**Detecting mirror matter on Earth**

**via its thermal imprint on ordinary matter**

1.2cm

R. Foot$^{a}$ and S. Mitra$^{b}$[^1]

0.7cm

*$^{a}$ School of Physics,*

*University of Melbourne,*

*Victoria 3010 Australia*

0.5cm

*$^{b}$ Instituut voor Theoretische Fysica,*

*Unversiteit van Amsterdam, 1018 XE Amsterdam,*

*The Netherlands*

Mirror matter type dark matter can exist on the Earth’s surface, potentially in enhanced concentrations at various anomalous impact sites. Mirror matter fragments can draw in heat from the ordinary matter environment, radiate mirror photons and thereby cool the surrounding ordinary matter. We quantify this effect and suggest that it could be used to help locate mirror matter deposits. This method, together with the centrifuge technique, seems to provide the most promising means to experimentally detect mirror matter type dark matter in the Earth.

One of the most fascinating candidates for the non-baryonic dark matter in the Universe is mirror matter[@ly; @flv; @blin]. From a particle physics point of view, mirror matter is a natural candidate for dark matter because:

-   It is theoretically well motivated since it is predicted to exist if parity and time reversal symmetries are exact, unbroken symmetries of nature.

-   It is necessarily dark and stable: Mirror baryons have the same lifetime as ordinary baryons and couple to mirror photons instead of ordinary photons.

-   Recent observations[@wmap] suggest that the cosmic abundance of non-baryonic dark matter is of the same order of magnitude as ordinary matter $\Omega_{\text{b}} \sim \Omega_{\text{dark}}$. A result which can naturally occur if dark matter is identified with mirror matter [@dark].

It is important to realize that mirror matter fragments can interact with ordinary matter non-gravitationally via the photon-mirror photon kinetic mixing interaction [@flv; @km]: $$\begin{aligned}
\mathcal{L} = \frac{\epsilon}{2}
F^{\mu \nu} F'_{\mu \nu},\end{aligned}$$ where $F^{\mu \nu}$ ($F'_{\mu \nu}$) is the field strength tensor for electromagnetism (mirror electromagnetism). One effect of the above interaction is that it gives the mirror charged particles a small electric charge, $\epsilon e$ [@km; @flv].

If there is some amount of mirror matter in the solar system then the photon-mirror photon kinetic mixing interaction will cause unique effects when mirror matter space-bodies collide with the planets/asteroids [@footyoon1; @footyoon2; @eros] and also when ordinary spacecraft propagate out through the solar system [@fvpioneer]. Motivation to study such effects comes from observations of anomalous (Earth) impact events on both small and large scales [@small; @tunguska; @haines], as well as the observed anomalous slow-down of [*both*]{} Pioneer spacecraft [@study]. Also, the remarkable observations of the NEAR Shoemaker space-probe [@near1] also seem to support the mirror matter hypothesis.

It turns out that[@eros] these observations can be explained if the photon-mirror photon kinetic mixing is in the range: $$\begin{aligned}
10^{-9} \lesssim {\left | \epsilon\right |} \lesssim 10^{-6}.
\label{zzz}\end{aligned}$$ Interestingly, if $\epsilon$ is in the above range, mirror matter should exist on the Earth’s surface since the photon-mirror photon kinetic mixing interaction is then strong enough to oppose the force of gravity[@centrifuge]. This means that mirror matter can potentially be experimentally discovered in the ground, at various anomalous impact sites.

One way to detect the existence of mirror matter in a sample of earth is to use a centrifuge[@centrifuge]. Centrifuging a sample containing mirror matter will expel mirror matter fragments thereby leading to a potentially observable weight loss of the sample. Another complementary way - the subject of the present paper – is to search for its thermal imprint. The idea is that a piece of mirror matter in the ground would draw in heat from the surrounding ordinary matter and radiate it away as mirror photons, thereby cooling the surrounding ordinary matter.

