We are studying the effects of SelfInteracting dark radiation SIdr on the evolution of the universe. Our main focus is on the cosmic microwave background CMB and how SIdr could potentially help resolve the Hubble tension. We are looking into different scenarios by mixing SIdr with FreeStreaming dark radiation FSdr or not to determine whether SIdr can indeed contribute to solving the Hubble tension. We find that SIdr alone can increase the Hubble constant inlineformula idIEq1mmlmathmmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula to inlineformula idIEq2mmlmathmmlmrowmmlmn70mmlmnmmlmo.mmlmommlmsubsupmmlmn1mmlmnmmlmrowmmlmommlmommlmn1.6mmlmnmmlmrowmmlmrowmmlmommlmommlmn1.3mmlmnmmlmrowmmlmsubsupmmlmommlmommlmtextkmsMpcmmlmtextmmlmrowmmlmathinlineformula with a value of inlineformula idIEq3mmlmathmmlmrowmmlmsubmmlmiNmmlmimmlmtexteffmmlmtextmmlmsubmmlmommlmommlmn3mmlmnmmlmo.mmlmommlmsubsupmmlmn27mmlmnmmlmrowmmlmommlmommlmn0.31mmlmnmmlmrowmmlmrowmmlmommlmommlmn0.23mmlmnmmlmrowmmlmsubsupmmlmrowmmlmathinlineformula. However including FSdr disfavors the existence of SIdr inlineformula idIEq4mmlmathmmlmrowmmlmsubmmlmover accenttruemmlmiNmmlmimmlmo stretchyfalsemmlmommlmovermmlmtextsimmlmtextmmlmsubmmlmommlmommlmn0.37mmlmnmmlmrowmmlmathinlineformula. Even though the Hubble constant is increased compared to the predicted value it entails inlineformula idIEq5mmlmathmmlmrowmmlmsubmmlmiNmmlmimmlmtexteffmmlmtextmmlmsubmmlmommlmommlmn3.52mmlmnmmlmommlmommlmn0.25mmlmnmmlmrowmmlmathinlineformula. Finally we implement the Fisher method for future experiments and a inlineformula idIEq6mmlmathmmlmrowmmlmn7.64mmlmnmmlmimmlmimmlmrowmmlmathinlineformula measurement of inlineformula idIEq7mmlmathmmlmsubmmlmover accenttruemmlmiNmmlmimmlmo stretchyfalsemmlmommlmovermmlmtextsimmlmtextmmlmsubmmlmathinlineformula will be obtained when combing data from Planck AliCPT and CMBS4.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Theory']
Testing the coupling of dark radiations in light of the Hubble tension
{'Testing the coupling of dark radiations in light of the Hubble tension': 'Zhiyu Lu, 1,2,3* , Batool Imtiaz 1,2,3,5 , Dongdong Zhang 1,2,3,4 , Yi-Fu Cai 1,2,3  \n- 1*  \nDepartment of Astronomy, School of Physical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China. 2 School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China. 3 CAS Key Laboratory for Researches in Galaxies and Cosmology, University of Science and Technology of China, Hefei, Anhui 230026,  \nChina.  \n4 Kavli IPMU (WPI), UTIAS, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Chiba, Japan. 5 Department of Computer Science, Mohammad Ali Jinnah University, Block-6, P.E.C.H.S, Karachi-75400 Sindh, Pakistan.  \n*Corresponding author(s). E-mail(s): zhiyulu@mail.ustc.edu.cn; Contributing authors: batool24@mail.ustc.edu.cn; don@mail.ustc.edu.cn; yifucai@ustc.edu.cn;', 'Abstract': 'We are studying the effects of Self-Interacting dark radiation (SIdr) on the evolution of the universe. Our main focus is on the cosmic microwave background (CMB) and how SIdr could potentially help resolve the Hubble tension. We are looking into different scenarios by mixing SIdr with Free-Streaming dark radiation (FSdr) or not to determine whether SIdr can indeed contribute to solving the Hubble tension. We find that SIdr alone can increase the Hubble constant ( H 0 ) to 70 . 1 +1 . 3 -1 . 6 , km/s/Mpc with a value of N eff = 3 . 27 +0 . 23 -0 . 31 . However, including FSdr disfavors the existence of SIdr ˜ N si ≈ 0 . 37 . Even though the Hubble constant is increased compared to the predicted value, it entails N eff = 3 . 52 ± 0 . 25 . Finally, we implement the Fisher method for future experiments and a 7 . 64 σ measurement of ˜ N si will be obtained when combing data from Planck, AliCPT, and CMB-S4.', '1 Introduction': "The standard ΛCDM cosmological model has achieved remarkable success in describing the Universe's evolutionary history, encompassing structure formation and elemental synthesis. Nevertheless, in the era of precision cosmology, it is crucial to assess both the theoretical and experimental self-consistency of the ΛCDM model. Despite its overall success, there are indeed hints of tensions within the observed data.  \nOne of the most pressing challenges is the discrepancy between the universe's expansion rate as measured by the cosmic microwave background (CMB) experiments and local (low redshift) measurements [1], commonly referred to as the Hubble tension [2-4]. The observed discrepancy in the Hubble constant ( H 0 ) between the Planck satellite's estimation of 67 . 27 ± 0 . 60 km s -1 Mpc -1 [5] and the measurement of 73 . 24 ± 1 . 74 km s -1 Mpc -1 [1] by the SH0ES collaboration amounts to more than 3 σ [6] (3 to 5 sigma tension based on different datasets). If it exists, this tension strongly indicates the presence of new physics beyond the ΛCDM model.  \nEstablishing a theoretical framework to resolve the Hubble tension is a daunting task. From the perspective of CMB power spectrum analysis, the precise determination of the acoustic peak angular scale θ s is crucial, as it determines both the sound horizon at decoupling r s and the distance D A to the CMB surface of last scattering via θ s = r s /D A . One class of models aims to reduce the sound horizon to alleviate the Hubble tension. For example, the early dark energy (EDE) model introduces a dark-energylike component that becomes active around the matter-radiation equality and dilutes thereafter. Though EDE has shown promise in mitigating the tension, it has introduced new challenges, such as worsening the S 8 tension and fine-tuning problems.[7, 8]. Moreover, some efforts have been made to investigate non-standard scenarios, e.g. dark matter-radiation interaction [9-11], modified gravity paradigm [12, 13], ultra-light primordial black holes [14].  \nAnother way to reduce sound horizon is the addition of dark radiations before recombination. This can lead to differences in the effective number of relativistic degrees of freedom N eff from standard model predictions, providing a clue to new physics. N eff is a key probe of the temperature evolution of the early Universe and characterizes the energy density of relativistic species, such as neutrinos. The energy contribution by photons can be determined by CMB temperature today. The remaining radiation species contributing to total radiation energy is given by  \nρ r ( T ≤ 1MeV) = [ 1 + 7 8 ( 4 11 ) 4 3 N eff ] ρ γ , (1)  \nwhere ρ γ = ( π/ 15) T 4 γ is the energy density in CMB photon with T γ = 2 . 725 K and N eff = 3 . 046 [15] 1 . The Planck bestfit to ΛCDM gives N eff = 2 . 99 ± 0 . 17 at 68% CL  \n[5]. To reconcile the indirect measurements of H 0 from the CMB, Baryon Acoustic Oscillations (BAO), Type Ia Supernovae, and the direct measurements from the local distance ladder in physical theories, one may require N eff ≃ 3 . 95 [17].  \nIf we overlook the neutrino decoupling process, we can obtain N eff = 3 by considering the conservation of entropy from T ≡ 10 MeV and T ≤ m e [18, 19]. The deviation from the integer value of three occurring in N eff is due to various factors in the early universe. For instance, neutrinos continue to interact with the primordial plasma during electron-positron annihilation, affecting their temperature. Additionally, the energy dependence of neutrino interactions allows those at the high-momentum tail to interact with SM particles, influencing the energy spectrum of the neutrino gas [20]. Thus, a high degree of confidence in the neutrino's effective degree of freedom consistent with 3 . 046 can provide robust evidence for the existence of SM neutrinos. On the other hand, a value higher than 3 . 046 could indicate the existence of additional radiations.  \nIt has also been shown in [21] that both the damping tail and phase shift allow us to jointly constrain the amount of free-streaming relativistic particles N eff , and the number of relativistic particles that are fluid-like and tightly coupled. Likewise, the analysis of large-scale structure information provides insight into the phase of the acoustic oscillations, allowing constraints on N eff [22-24]. Though ΛCDM has predicted three flavors of neutrinos [25], several terrestrial neutrino experiments have found anomalies against the standard picture, which may be an indication of the existence of dark radiation [26]. Cosmological data also indicates additional radiations [27].  \nIn fact, any physical mechanism capable of generating additional radiation will result in the same impacts on the background expansion as extra neutrinos do, leading to an increased N eff value in observations. Given the numerous models that can boost N eff , it's crucial to explore methods for distinguishing among them such as sterile neutrinos [28, 29], self-interacting neutrinos [30, 31] and ultra-light axion-like fields [32, 33]. If the extra radiation consists of relativistic particles like sterile neutrinos, it should exhibit behaviour similar to that of regular neutrinos in terms of perturbation theory.  \nSome recent work investigates the effect of changing the nature of radiation. Motivated by this, a recent study [30] proposed a framework involving self-interacting neutrinos with a non-vanishing sum of their masses. They suggest a significant increase in the value of Hubble constant H 0 = 72 . 3 ± 1 . 4km / s / Mpc, along with a lower value of the matter fluctuations, σ 8 = 0 . 786 ± 0 . 020, while preserving the fit to the CMB damping tail. This study investigates N eff = 4 . 02 ± 0 . 29. Das [34] considered the neutrino coupling between flavors, and a strong interacting mode can be a better fit with CMB data. In their subsequent work, ACT data is included and supports a notable level of neutrino self-interaction, causing a delay in neutrino-free streaming until right before matter-radiation equality[35]. However, the inclusion of Planck 2018 polarization data diminishes the inclination towards a strong interaction.  \nIn this study, our primary focus is the physics of dark radiation components. Dark radiation can be categorized as either coupled or decoupled. When decoupled, we assume it shares the same properties as neutrinos and is noted as FSdr (Free-Streaming  \ndark radiation for short). Going beyond the same property assumption is straightforward. For the coupled case, we focus on the self-interaction, which we refer to as SIdr. These distinctions are clearly outlined in Table 1. Though neutrinos are not 'dark', we include them in our FSdr framework to incorporate the truly dark ones as they exhibit the same behaviour under our assumptions.  \nTable 1 The Categories and conventions used in this work.  \nIn Section 2, we present the physical impact of SIdr on the power spectra. In Section 4, we constrain the parameters using the latest CMB data. Section 5 summarises the Fisher information matrix method we use in our forecasting analysis for next-generation experiments. Section 6 is dedicated to the conclusion and outlook.", '2 Impact on cosmological perturbations and CMB power spectra': "The self-interacting model among dark radiations intervened by a heavy mediator was developed in [36]. Those interested in details can find references therein. This section will explore the two primary factors contributed by extra radiations and SIdr that can influence the universe.  \nCosmological experiments, such as CMB polarization detection, provide valuable insights into the properties of radiations through two well-defined physical processes. Firstly, Thomson scattering depends on the total energy density of radiations. Before matter-radiation decoupling, over a broad span of angular scales, CMB polarization data is attenuated by the finite width of the last scattering surface in the presence of the density fluctuations with adiabatic conditions[37].  \nSecondly, since the time when neutrinos decouple from hot plasma, the whole universe has been transparent to them except for gravitational influence. The neutrinos will pull the photon-electron sound wave as they travel at the speed of light, while the sound speed is 1 / √ 3 times lower. Obviously, the effect of this 'pull' will be stronger if there are more neutrinos or to say FSdr. Thus, the amount of free-streaming particles will introduce a phase shift and diminish the amplitude of the acoustic peaks observed in the CMB [38-41]. The phase shift strongly indicates the existence of FSdr [21, 23]. These effects are quantified through the parameter R fs , which represents the fractional energy density contributed by the free-streaming part. It is given by  \nR fs = ρ fs ρ r , (2)  \nwhere ρ fs is the energy density of FSdr, and ρ r is the total energy density in radiations including photons, neutrinos and other possible dark radiations.  \nFig. 1 Left : The degeneracy relation between temperature ratio ξ si and flavor number N si . Right : The degeneracy relation between FSdr number N fs and SIdr ˜ N si .  \n<!-- image -->  \nSince we don't distinguish free-streaming dark radiation with neutrinos, the following steps are direct,  \nR fs = ρ fs ρ γ + ρ fs + ρ si = 7 8 N fs ξ 4 fs 1 + 7 8 N fs ξ 4 fs + 7 8 N si ξ 4 si = N fs ( 7 8 ξ 4 fs ) -1 + N eff , (3)  \nand the effective number of species [42] is  \nN eff ≈ N fs + N si ξ 4 si /ξ 4 fs ≡ N fs + ˜ N si . (4)  \nHere, 7 / 8 stands for the statistical factor for the fermionic particles, and ξ si = T si /T γ is the temperature ratio of SIdr to photons. ξ fs = (4 / 11) 1 / 3 is the temperature ratio between FSdr and photons. We also define the effective species number for SIdr, including the temperature effect ˜ N si . Eq. (4) reveals the degeneracy between the temperature and the flavors of the SIdr. As depicted in Fig 1, SIdr can explain the entire dark radiation contribution when N si = 3, ξ si = ( 4 11 ) 1 / 3 , and N fs = 0, yielding N eff ≈ 3 . 046. However, the scenario can also be interpreted with N si ≈ 4 and ξ si ≈ 0 . 68. Although Eq. (4) can't capture the contribution from non-instant decoupling, the error is well within the margin of error for the current experiment capacity.  \nIn the standard model, we have three species of FSdr (three neutrinos) and no SIdr, i.e., R SM fs = 0 . 408. Also, if all radiations are tightly coupled, then R fs = 0. Thus ( N eff , R fs ) can be parameterized for the properties of dark radiations up to the effective species ˜ N si . To break the degeneracy, one of the prior knowledge on ( N si , ξ si , G eff ) is required. 2 .  \nThe presence of SIdr perturbs the CMB peaks and amplitudes. In the sub-horizon region and deep in the radiation-dominated epoch, the baryon density is negligible,  \nFig. 2 Left : the Weyl potential evolution along scale factor at scale k = 0 . 11 / Mpc; Right : the monopole of photon perturbation transfer function for various R fs . We compare our analysis with ΛCDM by varying R fs from 0 to 0 . 41  \n<!-- image -->  \nand the monopole of photon perturbation can be expressed as:  \n[Θ 0 -Φ]( τ ) = [Θ 0 -Φ](0) cos( kc s τ ) -k √ 3 ∫ τ 0 dτ ' [Φ + Ψ]( τ ' ) sin[ kc s ( τ -τ ' )] , (5)  \nwhere (Φ+Ψ) / 2 is the Weyl potential, Θ 0 is the monopole of temperature fluctuation. The Weyl potential vanishes once the photons enter the sound horizon, leading to the radiation-driven effect of acoustic oscillations.[43, 44]. This extra source can carry a phase shift if the timing of the decay is modified, and the amplitude of the radiationdriving effect depends on the initial value of the Weyl potential  \nΨ+Φ= -(1 + 5 15 + 4 R fs ) R . (6)  \nHere, Φ , Ψ are the metric perturbations, and R is the comoving curvature perturbation. One can observe that the decrease in fractional energy density enhances the radiation-driving effect through the shift in initial value.  \nThe same idea can apply to the polarization multiples through the equation [39]  \nΘ P,ℓ ( τ 0 ) ≃ 5 18 ˙ d γ ( k, τ rec ) ˙ κ -1 ( τ rec ) ( 1 + ∂ 2 ∂ ( kτ 0 ) 2 ) j ℓ ( kτ 0 ) . (7)  \nWe can observe that polarization multiples are proportional to the photon temperature fluctuations, i.e., Θ P,ℓ ∝ ˙ d γ , causing them to be influenced by neutrinos in a manner similar to temperature fluctuations. Additionally, the time derivative of d γ does not affect the phase-amplitude shift. As a result, the free-streaming radiations imprint a net phase and amplitude shift in the CMB power spectra. The shifted phase and amplitude are given by [38]  \nϕ fs ≈ 0 . 19 πR fs , ∆ fs ≈ -0 . 27 R fs . (8)  \nThe presence of self-interaction between dark radiations prevents them from freestreaming and delays the decoupling time until a low redshift z dec , depending on  \nFig. 3 Same as Fig 2, but with a displacement of the influence on CMB power spectra.  \n<!-- image -->  \nthe strength of the interaction. As a result, the free-streaming neutrino fraction R fs decreases relative to ΛCDM. The amplitude of the SIdr scenario in the CMB temperature power spectrum is higher compared with ΛCDM, and the peaks are also shifted to higher ℓ [38]. These self-interacting dark radiations can be described by the following Boltzmann equation in the Newtonian gauge [45-47]  \n˙ δ si + 4 3 θ si -4 ˙ Φ = 0 , ˙ θ si + 1 2 k 2 ( F si , 2 -1 2 δ si -2Φ) = 0 , ˙ F si ,ℓ + k 2 ℓ +1 ( ℓ +1) F si ,ℓ +1 -ℓF si ,ℓ -1 = α ℓ ˙ τ si F si ,ℓ , ℓ ≥ 2 . (9)  \nHere, F si ,ℓ is SIdr's perturbation expanded in multiple space, F si , 0 ≡ δ si and F si , 1 ≡ 4 3 k θ si . In the above set of equations, α ℓ are ℓ -dependent O (1) angular coefficients that depend on the nature of the interacting model. Due to energy and momentum conservation, α ℓ = 0 for ℓ = 0 , 1 while, for ℓ ≥ 2, α ℓ ≈ O (1). The subsequent opacity from dark radiation self-scattering is ˙ τ si ∝ G 2 eff T 5 si , determined by the interacting strength and temperature of SIdr. During a frequently interacting period, the high moments are suppressed, and these equations then behave like a fluid. When ˙ τ si → 0, this causes Eq. (9) to mimic free streaming radiations. The transition time is determined by G eff . We use the publicly available modified CLASS [48] code CLASS SInu 3 to solve above set of equations.  \nIn Fig 3, we illustrate the evolution of the perturbation effects of net monopole, the Weyl potential and CMB TT and EE power spectra by keeping N eff = 3 . 046 fixed and varying the free-streaming fraction R fs from 0 to 0 . 