Axi-Higgs Cosmology Leo WH Fung1;a, Lingfeng Li1;b, Tao Liu1;c, Hoang Nhan Luu1;d, Yu-Cheng Qiu1;e, S.-H. Henry Tye1;2;f 1 Department of Physics and Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong S.A.R., China 2 Department of Physics, Cornell University, Ithaca, NY 14853, USA Email: a [email protected], b iaslfl[email protected], c [email protected], d [email protected], e [email protected], f [email protected] Abstract If the electroweak Higgs vacuum expectation value v in early universe is ∼ 7 1% higher than its present value v0 = 246 GeV, the Li puzzle in BBN and the CMB/ΛCDM tension with late-universe measurements on Hubble parameter are mitigated. We propose a model of an axion coupled to the Higgs field, named \axi- −30 −29 17 18 Higgs", with its mass ma ∼ 10 −10 eV and decay constant fa ∼ 10 −10 GeV, to achieve this goal. The axion initial value aini yields an initial ∆vini=v0 ∼ 0:01 throughout the BBN-recombination epoch and a percent level contribution to the total matter density today. Because of its very large de Broglie wavelength, this axion matter density !a suppresses the matter power spectrum, alleviating the arXiv:2102.11257v3 [hep-ph] 3 Sep 2021 CMB/ΛCDM S8/σ8 tension with the weak-lensing data. It also explains the re- cently reported isotropic cosmic birefringence by its coupling with photons. Adding the axion (m ∼ 10−22 eV) in the fuzzy dark matter model to the axi-Higgs model allows bigger ∆vrec and !a to address the Hubble and S8/σ8 tensions simultane- ously. The model predicts that ∆v may be detected by the spectral measurements of quasars, while its oscillation may be observed in the atomic clock measurements. Contents 1 Introduction 1 2 Big-Bang Nucleosynthesis 7 3 Hubble Tension 11 3.1 Standard ΛCDM Model . 12 3.2 ΛCDM Model with δvrec 6=0............................. 13 4 Axi-Higgs Model 20 4.1 Single-Axion Model . 22 4.2 Two-Axion Model . 26 5 S8/σ8 Tension 29 5.1 Matter clustering in ΛCDM and observations . 29 5.2 Variations of S8/σ8 .................................. 30 6 Hubble Tension versus S8/σ8 Tension 33 7 Isotropic Cosmic Birefringence 35 8 Testing the Axi-Higgs Model 37 9 Conclusions 40 A Redshifts of Recombination and Baryon Drag 41 B More Details of Calculating YjX 42 1 Introduction Cosmology has made tremendous progress since the mid-20th century, moving from a specula- tive to a precision science. The inflationary universe scenario, big bang nucleosynthesis (BBN), cosmic microwave background (CMB) and structure formation have merged theory and obser- vational data into a generally accepted picture of our universe. Two prominent successes in precision cosmology are the measurement of BBN and the deter- mination of Hubble parameter H0. However, as more and better data becomes available while theoretical understanding is progressing, tensions (or frictions/conflicts) emerge. They include in particular the four cases listed below. 1 1. While theoretical estimates for the primordial abundances of helium 4He and deuterium D in BBN are consistent with the observational data, the theoretical prediction for the primordial Lithium abundance, 7Li=H = (5:62 ± 0:25) × 10−10, is too big compared to its observed value 7Li=Hobs = (1:6 ± 0:3) × 10−10. This ∼ 9σ discrepancy is known as the 7Li puzzle [1]. 2. The determination of the Hubble parameter value from the CMB measurement in Planck 2018 (P18) within the Λ cold dark matter (ΛCDM) model (early universe), namely H0;P18 = 67:36 ± 0:54 km/s/Mpc [2], is smaller than H0;late = 73:3 ± 0:8 km/s/Mpc, the Hubble parameter value obtained from late-time (with redshift z < 2) measurements [3]. This ∼ 4 − 6 σ discrepancy is referred to as the Hubble tension. 3. Recently, a measurement of isotropic cosmic birefringence (ICB) was reported, based on EB the cross-power (parity-violating) Cl data in CMB [4]. It excludes the null hypothesis at 99.2% confidence level (C.L.). This needs to be explained too. 4. The weak lensing measurement of S8 together with the clustering parameter σ8 [5] yields a value smaller than that given by the CMB/ΛCDM value. This ∼ 2 − 3 σ [6,7] discrepancy poses another problem to our understanding of the universe. In this paper, we present a simple model, with an axion coupled to the Higgs field and hence named \axi-Higgs", to solve or alleviate these four tensions. Let us consider the possibility that the Higgs vacuum expectation value (VEV) in the standard model (SM) of particle physics, 1 v0 = 246 GeV today, is ∼ 1% higher in the early universe, i:e:, δvini = (vini − v0)=v0 ∼ 1% . If the massive gauge bosons, quarks and charged leptons in the SM all have masses of about δvini higher than their today's values, the discrepancies in the first two cases will be substantially reduced. We propose that a δv > 0 is the leading effect in modifying the ΛCDM model in the early universe. 