This idea was mentioned earlier in Ref.[@footyoon2] where the temperature imprint on the surrounding ordinary matter was evaluated in terms of the surface temperature of the mirror body. Thus, an important issue is to calculate the temperature of a mirror matter body embedded in ordinary matter. As we will show in this paper, for $\epsilon$ in the entire range of interest, Eq.(\[zzz\]), and assuming Earth-like temperatures, the interaction between ordinary and mirror matter is strong enough for thermal equilibration to occur. This result is quite important, since it means that any significant mirror matter deposit could be detected by its thermal imprint on the surrounding ordinary matter.

Consider a mirror matter fragment at temperature $T'$ embedded within (solid) ordinary matter at temperature $T>T'$. \[Mirror matter would necessarily be embedded [*within*]{} ordinary matter if $\epsilon < 0$ since the electrostatic force between ordinary and mirror matter is attractive [@footyoon2]\]. The mirror atoms in the mirror matter fragment undergo frequent collisions with ordinary atoms and, on average, gain energy. If the temperature is not too low[^2], classical statistical mechanics can be applied to calculate the heat transfer.

The heat transfer per unit volume, $dQ/dV$, can be expressed as the product (appropriately averaged over the Maxwellian velocity distributions) of the collision rate $\Gamma$, number density $n'$ and energy transfer per collision $\Delta E_{\text{kin}}$: $$\begin{aligned}
\frac{dQ}{dV} &=& \langle \Gamma n' \Delta E_{\text{kin}} \rangle \nonumber \\
&=& \langle \sigma v_{\text{rel}} n n' \Delta E_{\text{kin}} \rangle .
\label{mon1}\end{aligned}$$ In Ref.[@centrifuge] it was shown that for typical thermal velocities at room temperature the cross section is approximately isotropic and given by: $$\begin{aligned}
\sigma &\sim & 10^{-2} \epsilon^2 \text{ cm}^2 \text{ for } |\epsilon |
\lesssim 10^{-8} \nonumber \\
\sigma &\sim & 10^{-18}\text{ cm}^2 \text{ for } |\epsilon | \gtrsim
10^{-8}. \label{sigma}\end{aligned}$$ Since the collisions are approximately isotropic, the energy transfer per collision is roughly: $$\begin{aligned}
\langle \Delta E_{\text{kin}} \rangle \sim \frac{m}{2} (v^2 - v'^2)
\sim \frac{3}{2} k(T-T'),\end{aligned}$$ where we assume for simplicity that the ordinary and mirror material is composed of atoms of mass $m$. Using the above, Eq.(\[mon1\]) can be solved (a more precise computation is given in the appendix) leading to: $$\label{result}
\frac{dQ}{dV}=
4\pi^{-1/2}m^{-1/2}k^{3/2}\sigma
n'n{\left(\frac{T'+T}{2}\right)}^{1/2}{\left(T-T'\right)} .$$ This is the rate which heat is transferred from ordinary matter to mirror matter. We now consider the effect of this on a fragment of mirror matter embedded in ordinary matter.