408. The left panel displays the variation in the net monopole of the photon perturbation as a function of R fs . In comparison to the ΛCDM scenario, the presence of SIdr exhibits a higher amplitude of acoustic oscillations, and the peaks are shifted towards smaller scales. The middle panel demonstrates that the Weyl potentials have a larger initial value in scenarios with smaller R fs . Consequently, the radiation's driving effect is intensified, leading to a delay in their decay and a phase and amplitude shift in the photon transfer function.  \nFig. 4 The expected Hubble constant distribution in the SIdr scenario neglecting the parameter degeneracy. This distribution is based on the Planck measurement with possible mixing o FSdr and SIdr to achieve N eff ∼ 3 . 044. The grey shadow indicates the 1 σ region allowed by SH0ES.  \n<!-- image -->  \nThe resulted changes in the CMB TT and EE spectra are depicted in the right panel. We observe amplified amplitudes of the power spectra and a shift of the peak positions towards larger ℓ values as R fs decreases. These effects reach their maximum as R fs approaches 0.", '3 Cosmological probe: The Hubble tension': "As explained in Sec 2, even when N eff is fixed, one can manipulate the phases and amplitudes of the CMB spectra by tuning the parameter R fs . This manipulation creates room for a larger Hubble constant.  \nThe power of the SIdr model becomes apparent when considering the observed peak of the CMB multipoles l peak with corresponding wave numbers k peak , which can be determined using the approximation equation [11]:  \nℓ peak = k peak D A ≈ ( mπ -ϕ ) D A r s , (10)  \nwhere m ≥ 1 is an integer number referring to the index of the peak; ϕ is the phase, as discussed in Eq. (8); r s is the radius of the sound horizon. Additionally, D A is the comoving angular diameter distance to the redshift of recombination z ls  \nD A = 1 H 0 ∫ z ls 0 dz [ ρ ( z ) /ρ 0 ] 1 / 2 . (11)  \nρ ( z ) corresponds to the energy density of the universe at redshift z , while ρ 0 stands for present-day value. We can assert that H 0 is directly related to the function of phase suppressed by the sound horizon and influenced by late-time physics involved in the integration process:  \nH 0 = mπ -ϕ ℓ peak r s / ∫ z ls 0 dz ( ρ ( z ) /ρ 0 ) -1 / 2 . (12)  \nIf SIdr exists when N eff is fixed, according to Eq. 8, there is ϕ SI < ϕ ΛCDM . As a result, the numerator of Eq. (12) becomes smaller without introducing new energy components such as early dark energy or additional relativistic degrees of freedom. This naturally requires a larger H 0 while keeping the observed ℓ peak fixed.  \nThe Hubble constant proposed by SIdr model compared with ΛCDM model is  \nH SI 0 H ΛCDM 0 = π -0 . 19 πR SI fs π -0 . 19 πR ΛCDM fs . (13)  \nWe take H λ CDM 0 and N eff as independent Gaussian variables from Planck best-fit H 0 = 67 . 27 ± 0 . 60km / s / Mpc and N eff = 2 . 99 ± 0 . 17. Since a lower value of R fs will induce a higher H 0 , we don't consider cases R fs > 0 . 41. Assuming R SI fs has a uniform distribution between 0 to 0.41 gives us the distribution of H 0 in view of self-interacting radiations, as shown in Fig 4 Once R fs is less than 0.1, the CMB deduced H 0 will be within the 2 σ regime of the SH0ES prediction.  \nNote that Eq. 12 is approximately valid since we neglect the complicated influence of SIdr on the denominator part. Eq .13 is only used for illustrative purposes, as it couldn't capture the degeneracy between SIdr and cosmological parameters.", '4 Methodology': "In this work, we consider the contribution of SIdr to N eff by introducing the temperature ratio ξ si between SIdr and photons as a free parameter. Since it is the value N eff that directly affects observations, the combination of ξ si and the flavor of the SIdr N si introduces degeneracy, as discussed in Section 2, we therefore work with N si = 4 throughout this paper, inspired by the flavors of active and sterile neutrinos. N si shouldn't be confused with ˜ N si , the latter is the effective species number considering the temperature difference. N si can be set arbitrarily, as the data can only constrain the combination of N si and ξ si Another free parameter in our analysis is the effective self-interacting strength log 10 [ G eff / MeV -2 ], appearing in Eq. (9) through ˙ τ si . Note that because of the degeneracy between G eff & ξ si and N si & ξ si , when comparing models with different N si , the G eff should be scaled according to [35]  \n˜ G eff = G eff ( T si T fs ) 5 / 2 . (14)  \nTo test the importance of SIdr, we investigate different scenarios as summarized in Table 2. In cases with non-zero ˜ N si , we always allow G to vary freely. The motivation and physical interpolation is clear: case A tests whether SIdr only can account for all the observations; compared with case D where we fix the overall radiations N eff = 3 . 044 and only vary the interacting strength G , we can test the importance of self-interaction and extra-radiation in solving the H 0 tension. Case B tests which are the data required, SIdr or FSdr; assisted by case C, we then get a better understanding of the roles of G and N eff . In our analysis, we assume massless radiations since the impact of SIdr's mass is considered negligible, as validated by the study in Ref [35].  \nTable 2 The different scenarios in this work. A number means the parameter is fixed, and 'free' means the parameter can vary.  \nFor our analysis, we employ the Markov Chain Monte Carlo (MCMC) method, using the publicly available code Cobaya [49] and Montepython [50, 51]. We sample from the posterior distributions using the Metropolis-Hastings algorithm implemented in Cobaya , and we adopt the Gelman-Rubin convergence criterion [52].", '4.1 Data': 'In our analysis, we make use of the following datasets:  \n- · Planck : We utilize the full temperature and temperature-polarization power spectra measured by the Planck collaboration. The likelihood includes low ℓ -TT from Commander , 2 ≤ ℓ ≤ 29; low ℓ -EE from Simall , 2 ≤ ℓ ≤ 29; hith ℓ -TT from PlikHM , 30 ≤ ℓ ≤ 2508; high ℓ -TTTEEE from PlikHM , 30 ≤ ℓ ≤ 1996. Further details regarding these likelihoods can be found in [53].