Detecting chameleons through Casimir force measurements

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 masks 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 behavior 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 models with both present and future Casimir experiments. We find that present-day 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.

Figure 1

The dependence of the chameleonic pressure, $F_{\varphi }/A$, between two parallel plates on separation, $d$. We have taken $V(\varphi )={\Lambda }_0^4(1+{\Lambda }^n/{\varphi }^n)$ and fixed $m_c/m_b={10}^6$. The three plots show the behavior of $F_{\varphi }/A$ for a theory with $n=1$, $n=4$, and $n=-8$. Each of these are, respectively, representative of theories with $0<n\le 2$, $n>2$, and $n\le -4$. Three types of behavior are clearly visible in these plots. For $d\lesssim m_c^{-1}$, $F_{\varphi }/A\approx V_c-V_b$ which is independent of $d$: this is the “constant force behavior.“ For $m_c^{-1}\ll d\ll m_b^{-1}$, $F_{\varphi }/A\propto 1/d^p$ for some $p$. Theories with $0<n\le 2$ have $0<p\le 1$. If $n>2$ then $1<p<2$ and if $n\le -4$ we have $2<p\le -4$. This is the “‘power-law behavior.” Finally when $d\ll m_b^{-1}$, $F_{\varphi }/A\propto \exp (-m_{b}d)$, i.e. we have “exponential behavior.“ Note that in a standard Yukawa scalar field theory (where $m_{\varphi }=\mathrm{const}$) one would have $F_{\varphi }/A\approx \mathrm{const}$ for $d\ll m_{\varphi }^{-1}$ and an exponential dropoff for $d\gtrsim m_{\varphi }^{-1}$; however, there would be no region of power-law behavior.

Figure 2

The solid lines in (a) show the predicted chameleonic pressure between two parallel plates for $V={\Lambda }_0^4(1+{\Lambda }^n/{\varphi }^n)$, $n>0$, 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. 11, 12, 13, 15, respectively. The predictions shown in (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 (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in (a) are applicable to the most recent experiment conducted by Decca et al., labeled Indiana07 in (a).

Figure 3

The solid lines in (a) show the predicted chameleonic pressure between two parallel plates for $V={\Lambda }^4+{\Lambda }^{4+n}/{\varphi }^n$, $n\le -4$, 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. 11, 12, 13, 15, respectively. The predictions shown in (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 (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in (a) are applicable to the most recent experiment conducted by Decca et al., labeled Indiana07 in (a).

Figure 4

The solid lines show the predicted chameleonic pressure between two parallel plates for $V={\Lambda }_0^4\exp {\Lambda }^n/{\varphi }^n$ 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. 11, 12, 13, 15, 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 (a) show $(F_{\varphi }^{\mathrm{tot}}(d)-F_{\varphi }^{\mathrm{tot}}(d_{\mathrm{cal}}))/R$ where $F_{\varphi }^{\mathrm{tot}}$ is the predicted chameleonic force between a sphere and a plate, for $V={\Lambda }_0^4(1+{\Lambda }^n/{\varphi }^n)$, $n>0$, and $\Lambda ={\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. $R$ is the radius of the sphere and we have taken $d_{\mathrm{cal}}=10\mu \mathrm{m}$. The dotted line shows the current best experimental constraint on any such force pressure, which comes from Ref. 9. The predictions shown in (a) only apply when the test masses have thin shells, $m_{c}d\gg 1$ and $m_b{d}_{\mathrm{cal}}\ll 1$; $m_c$ is the chameleon mass inside the test masses, and $m_b$ is the chameleon mass in the background. The white region in (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in (a) are applicable to the 1997 Casimir force measurement performed by Lamoreaux [9].

Figure 6

The solid lines in (a) show $(F_{\varphi }^{\mathrm{tot}}(d)-F_{\varphi }^{\mathrm{tot}}(d_{\mathrm{cal}}))/R$ where $F_{\varphi }^{\mathrm{tot}}$ is the predicted chameleonic force between a sphere and a plate, for $V={\Lambda }_0^4(1+{\Lambda }^n/{\varphi }^n)$, $n\le -4$, and $\Lambda ={\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. $R$ is the radius of the sphere and we have taken $d_{\mathrm{cal}}=10\mu \mathrm{m}$. The dotted line shows the current best experimental constraint on any such force pressure, which comes from Ref. 9. The predictions shown in (a) only apply when the test masses have thin shells, $m_{c}d\gg 1$ and $m_b{d}_{\mathrm{cal}}\ll 1$; $m_c$ is the chameleon mass inside the test masses and $m_b$ is the chameleon mass in the background. The white region in (b) shows the values of the chameleon to matter coupling, $M$, for which the predictions shown in (a) are applicable to the 1997 Casimir force measurement performed by Lamoreaux [9].