Consider a spherical mirror matter ‘rock’, of radius $R'$ completely embedded within ordinary matter. If its surface temperature is $T'_{R'}$, then it will emit mirror photons, with energy loss $\mu \sigma {T'}_{R'}^4 4\pi R'^2$ (where $\mu \le 1$ is the emissivity of the surface). Meanwhile, energy is absorbed from ordinary matter. The energy absorbed depends on the rate of energy transfer, $\frac{dQ}{dV}$, so that $$\begin{aligned}
\int^{R'}_{0}\frac{dQ}{dV} 4\pi r^2 dr = \mu \sigma {T'}_{R'}^4 4\pi
R'^2 .
\label{1}\end{aligned}$$ Putting the numbers in (taking $m \sim 20 m_{\text{proton}}$, $n \sim n' \sim 10^{23}/\text{cm}^3$) and defining $u \equiv (T-T')/T$, we find: $$\begin{aligned}
\frac{dQ}{dV} \approx  \Lambda 
{\left(\frac{T}{300 K}\right)}^{3/2} {\left(1-\frac{u}{2}\right)}^{1/2}u \ 3\times
10^{12}\ \text{Joules/}(\text{cm}^{3}\text{.s})\end{aligned}$$ where $\Lambda = {\left( \frac{\epsilon}{10^{-8}} \right)}^2
$ for $|\epsilon | \lesssim 10^{-8}$ and $\Lambda = 1$ for $|\epsilon | \gtrsim 10^{-8}$. Thus, Eq.(\[1\]) becomes (after dividing both sides of the equation by $R'^2$): $$\begin{aligned}
\Lambda
\left( \frac{R'}{\text{cm}}\right)
{\left\langle u{\left(1-\frac{u}{2}\right)}^{1/2}\right\rangle}
\left( \frac{T}{300 K}\right)^{3/2}
10^{13}
= 0.6 \mu \left( \frac{{T'}_{R'}}{300 K}\right)^4 , 
\label{3}\end{aligned}$$ where we have defined the average of function $f(r)$ inside the region $r<R'$ as $$\begin{aligned}
{\left\langle f\right\rangle} \equiv 
\frac{3}{R'^3}\int^{R'}_0 f(r) r^2 dr .\end{aligned}$$ Clearly, Eq.(\[3\]) shows that for an Earth-like environment ($T \sim
300 K$), $|\epsilon | \gtrsim 10^{-12}$ \[which includes the entire range of most interest, Eq.(\[zzz\])\] and for all $R'$ of interest (including microparticle sizes, $R' \sim
10^{-4}\text{ cm}$), ${\left\langle u\right\rangle}\ll 1$. This implies that $$\begin{aligned}
{\left\langle T'\right\rangle} \simeq T\end{aligned}$$ which means that the mirror body thermally equilibrates with its ordinary matter surroundings.

If the mirror thermal conductivity ($\kappa'$) were very poor, one could still have ${T'}_{R'} \ll T$ (which is technically the case for $\kappa' \to 0$). However, for an Earth-like environment, this effect is negligible. To see this, consider a spherical shell $r_1 < r < R'$. In a steady-state situation, the energy going into this shell must equal the energy going out, which translates into the following equation: $$\begin{aligned}
\int^{R'}_{r_1} \frac{dQ}{dV} 4\pi r^2 dr - 
\kappa' \frac{\partial T'}{\partial r} 
4\pi r^2_1 = \mu \sigma {T'}_{R'}^4 4\pi R'^2 .\end{aligned}$$ \[Of course this equation reduces to the earlier one, Eq.(\[1\]), for $r_1 = 0$\]. For $r_1$ sufficiently close to $R'$, the thermal conductivity term (i.e. second term on the left-hand side of the above equation) dominates over the first term. In this case, we can evaluate the mirror temperature gradient near the surface of the mirror body: $$\begin{aligned}
{\left.\frac{\partial T'}{\partial r}\right |}_{r=R'} \approx  
-3 {\left( \frac{{T'}_{R'}}{300 K}\right)}^4 
{\left( \frac{0.016 J/(K.s.cm)}{\kappa'}\right)}
K/\text{cm}\end{aligned}$$ where, for convenience, we have compared $\kappa'$ to the average value of $\kappa$ for the Earth’s crust ($\langle \kappa \rangle \approx  0.016 \text{ J/K.s.cm}$). This is a very small gradient (for ${T'}_{R'} \lesssim 300 K$) which means that the mirror thermal conduction term is negligible in comparison to the ordinary-mirror matter energy transfer term.

The conclusion is, that for Earth-like temperatures ($T
\lesssim 300 K$), a mirror matter body would equilibrate in temperature with the ordinary matter surroundings and emit mirror radiation from its surface, thereby cooling the ordinary matter.

