Exploring the evidence for a large local void with supernovae Ia data
Vladimir V. Luković Balakrishna S. Haridasu Nicola Vittorio
astro-ph.CO




















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        red-.5ex.3ex]Exploring the evidence for a large local void with supernovae Ia dataV. V. Luković et al.]
            Vladimir V. Luković,^1,2E-mail: vladimir.lukovic@roma2.infn.it
            Balakrishna S. Haridasu,^1,2E-mail: haridasu@roma2.infn.it
            and Nicola Vittorio^1,2E-mail: nicola.vittorio@roma2.infn.it
        
            ^1Dipartimento di Fisica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133, Roma, Italy
        ^2Sezione INFN, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133, Roma, Italy
        
        
        2019firstpage–lastpage[
            [
            Accepted 2019 October 25. Received 2019 October 22; in original form 2019 July 25
        =====================================================================================
        
        
        In this work we utilise the most recent publicly available type Ia supernova (SN Ia) compilations and implement a well formulated cosmological model based on Lemaître-Tolman-Bondi metric in presence of cosmological constant Λ (ΛLTB) to test for signatures of large local inhomogeneities at z≤0.15. Local underdensities in this redshift range have been previously found based on luminosity density (LD) data and galaxy number counts. Our main constraints on the possible local void using the Pantheon SN Ia dataset are: redshift size of z_ size=0.068^+0.021_-0.030; density contrast of δΩ_0/Ω_0=-10.5_-7.4^+9.3% between 16th and 84th percentiles. Investigating the possibility to alleviate the ∼9% disagreement between measurements of present expansion rate H_0 coming from calibrated local SN Ia and high-z cosmic microwave background data, we find large local void to be a very unlikely explanation alone, consistently with previous studies. However, the level of matter inhomogeneity at a scale of ∼100Mpc that is allowed by SN Ia data, although not expected from cosmic variance calculations in standard model of cosmology, could be the origin of additonal systematic error in distance ladder measurements based on SN Ia. Fitting low-redshift Pantheon data with a cut 0.023<z<0.15 to the ΛLTB model and to the Taylor expanded luminosity distance formula we estimate that this systematic error amounts to 1.1% towards the lower H_0 value. A test for local anisotropy in Pantheon SN Ia data yields null evidence. Analysis of LD data provides a constraint on contrast of large isotropic void δΩ_0/Ω_0=-51.9%±6.3%, which is in ∼4σ tension with SN Ia results. More data are necessary to better constrain the local matter density profile and understand the disagreement between SN and LD samples.
            
                supernovae – cosmological parameters – large-scale structure of Universe 
            
        
        § INTRODUCTION
        
            The most reliable standard candles and one of the fundamental low-redshift astrophysical observables <cit.>, type Ia supernovae (SN Ia), also provide a cosmology-independent measurement of present expansion rate through the local distance ladder <cit.> (hereafter R16).
            In concert with observations of baryon acoustic oscillation (BAO) peak from galaxy redshift surveys <cit.> and cosmic microwave background (CMB) radiation data <cit.>, SN Ia contributed to establishment of the standard model of cosmology.
            Astonishing concordance between different cosmological observables brought by technical advancements in the last decades was followed by a number of inconsistent and/or unexpected results, e.g. 
            estimated values of matter clustering parameter, S_8, from low and high redshifts <cit.>, CMB parameter constraints in curvature extension of base Λ cold dark matter (ΛCDM) model <cit.> (hereafter P18), as well as several unexplained CMB anomalies <cit.>, etc.
            The best-known amongst ΛCDM tensions and one of the biggest challenges in modern cosmology is related to the H_0 value.
            Defining the scale of extra-galactic distance through the Hubble radius, as well as the present expansion rate, H_0 is a fundamental cosmological parameter both for CMB and low-redshift data <cit.>.
            The most stringent ΛCDM estimate coming from P18, H_0=67.36±0.54, is in 4.4σ tension with the most recent local measurement based on calibrated SN Ia, H_0= 74.03 ± 1.42, by <cit.> (hereafter R19).
            In comparison, H_0 measurements from the time delay of strongly lensed quasars support a higher value <cit.>; using differential age method with cosmic chronometers <cit.> provides an estimate in the middle of two results in tension <cit.>; while the constraints coming from gravitational waves are still fairly weak <cit.>.
            Although the quality of BAO data has substantially improved over the last decade, providing some of the most stringent cosmological constraints on expansion dynamics <cit.>, their H_0 estimate is degenerated with the sound horizon parameter, whose value relies on CMB observations and early Universe physics <cit.>.
            
            The intercept of SN Ia magnitude-redshift relation is determined by their absolute magnitude (M_B) and the Hubble radius, resulting in two degenerate parameters in SN Ia analyses.
            While the high-redshift SN Ia are necessary for assessing the underlying physics of dark energy, the low-redshift end (z≤0.15) of SN Hubble diagram can be analysed with minimal assumptions about properties of the cosmic fluid and expansion dynamics related only to the well-accepted cosmic acceleration <cit.>.
            Indeed, calibrating SN Ia absolute magnitude with the use of Cepheid variable stars and fixing the local distance-redshift relation (at z≤0.15) leads to a remarkable cosmology-independent measurement of H_0<cit.>.
            Different calibration techniques consider the usage of strong gravitational lensing <cit.>, tip of the red giant branch <cit.>, H II regions <cit.>, etc.
            Due to the strong correlation of cosmic expansion dynamics and H_0 value, the model-independent local estimates remain crucial for the study of late-time cosmic evolution [see also <cit.>].
            
            As is well-known, a large inhomogeneity in local matter density distribution can affect the geometry of local Universe and the distance measures, consequently biasing the resulting model inferences and the H_0 estimate. 
            A few independent groups used galaxy survey catalogues to test the local matter distribution, reporting hints of a large local void <cit.>.
            Concretely, <cit.> (hereafter KBC13) used UKIDSS Large Area Survey <cit.>, Galaxy and Mass Assembly (GAMA) <cit.> and 2M++ catalogue <cit.> to construct luminosity density sample (hereafter LD) over the redshift range 0.01<z<0.2.
            Relating it to the total matter density in the local Universe, they find evidence for an underdense region and suggest a void of about 300h_70^-1Mpc in size and -30% contrast with respect to the background.
            <cit.> (hereafter WS14) used 6dFGS, SDSS, and GAMA surveys to examine the local galaxy density field over 20% of the sky and up to 450h_70^-1Mpc in depth.
            Both number counts and peculiar velocity fields show evidence for at least 200h_70^-1Mpc large local underdensity with -15%±3% average contrast, ranging between -40% and -5% in different directions.
            Analysing a catalogue of 1653 galaxy clusters detected by their X-ray emission <cit.> report evidence for ∼140 Mpc large local underdensity with density contrast of 20%±10%.
            
