Null tests of the standard model using the linear model formalism
Valerio Marra Domenico Sapone
astro-ph.CO

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    Núcleo Cosmo-ufes & Departamento de Física, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, BrazilDepartamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
    We test both the FLRW geometry and ΛCDM cosmology in a model independent way by reconstructing the Hubble function H(z), the comoving distance D(z) and the growth of structure fσ_8(z) using the most recent data available.
    We use the linear model formalism in order to optimally reconstruct the above cosmological functions, together with their derivatives and integrals.
    We then evaluate four of the null tests available in literature that probe both background and perturbation assumptions.
    
    For all the four tests we find agreement, within the errors, with the standard cosmological model.
    Null tests of the standard model using the linear model formalism
        Domenico Sapone
        December 30, 2023
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    § INTRODUCTION
    
    
    
    The late-time accelerated expansion of the Universe has led cosmologists to revise the theory according to which the cosmos evolves either by introducing a new form of matter called dark energy <cit.> or by modifying directly the laws of gravity <cit.>. 
    Within the framework of Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies, a phase of accelerated expansion can be produced by a simple cosmological constant Λ.
    Although the above gives rise to severe coincidence and fine-tuning problems, observations seem in agreement with such an explanation <cit.>.
    
    
    In order to make progress and understand what is causing the universe to accelerate one has, generally speaking, two options.
    One can assume that the ΛCDM model is correct and work out its consequences. If inconsistencies with data are found, then physics beyond the standard model is necessary.
    The other complementary approach is to assume a specific model beyond ΛCDM and work out if it has advantages with respect to the standard paradigm.
    The second approach may be more powerful and may trigger new ideas and methodologies.
    The first approach is simpler so that it is possible to study and understand the model phenomenology in much greater details.
    The methodology of this paper belongs to the first approach.
    In particular, we will analyze null tests of the standard model.
    
    
    Null or consistency test analyses do not aim at finding the parameters of the model in question. Rather, they aim at uncovering possible tensions in data, which could be due to unaccounted-for systematics or a failure of the model itself.
    These tests are model independent in the sense that they use directly the data, and have the added advantage that, if violated, one knows which set of theoretical assumptions have to be reanalyzed.
    Equivalently, null tests have the ability to extract information that one may miss if restricting to parameter estimation.
    It is worth stressing that it is imperative to corroborate the underlying cosmological model in such a way in view of future experiments that will collect an enormous amount of data, spanning over a wide range of redshift, see, for example, DES <cit.>, eBOSS <cit.>, J-PAS <cit.>, DESI <cit.>, LSST <cit.>, Euclid <cit.>, SKA <cit.>.
    
    
    
    Here we consider four null tests that have been proposed during the past 10 years: the Om_1 diagnostic by Sahni et al. <cit.>, the Om_2 diagnostic by  Zunckel & Clarkson <cit.>, the Ok diagnostic by Clarkson et al. <cit.>, and the ns diagnostic by Nesseris & Sapone <cit.>.
    In order to evaluate these tests, we reconstruct the Hubble function H(z), the comoving distance D(z) and the growth of structure fσ_8(z) using the most recent data available.
    We use the linear model formalism in order to optimally reconstruct the above cosmological functions.
    This method is simple and powerful as one can obtain an exact statistical description of the reconstructed functions, including their derivatives and integrals. Furthermore, it is an analytical approach which is suitable to be used with large datasets.
    Therefore, we propose this method as a possible alternative to methods previously used in the literature, to list a few: binning data and discrete derivatives, principal component analysis, genetic algorithms, Padé approximation, Gaussian processes, non-parametric smoothing, machine learning <cit.>.
    
    This paper is organized as follows.
    In Section <ref> we review basic equations and the notation adopted in this paper, and in Section <ref> we briefly present the four null tests considered in this work.
    We describe the data we use in Section <ref> and detail our methodology in Section <ref>. The results of Section <ref> show that the standard cosmological model passes all the tests.
    The busy reader can jump to Figure <ref>.
    We conclude in Section <ref>.
    