The amount of radiation emitted by a mirror body depends on its size as well as its surface temperature. Since the body equilibrates with the ordinary matter surroundings, we can, to a good approximation, define a common temperature for the body, $T' \simeq T$. The amount of energy emitted by the body is then [@dust]: $$\begin{aligned}
E_{\text{emission}} &=& \int^{\infty}_0 4\pi^2 {R'}^2 Q_{R'} (R',\lambda) 
B_{\lambda} (T)
d\lambda
\nonumber \\
&=& \mu (T,R') \sigma T^4 4\pi R'^2\end{aligned}$$ where $$B_{\lambda}(T)=
\frac{2hc^{2}}{\lambda^{5}}\frac{1}{\exp{\left(hc/(\lambda k T)\right)}-1} .$$ For a perfect black-body emitter, $Q_{R'} (R',\lambda) = 1
\Rightarrow \mu = 1$. However for a finite sized solid body, we have: $$\begin{aligned}
Q_{R'} (R',\lambda) &\approx & 1, \text{ for } \lambda \ll R' \nonumber \\
Q_{R'} (R',\lambda) &\sim & R'/\lambda \text{ for } \lambda \gg R'\end{aligned}$$ Thus, the net effect is: $$\begin{aligned}
\mu (T,R') &\approx & 1 \text{ for } \ R' \gtrsim 
\text{few} \times 10^{-3}
\left( \frac{300 K}{T}\right) \text{ cm}
\nonumber \\
\mu (T, R') &\to & 0 \text{ for } \ R' \ll 10^{-3} \left(\frac{300 K}{
T}\right)\text{ cm}\end{aligned}$$

Summarizing things, we have shown that a mirror matter body embedded within ordinary matter (expected if $\epsilon < 0$) equilibrates and efficiently emits mirror radiation provided that the body is greater than about $10^{-3}$ cm in size. Of course, we cannot detect the mirror radiation directly, however we can infer the existence of an energy loss by looking for its thermal effect on the surrounding ordinary matter. The point is that as the mirror body radiates its heat away into mirror photons, the energy is replaced from the ordinary matter surroundings which will become cooler as a result.

To examine the size of the effect, consider a spherical body of radius $R'$ embedded within a much larger volume of ordinary matter. In this case, the energy lost per unit time to mirror radiation is approximately, $$\begin{aligned}
Q_{\text{rad}} = \sigma T^4 4\pi R'^2 \ \text{ if } R' \gtrsim 10^{-3}\text{ cm}\end{aligned}$$ As energy escapes, a temperature gradient in the surrounding ordinary matter is created, as heat is thermally conducted replacing the energy lost into mirror radiation. Thus, considering a shell of radius R surrounding the body ($R > R'$), then the heat crossing this surface per unit time is: $$\begin{aligned}
Q_c = \kappa \frac{\partial T}{\partial R} 4\pi R^2 .\end{aligned}$$ In the steady state situation, $Q_c = Q_{\text{rad}}$, which implies the temperature gradient, $$\begin{aligned}
\frac{\partial T}{\partial R} = \frac{\sigma T^4 R'^2}{\kappa R^2} .\end{aligned}$$ This means that the change in temperature of the ordinary matter surrounding a spherical mirror body at a distance $R$, $\delta T(R)$, is given by: $$\begin{aligned}
\delta T(R) &=& \frac{-\sigma T^4 R'^2}{\kappa R}
\nonumber \\
& \simeq & -\left( \frac{T}{270 K}\right)^4 \left( \frac{R'}{5
\text{ cm}}\right)^2 \left( \frac{50\text{ cm}}{R}\right)\left( \frac{0.016
\ \text{J/(K.s.cm)}}{\kappa}\right)\ K
\nonumber
\\
.\end{aligned}$$ This is clearly a significant thermal imprint which should be detectable if large ($R' \gtrsim 5\text{ cm}$) fragments (or numerous small fragments) exist close to the Earth’s surface. In other words, one could use a thermal imaging detector (or even an old thermometer!) as a mirror matter detector. But where should one look?