            The results of KBC13, WS14 and <cit.> go against the standard cosmological paradigm and the assumption of large-scale homogeneity.
            In fact, indications for a few hundred Mpc large local underdensity encouraged renewed popularity of  void models based on Lemaître-Tolman-Bondi (LTB) metric, also motivated as a remedy for the H_0 tension <cit.> and even as a possible explanation for the CMB cold spot <cit.>.
            Contrasting the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) models, the LTB metric <cit.> describes an isotropic but inhomogeneous dust system, which on cosmological scales can be used for studying cold dark matter density distribution.
            Since the discovery of cosmic acceleration, LTB metric was used to construct toy models that challenge the cosmological constant paradigm with an alternative scenario in which the apparent accelerated expansion is a result of the strongly underdense local Universe that smoothly converges to Einstein de Sitter solution on higher redshifts  <cit.>.
            Since the deceleration-acceleration transition redshift is well constrained at z_t≈0.6 [see e.g. <cit.> for model-independent estimates], this alternative explanation requires a giant (≈3Gpc) isotropic void that proved to be very unlikely compared to the standard cosmological model <cit.>.
            However, extending the LTB model with the addition of cosmological constant Λ, one can describe large (order of 100Mpc) local inhomogeneous matter distribution inside a ΛCDM background <cit.>.
            
            While the level of cosmic variance in matter distribution expected for standard model can only partially relieve the Hubble constant problem <cit.>, <cit.> found that a strongly underdense local inhomogeneity with non-vanishing cosmological constant, modeled with an approximate numerical solution for LTB metric, can fully coincide the CMB constraints with the local H_0 measurement.
            Similarly, <cit.> (hereafter HB17) extended the work of KBC13 by testing the consistency of their earlier findings with constraints coming from SN Ia and linear kinematic Sunyaev-Zel13dovich effect and showed that local void found by KBC13 could reduce the H_0 tension.
            A recent work of <cit.> argues that the combination of local inhomogeneity effect together with the re-calibration of Cepheids and local SN Ia distances using the recent parallax measurements from Gaia mission, is sufficient to completely remove the tension [see also <cit.> for follow-up discussions].
            <cit.> (hereafter KSR19) find that any inhomogeneity with contrast> 20% is strongly inconsistent with SN Ia data, consequently reassuring the ability to measure the distance and Hubble constant with locally calibrated SN to a 1% precision.
            
            The goal of this work is to probe the local matter density using the SN Ia as one of the most reliable low-redshift astrophysical observables, constraining size and amplitude of possible matter inhomogeneity at our local position.
            We investigate the findings of KBC13 and KSR19 by using the complete analytic description for treating the LTB metric in presence of cosmological constant (hereafter ΛLTB) introduced in <cit.>.
            Extending the earlier works, we directly fit the ΛLTB model to two biggest publicly available SN Ia samples - Joint light curve analysis <cit.> (hereafter JLA)  and Pantheon compilation <cit.> (hereafter Pan), consisting of 740 and 1048 SN Ia, respectively.
            We also utilise the luminosity density data from KBC13 within the given formalism to assess the (dis)agreement.
            In <ref> we review the theoretical description of LTB metric in presence of cosmological constant and proceed by presenting the effect that a local void can have on observable physical quantities in <ref>.
            Constraints on the void contrast and size coming from fitting JLA, Pantheon and KBC13 datasets are reported in <ref>, together with the revision of their consistency.
            Additionally, we discuss on the possible effect of local isotropic inhomogeneity on the H_0 measurement, and explore the level of anisotropy allowed by the data.
            Summarising our results in <ref>, we examine the differences compared to earlier findings by other teams.
            Finally, in the <ref> we provide a simplified void model capable of correctly reproducing our results based on complete ΛLTB description to a good approximation.
            
            Throughout the paper we use the dimensionless variable h_70=H_0/70 for distance measures; acronym w.r.t. stands for `with respect to'.
            
            
            
        
        § LOCAL MATTER DENSITY IN ΛLTB MODEL
        
            The simplest extension of standard homogeneous cosmological model in presence of large matter inhomogeneity is done using the LTB metric <cit.> that describes spherically symmetric pressureless cosmic fluid
             s^2=c^2  t^2-R_r(t,r)^2/1-k(r)r^2 r^2-R(t,r)^2 Ω^2.
            
            Here R(t,r) has the units of length and defines all physical distance measures as well as the expansion history in this model.
            The second free function of the metric, k(r) is dimensionless and defines the curvature within the shell of a radius r.
            We note that radial coordinate r is also dimensionless, with no physical meaning, and it should be considered only as a flag coordinate of different shells all centred at r=0.
            Using the above metric and the Einstein field equations (FE) for pressureless inhomogeneous matter fluid in presence of cosmological constant Λ, one can derive the generalised Friedmann-Lemaître equation as
            (R_t/R)^2=2 G M/R^3-c^2 k r^2/R^2+c^2Λ/3.
            
            Here M(r) is an integration constant, whose physical meaning is the total mass enclosed inside a sphere with radial coordinate r,
            
            M(r)=∫_0^r 4π R(t,r)^2ρ_m(t,r)R_r(t,r)  r .
            
            Inhomogeneous cosmic matter density profile, ρ_m(t,r), satisfies the conservation law coming from FE, ensuring that integrated mass M(r) remains constant in time.
            In inhomogeneous cosmology we differ two expansion rates, namely transverse and radial:
            
            H_⊥(t,r)   =R_t/R,
        
            H_||(t,r)   =R_t r/R_r.
        
        Hence, the one appearing in Friedmann-Lemaître <ref> is the transverse expansion rate.
        As usual, we denote t=t_0 as present age of the Universe and all the physical quantities with subscript "0" represent their values today, while H_0(r)=H_⊥(t_0,r) is used for the transverse expansion rate.
        
        Introducing scale factor, the Friedmann-Lemaître equation can be rewritten in a simpler form, although the definition of scale factor a(t,r) is not trivial nor customary as in the homogeneous cosmology.
        The value of flag coordinate r is commonly defined through a gauge with some other physical variable. 
        The choice of this gauge is arbitrary and has no implications on the model predictions nor on the results of data analysis.
        The most often used gauge <cit.> in LTB framework is:
        
        R_0(r)= c t_0 r  ⇒   r≡R_0(r)/c t_0 .
        
        However, in this work we use an alternative definition which turns out to be more convenient choice for the latter calculations of elliptical integrals present in the geodesic equations and distance variables.
        Following <cit.>, we define the dimensionless radial coordinate r through the relation with the total mass enclosed in a given shell as
        
        M(r)=4π/3C_Mr^3  ⟹   r≡√(3/4πM(r)/C_M),
        
        where C_M is a normalisation constant in units of mass whose value will be chosen at a later stage.
        Commonly, in both gauges the scale factor can be defined as
        
        a(t,r)≡R(t,r)/c t_0 r.
        
        This function clearly converges to the usual scale factor definition and the spatially uniform values in the limit of homogeneous FLRW metric.
        Generalised Friedmann-Lemaître <ref> can be rewritten in terms of three  dimensionless density parameters normalised to the present time,
        
        H_⊥(t,r)^2=H_0(r)^2(Ω_m(r)R_0(r)^3/R(t,r)^3+Ω_k(r)R_0(r)^2/R(t,r)^2+Ω_Λ(r)),
        
        These can be evaluated combining <ref> as,
        Ω_m   =1/H_0^22 G M/R_0^3=1/H_0^28 π  G C_Mr^3/3/c^3t_0^3a_0^3r^3=M̅/t_0^2H_0^2a_0^3 ,
        Ω_k   =-1/H_0^2c^2k r^2/R_0^2=-k/t_0^2H_0^2a_0^2 ,
        Ω_Λ   =1/H_0^2c^2 Λ/3=Λ̅/t_0^2H_0^2.
        
