An important scientific question. In recent years, there has been an increasingly serious deviation between the local direct measurements of the Hubble constant and the fitted values of the global model, in which the local direct measurements are derived from the local distance ladder measurements of the late universe, and the fitting values of the global model are derived from the observation limitations of the microwave background radiation of the early universe on the cosmological Standard Model. If this deviation is not caused by observational and systematic errors in any of these means of observation, then it is likely to mean that there is a new physics beyond the standard model of cosmology. In this paper, the problem of the Hubble constant crisis is briefly described from the aspects of observation and model, and the prospect is made from the aspects of observation and model based on the author's recent research on this problem.
Written by | Ronggen Cai (School of Physical Science and Technology, Institute of Theoretical Physics, Chinese Academy of Sciences, School of Physical Science, University of Chinese Academy of Sciences, Hangzhou Advanced Research Institute, University of Chinese Academy of Sciences), Li Li (Institute of Theoretical Physics, Chinese Academy of Sciences, School of Physical Sciences, University of Chinese Academy of Sciences, Hangzhou Advanced Research Institute, University of Chinese Academy of Sciences), Shaojiang Wang (Institute of Theoretical Physics, Chinese Academy of Sciences)
01 Introduction
The Hubble constant was first discovered in 1929 by American astronomer Edwin Hubble in his estimation of the receding velocity of nearby galaxies. He boldly guessed that the receding velocity vr of the nearest neighbor galaxy is proportional to its luminosity distance DL, i.e., vr=H0DL, where the scale coefficient H0≡100h km/(s· MPC) was later called the Hubble constant, and h is a dimensionless constant. Hubble's discovery heralded the expansion of the universe. In fact, two years before Hubble's discovery, the Belgian astronomer Georges Lemaître had written about a similar idea. The Hubble-Lemaître law was the first observational evidence of the expansion of the universe, and it directly prompted Einstein to abandon his obsession with introducing cosmological constants to obtain a static universe. We now know that H0 is the ratio of the time derivative of the scale factor of the current universe expansion to that scale factor, which measures the magnitude of the current rate of expansion of the universe. However, due to the state of observation technology at that time, the initial estimate of H0 (H0=500km/(s⋅Mpc)) was very crude. After nearly 100 years of development, the measurement accuracy of H0 has reached 1%. Recently, however, there has been a difficult rift between the values of the Hubble constant, measured by different measurements. The most prominent conflict arises from two methods of measurement in the early and late universes: the observational limitations obtained by the global fitting of the cosmological Standard Model from the cosmic microwave background radiation data from the photon decoupling from the early universe recombination period from the last scattering surface to the present, and the direct measurement of type Ia supernovae on the calibrated Hubble flow of Cepheid variables by local distance ladder ranging. For the former, the global fitting value of the Hubble constant in 2018 was H0=(67.27±0.60) km/(s⋅Mpc)[1]. For the latter, the SH0ES (supernova H0 for the equation of state) consortium led by Nobel Laureate Adam Reiss directly measured the Hubble constant in 2022 as H0=(73.04±1.04) km/(s⋅Mpc)[2]. It is easy to see a deviation of up to nearly 5 standard deviations between them (see Figure 1). If this deviation cannot be explained by the observational and/or systematic errors of the two measurements, then it undoubtedly poses a serious challenge to the current Standard Model of cosmology. This is the Hubble constant crisis [3-14], and its solution may require new physics beyond the current Standard Model of cosmology.
Fig. 1 Hubble constant crisis: up to nearly 5σ deviation between the H0 limit (blue) from CMB-Planck+ΛCDM and the H0 measurement (green) from the SH0ES co-group distance ladder SNe+Cepheid. Image from Ref. [2]
The paper is organized as follows: Section 2 presents the various observations, of which Section 2.1 presents observations from the early universe (including measurements related and unrelated to cosmic microwave background radiation) and Section 2.2 presents observations from the late universe (including measurements related and unrelated to local distance step ranging); Section 3 introduces various model constructions, of which Section 3.1 introduces the model construction of the early universe (including modified recombination history and early expansion history), and Section 3.2 introduces the model construction of the late universe (including uniform and non-uniform modifications to the late universe); Section 4 looks forward to the problem of the Hubble constant based on the author's recent research work, in which Section 4.1 looks forward from an observational perspective (including local and non-local cosmological variances), and Section 4.2 looks forward from a theoretical perspective (including a discussion of chameleon dark energy and scale-dependent dark energy). Section 5 summarizes the full text.