    
    
    
    
    
    
    § BASIC EQUATIONS
    
    
    
    
    Here we review the basic equations upon which all the null tests are built. The evolution of the dark energy component can be expressed in terms of its present energy density parameter Ω_de_0 and its equation of state parameter w(z) = p/ρ, being p and ρ the pressure and energy density, respectively.
    The subscript 0 will denote the present-day value of the corresponding quantity.
    The Hubble parameter is then:
    H^2(z)/H_0^2 = Ω_m_0(1+z)^3 +Ω_de_0(1+z)^3(1+ŵ) +Ω_k_0(1+z)^2
    
    where
    ŵ(z) = 1/log (1+z)∫_0^zw(z')/1+z' z' ,
    
    and Ω_m_0 and Ω_k_0 are the matter and curvature density parameters, respectively.
    If we are dealing with the cosmological constant, then w(z) = -1 and ŵ(z)=-1 at all redshifts. 
    Furthermore, the relation Ω_m_0+Ω_de_0+Ω_k_0=1 has to be satisfied.
    
    
    In a general FLRW model with curvature, the angular diameter distance can be written as:
    
    D_A(z) = c/1+z1/H_0√(-Ω_k_0)sin(√(-Ω_k_0)∫_0^z z'/H(z'))  .
    
    The angular diameter distance is related to the luminosity distance D_L and the dimensionless comoving distance D by the relations:
    
    
        D_L(z)   =(1+z)^2 D_A(z)  , 
        
        D(z)    =H_0/c  (1+z) D_A(z)  .
    
    In the Universe matter clusters forming perturbations δρ(t,x) to the underlying background energy density ρ(t). The growth of matter perturbation in the ΛCDM model is given by (assuming homogeneity and isotropy)
    δ”(z)+(5/1+z-H'(z)/H(z))δ'(z) -3/2Ω_m(z)/(1+z)^2δ(z)=0
    
    where a prime refers to the derivative with respect to z, and the time evolving matter energy density is 
    Ω_m(z) = Ω_m_0(1+z)^3/H^2(z)/H_0^2 .
    
    
    
    
    § NULL TESTS OF THE STANDARD MODEL
    
    
    
    In this section we list the null tests that we consider in this paper; we refer to the corresponding literature for further details. 
    
    
    
    
     §.§ Om diagnostic
    
    
    The Om diagnostic was introduced to test deviations with respect to the flat ΛCDM scenario.
    Setting w=-1 and Ω_k_0=0 in Eq. (<ref>), one can solve for Ω_m_0 and obtain the following diagnostic <cit.>:
    
    
        Om_1(z) = H(z)^2/H^2_0-1/(1+z)^3-1 .
    
    Equivalently, one can solve for Ω_m_0 in Eq. (<ref>) and obtain the alternative diagnostic <cit.>:
    
        Om_2(z) = 1/D'(z)^2-1/(1+z)^3-1 .
    
    Clearly, within the flat ΛCDM model one has:
    
    
        Om_1(z)=Om_2(z)=Ω_m_0 ,
    
    which has to be valid at any redshift.
    Any violation of the above relation will falsify the flat ΛCDM model.
    As discussed below, we will obtain the luminosity distance D_L(z) (and so D) from supernova Ia data, and the Hubble function H(z) from the cosmic chronometer data.
    
    
    
    
     §.§ Ok diagnostic
    
    
    
    The following constant-curvature test can falsify not only the flat ΛCDM model but all the FLRW models at once.
    The Ok diagnostic is defined according to <cit.> (see also <cit.>):
    
    
        Ok(z)=D'(z)^2 H(z)^2/H^2_0 -1/D(z)^2 .
    
    If the FLRW models are the correct background models, one has:
    
        Ok(z) = Ω_k_0 .
    
    Any violation not caused by standard-model perturbations
    [Standard-model perturbations can produce an additional systematic error on H_0 <cit.> and on the dark-energy equation of state <cit.>.]
    would have profound implications as the FLRW model is at the basis of almost any cosmological model (except inhomogeneous <cit.> and backreaction models, see the CQG special issue <cit.>, and also <cit.>).
    
