Dark matter vs. modified gravity theory, has the ultimate showdown begun?

Dark matter has always been a frontier topic in physics research. This concept, which was born out of abnormal astronomical observations, can explain many strange phenomena. However, we have not yet found any trace of it, so some scholars believe that dark matter does not exist. For many astronomical phenomena, it is only necessary to modify Newton's theory of gravity (MOND theory) under weak gravitational fields to explain the observation results. In terms of testing gravitational theories, astronomers have found through observations of wide-spaced binary stars that MOND and Newton's gravity are not superior, and even give opposite conclusions for the same type of samples. This article will try to explain the reason for this phenomenon.

Written by Tian Haijun (Hangzhou Dianzi University)

An Overview of Gravitational Theory

Isaac Newton first proposed the theory of universal gravitation in his book "Mathematical Principles of Natural Philosophy" published in 1687. This theory is a major breakthrough in the history of mechanics and has played an extremely important role in the process of human cognition of the laws of cosmic motion. In 1844 and 1846, British mathematician John Couch Adams and French mathematician Urbain Le Verrier respectively used this theory and combined it with the abnormal orbital motion of Uranus to calculate the existence of Neptune, the eighth planet in the solar system, and accurately predicted its position. This result was later confirmed by the observation of German astronomer Johann Gottfried Galle of the Berlin Observatory, making Newton's theory of universal gravitation famous in the world. In 1859, Le Verrier discovered that the trajectory of Mercury also deviated from the prediction of Newton's gravitational theory, and Mercury showed strange orbital precession characteristics at perihelion. Therefore, he believed that there was an unknown planet inside the orbit of Mercury that affected the orbit of Mercury. However, until Le Verrier died in 1877, people had not found this unknown planet. In fact, such a planet does not exist. At the perihelion, the strong gravitational field makes Newton's theory of gravity show inaccurate defects. It was not until 1915 that Einstein proposed the general theory of relativity, which almost perfectly explained the problem of Mercury's perihelion precession.

General relativity provides a self-consistent and rigorous theoretical description of spacetime and gravity. The predictions based on general relativity (such as gravitational lensing, gravitational redshift, black holes, gravitational waves, etc.) have all been verified in subsequent observations or experiments, which in turn proves the correctness of general relativity. Therefore, general relativity is considered to be the cornerstone of modern physics theory. Newtonian mechanics is a first-order approximation of general relativity. Under high speed (close to the speed of light) or strong gravitational field (large spacetime curvature), the movement of objects will show significant relativistic effects, and Newtonian mechanics is no longer valid. Under low-speed and weak field conditions, there is no significant difference between Newtonian mechanics and general relativity.

As observation capabilities continue to improve, scientists have discovered some difficult-to-understand observational phenomena, such as the “flat rotation curve” of galaxies[1]. Whether in the framework of Newton’s or Einstein’s gravitational theory, these phenomena will lead to the problem of missing gravity, that is, the gravity generated by observable ordinary matter cannot restrain the high-speed rotation of matter (stars or gas) far away from the center of the galaxy.

As early as the 1930s, Swiss astronomer Fritz Zwicky proposed the concept of dark matter to make up for the lack of gravity. That is, there are some non-luminous and non-observable substances in space. Although these substances do not participate in electromagnetic interactions, they have mass and can generate gravity. Current research shows that dark matter accounts for about 25% of the total density of the universe, while ordinary matter, which we are familiar with, accounts for only about 4.7%. In other words, dark matter is everywhere around us. Although many detectors have been deployed internationally (for example, the Alpha Magnetic Spectrometer (AMS) led by Nobel Prize winner Professor Ting Zhaozhong and my country's dark matter particle detection satellite "Wukong") to study dark matter, we still know very little about this substance except for its gravitational effect. Therefore, dark matter is often considered to be one of the two "dark clouds" currently hanging over human heads (the other is "dark energy". For the concept of dark energy, please refer to the article by Researcher Gong Yan [2]; Editor's note: See "Dark Energy: A Ghost in the Universe?").

On the other hand, some international scholars believe that, similar to the "ether" that people tried hard to find in the 19th century, dark matter does not actually exist, but the Newtonian theory of gravity that we recognize needs to be modified in some cases. The representative of this school of thought is Israeli physicist Mordehai Milgrom, who proposed modifying Newton's second law in 1983. Later, this theory was called modified Newtonian dynamics (MOD[3]). The MOND theory believes that Newtonian gravity needs to be modified in extremely weak gravitational fields. In this theory, the acceleration g is expressed as follows:

According to the above formula, the MOND radius of a star with the mass of the sun is about 7000 AU (1AU is 1 astronomical unit, which is the average distance between the sun and the earth). That is to say, beyond 7000 AU from the sun, the gravitational force of the sun on celestial bodies no longer obeys Newton's gravitational theory, and the MOND theory needs to be used to correct it.