02 Observation
The Hubble constant crisis is not only observed by the Planck and SH0ES collaborations as far as the deviation of the Hubble constant by nearly 5σ, but also by the fact that the Hubble constant measured directly by the late universe is systematically lower than the Hubble constant by the global fit of the early universe (see Fig. 2).
Fig. 2 The Hubble constant crisis: indirect fitting from the early universe and direct measurements from the late universe. Image from Ref. [8]
2.1 Early space
Although observations of the early universe were made in the current late universe, the data obtained actually reflect information about the early universe. However, this information cannot be directly used to measure the Hubble expansion rate of the current universe (i.e., the Hubble constant), so it is necessary to map the information extension of the early universe to the late universe with the help of specific cosmological models. The resulting Hubble constant value is often referred to as the global fit value for the model parameters given data.
2.1.1 Cosmic microwave background radiation
In the history that has been confirmed by observations under the framework of the Standard Model of particle physics combined with the Standard Model of cosmology, the annihilation of positrons and positrons began soon after the birth of the early universe 1 s, that is, neutrinos began to decouple, and 3 minutes later, as the universe gradually cooled, the synthesis of light elements began, which is the primordial nucleosynthesis (BBN). After 60,000 years, the amount of radiation and matter in the universe is roughly the same, and then the reverse process of electrons and protons synthesizing hydrogen and emitting photons begins to be difficult to compensate for the loss of electrons in the positive process, resulting in the Thomson scattering process of electrons and photons that is difficult to maintain when the age of the universe reaches 380,000 years, resulting in the decoupling of photons from the background plasma fluid and the beginning of free streaming to form the final scattering surface, which is the cosmic microwave background radiation (CMBR). microwave background radiation)。 After that, these CMB photons pass through the gravitational potential well of the material structure, and finally, a portion of the CMB photons reach the vicinity of the Earth and are observed by us.
As well as their joint limits, they can be used to measure the basic physical image of the Hubble constant.
Fig. 3 Comparison of the limits given by the combination of BBN with galaxies BAO (blue) and Lyman-αBAO (green) with the limits of Planck 2018 (purple) and the SH0ES group (orange). Image from Ref. [18]
BAO observations come from large-scale structural galaxies surveys, which record data such as the redshift of galaxies (converted into distances by a given reference model) and azimuth and other photometric or spectral data to determine the position of individual galaxies under a given reference model. The position arrangement of galaxies is not completely random, because the original perturbation induces a density perturbation of the baryon-photonic fluid after entering the event horizon, and the density perturbation propagates outward at the speed of sound at various points in space, but when the photons are decoupled from the baryon-photonic fluid, the original baryon matter part cannot maintain the continued propagation of the acoustic oscillation, so the density perturbation is frozen, and its co-dynamic scale is about rs≈150Mpc. Subsequently, the baryonic matter falls into the gravitational potential well formed by the dark matter to form galaxies, and the two-point correlation function of the positions of these galaxies is locally exceeded at rs compared to the case of a completely random distribution. Thus, while the BAO data comes from observations of the distribution of galaxies in the late universe, the information it records comes directly from the acoustic horizon information left by the early universe on the final scattering surface (or, more precisely, the baryon drag period). However, BAO does not measure this acoustic horizon directly, but rather its deviation from the following combinations predicted by the benchmark model in the parallel and perpendicular gaze directions, respectively:
2.2 Late Universe
Unlike the aforementioned observations of the early universe, observations of the late universe appear to be a direct measure of the Hubble expansion rate of the current universe, i.e., the Hubble constant itself. However, due to the nonlinear evolution of the material perturbation growth in the late universe, the measurement of the local universe itself will be affected by many systematic errors, so it is difficult to extract the contribution from the global background expansion part of the late local universe.
2.2.1 Distance ladder ranging
The key to measuring the Hubble constant in the late local universe is to measure the distance-redshift relationship by ranging:
E(z)≡H(z)/H0 depend on the specific cosmological model parameter inputs. However, the scope of application of different ranging methods is different, so it is necessary to link different ranging methods to form a distance ladder. The lowest level of the distance ladder is to use some geometric ranging methods (such as trigonometric parallax, vein, non-eclipse binary method, etc.) to calibrate the luminosity distance indicators at some medium distances (such as Cepheid variables, apex of red giant branches, surface brightness fluctuations, Miras, etc.); On the second level of the distance ladder, these medium-range luminous distance indicators can be used as calibrators to further calibrate some of the more distant luminous distance indicators (e.g., type Ia supernovae, type II supernovae, HII galaxies, etc.); Eventually, on the third stage of the distance ladder, these calibrated long-range photometric distance indicators can be used to measure the Hubble constant on the local cosmic Hubble flow, with the best observed long-range photometric distance indicator being the type Ia supernova that acts as standard candlelight. A schematic diagram of the three-stage distance ladder used by the SH0ES cooperative group is shown in Figure 4.