        Here M̅=8π GC_M/(3c^3t_0) and Λ̅=1/3Λ c^2t_0^2 are dimensionless constants, whereas k(r), H_0(r) and a_0(r) are not uniform in LTB metric.
        The introduced dimensionless density parameters are different from one shell to another, but do not depend on time.
        Since Ω_m(r) is proportional to the total mass enclosed in a shell, M(r), it can be represented as the ratio of  average and critical matter densities of each shell.
        Starting from <ref> and then using <ref> one can define average matter density:
        <ρ_m>(t,r)=M(r)/4π R(t,r)^3/3=C_M/c^3t_0^31/a(t,r)^3 .
        
        Then, definition of critical matter density together with <ref> straightforwardly confirm:
        ρ_ crit≡3H_⊥^2/8π G  ⟹  Ω_m(r)=<ρ_m>(t_0,r)/ρ_ crit(t_0,r) .
        
        Friedmann-Lemaître <ref> can also be rewritten in dimensionless form using <ref>t_0^2H_⊥(t,r)^2=M̅/a(t,r)^3-k(r)/a(t,r)^2+Λ̅ .
        
        In terms of the Birkhoff theorem, <ref> dictate every shell in ΛLTB model to evolve as an independent homogeneous FLRW model with a specific curvature k(r), a total mass enclosed M(r) and the same value of cosmological constant Λ for all shells, determining its transverse expansion rate H_⊥(t,r).
        Each shell is described with relative density parameters Ω_i(r), present expansion rate H_0(r), and evolving scale factor a(t,r)/a_0(r).
        
        Cosmological constant, included in the Einstein field equations, is uniform over space and constant in time due to its physical nature, which is also true for the dimensionless constant Λ̅ that we introduced.
        The density parameter corresponding to the cosmological constant, Ω_Λ(r), can  not be uniform in ΛLTB model, as this would imply that Λ is inhomogeneous and coupled to the cold dark matter density profile.
        This approach is reflected in a number of recent studies <cit.>, while not fully followed by HB17 who assume cosmic density parameter Ω_Λ to be constant throughout the underdense region and at the background.
        Similarly, the methods of inverse reconstruction of matter density profile in presence of cosmological constant must be constructed such that FE are satisfied (c.f. <cit.>).
        
        The gauge choice for the flag coordinate r, which is set by <ref>, enabled us to rewrite the Friedmann-Lemaître <ref> in a more elegant analytic form, i.e. <ref>.
        The remaining degree of freedom in the scaling constant C_M, or equivalently M̅, can be used to normalise the scale factor a(t,r).
        We set M̅ such that a_*=a(t_0,r_*)=1 for a chosen shell r=r_* and at the present time t_0.
        For example, one can choose r_*=0 to normalise the scale factor at the observer's position or r_*=+∞ to normalise at the background, instead.
        We follow the normalisation of scale factor on the background, setting the numerical value of M̅ such that a_*=a(t_0,r_*=+∞)=1 for the background shells.
        In particular, normalisation gives the physical meaning to the scaling constant, i.e. <ref> yields
        ρ_m^ bg(t)=ρ_m^ bg(t_0)/a^ bg(t)^3  ⟹  ρ_m^ bg(t_0)=C_Mc^3t_0^3 .
        
        Therefore, normalising far from the inhomogeneity, which is for any large enough r_*, implies that background scale factor and <ref> converge to the usual forms that we have in homogeneous FLRW model.
        It is easy to see that at present time t=t_0 the background quantities satisfy
        
        t_0H_0^ bg=√(M̅-k^ bg+Λ̅) .
        
        Partial differential <ref> can be used to express the derivative ∂ t/∂ a and integrate from the Big Bang up to scale a:
        t(a,r)-t_BB(r)/t_0=∫ _0^a√(α)/√(Λ̅ α^3-k(r) α+M̅)α .
        
        All variables on the right hand side are dimensionless and this elliptical integral is analytically solvable (for details see <cit.>).
        The Big Bang time t_BB(r) is an integration constant that may vary in space, affecting the age difference between the shells.
        Nevertheless, one does not expect large variation of t_BB(r) in the case of a small local inhomogeneity.
        Non-uniform Big Bang time is also related to instabilities of matter perturbations <cit.>.
        Hence, in this work we consider only the models with homogeneous Big Bang time t_BB(r)≡ const=0.
        
        Integrating <ref> on the background (r=r_*) from the Big Bang a=0 up to today a(t=t_0,r_*)=1 we have
        
        1=t(a=1,r_*)/t_0=∫ _0^1√(α)/√(Λ̅ α^3-k^ bg α+M̅)α,
        
        which is the numerical equation defining a value of M̅, necessary for the normalisation.
        At any other point of interest <ref> can be seen as the solution for t(a,r) or, after invertion, as a(t,r).
        Hence, calculating numerically the scale factor at any spatial position and time gives us the angular diameter distance R(t,r) and other physical observables of interest.
        
        
        
        § EFFECT ON ASTROPHYSICAL OBSERVABLES
        
        The high-redshift background observables  are not affected by the local isotropic inhomogeneity.
        Nevertheless, the analysis of local observables can yield different constraints on cosmological parameters from those obtained with FLRW models.
        Examples presented in this section form the basis for analysing observable data, but do not immediately represent the effect on cosmological inferences.
        We use a specific analytic form of the curvature, parametrised with k^ bg,k^ loc,r_0,Δ r and given as García-Bellido-Haugbølle (GBH) profile <cit.>:
        
        k(r)=k^ bg - (k^ bg - k^ loc) 1 - tanh(r - r_0/2 Δ r)/
        1 + tanh(r_0/2 Δ r) .
        
        Our focus is the case of ΛCDM background, i.e. k^ bg≡0, which we call ΛLTB model.
        The analytic form of k(r) together with the value of dimensionless cosmological constant Λ̅ define the evolution of expansion and all physical quantities in LTB metric.
        In the limit of Δ r → 0, a simpler top-hat profile (TH) is recovered,
        
        k(r)   =k^ loc         for  r<r_0 ,
        
        k(r)   =k^ bg         for  r≥ r_0 .
        While GBH form describes a smooth matter density profile, TH profile has one parameter less and is trivial to use for data fitting as two joint homogeneous models.
        Care must be given to relation between the physical quantities in two homogeneous regions.
        
        
         §.§ Reconstruction of geodesic
        
        The LTB metric describes an inhomogeneous dust cloud with spherical symmetry, indicated  by the angular term in <ref>.
        Hence, the observer located at the centre of this symmetry has an isotropic view.
        The assumption of isotropy gives us only the first approximation for treating the large cosmological inhomogeneities.
        To quantify the overall effect of large local void on cosmological inferences, we limit ourselves to the on-centre model that considers an observer located at the centre of symmetry in the local matter inhomogeneity.
        The assumption of this very special position can be avoided, although the geodesic equations for an off-centre observer become increasingly complex, affecting the feasibility of data analysis.
        We find the on-centre model as adequate for presenting all the relevant points in this work, but we extend the discussion about observer's spatial position later on.
        