At present, although the dark matter theory is in the mainstream, the MOND theory is better at explaining some observational phenomena on the galaxy scale, such as the Tully-Fisher relationship[5], which makes the MOND theory and the dark matter theory a pair of competing scientific theories (for details, please see the article by Researcher Chen Xuelei[6]; Editor's note: see "A Pair of Competing Scientific Theories: Dark Matter and Modified Gravitational Theory"). In extremely weak gravitational fields, a series of questions such as whether Newton's theory of gravity needs to be revised and whether the MOND theory is correct are currently frontier issues of great concern internationally.

Two wide-spaced binary stars

There are many celestial systems in the universe that are suitable for testing gravitational theories, such as wide-space binaries. Wide-space binaries are the simplest, smallest, and most fragile gravitationally bound systems, and they are ubiquitous in the universe (as shown in Figure 1). Because the member stars are far apart (up to 100,000-200,000 AU), the gravitational interactions between the member stars are extremely weak. Therefore, wide-space binaries are considered to be powerful probes for testing gravitational theories on a small scale.

Figure 1. A pair of wide binary stars about 150 light years away from us and about 8824 AU apart. Image source: Sloan Digital Sky Survey (SDSS)

However, due to some observational effects, the process of using wide-space binaries to test gravitational theories is relatively complicated. Many factors can lead to strong uncertainty in the results, or even completely opposite conclusions. These observational effects mainly include:

1. Projection effect of wide-spaced binary stars

In observation, it is difficult to measure the angle between the line connecting the two sub-stars and the line of sight, which makes it difficult for us to obtain the physical distance between the two sub-stars. We can only obtain the projected distance of the two sub-stars in the direction perpendicular to the line of sight. It can be assumed that the angle between the binary star and the observer's line of sight is randomly and uniformly distributed in the range of 0 to 360 degrees. Therefore, for large sample statistics, there is a simple linear relationship between the projected distance (sp) of wide-spaced binary stars and their semi-major axis (a, half of the physical distance) [8], that is,

(4)

Based on this linear relationship, we can directly use the projected distance instead of the physical distance of the binary star to carry out relevant scientific research, such as the mutual influence between member stars in the evolution of a binary star[9], and the examination of dark matter in the galactic halo[10].

However, when using wide-spaced binaries to test gravitational theories, significant statistical deviations will occur if the projection distance is used, especially when the binaries have no radial velocity and the projection distance is large. In 2019, El-Badry Kareem[11], who was then pursuing a doctorate at the University of California, Berkeley, simulated the impact of the projection effect of wide-spaced binaries on the relative velocity (∆V) of the two sub-stars through a computer, as shown in Figure 2 (left). If the binaries have no radial velocity and only two-dimensional proper motion (the other two motion components perpendicular to the radial velocity), when the projection distance of the binaries is greater than 0.1 pc (pc is another commonly used distance unit in astronomy, 1 pc is about 3.26 light years), ∆V (the black curve is the theoretical calculated value, the blue is the actual measured value) affected by the projection effect begins to deviate seriously from the true value (red dotted line).

Figure 2. Projection effect of a wide-space binary star[10]

How to crack this projection effect? ​​There are usually two ways:

(1) Provide accurate three-dimensional velocities for the binary star, and preferably provide accurate radial velocities for both sub-stars at the same time. Convert the radial velocities and proper motions of the two sub-stars into three-dimensional velocities in a Cartesian coordinate system, and calculate the velocity differences of the three components to calculate the relative velocity of the two sub-stars. This method can eliminate the influence of the projection effect, as shown in Figure 2 (right). However, the observation requires an accuracy higher than 0.2 km/s, which is difficult to achieve for low-resolution spectral surveys.

(2) Assuming that the orbit of the binary star satisfies an elliptical orbit, the angle between the binary star and the line of sight is randomly distributed, and given the statistical distribution of the eccentricity of the elliptical orbit (this distribution is given in the relevant literature [12]), the two-dimensional projection spacing and projection velocity can be restored to three-dimensional space through the Monte Carlo random point method, thereby eliminating the projection effect.

2. Perturbations of an unresolved companion in a binary system

Three-body and multi-body systems are very common in the universe. For example, the nearest star to us (only 4.244 light years away) - Proxima Centauri is in a triple star system called Alpha Centauri (the "solar system" where the Trisolarians in the famous science fiction novel "The Three-Body Problem" live), in which the distance between substars A (1.09 M⊙) and B (0.9 M⊙) is only 11.2 AU at the closest, and the angular separation is only about 4 arc seconds; the distance between substar C (i.e. Proxima Centauri, with a mass of only 1/10 of the sun) and the AB binary is about 15,000 AU; the AB binary can be clearly distinguished using a space telescope, as shown in Figure 3. However, for wide-distance binaries that are very far away from us, it is not easy to figure out whether they hide companion stars that are difficult to observe.