Fig.4 Three-level distance ladder adopted by the SH0ES cooperative group. Image from Ref. [2]
Type Ia supernovae can be used as a standard candle because it comes from the remnants of carbon-oxygen white dwarfs in binary systems that accretion of material from their companion stars (e.g., main-sequence stars, subgiants, red giants, or helium stars) until they reach the Chandrasekhar limit (1.44 solar masses), thus reaching the ignition temperature to restart carbon fusion, which in turn induces the remnants of the explosion of white dwarfs, so that their light curves are almost equal in absolute luminosity (denoted M) at their maximum. According to the definition of the distance modulus μ≡m−M, the fluorosis of a type Ia supernova is
where ⟨c⟩ is the purely numerical part of the speed of light in km/s, and ⟨H0⟩≡100h is the Hubble constant H0 in km/(s· MPC). Thus, once the absolute luminosity of a type Ia supernova is determined through the first and second order distance ladders
In fact, as long as the absolute luminosity of type Ia supernovae is around M=−19.2 as measured by the SH0ES group, the Hubble constant measured by most late universe models is around 73-74 km/(s⋅Mpc) [27]. Therefore, the real difference in the value of the Hubble constant in late local measurements is the calibration of the absolute luminosity of type Ia supernovae, such as the Hubble constant measured by calibrators such as Cepheid variables/apex of red giants/surface brightness fluctuations/Miras
H0=(73.04±1.04) km/(s⋅Mpc)[2]，
H0=(69.8±1.7) km/(s⋅Mpc)[28]，
H0=(70.50±4.13) km/(s⋅Mpc)[29]，
H0=(73.3.±4.0) km/(s⋅Mpc)[30]。
正因如此，哈勃冲突(H0 tension)有时也被称为MB tension。
2.2.2 Distance-independent ranging
Since the construction of the multi-level distance ladder requires the calibration and calibration of two luminosity distance indicators of the adjacent two-level distance ladder on the anchored galaxy, it inevitably leads to the transmission of calibration errors in each distance ladder, thus introducing considerable observational and systematic errors in the last level of the distance ladder. Although the SH0ES experimental group has made extraordinary efforts to reduce the sum of the calibration errors of the various levels of the distance ladder to less than 1% over the past two decades, if there is a way to avoid the construction of the distance ladder and realize long-distance direct ranging, then the observation and systematic errors of the Hubble constant in late local cosmic measurements, such as veins, surface brightness fluctuations, Tully-Fisher relations, strong gravitational lensing time delays, and gravitational wave standard whistles, will be significantly reduced.
The strong gravitational lensing time delay is measured by measuring the time difference between the different lenses of the strong gravitational lensing system reaching us. Usually the lens source (background object) of a strong gravitational lensing system is a quasar or even a supernova, while the lens (foreground object) is a galaxy. When light from a lens source passes through a lens, due to the light deflection effect of the gravitational potential well, when these rays reach us are traced, multiple images corresponding to that lens source are found. In most cases, these multiimages exhibit an asymmetrical arrangement, so that different optical paths take different distances (called geometric time delays), and the general relativity effect also introduces Shapiro time delays due to changes in the equivalent propagation velocity of light.
Fig.5 Comparison of the Hubble constant by the time delay of a strong gravitational lens independent of the distance ladder, image from Ref. [18]
The gravitational wave standard whistle utilizes the gravitational wave waveform radiated during the spiral phase of a dense binary star system
Fig. 6 Dark whistle restriction of the Hubble constant from the networking of LISA and Taiji space gravitational wave detectors. Image from Ref. [40]
03 Models
Although observations from the early and late universes have different limits on the Hubble constant, there is a tendency to ignore that the direct measurements of the Hubble constant in the late universe are systematically higher than the global fit from the early universe. It is difficult to imagine that there is some common observational and systematic error that leads to such systematic deviations, as the observational and systematic errors of various means of observation are not the same. This possibility will be discussed in Section 4.1, which will assume that this systematic deviation comes from some new physical model.
The simplest construction of the new physical model comes from simple and straightforward extensions of the Standard Model of cosmology, such as the introduction of tiny spatial curvature, the introduction of CPL (Chevallier-Polarski-Lin) parameterized kinetic dark energy in the late universe, the introduction of tiny new neutrino-like relativistic degrees of freedom before BBN, and the permutations and combinations of the above extensions. However, numerous studies (e.g., Ref. [41,42]) have shown that a simple extension of the Standard Model of cosmology only increases the uncertainty of the model parameters, but is not enough to completely solve the Hubble constant crisis.
