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The Einstein Equivalence Principle (EEP) carries a pivotal role in understanding theory of gravity and spacetime. It guarantees the gravity to be understood as geometric phenomenon. Considering gravitational coupling of matter in the standard model extension, we propose a novel scheme using frequency measurements to limit the equivalence principle violations in normal matter. The proposal consists of the comparison of high-precision clocks, comoving with the freely falling frame. The experimental comparison of identical kind of clocks on Earth surface can be used to carry out the proposed test, which allows performing simultaneous tests of weak equivalence principle and gravitational redshift. From the existing experiments of Sr optical clocks, we present a simultaneous determination of Earth-dependent parameter $\beta_{\text{E}}$ and clock-dependent parameter $\xi_{\text{Sr}}$ at the level of $10^{-5}$, and in combination with the gravitational redshift experiments and lunar laser ranging, we also obtain a limit on standard model extension coefficients for Lorentz violation. This work provides another important fundamental physics application for the continuous-improvement accuracy of atomic or optical clocks., Comment: 11 pages, 5 figure, 3 tables

Experiments measuring the Newtonian gravitational constant $G$ can offer uniquely sensitive probes of the test of the gravitational inverse-square law. An analysis of the non-Newtonian effect in two independent experiments measuring $G$ is presented, which permits a test of the $1/{r}^{2}$ law at the centimeter range. This work establishes the strongest bound on the magnitude $\ensuremath{\alpha}$ of Yukawa-type deviations from Newtonian gravity in the range of 5--500 mm and improves the previous bounds by up to a factor of 7 at the length range of 60--100 mm.

Given the high-precision modern space mission, a precise relativistic modeling of observations is required. By solving the eikonal equation with the post-Newtonian approximation, the light propagation is determined by the iterative method in the gravitational field of an isolated, gravitationally bound N-body system. Different from the traditional $N$ bodies that are independent with each other in the system, our system includes the velocities, accelerations, gravitational interactions and tidal deformations of the gravitational bodies. The light delays of these factors then are precisely determined by the analytical solutions. These delays are significant and are likely to reach a detectable level for the \emph{strong} gravitational fields, such as binary pulsars and some gravitational wave sources. The result's application in the vicinity of the Earth provides a relativistic framework for modern space missions. From the relativistic analysis in the TianQin mission, we find the possible tests for the alternative gravitational theories, such as a possible determination for the post-Newtonian parameter $\gamma$ in the level of some scalar-tensor theories of gravity.

The Newtonian gravitational constant, G, is one of the most fundamental constants of nature, but we still do not have an accurate value for it. Despite two centuries of experimental effort, the value of G remains the least precisely known of the fundamental constants. A discrepancy of up to 0.05 per cent in recent determinations of G suggests that there may be undiscovered systematic errors in the various existing methods. One way to resolve this issue is to measure G using a number of methods that are unlikely to involve the same systematic effects. Here we report two independent determinations of G using torsion pendulum experiments with the time-of-swing method and the angular-acceleration-feedback method. We obtain G values of 6.674184 × 10−11 and 6.674484 × 10−11 cubic metres per kilogram per second squared, with relative standard uncertainties of 11.64 and 11.61 parts per million, respectively. These values have the smallest uncertainties reported until now, and both agree with the latest recommended value within two standard deviations.

Traditional torque type gravity gradiometer has an important pole in gravity gradient measurements, while it is relatively inefficient and with the precision of about 1 E mainly caused by the static operating mode. In this paper, we develop an improved torque type gravity gradiometer to improve the measuring efficiency, which is based on the dynamic modulation. The dynamic modulation keeps the gradiometer rotating on a turntable steadily, measures the deflection angle of the torsion pendulum continuously and then obtains the gravity gradients. The result shows that after using the improved gradiometer, the gradients W xz and W yz are obtained with precisions of 0.45 E and 0.32 E respectively in a cycle of 20 min.

In the recent CODATA-2014 adjustment, the $G$ values obtained by different groups are not in good agreement with each other within respective uncertainties, which may be attributed to the possible unknown or incorrectly evaluated systematic errors in different methods. This paper proposes a scheme to measure $G$ with two independent methods. The first method is the extraction of the $G$ value from the fundamental periods of the torsion pendulum, with the source masses at two positions; the other one determines the $G$ value on the basis of the amplitude of the third-order harmonic component of the torsion pendulum. Both methods are performed with the same setup by using the large-amplitude torsion pendulum. Through theoretical derivation and simulation, the uncertainties of the $G$ values obtained by the two methods are both expected to be within 20 ppm. By comparing the $G$ values from the two methods, we can explore the potential systematic errors in the measurement.