    
    
    
    
    
     §.§ ns diagnostic
    
    
    While the previous tests probe the background structure of the universe, the test proposed in <cit.> (and deeply investigated in <cit.>) is sensitive to both background and perturbation observables. 
    The ns diagnostic is able to test the validity of the growth of matter perturbations under the assumption of an FLRW metric with a cosmological constant (which has no perturbations). In brief, the ns diagnostic is obtained from Eq. (<ref>): first, the corresponding Lagrangian is found, then, with the help of Noether's theorem, the associated conserved quantity is obtained.
    The conserved quantity is the ns diagnostic that is found to be: 
    
    
        ns(z)=1/1+zH(z)/H(0)fσ_8(z)/fσ_8(0)×
              exp[3·10^4/2ω_m∫_0^z(1+x)^2σ_8,0-∫_0^xfσ_8(y)/1+ydy/H(x)^2fσ_8(x) x]
    
    where σ_8,0 is the normalization of the power spectrum (present day mass fluctuation on a scale of 8 h^-1Mpc, this refers to linear
    perturbation theory), and:
    
    
        fσ_8(z)= f(z) G(z) σ_8,0 ,
    
    in which G(z) is the growth function of matter perturbations normalized to unity today (G(z)=δ(z)/δ(0), from (<ref>)) and the growth factor is f= ḷṇG/ḷṇ(1+z)+1.
    
    
    In order to be robust against systematics of a particular experiment, we need to reconstruct the ns(z) test by using four independent observables: the Hubble parameter H(z), the growth of structure fσ_8(z), σ_8,0 and ω_m = Ω_m_0h^2.
    As discussed below, we will obtain fσ_8 from the RSD measurements collected by different experiments; the Hubble data come from the cosmic chronometers; instead, for σ_8,0 we will use alternatively the results from the SDSS-III BOSS <cit.> and KiDS <cit.> surveys, and for ω_m the results from the Planck satellite <cit.>. 
    
    Within any (not necessarily flat) ΛCDM model one has:
    
    
        ns(z)=1  .
    
    Any violation of the above relation may imply a deviation from the FLRW model, nonzero dark-energy perturbations and/or a deviation from GR.
    
    
    
    
    
    
    
    
    
    § DATA
    
    
    
    
    The four tests above use three cosmological functions: H(z), D(z) and fσ_8(z). We will estimate these functions using three distinct datasets. We will use the full covariance matrix when available.
    
    
    
     §.§ Cosmic chronometers
    
    
    The so-called “cosmic chronometers” are passively evolving galaxies from which it is possible to obtain in a model-independent way the Hubble parameter H(z) at various redshifts <cit.>.
    Here, we use the 31 independent data points given in Table <ref>.
    This is the most up-to-date collection of H(z) data: see Figure <ref> for a plot (where also shown is the corresponding linear model best fit, see Section <ref>).
    
    
    
    
    
     §.§ Supernovas Ia
    
    
    We obtain the luminosity distance (in Mpc) from the distance modulus μ:
    
    
        D_L(z)=10^μ(z)-25/5 .
    
    The distance modulus can be obtained from supernova Ia data. We will consider the compressed JLA compilation <cit.> so that:
    
    
        μ(z) = μ_b(z)-M  ,
    
    where μ_b(z) is the binned distance modulus and M is an unknown offset (a nuisance parameter). Fig. <ref> shows the JLA data.
    
    
    
     §.§ Redshift space distortion data
    
     
    
    In order to reconstruct fσ_8(z) we will consider the robust and independent measurements given in the “Gold” RSD compilation from <cit.>, see Fig. <ref>.
     
     
    
    The three sets of data used in this paper come from several surveys that span different angular and redshift ranges. Therefore, it should be justified to assume that we are considering a fair sample of the universe.
    
    
    
    
     §.§ Addition parameters
    
    
    
    For the parameter ω_m, necessary for the ns test, we use the results from Planck <cit.>:
    
    
        ω_m     = 0.1427 ± 0.0014  .
    