Therefore, it is necessary to make some highly specific modifications to the Standard Model of cosmology, such as introducing new energy density components, new forms of interaction, new modifications of gravitational effects, etc., and even attempts to modify the evolution of fundamental physical constants over time, or even to shake the basic principles of cosmology. Since the Hubble constant crisis can be roughly thought of as a contradiction between current observations of the early and late universes, its model construction can also be roughly divided into modifications of the evolution of the early universe and modifications of the evolution of the late universe.
3.1 Early space
Modifications to the early universe need to at least conform to the limits of the existing CMB and BAO, which essentially measure the Lord
There are two ways to do this: one is to reduce the time it takes for the sound wave to propagate, and the other is to directly reduce the speed of sound itself. Reducing the propagation time of sound waves can reduce the acoustic horizon by modifying the rerecombination history in the process of photon decoupling, thereby advancing the recombination period. Reducing the speed of sound can modify the relative magnitude of the radiation and (baryon) matter in the gravity ratio by modifying the expansion history before the photon decouples.
3.1.1 Modify recombination history
Modifying the recombination history can be achieved by adding the original magnetic field [43] or by allowing a non-standard recombination history [44]. Taking primordial magnetic fields as an example, current astronomical and cosmological observations (e.g., galaxies, galaxy clusters, and holes) often encounter magnetic environments [45], and its origin is still a mystery, but it is generally believed that it may have arisen from the early universe (e.g., weak phase transitions or inflation). Such a primordial magnetic field induces small-scale non-uniformity, forcing the baryon to move along the magnetic field to a region with a lower energy density of the magnetic field, thereby accelerating the recombination process, thereby reducing the acoustic horizon and raising the Hubble constant. However, there is no evidence of baryon clumping at small scales from the CMB data [46,47], so the primordial magnetic field scheme does not solve the Hubble constant problem. Similarly, CMB data do not support non-standard recombination histories [48], unless a specific new physics alters the evolution of atomic physical constants (e.g., hydrogen atomic ionization energy) or fundamental physical constants (e.g., electron mass) in the early universe [49,50].
3.1.2 Modify the early bloat history
Modifying the early expansion history can be achieved by injecting new energy components into the early universe, such as dark radiation and early dark energy. Let's start with dark radiation: since the BBN has strongly limited the relativistic number of effective degrees of freedom of the neutrinos before the BBN, dark radiation must be introduced after the BBN so as not to break the previous limit on the number of effective degrees of freedom of the BBN. If the introduced dark radiation is a photon-like radiant component that can be free streaming, then it will wash away the small-scale radiative perturbations, thus changing the Silk damping scale of the CMB power spectrum at the small scale. In fact, it is not possible to introduce free-flowing dark radiation while maintaining both the acoustic peak and the Silk attenuation scale. Therefore, only neutrinos with strong self-interaction can be introduced, such as neutrinos with strong self-interaction, but this will result in the CMB polarization characteristics not matching the CMB data [52].
Early dark energy is essentially a type of dark radiation: the simplest example is the axon field [53]. The shape of the axion potential function is adjusted so that the axion mass is much smaller than the Hubble parameter at that time, so that the axion field will be subjected to the background Hubble resistance, causing it to freeze at a certain position in the potential function for most of the time before the CMB, which is the early dark energy as an effective cosmological constant. As the universe expands, when the Hubble parameter drops to a mass comparable to that of the axle, the axle rolls off its potential function and begins to oscillate and decay. By selecting a suitable shape of the potential function, the energy density decay rate corresponding to the oscillation can be equal to or even faster than the radiative decay rate [54], which in turn allows us to set a larger initial value of early dark energy from the beginning, thereby significantly changing the early expansion history. Analysis of the model's data shows that early dark energy needs to reach about 5% of the total energy at that time slightly earlier than the time when the radiating matter is equal, and then decay in a faster way than radiation.
However, such a simple model has three fatal problems: first, the fine-tuning problem, in order to make the early dark energy reach the ratio of 5% slightly earlier than the period when the radiant matter is equal, the initial value of the axial subfield must be carefully fine-tuned; Second, there is the problem of coincidence, the time when the accumulation and rapid decay of early dark energy must occur slightly earlier than the period when the radiating matter is equal; Third, the S8 problem, since the introduction of early dark energy inhibits the growth of early matter disturbances, the amount of matter must be increased at the same time to counteract this effect. However, the increased amount of matter will increase the material perturbation at the minimum linear scale (i.e., S8) in the late period, which is inconsistent with the limitation of the material perturbation on the late large-scale structural survey. In fact, the discussion of question 3 above also applies to almost all modified models of the early universe [12], which either do not correspond to the observed nature of galaxy clusters or to the limitations of weak gravitational lensing of galaxies (see Figure 7).