We improve the test of the gravitational inverse-square law at the submillimeter range by suppressing the vibration of the electrostatic shielding membrane to reduce the disturbance coupled from the residual surface potential. The result shows that, at a 95% confidence level, the gravitational inverse-square law holds ($|\ensuremath{\alpha}|\ensuremath{\le}1$) down to a length scale $\ensuremath{\lambda}=48\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$. This work establishes the strongest bound on the magnitude $\ensuremath{\alpha}$ of the Yukawa violation in the range of $40--350\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$, and improves the previous bounds by up to a factor of 3 at the length scale $\ensuremath{\lambda}\ensuremath{\approx}70\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$. Furthermore, the constraints on the power-law potentials are improved by about a factor of 2 for $k=4$ and 5.

The Newtonian gravitational constant G, which is one of the most important fundamental physical constants in nature, plays a significant role in the fields of theoretical physics, geophysics, astrophysics and astronomy. Although G was the first physical constant to be introduced in the history of science, it is considered to be one of the most difficult to measure accurately so far. Over the past two decades, eleven precision measurements of the gravitational constant have been performed, and the latest recommended value for G published by the Committee on Data for Science and Technology (CODATA) is (6.674 08 ± 0.000 31) × 10−11 m3 kg−1 s−2 with a relative uncertainty of 47 parts per million. This uncertainty is the smallest compared with previous CODATA recommended values of G; however, it remains a relatively large uncertainty among other fundamental physical constants. In this paper we briefly review the history of the G measurement, and introduce eleven values of G adopted in CODATA 2014 after 2000 and our latest two values published in 2018 using two independent methods.

The theory of general relativity has an ultraviolet(UV) problem that can be ameliorated by gravity with higher derivatives. The four-derivative gravity as an effective gravitational theory at UV scalars has two Yukawa-like corrections with parameters ${\ensuremath{\lambda}}_{0}$ and ${\ensuremath{\lambda}}_{2}$. By the analysis of experiments of HUST-2015, Newman's group, lake experiment, and Cassini spacecraft, we obtain the strong-bound regions for these parameters at submillimeter-to-millimeter, centimeter-to-meter, tens-of-meter, and solar-system scales, in which the properties of potential are clearly shown. Recently, the ghostfree and singularityfree gravity, a more potential modified gravity theory, introduces the novel conception of nonlocality. We test the scale of gravitational nonlocality ${\ensuremath{\lambda}}_{\mathrm{m}}l2.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{m}$ by the torsion pendulum experiment HUST-2015. The predicted decaying spatial oscillations are visible meaning possible violation at shorter ranges. Our result provides useful information for gravitational interaction at microscopic ranges.

Improving the precision of current tests of the equivalence principle with a rotating torsion pendulum requires a more complete analysis of systematic effects. Here, we discuss in detail one of the important systematic effects, the influence from the tilt error motion of the rotation axis of a rotary stage, namely, wandering of the instantaneous rotation axis around its average direction. Its influence on the rotating torsion pendulum is modeled phenomenologically, and the parameters in the model are calibrated. It is shown that the influence can contribute a correction of η ≈ 5 × 10−13 to the equivalence-principle violating parameter for a rotary stage whose tilt error motion of interest is about 31 nrad in magnitude. We also show that such an influence can be reduced to the level of η ≈ 1 × 10−14 by means of active compensation of the tilt error motion using a set of piezoelectric actuators placed under the stage stator.

Although the relativistic manifestations of gravitational fields in gravimetry were first studied 40 years ago, the relativistic effects combined with free-fall absolute gravimeters have rarely been considered. In light of this, we present a general relativistic model for free-fall absolute gravimeters in a local-Fermi coordinates system, where we focus on effects related to the measuring devices: relativistic transverse Doppler effects, gravitational redshift effects and Earth's rotation effects. Based on this model, a general relativistic expression of the measured gravity acceleration is obtained.

