A comparison of the R_h=ct and LCDM cosmologies based on the observed halo mass function
Manoj K. Yennapureddy Fulvio Melia













=1mm












    ℒ d  d Lips·iRχ T Rχ T⟨⟨||⟩⟩
            1.5pt< -10.5pt3.5pt
                          ∼ 1.5pt 1.5pt> -9.5pt3.5pt
                          ∼ 1.5pt^1Department of Physics, The University of Arizona, Tucson, AZ 85721, manojy@email.arizona.edu
                ^2Department of Physics, the Applied Math Program, and Department of Astronomy,
                 0.1in The University of Arizona, Tucson, AZ 85721, fmelia@email.arizona.eduYennapureddy & MeliaThe Observed Halo Mass FunctionThe growth of structure may be traced via the redshift-dependent halo mass function.
                        This quantity probes the re-ionization history and quasar abundance in the Universe,
                        constituting an important probe of the cosmological predictions. Halos are
                        not directly observable, however, so their mass and evolution must be inferred
                        indirectly. The most common approach is to presume a relationship with galaxies and halos.
                        Studies based on the assumption of a constant halo to stellar mass ratio
                        M_h/M_* (extrapolated from z<4) reveal significant tension with ΛCDM—a failure
                        known as “The Impossibly Early Galaxy Problem". But whether this ratio evolves or remains
                        constant through redshift 4<z<10 is still being debated. To eliminate the tension with
                        ΛCDM, it would have to change by about 0.8 dex over this range, an issue that 
                        may be settled by upcoming observations with the James Webb Space Telescope. In 
                        this paper, we explore the possibility that this major inconsistency may instead be an 
                        indication that the cosmological model is not completely correct. We study this problem 
                        in the context of another Friedmann-Lemaître-Robertson-Walker (FLRW) model known
                        as the R_ h=ct universe, and use our previous measurement of σ_8
                        from the cosmological growth rate, together with new solutions to the
                        Einstein-Boltzmann equations, to interpret these recent halo measurements. We
                        demonstrate that the predicted mass and redshift dependence of the halo
                        distribution in R_ h=ct is consistent with the data, even
                        assuming a constant M_h/M_* throughout the observed redshift range
                        (4≲ z≲ 10), contrasting sharply with the tension in ΛCDM.
                        We conclude that—if M_h/M_* turns out to be constant—the massive
                        galaxies and their halos must have formed earlier than is possible in ΛCDM.A comparison of the R_ h=ct and ΛCDM cosmologies 
                
                based on the observed halo mass function
                    Manoj K. Yennapureddy^1 and Fulvio Melia^2John Woodruff Simpson Fellow.
                    June 5, 2019
                ===========================================================================================
                
                
                
                § INTRODUCTION
                
                        The central principle behind the theory of structure formation is that large-scale
                        assemblies, such as galaxies and clusters, formed via the growth of gravitational
                        instabilities in the primordial density field, comprised of dark matter, radiation
                        and baryonic matter. By assumption, dark matter is weakly interacting, so it
                        decoupled from the radiation quite early and its fluctuations grew gravitationally
                        to form the halos. Baryonic matter subsequently accreted into these potential wells
                        once it also decoupled from the radiation, forming bound objects that would become
                        stars and galaxies. Although this latter process is not yet fully understood, there is
                        better consensus concerning the halo evolution itself, codified through the so-called
                        halo mass function <cit.>.
                        As it turns out, the halo mass function is highly sensitive to the
                        cosmological parameters in ΛCDM, including the mass fraction Ω_ m,
                        the dark energy equation of state parameter w_ de, and σ_8<cit.>—at least at lower redshifts (i.e., z≤ 2), where it plays a vital role
                        in constraining standard cosmology. At higher redshifts, the halo mass function plays a
                        vital role in probing the re-ionization history of the Universe <cit.>
                        and the quasar abundance and formation sites <cit.>.
                        It goes without saying that constraining and evaluating the halo mass function is
                        therefore critical to the evaluation of structure formation in the Universe.
                
                        The standard model predicts a rapid evolution in the number density of
                        massive halos throughout the redshift range 10≳ z≳ 4. If a strong connection
                        exists between the halos and galaxies they host, one should expect to see a comparably
                        rapid evolution in the number density of galaxies via their luminosity and mass distributions, implying that one should see in ΛCDM a sharp decline in the number density of luminous galaxies at constant luminosity, or a rapid decrease in luminosity for a fixed number density, towards high redshifts. Although the halo mass function has been evaluated using numerical and N-body simulations at high redshifts, only recently has it been tested observationally. The halos themselves are not directly observable, so they must be probed indirectly, e.g., through the measured galaxy mass distribution assuming a close relationship between the two.
                
