3>
              <h4> • Allows GPS</h4>
              <h4> • Underpins modern geodesy</h4>
            </div>

</div>

<div class="fragment " data-fragment-index="0" style="text-align: center; float: right; width: 30%">
            <br>
            <img src="img/HuygensPendulumBW.png" width="85%"><br>
            <font size="3">[Huygens 1673]</font>
          </div>

# Clock: a thing that ticks

### Predictable, period motation + method to count ticks

<br>
          <div>
            <div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
              <img src="img/StoneHenge.png" width="86%"><br>
              <font size="3">[Getty]</font>
              <br>
              <h3>
                Earth's orbit/rotation
              </h3>
            </div>

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
              <img src="img/H1.png" width="96%"><br>
              <font size="3">[H1, Royal Museums Grenwhich]</font>
              <br>
              <h3>
                Swinging pendulum
              </h3>
            </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 32%">
              <img src="img/laser-JILA.jpg" width="100%">
              <font size="3"> [JILA]</font>
              <br>
              <h3>
                Oscillating electromagnetic wave
              </h3>
            </div>
          </div>

<div class="fragment fade-in" style="text-align: left; float: left; width: 100%">
            <br>
            <br>
            <br>
            <h3>
              • Pendulums (+all kinematic clocks): depend on materials, location, temperature etc.
              <br>
              • Earth rotation: unstable at ~5 ms per day $10^{-7}$
              <br>
              • Atomic clock: universal timekeeping based on fundamental physics
            </h3>
          </div>

## Lord Kelvin in 1879 (attributed to Maxwell):

<div style="text-align: left; float: left; width: 70%">
            <div style="text-align: center; float: center; width: 100%">
              <font size="18">
                The recent discoveries ...
                
                indicate to us natural standard <br>pieces of matter such as <b>atoms of hydrogen or sodium,<br>
                  ready made in infinite numbers, <br>all absolutely alike in every physical property</b>.<br><br>
              </font>
            </div>

<div class="fragment fade-in" data-fragment-index="1"
              style="text-align: center; float: center; width: 100%">
              <font size="18">
                The time of vibration of a sodium particle corresponding to <br>any one of its modes of
                vibration is <b>known to be absolutely independent of its position in the Universe</b>...
              </font>
            </div>

<div class="fragment fade-in" data-fragment-index="1" style="text-align: center; float: center; width: 90%">
              <font size="4">
                Thomson, W., and P. G. Tait, 1879, Elements of Natural Philosophy (Cambridge University Press,
                Cambridge,
                England)</font>
            </div>

</div>
          <div style="text-align: left; float: right; width: 30%">
            <img src="img/Maxwell.png" width="100%"><br>
            <font size="3"> [R. H. Campbell | Institution of Electrical Engineers]</font>
          </div>

# Atomic Clocks 101:

<br>

### Frequency reference: atomic transition

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
            <img src="img/Bohr.png" width="70%"><br>
            <font size="3"> JabberWok [Wikimedia Commons]</font>
            <br><br><br>
            <h3>
              Atomic transition: "perfect*" frequency reference
            </h3>
          </div>

<div class="fragment fade-in" style="text-align: center; float: left; width: 32%">
            <img src="img/Wcislo2016.png" width="95%">
            <font size="3">[Wcislo, Nature Astronomy (2016)]</font>
            <br><br><br>
            <h3>
              Compare frequency of oscillator to atomic transition
            </h3>
          </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 32%">
            <img src="img/Al+Choi2010.png" width="95%">
            <font size="3">[Chou et al., Science 329, 1630 (2010)]</font>
            <br><br><br>
            <h3>
              Lock on: Adjust local oscillator to maximum transition rate
            </h3>
          </div>

# Optical atomic clocks

<div class="fragment fade-in" style="text-align: left; float: left; width: 55%">
            <img src="img/Stability.png" width="95%"><br>
            <font size="3"> [M. Safronova]</font>
            <br>
            <br>
            <br>

* Requires ultra-stable laser ("see" narrow line)
            * Frequency comb ("listen" to high frequency)

</div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 45%">
            <img src="img/basicclock2.png" width="95%">
            <font size="3">[Centre for Quantum Technologies | U. Singapore]</font>
            <br>
            <br>
            <br>
            <img src="img/Diddams-Comb.png" width="95%">
            <font size="3">[Diddams <i>et al.</i>, Science <b>369</b>, 3676 (2020)]</font>
          </div>

* Accurate to parts in $10^{18}$
            * better than 1 s over lifetime of the universe!
          </div>