The chances of discovering a significant mirror matter deposit in a random piece of earth would be expected to be quite low. However, any mirror matter deposit near the Earth’s surface should be associated with the impact of a mirror matter space-body. Since the impact of a mirror matter body with the Earth has important observational differences compared with an ordinary body impact it should be possible to identify candidate sites if they exist. Remarkably, there is significant evidence for such anomalous impact sites (see Ref.[@footyoon1; @footyoon2; @eros] for a more detailed discussion):

-   On small scales ($D_{\text{SB}} \sim 5$ meters), there are relatively frequent events, some of which have been quite well studied[@small]. At least one event of this type, the 2001 Jordan event, is particularly interesting because the ‘body’ was observed to hit the ground. If caused by the impact of a mirror matter body, mirror matter fragments should exist right on the ground and in the top surface layers ($\sim$ cm depth). One would expect a sizable thermal imprint at this impact site if caused by the impact of a mirror matter body.

-   On large scales ($D_{\text{SB}} \sim 100$ meters), we have the famous 1908 Tunguska event[@tunguska]. In this case the body exploded in the atmosphere at an altitude of about 8 km. If caused by a mirror matter space-body, one would expect fragments on or near the surface, potentially detectable by their thermal imprint.

-   On even larger scales ($D_{\text{SB}} \sim 500$ meters), events should be quite infrequent occurring on time scales much greater than $10^3$ years. Yet, there is interesting evidence that such anomalous impacts occur. One such event is the Edeowie glass field in South Australia[@haines]. This event is characterized by a field of melted rocks (some of which are actually melted in situ) on a flat desert plain over an area of about $10^2 \text{ km}^2$. This event could be caused by a large mirror body hitting the ground at cosmic velocity $v \sim 30$ km/s (which is quite unlike the small Jordan event and Tunguska event). In this case a large quantity of mirror matter should exist, but could be buried some distance below the surface because of the relatively large impact velocity (the stopping distance can be obtained from Refs.[@footyoon1; @eros] and is about $L \sim 10 (10^{-8}/\epsilon)^2 (v/30 \text{km/s})^4$ meters)[^3].

The above anomalous impact sites should be a good place to look for mirror matter via its thermal imprint on the surrounding ordinary matter. Of course, once located it can be further tested via the centrifuge technique[@centrifuge].

In conclusion, we have shown that if the photon-mirror photon kinetic mixing parameter $\epsilon$ is in the range of interest, Eq.(\[zzz\]), mirror matter grains embedded within ordinary matter will thermally equilibrate with the ordinary matter environment. Consequently, a mirror matter grain will cool the surrounding ordinary matter by drawing in heat and radiating it away in the form of mirror photons. We have shown that this cooling effect is significant enough to be detected for fragments larger than about a centimeter in size (or numerous smaller fragments). Thus, a mirror matter deposit could be located on Earth by its thermal imprint on the ordinary matter surroundings. This method, together with the centrifuge technique seems to provide the most effective means to experimentally detect mirror matter on Earth.