                        Data gathered recently with the Cosmic Assembly Near-Infrared Deep Extragalactic
                        Survey (CANDELS; <cit.>), and the Spitzer Large Area Surveys (SPLASH; <cit.>), allow us to now infer the halo mass function and its evolution with redshift. Past analyses of the halo mass function and galaxy luminosity function from these surveys have generated significant tension between the observations and predictions in the context of ΛCDM
                <cit.>. These studies derived the halo mass from the UV luminosity function by assuming a relationship between UV luminosity and stellar mass which was then used to infer the halo mass function assuming another relationship between the stellar mass and halo mass. The outcome of this work <cit.> indicated that the halo to stellar mass ratio is constant throughout the redshift range 0<z<4, but it is not yet clear whether this ratio evolves or remains constant at z>4.
                
                In their analysis, Behroozi et al. <cit.>, Behroozi & Silk <cit.> and
                Finkelstein et al. <cit.> concluded that in order to alleviate the tension with ΛCDM,
                this ratio needs to evolve by
                as much as ∼ 0.8 dex. An opposing view <cit.> maintains that such an evolution is not supported by existing data, and that this ratio is instead roughly constant at redshifts 4<z<10, continuing the trend seen at z<4. In this case, the halo distribution would be inconsistent with ΛCDM by at least 2 to 4 orders of magnitude at redshifts 4<z<10, a disparity termed as “The Impossibly Early Galaxy Problem" <cit.>. It is anticipated that future observations with the NIRCam and NIRspec on the James Webb Space Telescope may settle this debate.
                
                        In this paper, we consider what would happen if this problem turns out to be
                        real and provide a possible solution using a recently completed study of the perturbation growth in the R_ h=ct universe <cit.> to describe and report the growth of structure from redshift z∼ 10^11 to 0 in this alternative Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology. We shall summarize the essential features
                of this alternative model in  2, and then derive the growth equations in  2. We shall evaluate the halo mass
                function in  3 and 4, and end with our conclusions in  5. 
                
                
                
                § THE R_  H=CT UNIVERSE
                
                The R_ h=ct universe <cit.> 
                has thus far been tested using a variety of observations. For a summary, see Table 2 in ref. <cit.>. The principal 
                difference between ΛCDM and R_ h=ct is that the latter model is constrained by the equation of state ρ+3p=0, 
                i.e., the so-called zero active mass condition in general relativity. In the standard model, radiation was dominant early, 
                followed by matter and dark energy at later times, whereas dark energy has always been present in R_ h=ct, with a 
                significant component of radiation early on, followed by matter towards lower redshifts. Also, the dark-energy equation 
                of state is w_ de=-1 in ΛCDM, while it is w_ de=-1/2 in R_ h=ct (see 
                ref. <cit.> for further details).
                
                Some additional support for the R_ h=ct cosmology, based on an alternative 
                theoretical concept, may also be found in ref. <cit.> and the updated 
                discussion in ref. <cit.>. But in spite of the success this model has enjoyed 
                thus far in accounting for many observations as well, if not better, than the standard
                model, some counter claims have also been published in recent years, so the issue
                of whether or not it is the correct cosmology still needs to be resolved. This
                is a principal reason for continuing to test it as we do in this paper. Our
                analysis here advances this discussion significantly by providing new insights
                and an important new comparison between R_ h=ct and ΛCDM
                using observations over an unusually large redshift range (see figs. 2-7 below).
                
                As noted above, over the past decade, R_ h=ct has been compared to ΛCDM 
                using data across a broad redshift range, using integrated distances and the
                redshift-dependent Hubble parameter, among various other measures. Still,
                some of these data are often associated with unknown systematics and, worse, 
                are often dependent on the presumed background model. The analysis of Type Ia
                SNe is a well-known example in which the lightcurve is characterized by at
                least 3 `nuisance' parameters that need to be optimized along with those in the
                cosmological model. A different choice of assumptions (e.g., the unknown 
                intrinsic dispersions) and techniques (e.g., χ^2 minimization versus 
                maximization of a likelihood function and/or model selection with information 
                criteria), can sometimes produce varying outcomes in these tests.
                
                For example, Type Ia SNe are challenging to use for model testing when various 
                subsamples are merged together to improve the statistics, since one must deal
                with different unknown systematics in each case. In his assessment, Shafer 
                <cit.> merged the Union2.1 and JLA samples and found that this
                compilation favours the standard model. In his analysis, however, he avoided 
                the unknown intrinsic dispersions by instead constraining the reduced χ^2
                to be 1 in each subsample. In recent years, a superior statistical approach 
                has been developed <cit.> in which these unknowns
                are instead estimated by maximizing the overall likelihood function. The
                outcome of which cosmology is preferred by the SN data changes depending
                on which of these assumptions and methods are chosen.
                
