Plasma noise in TianQin time delay interferometry

The starting point to properly simulate plasma noise is the generation of real-time plasma density data. Here we use the Space Weather Modeling Framework (SWMF) model, which comes from the NASA Community Coordinated Modeling Center (CCMC), to generate electron number density data with a time resolution of $60~{}\rm{s}$ [41]. Taking the real space plasma observation data of the ACE satellite as inputs, we use the SWMF model to obtain the simulation data by solving the magnetohydrodynamics (MHD) equations. By combining the orbital data in Appdendix A, the plasma noise of each link is illustrated in Fig. 1, which includes a magnetic storm ($K_{P}=5$) and in this way gives a noise level higher than normal. Here $K_{P}$ is an index to describe the intensity of magnetic disturbances in near-Earth space [42, 43]. Details on the simulation procedure can be found in Refs. [25, 27]. Due to the short light travel time between two satellites ($\sim 0.5$ s), the difference between $s_{ij}(t)$ and $s_{ji}(t)$ is very small (cf. Fig. 1). Eq. (4) shows that this difference is about $1\times 10^{-5}~{}\rm{pm}$, attributed to the fact that the velocity of TianQin satellites is about $2~{}\rm{km/s}$. As show in Fig. 1, the dotted and solid lines are overlapped. Therefore we can treat the plasma noises along different propagation directions between a pair of spacecraft as equal. The noise ASD curves of the three arms are shown in Fig. 2.

It is important to emphasize that the spectral density can only reach $8~{}\rm{mHz}$ due to the limitation of temporal and spatial resolution (the finest spatial resolution of the SWMF model is $0.25~{}\rm{R_{E}}$, and $\rm{R_{E}}$ stands for the Earth radius). However, for TianQin, we are interested in the behaviour of plasma noise in the range of $10^{-4}-1~{}\rm{Hz}$. Therefore, in this study we further generate data with a time resolution of $5~{}\rm{s}$ using the PPMLR-MHD model developed by Hu et al. [44]. Adopting the same methodology as in Ref. [25], we have obtained the results shown in Fig. 3. The ASD of the plasma noise of the three arms is shown in Fig. 4. The PPMLR-MHD model has the finest spatial resolution of 0.1 Earth radius in the near Earth region of $-10~{}\rm{R_{E}}\leq x,y,z\leq 10~{}\rm{R_{E}}$ in the GSE (Geocentric solar ecliptic) coordinate system, and this model has been widely used to study solar wind-magnetosphere-ionosphere system related phenomena [45, 46, 47]. Note that, unfortunately, due to high cost of such MHD simulations, the 5 s step size and the 900 s data length set the current limits that are available to us. Nevertheless, we deem it sufficient to demonstrate a general trend in the plasma noise spectra.

Overall, in time domain, we can see that the plasma-induced optical path-length variation along a single link can reach 1 pm level. In addition, as shown in Fig. 2 and Fig. 4, we notice that the ASD of the plasma noise decreases with increasing Fourier frequency. In the following subsection III.2, we show this conclusion is valid in general for plasma noise in the millihertz band. Moreover, the patterns in the Fig. 1 and the Fig. 3 suggest correlation among the plasma noises along different arms. Indeed, the Table 1 below gives the correlation coefficients calculated for the two figures, and the numerical values indicate significant positive correlation among different arms.

Solar wind plasma is usually in a turbulent state, and it is hard to describe its properties in time and space accurately. However, a theory that is widely accepted, and consistent with observations is Kolmogorov’s statistical theory of turbulence [31, 32]. The theory suggests that large eddies in a turbulent flow constantly split into smaller ones, and transfer energy in the process. Based on Kolmogorov’s theory, the three-dimensional spectral density of the refractive index coefficient can be derived as [32]

where $k$ is the wave number, and $C^{2}_{n}$ is the refractive-index structure parameter. Similarly, the electron number density variation also follows the power spectral density (PSD) with a $-\frac{5}{3}$ spectral index in one dimension [30, 29].