    For the parameter σ_8,0, also necessary for the ns test, we use three different values from two surveys. Indeed, the Planck cosmological constraints on σ_8,0 are in
    tension with those from Planck clusters <cit.> and from weak lensing measurements, and it is important to test how  different values of σ_8,0 impact our results.
    We will consider results from the SDSS-III BOSS <cit.> and KiDS <cit.> surveys, which are independent from Planck. SDSS is a spectroscopic galaxy survey while KiDS is a survey that uses cosmic shear, galaxy-galaxy lensing and redshift-space distortion galaxy clustering measurements. The values we consider are:
    
    
        σ_8,0   = 0.804 ± 0.051  SDSS-III BOSS , 
        σ_8,0   = 0.832 ± 0.080  KiDS , 
        σ_8,0   = 0.747 ± 0.109  KiDS (conservative cut) .
    
    The Hubble constant H_0 can be determined using the reconstructed H(z) function at z=0.
    However, it may also be interesting to consider an independent prior on H_0. This could be useful to test for systematics. We will consider results from local measurements <cit.> and from the CMB <cit.>, respectively:
    
    
        H_0,loc   = 73.24 ± 1.74  km/s/Mpc , 
        
        H_0,cmb   = 67.27 ± 0.66  km/s/Mpc .
    
    The parameter M enters in the determination of the dimensionless comoving distance D that is used in the tests Om_2 and Ok. As M is degenerate with -5 log_10 H_0, we leave M free.
    
    
    
    
    
    
    
    
    
    
    § METHOD
    
    
    
    In order to reconstruct the cosmological functions, and also their derivatives and integrals, we will use the linear model formalism; see, for instance, <cit.>.
    
    
    
    
     §.§ Linear model analysis
    
    
    
    
    Let us choose a set of base functions g_α(z) whose  linear combination will constitute the template function t(z,c_α):
    
    
        t(z,c_α) = ∑_α=0^α_ max c_α  g_α(z)  ,
    
    where α is an integer.
    The assumption is that t(z,c_α) can describe the actual functions that we want to reconstruct: H(z), μ(z) or fσ_8(z). 
    Clearly, this is conditional to an appropriate choice of g_α(z) and α_ max, for each of the functions H(z), μ(z) and fσ_8(z).
    We call α_ max the “order” of the template, which will then have α_ max+1 coefficients.
    
    Let us then assume that the data are given by:
    
    
        d_i = t_i + e_i  ,
    
    where t_i=t(z_i,c_α) and e_i are Gaussian errors with covariance matrix C_ij.
    
    
    Next we fit the template t to the data and use the linear model formalism to calculate the Fisher matrix relative to the parameters c_α.
    This gives an exact description of the likelihood as the template is linear in its parameters.
    The Fisher matrix is:
    
    
        F_αβ = g_β i C^-1_i j g_α j
    
    where g_α i= g_α(z_i), and the best-fit values of c_α are:
    
    
        c_α,  bf= F^-1_αβ B_β ,
    
    where B_α = d_i C^-1_i j g_α j.
    
    Summarizing, we have propagated the covariance matrix C_ij into the covariance matrix F^-1_αβ on the parameters.
    
    
    
    
     §.§ Error propagation
    
    
    Let us denote with ϕ(z,θ_α) either Om_1, Om_2, Ok or ns. ϕ(z,θ_α) will be a nonlinear function of the various templates t(z,c_α) (one for each H(z), μ(z) and fσ_8(z)) and their derivatives and integrals.
    The parameter vector {θ_α} comprises the template parameters of Section <ref> and the additional parameters of Section <ref> that enter ϕ(z,θ_α).
    The corresponding covariance matrix Σ_αβ is obtained by forming an appropriate block diagonal matrix using the covariance matrices of the corresponding parameters.
    We have chosen independent data (see Section <ref>) so that correlations among different sets of data are not expected to be important.
    
    
    
    In order to compute the error on ϕ(z,θ_α) due to the uncertainty encoded in the covariance matrix Σ_αβ, a straightforward approach is to apply a change of variable from {θ_α} to ϕ.
    At the first order, the error is then given by:
    
    
        σ^2_ϕ = J_αΣ_αβ J_β ,
    
    where
    
    
        J_α =. ∂ϕ(z,θ_α)/∂θ_α|_θ_α,  bf .
    