                Another recent test <cit.> used local probes, combining SN data with 
                measurements of the Hubble parameter H(z) and baryon acoustic oscillations 
                (BAO). This analysis also showed that ΛCDM is favoured over R_ h=ct,
                contrasting with other work where the opposite was reported <cit.>.
                The difference results may be traced to the choice of data sets in the two studies. 
                As is the case for SN measurements, the BAO also do not provide model-independent 
                information since the location of the BAO peak cannot easily be distinguished from 
                redshift space distortions (RSD). As of today, only 3 such measurements have provided
                a clean peak location. In other cases, a cosmology must be preassumed in order
                to model the RSD, rendering the data highly model-dependent. Any use of these
                BAO data, and the of H(z) measured from them, produces a biased outcome. 
                In their assessment, Lin et al. <cit.> used all the data and concluded,
                not surprisingly, that they favour ΛCDM because the standard model
                was used to estimate the RSD. When only model-independent data are used 
                instead, however, one reaches the opposite conclusion <cit.>.
                
                As we discuss elsewhere in this paper, the inferred halo mass function is itself 
                subject to an important unknown: the redshift dependence of the halo to stellar 
                mass ratio M_h/M_*, so our conclusions may also require revision once new data
                will have been acquired. Depending on whether or not this ratio changes by roughly
                an order of magnitude between redshifts 4 and 10, a factor yet to be resolved 
                observationally, ΛCDM may or may not be favoured over R_  h=ct. 
                Nonetheless, providing one more important comparison between these two models 
                is essential in establishing the conditions that must be met in order for 
                R_ h=ct to be viewed as a viable alternative to the standard model.
                
                        
                
                § THE EINSTEIN-BOLTZMANN EQUATIONS FOR DARK MATTER AND ENERGY FLUCTUATIONS
                
                        To obtain the complete evolution of density fields starting from initial perturbations, one must solve the
                        coupled Boltzmann/Einstein equations (see ref. <cit.>.
                        The baryons are strongly coupled with radiation until decoupling and therefore do not
                        contribute to the growth of structure during this epoch. Once they decouple from the
                        radiation, baryons follow the evolution of dark matter, which has preceded them in
                        forming bound systems. Hence, the initial growth of structure is dominated by dark
                        matter. For this paper, which is focused on the question of halo growth, we
                        therefore concentrate solely on the growth of dark matter perturbations. Other
                        aspects of structure growth will be presented elsewhere <cit.>. Thus,
                        since we are not interested here in temperature fluctuations of
                        the radiation field or acoustic oscillations, we justify the use of
                        Einstein-Boltzmann equations customized solely for the purpose of describing
                        the growth of dark matter perturbations, which we derive as follows.
                
                The distribution function for any species (i.e., dark matter, baryons, radiation, etc.) may
                depend on the coordinates (x^μ) and momentum (P^μ) 4-vectors, resulting in an
                8-dimensional phase space. An additional constraint emerges, however, from the invariant
                contraction of the momentum, g_μνP^μ P^ν=-m^2, which reduces the phase space to
                7-dimensions. So we choose x^μ, p≡ |p⃗| and the direction of the momentum,
                p̂^i, as our independent variables. Louisville's theorem produces the equation
                         
                    df_s/dλ=∂ f_s/∂ x^0∂ x^0/∂λ+
                             ∂ f_s/∂ x^i∂ x^i/∂λ+
                             ∂ f_s/∂ p∂ p/∂λ+
                             ∂ f_s/∂p̂^i∂p̂^i/∂λ=C[f_s] ,
                
                where f_s(x^μ,p,p̂^i) is the distribution function for any species `s' (i.e., dark matter,
                dark energy, baryons, etc.), λ is the affine parameter, and C[f_s] is a collision/source
                term for this species. We define P^μ=dx^μ/dλ, so that dx^0/dλ=P^0.
                Dividing the above equation by P^0, and neglecting the fourth term that is of second order,
                gives
                         
                    df_s/dη=∂ f_s/∂η+∂ f_s/∂ x^iP^i/P^0+∂ f_s/∂ p∂ p/∂η=
                                 C[f_s]/P^0 ,
                
                where η is now the conformal time, dη≡ dt/a(t), in terms of the expansion factor
                a(t) and cosmic time t in the Friedmann-Lemaître-Robertson-Walker metric.
                
                In the above equation, we may write P^i/P^0=(p/E)p̂^i (where E is the energy) and,
                using the geodesic equation, we get
                         
                    dp/dη=-ℋp+Ep̂^l∂_lh_00/2-p/2dh_ij/dηp̂^ip̂^j ,
                
                where ℋ is the Hubble parameter written in terms of η, and h_αβ
                are the perturbed metric coefficients. Substituting Equation (3) into Equation (2),
                we get
                         
                    df_s/dη   +   pp̂^i/E∂ f_s/∂ x^i+p(-ℋ+
                             E/pp̂^l∂_lh_00/2-1/2h_ij^'p̂^ip̂^j)
                             ∂ f_s/∂ p=
                                  a/E(1-Φ)C[f_s] ,
                
                where we have substituted P_0=E/a(1+Φ), in terms of the perturbed gravitational
                potential Φ. We now separate the distribution function into its unperturbed component,
                f̅_s, and the perturbed contribution, ℱ_s, such that
                         
                    f_s(η, x^i, p, p̂^i)=f̅_s(η, x^i, p, p̂^i)+
                                 ℱ_s(η, x^i, p, p̂^i) .
                