        Geodesic equations can be derived following <cit.>:
         r/ z   =1/1+zc√(1-k r^2)/R_t r ,
         t/ z   =-1/1+zR_r/R_t r .
        
        where z is the cosmological redshift.
        This system of differential equations must be solved numerically along the geodesic with help of <ref>, in the entire redshift range.
        In order to derive distance-redshift relations one needs to numerically reconstruct the function R(z), or equivalently a(z), which can be obtained by numerical inversion of <ref> at every z along the geodesic <cit.>.
        However, we find it easier and computationally faster to use the geodesic equations and all physical variables of interest rewritten with (a,r) as independent fundamental coordinates, after the change of variables from (t,r) a/ z=a_t t/ z+a_r r/ z.
        
        
        Adding the assumption of homogeneity, geodesic equations reduce to the standard form; using <ref> we have:
         t/ z=-1/1+za+r a_r/a_t+r a_t r=-1/1+za/a_t  ⇒   a/a=- z/1+z .
        
        We note that this relation is valid on all shells where spatial derivative is null, i.e. in every homogeneous subregion along the matter density profile.
        
        
         §.§ Density contrasts
        
        It is useful to define the relative contrasts for quantities of interest, such as physical matter density, dimensionless density parameter Ω_m, and expansion rates.
        The contrast can be defined at any time and any spatial point inside inhomogeneous region. 
        Consequently, for physical matter density we have
        δρ_m/ρ_m=ρ_m(t,r)/ρ_m^ bg(t)-1
        
        The evolution of matter density contrast is shown in <Ref> for a ΛLTB void model with specific parameters determining cosmological constant Λ̅, central curvature k^ loc, size and steep transition zone, given with r_0 and Δ r.
        As shown in this case, the physical matter density ρ_m will have compensated radial profile in the case of small Δ r and/or large central contrast.
        Since we use the comoving radial distance, the physical expansion of void size can not seen on <Ref>.
        
        Most often we consider the present (t=t_0) contrast between observer's position (r=0) and background (r=+inf) and we denote it with subscript 0.
        The contrast of dimesionless matter density parameter Ω_m(r) and both expansion rates are defined in the same manner as for physical matter density in <ref>.
        We will focus on present central contrasts which can be derived using <ref>:
        δρ_0ρ_0=   ρ_m^ loc(t_0)/ρ_m^ bg(t_0)-1=(1/a_0^ loc)^3-1 ,
        δΩ_0Ω_0=   Ω_m^ loc/Ω_m^ bg-1=(H_0^ bg/H_0^ loc)^2(1/a_0^ loc)^3-1 .
        
        Notably, these two contrast are different due to the induced spatial inhomogeneity of the expansion rates,
        δ H_0 H_0=H_0^ loc-H_0^ bg/H_0^ bg .
        
        In conclusion, relation |δρ_0/ρ_0|<|δΩ_0/Ω_0| holds both in the case of local underdensity as well as overdensity.
        Writing the equivalent of <ref> at a(t=t_0,r=0)=a_0^ loc and after some simple calculations, one can derive analytic expressions that relate these three contrasts.
        
        We would like to note that the current ΛLTB modelling can be equivalently parametrised with Ω_m^ bg, δρ_0/ρ_0, z_ size and Δ z, which are derived from Λ̅, k^ loc, r_0 and Δ r.
        Approximate series expansion formulae, valid for ΛCDM background, are provided in <ref>.
        
        
         §.§ Scale factor
        
        The radial dependence of scale factor in LTB models can be observed only along the geodesic as a(z)=a(t(z), r(z)) (see <Ref>).
        Moreover, modelling a(z) can be used a posteriori to inversely reconstruct the matter density profile <cit.>.
        
        Inside homogeneous region close to the centre of GBH profile and at the background, scale factor satisfies <ref> and, hence:
        
        a(z)=   a_0^ loc1+z    for z<<z_ size ,
        
        a(z)=   11+z    for z>>z_ size .
        
        In fact, redshift size of an inhomogeneous matter profile can be defined through the equality
        
        2 (1+z_ size) a(z_ size)=a_0^ loc+1 .
        
        
         §.§ Local expansion rate
        
        As explained in previous section, LTB metric is characterised with two expansion rates, shown in <Ref>. 
        The transverse expansion rate given by Friedmann-Lemaître <ref> can be used together with density parameters Ω_i to construct the equivalent FLRW model of each shell.
        On the contrary, radial expansion rate is related to the spatial derivative along the line of sight (cf. <ref>).
        Certainly, all physical quantities defined in ΛLTB model converge to their ΛCDM form at high redshift limit as a consequence of the Birkhoff theorem for the local spherical inhomogeneity (see <Ref>).
        
        The intercept of SN Ia m(z) relation is determined by the absolute magnitude of SN and the local expansion rate.
        Calibrated SN Ia are able to measure the H_0 only after fixing the slope of distance-redshift relation as presented in <cit.>.
        However, if the inhomogeneity extends up to z ∼ 0.1, not accounting for its effect properly can represent an important systematic in H_0 measurement.
        
        
        § RESULTS AND DISCUSSION
        
        In this section we present our main findings in the analysis of LD data from KBC13 and the SN Ia data from JLA and Pantheon[We use the latest release of Pantheon dataset from Nov 2018, while verifying that using the SN Ia redshifts published earlier has minimal to no effect on the likelihood in all fits performed here (see also <cit.>)] samples <cit.>.
        The likelihoods for SN Ia datasets are implemented as in their respective releases.
        Results are presented for the derived parameters: relative matter density at the background Ω_m^ bg, central contrast δΩ_0/Ω_0, inhomogeneity size z_ size and density profile shape Δ z.
        We use a prior of Ω_m^ bg=0.3153±0.0073 taken from CMB TT,TE,EE+lowE+lensing data analysis for the ΛCDM background <cit.>, which appropriately aids low-redshift SNe to constrain the local matter density profile (hereafter indicated as JLA+P and Pan+P).
        Other parameters are sampled from wide flat prior ranges: -0.4≤δΩ_0/Ω_0≤0.2, z_ size≤0.13 (≈ 0.5Gpc in comoving distance) and Δ z≤0.6 z_ size, for which we verify a posteriori that relaxing the priors does not affect our results.
        
        As elaborated in <ref>, the large local void is expected to affect the magnitude-redshift relation of SN Ia inside the void, due to the higher transverse expansion rate and lower values of relative density parameters Ω_m(z) and Ω_Λ(z) w.r.t. ΛCDM constraints.
        We start our analysis by testing for redshift dependence of SN Ia magnitude parameter
         M=M_B+5Log_10(c/H_0).
        
        To this end, we use Pantheon SN Ia whose apparent magnitudes have already been corrected with a cosmology-independent method which marginalises the effect of all magnitude correction parameters and leaves only M as a free parameter, besides the cosmological model (see <cit.>).
        