    Eq. (<ref>) is exact if ϕ(z,θ_α) is t(z,c_α), its derivative or integral. Indeed in this case it will depend linearly on the parameters {θ_α}. 
    
    
    
    
    
    
    
    
    
     §.§ Choice of base functions
    
    
    We will adopt the following base functions:
    
    
        H(z)     ⟶ g_α(z) = z^α    with  0≤α≤α_ max , 
        μ(z)     ⟶ g_β(z) = (ln z)^β    with  0≤β≤β_ max , 
        
        fσ_8(z)    ⟶ g_γ(z) = (1+z)^γ    with  0≤γ≤γ_ max .
    
    We have chosen the latter as they can reproduce the fiducial ΛCDM functions with not too high orders α_ max, β_ max and γ_ max, that is, with not too many parameters.
    Furthermore, the base functions chosen for fσ_8(z) allow us to compute analytically the inner integral in Eq. (<ref>).
    
    
    
    In order to choose the orders α_ max, β_ max and γ_ max on which the template functions t(z,c_α) depend, see (<ref>), we use the following strategy:
    
      *  We pick values for {α_ max, β_ max, γ_ max}. Each order has to be within 0 and N-1, where N is the number of data of the corresponding catalog.[With N points one can fit up to N coefficients so that, for example, the order α_ max cannot exceed N-1.]
      *  We reconstruct H(z), μ(z) and fσ_8(z) using the method of Section <ref> with mock catalogs created using a fiducial ΛCDM cosmology.
    These mocks share the same redshift values and covariance matrix of the real data but do not have fluctuations. In other words, the corresponding H(z), μ(z) and fσ_8(z) values are exactly the fiducial one.
    Mock catalogs without fluctuations are often used when one wants to consider the average behavior of a large number of “real life” mocks with fluctuations.
    
    
    
    
      *  We evaluate the four null tests of Section <ref>.
    
    
      *  We obtain the sigma bands of the null test functions using the results of Section <ref>.
    
    
      *  We calculate the following vector:
    
    
        Δ_ϕ= {|ϕ(z_i,θ_α,  bf)-ϕ_ fid(z_i)|/σ_ϕ(z_i,θ_α,  bf)} ,
    
    with i=1, …, N.
    
      * 
    We set a reconstruction accuracy Q and we repeat the steps <ref>-<ref> until we find the smallest values {α_ max, β_ max, γ_ max} so that the following  condition is satisfied:
    
    
        max{Δ_Om_1,Δ_Om_2,Δ_Ok,Δ_ns} < Q  .
    
    By setting Q=0.1, we find:
    
    
        {α_ max, β_ max, γ_ max}= { 3, 6,4} .
    
    Figures <ref>-<ref> show the reconstruction of the cosmological functions that we obtained.
    
    The above strategy guarantees that the template functions can replicate the fiducial ΛCDM functions without inserting modeling biases in the analysis while, at the same time, keeping the template order as low as possible.
    It is worth stressing that within our methodology a unnecessarily high order would lead to higher uncertainties in the Fisher matrix of equation (<ref>) thus degrading the constraining power of the null tests.
    
    
    
    
    
    
    
    § RESULTS
    
    
    
    
     §.§ Om diagnostic
    
    
    
    Figure <ref> (left panels) show the reconstruction of the diagnostics Om_1 and Om_2.
    The 1σ gray bands are obtained using equation (<ref>).
    The Om_1 test is, within its error, consistent with the reference value of about 0.3.
    Figure <ref> was obtained adopting the H_0,cmb value of equation (<ref>) (solid line and gray bands), H_0= 70  Km s^-1Mpc^-1 (dashed line) and the H_0,loc value of equation (<ref>) (dot-dashed line).
    