# Example: gravitational redshift

<br>

$$ \Huge {\color{orange}
          \frac{\Delta f}{f} = -\frac{\Delta U}{c^2} \sim 10^{-16} \frac{\Delta r}{{\rm m}}
          }
          $$

<br>

<div class="fragment fade-in" style="text-align: center; float: left; width: 60%">
            <img src="img/Chou-A.png" width="50%">
            <img src="img/Chou-B.png" width="45%"><br>
            <font size="4">30 cm [Chou et al., Science <b>329</b>, 1630 (2010)]</font><br><br>

* $\sim$ 30 cm
          </div>

<div class="fragment fade-in" style="text-align: center; float: right; width: 40%">
            <img src="img/Ye-Redshift.png" width="85%">
            <br>
            <font size="4">[Bothwell <i>et al.</i>, Nature <b>602</b>, 420 (2022)]
            </font><br><br>

* $\sim$ 1 mm
          </div>

* Gravitational red-shift is one of the biggest systematic which must accounted for in clock comparisons --
            even within the same lab!
          </div>

# Unexplained mysteries

<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>What is Dark Energy?</h2>
              <h3> • Cosmologists: Why is Λ so large?
                • Particle theorists: Why is Λ so small?
              </h3>

</div>

<h2>What is Dark Matter?</h2>
              <h3> • Missing mass from the universe
                • Matter that interacts <em>only</em> gravitationally? </h3>

</div>

<br>
            <div class="fragment" data-fragment-index="3" style="text-align: left;" data-markdown>
              <h3> 95% of universe is "dark"; we can only use the 5% to learn about the rest! </h3>
            </div>

</div>

# Dark Matter: Galactic Rotation Curves

<div style="text-align: center; float: left; width: 70%">
            <br><br>
            <img
              src="img/RotCurve.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>

# Evidence builds up

<br>
          <div style="text-align: left; float: left; width: 50%">
            <div>
              <h2> Rotation curves</h2>
              <h3> • Missing galactic matter</h3>
            </div>
            <br>
            <div class="fragment"  data-fragment-index="1">
              <br>
              <h2> Gravitational lensing</h2>
              <h3> • Gravitational bending of light</h3>
            </div>
            <br>
            <div class="fragment"  data-fragment-index="2">
              <br>
              <h2> Cosmic Microwave Background</h2>
              <h3> • Gravitational imprint on background radiation</h3>
            </div>
            <br>
            <div class="fragment"  data-fragment-index="3">
              <br>
              <h2> Large-Scale Structure Simulations</h2>
              <h3> • Dark Matter required to match observations</h3>
            </div>

<div class="fragment"  data-fragment-index="4">
                •  Ωc = 0.2589(57) [Plank]
              <br>
                $$
                  \rho_{\rm DM} \approx 0.4 ~ {\rm GeV}/{\rm cm}^3
                $$
            </div>

</div>
          <br>
          <div style="text-align: center; float: right; width: 50%">

<div class="r-stack">
              <img class="fragment fade-out" data-fragment-index="1" src="img/RotCurve.png" width="100%">
              <div class="fragment fade-in-then-out" data-fragment-index="1" >
                <img src="https://upload.wikimedia.org/wikipedia/commons/a/a8/1e0657_scale.jpg" width="100%">
                <br><font size="3">[NASA/CXC/M. Weiss - Chandra X-Ray Observatory]</font>
              </div>
              <div class="fragment fade-in-then-out" data-fragment-index="2">
                <img src="img/CMB.png" width="85%">
                <img src="img/CMB-PowerSpectrum-Planck.png" width="75%">
                <br><font size="3">[ESA/Planck Collaboration (2013)]</font>
              </div>
              <div class="fragment fade-in" data-fragment-index="3" >
                <img src="img/Cosmic_web.jpg" width="100%">
                <br><font size="3">[Volker Springel/MPI, Wikimedia Commons]</font>
              </div>
            </div>

</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 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
                 <font size="6">• e.g., Ashlee Caddell, Flambaum, BMR, <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030">Phys. Rev. D <b>108</b>, 083030 (2023)</a></font></h3>
            </div>

<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>

# "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>
              \[\huge
                \frac{\delta f}{f} = \kappa \, \phi_\text{DM}(\vec{r},t)
              \]
            </div>
            <br>
            <div class="fragment fade-in">
              <h4>
                 • $\kappa$ - model dependent parameter
              </h4>
              <h4>
                 • Calculated: high-precision atomic theory
              </h4>
            </div>

