A rapid transition of G_{\rm eff} at z_t \simeq 0.01 as a solution of the Hubble and growth tensions
AA rapid transition of G eff at z t ’ .
01 as a solution of the Hubble and growth tensions
Valerio Marra
1, 2, 3, ∗ and Leandros Perivolaropoulos † Núcleo de Astrofísica e Cosmologia & Departamento de Física,Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brazil INAF – Osservatorio Astronomico di Trieste, via Tiepolo 11, 34131, Trieste, Italy IFPU – Institute for Fundamental Physics of the Universe, via Beirut 2, 34151, Trieste, Italy Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece
The mismatch in the value of the Hubble constant from low- and high-redshift observations maybe recast as a discrepancy between the low- and high-redshift determinations of the luminosityof Type Ia supernovae, the latter featuring an absolute magnitude which is ≈ . µ G ≡ G eff G N at z t ’ .
01 could explain the lower luminosity (higher magnitude) of local supernovae,thus solving the H crisis. A model that features µ G = 1 for z (cid:46) .
01 but µ G ’ . z (cid:38) . z (cid:38) .
01 by the required valueof ≈ . Introduction. —The mismatch in the determina-tion of the Hubble constant using two different well-understood calibrators – the Cepheid-calibrated super-novae Ia (SnIa) and the CMB-calibrated sound horizonat last scattering – is perhaps the most important openproblem of modern cosmology. The latest CMB con-straint from the Planck Collaboration [1, table 2] is: H P180 = 67 . ± .
54 km s − Mpc − , (1)while the latest local determination by SH0ES reads [2]: H R200 = 73 . ± . − Mpc − . (2)This 9% discrepancy, corresponding to more than 4 σ , isthe so-called H crisis. Mismatch in the SnIa absolute magnitude. —Aspointed out in [3], it is useful to look at the H crisisvia the corresponding constraints on the absolute magni-tude M B of SnIa. Propagating, via BAO measurements,the CMB constraint on the recombination sound hori-zon r d to the SnIa absolute magnitude M B using theparametric-free inverse distance ladder of [4], one obtains: M P18 B = − . ± .
027 mag , (3)while, using the demarginalization method of [5], the con-straint of equation (2) implies: M R20 B = − . ± .
037 mag . (4)These two determinations are in tension at 3 . σ . Notethat these values are relative to the calibration of thePantheon supernova dataset [6]. We remind the readerthat M B is not a fundamental parameter as it dependson the method used to fit SnIa light curves and its pa-rameters.The local constraint of equation (4) comes from the as-trophysical properties of anchors, Cepheids and the white dwarfs responsible for the SnIa explosions. The CMB de-termination of equation (3) comes instead from the com-bined constraining power of CMB, BAO and SnIa obser-vations. More precisely, first, CMB and BAO constrainthe luminosity distance-redshift relation d L ( z ) and so thedistance modulus µ ( z ): µ ( z ) ≡ m B ( z ) − M B = 5 log (cid:20) d L ( z )10pc (cid:21) . (5)Then, the Pantheon dataset, by constraining the SnIaapparent magnitudes m B , produces a calibration of M B .Here, one exploits the fact that SnIa are standard candleswith an a priori unknown M B .Seen from this point of view, the H crisis is more of anastrophysical problem than a cosmological one. Indeed,as the local calibration is performed via Cepheid stars atredshifts less than 0 .
01, it seems that a solution to thiscrisis could be the existence of a mechanism that changesthe physics of SnIa so that they are dimmer at z < . z .A simple phenomenological approach is to assume atransition in the value of the absolute magnitude M B ata redshift z t ’ .
01 [7]: M B ( z ) = M R20 B if z ≤ z t M R20 B + ∆ M B if z > z t , (6)where the needed gap in luminosity is approximately re-lated to the corresponding gap in H according to (seeequation (5)):∆ M B ≡ M P18 B − M R20 B ≈ H P180 H R200 ≈ − . . (7)Figure 1 illustrates this scenario showing the inferredSnIa absolute magnitudes M B,i = m B,i − µ ( z i ) of the a r X i v : . [ a s t r o - ph . C O ] F e b - - - - - - M B M B R20 =- M B P18 =- Figure 1. The inferred SnIa absolute magnitudes M B,i = m B,i − µ ( z i ) of the binned Pantheon sample [6] under the assumptionof a Planck/ΛCDM luminosity distance. The data are inconsistent with the M R20 B of equation (4) but they become consistentif there is a transition in the absolute magnitude with amplitude ∆ M B ≈ − . binned Pantheon sample [6] for a Planck/ΛCDM lumi-nosity distance. The data are inconsistent with the de-termination of equation (4) but they become consistentif there is the M B transition proposed in equation (6). Inother words, a model which features this transition andwith a phenomenology similar to the one of the stan-dard ΛCDM model would be compatible with the localCepheid calibration, SnIa, BAO and CMB. The gravitational transition model and the H crisis. —As we will now argue, a sudden change in thevalue of the effective gravitational constant G eff couldinduce the M B transition of equation (6) and explain, inaddition, the reduced growth rate of perturbations thatis suggested by the σ − Ω m tension.By “effective gravitational constant” we refer to thestrength of the gravitational interaction and not to thePlanck mass related G ∗ which determines the cosmolog-ical expansion rate [8]. In other words, we are consid-ering a scenario in which perturbations are suppressedbut the background expansion does not deviate fromthe one of the phenomenologically successful standardΛCDM model, for which the low-redshift expansion rate H ( z ) is of the form H ( z ) = H P180 p Ω m (1 + z ) + 1 − Ω m . (8)A SnIa explosion occurs when the mass of a white dwarf reaches the critical Chandrasekhar mass m c byaccreting matter from a companion. The constancy intime of this critical mass is the cornerstone on which the‘SnIa standard candle’ hypothesis is based. Even thoughSnIa at z > .
