A Sequestered Axion-Dilaton Unification: Heterotic Kalb-Ramond Flux as the Common Origin of the Dark Sector, Big Bounce, Cosmic Birefringence and Dipole

001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 A Sequestered Axion-Dilaton Unification: Heterotic Kalb-Ramond Flux as the Common Origin of the Dark Sector, Big Bounce, Cosmic Birefringence and Dipole Pierre Moutounet-Cartan1 1Independent Researcher, Alumni, Department of Mathematics, Imperial College London, United Kingdom. Contributing authors: [email protected]erial.ac.uk; Abstract We develop a parameter-sparse cosmology in which the ten-dimensional dilaton and Kalb-Ramond compactify to a four-dimensional dilaton ϕ(acting as dark energy) and an axion ac(acting as dark matter) whose dynamics are locked to a single, gauge-sequestered trajectory in moduli space. At very early times, we show that the axion-fermion interaction results in a Big Bounce due to the axial chemical potential that resolves the Big Bang singularity. Post-drag, the integrated transfer implies a canonical dilaton excursion that yields upward shift in H0against ΛCDM and a suppression of late growth that lowers S8, as well as yielding a slightly phantom effective equation of state today of weff ϕ≃ −1.02. As the axion reservoir empties, the energy transfer from axionic dark matter to the dark energy dilaton self-terminates and the Universe approaches the Copeland-LiddleWands dust tracker with an asymptotically Minkowski, horizon-free, Einstein-de Sitter-like future. The Green-Schwarz coupling rotates the CMB polarisation proportionally to the dilaton excursion and the holomorphic one-loop threshold, yielding the hard prediction β≃0.29◦±0.06◦to solve for the current Hubble tension, in excellent agreement with current birefringence measurements. A correlated dipolar pattern is predicted when a super-horizon dilaton mode is present. Keywords: quantum cosmology, Kalb-Ramond flux, axion, dilaton, dark energy, dark matter, cosmic microwave background, birefringence, Universe, cosmic acceleration 1 Heterotic String, Compactification, & Sequestration 1.1 State of the art Heterotic string theory generically yields a 2-form Kalb-Ramond (KR) field with 3-form flux H3and a dilaton Φ. Upon compactification, the 4D dual of H3is an axion-like field with derivative couplings to the axial current of fermions, (∂µa/fa)Jµ 5[1,2]. In heterotic compactifications, the Green-Schwarz mechanism yields B∧(TrF∧F−TrR∧R), so the 4D gauge kinetic function is holomorphic, fg(Z), with ℜ(fg) controlling the gauge coupling and ℑ(fg) sourcing F˜ F; one-loop thresholds introduce modular dependence [3–6]. Axions as dark matter and ultralight condensates are well studied [7,8]. Quintessence with exponential potentials admits tracking solutions, but it alone does not accelerate at late times without field coupling with 1 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 matter [9]. The latest DESI Collaboration analysis has confirmed that the evidence for dynamical dark energy, particularly at low redshift (z≲0.3), is robust and stable under different modelling choices [10]. This suggests that the current ΛCDM model might have finally reached its limits of applicability, signalling the onset of a Kuhnian crisis. We propose a framework where the KR flux sources an axion (dark matter) that slowly decays its energy into the dilaton (dark energy) which drives an extra-nudge to cosmic acceleration. 