Fig.7 Limitations of the early universe model (right) by weak gravitational lensing of galaxies (top left), H0 measurements by the SH0ES group (middle left), and observations of baryon acoustic oscillations (bottom left). Image from Ref. [12]
3.2 Late Universe
Modifications to the late universe can be broadly divided into two categories: uniformity modification and non-uniform modification, depending on whether the new physical model of the late universe is spatially dependent.
3.2.1 Uniformity Modification
Modifications to new physical models of the late universe are homogeneous if they are only time-dependent, such as most kinetic dark energy models. However, almost all uniformity-modified models of the late universe are strongly constrained by the reverse distance ladder. Unlike normal distance ladders (e.g., type Ia supernovae calibrated by Cepheid variables), the reverse distance ladder [55-57] uses uncalibrated type Ia supernovae on the Hubble stream combined with BAO data to form a reverse distance ladder from low redshift (z≈0.1) to high redshift (z≈1), and is calibrated at high redshift (usually observed by CMB as a priori calibrated by the acoustic horizon under the constraints of the Cosmological Standard Model [58-60]). It is precisely because the reverse distance ladder requires only acoustic visual markers from the early universe that it does not depend on the late universe model, so that model-independent constraints can be given to the late universe model, and the Hubble constant values given by these reverse distance ladder limits are biased from the measurements of the early universe [21,61-65], unless the acoustic horizon prior given by the early universe model is changed, which in turn supports that the model modification should be from the early universe. Even if the high-redshift calibration of the reverse distance ladder is replaced from the acoustic horizon given by the CMB observations with other high-redshift observations, such as strong gravitational lensing time-delay observations [66,67] and gravitational wave standard whistles [68], the resulting Hubble constant limit is still biased towards measurements from the early universe. Therefore, a homogeneity modification model for the late universe also does not seem to fully solve the Hubble constant problem.
One model construct that may escape the constraint of the reverse distance ladder comes from a modification of the very late universe, where the very late universe means that its deviation from the Cosmological Standard Model occurs within the lower bound of the Hubble flow redshift (i.e., z≲0.01), such as the Phantom dark energy transition model that occurred in the very late stage, and the equation of state parameter of the dark energy crosses the ghost transition point w=−1 at the very late stage [69]. Since this ghost dark energy model is consistent with the Cosmological Standard Model above the Hubble Flow upper limit redshift, it does not break the early cosmic observation limit or even the reverse distance ladder limit. However, when the forward distance ladder is combined with the reverse distance ladder, the internal inconsistencies of this model are revealed [27,70-72]. Specifically, the Hubble constant H0 and the absolute luminosity M of supernovae using the uncalibrated type Ia supernova as the positive distance ladder limit are in conflict with the H0 and M limited by the reverse distance ladder of CMB calibration, and even if the H0 obtained by the reverse distance ladder limit M is used to calibrate the type Ia supernova in the forward distance ladder, there is still a conflict with the H0 of the forward distance ladder limit. So no matter what overall uniformity modifications are made to the late universe, the Hubble constant crisis remains.
In two recent papers [13,14], we have further strengthened this conclusion by modifying the traditional reverse distance ladder. The traditional reverse distance ladder requires a calibration at a high redshift, usually the acoustic horizon of the CMB under a given early universe model, thus obtaining a restriction that is not related to the late universe model, but it is also clearly dependent on the early universe model. We chose the cosmological age (CC) cosmic chronometer measurement [73] as the high redshift scaler, which directly measures the Hubble expansion rate of a class of slowly evolving galaxies by continuously tracking the age-redshift relationship.
Therefore, the integration of the cosmological model H(z) in the traditional ranging method is avoided, and thus it is irrelevant to any cosmological model. In order to fit in with the use of cosmological standard clock data, we further adopted the PAge (parametrization based on the cosmic age) model, a cosmological age-based parametric method [74-76]. Based on the fact that the age of the universe is mainly derived from the period of matter dominance, the model expands Ht to the second order of t
The deviation of the period of the universe's age relationship Ht=2/3 ignores the small contribution of the radiation-dominated period to the age of the universe. The PAge model is superior to other model parameterization methods in that it is an economical and global parameterization, which uses only two parameters to accurately and faithfully express the evolutionary behavior of various late models in the full redshift range, while other parameterization methods (e.g., Taylor expansion by redshift z or y=1−a) have seriously deviated from its parameterized cosmological model at the medium-high redshift (e.g., z≈1) end (see Fig. 8). With the help of this modified version of the reverse distance ladder [13,14], we find strong evidence (BIC criterion greater than 10) that the new physical model parameterized by the PAge model is not superior to the cosmological standard model ΛCDM model, that is, the uniformity modification model of the late universe does not solve the Hubble constant crisis better than the ΛCDM model.