    
    
    
    
    
    
    In order to evaluate the Om_2 test we need to specify the value of the nuisance parameter M, on which D depends. As discussed in Section <ref>, M is degenerate with -5 log_10 H_0. Therefore, we can fix H_0 to an arbitrary value such as H_0= 70  Km s^-1Mpc^-1 and adopt the value of M for which Om_2 is closest to a constant. If such a value of M does not exist, then the flat ΛCDM model is falsified.
    The lower left panel of Figure <ref> shows the Om_2 diagnostic for the value M=0.
    Also this test is passed.
    It is worth stressing that this test (and also the Ok test discussed in the next section) will be more constraining when the comoving distance D will be reconstructed using data which does not need unconstrained nuisance parameters such as M.
    Future BAO data could be useful in this respect.
    
    
    
    The Om_1 test (and also the Ok test) diverges at z=0 as it involves a 0/0 limit.
    Because of the noise in the data, this limit is not well behaved and it does not follow the theoretical behavior.
    Nonetheless, this is not a problem as also the errors diverge so that this singular behavior is correctly taken into account and does not bias the result.
    
    
    
    
    
    
    
     §.§ Ok diagnostic
    
    
    In Fig. <ref> (top right panel) we show the results of the reconstruction of the Ok diagnostic of Eq. (<ref>).
    As with the Om_2 test of the previous section, one has to specify the value of the nuisance parameter M.
    Moreover, in this case, the Hubble constant H_0 appears directly in the equation (<ref>).
    In the plot we show the Ok diagnostic that is obtained when adopting the H_0,cmb value of equation (<ref>) (solid line and gray bands), H_0= 70  Km s^-1Mpc^-1 (dashed line) and the H_0,loc value of equation (<ref>) (dot-dashed line).
    It is clear that the reconstruction is consistent with Ω_k_0 = 0. 
    The results found are consistent with those presented in <cit.>.
    
    
    
    
    
     §.§ ns diagnostic
    
    
    In Figure <ref> (lower right panel) we show the results relative to the ns null test of equation (<ref>).
    Present-day data again confirm within 1σ errors that the ΛCDM model is viable.
    In this case, for consistency <cit.>, H(0) is H(z=0), where the latter is the Hubble function reconstructed from cosmic chronometer data.
    We could also use the H_0 values used for the other tests; however, the effect is an overall shift of the entire curve as H(0) enters as a multiplicative factor.
    The σ_8,0 adopted (black solid line) is from the SDSS-III BOSS survey but we also plotted the ns test for the σ_8,0 values from KiDS, see Section <ref>.
    
    
    Next-generation surveys are expected to be able to accurately reconstruct this test: for example, H(z) data will be obtained via galaxy clustering and fσ_8(z) data via independent weak lensing observations.
    
    
    
    
    
    
    
    
    
    § CONCLUSIONS
    
    
    
    
    
    In this paper, using H(z), Supernova Ia and fσ_8 data we have reconstructed four null-tests, that can be used to probe deviations from either ΛCDM or the assumption of homogeneity and isotropy in the universe. The reconstruction has been performed by fitting the data with the linear model formalism which provides an exact statistical description of the reconstructed functions together with their derivatives and integrals.
    We find that all the four tests are in agreement with the standard cosmological model and no interesting deviations were found.
    This also implies that we have not found any tension on the data. 
    However, a special attention goes to the value of H_0 used. As shown, the Om_1 test is the most sensitive to H_0 due to its direct dependence; from Fig. <ref> we understand that lower values of H_0 favor a larger value of Om_1. The other tests are less affected to the value of H_0.
    
    Current data give results that are far from the cosmic variance limit and future data will revolutionize the usefulness of the null tests: hundreds of thousands of supernovas Ia, hundreds of millions of galaxy spectra and even more shape measurements, 21-cm data to high redshifts will enable us to accurately use the null tests to assess the viability of the standard model of cosmology. In order to be ready for these large datasets it is important to develop and explore alternative methodologies to reconstruct the relevant cosmological functions.
    
    
    
    
    
    
    It is a pleasure to thank Michele Moresco for clarifications regarding cosmic chronometer data.
    VM thanks CNPq and FAPES for partial financial support. 
    DS acknowledges financial support from the Fondecyt project number 11140496.
    
    
    
    utphys