1.2 Compactification in an Einstein frame We work at tree level in the 10-D string frame with the NS-NS sector (GMN , Φ, BMN ) and gauge fields AM. The heterotic bosonic action schematically is [3,11,12]: S10 =1 2κ2 10 Zd10X√−Ge−2Φ R+ 4(∂Φ)2 −1 12HMNPHMNP! −α′ 8κ2 10 Zd10X√−Ge−2Φ Tr FMN FMN +α′ 4κ2 10 ZB∧(Tr F∧F−Tr R∧R) + ··· , with H=dB −α′ 4(ΩYM 3−ΩL 3). We compactify on a 6D manifold M6. The 4D Einstein frame metric is g(E) µν =e−2Φg(4) µν so that the 4D Planck mass in an Einstein-frame is M2 Pl =V6 κ2 10 . We define the heterotic axion-dilaton with K¨ahler potential K=−ln(S+¯ S)−3 ln(T+¯ T). Start from the string-frame kinetic term S10 ⊃ − 1 12κ2 10 V6Zd4x√−g4e−2ΦHµνλHµνλ. Imposing the Bianchi identity, the axion is given by the KR duality Hµνλ =e2Φ fa ϵµνλσ∂σac ⇒HµνλHµνλ =6e4Φ f2 a ∂µac∂µac. Note that the Einstein-frame metric is given by g(E) µν =e−2Φg(4) µν , with a determinant √−g4= e4Φ√−gEand an inverse metric gµν (4) =e−2Φgµν (E). Therefore, the kinetic term becomes: S4=−1 2 V6 κ2 10f2 a e4Φ Zd4x√−gE∂µac∂µac. Canonical normalisation yields fa(Φ) = √V6 κ10 e2Φ =MPl e2Φ.(1) Similarly, one can derive Vϕ(Φ) = V6⟨H2⟩ 12κ2 10 e−4Φ =⟨H2⟩ 12 M2 Pl e−4Φ. In the 10D string frame, the gauge kinetic term is given by S10 ⊃α′ 8κ2 10 Zd10X√−Ge−2Φ Tr FMN FMN . Compactifying on a 6D manifold of string frame volume V6, keep only the 4D components Fµν and integrate over the internal space, so that in an Einstein-frame by the usual Weyl rescaling of the 4D metric we get S4⊃ −1 4ℜ(fg(ϕ)) Zd4x√−gETr FµνFµν E, with the real part of the Einstein-frame gauge kinetic function ℜ(fg(Φ)) = Cg α′V6 κ2 10 e−2Φ =Cgα′M2 Ple−2Φ, where Cgis a numerical constant. Along the gauge-sequestered direction defined below, the real part of fgis constant, i.e., ∂ϕℜ(fg)≃0. 2 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 1.3 Gauge sequestration Let us derive the axion decay fain a no-gaugeeffect basis, i.e. along the modulus direction where the 4D gauge coupling (hence any confinement scale Λ) is constant, while the axion decay constant favaries. Consider the two dimensionless scalars x≡Φ, y ≡ln V6. Thus, in the Einstein frame we derived ln fg=y−2x+··· ,ln fa= 2x+1 2y+··· . A displacement (δx,δy) changes these as δln ℜ(fg) = −2δx +δy, δ ln fa= 2δx +1 2δy. The sequestered path means keeping the gauge coupling (hence ℜ(fg)) fixed, i.e., δln ℜ(fg)=0⇒δy = 2δx. Along this direction, δln fa= 2δx +1 2(2δx)=3δx. Hence, at the level of coordinates (x, y), fa grows exponentially with slope 3 per unit along the sequestered path. To fix the normalisation without free parameters, we use the standard 4D N= 1 K¨ahler potential for the universal moduli, K=−ln(S+¯ S)−3 ln(T+¯ T) with s∝exp(−2x+y), t ∝exp(y/3). (2) Thus, this gives the kinetic term dln s=−2dx+dy, d ln t=1 3dy ⇒ Lkin =M2 Pl 2GIJ ∂µφI∂µφJ with the matrix given by G=2−1 −12 3, φI= (x, y). The sequestered tangent is v= (1,2). Its norm in the canonical metric is Gvv =vTGv = 2 − 4+8/3 = 2/3. Define the canonically normalised rolling field along the path by dϕ =MPlpGvv ds =MPlr2 3ds ⇒∂ϕln fa(ϕ) = 3 MPlp2/3=9 √6 1 MPl . The NS-NS flux potential scales as e−4Φ thus along v: ∂ϕln V(ϕ) = −2√6 MPl . Geometrically, the pair (ϕ, ac) forms a warped cylinder (an S1fibre over the dilaton base): ds2=dϕ2+f2 a(ϕ)da2 c, fa(ϕ)∝exp 9 √6 ϕ MPl . Here acis the circle angle (the axion), while ϕis the fibre radius set by the dilaton. Coherent axion dark matter is just uniform circular motion on that fibre. Dark matter (axion) is angular momentum stored on the axion circle, while dark energy (dilaton) is the slow radial expansion of that circle. 