\n- · CMB Lensing : We consider the Planck reconstructed CMB lensing power spectrum [53]. The CMB lensing power spectrum probes structures over a broad range of redshift, with a peak at z ≈ 1 -2. The scale cuts used in the Planck lensing power spectrum likelihood include modes with 8 ≤ L ≤ 400, for which non-linear corrections are negligible.\n- · Riess2018a : We include the local measurement of the Hubble constant by the Hubble Space Telescope, H 0 = 73 . 48 ± 1 . 66 [54].\n- · Pantheon : Additionally, we incorporate the Pantheon dataset, which is a compilation of Type Ia supernovae measurements [55].\n- · BAO : We also include measurements of the BAO from the 6dF Galaxy Survey [56], SDSS MGS [57], and SDSS DR12 [58].  \nOur baseline parameter space consists of nine parameters: the self-interacting strength G eff , the temperature ratio of the dark radiations ξ si , the Hubble constant H 0 , the physical energy density for cold dark matter and baryons Ω c h 2 and Ω b h 2 respectively, the reionization optical depth τ reio , the spectrum amplitude A s , and the spectrum index n s . Since BBN is affected by N eff , we take Y p as another free parameter for possible N eff ≫ 3. We use the priors suggested by Cobaya 4 . The priors for SIdr are ξ si ∼ N (0 . 7 , 0 . 001) , log 10 [ G eff / MeV -2 ] ∼ N ( -1 , 0 . 1). These selections are under the following considerations: SIdr is likely to share the same temperature with neutrinos, e.g. sterile neutrino, and a large self-interacting strength is supported by previous studies.', '4.2 Results': "Our analysis explores two distinct interacting models depending upon their coupling strength: the strong-interacting model with log 10 [ G eff / MeV -2 ] > -2 . 5 and the medium-interacting model with log 10 [ G eff / MeV -2 ] < -2 . 5. The impact of these models on various cosmological parameters can be observed in Fig 5, Fig 6 and Fig 7. Due to the parameter degeneracy present in the formation of the power spectrum, the value of log 10 [ G eff / MeV -2 ] and ˜ N si affect parameters such as H 0 , A s , n s , τ , Ω m , and N eff .  \nOne can expect a direct effect that, for the models with only SIdrs, the phase shift in Eq. (10) is cancelled. To keep the observations unchanged, a larger H 0 is required. On smaller scales, this shift is enhanced by the spectral index n s . Additionally, the SIdr induces an amplification of the amplitude, resulting in a smaller spectrum amplitude A s .  \nFig. 5 One-dimensional posterior distribution for log 10 [ G eff / MeV -2 ], ˜ N si , N fs and N eff are presented. The two scenarios are labelled by the parameters N fs , ˜ N si , either fixed or free: SIdr with FSdr (blue) and SIdr only (violet). The violet curves favour the strong coupling and medium coupling between SIdr, as shown in the upper-left panel, while the inclusion of FSdr alters this preference, as indicated in the upper-right panel. Both scenarios exhibit similar predictions for the overall radiation component, as demonstrated in the lower-right panel.  \n<!-- image -->  \nFig. 6 Similar to Fig. 5, but now considering the parameters Ω m , H 0 , and S 8 , and incorporating the ΛCDM results. Notably, the Hubble parameter in these two scenarios is increased, alleviating the tension with local measurements, while the tension in S 8 is slightly enhanced. These changes result from the inclusion of SH0ES data and the consideration of N eff as a freely derived parameter.  \n<!-- image -->  \nFig 5 demonstrates that if SIdr consists of all the dark radiation (the violet curve), there will be a strong coupling region, highly preferred over medium coupling. While when possible FSdr exists, the SIdr components are decreased to around zero and couldn't distinguish strong or medium coupling. When we focus on ˙ τ si for both cases, we find a consistent value due to the degeneracy between interacting strength and temperature, i.e., ˙ τ si ∝ G 2 eff T 5 si .  \nIn the subsequent panels of Fig 5, it becomes evident that FSdr plays a vital role, constituting the majority of the dark radiations with a value of R fs ≈ 0 . 392, or equivalently N fs = 3 . 08. Considering N fs as a free parameter with a uniform prior strongly favors their existence over SIdr, ˜ N si = 0 . 37. Furthermore, we observe that in either case, the overall radiation species number is similar N eff ≈ 3 . 45. This leads to a higher and consistent H 0 ≈ 70 . 2km / s / Mpc estimation in both cases in Fig 6, although the underlying mechanisms differ significantly. One is that the SIdr leads to an increase of H 0 , but additional radiation is required to fix the amplitude and phase in high multipoles. On the other hand, when FSdr are considered, because of the prior from local H 0 measurement, the N eff should be increased. Therefore, once N eff > 3 . 044 is not allowed, SIdr is inadequate for resolving the Hubble tension, though the tension for now is reduced to 1 . 39 σ . More importantly, a similar outcome is achieved by allowing additional FSdr; thus, the SIdr scenario thus has no advantage.  \nIn Fig 6, both models predict the same value of S 8 , supporting the existence of tension in S 8 . Notably, the best-fit value of S 8 is higher than what is predicted by ΛCDM, a common issue encountered by solutions to reduce the sound horizon.  \nThe Fig 7 illustrates the contour plots for different parameters. We observe that n s and A s are the parameters mostly affected by G eff , while other parameters do not show obvious separated regions as expected in the earlier paragraph. The significance of the strong interaction is smaller compared to the medium interaction concerning A s and n s . Most parameters are consistent with ΛCDM predictions, except for the  \nFig. 7 One-dimensional and two-dimensional posterior distributions for the full cosmological parameter space with 1 -σ and 2 -σ intervals are represented by dark and light-shaded contours, respectively.  \n<!