Fig.8 Comparison of BAO characteristic scales (red, blue, green) and BAO observations under the ΛCDM model and its PAge/MAPAge parameterized model, as well as the Taylor expansion approximation according to redshift z and y=1−a. Image from Ref. [14]
3.2.2 Non-uniformity modification
If the modification of a new physical model of the late universe also has a dependence on space, it is considered a non-uniform modification, such as the interactive dark energy model [77]. The model introduces the interaction between dark energy and dark matter, so that a part of dark matter can decay into dark energy. Since dark matter itself fluctuates in space, the dark energy that interacts with it also creates a dependence on space. The interactive dark energy model increases the proportion of late dark energy, which in turn directly pushes up the Hubble expansion rate. The decrease of dark matter itself is at the cost of increasing the Hubble constant in order to maintain the physical proportion of dark matter energy density
In order not to be strongly limited by the aforementioned reverse distance ladder to the late uniformity model. Thus, in all respects, the interactive dark energy model is a potential candidate to solve the Hubble constant crisis [78].
Another non-uniform modification of the late universe comes from questioning the principles of late cosmology [79], such as that we are in a local cosmological void (i.e., cosmic void). The distribution of galaxies in such holes is very sparse, so the proportion of matter density in the holes is very low compared to other regions of the universe, and the distribution of dark energy is correspondingly higher, so the local Hubble expansion rate is also larger. It is worth noting that as early as the mid-90s of the 20th century, mainland scholars were the first in the world to raise the problem of using local cosmic holes to explain the overestimation of the Hubble constant [80,81]. Recent galactic survey observations [82] seem to support that we are in a large local cosmological hole with a radius of 300 Mpc and a depth of −30% (i.e., the Keenan-Barger-Cowie hole [83]) of the KBC, so it has been speculated that the hole is the cause of the Hubble constant crisis [84]. However, if a type Ia supernovae were used to trace the Hubble expansion rates at different redshifts, the claimed low-density range would be found to be incompatible with observations beyond their radii [85,86]. Therefore, there is no large enough hole in local cosmology that is large enough deep enough to solve the Hubble constant problem [87].
04 Lookout
So far, the Hubble constant problem has been briefly reviewed from the aspects of observation and model, in which the overall trend of the early observations systematically lower than the late observations shows that the Hubble constant problem is more serious to a certain extent, and the strong constraints from various observations that both the early and late model constructions will face in the model also show the difficulty of solving the Hubble constant problem. Only with the author's limited knowledge and biased view
However, the specific form of interaction is still unknown, and needs to be tested and explored in more detail in the future. In this section, we will introduce a special interaction model [88] and its observational evidence in the light of the authors' recent research work [89].
4.1 Observation
Observations from the early universe have less error and are relatively centrally distributed, and it is generally believed that if the source of the Hubble constant problem is indeed from observational and systematic errors, it is more likely to come from measurements of the late local universe. However, the measurements from the late local universe are widely diffuse, and although systematically higher than the earlier observations, it is difficult to explain them with a single systematic error. However, if it can be explained by a single systematic error, then there must be a new physics that has not been theoretically modeled by the systematic error. For this reason, it is still necessary to examine systematic errors with caution.
The error composition of the distance ladder measurement of type Ia supernovae is mainly divided into three parts: the first part is the calibration error from different distance ladders; The second part comes from the error of supernova standard cangification; The third part is from the cosmological variance of supernova samples. Among them, the distance ladder calibration error has been reduced to less than 1%, so it will not be discussed further. The standard cannellization error of supernova comes from the reality that although supernova is theoretically the ideal standard candle, it will be affected by the supernova predecessor star (white dwarf accretion model or white dwarf merger model, etc.) and its local environment in actual observation, so its light curve has a certain dispersion, and various corrections need to be made to become standard cangification.