1.4 Dark sector energy transfer In the KR-dilaton-axion sector, energy transfer from the axion to the dilaton arises whenever axion parameters inherit dilaton dependence from compactification. The KR three-form sets the axion decay constant through the internal volume and the dilaton (so that fa(ϕ) varies in the Einstein frame), and the same moduli control the axion kinetic metric and the non-perturbative scale that generates its potential. Consequently, the homogeneous background obeys split continuity equations with a source term Qa↔ϕ. A rolling dilaton therefore sources an energy transfer from axion (dark matter) into the dilaton (dark energy), providing a natural and minimal mechanism to sustain a large dilaton energy density even when the bare Einstein-frame potential is steep. The total canonical Einstein frame Lagrangian is given by L=1 2(∂ϕ)2−Vϕ(ϕ) | {z } Dilaton ≡Lϕ (3) +1 2Ka(ϕ)(∂ac)2−Va(ac, ϕ) | {z } Axion ≡La (4) −1 fa(ϕ)(∂µac)Jµ 5 | {z } Interaction ≡Laϕ +··· (5) 3 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 where Kais the compactification axion kinetic metric. We split stress-energy of the axion sector such that ∇µTµν a= (∂ϕLa+∂ϕLaϕ)∂νϕ=−∇µTµν ϕ. Thus, Qa↔ϕ=˙ ϕ ∂ϕ(La+Laϕ) (6) =˙ ϕ1 2(∂ϕKa)(∂ac)2−∂ϕVa+(∂ϕfa) f2 a (∂µac)Jµ 5. (7) For axionic dark matter in a quadratic well, ac(t)≃a0cos(mat+θi), with negligible spatial gradients. Over many oscillations: ⟨pa⟩ ≃ 0,⟨(∂ac)2⟩ ≃ ⟨˙ a2⟩ ≃ ρa,⟨a2⟩ ≃ ρa m2 a . Thus, for the axionic potential, we also have along the gauge sequestered vector ∂ϕVa=∂ϕ(m2 aa2 c)/2 which yields ⟨∂ϕVa⟩ ≃ (∂ϕln ma)ρa=−(∂ϕln fa)ρa. On the other hand, we can parametrise the axial current in the co-moving frame as Jµ 5= (n5,j5) with ⟨j5⟩= 0 by isotropy. Thus, by neglecting axion spatial gradients, we have that Jµ 5≃˙ acn5.Furthermore, by using the derived expression for fa(1), we have ∂ϕln fa=9 √6MPl . Finally, since we have chosen the canonical axion, it is trivial by definition that ∂ϕKa= 0, and thus along the gauge sequestered vector, we have Qa↔ϕ≃9 √6 ˙ ϕ MPl ρa+˙ acn5 fa.(8) 2 From Big Bounce to Late Universe 2.1 The Big Bounce Note that the derivative fermion-axion coupling yields an axial chemical potential given by µ5=˙ ac fa . For relativistic fermions (per degrees of freedom), we have [13]: n5=g5 6µ5T2+Oµ3 5≃g5 6 ˙ ac fa T2 where g5is the axially active degrees of freedom. The anti-friction can pump ˙ acup to the energy cap [14], i.e., ˙ a2 c≲2ρrad = 2h∗T4, h∗=π2 30g∗ where g∗is the effective relativistic degrees of freedom. Thus, at saturation ˙ acn5 fa =g5 3h∗ T6 f2 a⇒Tb=2Vϕf2 a g5h∗1/6 . A conservative estimate here is to cap the potential by the bath at the stall, that is Vϕ(ϕb)≤ρrad(Tb) = h∗T4 b. If we saturate this bound, we find Tb=r2 g5 fa.(9) At a momentary dilaton stall, the fourdimensional axion–fermion derivative coupling obtained by compactifying the Kalb-Ramond sector is equivalent—after integrating out the axion and the non-propagating torsion—to the Einstein-Cartan axial–axial four-fermion interaction. Indeed, this is essentially a bona fide auxiliary map. Let ϕb=ϕ(tb) be the dilaton at the bounce time tb. The following three conditions are met (i) the dilaton stalls, i.e., ˙ ϕb= 0 so that ∂µfa(ϕb) = 0 thus, the axion decay is constant momentarily; (ii) fermions are effectively chiral, 4 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 hence, ∂µJµ 5= 0, thus, anomalies are momentarily negligible; (iii) the axion mass is irrelevant on that timescale. Let us start from the derivative coupling to the total axial current and complete the square, so that La=1 2(∂ac)2−1 