Figure 3. Alpha Centauri triple system. Proxima appears darker due to its small mass, while the other two brighter companion stars (AB) form a close binary due to their proximity. Image source: Jan Hattenbach (wide-angle shot), Jared Males (Inset)[13]

If there is an indistinguishable companion star in the wide-space binary sample, there will be two effects that interfere with the observation of wide-space binaries: one is the increase in the total mass of the system; the other is the "recoil" velocity, which increases the velocity difference between the two component stars. Both of these effects will have a serious impact on the test of gravitational theory.

The most direct way to eliminate these effects is to select pure samples of wide-space binary stars and strictly control the inclusion of triple or multiple star systems. This requires that the observation accuracy of each sub-star in the binary must be high enough in terms of physical quantities such as distance, proper motion, and radial velocity. This often greatly reduces the number of samples and ultimately affects the statistical nature of the results. Another way is to allow the inclusion of triple or multiple star systems when selecting binary star samples, and consider factors such as the inclusion ratio, mass ratio distribution, and semi-major axis distribution when mathematically modeling. The relevant parameters can be obtained through multi-parameter fitting [14]. The advantage of this approach is that the statistical samples are relatively sufficient, but the modeling process is complicated and the results are easily affected by many factors.

In addition to the two observational effects mentioned above, there are also factors such as the extinction effect of the interstellar medium and the high contamination rate of large-distance false binary stars, which may affect the effectiveness of wide-distance binary star tests of gravitational theories. Because of this, there is a lot of "battle space" left for Newtonian gravitational theory and MOND, and it is not surprising that they come to completely different conclusions.

Three: Right and Wrong

In the Alpha Centauri triple system mentioned above, the distance between Proxima Centauri and the AB binary star is far enough that the gravitational force exerted on it by the AB system is extremely weak. Moreover, since they are the closest star systems to us, their physical parameters such as three-dimensional velocity, spatial position, and mass are relatively easy to measure. Therefore, as early as 15 years ago, scientists tried to test MOND and Newton's theory of gravity through the orbital motion of Proxima Centauri[15, 16]. However, because the AB binary star would complicate the orbital motion of Proxima Centauri and require extremely high astronomical measurement accuracy (such as 0.5 microarcsecond accuracy[17], far exceeding the current human observation capability), they ultimately failed to give clear test results. Subsequently, scientists also tried to use the wide-distance binary star samples of the most successful sky survey projects at the time, such as the Hipparcos satellite and the Sloan Digital Sky Survey (SDSS)[18, 19], to test the gravitational theory, but due to problems such as insufficient sample size and measurement accuracy, only some signs of MOND signals were found in extremely weak gravitational fields[20].

On December 19, 2013, the European Space Agency developed and launched a space telescope, the Gaia satellite probe (the successor to the Hipparcos satellite). The main goal of this probe is to conduct multiple observations of more than one billion stars in the Milky Way with unprecedented accuracy, measuring their positions, distances, motions, and other information. After nearly a decade of observations, the Gaia satellite has observed and published astrometric parameters for nearly 1.6 billion stars, of which the radial velocities of more than 33 million stars have been measured. For brighter celestial bodies, the astrometric accuracy can reach 0.02 milliarcseconds (for stars brighter than 15 magnitudes), and the radial velocity measurement accuracy can reach 0.3 km/s (for stars brighter than 8 magnitudes). Although these parameters are still somewhat different from our expected measurement accuracy (0.5 microarcseconds), we can select a sample of millions of wide-space binary stars from the massive Gaia catalog data [10, 21]. Such a rich sample data enables us to obtain statistically significant results when conducting gravitational theory tests.

Figure 4. The team led by Professor Hongsheng Zhao of the University of St. Andrews in the UK further refined the data used by Professor Kyu-Hyun Chae of Sejong University in South Korea and obtained the results using their own statistical methods. In the image, the ordinate is defined by formula (3), the abscissa rM is defined by formula (2), and rsky is sp in this paper. It is obvious that the observed data curve (solid lines of different colors) does not conform to the prediction of MOND theory (dashed line) when the spacing is greater than rM.

In the past year of 2023, several international research teams released the results of testing the gravitational theory using the latest Gaia wide-space binary star samples. The more representative teams are from the University of St Andrews in the UK[14], Sejong University in South Korea[22], the National Autonomous University of Mexico[23] and Queen Mary University of London in the UK[24]. These four research groups selected different numbers of wide-space binary star samples in the solar neighborhood from the Gaia star catalog and independently carried out testing studies on the gravitational theory, and finally came to very clear conclusions.