It is important to note that, according to the theory of galaxy formation, larger galaxies form in larger dark matter halos, and larger dark matter halos are more likely to be distributed in a denser environment [96], so it is safe to speculate that the Hubble residuals of type Ia supernovae are also related to the material density environment in which the host galaxy is located. Before introducing this association, let's review the correlation between cosmological variance and observer localized density when measuring the Hubble constant with supernova samples.
4.1.1 Local cosmological variance
For supernovae in the Hubble flow range, the Hubble constant can be passed by a simple break between distance and regression velocity in Hubble's law
So, if an observer is in a local hole because its local density exceeds is negative, then the observer will always tend to overestimate its Hubble constant, which is why it is considered that the local hole can be used as an explanation for the Hubble constant problem. However, further calculations show that the standard deviation of the Hubble bias decreases with radius for supernova samples distributed across a certain radius around the observer [98]. Therefore, it is only necessary to select a supernova sample far enough away (e.g., the lower limit of the Hubble flow redshift z≳0.023) to control the standard deviation of the Hubble deviation contributed by the observer to the Hubble constant measurement to less than 1% even if it is in a local hole. So, a local hole smaller than the Hubble redshift range (also known as the Hubble bubble) does not solve the Hubble constant crisis we face (note that this is not the same as the aforementioned case of a void that excludes cosmological size).
4.1.2 Non-local cosmological variance
For the Hubble bias for arbitrary sample distributions obtained in our recent work [89], there is a very special case, that is, the host galaxy of the selected supernova sample is in the same local R-scale mean mass density exceedance
, then the Hubble bias corresponding to the supernova sample will also be negatively correlated with the local R-scale average material density excess of the supernova host galaxy:
The angle brackets represent the averaging of all these samples for supernovae. As you can see, this is different from the local cosmological variance relation (12) in Section 4.1.1, where the local density associated with the Hubble bias is no longer the observer's local density, but the local density of the sample supernova host galaxy. Therefore, we refer to this association as non-local cosmological variance.
Surprisingly, when the actual observational data were used to directly test the above-mentioned non-local cosmological variance relationship, we found that there was also a non-negligible conflict between the observations and the theoretical predictions. Specifically, the material density field reconstructed from the BOSS DR12 (baryon oscillation spectroscopic survey data release 12) data is used to estimate the arbitrary R-scale average matter density excess of the host galaxy of the Pantheon(+) supernova sample, and then the supernovae with the same local material density are selected as a group to fit their Hubble constants. It has been found that the higher the density of the supernovae, the greater the fitting of the Hubble constant. This is contrary to the negative correlation trend expected by the non-local cosmological variance relationship, and the degree of contradiction reaches a conflict of nearly 3σ at the scale R=60Mpc/h. We call this conflict the Hubble bias conflict, and unlike both the Hubble conflict and the S8 conflict, it is a new cosmological conflict that reveals at a deeper level the possibility of a new physics beyond the current standard model of cosmology.
4.2 Models
Since it has been observed that the measured Hubble constant always has a systematic error from the supernova sample, and the systematic error is also related to the material density environment in which the supernova host galaxy is located, a natural question is whether the systematic error exists in other observations. It can be seen that almost all early global background measurements, including supernova observations, systematically have smaller Hubble constants than late local measurements, due to the greater increase in the material density of the late local universe and the fact that various late luminosity distance indicators are located in such a high material density environment (i.e., galaxies or their faint halos).
Not only the distance indicator, but also the calibrator used to calibrate the distance indicator, for example, the Hubble constant measured with a supernova calibrated at the tip of the red giant branch is always smaller than that measured by a supernova calibrated with a Cepheid, because the tip of the red giant branch is usually located in a low-density halo outside the galactic disk, while Cepheids are usually located on a high-density galactic disk.
In addition, the Hubble constant measured by the strong gravitational lensing, which is also a late local measurement method, is biased towards the early cosmic observations, because the lens samples are deliberately selected when selecting the lens galaxy samples that are far away from the galaxy cluster, so the strong gravitational lens samples used to measure the Hubble constant are naturally in a low material density environment.
Based on these observations and the laws they present, it is reasonable to assume that there is a positive correlation between the Hubble expansion rate and the local density.
4.2.1 Chameleon dark energy model
One theoretical model that naturally produces this positive correlation between the Hubble expansion rate and the local density is the so-called chameleon dark energy [88], which originates from the chameleon mechanism. The motivation for proposing the chameleon mechanism was originally for the purpose of shielding modified gravitational effects in high-density environments on small scales. The mechanism assumes that a scalar field is coupled with the local density of matter in a specific way, so that the effective mass of the scalar field is greater in the high-density environment, that is, the conduction of the fifth force range is shorter, so as to achieve the effect of shielding the fifth force. However, the chameleon mechanism also has a concomitant effect, that is, the potential energy of the vacuum expectation at the scalar site is also higher at high density, which means that the effective cosmological constant is also larger, and thus the local Hubble expansion rate is also greater (see Figure 9). Therefore, in this model, the effective cosmological constant changes with the fluctuation of the density of matter at different scales, but at a fixed scale it is equivalent to the cosmological Standard Model.
