The Newtonian gravitational constant, G, is one of the most important fundamental constants of nature, however, it still represents one of the mysterious constants in Universe as the gravitational force remains resistant to the standard model of physics and disconnected from quantum theory [1] . Given its critical roles in many fields including theoretical physics, geophysics, astrophysics and astronomy, although the gravitational constant is most difficult to measure accurately [2] , more than 200 experiments have been performed to identify the precise value of G since Henry Cavendish performed the first one more than 100 years ago [3] . As a result, the measured values of G are relatively stable from 6.66 to 6.7559 × 10⁻¹¹ m³·kg⁻¹·s⁻² [3] . Meanwhile, some modified gravity theories have been proposed [4] - [8] . Although non-Newtonian components in short range were suggested by some studies [9] [10] , but there is still no clear evidence to support this opinion. More recently, a measurement between millimetre-sized masses (two gold spheres of 1 millimetre radius) was performed and determined a G value of (6.04 ± 0.06) × 10⁻¹¹ m³·kg⁻¹·s⁻² (Westphal et al. Nature 2021) [11] , which deviates from the recommended CODATA value (G_CODATA = 6.67430 (15) × 10⁻¹¹ m³·kg⁻¹·s⁻²) by ~9%. This deviation is quite significant when compared with the measured values ( Figure 1 ), even the values measured 100 years ago (6.67 × 10⁻¹¹ m³·kg⁻¹·s⁻² in 1873, deviates by only ~0.06%; 6.66 × 10⁻¹¹ m³·kg⁻¹·s⁻² in 1895-1897, deviates by only ~0.2%). It is well known that these historical experiments have been performed using macroscopic masses at the kilogram scale and beyond. For this tension, the authors then explained “This offset is fully covered by the known systematic uncertainties in our experiments, which include unwanted electrostatic, magnetic and gravitational influences from the masses and supports, as well as geometric uncertainties in the centre-of-mass distance due to the actual shape of the masses”. However, after carefully checking Westphal et al.’s experimental protocol and comprehensively surveying the historical G values, we found that Westphal et al.’s experiment was designed very well and it seems impossible that such a big deviation is resulted from “the known systematic uncertainties”, as Westphal et al. considered. We previously proposed an equation that can precisely define G with cosmic microwave background (CMB) [12] . Based on this observation, here we propose a hypothesis that G may significantly decrease between very small-sized masses as single-slit diffraction may occur for CMB travel across these masses. As a result, the proposed theory explains the tension well.

To address the above tension, here we proposed the possibility that the value of G measured by Westphal et al. could be the true value at that scale of 1 millimetre radius mass. If this hypothesis is true, obviously new theory is needed.

We previously revealed a quantitative relation between the Newtonian gravitational constant G and the temperature T of the cosmic microwave background (CMB), by which G can be precisely determined by the temperature T of CMB as the following equation [12] [13] .

$G_T=\frac{T^2}{T_0^2}{G}_0$ (1)

where G₀ is the gravitational constant at present space-time with the CMB temperature of T₀, whereas G_T is the gravitational constant at the space-time with a CMB temperature of T. It is well known that CMB belongs to blackbody radiation. Thus, according to the following Planck distribution function,

$u\left(\lambda \right)=\frac{8\pi h}{{\lambda }^3}\frac{1}{\exp \left(\frac{hc}{\lambda kT}\right)-1}$ (2)

where h, c, k are Planck’s constant, speed of light in vacuum, and Boltzmann’s constant, and u(λ) is the energy density of CMB radiation in the wavelength of λ, thus, the curve of the energy density of CMB radiation of the present space-time can be shown as Figure 2(a) . In Figure 2(a) , the x axis is the wavelength of various CMB electromagnetic wave components and the y axis is the corresponding energy density of specific electromagnetic wave component. The result showed that the peak energy density is located in ~0.005 metres (~5 millimetres) and the curve decreases sharply for CMB electromagnetic wave components with bigger or smaller wavelength ( Figure 2(a) ).