                Then, multiplying Equation (4) by E(p), and integrating over momentum space, collecting the
                zeroth-order terms, gives
                         
                    ∫d^3p/(2π)^3E(p)df̅_s/dη-∫d^3p/(2π)^3ℋpE(p)
                             ∂f̅_s/∂ p=∫d^3p/(2π)^3aC[f_s] .
                
                In this expression, C[f_s] is zero in the context of ΛCDM because the particle number is
                conserved during this phase of the fluctuation growth. But this is not the case in R_ h=ct.
                The early universe in this model contains approximately 80% dark energy and approximately
                20% radiation, with a small contamination of matter <cit.>. At late times,
                the R_ h=ct universe contains approximately 70% dark energy and 30% matter. A coupling
                therefore exists between dark matter and dark energy, such that the particle number for each
                individual species is not conserved in this model. The right-hand side of Equation (6) is
                therefore not zero in R_ h=ct. Integrating the second term on the left-hand side by parts
                and neglecting the boundary term, we arrive at the expression
                         
                    ∫d^3p/(2π)^3E(p)df̅_s/dη   +   3ℋ∫d^3p/(2π)^3(E+p^2/3E)f̅_s=
                                  ∫d^3p/(2π)^3aC[f_s] .
                
                
                This equation may be further reduced by using the following definitions for the (background)
                density and pressure:
                         
                    ρ_s=∫d^3p/(2π)^3E(p)f̅_s ,
                
                and
                         
                    𝒫_s=∫d^3p/(2π)^3p^2/3Ef̅_s .
                
                When applied to dark matter, Equation (7) may thus be written as follows
                         
                    dρ_ dm/dη+3ℋ(ρ_ dm+𝒫_ dm)=
                                ∫d^3p/(2π)^3aC[f_ dm] .
                
                For this particular species (i.e., dark matter), we may also put 𝒫_ dm=0. In
                addition, we use an approximate empirical expression, ρ_ dm=(ρ_c/3a^2)
                exp(-a_*/a(1-a)/(1-a_*)), to model the transition from a
                radiation/dark-energy dominated early universe to a matter/dark-energy dominated universe
                at late times, where ρ_c is the critical density today, and a_* represents the
                scale factor at matter radiation equality. Note that we are also normalizing a(t_0) to
                be 1 today, which is possible in a spatially flat Universe. We infer the required
                collision/source term in Equation (10) by using this empirical expression for ρ_ dm,
                which yields
                         
                    ∫d^3p/(2π)^3aC[f_ dm]=ℋρ_ dm+Hρ_ dm/a(a_*/1-a_*) .
                
                It is not difficult to see that the above equation is satisfied to zeroth order only if
                         
                    C[f̅_ dm]=ℋE/af̅_ dm+ℋE/a^2f̅_ dm(a_*/1-a_*) .
                
                This collision/source term explicitly shows the interactions between dark energy and dark
                matter required to sustain the zero active mass condition described above. That is, in order
                for the partitioning of 80% dark energy plus 20% radiation in the early Universe to
                transition to a balance of 70% dark energy plus 30% matter today, a fraction of the
                dark energy must decay/evolve into dark matter. As such, the collision/source for dark
                energy must be the negative of Equation (12), so that
                
                    dρ_ de/dη   +   3ℋ(ρ_ de+𝒫_ de)=
                          -0.3in -∫d^3p/(2π)^3[ℋEf̅_ dm+ℋE/af̅_ dm(a_*/1-a_*)] ,
                
                where
                
                    C[f̅_ de]=-ℋE/af̅_ dm+ℋE/a^2f̅_ dm(a_*/1-a_*) .
                
                
                Returning now to Equation (4), and using Equation (12), we find for dark matter that
                         
                    df_ dm/dη   +   pp̂^i/E∂ f_ dm/∂ x^i+
                                (-ℋ+E/pp̂^l∂_lh_00/2-
                          -0.7in 1/2h_ij^'p̂^ip̂^j) ∂ f_ dm/∂ p=
                                [ℋf_ dm+ℋ/af_ dm(a_*/1-a_*)](1-Φ)
                
                We again multiply this equation by E(p) and integrate over momentum space, but now
                collecting first order terms, finding that
                         
                    -0.1in d(δρ_ dm)/dη   +   (ρ_ dm+𝒫_ dm)
                                ∂_iv^i_ dm
                                +3ℋ(δρ_ dm+δ𝒫_ dm)+
                          -0.7in  3(ρ_ dm+ 𝒫_ dm)dΦ/dη= 
                                [ℋ+ℋ/a(a_*/1-a_*)](1-Φ)δρ_ dm
                
                where, as always, Φ is the gravitational potential. Then, defining
                δ_ dm≡δρ_ dm/ρ_ dm for the dark-matter perturbation,
                with 𝒫_ dm=δ𝒫_ dm=0, we may write
                         
                    dδ_ dm/dη=1/ρ_ dmd(δρ_ dm)/dη-
                             δ_ dm/ρ_ dmdρ_ dm/dη .
                