Recently, in Ref. [30], the influence of solar wind plasma on LISA science measurements is analyzed based on Kolmogorov’s theory. The variation of optical path length on a single link caused by electron number density fluctuation has been obtained [30]. The electron density spectrum is given by

where $P_{0}$ and $f$ are the amplitude and frequency of the electron number density fluctuations, and $k_{0}$ the wavenumber corresponding to the amplitude $P_{0}$, and $V$ the velocity of solar wind. This model of electron density fluctuations in Eq. (7) has been verified with observation data coming from the WIND/SWE instrument. The results have shown that the spectral density in the $10^{-4}-10^{-2}~{}\rm{Hz}$ range agrees well with the Kolmogorov power-law spectrum of Eq. (6). Further results are shown in Fig. 5 of Ref. [30], which presents the plasma noise of single links, and they all indicate that each of the 572 daily spectra becomes smaller with increasing Fourier frequency in the range of $10^{-4}~{}\rm{Hz}-10^{-1}~{}\rm{Hz}$.

In Ref. [48], two different electron number density spectra were observed in the $10^{-3}-1~{}\rm{Hz}$ range based on real observations from the ISEE 1-2 spacecraft. In the range of $10^{-3}-6\times 10^{-2}~{}\rm{Hz}$, the spectral index is $-1.67\pm 0.05$, which satisfies the Kolmogorov spectrum. Above $6\times 10^{-2}~{}\rm{Hz}$, the spectral index is $-0.9\pm 0.2$ [49]. In Ref. [50], based on the data from the Magnetospheric MultiScale mission (MMS), a spectral index of $-0.86\pm 0.03$ was observed in the range $0.039-0.97~{}\rm{Hz}$. In addition, there is extensive literature showing that other parameters of solar wind turbulence follow the Kolmogorov spectrum below $0.1~{}\rm{Hz}$ [51, 52, 53]. These observations are consistent with our MHD simulation results in III.1.

From the above discussion, it can be seen that the spectrum of electron density fluctuations in the solar wind is roughly composed of two different power-law spectra within $10^{-4}-1~{}\rm{Hz}$. At the MHD scale (also referred to as inertial range), i.e. in the range of $\sim 10^{-4}-10^{-1}~{}\rm{Hz}$, density fluctuations follow a Kolmogorov spectrum. At the ion scale corresponding to $\sim 10^{-1}-1~{}\rm{Hz}$, the density spectrum becomes slightly flatter, but still decreases with increasing Fourier frequency. [48, 49, 50, 51, 52, 53]. In Sec. IV.2, we will use these properties of plasma noise, together with MHD simulation data to generate simulated data in the entire observational frequency band of $10^{-4}-1~{}\rm{Hz}$.

In the previous section, we have analyzed the plasma noises of single laser links through MHD simulations and turbulence theory, and the results have shown that the effect has a non-negligible contribution to the noise budget of TianQin. However, in current design of space-based GW detectors, it is necessary to combine multiple measurements from different arms to reduce laser frequency noise in data post-processing, i.e. the TDI technique [16, 54, 55, 17]. Therefore, we would like to study how the plasma noises of single links are transferred through TDI combinations, and to derive the corresponding analytical expressions for the residual plasma noises. In the following, for the sake of simplicity, we assume that the laser frequency noise, optical bench displacement noise, etc., can be suppressed below the secondary noise level by TDI combinations.

As shown in Fig. 5, the TianQin constellation is formed by three identical drag-free spacecraft orbiting around the Earth. Each satellite has two test masses, two optical benches, and two lasers. The distance between each pair of satellites is about $1.7\times 10^{5}~{}\rm{km}$ [8, 56]. The following notation is employed, according to the TianQin detector configuration as shown in Fig. 5 [54, 55, 17]. The three spacecraft are marked as S/C $i$, and each satellite carries two identical optical benches ($i$, $i^{\prime}$) connected by optical fibers. The arm lengths of the opposite side of each satellite $i$ are denoted by $L_{i},L_{i^{\prime}}$, corresponding to the light travel time $t_{i},t^{\prime}_{i}$, where the unprimed number indicates that the laser beam travels in a clockwise direction, and the primed number a counterclockwise direction. The unit vectors $\vec{\textbf{n}}_{i}$, $\vec{\textbf{n}}_{i^{\prime}}$ are along each arm of the triangular configuration, respectively, and $\vec{\textbf{n}}_{i}=-\vec{\textbf{n}}_{i^{\prime}}$ for each $i$.