We construct a quasi-inertial reference system for the atom interferometer in terms of the light-field compensation, which is equivalent to the establishment of a quantum drag-free system. The compensation considered in our approach involves feedbacks of both classical and quantum nature. As a result, it may significantly reduce the dependence of the interferometric phase shift on the atomic initial position and velocity. As the error bound for gravity experiments originates primarily from the uncertainties in the initial conditions, the proposed quasi-inertial reference system might lead to substantial improvement in this regard. To be specific, the refined process introduced in the present study gives rise to the possibility of the measurement of the Newtonian gravitational constant G beyond the precision level of 10−6, as well as the test of the weak equivalence principle beyond 10−14. Moreover, when compared with the conventional quasi-inertial reference based on macroscopic objects, additional quantum feedback plays an essential role concerning both theoretical rigor and practical performance enhancement. We further explore the possibility of implementing the proposed quasi-inertial reference system in ongoing experimental developments.

Precision measurements of the inverse-square law via experiments on short-range gravity provide sensitive tests of Lorentz symmetry. A combined analysis of data from experiments at the Huazhong University of Science and Technology and Indiana University sets simultaneous limits on all 22 coefficients for Lorentz violation correcting the Newton force law as the inverse sixth power of distance. Results are consistent with no effect at the level of $10^{-12}$ m$^{4}$., 7 pages two-column REVTeX, accepted for publication in Physical Review Letters

Here we present a new test of the equivalence principle designed to search for the possible violation of gravitational parity using test bodies with different chiralities. The test bodies are a pair of left- and right-handed quartz crystals, whose gravitational acceleration difference is measured by a rotating torsion pendulum. The result shows that the acceleration difference towards Earth $\mathrm{\ensuremath{\Delta}}{a}_{\text{left}\text{-right}}=[\ensuremath{-}1.7\ifmmode\pm\else\textpm\fi{}4.1(\text{stat})\ifmmode\pm\else\textpm\fi{}4.4(\text{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{\ensuremath{-}2}$ ($1\text{\ensuremath{-}}\ensuremath{\sigma}$ statistical uncertainty), correspondingly the E\"otv\"os parameter $\ensuremath{\eta}=[\ensuremath{-}1.2\ifmmode\pm\else\textpm\fi{}2.8(\text{stat})\ifmmode\pm\else\textpm\fi{}3.0(\text{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$. This is the first reported experimental test of the equivalence principle for chiral masses and opens a new way to the search for the possible parity-violating gravitation.

Although frequency-shift gravity-gradient compensation is currently the technique with the most potential to test the weak equivalence principle with atom interferometry at high precision, it faces an unavoidable obstacle on how to cancel the coupling between the Raman-pulse-duration effect and the gravity gradient related to the atomic initial kinematics. According to the Einstein equivalence principle, we develop a method to calculate this coupling effect in a space-stretching free-falling frame, based on which an improved frequency-shift gravity-gradient compensation technique is put forward to make high-accuracy weak-equivalence-principle tests with the microscopic particles, such as at the level of ${10}^{\ensuremath{-}14}$, more possible.

TianQin is a space-based laser interferometric gravitational wave detector aimed at detecting gravitational waves at low frequencies (0.1 mHz -- 1 Hz). It is formed by three identical drag-free spacecrafts in an equilateral triangular constellation orbiting around the Earth. The distance between each pair of spacecrafts is approximately $1.7 \times 10^{5} ~\rm{km}$. The spacecrafts are interconnected by infrared laser beams forming up to three Michelson-type interferometers. The detailed mission design and the study of science objectives for the TianQin project depend crucially on the orbit and the response of the detector. In this paper, we provide the analytic expressions for the coordinates of the orbit for each spacecraft in the heliocentric-ecliptic coordinate system to the leading orders. This enables a sufficiently accurate study of science objectives and data analysis, and serves as a first step to further orbit design and optimization. We calculate the response of a single Michelson detector to plane gravitational waves in arbitrary waveform which is valid in the full range of the sensitive frequencies. It is then used to generate the more realistic sensitivity curve of TianQin. We apply this model on a reference white-dwarf binary as a proof of principle., Comment: 19 pages, 4 figures, to be published in CQG

Local Lorentz invariance is an important component of General Relativity. Testing for Local Lorentz invariance can not only probe the foundation stone of General Relativity but also help to explore the unified theory for General Relativity and quantum mechanics. In this paper, we search the Local Lorentz invariance violation associated with operators of mass dimension d = 6 in the pure-gravity sector with short-range gravitational experiments. To enlarge the Local Lorentz invariance violation signal effectively, we design a new experiment in which the constraints of all fourteen violation coefficients may be improved by about one order of magnitude.

