CN115561824B  Space gravitational wave detection sensitivity curve calculation method based on time delay  Google Patents
Space gravitational wave detection sensitivity curve calculation method based on time delay Download PDFInfo
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 CN115561824B CN115561824B CN202211279268.4A CN202211279268A CN115561824B CN 115561824 B CN115561824 B CN 115561824B CN 202211279268 A CN202211279268 A CN 202211279268A CN 115561824 B CN115561824 B CN 115561824B
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Abstract
The invention relates to a time delaybased space gravitational wave detection sensitivity curve calculation method, which is used for obtaining a detection sensitivity curve of a space gravitational wave detector, wherein the space gravitational wave detector takes three spacecrafts as vertexes to form a triangle, each spacecraft is provided with two testing quality and two lasers, and the three spacecrafts are connected by laser beams emitted by the lasers; the method comprises the following steps: s1, taking frequency points at certain intervals in a certain frequency interval, and storing the frequency points into a frequency list; s2, defining a transfer function, and calculating to obtain transfer function values at all frequency points; s3, calculating to obtain a test quality noise item and a shot noise item of Michelson X combination at each frequency point by using a time delay interferometry technology; s4, defining a sensitivity function, calculating the sensitivity value at each frequency point by using the values calculated in the steps S2 and S3, and storing the sensitivity value in a sensitivity list; s5, drawing a sensitivity curve by taking each frequency point as an abscissa and the sensitivity value at each frequency point as an ordinate.
Description
Technical Field
The invention relates to the gravitational wave detection field, in particular to a time delaybased space gravitational wave detection sensitivity curve calculation method.
Background
In the beginning of 2016, the us land laser interference gravitational wave astronomy (LIGO) published results of direct detection of gravitational waves, confirming the prediction of einstein about the existence of gravitational waves one hundred years ago, and opening up a new research direction, gravitational wave astronomy. Unlike electromagnetic waves, gravitational waves will create a completely new window for us to observe the universe. From a large explosion in the universe to a black hole, from a merging of the intermediate stars to a double star of the white dwarf, from a topological defect in the universe to dark energy of a dark substance, and the like.
The ground gravitational wave measurement is limited by the influence of the earth gravitational force and the size of the scale, and the detected gravitational wave range is mainly concentrated at high frequency. Spatial gravitational wave detection is not an option, as it is understood that a wider range of gravitational waves and wave sources are of a nature. As a representative of the spatial gravitational wave detection items, LISA (Laser Interferometer Space Antenna) describes clear paths and platforms for the spatial gravitational wave detection subjects at the task concept level, provides references for the design of other spatial laser interference gravitational wave detection items such as tai chi plan, tian qin plan and the like in the world, particularly promotes the development of the spatial gravitational wave detection subjects in the aspects of wave source analysis, data processing, relevant wave source astronomy, relative theory itself and the like, and promotes the spatial gravitational wave detection to gradually form a subject research field.
Since the 80 s of the 20 th century, attempts have been made to measure the gravitational wave directly by means of a Michelson laser interferometer based on the recognition that the gravitational wave causes different changes in the arm lengths of the two working arms of the long baseline laser interferometer, resulting in a change in interference fringes. The laser interferometer has the advantages of wide measurement frequency band and high sensitivity, so that the laser interferometer is a mainstream detection means for detecting gravitational waves quickly. The geometric shape of the space gravitational wave detector is triangular by taking three spacecrafts as vertexes, each spacecraft is provided with two testing quality and two lasers, and each side is an arm. In order to determine whether a particular gravitational wave source can be detected by the gravitational wave detector, it is necessary to know the sensitivity limit of the instrument, the detector sensitivity reflecting the sensitivity of the detector to the target signal. The sensitivity of these instruments is typically described by plotting the minimum amplitude of the gravitational wave (average source position and polarized) versus the frequency of the gravitational wave that the instrument can detect. In order to increase the detectable intensity, efforts should be made to reduce the effects of different noise processes. Typically, the detector laser phase noise is quite large. For a floor interferometer, the fixed lengths of the two arms are the same, so that the laser experiences the same delay, and therefore, the laser phase noise can be well eliminated. However, for a spatial interferometer, it is not possible to keep the length of each arm constant. This results in lasers in different arms having different delays, and residual laser phase noise greatly affects the detection sensitivity. To address this problem, tinto et al first proposed a Time Delay Interferometry (TDI) technique with different combinations of measurement data to eliminate noise.