<br>
            <div class="fragment fade-in">
              <h3>Example: variation of fundamental constants</h3>
              <br>
              $$
                \large
                \mathcal{L} = \phi F_{\mu\nu}F^{\mu\nu}
                \implies \alpha\to\alpha[1+\phi_\text{DM}(\vec{r},t)]
              $$
              <br>
              $$
                \large
                \kappa = \frac{\partial (f/f_0)}{\partial (\alpha/\alpha_0)}
              $$
            </div>

</div>

# Variation of fundamental constants
          ### Example: fine structure constant: $\alpha$

<div>
            \[\large
            \omega^{A} = \underbrace{F_A(\alpha)}_{\text{Transition-specific}}
            \times \underbrace{m_e c^2\alpha^2}_{\text{Units}}
            \qquad\qquad
            \alpha = \frac{e^2}{4\pi\epsilon_0\,\hbar c}
            \]
          </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)
          </div>

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

# Spatial variation

* Observe spectra from distant stars
            * Compare to measurements on Earth

</div>

### Possible non-zero result

<img src="img/AusDipole2-Murphy12.png" width="90%"><br>
            <font size="3">[M. Murphy 2012]</font>
            <br>
            <br>

* 3 $\sigma$ result: hard to quantify systematics
            * Webb _et al._, PRL **87**, 091301 (2001); **107**, 191101 (2011)

* Over year, earth moves through gradiant

$$
              \delta\alpha/\alpha_0 \sim 10^{-20} / \text{year}
              $$

* Very near-future clocks can probe this

</div>
          </div>

# Oscillating signals

<div style="text-align: left; float: left; width: 50%">
            <br>
            <h3>Case: oscillating field:</h3>
            \[\huge
              \phi_{DM} = \phi_0 \cos({m_\phi t})
            \]
            <br>
            \[\huge
              \frac{\delta f}{f} = \kappa \phi_0 \cos({m_\phi t})
            \]
            <br>
            <br>
            <div class="fragment fade-in">
              <h3>• Constraints from clock-clock comparisons</h3>
              <br>
              <h3>• ~ $10^{-2}$ Hz (optical clocks)</h3>
              <h4>
                  • e.g., Wcislo <i>et al.</i> Science Advances <b>4</b>, 4869 (2018)
              </h4>
              <br>
              <h3>• ~ $10^{-6}-10^{-2}$ Hz (microwave clocks)</h3>
              <h3>• $\implies m = 10^{-21}-10^{-18}$ eV</h3>
              <h4>
                  • e.g., Hees <i>et al.</i> Phys. Rev. Lett. <b>117</b>, 061301 (2016)
              </h4>
            </div>
          </div>

<div style="text-align: left; float: right; width: 50%">
            <br> <br>
            <div class="fragment fade-in">
              <h3>The DAMNED experiment:</h3>
              <img src="img/damned.png" width="90%"><br>
              <p>(DArk Matter search using Non-Equal Delay lines)</p><br>
              <div class="fragment fade-in">
                <h3>• Modifies length + refractive index</h3>
                <h3>• 10-100 kHz range ($m=10^{-10}$ eV)</h3>
                <h3>• Led by Etienne Savalle (SYRTE Paris)</h3>
                Savalle, Hees, Frank, Cantin, Pottie, <b>BMR</b>, Cros,
                McAllister,
                Wolf, <br>
                <a href="https://link.aps.org/doi/10.1103/PhysRevLett.126.051301">
                  Phys. Rev. Lett. <b>126</b>, 051301
                  (2021)</a>
              </div>
            </div>
          </div>

# GPS.DM

<div style="text-align: left; float: left; width: 35%">
            <br>
            
            
            <h3>• 30 Cs, Rb atomic clocks</h3>
            <br>
            <h3>• Over 20 years of high-quality data</h3>
            <br>
            <h3>• Publicly available (JPL)</h3>
            <br>
            <h3>• 50,000 km Dark Matter observatory</h3>
            <br>
            
            <div class="fragment fade-in">
              <video width="75%" data-autoplay muted>
                <source src="img/GPSDM.mp4">
              </video><br>
              <font size="3">[Video: Conner Dailey (honors student)]</font>
            </div>
          </div>

<div style="text-align: center; float: right; width: 65%">
            
            <img div class="fragment fade-in" src="img/gps-data.png" width="45%">
            <br>
            <br>
            <br>
            <div class="fragment fade-in">
              <img src="img/GPSDM-results.png" width="98%">
              <b>BMR</b>, Blewitt, Dailey, Murphy, Pospelov, Rollings,
              Sherman, 
              Williams, Derevianko, <br>
              <a href="http://www.nature.com/articles/s41467-017-01440-4">
                Nature Comms. <b>8</b>, 1195 (2017)</a>