The propagation time delay due to the finite speed of light (FSL) in atom gravimeters introduces a bias in the gravity measurement, as well as that in classical free-falling corner-cube gravimeters, which is usually termed the FSL effect. For a typical atom gravimeter, the FSL time delay is about several nanoseconds, resulting in the FSL effect, a non-negligible bias in the gravity-acceleration measurement. However, a time delay of about several microseconds, achieved by controlling the Raman-pulse timing directly, contributes a negligible effect. This interesting phenomenon motivates us to make clear two questions: first, what are the origins of the FSL effect in atom gravimeters, and second, what is the difference between the two time delays? Our analysis shows that the FSL effect in atom gravimeters is not just a matter of FSL time delay to a great extent but also the change in the effective wave vector; moreover, the FSL time delay can be quantitatively regarded as the same as the pulse time delay since both actually affect the gravity measurement by changing the two interferometer pulse separations.

The phase function is a useful tool to study all observations of space missions, since it can give all the information about light propagation in a gravitational field. For the extreme accuracy of the modern space missions, a precise relativistic modeling of observations is required. So, we develop a recursive procedure enabling us to expand the phase function into a perturbative series of ascending powers of the Newtonian gravitational constant. Any $n$th-order perturbation of the phase function can be determined by the integral along the straight line connecting two point events. To illustrate the result, we carry out the calculation of the phase function outside a static, spherically symmetric body up to the order of ${G}^{2}$. Then, we develop a precise relativistic model that is able to calculate the phase function and the derivatives of the phase function in the gravitational field of rotating and uniformly moving bodies. This model allows the computing of the Doppler, radio science, and astrometric observables of the space missions in the Solar System. With the development of space technology, the relativistic corrections due to the motion of a planet's spin must be considered in the high-precision space missions in the near future. As an example, we give the estimates of the relativistic corrections on the observables about the space missions TianQin and BEACON.

We investigated Lorentz violation through anisotropy of gravity using a worldwide array of 12 superconducting gravimeters. The Lorentz-violating signal is extracted from the difference between measured gravity and a tidal model. At the level of sensitivity we reach, ocean tides start to play an important role. However, most models available that include ocean tides are empirically based on measured gravity data, which may contain Lorentz-violating signal. In this work we used an ocean tides included tidal model derived from first principles to extract Lorentz-violating signal for the first time. We have bounded space-space components of gravitational Lorentz violation in the minimal standard model extension (SME) up to the order of $10^{-10}$, one order of magnitude improved relative to previous atom-interferometer tests., 5 pages, 1 figure

Atom interferometry is currently developing rapidly, which is now reaching sufficient precision to motivate laboratory tests of general relativity. Thus, it is extremely significant to develop a general relativistic model for atom interferometers. In this paper, we mainly present an analytical derivation process and first give a complete vectorial expression for the relativistic interferometric phase shift in an atom interferometer. The dynamics of the interferometer are studied, where both the atoms and the light are treated relativistically. Then, an appropriate coordinate transformation for the light is performed crucially to simplify the calculation. In addition, the Bord\'e $ABCD$ matrix combined with quantum mechanics and the ``perturbation'' approach are applied to make a methodical calculation for the total phase shift. Finally, we derive the relativistic phase shift kept up to a sensitivity of the acceleration $\ensuremath{\sim}1{0}^{\ensuremath{-}14}\text{ }\text{ }{\mathrm{m}/\mathrm{s}}^{2}$ for a $10\text{\ensuremath{-}}\mathrm{m}$-long atom interferometer.