fa (∂ac)·J5 =1 2∂ac−J5 fa2 −1 2f2 a J5µJµ 5 | {z } L(KR) b . We can match the terms of L(KR) b=L(EC) b which yields the equivalence: fa(ϕb) = r8 3MPl =r1 3πG (10) in natural units (ℏ=c= 1). Substituting the result of the (10) into (9) yields Tb=r2 g5 fa(ϕb) = 4 √3g5 MPl = 0.34 MPl 45 g51/2 . (11) Physically, this means that the derivative axion–fermion coupling required to halt the dilaton at the same epoch as the Einstein-Cartan bounce is set by a decay scale at the Planck scale. The sequestered string-motivated axion–dilaton model reproduces the Einstein–Cartan bounce scale, aligning a stringy derivative interaction with the EC spin–torsion mechanism for the very early Universe. Interestingly, away from the exact bounce, the first deviations are controlled by ϵϕ=∂µfa faH FRW −−−→ ϵϕ∝|˙ ϕ| H, ϵa=m2 a H2 ∂ϕln Λ −−−−→ =0 ϵa∝1 (Hfa)2. With entropy per co-moving volume conserved, we have the scale factor at bounce given by [14]: ab a0 =T0 Tbg∗S(T0) g∗S(Tb)1/3 =T0 MPl √3g5 4g∗S(T0) g∗S(Tb)1/3 . By using the Standard Model degrees of freedom at high temperatures, and the present particle-horizon radius as the co-moving reference, this yields ab= 42.8µma0 4.6·1026 mg5 451/2 g∗S(T0) 3.91 1/3106.75 g∗S(Tb)1/3 T0 2.725 K2.44 ·1018 GeV MPl . Hence, the Universe scale factor was about 40 microns at the bounce. This is fairly close to the derivation of Pop lawski [15] who found about 50 microns purely in the context of the EinsteinCartan gravity. If right-handed states do not participate (e.g. some suppressed Yukawa/chirality flips at very high T), then only left-handed fermions are axially active, thus min g5= 24. If the Standard Model is extended to include right-handed neutrinos and if we assume that all chiralities are fully thermalised, then max g5= 48. Thus, the Universe scale factor at the dilaton stall is in the range 31.3µm≲ab≲44.2µm. If the collapsing universe (where t<tb) had a similar ”set-up” as our Universe, one would not expect it to collapse due to its accelerating nature. Pop lawski [15], Dymnikova [16], Gazta˜naga et al. [17] have theorised that the ”Big Bounce” represents the white hole side of an Einstein-Rosen bridge. It is a natural choice, as the extreme matter densities can arise from a black hole, generating non-negligible axial chemical potential. In our KR axion–fermion bounce, the asymptotic infinite limit for His never reached, and at the bounce we have that H= 0 because the KR negative-stiff piece cancels the blue-shifted fluids exactly at the bounce. Immediately after the bounce, Hrises extremely rapidly until it reaches a maximum. Once radiation takes over, Hstarts decreasing. Thus, momentarily, at and around tb (at the throat of the Einstein-Rosen bridge) the Universe is in a de Sitter solution (wϕ=−1) – the de Sitter skin or core. The characteristic skin Hubble scale Hsis given by Hs=1 MPl rVϕ(ϕb) 3. 5 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 By using the saturation result at the stall, this yields Hs=16√3h∗ 9g5 MPl = 0.405 MPl h∗ 35.121/245 g5 which is approximately 1.5·1042s−1. The deSitter-skin bounce happens so early and fast that CMB scales were enormously super-horizon then, so causal microphysics at the skin cannot imprint observable non-Gaussianity in the CMB. The skin needs ∆tskin ∼N/Hs(with N= O(1)) of proper time to establish the required n5 at bounce. The proper time from the horizon to the high-curvature core for a radial geodesic in Schwarzschild is τff∼πGM/c3where Mis the mass of the parent black hole. Requiring τff≳ N/Hsyields M≳4g5N