Disclosure of Invention
In order to further solve the problems, when the spatial gravitational wave detection sensitivity calculation is performed, the invention adopts different combinations of TDI to describe each noise item, and compared with the traditional spatial gravitational wave detection sensitivity calculation, the invention can provide more comprehensive technical support for auxiliary spatial gravitational wave detection task demonstration work, namely, can support detection system index analysis and scientific target analysis, and can provide technical support for TDI algorithm analysis work before simulation data or detection data are provided.
The invention aims to calculate a space gravitational wave detection sensitivity curve based on a link combination mode of a first generation time delay interference method, and provide support for performance analysis of space gravitational wave detection tasks.
In order to achieve the above purpose, the present invention is realized by the following technical scheme.
The invention provides a time delaybased space gravitational wave detection sensitivity curve calculation method, which is used for obtaining a detection sensitivity curve of a space gravitational wave detector, wherein the geometrical shape of the space gravitational wave detector is as follows: forming a triangle by taking a first spacecraft, a second spacecraft and a third spacecraft as vertexes, wherein each side of the triangle is an arm, each spacecraft is provided with two test quality and two lasers, and the three spacecrafts are connected by laser beams emitted by the lasers; the method comprises the following steps:
s1, frequency points are taken at certain intervals in a certain frequency interval and stored;
s2, defining a transfer function, and calculating to obtain transfer function values at all frequency points;
s3, calculating to obtain a test quality noise item and a shot noise item of Michelson X combination at each frequency point by using a time delay interferometry technology;
s4, defining a sensitivity function, calculating the sensitivity value at each frequency point by using the values calculated in the steps S2 and S3, and storing the sensitivity value;
s5, drawing a sensitivity curve by taking each frequency point as an abscissa and the sensitivity value at each frequency point as an ordinate.
As one of the improvement of the technical scheme, the triangle formed by using the first spacecraft, the second spacecraft and the third spacecraft as vertexes has three arms with the same length.
As one of the improvements of the above technical solution, in the step S2, the expression of the transfer function R (u) is:
where the variable u=2pi fL/c, c is the speed of light, f is the gravitational wave frequency, L is the arm length, ci (·) is the cosine integral function, si (·) is the sine integral function.
As an improvement of the foregoing technical solution, in the step S3, the michelson X combination has the following expression:
wherein E is _{13} A time delay operator representing the first spacecraft to the third spacecraft; e (E) _{31} Representing a third spacecraft to first spacecraft time delay operator; e (E) _{12} A time delay operator representing a first spacecraft to a second spacecraft; e (E) _{21} Representing a time delay operator of the second spacecraft to the first spacecraft.
As one of the improvements of the above technical solution, in the step S3, the michelson X combined test quality noise term N at each frequency point _{pf} The calculation formula of (2) is as follows:
wherein i represents a summation variable; p is p _{i} Entries representing the first row and the ith column in the michelson Xcombination matrix; q _{i} Entries representing the second row, ith column, in the michelson Xcombination matrix; r is (r) _{i} Entries representing the third row, ith column, in the michelson Xcombination matrix;
S _{pf} the power spectrum density of the test mass acceleration noise is expressed as follows:
wherein s is _{a} Representing the test mass acceleration noise amplitude spectral density.
As one of the improvements of the above technical solution, in the step S3, the shot noise term N of michelson X combination at each frequency point _{shot} The calculation formula of (2) is as follows:
wherein S is _{shot} The power spectral density of shot noise is expressed as:
wherein s is _{x} Representing the amplitude spectral density of shot noise.
As one of the improvements of the above technical solution, in the step S4, the expression of the sensitivity function S (f) is:
wherein, alpha is a coefficient related to configuration and observation time, R (f) represents a transfer function taking gravitational wave frequency f as an independent variable;
the power spectral density of the total noise representing the michelson combination is expressed as:
as one of the improvements of the above technical solution, the expression of the sensitivity function S (f) is specifically:
where T represents the observation time and 5 is the signal to noise ratio of the 1 year observation time.
Compared with the prior art, the invention has the advantages that:
PSD and N of noise in a computational sensitivity expression _{pf} And N _{sh} When the sum of the ot is over, a first generation TDI link combination mode is used to eliminate laser phase noise.
Drawings
FIG. 1 is a schematic illustration of the geometry of a spatial gravitational wave detector;
FIG. 2 is a flow chart of the method of the present invention;
the solid line below fig. 3 is the sensitivity curve for an arm length of 250 ten thousand kilometers;
the solid line below fig. 4 is a sensitivity curve for an arm length of 300 tens of thousands of kilometers;
the solid line below fig. 5 is the sensitivity curve for an arm length of 500 tens of thousands of kilometers.
Detailed Description
The technical scheme provided by the invention is further described below by combining with the embodiment.