        Full Pantheon dataset, along with the CMB prior, is fitted to standard ΛCDM model, allowing for three binned values of M in redshift ranges z≤0.08 (green), 0.08<z≤0.15 (purple) and z>0.15 (orange), with 194, 103 and 751 SNe, respectively.
        The resulting 1σ confidence interval for M^ bg=-1.184±0.007 is in excellent agreement with M^ tot=-1.185±0.005 from the standard non binned fit shown as grey stripe in <Ref>.
        The visible inconsistency in the first two bins, M^1=-1.210±0.015, M^2=-1.153±0.012 might be hinting to possible local geometry effects.
        Fitting the Pantheon SN Ia in the redshift range 0.0233<z<0.15 to the FLRW series expansion (SE) formula for luminosity distance with q_0=-0.55, as presented in R16, we obtain M( SE)=-1.182±0.010, which is also different from values obtained in the first two bins.
        Various works in the literature, including R16 and their previous articles, introduce a parameter a_B that can be derived from the intercept magnitude M as
        
        a_B=-0.2 M -5 + log_10c ,
        
        where c is the speed of light in km/s.
         
        Results from the simple binned analysis already provide a strong motivation to study a more physical, inhomogeneous cosmological model, characterised by varying H_0(r) and matter density profile Ω_m(r).
        Although an extended analysis of the M evolution at higher redshifts could have important cosmological implications (see e.g.  <cit.> for effect on cosmic acceleration), here we looked for possible local variation in M as a hint of local matter inhomogeneity, while our background cosmology is assumed to be flat ΛCDM.
        
        
        
         §.§ ΛLTB analysis with LD data:
        
        KBC13 summarises 22 LD data points obtained from several surveys, of which 10 are their original data points. 
        Due to the unknown correlations that could arise from the overlap between different redshift surveys we opt to utilise only these 10 data points to constrain local matter density profile. 
        They were obtained using data from 35,000 galaxies divided in 10 redshift bins in the range 0.005 < z < 0.20. 
        Using LD points, KBC13 report the existence of a large local void with a luminosity density contrast of about -30% in the inner region, which is expected to be proportional to the contrast of total matter density δρ_0/ρ_0 (see <Ref>).
        We initially perform a simple reconstruction of the density profile using the Gaussian process – a model-independent technique for data analysis (see e.g. <cit.>).
        The reconstructed 1σ region shown in grey on <Ref> is obtained by imposing homogeneity both inside and outside the expected void size.
        As such, it is allowed to predict no evidence of a void, but our resulting posterior of the reconstructed region shows definite indications of the local underdensity with δρ_0/ρ_0∼-40%.
        We proceed by fitting LD data points to the full ΛLTB model and confirm that the conservative density contrast of δρ_0/ρ_0∼-30%, suggested by KBC13 and later utilised in HB17 and KSR19, appears to be an underestimate w.r.t. our best-fit shown in <Ref>.
        
        Due to the small amount of data, we test our findings by performing a leave one out (LOO) analysis, implemented as random removal of each of the 10 points from the fit.
        It clearly shows the stability of the estimated redshift size, while the value of the profile contrast δρ_0/ρ_0∼-44.0± 6.0 is found to be dominated by one stringent measurement at z = 0.034.
        Eliminating this data point coming from 2MASS survey, the remaining 9 points from UKIDSS and GAMA surveys estimate the contrast to be δρ_0/ρ_0∼-32.0± 9.5, in consistence with the conservative value suggested by KBC13.
        A direct comparison of our best-fit profile (blue) and the profile with δρ_0/ρ_0=-30% (green) in <Ref> clearly shows the importance of 2MASS data point at z = 0.034.
        Nevertheless, the induced contrast of fractional matter density δΩ_0/Ω_0 coming from the fit to 9 points (quoted as LD^* in <Ref>) is still higher than the one used in previous works where the difference between δΩ/Ω and δρ/ρ contrasts is not considered (c.f. KSR19, HB17, <cit.>).
        
        The primary inferences of this analysis are verified to remain unaltered changing the assumed analytic form of matter density profile. 
        Taking the result in <Ref> at face value, would indicate an extreme tension with a homogeneous matter density scenario.
        While the standard model of cosmology expects variation of the local matter density and velocity fields, a more pressing problem arising from this result is the size of the estimated inhomogeneity.
        Based on ΛCDM matter power spectrum constrained by CMB, density perturbations of size z_ size≈0.08 are expected to have the variance of δρ_0/ρ_0≈1.5-2% (corresponds to δΩ_0/Ω_0≈2-3%) at 68% confidence level <cit.>.
        Hence, this expectation is at ∼ 8σ tension with the result of the fit to complete LD data, while the exclusion of 2MASS data point at z =0.034 relaxes deviation to a however large ∼ 4σ tension.
        
        The best-fit ΛLTB (blue) curve in <Ref> is closely following the predicted model-independent reconstruction (grey), except for the short overdense region, which is an expected compensation of a large and steep underdense matter profile.
        In fact, the data point at z=0.081 might be a signature of the same. 
        Although not statistically significant to draw conclusions for an overdensity, it aids to constrain the void size.
        The specific interplay between the model description and the fact that only 10 data points are available for constraining 4 profile parameters, results in over-fitting difficulties. 
        For this reason we perform a Gaussian approximation of the posterior likelihood distribution before estimating the confidence regions presented in <Ref>, which does not alter our primary inferences.
        
        Given the lack of consistency in reported results based on LD data, limited sky coverage of the data, reports about angular variation (WS14), and possible effects of binning we also deem it important to estimate the level of (dis)agreement with the SN Ia data.
        
        
        
         §.§ ΛLTB analysis with SN Ia datasets:
        
        Probing for signatures of large local inhomogeneity (underdense or overdense) in the SN Ia datasets, we fit the local matter density profile simultaneously with the background cosmology (Ω_m^ bg) which is, nevertheless, dominantly guided by the Planck CMB constraint.
        Since the variation of H_0(r) is modelled, in ΛLTB we define the SN Ia intercept magnitude parameter using background expansion rate
         M=M_B+5Log_10(c/H_0^ bg).
        
        In the case of JLA dataset, fitting χ^2 also depends on SN Ia magnitude correction parameters for lightcurve stretch, colour and host galaxy mass, which are later marginalised as nuisance parameters <cit.>.
        
        The SN Ia constraints on local matter density profile presented in the form of confidence regions in <Ref> and <Ref> show high consistency with homogeneous ΛCDM fit, characterised by δΩ_0/Ω_0=0.
        The analysis of JLA dataset yields two distinguished local χ^2 minima, emphasised with dashed lines in <Ref>. The underdensity reported by WS14, with the size of ≈ 215h_70^-1Mpc and the average contrast δρ_0/ρ_0=-15±3% (corresponds to δΩ_0/Ω_0=-19±4%) is in between the two minima we find for JLA dataset.
        Interestingly, the second local minimum specifies a void with the physical size in line with the findings of KBC13, d_A^(c)=300h_70^-1Mpc, but with a much weaker contrast.
        Indeed, the large local void reported by KBC13 is rejected at ≳ 4σ confidence by JLA dataset.
        However, our analysis of reduced LD^* data points gives less stringent constraint on the void contrast which is no longer in tension with our JLA result (cf. <Ref>).
        