                Substituting for d(δρ_ dm)/dη in Equation (16), and isolating the
                Fourier mode k, we find that
                        
                    dδ_ dm,k/dη =-ku_k -3dΦ_k/dη-ℋ[1+a_*/a(1-a_*)]Φ ,
                
                where we have written ∂_iv^i_k=ku_k, in terms of the velocity perturbation u_k
                of the dark matter. Finally, we take the second moment of Equation (15),
                multiplying it by pp̂^i and contracting it with ik̂_i. Then integrating over
                momentum space, and collecting first order terms, we get
                
                    d(ρ_ dmu_ dm,k)/dη   +   4ℋρ_ dmu_ dm,k+kΦρ_ dm=
                          ℋ[1+a_*/a(1-a_*)]ρ_ dmu_ dm,k ,
                
                where u_ dm,k is the k^ th velocity perturbation of dark matter. Substituting for
                dρ_ dm/dη in the above equation we thus get
                        
                    du_ dm,k/dη=-1/ada/dηu_ dm,k-kΦ_k .
                
                
                Turning now to the dark-energy perturbations, we begin with Equation (4) and the interaction term
                in Equation (14), and find that
                
                    df_ de/dη   +   pp̂^i/E∂ f_ de/∂ x^i+p(-ℋ+
                    E/pp̂^l∂_lh_00/2-
                          -0.7in1/2h_ij^'p̂^ip̂^j)
                    ∂ f_ de/∂ p=
                    -ℋ(1-Φ)[1+a_*/a(1-a_*)]f_ dm ,
                
                where f_ de is the distribution function for dark-energy. Thus, partitioning f_ de into
                its unperturbed (f̅_̅ ̅d̅e̅) and perturbed (ℱ_ de) components, as was done
                in Equation (5), we can collect the first-order perturbed terms to find that
                
                    dℱ_ de/dη   +   pp̂^i/E∂ℱ_ de/∂ x^i-
                    ℋp∂ℱ_ de/∂ p+p(E/pp̂^l∂_l
                    h_00/2-
                          -0.5in1/2h_ij^'p̂^ip̂^j)∂f̅_ de/∂ p= -ℋ[1+a_*/a(1-a_*)](ℱ_ dm-
                    Φf̅_ dm).
                
                Multiplying this equation by E(p) and integrating over the momentum space then gives
                
                    -0.2ind δρ_ de/dη   +   ρ_ de(1+w_ de)ku_ de,k+3ℋδρ_ de(1+δ𝒫_ de/δρ_ de)+
                          -0.6in 3 dΦ/dηρ_ de(1+w_ de)=
                    ℋ[1+a_*/a(1-a_*)](ρ_ dmΦ-δρ_ dm).
                
                Defining δ_ de=δρ_ de/ρ_ de, and using
                𝒫_ de=-ρ_ de/2 (Melia & Fatuzzo 2016) we may write
                
                    dδ_ de/dη=1/ρ_ ded δρ_ de/dη-
                    δρ_ de/ρ_ de^2dρ_ de/dη ,
                
                so that with Equations (13) and (23), we find that
                
                    dδ_ de/dη   =   -k/2u_ de,k-3ℋδ_ de(1/2+δ𝒫_ de/δρ_ de)-
                    3/2dΦ/dη+
                          ℋ[1+a_*/a(1-a_*)]
                    ρ_ dm/ρ_ de(δ_ de-δ_ dm+Φ) .
                
                
                The sound speed for our coupled dark matter/dark energy fluid is not known yet, so
                we write it as follows
                
                    c_s^2≡δ𝒫/δρ=δ𝒫_ de/δρ_ dm+δρ_ de=δ𝒫_ de/δρ_ de/(1+δρ_ dm/δ+
                    ρ_ de) ,
                
                analogously to what is commonly done with the coupled baryon-radiation fluid in the
                standard model. And following the conventional approach of assuming adiabatic fluctuations,
                we also write
                
                    δ𝒫_ de/δρ_ de=c_s^2[1+2ρ_ dm/ρ_ de] .
                
                
                For the sake of simplicity, we assume the sound speed to be a constant delimited to the range
                0<(c_s/c)^2<1. We have found that the actual value of this constant has a negligible impact
                on the solutions to the above equations since the ratio of dark matter density to dark energy
                is always much less than 1 in the R_ h=ct universe, and we therefore adopt the simple
                fraction c_s^2=c^2/2 throughout this work. Thus, using Equations (25) and (27), we get
                
                    dδ_ de/dη   =   -k/2u_ de,k-δ_ de(
                    3ℋ/2+3ℋc_s^2+6ℋc_s^2ρ_ dm/ρ_ de)-
                          -0.4in3/2dΦ/dη+ ℋ[1+
                    a_*/a(1-a_*)]ρ_ dm/ρ_ de(δ_ de-δ_ dm+Φ) .
                