7 That a δvBBN & 1% at BBN time solves the Li problem is known [1,8{17]. That an electron mass me / v about 1% higher at recombination time (i:e:, δme ' δvrec) has been suggested to alleviate the Hubble tension [18, 19]. To implement both, the Higgs VEV with δv ∼ 1% needs to stay throughout the BBN-recombination epoch (from seconds/minutes to 380,000 years after −16 −1 the Big-Bang) and then drops to its today's value where its drift rate is . 10 yr , to satisfy the observational bounds [20{22]. Such a setup can be naturally achieved in string theory. Consider the scenario of brane world in Type IIB string theory, where anti-D3-branes span our three spacial dimensional universe. The SM particles are open-string modes inside the branes. It is known that the electroweak- scale interactions will shift the cosmological constant Λ by many orders of magnitude above 1 X−Xref In this paper, we will take a set of shorthand notations, including ∆X = X − Xref , δX = d ln X = Xref @ ln Y d ln Y and YjX = @ ln X , YjjX = d ln X , unless otherwise specified. If X = !b, the notations of YjX and YjjX will be further simplified as Yjb and Yjjb etc. 2 its exponentially-small observed value, so fine-tuning is needed to have the right value. In 2 the supergravity (SUGRA) model proposed recently [23], a superpotential W = X(msF (A) − κHuHd) + ··· is introduced. Here A stands for complex-structure (shape) moduli and dilaton that describe the compactification of extra dimensions and X is a nilpotent superfield which projects the two electroweak Higgs doublets Hu, Hd to the single Higgs doublet φ. This leads to the axi-Higgs model, 2 2 2 a 2 y 2 a V = mafa 1 − cos + msF (a) − κφ φ ; with F (a) = 1 + C 2 : (1.1) fa MPl In this model, the axion-like field a is a pseudo-scalar component in A. This axion starts with an initial value a in the early universe. We normalize F (a) to be F (a = 0) = 1, such that the inip p p Higgs VEV v0 = 2ms= κ = 246 GeV and the Higgs boson mass mφ = 2ms κ = 125 GeV. So this model is characterized by four parameters, namely ma; fa;C and aini. The perfect square form of the Higgs potential, where the Higgs contribution to Λ is completely screened by the 4 supersymmetry (SUSY) breaking anti-D3-brane tension ms, allows a naturally small Λ [24,25]. Notably, this perfect square form of the Higgs potential, together with the damping effect of the Higgs decay width (Γφ ' 4 MeV), is crucial in yielding the desirable feature of the model: the effect of the Higgs field evolution is totally negligible in the axion evolution, but the axion evolution significantly affects the evolution of the Higgs VEV 2. Note that all parameters in the standard electroweak model are unchanged. In particular, the electron Yukawa coupling is unchanged, so δv = δme. 2 2 Starting with an initial δvini = Caini=2MPl for aini 6= 0, via the mis-alignment mechanism [27{ 3 29], δv evolves after the recombination epoch (z ∼ 10 ) when 3H(t) drops below ma. We find the favored axion mass −30 −29 ma ∼ 10 − 10 eV : (1.2) Here the upper limit of ma is determined by whether δv will drop too much by the time of −29 recombination, which happens for ma > 3:3 × 10 eV. The lower limit of ma, instead, is set by the late-time measurements of δv(t) or its drift rate. The current atomic clock (AC) −30 measurements on d(δv)=dtjt0 [22] excludes ma . 1:6 × 10 eV at 95% C.L. Such a mass scale is compatible with string theory and typical axion masses [30, 31]. Note that it is very difficult to satisfy the AC bound today if we introduce a scalar field ' instead, as F ('), a counterpart of F (a) in Eq. (1.1), will contain a linear term with a coefficient too big in the absence of fine-tuning. Physically, an upward variance of the Higgs VEV will reduce Yp but raise D/H. The current experimental bounds on Yp and D/H are still compatible with a change of percent level in v if η is also 1 − 2% larger than its reference value 6:127 × 10−10 [2, 32]. Beyond that, it is suggested 2The name "axi-Higgs" has also been used to refer to a boson in a model [26] different from the one described by Eq.