        
        Even though the analysis of Pantheon dataset does not recover the primary minimum obtained for JLA supernovae, the resulting confidence regions are of high similarity.
        Our Pantheon and Pan+P constraints for ΛLTB model shown in <Ref> include inside 1σ confidence region voids with contrast δΩ_0/Ω_0≈-20%, such as the representative void of WS14.
        The constraints from LD data of KBC13 (blue shaded ellipses in <Ref>) are pointing to much deeper void, although the redshift size of the two best-fits are aligned.
        In fact, the estimated contrasts from Pan+P and LD data are at ∼ 4σ tension (see <Ref>).
        The nature of inferences resulting from LD data, their disagreement with the expected cosmic variance and the SN Ia constraints, further strengthens our motive to perform LOO analysis to test this data sample.
        LOO analysis led us to consider the fit to the reduced LD^* dataset, which is shown with blue dashed contours in <Ref>.
        As mentioned earlier, the LD^* constraint on void size remains unaltered, while the estimated void contrast shifts towards a shallow inhomogeneity that has  no tension with the Pantheon results. 
        
        Our ΛLTB inferences coming from Pantheon dataset are contrasting the recent result of KSR19 which reports a very high significance of ∼ 5σ against any large local void with contrast |δρ_0/ρ_0|>20%.
        The higher amount of SN Ia data points that they used, 1295 compared to 1048 available in Pantheon, are additionally populating the low-redshift range and are expected to improve the error bars of profile parameters by ≳ 50%.
        However, we suspect that theoretical modelling is also contributing to the difference w.r.t. our result.
        While the newer data is not publicly available at this moment, we intend to extend the current analysis in a future communication.
        In <ref> we present the effects of simplifying the density profile from GBH to TH solution and fixing the model parameters such as Ω_m^ bg and z_ size, instead of marginalising over them to obtain the constraints on void contrast.
        
        
        We chose to report the results of SN Ia analysis with and without the Planck CMB constraint on background matter density.
        Both SN Ia datasets provide excellent fits in the case of a local underdensity, but do not exclude the possibility for a z_ size>0.1 overdensity (see <Ref>).
        The Planck prior on background cosmology reduces the degeneracy between Ω_m^ bg and δΩ_0/Ω_0 parameters in SN Ia likelihood, providing tighter constraints.
        Nevertheless, the main inferences presented here are not strongly affected by this prior due to the excellent agreement between the Planck value and our SN Ia best-fits for Ω_m^ bg.
        Examples presented in <ref> also explore the case of using a fixed value for the relative matter density at the background, as contrary to sampling this likelihood parameter.
        
        Compared to homogeneous model, the additional three parameters that characterise the void profile afford better fits to the SN Ia data, namely χ^2_ JLA=678.30 and χ^2_ Pan=1020.72 for ΛLTB model in contrast to χ^2_ JLA=682.94 and χ^2_ Pan=1026.05 for ΛCDM model.
        We quantify this comparison with Akaike information criteria <cit.> that penalises the usage of three extra parameters as Δ AIC=Δχ^2 + 2*3.
        All fits reported in <Ref> have Δ AIC<2 implying no preference between the two models.
        
        
        
         §.§ Anisotropy considerations
        
        Using galaxy redshift distribution and number counts up to the depth of z<0.1, WS14 observed 215h_70^-1Mpc local underdensity and an angular variation of its estimated density contrast in the range -4% to -40%, with the mean of -15% (corresponds to δΩ_0/Ω_0=-19% shown in <Ref>).
        Likewise, the Fig. 11a in KBC13 clearly shows discrepancy between LD points observed in different regions on the sky, again pointing towards a possibly anisotropic local density profile.
        Hence, an important extension to the analysis presented here can be done by taking into account the anisotropy of the data, as well as the matter density profile and the observer's position inside of the considered inhomogeneity (see e.g. <cit.>).
        If there was any anisotropy present in the matter distribution, it would be averaged out in the analysis that uses isotropic model like ours, leaving us with the expectation to find angular variation of density contrast around this average when modelling for anisotropic matter profiles.
        Be that as it may, the statistical significance in favour of the observed anisotropy in LD data is limited by the sky coverage of the used surveys, which amounts to ∼ 15% for KBC13 and ∼ 22% for WS14, spread over three distant patches on the celestial plane.
        
        Several groups performed tests on SN Ia data looking for features of local and/or global anisotropy <cit.>, also in relation to the local matter distribution <cit.>.
        These anisotropy estimates are also impaired by the inhomogeneous angular distribution of the observed SN on the sky, which is particularly visible on medium and high redshifts due to the narrow fields of the surveys, such as SDSS.
        Probing low-redshift SN Ia for signatures of anisotropy, we estimate the statistical variance of model parameters over angular directions.
        Concretely, the two parameters of interest, M and δΩ_0/Ω_0 in ΛLTB model, are tested separately for dipole anisotropy (see also e.g. <cit.>).
        The magnitude dipole is modelled as a vector of intensity M_d and direction n_d.
        Then, the magnitude parameter M^(i) of each SN Ia inside the void becomes  
         M^(i)   =M̅+ M_dcos( n^(i), n_d)      for z^(i)≤ z_ size ,
         M^(i)   =M̅      for z^(i)> z_ size .
        
        where M̅ is the average intercept magnitude, while the normalised vector n^(i) represents the angular position of each SN Ia.
        Although we fitted the full Pantheon dataset, the dipole is constrained mainly by 190 SN Ia under z≤0.08, which are also more homogeneously distributed on the celestial plane compared to the full dataset.
        Here we used a TH matter density profile for simplicity (see <ref>).
        Finally our estimate for the magnitude dipole
         M_d=0.011^+0.026_-0.031
        would produce the angular variation
        Δ M_ r.m.s.=√(Var(Δ M))=√(2)2 M_d=0.008±0.018 .
        
        This angular variance of magnitude inside the void can be related to the anisotropy of local expansion rate as
        (Δ H_0^ loc/H_0^ loc)_ r.m.s.≈0.2ln(10)  Δ M_ r.m.s.=0.4±0.8%
        Our result for the level of local anisotropy in Pantheon SN Ia is consistent with previous works that find null evidence against assumption of the isotropy <cit.>, although less stringent compared to global anisotropy constraint found by <cit.>.
        Analogously to <ref> we statistically probed the dipole anisotropy of the estimated void contrast and found that it could account for the angular variance
        Δ(δΩ_0/Ω_0)_ r.m.s.=3.7±4.1% .
        
        While this result at face value shows that Pantheon dataset does not exclude the angular variation of estimated void contrast Δ(δΩ_0/Ω_0)∼ 10% at 2σ confidence level, we still find it difficult to coincide our constraints on matter density profile from Pantheon SN Ia and LD data, or explain the level of anisotropy observed by WS14.
        We would like to note that two former examples should be considered only as rough estimates of anisotropy level since: the dipole model does not represent a complete relativistic solution of FE, the variation of profile size is not considered, and the Pantheon SN Ia have had the apparent magnitudes corrected using the BBC method which relies on the assumption of isotropy <cit.>.
        Although crucial in future studies, the present data is not quantitatively sufficing to benefit from the full anisotropic analysis given as an extension of the cosmological metric and theoretical model for the anisotropic density profile.
        