-- image -->  \nphysical abundance of dark matter Ω c h 2 , which determines the equality epoch. The rise in ω c is attributed to the increase in N eff as discussed in [59]:  \n∆ ω c ≈ ω ΛCDM c 1 + 7 8 ( 4 11 ) 4 / 3 ∆ N eff 1 + 7 8 ( 4 11 ) 4 / 3 N ΛCDM eff , (15)  \nwhere ∆ gives the difference compared to the value in ΛCDM.  \nBecause most of the information of τ reio comes from large-scale data, the parameter τ reio remains unaffected by the existence of SIdr. Large-scale perturbations enter the horizon at a late time when the nature of the dark radiation doesn't matter 5 .  \nIn Fig 8, we delve into the details of strong and medium self-interactions. The results suggest that the parameter N fs aligns with the ΛCDM in both cases. This implies the presence of FSdr necessary to account for the phase shift in the power spectra. It also indicates a strong tendency toward ΛCDM, allowing for a fraction of SIdr approaching zero, i.e., ˜ N si = 0 . 31 +0 . 12 -0 . 35 ,  \nFig. 8 Same as Fig 6 but for ˜ N si , N fs and N eff , we split into strong coupling (violet-dashed curve) and medium coupling (purple-dashed curve). In either case, the existence of SIdr is limited compared with FSdr.  \n<!-- image -->  \nFig. 9 The fractional difference in TT, EE, TE power spectra in terms of cosmic variance. We show the best-fit results for both strong (solid curves) and medium interacting (dashed curves) regions. The gray error bars show Planck 2018 measurements. The labels are the same as Table 3.  \n<!-- image -->  \nWe compare the best-fit power spectrum with that of ΛCDM in terms of the cosmic variance. The oscillations observed in Fig 9 indicate a phase shift compared to the best-fit power spectrum of the ΛCDM. The cosmic variance can be calculated using  \nthe power spectrum C ℓ :  \nσ CV =                  √ 2 2 ℓ +1 C TT ℓ TT ; √ 2 2 ℓ +1 C EE ℓ EE ; √ 1 2 ℓ +1 √ C TT ℓ C EE ℓ +( C TE ℓ ) 2 TE . (16)  \nAs depicted in Fig 9, the model without FSdr remains consistent with the best-fit power spectrum from the Planck data. The differences between the models are not apparent until the deep damping tail for ℓ > 2000. Therefore, high-resolution experiments will be crucial in distinguishing between these two models. On the other hand, when FSdr are included, the peaks exhibit more out-of-phase oscillations as shown in Fig 9 because of the different N eff from ΛCDM. Additionally, in the second panel, we observe that the E-polarization data from Planck provides limited constraints on the models, especially in the case including FSdr. In Appendix A, we show the result when including ACP and SPT polarization data.  \nTable 3 The difference in χ 2 between the ΛCDM model is calculated. In this context, 'S' represents 'Strong' and 'M' represents 'Medium', indicating the interacting strength. 'A' and 'B' refer to the scenarios outlined in Table 1. For example, 'SA' denotes the strong interacting region for the SIdr only case ( N fs = 0 , ˜ N si ).  \nWe have summarized the statistics in Table 3. The χ 2 values have been calculated compared to the ΛCDM model using the baseline dataset. Here we note the χ 2 in CMB data as χ 2 Planck . Specifically, we test the strong and medium interacting regions in cases A and B. In case A, neither the strong nor medium interacting regions provide a better fit for the CMB data. However, in the baseline dataset, we observe improvement, with medium interaction being more optimal than strong interaction. A similar pattern is seen in case B, which shows a better fit due to adding a new free parameter. The small ∆ χ between strong and medium interaction indicates the limited influence of SIdr. When comparing cases A and B, we find that a better fit is achieved when FSdr are included, following the trend 'strong interaction < medium interaction < no interaction'. These results suggest the limited significance of SIdr, at least within the Planck observations.", '4.3 What CMB and SH0ES Data Respectively Tell Us': "In the previous section, our baseline dataset included both high-redshift and lowredshift data, and we found no evidence for SIdr. However, the conflicting predictions  \nFig. 10 The one-dimensional posteriors for H 0 , N eff , and log 10 [ G eff / MeV -2 ] are presented. The solid curves depict predictions based on Planck data, whereas the dashed curves incorporate the SH0ES measurements. The grey-shaded region represents the 2 σ confidence interval permitted by SH0ES. The violet, blue, green, and orange curves correspond to scenarios A, B, C, and D, respectively, as detailed in Table 1.  \n<!-- image -->  \nfor H 0 from SH0ES and CMB data prevent drawing definitive conclusions by just using the baseline dataset. In this subsection, we analyze models with and without the SH0ES results. Our primary focus is on the CMB data from Planck2018, covering temperature, E-mode polarization, and lensing, as well as the H 0 prior from SH0ES. Details are provided in Section 4.1. As shown in Figure 10, we evaluate four scenarios using only Planck data (solid curves) and Planck combined with SH0ES (dashed curves).  \nStarting with the solid green curves, this represents the N eff extension to ΛCDM, which yields a low value of H 0 . When local data is included, the dashed green curve shows H 0 ∼ 70 and an increase in N eff . The solid orange curve predicts a weak interacting strength and a low H 0 value, indicating that the CMB data alone does not favour SIdr if we fix the radiation species. However, with the inclusion of local data (dashed orange curve), the interacting strength becomes significant, leading to a higher H 0 value. This behaviour shows the virtue of the SIdr model, reducing the tension to 1 . 8 σ . However, from the orange solid and dashed curves, we see the trouble of SIdr that CMB data doesn't automatically prefer.  \nThe most surprising result comes from the pink curves. The solid pink curve shows N eff ∼ 3 while reducing the H 0 tension to 2 . 4 σ . This reduction in tension is a combined  \nresult of the self-interacting nature of SIdr and the increased error in N eff . However, including local data does not shift the interacting strength, as shown by the overlapping pink dashed and solid curves in the third panel. We still observe a higher H 0 value, consistent with the increase in N eff , supporting our previous analysis that SIdr alone does not resolve the tension.  \nFinally, the blue solid curves in H 0 and N eff closely follow the green ones but with minor features similar to the violet curves, indicating that FSdr dominates over SIdr, further supporting our conclusions.  \nIn conclusion, without a prior on H 0 , the CMB data does not necessitate SIdr to be self-interacting. However, moderate interacting strength helps to reduce the tension to 3 σ . When SH0ES data is included, the FSdr model is preferred over the SIdr model.", '5 Fisher Forecast': "We have observed the significance of polarization data in constraining the properties of dark radiations. Thus, it becomes crucial to achieve a more precise detection of the CMB sky map, which improves the determination of peak amplitudes and phases and reduces the marginalised error. These advancements will greatly enhance our ability to constrain dark physics. In this section, we employ the Fisher forecast method to assess the potential of upcoming CMB experiments in constraining dark physics.  \nThe Fisher matrix formalism is a widely used tool for predicting the statistical capabilities of future experiments in measuring cosmological parameters. It is defined as the expectation value of the second derivative matrix of the logarithm of the likelihood function with respect to the parameters of interest:  \nF ij = -〈 ∂ 2 ln L ∂θ i ∂θ j 〉 , (17)  \nwhere the average is taken over all possible realizations of the data assuming a certain fiducial model. The errors in the parameters can be obtained from the inverse of the diagonal Fisher information matrix:  \nσ ( θ i ) = ( F -1 ) 1 / 2 ii . (18)  \nThe Cram'er-Rao bound states that the errors obtained from the Fisher information matrix represent the smallest achievable errors for unbiased estimators. In the case of a Gaussian likelihood, the components of the Fisher matrix are given by [60]:  \nF ij ≡ ∑ ℓ F ij l = ∑ ℓ 2 ℓ +1 2 f sky Tr ( C -1 ℓ ( ⃗ θ ) ∂ C ℓ ∂θ i C -1 ℓ ( ⃗ θ ) ∂ C ℓ ∂θ j ) . (19)  \nwhere we have defined the Fisher information for single ℓ -component F ij ℓ and f sky is the sky coverage. C ℓ is a matrix that containing various observables:  \nC ℓ ≡     C TT ℓ + N TT ℓ C TE ℓ C Tϕ ℓ 0 C TE ℓ C EE ℓ + N EE ℓ 0 0 C Tϕ ℓ 0 C ϕϕ ℓ + N ϕϕ ℓ 0 0 0 0 C BB ℓ + N BB ℓ     . (20)  \nIn this section, ϕ is the lensing potential [61]. We extend Eq. (20) by including the B-mode power spectrum. We assume a white-noise power spectrum N XX ' ℓ for the effective noise, where XX ' ∈ { TT,EE,TE,BB } , is given by;  \nN XX ' ℓ = s 2 exp ( ℓ ( ℓ +1) θ 2 FWHM 8 log 2 ) , (21)  \nwhere θ FWHM represents the experimental resolution, s is the instrumental noise in temperature, and √ 2 s is the noise in polarization. The noise for lensing potential is reconstructed according to [61]. The Fisher forecast is performed through a modification of the public code Fishchips [62].  \nIn this study, we compare three categories of experiments: the current state-ofthe-art CMB data provided by the Planck satellite, and the improved large-scale polarization CMB measurements from the scheduled experiment in Tibet 6 ; the Stage4 CMB experiment (CMB-S4) [76]. Details of the experimental configurations we use is summarized in Table 4.  \nTable 4 The experimental parameters used for the fisher analyses.  \nWe assume equal ℓ -band coverage across different channels, including temperature, polarization-E, and lensing potential. The fiducial cosmology model we used is based on the best-fit parameters in Section 4.2 for the model N fs , ˜ N si in the strong self-interacting region. The parameters we're interested in this section are those related to dark radiations, so we vary the following parameter set { log 10 [ G eff / MeV -2 ] , ˜ N si , N fs , H 0 } . Furthermore, we conducted an analysis to evaluate the potential improvement achieved by incorporating polarization B-mode data. However, our findings indicate that, even in CMB-S4 experiments, the contribution from polarization B-mode data is consistently limited, as illustrated in Fig 11 and Fig 12.", '5.1 Results': 'First, we present the Fisher information for the Hubble constant across all ℓ bands defined in Eq. 19, neglecting the covariance between different parameters (see Fig 11).  \nOn larger scales, the AliCPT could provide more information compared to Planck due to its lower noise level, depending on the sky coverage. However, f sky < 0 . 2 yields less information than Planck data. On smaller scales, the CMB-S4 experiment dominates due to its high resolution. The inclusion of polarization B-mode data does not significantly improve the results on H 0 but contributes at the level of O (0 . 1) to the interacting strength log 10 [ G eff / MeV -2 ] as indicated by the blue curves.  \nFor Planck data, the information is primarily concentrated in the middle bands 300 < ℓ < 1800, beyond which it decreases rapidly. Due to improved noise control in CMB-S4, it provides abundant information in the far damping tail for ℓ > 1800. We can propose that CMB-S4 alone cannot provide information about large scales. Thus, combining these three experiments will have the best capacity over the entire ℓ band.  \nFig. 11 Fisher information for H 0 and log 10 [ G eff / MeV -2 ] across ℓ bands considering CMB experiments. For the AliCPT configuration, we varied the sky coverage from 0.1 to 0.4 (dashed lines).  \n<!-- image -->  \nAfter having constructed the Fisher matrix in Eq.19, we can obtain an expected error covariance matrix on the parameter space by inverting the Fisher matrix.  \nC ij = ( F -1 ) ij . (22)  \nWe present the result of Eq. 22 in Fig 12 Different configurations have been applied: the CMB-S4 experiment using CMB temperature, polarization E, and lensing potential power spectrum, the same configuration including polarization B power spectrum, a combination of Planck, AliCPT (assuming f sky = 0 . 