Examples
As shown in fig. 1, a schematic diagram of the geometry of the spatial gravitational wave detector is shown; three spacecrafts (SC 1, SC2, SC 3) are triangular, wherein the angle γ of SC2, SC3 to SC1 is arbitrary; each SC contains two test masses and two lasers, mounted on an optical bench, Z ^{i} Representing minor frequency fluctuations of the two laser beams exchanged between the optical stations; three SCs connected by a laser beam, a time delay operator E for f (t) from SCi to SCj _{ij} Defined as E _{ij} f(t)≡f(tL _{ij} C), f (t) is a continuous function, t is time, L _{ij} The arm length from SCi to SCj, and c is the speed of light.The SCi laser phase for SCj and the noise of the optical bench are shown. From U ^{i} And V ^{i} (i=1, 2, 3) represents the minute frequency fluctuations of the light beams propagating to SCi by the other two SCs, respectively.
For convenience we consider only laser phase, test quality and shot noise. The data flow arriving at SC1 can be expressed as
Here, V ^{1} ，U ^{1} Minute frequency fluctuating data streams, Z, of SC3 and SC2 arrival at SC1, respectively ^{1} Representing the minute frequency fluctuation data stream of the two laser beams exchanged between SC1 and the other two optical stations. Delta _{ij} Is the fluctuation caused by random velocity noise of the test mass,indicates GW signal on SC1 to SC3 measuring arm,>denote GW signals on SC1 and SC2 measuring arms, Y ^{shot} Is fluctuation caused by shot noise, +.>Representing fluctuations caused by shot noise between SC1 and SC3, < >>Representing fluctuations caused by shot noise between SC1 and SC 2. For noise combination>Sequence (p) _{i} 、q _{i} 、r _{i} The Power Spectral Densities (PSDs) of the test mass acceleration noise and shot noise according to the formula (3) matrix onetoone correspondence given below can be expressed as
Wherein N is _{pf} Test quality noise term for Michelson X combination, N _{shot} Shot noise term for Michelson X combination, S _{pf} To test the mass acceleration noise power spectral density, S _{shot} The power spectrum density of shot noise, f is the detection frequency, s _{a} Sum s _{x} The Amplitude Spectral Densities (ASDs) of the test mass acceleration noise and shot noise, respectively.
For gravitational wave detection, different combinations of data may result in different link configurations. The more links, the more experimental data is utilized, and the better the sensitivity. However, to derive an analytical expression for the sensitivity function, we typically start with a fourlink michelson data combination, as the computation is not so complex. For michelson data combining, by assuming E _{ij} ＝e ^{iΩL/c} (where Ω is 2pi f, i is imaginary unit above e), and all arm lengths are L, the calculation of the frequency domain can be further simplified.
As above, i.e. michelson Xcombination, the Power Spectral Density (PSD) of the total noise of michelson combination can ultimately be given, since such data combination can cancel the noise of the laser phase and the optical bench.
And f is the gravitational wave frequency. The superscript X represents the michelson X combination and N represents the noise term.
The detection sensitivity curve is defined as:
where α is a coefficient related to configuration and observation time, f is the gravitational wave frequency, and R is the transfer function.
Based on the discussion above, we tried to further calculate the sensitivity function and apply it to a typical space probe as follows: LISA.
The technical implementation flow chart of the embodiment of the invention is shown in fig. 2, and specifically comprises the following steps:
the frequency interval of 0.00001 to 1 takes frequency points with 0.001HZ as interval and stores the frequency points in a frequency list.
And calculating corresponding transfer function values at each frequency, and storing the values into a transfer function list.
TDI was added at each frequency according to the formula: calculating N according to formulas 1,2,3 _{pf} And N _{sh} ot。
And (4) calculating corresponding sensitivity values according to the formula (4), and storing the corresponding sensitivity values into a sensitivity curve list.
And drawing a sensitivity curve by taking the frequency as an abscissa and the sensitivity value at each frequency point as an ordinate.
In this embodiment, an example uses the transfer function as:
wherein u is 2 pi fL/c, c is the light speed, L is the arm length, ci is the cosine integral function, si is the sine integral function, and the included angle between two arms is pi/3.
An example use sensitivity function is:
where 5 is the signal to noise ratio for a 1 year observation time. It should be specifically noted that the sensitivity function used herein is only one example, and is used as a basis for calculating the data shown in fig. 35 according to the embodiment. The detection sensitivity function is different in different cases.
The solid line below fig. 3 is the sensitivity curve for an arm length of 250 ten thousand kilometers.
The solid line below fig. 4 is the sensitivity curve for an arm length of 300 tens of thousands of kilometers.
The solid line below fig. 5 is the sensitivity curve for an arm length of 500 tens of thousands of kilometers.