        
        
         §.§ Distance ladder measurements
        
        One of the most important implications of a large local void is the effect on direct measurement of present cosmic expansion rate by the means of low-redshift SN Ia.
        Consider a spherical matter inhomogeneity, characterised with a density contrast of e.g. δΩ_0/Ω_0=-25% which is inside the 1σ confidence region of Pan+P constraints, presented in <Ref> and <Ref>.
        This matter contrast will also induce a higher local present expansion rate by δ H_0/H_0≈3.6% w.r.t. the background expansion rate (see <Ref>.
        Hence, the luminosity distance of a SN located on any shell inside the local void is higher than what one would estimate using the background metric.
        This can affect the analysis of SN Ia data in the redshift range inside the local void as well as the resulting model inferences.
        
        In light of everything said above, let's assume our local matter distribution has indeed an isotropic underdense profile.
        Analysing SN Ia data from the redshift range inside this local void with a simpler homogeneous model would provide a different constraint on H_0^ bg than the LTB model.
        In fact, the assumption of locally homogeneous cosmic expansion (H_0^ loc=H_0^ bg) can overestimate the value of background expansion rate, while underestimating H_0^ loc, by ∼1/2δ H_0/H_0 and affect the inferences on other background cosmological parameters.
        
        As discussed in <Ref> theoretical expectations for the large local void based on ΛCDM model, and its effect on H_0 measurement are small, which is reflected in results presented by recent studies.
        On one side, the level of expected cosmic variance in local matter density for the Planck ΛCDM model is estimated to impact the systematic error of H_0^ bg at ≳ 2% when using SN Ia from z≥0.01 and at ≳ 1% when using SN Ia from z≥0.0233<cit.>.
        On the other side, the large-volume cosmological N-body simulations based on the Planck ΛCDM model, such as e.g. in <cit.>, were used to quantify effects of both the cosmic variance and the SN Ia sample selection on H_0^ bg value - <cit.> estimate the expected variance level of measured H_0^ bg to be ∼ 0.5% for 0.0233<z<0.15 cut (c.f. an earlier work by <cit.>).
        In order to minimise the effect of local matter distribution, R16 (including their previous works) use  only the SN Ia above z>0.0233 for the model-independent estimate of H_0^ bg<cit.>.
        In consistence with <cit.>, KSR19 report no evidence for strong variation of matter density profile in the range z<0.15 from SN Ia data, securing the ability to measure H_0^ bg value to a percent value.
        
        The measurement of background expansion rate based on low-redshift SN Ia is obtained from the intercept magnitude parameter M (see <Ref>).
        While the exact value of H_0^ bg relies on local SN Ia luminosity calibration, we are mainly interested at the relative change of this estimate due to the effect of a large local void, which can be seen as the shift in estimated value of M.
        For this purpose, and only in this subsection, we analyse 237 Pantheon SN Ia from 0.0233< z<0.15 redshift cut.
        Fitting them to FLRW series expansion (SE) formula for luminosity distance and using a fixed value of q_0=-0.55, as presented in R16, we obtain M( SE)=-1.1825±0.0094.
        In low redshift range, this fit is equivalent to ΛCDM fit with Ω_m=0.3, which we will use for comparison to underdense scenario.
        The same Pantheon cut is fitted to ΛLTB model with background cosmology prior Ω_m^ bg=0.3 (q_0^ bg=-0.55), density contrast prior range -50%≤δΩ_0/Ω_0≤5%, void size 0.025≤ z_ size≤0.12, and shape parameter 0≤Δ z≤ z_ size.
        In order to ensure enough SN Ia that constrain the background, we consider only profiles that reach the background regime up to z≤0.13 by additionally removing the region of parameter space with such combination of large z_ size and Δ z values.
        After marginalising the matter density profile out of the low-redshift Pantheon likelihood, we estimate the intercept SN magnitude parameter[Confidence intervals are given between 16th and 84th percentiles] to be
        M(Λ LTB)=-1.1577_-0.0148^+0.0158.
        The remaining parameter constraints are presented in the last row of <Ref>.
        Regardless the three additional parameters that model the local matter underdesnity, the ΛLTB fit to low-redshift Pantheon SN Ia reveals an improvement of the best-fit by ΔAIC=-3.3.
        
        The local matter underdensity affects the estimated value of M due to its degenearcy with density constrast.
        This can be seen on <Ref>, where we show the low-redshift Pantheon constraints in δΩ_0/Ω_0- M plane.
        The best-fit of ΛCDM model (equivalent to SE) can be found on vertical axis for zero void contrast and is marked as a red point in <Ref>, which is 1.7σ away from our constraints on ΛLTB model.
        Both values for the intercept magnitude parameter M have been estimated fitting the same 237 Pantheon SN Ia to two different models and are, hence, fully correlated.
        Taking into account this correlation, the difference between two values is evalueted to be Δ M=0.0247^+0.0065_-0.0055.
        As the ΛLTB model has more parameters in fitting, the error bar of resulting M(Λ LTB) is bigger than in the case of M(Λ CDM), and elseways, we note that if they were equivalent then the uncertainty on estimated shift Δ M would have been zero.
        Since this change of intercept magnitude value is coming only from the theoretical modelling and the two models have comperable AIC values (see <Ref>), it should be seen as a possible source of systematic error on M estimate.
        The positive shift of Δ M=0.0247 corresponds to 1.14% lower H_0^ bg obtained with ΛLTB model than with SE formula.
        Compared to the total error of M( SE), this value of Δ M is significant.
        However, the major contribution to the uncertainty of H_0 measurement is presently constricted to calibration of standard SN Ia absolute magnitude, M_B, that together with much smaller uncertainty on intercept magnitude M( SE) reaches the total error bar of 2.4% reported by R16.
        Adding the effect of large local matter inhomogeneity (z_ size>0.025) on estimate of SN Ia intercept magnitude could increase this uncertainty level to 2.7% on the lower end of H_0 measurement.
        The improvement in Cepheid and SN Ia calibation techniques (see R19), followed by reduction of uncertainty on M_B, can lead to a situation in which the effect of local matter inhomogeneity becomes more relevant for the measurement of background cosmic expansion rate.
        
        
        § CONCLUSIONS
        
        In the present work we have investigated the evidence for a large local void using a well formulated ΛLTB model against the most recent publicly available SN Ia compilations.
        In order to reduce the correlation with background cosmology, we have used the CMB constraint on Ω_m^ bg<cit.>, which aids to provide a slightly tighter constraints on the local void profile.
        The analysis on SN Ia datasets is also complimented with the luminosity density data obtained from galaxy surveys by <cit.>, which together with <cit.> played a key role in propelling the recent discussion about possible existence of a large local void and its effect towards resolving one of the most prominent discordances in modern cosmology - the H_0 tension.
        We summarise our primary results in the following.
        All reported confidence intervals (c.i.) are between 16th and 84th percentiles.
        