The first-generation TDI has different data combinations. We use $\alpha$ and $X$ as examples to calculate the residual plasma noise. The schematic optical paths of the two combinations can be found in, e.g., Ref. [54]. Accordingly, the plasma noise in the first-generation TDI $\alpha$ combination can be written as follows [26]

with the time-delay operators $\mathcal{D}_{i}f(t):=f(t-L_{i}/c)$ and the light speed $c$. Similarly, the $X$ combination is given by

Let us now calculate the residual plasma noise for these combinations[15]. First, we transform these expressions from the time domain to the frequency domain through Fourier transform. Then by calculating the expectation value, we obtain the spectral density of the noise. Therefore, the spectral densities of the residual plasma noises corresponding to $\alpha$ and $X$ are given by

where $S(f)$ is the spectral density of plasma noise of a single link. Here, we assume that the arm length does not change with time and is equal to $L$. Additionally, in the calculations of Eqs. (10) and (11) the following expression is used

where $<>$ denotes expectation values [57]. It means that the plasma noises of different arms are uncorrelated. Also recall that the plasma noises of the same arm with different propagation directions are nearly equal (cf. Fig. 1), and hereby we neglect the differences. In addition, the other extreme situation corresponds to when the plasma noises of different arms are perfectly correlated with one another. In this case, TDI would cancel the plasma noise in the same manner as it does to the laser frequency noise.

The results obtained from Eqs. (10) and (11) are illustrated in Fig. 6. A fitted plasma noise spectral density $S(f)$ is plugged into the residual plasma noise expression, and the former is obtained by extrapolating the spectrum of the MHD data in Fig. 2 to the entire frequency band. Fig. 6 shows that below $\sim 0.1~{}\rm{Hz}$ the TDI-combined plasma noise is smaller than the noise of a single link, while above $\sim 0.1~{}\rm{Hz}$, the former is larger than the latter. However, since the high-frequency plasma noise itself is quite small, the TDI-combined noise is still smaller than the secondary noise in the frequency band of $10^{-4}-1~{}\rm{Hz}$. As a comparison, we also draw the secondary noise for both TDI $\alpha$ and $X$ combinations. The secondary noises are given by the following formulas [15, 58, 59]

in which $S^{\rm{acc}}_{y}=\frac{\left(1\times 10^{-15}\rm ms^{-2}Hz^{-1/2}\right)^{2}}{(2\pi fc)^{2}}=2.8\times 10^{-49}(f/1\rm Hz)^{-2}\rm Hz^{-1}$ and $S^{\rm{opt}}_{y}=\frac{\left(1\times 10^{-12}\rm mHz^{-1/2}\right)^{2}\times(2\pi f)^{2}}{c^{2}}=4.4\times 10^{-40}(f/1\rm Hz)^{2}Hz^{-1}$ are the acceleration and displacement noises of a single link, respectively [8, 15].

Ref. [15] found that the first-generation TDI combinations cannot meet the noise requirement in some frequency bands after considering the numerically optimized orbits. In order to further suppress laser phase noise, TianQin may need second-generation TDI combinations. Therefore, it is desirable for us to analyze how the plasma noise of a single link is transferred in the second-generation TDI combinations to determine whether it affects the overall performance. According to Ref. [16], the mathematical expressions of the second-generation TDI $\alpha$ and $X$ combinations are given by the following formulas:

where the semicolon ’;’ indicates time delay that is time-dependent, i.e., $f(t)_{;j}=f(t-L_{j}(t)/c)$.

According to Refs. [15, 60], the spectral density of the residual plasma noises for second-generation TDI $\alpha$ and $X$ combinations are read off as

Eqs. (17) and (18) have the same mathematical form as the secondary noises after the second-generation TDI $\alpha$ and $X$ combinations, since we have treated the plasma noise in a similar way to shot noise and test mass acceleration noise. Inserting the expressions (13) and (14) into the above two equations gives the final residual noise, and the results are shown in Fig. 6. We can see that the residual noise after the second-generation combinations is lower than the secondary noise requirements. These analytic results are confirmed by numerical simulations, which will be described in details in the following subsection.