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[**Appendix**]{}

Calculation of Heat transfer
============================

If a mirror atom collides with an ordinary atom it can gain or lose kinetic energy, depending on the initial velocities and the direction in which it scatters. It is convenient to express the gain in kinetic energy of the mirror atom in terms of the initial velocities and scattering angles as follows. Denote the initial velocity of the mirror atom (ordinary atom) as $v'$ ($v$), and the final velocity of the mirror atom (ordinary atom) as $\tilde{v}'$ ($\tilde{v}$). Define x, y, z coordinates in velocity space as follows. Define the z-direction to be in the direction of $v'-v$. The x-direction is chosen to be in the direction of the projection of $v$ on the plane orthogonal to the z-direction. Finally, the y-direction is defined as the outer product of the z and x-directions. Now define the spherical coordinates $\theta$ and $\phi$ in the usual way. For any vector we define $\theta$ as the angle with the z-direction and $\phi$ as the angle the vector projected on the x-y plane makes with the x-direction. It follows from conservation of energy and momentum that if the final velocity of the mirror atom relative to the ordinary atom, $\tilde{v}'-\tilde{v}$, is in the direction ${\left(\theta,\phi\right)}$ then the gain in kinetic energy is $$\Delta E_{\text{kin}}{\left(v',v,\theta,\phi\right)}=\frac{m}{4}{\left(1-\cos{\theta}\right)}
{\left(v^{2}-v'^{2}\right)}+ mv'v\sin{\left(\alpha\right)}\sin{\left(\theta\right)}\cos{{\left(\phi\right)}}$$ where $\alpha$ is the angle between the initial velocities. To obtain the rate of heat transfer, we have to multiply this with the collision rate involving mirror and ordinary atoms with the initial velocities $v'$ and $v$ and integrate over all initial and final velocities. The velocity distribution of mirror atoms and ordinary atoms is given by the Maxwell distribution. We define $n'{\left(v\right)}$ and $n{\left(v\right)}$ as giving the number of mirror atoms respectively ordinary atoms per unit volume and per unit volume in velocity space: $$\begin{aligned}
n{\left(v\right)}&=&n{\left(\frac{m}{2\pi k T}\right)}^{\frac{3}{2}}\exp{\left(-\frac{m v^{2}}{2 k T}\right)}\\
n'{\left(v\right)}&=&n'{\left(\frac{m}{2\pi k T'}\right)}^{\frac{3}{2}}\exp{\left(-\frac{m v^{2}}{2 k T'}\right)}\end{aligned}$$ Here $n$ ($n'$) is the number density of the (mirror) atoms. In terms of the above defined functions the amount of heat absorbed by the mirror fragment per unit volume and per unit time, $\frac{dQ}{dV}$, can be expressed as: $$\label{qint}
\frac{dQ}{dV}=\int d^3 v'd^{3}v\sin{\left(\theta\right)}d\theta d\phi \Delta E_{\text{kin}}{\left(v',v,\theta,\phi\right)}
\frac{d\sigma}{d\Omega}{\left | v'-v\right |} n'{\left(v'\right)}n{\left(v\right)}$$ where $d\sigma/d\Omega$ is the elastic differential cross section. In [@centrifuge] it is shown that for typical thermal velocities at room temperature $d\sigma/d\Omega$ becomes approximately isotropic. We can therefore put $d\sigma/d\Omega=\sigma/(4\pi)$ in Eq.. The integrals in Eq. can be evaluated by substituting $v\rightarrow v'+u$, and performing the $v'$ integration first. The result is: $$\label{rslt}
\frac{dQ}{dV}=
4\pi^{-1/2}m^{-1/2}k^{3/2}\sigma n'n{\left(\frac{T'+T}{2}\right)}^{1/2}{\left(T-T'\right)} .$$

1.3cm [**Acknowledgements**]{} 0.5cm One of us (R.F.) would like to thank P. W. Haines for discussions regarding the Edeowie glass field[@haines] and T. L. Yoon for some comments on the paper. 0.8cm

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[^1]: E-mail address: foot@physics.unimelb.edu.au, saibalm@science.uva.nl

[^2]: For temperatures much lower than the Debye temperature, less than room temperature for most solids, the high frequency vibrational modes of the solid become frozen into the ground state, and classical statistical mechanics breaks down.

[^3]: For a large mirror matter deposit (which might be the case with the Edeowie glass field), another useful way to infer its existence would be to use gravimeters to seach for local variations in the Earth’s gravity.