                Finally, we take the second moment of Equation (21), multiply it by pp̂^i and contract it
                with ik̂_i. Integrating over momentum space, and collecting first order terms, we thus find that
                
                    du_ de,k/dη   =   -5ℋ/2u_ de,k-kΦ+2kc_s^2[1+2ρ_ dm/ρ_ de]δ_ de+ 
                          -0.2inℋ[1+a_*/a(1-a_*)]ρ_ dm/ρ_ de(u_ de,k-2u_ dm,k) .
                
                
                Our final equation comes from perturbing the FLRW metric in Einstein's Equations
                (see Ma & Bertschinger 1995), which gives
                        
                    k^2Φ_k   +   31/ada/dη(dΦ_k/dη+
                            1/ada/dηΦ_k)=
                                    
                            4π G a^2 [ρ_ mδ_ dm,k+ρ_ deδ_ de] .
                
                In arriving at Equation (30), we have chosen the Newtonian gauge for the primary reason that the independent
                components in this gauge have a direct correspondence to the gauge invariant Bardeen variables
                <cit.>.
                
                It is important to stress that the set of Equations (18, 20, 28, 29) in R_ h=ct
                differ from their counterparts in ΛCDM. This happens because dark energy and dark matter
                are coupled in R_ h=ct, while dark energy is simply a cosmological constant in the most basic
                Λ CDM model. The only expression that is formally common to both R_ h=ct and
                Λ CDM is Equation (20), though the dependence
                of ρ_ dm on a(t) is, of course, model dependent. Mathematically, this comes about
                because the collision/source term in Equation (12) actually cancels out in the perturbation
                Equation (19) for the velocity perturbations. The dependence on cosmology also enters
                into the growth of δ_ dm via the
                model-dependent ℋ and a(t) functions. These quantities change with time according to
                the background evolution, and are therefore strongly dependent on the chosen model.
                
                        
                
                More specifically, an inspection of these equations reveals that there are three principal differences
                between R_ h=ct and ΛCDM: i) the scale factor a(t) in R_ h=ct is given as
                a(t)=(t/t_0) at all epochs, whereas in ΛCDM it is proportional to t^1/2 and t^2/3
                during the radiation and matter dominated phases, respectively; ii) the matter density
                scales as ρ_ dm=(ρ_c/3a^2)exp(-a_*/a(1-a)/(1-a_*)) in
                R_ h=ct, whereas it is given as ρ_ m=(Ω_mρ_c/a^3) in ΛCDM; and
                iii) the various modes of the density field in ΛCDM exited the horizon during inflation,
                whereas none of the modes ever crossed the horizon in R_ h=ct<cit.>. In ΛCDM,
                small-scale modes re-entered the horizon while radiation was dominant, while larger-scale modes
                entered the horizon when matter dominated, which produces a late start for the growth of
                structure compared with what happens in R_ h=ct. This appears to be the principal
                reason why galaxies and supermassive black holes appeared earlier in R_ h=ct
                than in the standard model.
                
                        We shall formally introduce the growth function in Equation (31) below, and plot it
                        in figure 1. It is obtained by solving Equations (10–30) simultaneously (see
                        ref. <cit.> for more details). It is quite evident from this plot that the
                        growth factor in R_ h=ct is significantly stronger at large redshifts than that
                        in ΛCDM, in full agreement with the previous results of our analysis at lower
                        redshifts <cit.>. In contrast, the growth function
                        in ΛCDM indicates a strong evolution from z∼ 10 to z∼ 4. And since
                        galaxies typically form on a dynamical timescale ∼ 300 Myr <cit.> after
                        halo virialization, the rapid evolution in the number density of halos from
                        z∼ 8 to ∼ 4 predicted by ΛCDM corresponds to a rapid evolution
                        in the UV luminosity of galaxies at 6.0≳ z≳ 3.4. This is one
                        of the points of contention between the two camps, since this (required) rapid
                        evolution in the UV luminosity function conflicts with the observations <cit.>.
                        The observed UV luminosity evolves much more slowly than this prediction,
                        which would mean that these massive galaxies would have formed much earlier than
                        expected in ΛCDM.
                
                        
                