To verify the analytical studies described above, we further numerically calculate the residual plasma noise after different TDI combinations. We use MHD simulations and the Kolmogorov spectrum introduced in Sec. III to generate simulated data up to $1~{}\rm{Hz}$ in the frequency band, and the frequency domain results are shown in Fig. 7. The specific procedure is to perform a spectral fit of the MHD simulation data in Fig. 2 and Fig. 4, followed by an extrapolation to higher frequencies based on the Kolmogorov spectrum. In this way, we do not alter the trend of the MHD simulation data on a relatively long time scale ($f<8$ mHz), but add some fluctuations that satisfy the Kolmogorov spectrum on a relatively short time scale ($f>8$ mHz).

The time and frequency domain results after different TDI combinations with data at 60 s time resolution (see Fig. 1) are given in Fig. 8, Fig. 9 and Fig. 10, respectively. It can be seen from Fig. 8 that the residual plasma noise after the first- and second-generation TDI $\alpha$ combinations is within $\pm~{}6\times 10^{-15}~{}\rm{m}$. From Fig. 9, we see that the residual plasma noise after the first-generation TDI $X$ combination is approximately twice as high as the first-generation $\alpha$ combination, which is consistent with our analytical results, and also with the results in Ref. [26].

Furthermore, the frequency domain results indicate that the above numerical simulation results agree with the analytical results of Sec. IV, i.e., the residual noise after the TDI combinations is smaller than the secondary noise in the range of $10^{-4}-1~{}\rm{Hz}$. For each TDI combination, the residual plasma noise after the TDI combinations is larger than the plasma noise of a single link in the high-frequency band of $10^{-1}-1~{}\rm{Hz}$. However, since the plasma noise of a single link is relatively small at high frequencies, this amplification will not affect scientific measurements. It should also be remarked that a GW signal with $f_{\rm{GW}}=1~{}\rm{mHz}$ is included for testing purposes in Fig. 10.

Fig. 10 also shows that the numerical results are slightly smaller than the analytical results after TDI combinations. This is due to the fact that in the analytical formulas we assume no correlation between the plasma noises of different arms. To assess the suppressing effect of TDI for the plasma noise owing to noise correlation, we further calculate the ratio of the plasma noise and the secondary noise requirement before and after TDI $X$ combination, and the results are shown in Fig. 11. The results demonstrate that the TDI combination can moderately suppress the plasma noise at the low frequency band below $\sim 10^{-2}~{}\rm{Hz}$. Another way to see this suppressing effect is to look at the equivalent GW noise levels in the sensitivity diagram (see Fig. 12), where it shows that the plasma noise level below $\sim 10^{-2}~{}\rm{Hz}$ indeed becomes lower after TDI combination.

In the above simulation, we have obtained simulation data over the entire frequency band ($10^{-4}-1~{}\rm{Hz}$) based on Kolmogorov’s turbulent statistical theory by extending the spectrum of the MHD simulation data from $8~{}\rm{mHz}$ (time resolution $60~{}\rm{s}$) up to $1~{}\rm{Hz}$. However, there may be a difference between the high frequency information of the data obtained by this method and the real plasma noise. To be on the safe side, we further multiply the high frequency data in Fig. 10 by a factor of 5, and then the results after various TDI combinations are presented in Fig. 13. As it is apparent from the plot, the plasma noise after the combination is still smaller than the secondary noise in the frequency band from $10^{-4}-1~{}\rm{Hz}$ shown in Fig. 6.

To further verify our results above $\sim 10^{-2}$ Hz, we repeat the simulation process using the data from Fig. 3, i.e., using a time resolution of $5~{}\rm{s}$. The results are displayed in Fig. 14 and Fig. 15. The residual plasma noise after the first-generation TDI combinations is below $4\times 10^{-14}~{}\rm{m}$, as shown in Fig. 14. The results in frequency domain are consistent with the results in Fig. 10.

It should be emphasized that in the second-generation TDI simulations, we have used the following approximation

where we have ignored higher order corrections. Considering that the high-frequency component of the plasma noise is quite small, the approximation in Eq. (19) is reasonable. In the simulations, we have used a 31st-order Lagrange fractional delay filter [61].