01 seem consistent with the hypothesis ofa constant M B [9] and, therefore, a smooth variation of m c is unlikely, a sudden transition of m c at z < .
01 cannot be easily excluded.The Chandrasekhar mass depends on both G eff andthe mass per electron m according to [10]: m c ’ m (cid:18) (cid:126) cG eff (cid:19) / . (9)In the standard model, this result is independent of theaccretion history and of the white dwarf progenitor de-tails [11] and leads to m c ’ . M (cid:12) . A possible suddentransition of a fundamental constant would induce a cor-responding transition of both the Chandrasekhar massand therefore the SnIa peak absolute luminosity L [12]. Areasonable hypothesis is that L increases as m c increasesor equivalently as G eff (and/or m ) decreases [13].Assuming a fixed m (and fine structure constant), thismay be expressed via a power law dependence L ∼ G − α eff , (10)where α > O (1). The simplest hypothesis L ∼ m c leads to α = 3 /
2. This value of α will be assumed in thepresent analysis [13]. Since the SnIa absolute magnitudeis connected with the absolute luminosity via M B − M B = −
52 log LL , (11)where the index indicates the local (present) values, itis clear that a decrease of G eff would lead to an increaseof L and a decrease of M B .It then follows that the M B transition of equation (6),and so the H crisis, could be explained by the followingevolution of the effective gravitational constant: µ G ( z ) ≡ G eff G N = ≡ µ
52 log L P18 L R20 = 154 log µ >G , (13)where L P18 and L R20 are the CMB-calibrated andCepheid-calibrated SnIa luminosities, respectively, sothat: ∆ µ G = 10 ∆ M B − ≈ − . . (14) Addressing the growth tension. —Such a z > z t suppression of the gravitational interaction is triviallyconsistent with local constraints on Newton’s constantand it has the potential to explain the growth tension bydecreasing the growth rate of cosmological matter fluc-tuations δ ( z ) ≡ δρρ ( z ) according to the dynamical lineargrowth equation: δ + (cid:20) ( H ) H −
11 + z (cid:21) δ ≈ H H (1 + z ) µ G ( z ) Ω m δ, (15)where denotes derivative with respect to redshift z . Infact, a 12% reduction of µ G would fit the same data witha 12% larger Ω m .Dynamical probes of cosmological perturbations in-cluding cluster counts (CC) [14–17], weak lensing(WL) [18–25] and redshift-space distortions (RSD) [26–30] consistently favor a lower value of the matter den-sity parameter Ω m in the context of General Gelativ-ity (GR). This implies weaker gravitational growth ofperturbations than the growth indicated by GR in thecontext of a Planck18 / ΛCDM background geometry atabout 2 − σ level [27, 29, 30]. This weakened growthis quantified by considering the parameter σ , defined asthe matter density rms fluctuations within spheres of ra-dius 8 h − Mpc at z = 0. The value of σ is connected σ f σ Likelihood Contours for P18 Λ CDMP18 Λ CDM likelihood contours0.650.700.750.800.850.900.95 σ f σ Likelihood Contours for Δμ G =- Λ CDM likelihood contours0.1 0.2 0.3 0.4 0.5 0.6 0.70.650.700.750.800.850.900.95 Ω m σ f σ Likelihood Contours for w =- =- Figure 2. The σ − Ω m constraints (blue) from redshiftspace distortion data for Planck/ΛCDM (upper panel), the µ -transition model ( z t = 0 .
01 and ∆ µ G = − .