Surprisingly, the conclusions of these four research groups are inconsistent. Two teams from the UK [14, 24] believe that under extremely weak gravitational fields, the orbital motion of wide-separated binaries does not show anomalies, but is well consistent with Newtonian gravitational theory. In particular, the team led by Professor Zhao Hongsheng of the University of St. Andrews finally ruled out the MOND theory with a strong statistical confidence level (16σ) [14], as shown in Figure 4. The other two teams also claimed with a high statistical confidence level (10σ) [22] that the orbital motion of wide-separated binaries under extremely weak gravitational fields (i.e., when the distance is greater than 2000 AU) shows significant gravitational anomalies, and its characteristics are more consistent with the predictions of MOND theory, which means that Newtonian gravitational theory needs to be revised under extremely weak gravitational fields. Such completely opposite conclusions made this topic a hot topic internationally and attracted widespread media attention [25-27].

The inconsistency in the above conclusions is mainly due to the fact that the samples used by each team are contaminated to varying degrees. To eliminate these contaminations, complex statistical methods are required, which are themselves susceptible to interference from a variety of factors. When screening wide-space binaries, different teams need to consider both the number and quality of samples. An ideal sample requires sufficient quantity and high observation quality (including radial velocity), and must pass strict conditions to limit the contamination of unresolved companions or false binaries. However, in the current observational data, there are relatively few samples that meet the high-quality requirements. As shown in Table 1, Dr. Hernandez of the National Autonomous University of Mexico [23] set extremely strict sample screening conditions and ultimately obtained only 436 pairs of wide-space binary samples, of which only 87 pairs had a projected separation greater than 2000 AU. Too few samples will inevitably affect the statistical performance of the results.

The other three teams appropriately relaxed the conditions during the sample screening process to ensure that they obtained sufficient samples, but the samples were all contaminated to varying degrees, especially with the presence of triple or multiple star systems. The indistinguishable companion stars in these contamination sources will complicate the orbital motion of the binary star and affect the mass of the binary star. According to formula (3), these two physical quantities are the most important parameters for binary star tests of gravitational theory. In order to eliminate the influence of these unfavorable factors, it is usually necessary to build a very complex model. For example, the team at the University of St. Andrews built a model with seven free parameters to describe the proportion of triple or multiple star systems mixed in the wide-distance binary star sample, the distribution of binary orbital parameters, eccentricity, etc.

Table 1. Samples of wide-space binary stars selected by different teams in 2023

In order to simplify the statistical method and improve the reliability of the results, Professor Kyu-Hyun Chae[29] of Sejong University in South Korea formulated more stringent sample screening conditions based on his original sample[22]. For example, he limited the astronomical measurement accuracy of each star to be better than 0.005 milliarcseconds and the radial velocity of each sub-star to be higher than 0.2 km/s. In the end, he obtained a sample of 2463 pure wide-space binary stars, which was almost free of contamination from triple or multiple star systems and false binary stars. Therefore, there is no need to construct a complex statistical model to eliminate the influence of factors such as contamination rate and projection effect on the statistical results. The final results still show that when the projected spacing is less than 2000 AU, the orbital motion of the binary star satisfies Newton's gravitational theory very well; however, when it is greater than 2000 AU, the orbital motion of the binary star shows significant anomalies, and its anomaly characteristics are more consistent with the prediction of MOND theory, as shown in Figure 5. Even with the use of 40 pairs of pure wide-space binary star samples with extremely high observation accuracy, the conclusions finally obtained by the team remain unchanged. To this end, Dr. Hernandez and Dr. Chae worked together to write a review article specifically on the statistical methods and data processing process of the University of St. Andrews team [30], commenting on possible inappropriateness. For example, the University of St. Andrews team fitted the proportion of unresolved companion stars in the sample to be as high as nearly 70%, which is significantly higher than the generally known proportion (i.e., 30% to 50% [31-33]). In addition, another team from the University of Portsmouth in the UK also came to a conclusion inconsistent with the MOND theory expectations by analyzing the orbits of celestial bodies in the solar system [34].

Four conclusions

Although with the help of Gaia's high-precision measurement of a large sample of wide-space binaries, we have a deeper understanding of Newtonian gravity and MOND theory, but which team's results are more in line with the actual situation, scientists may have to wait until the Gaia satellite releases more and more accurate data in 2025, or until the Chinese Space Station Engineering Survey Telescope (CSST) is launched and provides high-precision astronomical measurements of a large sample of wide-space binaries, before they can give a convincing conclusion. Whether Newtonian gravity theory needs to be revised in extremely weak gravitational fields, whether MOND theory is correct, whether dark matter exists, what is its nature, etc., this series of questions all involve the foundation of the "building" of modern physics, and are major issues at the international forefront. With the continuous improvement of observation technology and the continuous improvement of statistical methods, we firmly believe that with the unremitting efforts of scientists, the answers to these questions will eventually come to light.

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This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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