c3r3 h∗ M3 Pl f2 a(ϕb) (10) −−→ 3 2r3 h∗ g5N c3MPl = 19.73 MPl 35.12 h∗1/2g5 45N 1 in natural units (c= 1). This is likely to be respected by the vast majority of black holes. Now, if we saturate the generalised second law at the bounce by setting the black hole BekensteinHawking entropy equal to the entire FRW entropy at ab[14,18]: SBH =πr2 s ℓ2 Pl := SFR =sb 4π 3a3 b∼g∗S(Tbab)3 where ℓPl is the Planck length. Thus, solving for rsyields rs∼ℓPl√g∗STbab TPlℓPl 3/2 . Hence, by plugging ab, this gives that our Universe’s parent black hole has a Schwarzschild radius rs∼1010 metres and mass M∼106M⊙. This parent black hole falls into the supermassive category, with a Planck-scale de Sitter core. 2.2 Cosmic acceleration Assume the axion potential arises from a ϕindependent microscopic amplitude in the angular variable, so that near the minimum ma∝1/fa. For coherent axion dark matter with ma≫H and slow roll of ϕ, the adiabatic invariant yields ρaa3fa= constant. Dark-sector energy exchange obeys (cycle-averaged) ˙ ρa+ 3Hρa=˙ ϕρa∂ϕln ma=−˙ ϕρa∂ϕln fa ˙ ρϕ+ 3H(1 + wϕ)ρϕ=−(˙ ρa+ 3Hρa). Integrating in a co-moving volume yields ρϕ ρat0 =Ωϕ Ωa = exp ZΦ ∂ϕln fa(ϕ)dϕ−1 (12) = exp 9 √6 ∆ϕ MPl −1.(13) Rearranging yields a dilaton roll of ∆ϕ MPl =√6 9ln 1 + Ωϕ Ωa(14) By using fiducial values from Ade et al. [19], we find ∆ϕ/MPl ∼0.3, which is consistent with the finding of Lodha et al. [10] who described ∆ϕ∼(0.2−0.4)MPl. If the dilaton has expanded by about 0.3MPl to generate the observed darkenergy to dark-matter abundance ratio, the Universe must have expanded slightly more than in the ΛCDM expectation where ρϕis constant. Let us keep the early-time physical densities fixed at their CMB values ωb, ωc. If the axion-to-dilaton transfer suppresses cold dark matter by a factor exp(−ξ) after the baryon drag epoch (zd), then fitting late-time distances gives τH≡H0 HΛ 0 =1−ωc ωb+ωc1−e−ξ(zd)−1/2 (15) where we define ξby the dilaton excursion ξ(z)≡9 √6 ∆zϕ MPl ,∆zϕ≡ϕ0−ϕz. The relationship 15 depends only on the total post-drag excursion ∆zdϕof the dilaton. Solving for ξand substituting yields ∆zdϕ MPl =√6 9ln 1 + τ2 H−1ωb+ωc ωc,(16) 6 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 which, by using the Hubble uplift between Aghanim et al. [20], Riess et al. [21] yields ∆zdϕ/MPl = 0.047 ±0.010. Studies which do not take into account the interaction will find a phantom effective equation of state. Indeed, weff ϕ=wϕ−Q 3Hρϕ≃ −1.0186 ±0.0040. Similarly, we have that ΩKR m= ΩΛ m−ωc(1 − e−ξ(zd)) = 0.285 ±0.007, and thus, the S8shift is given by SKR 8 SΛ 8 = 0.956 ±0.016 ⇒SKR 8= 0.775 ±0.017 This is in agreement of local observations, within <1σof Miyatake et al. [22], Heymans et al. [23], Abbott et al. [24]. 2.3 Final fate of the Universe Because the axion decay fagrows as the dilaton ϕrolls, the axion energy is additionally diluted relative to dust. In this case, the derivative coupling implies a homogeneous transfer Qa↔ϕat the background level (Eq. (8)). Thus, when the axion reservoir is exhausted, Qa↔ϕ→0 and the wellknown scaling solution of an exponential potential in a matter background is approached. On the gauge-sequestered trajectory one finds ws=∂ϕ ∂ln a a→∞ −−−→ r3 8⇒∂ln fa ∂ln a a→∞ −−−→ 9 4.(17) The decay constant grows as a fixed power of the scale factor: fa∝a9/4. This kinematically guarantees that the coupling-driven transfer weakens rapidly as the universe expands, because the axion parameters race away from their initial values. Thus fa∝a9/4,ma∝f−1 a∝a−9/4, and for oscillatory axionic dark matter, ρa∝a−21/4, Qa↔ϕ∼Hρa∝a−3/2a−21/4=a−27/4. (18) The system lands on the Copeland-Liddle-Wands exponential-potential tracker [9]. For an exponential V(ϕ) in a dust background (wm= 0), the scalar admits a scaling solution for λ= 2√6: Ωϕ→3 λ2=1 8, wϕ→0, H →0+.