Detecting Chameleons through Casimir Force Measurements

Abstract

The best laboratory constraints on strongly coupled chameleon fields come not from tests of gravity per se but from precision measurements of the Casimir force. The chameleonic force between two nearby bodies is more akin to a Casimir-like force than a gravitational one: The chameleon force behaves as an inverse power of the distance of separation between the surfaces of two bodies, just as the Casimir force does. Additionally, experimental tests of gravity often employ a thin metallic sheet to shield electrostatic forces, however this sheet mask any detectable signal due to the presence of a strongly coupled chameleon field. As a result of this shielding, experiments that are designed to specifically test the behaviour of gravity are often unable to place any constraint on chameleon fields with a strong coupling to matter. Casimir force measurements do not employ a physical electrostatic shield and as such are able to put tighter constraints on the properties of chameleons fields with a strong matter coupling than tests of gravity. Motivated by this, we perform a full investigation on the possibility of testing chameleon model with both present and future Casimir experiments. We find that present days measurements are not able to detect the chameleon. However, future experiments have a strong possibility of detecting or rule out a whole class of chameleon models.

Figures (15)

• Figure 1: The dependence of the chameleonic pressure, $F_{\phi}/A$, between two parallel plates on separation, $d$. We have taken $V(\phi)=\Lambda_{0}^4(1+\Lambda^{n}/\phi^n)$ and fixed $m_c/m_b= 10^{6}$. The three plots show the behaviour of $F_{\phi}/A$ for a theory with $n = 1$, $n = 4$ and $n = -8$. Each of these are respectively representative of theories with $0 < n \leq 2$, $n > 2$ and $n \leq -4$. Three types of behaviour are clearly visible in these plots. For $d \lesssim m_{c}^{-1}$, $F_{\phi}/A \approx V_{c}-V_{b}$ which is independent of $d$: this is the 'constant force behaviour'. For $m_{c}^{-1} \ll d \ll m_{b}^{-1}$, $F_{\phi}/A \propto 1/d^{p}$ for some $p$. Theories with $0 < n \leq 2$ have $0 < p \leq 1$. If $n > 2$ then $1 < p < 2$ and if $n \leq -4$ we have $2 < p \leq -4$. This is the 'power-law behaviour'. Finally when $d \ll m_{b}^{-1}$, $F_{\phi}/A \propto \exp(-m_{b}d)$, i.e. we have 'exponential behaviour'. Note that in a standard Yukawa scalar field theory (where $m_{\phi} = {\rm const}$) one would have $F_{\phi}/A \approx {\rm const}$ for $d \ll m_{\phi}^{-1}$ and an exponential drop-off for $d \gtrsim m_{\phi}^{-1}$; however there would be no region of power-law behaviour.
• Figure 2: The solid lines in Figure (a) show the predicted chameleonic pressure between two parallel plates for $V = \Lambda^4_0\left(1 + \Lambda^{n}/\phi^n\right)$, $n>0$ and $\Lambda = \Lambda_0 = 2.4 \times 10^{-3} \,\mathrm{eV}$. The dotted lines show the current experimental constraints on any such pressure. Sparnaay, Padova, Indiana03 and Indiana07 refer to Refs. sparnaay, Bressi, Decca1 and Decca3 respectively. The predictions shown in Figure (a) only apply when the test masses have thin-shells and for $m_c^{-1}\ll d \ll m_b^{-1}$; $m_c$ is the chameleon mass inside the test masses and $m_b$ is the chameleon mass in the background. The white region in Figure (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in Figure (a) are applicable to the most recent experiment conducted by Decca et al., labeled Indiana07 in Figure (a).
• Figure 3: The solid lines in Figure (a) show the predicted chameleonic pressure between two parallel plates for $V = \Lambda^4 + \Lambda^{4+n}/\phi^n$, $n\leq -4$ and $\Lambda = \Lambda_0 = 2.4 \times 10^{-3} \,\mathrm{eV}$. The dotted lines show the current experimental constraints on any such pressure. Sparnaay, Padova, Indiana03 and Indiana07 refer to Refs. sparnaay, Bressi, Decca1 and Decca3 respectively. The predictions shown in Figure (a) only apply when the test masses have thin-shells and for $m_c^{-1}\ll d \ll m_b^{-1}$; $m_c$ is the chameleon mass inside the test masses and $m_b$ is the chameleon mass in the background. The white region in Figure (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in Figure (a) are applicable to the most recent experiment conducted by Decca et al., labeled Indiana07 in Figure (a).
• Figure 4: The solid lines show the predicted chameleonic pressure between two parallel plates for $V = \Lambda^4_0 \exp \Lambda^n/\phi^n$ and $\Lambda = \Lambda_0 = 2.4 \times 10^{-3} \,\mathrm{eV}$. The dotted lines show the current experimental constraints on any such pressure. Sparnaay, Padova, Indiana03 and Indiana07 refer to Refs. sparnaay, Bressi, Decca1 and Decca3 respectively. The predictions shown above only apply when the test masses have thin-shells, $m_c^{-1}\ll d \ll m_b^{-1}$; $m_c$ is the chameleon mass inside the test masses and $m_b$ is the chameleon mass in the background.
• Figure 5: The solid lines in Figure (a) shows $(F_{\phi}^{\rm tot}(d) -F_{\phi}^{\rm tot}(d_{cal}))/R$ where $F_{\phi}^{\rm tot}$ is the predicted chameleonic force between a sphere and a plate, for $V = \Lambda^4_0(1 + \Lambda^{n}/\phi^n)$, $n > 0$ and $\Lambda = \Lambda_0 = 2.4 \times 10^{-3} \,\mathrm{eV}$. $R$ is the radius of the sphere and we have taken $d_{cal} = 10\,\mu\mathrm{m}$. The dotted line shows the current best experimental constraint on any such force pressure, which comes from Ref. lam97. The predictions shown in Figure (a) only apply when the test masses have thin-shells, $m_c d \gg 1$ and $m_b d_{cal}\ll 1$; $m_c$ is the chameleon mass inside the test masses and $m_b$ is the chameleon mass in the background. The white region in Figure (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in Figure (a) are applicable to the 1997 Casimir force measurement performed by Lamoreaux lam97.