4), and CMB-S4 experiments using the entire CMB power spectrum. We use equal bands for different channels according to the corresponding experiment configuration in Table 4. Although including the B power spectrum has limited improvement on a single parameter, it helps decrease the degeneracy between neutrinos and dark radiations.  \nFig. 12 Posteriors from a simplified Fisher forecast for CMB-S4 like measurements of the CMB temperature, E polarization and lensing information. We have considered the optimal case for the sky coverage in the AliCPT experiment, i.e., f sky = 0 . 4.  \n<!-- image -->  \nTable 5 The prediction errors of parameters corresponding to different CMB experiments based on Fisher analysis.  \nTo quantify the capacity for parameter constrain, we calculate the marginalized 1 σ error on a single parameter and compare the difference in this error between configurations. We have concluded all the results in Table 5. The detection significance can be easily calculated using z-score z = ( x -µ ) /σ , e.g. (2 . 49 σ, 6 . 19 σ, 6 . 22 σ, 7 . 64 σ ) for ˜ N si', '6 Conclusions and Outlook': "The existence of Self-Interacting dark radiations can have significant impacts on the universe's evolution. When SIdr enter the sound horizon, it remains coupled, thereby  \namplifying the radiation driving effects. Consequently, the phase and amplitude of these scales in the power spectrum are shifted. Therefore, SIdr becomes a possible method for addressing discrepancies between observations.  \nThis study introduces a temperature ratio as a free parameter, denoted as ξ si , between dark radiation and photons. This departs from previous work where the integer flavors of dark radiations were specified [34]. This parameter along with the interacting strength log 10 [ G eff / MeV -2 ] captures all the properties of SIdr. When neglecting parameter degeneracies, theoretical analysis shows that without explicitly incorporating additional radiation, N eff = 3 . 02 ± 0 . 27, and an increased Hubble constant H 0 = 69 . 3 ± 2 . 0km/s/Mpc naturally emerges when all dark radiations were are tightly coupled. This makes SIdr a hopeful solution to the Hubble tension.  \nWe employed the latest CMB data from the Planck experiment, BAO data from SDSS, supernova (SN) data from Pantheon, and local H 0 measurements from SH0ES to constrain the SIdr model and address the Hubble tension. The results have been further validated using data from the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT).  \nThe predictions for the most cosmological parameters are consistent for all the datasets, and an increased Hubble constant, i.e. approximately H 0 ≈ 70km / s / Mpc, has been achieved. The SIdr only (scenario A) aligns more closely with Planck results for parameters such as N eff = 3 . 23 ± 0 . 25, although it degrades the fits to both the CMB data and the baseline dataset. Furthermore, a strong interaction result does not fit well compared to a medium interaction, with the limitation being ΛCDM. When considering the inclusion of FSdr, the existence of SIdr is disfavored. Although incorporating SIdr leads to an increase in the Hubble constant, it also increases the effective number of neutrino species ( N eff ). Considering the results from Fig 5 and Fig 6, which illustrate scenarios A ( N fs = 0 , ˜ N si ) and B ( N fs , ˜ N si ), we observe that SIdr helps alleviate the tension. However, in the latter case, we get ˜ N si ∼ 0. This indicates that while SIdr may exist, it cannot dominate over FSdr in addressing the Hubble tension. To further investigate the evidence for SIdr, we compare the four scenarios A, B, C, D (discussed in Sec 4.3) using both Planck data alone and in combination with SH0ES data. Although SIdr can reduce the tension even without an H 0 prior, the inclusion of SH0ES data predicts the same interacting strength for SIdr, thereby diminishing confidence in SIdr as a solution to the tension, as opposed to additional radiation. This concern is further reinforced when considering FSdr. Therefore, no evidence supports the hypothesis that SIdr can resolve the H 0 problem.  \nAs another cross-check, we analyze the SPT and ACT data, which provide more moderate results. These datasets support the existence of approximately 1 flavor SIdr in the radiations component while raising N eff to 3.52. The all-coupling case is consistently disfavored by all the datasets since it degrades the fits. Although ACT-SPT data leaves some room for the existence of SIdr, it doesn't play the role that solves the Hubble tension.  \nLastly, we employed a Fisher forecast analysis to predict future constraints on the SIdr model. A 32% measurement on SIdr specie will be obtained using CMB-S4 experiment. When combined with AliCPT and Planck experiments, we will get a 26% measurement.  \nFSdr leaves another featured observation on Large Scale Structure (LSS), which includes more modes than the 2 dimension CMB map. Thus, researching dark radiations with LSS will give more information. We leave this topic in the near future.  \nAcknowledgments. The authors thank Pierre Zhang for the fruitful comments. Thanks the pioneering work by Das and Kreisch. Thanks to the public code cobaya, Class and Class SInu. This work is supported in part by the National Key R&D Program of China (2021YFC2203100), CAS Young Interdisciplinary Innovation Team (JCTD-2022-20), NSFC (12261131497, 11653002), 111 Project for 'Observational and Theoretical Research on Dark Matter and Dark Energy' (B23042), Fundamental Research Funds for Central Universities, CSC Innovation Talent Funds, USTC Fellowship for International Cooperation, USTC Research Funds of the Double FirstClass Initiative, CAS project for young scientists in basic research (YSBR-006). Kavli IPMU is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. We acknowledge the use of computing facilities of astronomy department, as well as the clusters LINDA & JUDY of the particle cosmology group at USTC.", 'Appendix A Other dataset': 'As the Placnk polarization data limits the existence of SIdr, we investigate the possible SIdr in ACT and SPT data. 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