                § HALO MASS FUNCTION
                
                        The halo mass function was first derived by Press and Schechter <cit.>  assuming
                        spherical collapse and a primordial Gaussian density field. When tested against numerical
                        simulations, however, it became evident that the Press-Schechter formalism
                        over-predicts the number of halos at the high mass end, and under-predicts at
                        the low mass end. This inconsistency was resolved by introducing ellipsoidal
                        collapse, rather than spherical, by Sheth-Tormen <cit.>. But the Bolshoi
                        simulations performed by Klypin, Trujillo-Gomez & Primack <cit.> several years later
                        indicated that, while discrepancies in the Sheth-Tormen mass function at z∼ 0
                        are less than 10% for halo masses in the range 5×10^9 - 5×10^14 M_⊙,
                        this prescription over-predicts the density by about 50% at z∼ 6 for masses
                        in the range 10^11-10^12 M_⊙, getting even worse (by an order of
                        magnitude) by z∼ 10. Unfortunately for the standard model, the inclusion
                        of corrections from the Bolshoi simulations actually exacerbates the discrepancy
                        between theory and observation. For this reason, and the fact that analogous
                        simulations to the Bolshoi calculations have not yet been carried out for
                        R_ h=ct, we won't include such adjustments in this paper. We
                        point out that if we were to add such corrections to R_ h=ct, the
                        comparison of this model's predictions with the data under the
                        assumption of a constant halo mass to stellar mass ratio would be even more
                        favourable than the Sheth-Tormen formulation on its own, as one may readily see in
                        figures 2-7. As such, our exclusion of these corrections produces an
                        effect more favourable to ΛCDM than R_ h=ct, even with this assumption,
                        which we do in order give the standard model as much benefit of the doubt as possible.
                
                        The Sheth-Tormen mass function is given as
                        
                    f(σ)=A√(2a/π)[1+(σ ^2/aδ_c^2)^p]
                            δ_c/σexp[-aδ_c^2/2σ^2] ,
                
                        where A=0.3222 is a normalization factor, and a=0.707 and p=0.3. Using this halo mass
                        function, one may obtain the number of dark matter halos per comoving volume with masses
                        less than M as follows:
                        
                    dn/dln M=ρ_0/M f(σ)|dlnσ/dln M| ,
                
                        where σ is defined according to the expression
                        
                    σ_R^2(R,z)=b^2(z)/2π^2∫_0^∞k^2P(k)W^2(k,R)dk ,
                
                        and P(k) is the power spectrum, W(k,R) is the top-hat filter and b(z) is the
                        growth factor shown in figure 1 for both ΛCDM and R_ h=ct.
                
                        
                
                § OBSERVED HALO MASS FUNCTION
                
                        The data used in this paper were assembled by Steinhardt et al. <cit.>,
                        based on measurements obtained using three different techniques, including
                        the clustering method <cit.> based on the
                        spatial distribution of galaxies to obtain the halo masses. This method doesn't
                        assume any physical properties of the galaxies themselves, but assumes a model
                        for the dark matter concentration. Other techniques include template fitting
                        <cit.>, that adopts a relationship between the luminosity and
                        stellar masses; the abundance matching technique <cit.>
                        that relates critical features in the galaxy luminosity or mass function,
                        such as a `knee', to crucial elements in the halo mass distribution, that
                        can then be used to match the galaxy and dark matter densities to infer the
                        halo mass function. The high redshift (z≥6) data points are derived from
                        the UV luminosity function, that yields halo masses by assuming that the
                        halo mass to light ratio obtained at lower redshifts persists to higher
                        redshifts. Most of the data used in this work were obtained
                        assuming a constant ratio of halo to stellar-mass. The two main principles
                        for arriving at this ratio are (i) that 10% of the baryonic matter
                        eventually condensed into stars <cit.> and (ii) the
                        observation of a 6:1 ratio of dark matter to baryonic matter <cit.>.
                
                        
                
                        It is quite obvious from the progression seen in figures 2–7 that the
                        observed halo mass function obtained via these different techniques
                        <cit.> is entirely
                        inconsistent with the distribution predicted by ΛCDM, if the halo to stellar-mass
                        ratio remains constant throughout the 4<z<10 redshift range <cit.>. 
                        Of course, the caveat is that these data were not measured directly, and
                        were obtained using relationships derived at low redshifts. Steinhardt et al. <cit.>
                        studied the possibility that these correlations could be breaking down
                        at high-z. Their investigation indicated, however, that the star-formation rate
                        vs. stellar mass of these high redshift galaxies lies on the extrapolation from
                        lower redshift galaxies. In addition, the ratio of stellar mass to halo
                        mass in these high redshift galaxies is similar to the standard value 30:1
                        seen at all redshifts. These two tests therefore indicate that the high redshift
                        galaxies are quite normal, implying that the problem is real.
                
                        In addition to this, Steinhardt et al. <cit.> determined that an evolution of
                        0.8 dex in M_Halo/L_UV is needed to mitigate this problem. Such a
                        change might occur if the stellar population in galaxies at z=8 is younger
                        than that at z=4. Steinhardt et al. <cit.> extensively investigated whether
                        this possibility could mitigate the disparity by modeling the halo mass to light
                        ratio from an initial stellar population assuming they formed in one rapid burst
                        at z=12 and then evolved along the main sequence until z=4-8, where they
                        were observed. This resulted in a star formation rate ∝ M_*^0.7, with
                        a stellar age asymptotically approaching 50-150 Myr, starting from an initially
                        small value. But this isn't sufficient enough to remove the problem and, worse,
                        the above approach isn't realistic considering a dynamical timescale of
                        300 Myr for star formation after virialization of the halo.
                