Recently, first limits on putative Lorentz invariance violation coefficients in the pure gravity sector were determined by the reanalysis of short-range gravity experiments. Such experiments search for new physics at sidereal frequencies. They are not, however, designed to optimize the signal strength of a Lorentz invariance violation force; in fact the Lorentz violating signal is suppressed in the planar test mass geometry employed in those experiments. We describe a short-range torsion pendulum experiment with enhanced sensitivity to possible Lorentz violating signals. A periodic, striped test mass geometry is used to augment the signal. Careful arrangement of the phases of the striped patterns on opposite ends of the pendulum further enhances the signal while simultaneously suppressing the Newtonian background., Accepted Phys. Rev. D

In the measurement of the gravitational constant G with angular acceleration method, the accurate estimation of the amplitude of the useful angular acceleration generated by source masses depends on the effective subtraction of the spurious gravitational signal caused by room fixed background masses. The gravitational background signal is of time-varying frequency, and mainly consists of the prominent fundamental frequency and second harmonic components. We propose an improved correlation method to estimate the amplitudes of the prominent components of the gravitational background signal with high precision. The improved correlation method converts a sinusoidal signal with time-varying frequency into a standard sinusoidal signal by means of the stretch processing of time. Based on Gaussian white noise model, the theoretical result shows the uncertainty of the estimated amplitude is proportional to σNT, where σ and N are the standard deviation of noise and the number of the useful signal period T, respectively.

In the measurement of the gravitational constant G with angular acceleration method, the equilibrium position of torsion pendulum with tungsten fiber undergoes a linear slow drift, which results in a quadratic slow drift on the angular velocity of the torsion balance turntable under feedback control unit. The accurate amplitude determination of the useful angular acceleration signal with known frequency is biased by the linear slow drift and the coupling effect of the drifting equilibrium position and the room fixed gravitational background signal. We calculate the influences of the linear slow drift and the complex coupling effect on the value of G, respectively. The result shows that the bias of the linear slow drift on G is 7 ppm, and the influence of the coupling effect is less than 1 ppm.

The finite-speed-of-light (FSL) effect is a systematic error in atom gravimeters arising from the time delay due to the propagation of the light. It includes the frequency-chirp-independent part and the frequency-chirp-dependent part, which were not considered completely. The FSL effect in atom gravimeters is different from that in corner-cube absolute gravimeters. In the past, this effect has been widely studied in corner-cube absolute gravimeters, whereas little has been discussed about and done with atom gravimeters. In this paper, we mainly propose a complete analytical study based on a coordinate transformation and on a ``perturbation'' approach to estimate this effect in an atom gravimeter. This also offers the potential to calculate the general relativistic effects in atom gravimeters. In addition, a comparison with a crude ``average-path'' analysis is given for a particular case of the FSL effect in atom gravimeters.

The Beyond Einstein Advanced Coherent Optical Network (BEACON) mission was designed to achieve an accuracy of $$10^{-9}$$ in measuring the Eddington parameter $$\gamma $$ , which is perhaps the most fundamental Parameterized Post-Newtonian parameter. However, this ideal accuracy was just estimated as a ratio of the measurement accuracy of the inter-spacecraft distances to the magnitude of the departure from Euclidean geometry. Based on the BEACON concept, we construct a measurement model to estimate the parameter $$\gamma $$ with the least squares method. Influences of the measurement noise and the out-of-plane error on the estimation accuracy are evaluated based on the white noise model. Though the BEACON mission does not require expensive drag-free systems and avoids physical dynamical models of spacecraft, the relatively low accuracy of initial inter-spacecraft distances poses a great challenge, which reduces the estimation accuracy in about two orders of magnitude. Thus the noise requirements may need to be more stringent in the design in order to achieve the target accuracy, which is demonstrated in the work. Considering that, we have given the limits on the power spectral density of both noise sources for the accuracy of $$10^{-9}$$ .

We consider the relativistic tidal effects on frequency shift of clock-comparison experiments. The relativistic formulation for frequency shift and time transfer is derived in the gravitational field of a tidal, axisymmetric, and rotating Earth. With the help of Love numbers describing the tidal response of solid Earth, we formulize the mathematical connection between tidal effects from the ground-based clock-comparison experiments and the local gravity tides from the gravimeters, which in turn provides us an approach to eliminate tidal influences on clock comparison with the local gravity tides data. Moreover, we develop a method of the perturbed Kepler orbit to determine relativistic effects of clock comparison for space missions, which allows more precise calculations comparing to the conventional method of unperturbed Kepler orbit. With this perturbed method, it can give the perturbation of relativistic effects due to the orbital changes under the influences of tidal forces, Earth's oblateness etc. In addition, as the applications of our results, we simulate tidal effects in frequency shift for the clock comparison on the ground and also give some estimates for TianQin mission and GPS.

