Figure 1

Left: The value of $\chi$ varies by an amount $\overline{\mathrm{\Delta }\chi }$ from its average value, ${\chi }_0$, within our observable Universe (the interior of the two vertical lines), due to the long-wavelength fluctuation (solid wave). The average within our observable Universe (long-dashed line) is not necessarily the same as the background value over the entire Universe (dotted horizontal line). Right: Exclusion plot for Eqs. (22) (dark blue), (23) (orange), (24) (green), (25) (red), (26) (purple), and (27) (brown), with $x=0.07$, $a=10$, $b=0.25$ and $c=1$. The red and brown lines are hard to see on this scale at the bottom of the plot. The allowed region is left white.