(19) The dilaton locks onto the dominant component and effectively becomes dust (wϕ→0). Because the dilaton has c2 s≃1, it is smooth on sub-horizon scales and does not contribute to clustering; the clustered fraction is Ωclust = 1 −Ωϕ= 7/8. Although the dilaton acts like dust in the background, it does not clump on small scales. Only the 1/8 matter gravitates into structures, which is why growth continues, but about 8% more slowly than in pure Einstein-De Sitter. Further, Copeland et al. [9] requires λ2>20 due to nucleosynthesis constraints, which are here matched as λ= 2√6⇒λ2= 24 >20. Note that the deceleration parameter is given by q=1 21−3Ωϕ+w2 s+ Ωra→∞ −−−→ 1 2. The background expansion is the classic EinsteinDe Sitter matter law a∝t2/3, thus acceleration is transient by construction. Finally, using the standard general relativity fitting form ς= ∂ln D/∂ ln a≃Ωγ m, and the coupled dark energy correction for a background-only energy transfer, we have today γKR 0≈ 31−wϕ+Q Hρϕ 5−6wϕ = 31+2wϕ−3weff ϕ 5−6wϕ≈0.56, which compares to γΛ 0≈0.55 – thus today the growth is similar (albeit slightly more suppressed in KR) in both models. Asymptotically, the growth rate ςtends to a constant given by the exact algebraic limit of the GR growth equation (with ∂ln H/∂ ln a=−3/2 and Ωm→7/8): ςt→∞ −−−→ 1 2 −1 2+r1 4+ 6Ω∞ m!=√22 −1 4 and thus γKR ∞≈logΩ∞ mς∞= log7/8 √22 −1 4!≈0.60. 7 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 Also note that R≡6( ˙ H+2H2)→0+,1 aH → ∞, a Z∞dt a(t)→ ∞ (20) Eq. (20) shows that the Ricci scalar decays to zero, indicating that spacetime curvature is washed away and the geometry asymptotes to Minkowski. The co-moving Hubble radius diverges and the future conformal time is infinite, which together imply the absence of a future event horizon. Physically, while bound structures persist locally, the large-scale universe becomes increasingly open to communication and causal influence, even as its contents grow dilute and cold. This makes this quintessence model attractive for quantum-relativity unification, as a finite de Sitter entropy hints at a finite number of states, clashing with standard QFT intuition and raising puzzles. 3 Cosmological Microwave Background Birefringence & Dipole 3.1 Birefringence from one-loop thresholds Before we continue on the derivations of the Green-Schwarz induced CP-odd angle, it is important to show that the logarithmic Dedekind slope is π/12, as this will be a result that will be used in the derivation of the CMB birefringence. Set q≡exp(2πiT) with ℑ(T)>0, so that Tlies in the upper-half plane. The Dedekind function has the product and logarithmic derivative η(T) = q1/24 ∞ Y n=1 (1−qn) ⇒ln η(T) = 1 24 ln q+ ∞ X n=1 ln(1 −qn) ⇒∂Tln η(T) = πi 12E2(T) where E2is the quasi-Eisenstein series of weight 2. Further note that as ℑ(T)≳1 (for large volume), q→0, and thus E2(T) = 1 + O(q). Hence, in that case ℑ(∂Tln η(T))ℑ(T)≳1 −−−−−→ π 12.