</div>

# Super-massive black hole at the Galactic Centre

<div style="text-align: left; float: left; width: 50%">
            <h3>• with UCLA Galactic Centre Group</h3>
            <h4>   • Observations led by Tuan Do</h4>
            <h4>   • Andrea Ghez: Awarded 2020 Nobel
              prize for discovery of black hole</h4>
            <br>
            <h3>• Keck telescope in Hawaii</h3>
            <br>
            <h3>• Motion of ∼1000 stars tracked</h3>
            <h4>   • Precise spectroscopy for many stars</h4>
            <br>
            <br>
            <div class="fragment fade-in">
              <h3>• High gravitational potential</h3>
              <h3>• Possibly large concentration of dark
                matter </h3>
              <h3>• (also other "exotic" physics)</h3>
            </div>
          </div>

<div style="text-align: center; float: right; width: 50%">
            <img src="img/keck-ao.jpg" width="95%">
            <font size="3">[Keck Observatory | Ethan Tweedie Photography]</font>
            <br>
            <br>
            
            <br>
          </div>

# Conclusion

<div>
          <div style="text-align: left; float: left; width: 50%">
            
            ### Standard Model + General Relativity

* Extremely successful, but incomplete
  
            <br>

### Atomic clocks: extraordinarily precise
  
            * Allows high-precision, low-energy search for new physics
  
              <br>
              <br>
  
              #### CSIRO Space & Astronomy Colloquium, Sydney, June 2024

<br>
    
              ### Benjamin M. Roberts
      
              _School of Mathematics and Physics,_

_University of Queensland, Australia_
    
              <br><br>
    
              * Slides [broberts.io/talks](https://broberts.io/talks)

* Caddell, Flambaum, BMR, [Phys. Rev. D **108**, 083030 (2023)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.083030)
            <br>
            * Savalle _et al._, [Phys. Rev. Lett. **126**, 051301 (2021)](https://link.aps.org/doi/10.1103/PhysRevLett.126.051301)
            <br>
            * BMR _et al._, [Nature Comms. **8**, 1195 (2017)](http://www.nature.com/articles/s41467-017-01440-4)
            <br>
            * BMR _et al._, [N. J. Phys. **22**, 093010 (2020)](https://iopscience.iop.org/article/10.1088/1367-2630/abaace)
            <br>
            * Hees, Do, BMR, Ghez _et al._, [Phys. Rev. Lett. **124** 081101
            (2020)](https://link.aps.org/doi/10.1103/PhysRevLett.124.081101)
            <br>
            * Filzinger _et al._, [arXiv:2312.13723](http://arxiv.org/abs/2312.13723)

<br><br>
            <br><br>
            Supported by: ARC and   <img src="img/BQI.png" width="40%">
          </div>

</div>

# Bullet Cluster

<div class="r-stack">
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                <img src="https://upload.wikimedia.org/wikipedia/commons/e/ea/Bullet_cluster.jpg" width="100%"><br>
                <font size="3">Chandra X-ray Observatory [NASA]</font>
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                <img src="https://upload.wikimedia.org/wikipedia/commons/a/a8/1e0657_scale.jpg" width="90%">
                <font size="3">NASA/CXC/M. Weiss - Chandra X-Ray Observatory</font>
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            <br>
            <h3>• Colliding galaxy clusters ~ 4 Gyr</h3>
            <br>
            <h3>• X-ray emission image:<br> distribution of gas (baryons)</h3>
            <br>
            <br>
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              <h3>• Composite image: X-ray + lensing</h3>
              <br>
              <h3>• Pink = gas (X-ray)</h3>
              <h3>• blue = mass (gravitational lensing)</h3>
              <br>
              <h3 class="fragment fade-in" data-fragment-index="1">
                • Frictionless dark matter: <br>
                passes straight through
              </h3>
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# WIMP miracle

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            <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>
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              • Universe cools: <br>
              Boltzmann suppression
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            <br>
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              • Universe expands: <br>
              Shuts off annihilation
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              <img class="fragment fade-in" data-fragment-index="0" src="img/YDM/Yeq.png" width="90%">
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            <h3 class="fragment fade-in" data-fragment-index="3">
              • Weak interaction ($G_F \simeq 10^{-5}\,\text{
              GeV}^{-2}$): new physics scale<br>
              • $m_\text{weak} \simeq 100\,\text{GeV}$ <br>
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              • $\langle\sigma v\rangle_{\rm weak} \simeq
              3\times10^{-26}\,{\rm
              cm}^3/{\rm s}$ !!!
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