Figure 7

The solid lines show $(F_{\varphi }^{\mathrm{tot}}(d)-F_{\varphi }^{\mathrm{tot}}(d_{\mathrm{cal}}))/R$ where $F_{\varphi }^{\mathrm{tot}}$ is the predicted chameleonic force between a sphere and a plate, for $V={\Lambda }_0^4\exp {\Lambda }^n/{\varphi }^n$ and $\Lambda ={\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. $R$ is the radius of the sphere and we have taken $d_{\mathrm{cal}}=10\mu \mathrm{m}$. The dotted line shows the current best experimental constraint on any such force pressure, which comes from Ref. 9. The predictions are valid for $m_c^{-1}\ll d<d_{\mathrm{cal}}\ll m_b^{-1}$; $m_c$ is the chameleon mass inside the test masses and $m_b$ is the chameleon mass in the background.

Figure 8

Current constraints from Casimir force measurements on chameleon theories with $V(\varphi )={\Lambda }_0^4(1+{\Lambda }^n/{\varphi }^n)$; ${\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. The cases $n=-8$, $n=-4$, $n=1$, and $n=4$ are shown above. Presently $\Lambda \approx {\Lambda }_0$ is only ruled out for theories with $n=-4$ and $n=-6$ (for which a plot is not shown). See text for further discussion.

Figure 9

Relative strengths of the chameleonic and Casimir forces between two parallel plates at zero temperature. We have taken $\Lambda =2.4\times {10}^{-3}\mathrm{eV}$. (a) is for $V={\Lambda }^4+{\Lambda }^{4+n}/{\varphi }^n$ and (b) for $V={\Lambda }^4\exp ({\Lambda }^n/{\varphi }^n)$ with $\Lambda =2.4\times {10}^{-3}\mathrm{eV}$. For most values of $n$ we see that $\epsilon (d)=1$ for $d\approx 30–40\mu \mathrm{m}$. At $d\approx 10\mu \mathrm{m}$, $\epsilon \approx 0.01–0.1$ in most cases.

Figure 10

Relative strengths of the chameleonic and the Casimir forces between two parallel plates at different temperatures. We have taken $V={\Lambda }_0^4(1+{\Lambda }^n)/{\varphi }^n$ and $\Lambda ={\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. (a) shows how $F_{\varphi }/(F_{\mathrm{cas}}+{\overline{F}}_{\mathrm{thermal}})$ at $d=10\mu \mathrm{m}$ depends on $T$, and (b) shows the temperature dependence of $F_{\varphi }/{\overline{F}}_{\mathrm{thermal}}$. If $n=1/2$ then we see that the chameleonic force only dominates over the thermal contribution to the Casimir force for $T\lesssim 2\mathrm{K}$. (c) shows how $F_{\varphi }/(F_{\mathrm{cas}}+F_{\mathrm{thermal}})$ at 300 K depends on separation, $d$. We see that at 300 K and with $d\approx 10\mu \mathrm{m}$, $\epsilon \approx 0.005–0.1$.

Figure 11

Relative strengths of the chameleonic and the Casimir forces between two parallel plates at different temperatures. We have taken $V={\Lambda }_0^4\exp ({\Lambda }^n/{\varphi }^{)}$ and $\Lambda ={\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. (a) shows how $F_{\varphi }/(F_{\mathrm{cas}}+{\overline{F}}_{\mathrm{thermal}})$ at $d=10\mu \mathrm{m}$ depends on $T$, and (b) shows the temperature dependence of $F_{\varphi }/{\overline{F}}_{\mathrm{thermal}}$. If $n=1/2$, the chameleonic force dominates over the thermal contribution to the Casimir force for $T\lesssim 5\mathrm{K}$. (c) shows how $F_{\varphi }/(F_{\mathrm{cas}}+F_{\mathrm{thermal}})$ at 300 K depends on separation, $d$. At 300 K and with $d\approx 10\mu \mathrm{m}$, $\epsilon \approx 0.002–0.02$.