It is well known that during passing an obstacle having similar size with the wavelength, diffraction would be occur for the electromagnetic wave. Based on this observation, we thus proposed the following hypothesis. For two small-size masses (e.g. ≤5 millimetre diameter spheres), the CMB electromagnetic wave components whose wavelength equal to or less than the mass size would totally contribute to gravity, while the ones whose wavelength larger than the mass size would contribute to gravity only using its central energy in the diffraction pattern. Moreover, it is known that the light intensity of single-slit diffraction pattern obeys the following equation (details can be found at https://openstax.org/books/university-physics-volume-3/pages/4-2-intensity-in-single-slit-diffraction ),

$I=I_0{\left(\frac{\text{sin}\left(\beta \right)}{\beta }\right)}^2$ (3)

where β is the angle of diffraction and I₀ is the central maximal density. Figure 2(b) shows the density distribution for β ranging from −10 to 10. According to Equation (3) and Figure 2(b) , it is clear that the density decreases rapidly to almost zero after −3π and 3π. It is thus not difficult to calculate the central energy would be ~93.4% of the total energy for one specific electromagnetic wave. Thus, the CMB energy E_d contributed to the gravity of the masses with size of diameter d can be described as

$E_d=\underset{\lambda <d}{{\displaystyle \sum }}\frac{8\pi h}{{\lambda }^3}\frac{1}{\exp \left(\frac{hc}{\lambda kT}\right)-1}+0.934\underset{\lambda \ge d}{{\displaystyle \sum }}\frac{8\pi h}{{\lambda }^3}\frac{1}{\exp \left(\frac{hc}{\lambda kT}\right)-1}$ (4)

Moreover, according to Boltzmann’s equation and Equation (1), the gravitational constant G_d for the masses with size of diameter d can be given as

$G_d={\left(\frac{E_d}{E_{total}}\right)}^2{G}_0$ (5)

where E_total is the total energy of all of the CMB electromagnetic wave components.

To confirm the above theory, we then calculated the relation of the gravitational constant G_d with the corresponding mass sphere diameter d ranging from 0.0002 to 0.2 metres using a widely accepted value of G (6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻²) as the reference G₀ value. As shown in Figure 2(c) , the value of G_d is quite stable and close to the reference G₀ value when the sphere diameter is greater than 0.02 meters, however, for mass spheres whose diameter less than 0.02 metres, G_d decreases rapidly ( Figure 2(c) & Figure 2(d) ). As a result, we calculated the theoretical value of G between 2 millimetre (0.002 metre) diameter spheres is G_theo = 5.96 × 10⁻¹¹ m³·kg⁻¹·s⁻², which matches the 13.5-h-long fitted value G_fit = (5.89 ± 0.20) × 10⁻¹¹ m³·kg⁻¹·s⁻² and the long-term combined value G_comb = (6.04 ± 0.06) × 10⁻¹¹ m³·kg⁻¹·s⁻² measured by Westphal et al. [11] very well ( Figure 2(d) ). Meanwhile, we provided the predicted G values between millimetre-sized masses from 1 mm diameter to 10 mm diameter in Table 1 and more details in Supplementary Table 1 (available at http://www.cuilab.cn/bigg ) which may be used to confirm the proposed theory or hypothesis in the future.

Moreover, according to the current Hubble’s constant (~70 km/s/Mpc), it is not difficult for this theory to predict that the G value is changing at a rate of −9.551246 × 10⁻²¹ m³·kg⁻¹·s⁻² per year, that is, $\stackrel{\dot{}}{G}/G=-1.431048\times {10}^{-10}$ yr⁻¹, which is quite close to that derived according to changes in the earth’s spin ( $\stackrel{\dot{}}{G}/G\propto -1.0\times {10}^{-10}$ yr⁻¹) [14] , and that derived by lunar tidal acceleration, $\stackrel{\dot{}}{G}/G=\left(-6.4\pm 2.2\right)\times {10}^{-11}$ yr⁻¹ [15] . These results further supports the proposed theory from an alternative view of point.

In summary, we proposed a theory which can perfectly explain the tension between the newly measured G value of millimetre-sized masses and the established values. However, more experiments are needed to support this theory. For example, according to this theory, the gravitational constant between two aluminium spheres of the same mass with the gold spheres used in Westphal et al.’s study (diameter will be ~0.00385 metres) is predicted to be G_theo = 6.17 × 10⁻¹¹ m³·kg⁻¹·s⁻². Similar experiments are needed to support this theory using multiple millimetre-sized and even submillimetre-sized masses. In addition, we only theoretically simulated diameter ranging from 0.0002 to 0.2 metres due to computing resource, theoretical simulation for the sub-atom scale (e.g. between electrons) is also necessary in the future. It should be noted that Fitzgerald et al. introduced 0.2 m as the length of small mass according to the maximum calculated gravitationally induced torque and torque due to Knudsen forces in their method for measuring G [16] .