12) that actu-ally resolves the H crisis (middle panel) and w CDM with w = − . H tension via asmooth deformation of H ( z ) (lower panel). with the amplitude of the primordial fluctuations and isdetermined by the growth rate of cosmological fluctua-tions. A useful bias-free statistic probed by RSD data isthe quantity f σ : f σ ( a ) = σ δ ( a = 1) a δ ( a, Ω m , µ G ) , (16)where δ is obtained by solving equation (15). This theo-retical prediction may now be used to constrain, via f σ data, the parameters Ω m , σ and µ G ( z ) in the context ofa specific background H ( z ) and a parametrization µ G ( z ).In the present analysis we assume the step-like transitionof µ g ( z ) proposed in equation (12) and either a ΛCDMbackground or a fixed dark energy equation of state w .In the context of a ΛCDM background H ( z ), RSD andWL data are well fit by [31–33]:Ω growth0 m = (1 + ∆ µ G ) Ω P180 m = 0 . +0 . − . , (17)which, adopting Ω P180 m = 0 . ± . µ G = − . ± . , (18)which is in good agreement with the value required to ex-plain the M B transition (see eq. (14)), and is compatiblewith CMB [1] and gravitational wave [34] constraints.In Figure 2 we show the σ − Ω m likelihood contours(blue) from RSD data for Planck/ΛCDM (upper panel),the µ G -transition model ( z t = 0 .
01 and ∆ µ G = − . H tension (middle panel) and w CDM with fixed dark enrgy equation of state param-eter w = − . H ten-sion via a smooth deformation of H ( z )[33] (lower panel).We adopted the conservative robust f σ dataset of [35,table 2], which is a subset of the up-to-date compila-tion presented in [32]. Superposed are the correspondingPlanck/ΛCDM CMB TT likelihood contours (red). Thegrowth tension is resolved only in the case of the µ G -transition model (middle panel), while the smooth H ( z )deformation approach (lower panel) worsens the growthtension [33] seen in the upper panel in the context ofPlanck/ΛCDM . Discussion. —We have showed that a rapid 10% in-crease of the effective gravitational constant roughly 150million years ago ( z t ’ .
01) can solve simultaneouslythe H crisis and the σ − Ω m growth tension. A phys-ical model where such a transition could be realized is ascalar-tensor theory with a step-like scalar field potential V ( φ ) and/or with an abrupt feature in the functionalform of the nominimal coupling function F ( φ ) [8]. Inthis theory the gravitational interactions are determinedby G eff = πF F +4 F ,φ F +3 F ,φ while the background expansionis controlled by the Planck mass which corresponds to G ∗ = πF . Alternatively, a quantum tunelling betweentwo distinct degenerate vacua of a scalar tensor potential V ( φ ) would also induce such a gravitational phenomeno-logical transition for an appropriate form of a nonmini-mal coupling F ( φ ).One could test the µ G transition model on cosmolog-ical and astrophysical scales. First, future surveys willtightly constrain the growth of perturbation at 0 < z < µ G that may rule outthe model. Furthermore, low-redshift gravitational waveobservations could not only rule out the mechanism hereproposed but, if detected, map it through redshift. Forexample, gravitational waves of merging binary neutronstars carry information about the value of G eff at the timethe merger took place. Thus bounds can be placed on thevariation of G eff between the merger time and the presenttime [34] assuming that the actual masses of the neu-tron stars are consistent with the theoretically allowedrange. Similarly, one could constrain G eff via the gap inthe mass spectrum of black holes due to the existence ofpair-instability supernovae [37]. Alternatively, one mayuse standard sirens to measure the low- z luminosity dis-tance and compare it with the corresponding luminositydistance obtained with SnIa standard candles where theChandrasekhar mass and G eff are assumed constant. Fu-ture standard siren gravitational wave measurements areexpected to constrain (or detect) such variations of G eff at a level of 1 .
5% [38].The µ G transition scenario may also be constrainedvia stellar constraints. Indeed, one expects the evolutionof a star to be strongly dependent on the strength ofthe gravitational interaction. A variation of G eff wouldindeed affect the hydrostatic equilibrium of the star andin particular its pressure profile and temperature. This,in turn, will then affect the nuclear reaction rates so thatthe lifetimes of the various stars will be modified [39].Finally, an important assumption made in our analy-sis is that the SnIa luminosity increases with the Chan-drasekhar mass. This is intuitively correct and consistentwith a large part of the literature [10, 13, 40, 41]. How-ever, a recent analysis [42], using a semi-analytical modelto calculate SnIa light curves in non-standard gravity hasclaimed that the SnIa luminosity may actually decreasewith the Chandrasekhar mass. If this was the case thenthe proposed model would require stronger gravity toresolve the Hubble crisis and would thus be unable toresolve simultaneously the growth tension. The clarifi-cation of this important issue is therefore an interestingextension of the present analysis. ACKNOWLEDGEMENTS
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