Figure 12

Shaded area is the region of $\Lambda$-$n$ parameter space that could potentially be detected or ruled out with 95% confidence by the new Grenoble experiment proposed by Lambrecht et al. in Ref. 25. (a) is for $V={\Lambda }_0^4(1+{\Lambda }^n)/{\varphi }^n$ and (b) for $V={\Lambda }_0^4\exp ({\Lambda }^n/{\varphi }^n)$ with ${\Lambda }_0=(2.4\pm 0.1)\times {10}^{-3}\mathrm{eV}$. If $\Lambda ={\Lambda }_0$, which would be natural if the chameleon field is responsible for the late-time acceleration of the Universe, then $\Lambda$ must lie between the two dotted lines i.e. $\Lambda =(2.4\pm 0.1)\times {10}^{-3}\mathrm{eV}$. It should be noted that for these constraints to apply the chameleon to matter coupling $\beta =M_{\mathrm{Pl}}/M$ must be large enough for test masses to have thin shells and small enough for the inverse chameleon mass in the background to be large compared with the separations probed i.e. $m_b^{-1}\gg 10\mu \mathrm{m}$. Both of these requirements will depend to varying degrees on the quality of the laboratory vacuum in which the experiment is performed.

Figure 13

Shaded area is the region of $\Lambda$-$M$ parameter space that could potentially be detected or ruled out with 95 confidence by the new Grenoble experiment proposed by Lambrecht et al. in Ref. 25 for two choice of the vacuum pressure. The plots shown are for $n=1$ with either $V={\Lambda }_0^4(1+{\Lambda }^n)/{\varphi }^n$ or $V={\Lambda }_0^4\exp ({\Lambda }^n/{\varphi }^n)$. The situation for other $\mathcal{O}(1)$ values of $n$ is similar. ${\Lambda }_0=(2.4\pm 0.1)\times {10}^{-3}\mathrm{eV}$. If $P_{\mathrm{vac}}={10}^{-3}\mathrm{torr}$ then for both choices of potential, the Grenoble experiment could detect all $n=1$ theories with $\Lambda =2.4\times {10}^8$ and ${10}^7\mathrm{GeV}\lesssim M\lesssim {10}^{16}\mathrm{GeV}$. If $P_{\mathrm{vac}}$ is lowered to ${10}^7\mathrm{torr}$, then values of $M$ smaller than ${10}^4\mathrm{GeV}$ could be detected.

Figure 14

Shaded area is the region of $M$-$\Lambda$ parameter space that could potentially be detected or rule out the new experiment proposed by Lamoreaux [24, 27] i.e. $\Delta F_{\varphi }^{\mathrm{tot}}(33\mu \mathrm{m})=F_{\varphi }^{\mathrm{tot}}(33\mu \mathrm{m})-F_{\varphi }^{\mathrm{tot}}(d_{\mathrm{cal}}=1\mathrm{cm})>0.1\mathrm{pN}$. The top plot left is for $V={\Lambda }_0^4(1+{\Lambda }^n)/{\varphi }^n$, ${\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$, and $n=1$, and the bottom one is for $V={\Lambda }_0^4\exp ({\Lambda }^n/{\varphi }^n)$, ${\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$, and $n=1$. The graphs for other values of $n$ close to 1 are similar.

Figure 15

Shaded areas show the contours of the predicted values $\Delta F_{\varphi }^{\mathrm{tot}}(82\mu \mathrm{m})=F_{\varphi }^{\mathrm{tot}}(82\mu \mathrm{m})-F_{\varphi }^{\mathrm{tot}}(d_{\mathrm{cal}}=1\mathrm{cm})$ in the new experiment proposed by Lamoreaux [24, 27]. We have taken $V(\varphi )={\Lambda }_0^4(1+{\Lambda }^n/{\varphi }^n)$ and $\Lambda ={\Lambda }_0=2.4\times {10}^{-3}\mathrm{eV}$. The experiment should be sensitive to $\Delta F_{\varphi }^{\mathrm{tot}}>0.1\mathrm{pN}$. At this separation the total Casimir force is $<7.6\times {10}^{-3}\mathrm{pN}$, and so, with improved sensitivity, it should ultimately be able to distinguish chameleonic forces as 0.01 pN from the Casimir background without an incontrovertible model for the thermal contribution to the Casimir force.