图9 变色龙暗能量机制示意图 (a) 变色龙暗能量有效势Veff(φ)=V(φ)+U(φ)，其中变色龙场势函数取 Peebles-Ratra 势函数V(φ)=αΛ4(Λ/φ)n，变色龙耦合项取伸缩子耦合U(φ)=exp(φ/Λ)
。 It is easy to see the density of the substance corresponding to the solid line
When the density of matter is greater than that of the dashed line, the value of the potential function (vacuum energy) corresponding to the vacuum expectation of the effective potential of the solid line is also greater than that of the dashed line. (b) The Planck 2018 measurement (red) is selected as the background cosmology, so that the local Hubble constant (horizontal axis) corresponding to the local mass density exceeding (vertical axis) can fit the SH0ES measurement (blue). Image from Ref. [88]
Thus, the local Hubble constant measured with a distance indicator sample in a high-density environment includes three contributions: one from our own material density environment (which is positively skewed on small scales); The other part comes from the sum of the density fluctuations between us and the distance indicator sample (if far enough away from the indicator sample, the sum of the density fluctuations of this part of the material density fluctuation should be close to zero); The last part comes from the density of matter in which the distance indicator sample is located (this contribution is generally positive), so the final measured Hubble constant will always contribute more to the true background expansion. In this physical image, the early measurement of the Hubble constant is smaller because the density of the earlier matter also fluctuates less, reflecting the part of the true background expansion.
In addition, the S8 problem can also be explained in this model, that is, the greater the growth of the matter perturbation, the faster the local Hubble expansion rate, which in turn dilutes the original material perturbation growth, and finally reaches the equilibrium state of S8 is naturally smaller than the cosmological standard model expects. In addition, because the higher the local material density fluctuation is diluted by the larger local cosmological constant in the later period, it allows for greater material density fluctuation at high redshift relative to the space-wide fixed cosmological constant (i.e., the cosmological Standard Model), which naturally explains the unexpected mass star coefficient density observed by JWST (James Webb space telescope) at high redshift. In the future, we will study this model in more detail at the perturbation level (as a special case of the interactive dark energy model).
4.2.2 Scale depends on dark energy
Thus, the very definition of the cosmological constant problem may also point us to its way out, i.e., the effective cosmological constant may be a scale-dependent physical quantity, which can be very large at a very small scale due to drastic changes in space-time; But on a very large scale, it averages to a very small value due to the homogeneity and isotropy of space-time, and also due to some mechanism (e.g., Ref. [99]). Our chameleon dark energy model also provides a similar picture to some extent, that the coupling of the chameleon field with different average material densities at different scales gives effective cosmological constants of different magnitudes, and the Hubble constant problem is a reflection of this physical image on two scales, i.e., the CMB scale and the local Hubble flow scale.
05 Conclusion
Modern cosmology has gone through historical stages such as thermal big bang cosmology, inflationary cosmology, and exact cosmology, and finally formed the standard model of cosmology with inflation, dark matter, and dark energy as elements, that is, the six-parameter ΛCDM model. The model is able to roughly fit almost all of the observational facts of the 10-billion-year cosmological history to date, from the galactic scale to the cosmological scale. However, as a phenomenological model, the theoretical origin of its elements is not yet known, and the Hubble constant crisis and S8 conflict have posed serious challenges to it in recent years. However, crises are also opportunities, and perhaps the Hubble constant crisis is one such historical opportunity to give us a glimpse into the (new) physics underlying the Standard Model of cosmology.
In this paper, we briefly review the observational evidence and model construction of the Hubble constant problem, as well as the implications from both observational and theoretical aspects. Our main conclusion is that the problem of the Hubble constant is not due to systematic errors in observations, but to some new physics that has not been theoretically modeled. However, most model constructions of the early and late universes are strongly constrained by various observations, and for the time being, it seems that only the interacting dark energy model has the potential to be a candidate to solve both the H0 and S8 conflicts. Finally, a special interactive dark energy model is introduced, and preliminary evidence for it is found from the observational data. In the future, large-scale sky surveys, next-generation CMB satellites, and the continental's space station telescope program will finally provide us with the opportunity to reveal the physical nature of the Hubble constant crisis.
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