Some parameters for plotting the sensitivity curve are calculated as follows: speed of light 300000000m/s; the laser power is 2W; telescope diameter 0.4m; laser wavelength 1.064×10 ^{6} m; test mass acceleration noise 3×10 ^{15} The method comprises the steps of carrying out a first treatment on the surface of the Shot noise 20×10 ^{} 12; the observation time was one year, 31536000 seconds; arm lengths were plotted at 250 tens of thousands of meters, 300 tens of thousands of meters and 500 tens of thousands of meters.
Wherein J0651+2844 and J0934+4411 are doublewhite dwarf systems. The wave sources are already marked in the figure.
As can be seen from the above detailed description of the invention, the invention realizes a link combination mode based on the first generation time delay interference method, calculates a space gravitational wave detection sensitivity curve, and provides support for the performance analysis of space gravitational wave detection tasks.
Finally, it should be noted that the above embodiments are only for illustrating the technical solution of the present invention and are not limiting. Although the present invention has been described in detail with reference to the embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made thereto without departing from the spirit and scope of the present invention, which is intended to be covered by the appended claims.
Claims (7)
1. The method is used for obtaining a detection sensitivity curve of a space gravitational wave detector, and the geometrical shape of the space gravitational wave detector is as follows: forming a triangle by taking the first spacecraft, the second spacecraft and the third spacecraft as vertexes; each side of the triangle is an arm, each spacecraft is provided with two test quality and two lasers, and the three spacecrafts are connected by laser beams emitted by the lasers; the method comprises the following steps:
s1, frequency points are taken at certain intervals in a certain frequency interval and stored;
s2, defining a transfer function, and calculating to obtain transfer function values at all frequency points;
s3, calculating to obtain a test quality noise item and a shot noise item of Michelson X combination at each frequency point by using a time delay interferometry technology;
s4, defining a sensitivity function, calculating the sensitivity value at each frequency point by using the values calculated in the steps S2 and S3, and storing the sensitivity value;
s5, drawing a sensitivity curve by taking each frequency point as an abscissa and the sensitivity value at each frequency point as an ordinate;
in the step S4, the expression of the sensitivity function S (f) is:
wherein, alpha is a coefficient related to configuration and observation time, R (f) represents a transfer function taking gravitational wave frequency f as an independent variable;
the power spectral density of the total noise representing the michelson X combination is expressed as:
2. the method for calculating the spatial gravitational wave detection sensitivity curve based on time delay according to claim 1, wherein the triangle formed by using the first spacecraft, the second spacecraft and the third spacecraft as vertexes has three equal arm lengths.
3. The method for calculating the spatial gravitational wave detection sensitivity curve based on time delay according to claim 2, wherein in said step S2, the expression of the transfer function R (u) is:
where the variable u=2pi fL/c, c is the speed of light, f is the gravitational wave frequency, L is the arm length, ci (·) is the cosine integral function, si (·) is the sine integral function.
4. The method for calculating a spatial gravitational wave detection sensitivity curve based on time delay as claimed in claim 3, wherein in said step S3, the expression of michelson X combination is:
wherein E is _{13} A time delay operator representing the first spacecraft to the third spacecraft; e (E) _{31} Representing a third spacecraft to first spacecraft time delay operator; e (E) _{12} A time delay operator representing a first spacecraft to a second spacecraft; e (E) _{21} Representing a time delay operator of the second spacecraft to the first spacecraft.
5. The method of calculating a spatial gravitational wave detection sensitivity curve based on time delay as claimed in claim 4, wherein in said step S3, a test quality noise term N of Michelson X combinations at each frequency point _{pf} The calculation formula of (2) is as follows:
wherein i represents a summation variable; p is p _{i} Entries representing the first row and the ith column in the michelson Xcombination matrix; q _{i} Entries representing the second row, ith column, in the michelson Xcombination matrix; r is (r) _{i} Entries representing the third row, ith column, in the michelson Xcombination matrix;
S _{pf} the power spectrum density of the test mass acceleration noise is expressed as follows:
wherein s is _{a} Representing the test mass acceleration noise amplitude spectral density.
6. The method of calculating a spatial gravitational wave detection sensitivity curve based on time delay as claimed in claim 5, wherein in said step S3, a shot noise term N of Michelson X combination at each frequency point _{shot} The calculation formula of (2) is as follows:
wherein S is _{shot} The power spectral density of shot noise is expressed as:
wherein s is _{x} Representing the amplitude spectral density of shot noise.
7. The method for calculating a spatial gravitational wave detection sensitivity curve based on time delay according to claim 1, wherein the expression of the sensitivity function S (f) is specifically:
where T represents the observation time and 5 is the signal to noise ratio of the 1 year observation time.
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