(21) Let H={T∈C:ℑ(T)>0}with the SL(2,Z) action γ:C→C, γ(T) = aT +b cT +d, ad −bc = 1. A half-integral weight-kautomorphic form ℘k with k∈R>0obeys [5,25]: ℘k(γ(T)) = ι(γ)(cT +d)k℘k(T) with a unitary multiplier ι(γ). The Dedekind function η(T) has k= 1/2 and the usual multiplier. Let Lk→Hbe the holomorphic line bundle whose local holomorphic frame ek(T) transforms as ek(γ(T)) = ι(γ)(cT +d)−kek(T). Then holomorphic sections sk(T)≡℘k(T)ek(T) are exactly weight-kforms. Equip Lkwith the SL(2,Z) invariant Petersson Hermitian metric ||sk(T)||2 P=ℑ(T)k|℘k(T)|2. For a holomorphic section sk=℘kek, it is convenient to work with the logarithmic version: A=∂ln ||sk||2 P−∂ln ||℘k||2 P=∂(kln ℑ(T))+∂ln ℘k(T). Thus the (1,0) connection acting on ℘kis Dk Tln ℘k(T)≡∂Tln ℘k(T) + k 2 1 T−¯ T =∂Tln ℘k(T) + k 4iℑ(T). This is the standard modular-covariant logarithmic derivative where the extra ∝k/i term is the Chern connection term coming from the Petersson metric. From (21), we have at k= 1/2: D1/2 Tln η(T) = iπ 12 ˆ E2(T),ˆ E2(T)≡E2(T)−3 πℑ(T) where ˆ E2is the non-holomorphic modular completion (weight 2) [26, p. 19]. This is the precise sense in which the Petersson completion cancels the quasi-modular anomaly: the right-hand side transforms covariantly (weight 2), so the 1-form Ais globally well-defined on the modular curve. Along a path in moduli space, the infinitesimal change of the CP-odd angle is encoded by the 8 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 imaginary part of the Chern connection pulled back to the path: ϖ≡ ℑh∂Tln pℑ(T)η(T)dTi=ℑ(AdT). (22) Using T−¯ T= 2iℑ(T), one may rewrite ϖin a purely geometric, modular-invariant form by expanding ˆ E2: ϖ=π 12 dT T−¯ T+d¯ T ¯ T−T+··· =π 12 d(T−¯ T) T−¯ T+··· (23) up to terms are the exponentially small q-series ripples inherited from (21); the leading term is fixed entirely by the Dedekind half-weight. Along the sequestered roll we keep ℜ(T) fixed and vary ℑ(T). Introduce the canonical radial coordinate in moduli space: u≡ln ℑ(T), uc≡r3 2u=ϕ MPl ,(24) so that duc=dϕ/MPl. Writing ϑ≡T−¯ T= 2iℑ(T), we have dϑ duc = 2iℑ(T)du duc = 2iℑ(T)r2 3,1 ϑ dϑ duc =r2 3. (25) Evaluating (23) on the unit tangent ∂ucgives the local slope ϖ(∂uc) = π 12r2 3+··· .(26) In ten dimensions the NS-NS 2-form BMN has gauge-invariant 3-form flux H=dB −α′ 4ΩYM 3−ΩL 3, dH =−α′ 4(Tr F∧F−Tr R∧R) implementing the Green-Schwarz mechanism. The relevant 10D terms in the string frame is SGS ∝α′ 4ZB∧Tr F∧F−Tr R∧R. Compactifying on a 6D manifold M6of stringframe volume V6and retaining the 4D gauge fields yields, in four dimensions, Lgauge =−1 4ℜ(fg)FµνFµν +1 4ℑ(fg)Fµν ˜ Fµν, where ℑ(fg) denote the physical CP-odd angle coupling to electromagnetism in the physical effective action, so that by definition dℑ(fg) = ϖ. Let θ≡ ℑ(fg). Integrating (26) from last scattering to today and using duc=dϕ/MPl yields up to q-ripples: ∆θ=Zuc,0 uc,zd ϖ(∂uc)duc=π 12r2 3Zϕ0 ϕzd dϕ MPl (27) =π 12r2 3 ∆zdϕ MPl .(28) Therefore the predicted cosmic birefringence in a large volume along the gauge-sequestered path is β=1 2∆θ(28) =π 24r2 3 ∆zdϕ MPl rad.(29) Thus, using this result, and substituting the equation of the post drag dilaton excursion with respect to the Hubble tension equation (16) yields β≃π 108 ln 1 + τ2 H−1ωc+ωb ωc.(30) Thus, using the fiducial values from Aghanim et al. [20], Riess et al. [21], this yields the hard prediction: β∗= 0.289◦±0.061◦.(31) Source β σ(β, β∗)Implied H0 [27] 0.35◦±0.14◦+0.40 73.7±2.8 [28] 0.30◦±0.11◦+0.09 72.7±2.2 [29] 0.34◦±0.09◦+0.47 73.5±1.8 [30] 0.35◦±0.70◦+0.09 73.7±13.5 [31] 0.20◦±0.08◦−0.88 70.9±1.6 [32] 0.30◦±0.05◦+0.14 72.7±1.1 Table 1 CMB birefringence observations βincluding the σwith β∗prediction using Aghanim et al. [20], Riess et al. [21]. The H0[km/s/Mpc] is inferred from each βvia the birefringence-Hubble mapping (30), using HΛ 0= 67.36 ±0.54 km/s/Mpc [20]. 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