The high precision test of the weak equivalence principle (WEP) using a rotating torsion pendulum requires thorough analysis of systematic effects. Here we investigate one of the main systematic effects, the coupling of the ambient magnetic field to the pendulum. It is shown that the dominant term, the interaction between the average magnetic field and the magnetic dipole of the pendulum, is decreased by a factor of 1.1 × 104 with multi-layer magnetic shield shells. The shield shells reduce the magnetic field to 1.9 × 10-9 T in the transverse direction so that the dipole-interaction limited WEP test is expected at η ≲ 10-14 for a pendulum dipole less than 10-9 A m2. The high-order effect, the coupling of the magnetic field gradient to the magnetic quadrupole of the pendulum, would also contribute to the systematic errors for a test precision down to η ∼ 10-14.

We present an atomic multiwave interferometer with magnetic sublevels to precisely determine the Aharonov-Casher (AC) geometric phase. Simulations show that this interferometer has sharper fringes than a normal two-wave interferometer, which means a higher phase resolution can be achieved. Moreover, atoms evolving in a single hyperfine structure state make the interferometer insensitive to the dc Stark phase shift. This dc Stark shift is one of the main noise sources in AC phase measurements. The constraint of the photon rest mass is also discussed when using this atomic interferometer to measure the Aharonov-Casher phase.

We report a test of the universality of free fall (UFF) by comparing the gravity acceleration of the $^{87}$Rb atoms in $m_F=+1$ versus that in $m_F=-1$, where the corresponding spin orientations are opposite. A Mach-Zehnder-type atom interferometer is exploited to sequentially measure the free fall acceleration of the atoms in these two magnetic sublevels, and the resultant E$\rm{\ddot{o}}$tv$\rm{\ddot{o}}$s ratio is ${\eta _S} =(0.2\pm1.2)\times 10^{-7}$. This also gives an upper limit of $1.1\times 10^{-21}$ GeV/m for possible gradient field of the spacetime torsion. The interferometer using atoms in $m_F=\pm 1$ is highly sensitive to the magnetic field inhomogeneity, and a double differential measurement method is developed to alleviate the inhomogeneity influence. Moreover, a proof experiment by modulating the magnetic field is performed, which validates the alleviation of the inhomogeneity influence in our test., Comment: 12 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1503.00433

TianQin is a proposal for a space-borne detector of gravitational waves in the millihertz frequencies. The experiment relies on a constellation of three drag-free spacecraft orbiting the Earth. Inter-spacecraft laser interferometry is used to monitor the distances between the test masses. The experiment is designed to be capable of detecting a signal with high confidence from a single source of gravitational waves within a few months of observing time. We describe the preliminary mission concept for TianQin, including the candidate source and experimental designs. We present estimates for the major constituents of the experiment's error budget and discuss the project's overall feasibility. Given the current level of technology readiness, we expect TianQin to be flown in the second half of the next decade., 32 pages, 4 figures, to be published in Class. Quantum Grav

By using a torsion pendulum and a rotating eightfold symmetric attractor with dual modulation of both the interested signal and the gravitational calibration signal, a new test of the gravitational inverse-square law at separations down to 295 μm is presented. A dual-compensation design by adding masses on both the pendulum and the attractor was adopted to realize a null experiment. The experimental result shows that, at a 95% confidence level, the gravitational inverse-square law holds (|α|≤1) down to a length scale λ=59 μm. This work establishes the strongest bound on the magnitude α of Yukawa-type deviations from Newtonian gravity in the range of 70-300 μm, and improves the previous bounds by up to a factor of 2 at the length scale λ≈160 μm.

Based on statistical properties, two typical models are considered to calculate the uncertainties for some random noise sequences on the period extraction of a torsion pendulum, which is important and instructive in the measurement of gravitational constant G with the time-of-swing method. An expression of the uncertainty for the period measurement is obtained, which is dependent on the ratio Δt/(1/λ) where Δt is the interval of the sample time and 1/λ is the length of the correlation time. The result of processing experimental data shows that as the interval of the sample time Δt gradually shortens, the uncertainty of the period becomes smaller, and further when the ratio Δt/(1/λ) is less than 1, the uncertainty remains substantially unchanged.