          *  Fitting ΛLTB model to LD data from KBC13, we find a void with density contrast of (δρ_0/ρ_0)^c.i. = -43.8± 6.0%, deeper than originally proposed δρ_0/ρ_0 = -30%.
        
          *  From SN Ia we do not find a strong evidence for a void or otherwise – a fit to Pan+P yields (δρ_0/ρ_0)^c.i. = -7.9_-5.8^+7.0% which corresponds to (δΩ_0/Ω_0)^c.i.=-10.5^+9.3_-7.4%, and a wide range for the redshift size z_ size^c.i.=0.068^+0.021_-0.030.
        However, Pantheon constraints do not exclude a void of e.g. z_ size≈0.075 and δΩ_0/Ω_0≈-25% at 1σ confidence level (c.f. <Ref>).
        
          *  JLA likelihood is in a good overall agreement with Pantheon, except in the range z≤0.04 (c.f. <Ref>). Both SN Ia datasets are at ≳ 3σ tension with our result obtained from KBC13 data, but in excellent agreement with parameters of the isotropic void proposed by WS14.
        
          *  Leave one out analysis of KBC13 data reveals that the corresponding estimated contrast is dominantly constrained by one stringent point.
        Removing this only data point from 2MASS survey and using the remaining 9 points from UKIDSS and GAMA surveys, relaxes their tension with SN Ia data.
        
          *  The model comparison with Akaike information criteria shows no preference between the tested models (c.f. <Ref>).
        
          *  Probing the level of statistical variance in ΛLTB model parameters over angular direction, we find null evidence for the dipole anisotropy.
        Its contribution to the angular variance is small: Δ M_ r.m.s.=0.008±0.018 for the SN intercept magnitude parameter or Δ(δΩ_0/Ω_0)_ r.m.s.=3.7±4.1% for the void contrast.
        
          *  In our analysis of SN Ia data we do not find any evidence for a large isotropic void that could resolve the 9% discrepancy between two H_0 estimates in tension, leaving this local effect alone to be a highly unlikely explanation.
        
          *  We fitted low-redshift range of Pantheon SN Ia data from 0.0233<z<0.15 to the FLRW series expansion formula for luminosity distance and to the ΛLTB model, comparing their estimates of SN Ia intercept magnitude, which is used for H_0 measurement.
        Adding the possibility of a large local void, we observe a small improvement in the fit by Δ AIC=-3.3 and a shift of the inferred intercept magnitude parameter Δ M=0.0247^+0.065_-0.055 that could be a source of additional systematic error in H_0^ bg of 1.14%.
        While the simpler model, which is based on series expansion formula of luminosity distance in FLRW metric, is only 1.7σ away from our ΛLTB constraints (see <ref>), this result is questioning whether the total error budget of distance ladder measurements based on low-redshift SN Ia should be reconsidered.
        
        
        Increase of presently available data, followed by more complex theoretical modelling, is necessary to better understand the observed disagreement between LD and SN Ia samples.
        As already mentioned, considering off-centre position of the observer would extend the present isotropic ΛLTB formalism, also allowing for an anisotropic point of view by construction.
        With increasing number of observations in future surveys, SN Ia may prove as effective tracers of local matter density distribution.
        
        
        
        § ACKNOWLEDGEMENTS
        
        The authors would like to thank Amy J. Barger, Benjamin L. Hoscheit, Adam Riess and Tom Shanks for useful discussions, and Caterina Traficante for contribution to this work during her thesis. Authors acknowledge financial support by ASI Grant No. 2016-24-H.0.
        mnras
        
        
        § TOP-HAT DENSITY PROFILE
        
        Our results for ΛLTB model can be reproduced to a good extent using a simpler TH density profile constructed as a two-step homogeneous kΛCDM model.
        The following approximate formulae are derived from ΛLTB model as third order series expansion in terms of present matter density contrast x=δρ_0/ρ_0.
        Given that Λ is constant, its dimensionless density parameter is not (see <ref>).
        δ H_0 H_0=   x(-0.17093-0.32158 (Ω_m^ bg-0.3)+0.24932(Ω_m^ bg-0.3)^2)
        
        +   x^2(0.03141+0.063 (Ω_m^ bg-0.3))
        
        +   x^3(-0.02237)
        δΩ_0Ω_0=   x(1.34186+0.64317 (Ω_m^ bg-0.3)-0.49863(Ω_m^ bg-0.3)^2)
        
        +   x^2(0.36669+0.84699 (Ω_m^ bg-0.3))
        
        +   x^3(+0.0563)
        δΩ_ΛΩ_Λ=   x(0.34186+0.64317(Ω_m^ bg-0.3)-0.49863(Ω_m^ bg-0.3)^2)
        
        +   x^2(0.02483+0.20382 (Ω_m^ bg-0.3))
        
        +   x^3(+0.03211)
        In the range 0.2<Ω_m^ bg<0.4 and -30%<δρ_0<30% these eqs. are correct with less than 1% error on the value.
        The constructed TH model can be characterised with three cosmological parameters: Ω_m^ bg, δΩ_0/Ω_0 and z_ size, as well as the intercept SN Ia magnitude parameter M.
        As usual, using a prior on Ω_m^ bg ensures tighter constraints on void parameters.
        Using <ref> together with two-step kΛCDM formalism, it is easy to evaluate all the distances of SN Ia inside and outside the void.
        
        Sampling Pantheon χ^2 for different model parameters' values, we immediately notice it can be very irregular along the redshift axis, since the top-hat profile does not have a smooth transition from local to background geometry (see <Ref>).
        We find that a TH void with a contrast δΩ_0/Ω_0 ≈ -25% has comparable χ^2 to the best-fit of ΛCDM model for specifically chosen values of z_ size≈0.075, although this fitting method has no way of accounting for the systematic error arousing with fixing the physical size of the void.
        To overcome this, one can perform smoothing of the sampled TH likelihood over redshift bins of size Δ z=0.01.
        Although short in size, these bins have ∼ 20 low-redshift SN each, regularising the average likelihood dependence on z_ size.
        The 1σ and 2σ confidence regions resulting from smoothed likelihoods are shown as solid lines in <Ref>.
        We note that contours obtained in this way for Pan+P data are substantially similar to what we got using the GBH profile (c.f. <Ref>).
        After properly marginalising out other parameters in the smoothed likelihood, we get a Pan+P constraint on contrast as (δΩ_0/Ω_0)^c.i.=-9.0_-6.8^+7.6%.
        The corresponding result in <Ref> has somewhat wider error bars since the smooth GBH function allows for more freedom in density profile than the TH form.
        
        The second and third panels in <Ref> show the importance of fitting the model parameters in this scheme as opposed to fixing them.
        Specifically, the last panel is obtained by
        sampling Pantheon χ^2 for contrast δΩ_0/Ω_0 and M parameters, while fixing the void size to z_ split and Ω_0^ bg=0.3.
        Hence, this method is unable to provide a constraint on void size, just as the constraint on density contrast depends on the prior choice of z_ split.
        The contours obtained in third panel of <Ref> are sequences of confidence intervals for δΩ_0/Ω_0 parameter,
        recognised by a stripe shape in z_ split-δΩ_0/Ω_0 plane.