                        Steinhardt et al. <cit.> considered this scenario and modeled the halo mass to
                        light ratio as described above, concluding that this too is insufficient to
                        reconcile the problem. Another possibility is that the halo mass to
                        stellar mass ratio evolves towards higher redshifts. An evolution of 0.8 dex
                        in this ratio would reconcile the problem. But such a modification is only
                        possible either by a complete absence of dark matter at redshift 8, or if
                        100% of the baryons condensed instantly into stars at high redshift upon
                        halo virialization, which is quite impossible. Hence, one may reasonably
                        conclude that this problem may be reconciled in ΛCDM only via the
                        introduction of implausible physics. When viewed in the context of other
                        “too early” types of problems, the disparity evident in figures 2-7 is
                        quite damning for the standard model. For example, the early appearance of
                        supermassive black holes at z∼ 6-7<cit.>
                        and galaxies at z∼ 10-12 (see references cited in <cit.>, argues in 
                        favor of these problems being real, presenting a challenge to any attempt to 
                        alleviate them in the context of ΛCDM.
                
                In contrast, the comparison between the Steinhardt et al. <cit.> data, under
                the assumption that the halo to stellar mass ratio is constant in the redshift range 4<z<10, and the
                predictions of R_ h=ct, is very favourable—except at the very high mass end of the halo mass
                distribution, as one may see in Figures 2-7. The standard model disagrees progressively more and more
                with this approach as the redshift increases, while R_ h=ct fits the data throughout the range
                10≳ z≳ 4 very well at the low and intermediate mass end, and overpredicts by one to two
                orders of magnitude at the high mass end. This over-prediction may be due to two possible reasons: (1) As
                noted earlier, the Bolshoi simulation <cit.> has indicated that the Sheth-Tormen mass function
                overpredicts the number of halos by at least 10% at redshift z∼ 0, and overpredicts by at least
                50% at redshift z ∼ 10. Although simulations similar to Bolshoi haven't yet been carried out
                for R_ h=ct, a trend analogous to this in the context of this model, would produce corrections
                that largely mitigate the problem at the high mass end; (2) This over-prediction may also be due in
                part to observational selection effects that may be `hiding' some of the sources. Some massive galaxies
                may have been missed due to extinction, which future observations might be able to address. Regardless
                of which, if any, of these mitigating factors are at play in R_ h=ct, none of them can resolve the
                disparity arising from the predictions of ΛCDM. The discrepancy seen in the standard model is
                extreme, ranging from one to over four orders of magnitude from low to high mass, throughout the redshift
                range 4≲ z≲ 10. The factors that may alleviate the high-mass end problem with R_ h=ct,
                actually makes the comparison much worse for ΛCDM, increasing the disparity between predictions
                and observations. The weaker evolution in growth rate predicted by R_ h=ct is the vital reason for
                its success, indicating that massive galaxies must have formed earlier than predicted in the standard model,
                consistent with the observations.
                
                        The problem in ΛCDM may instead be reconciled with an evolution in the halo
                        mass to light ratio, which could happen, e.g., if
                        the initial mass function were top-heavy. Studies have shown, however, that
                        this function should be the same at all redshifts z≲ 8<cit.>. Hopefully, this conclusion can be tested using
                        supernova rates in the future, which may eliminate even this last
                        possible caveat for the significant tension between the observed
                        halo mass function and ΛCDM. On the flip side, if it turns
                        out that future observations with JWST support an evolution in the halo to stellar
                        mass ratio of at least ∼ 0.8 dex between z∼ 4 and 10, validating the
                        predictions of ΛCDM, the inferred halo distribution will be in tension
                        with the predictions of R_ h=ct. The differences are so significant (at least
                        several orders of magnitude) that a refinement of the halo distribution may produce
                        one of the most robust comparative cosmological tests of these models.
                
                        
                
                § CONCLUSION
                
                        In this paper, we have discussed an ongoing debate concerning the
                        early appearance of massive galaxies (and their halos), which may challenge the formation
                        of structure predicted by ΛCDM if the halo to stellar mass ratio is roughly constant
                        in the redshift range 4<z<10. This difficulty could be mitigated with a refinement of the
                        underlying theory of star formation and galaxy evolution, but appears to require implausible
                        modifications to the physics underlying these phenomena (Steinhardt et al. 2016). Some
                        support for the existence of a real problem is provided by other types of “too early”
                        problems, such as the premature appearance of supermassive black holes at z∼ 6-7<cit.>.
                
                        Combining our earlier measurement of σ_8 at redshift 0 <cit.>
                        with our recently completed calculation of the growth function using the coupled Boltzmann
                        and perturbed Einstein equations, we have re-analyzed
                        “The Impossibly Early Galaxy Problem" in the context of R_ h=ct
                        and showed that this problem virtually disappears in this cosmology
                        even if the halo to stellar mass ratio is constant. Although, the R_ h=ct
                        universe overpredicts the number density of halos by one to two orders of magnitude at the very
                        high mass end, this problem may be mitigated by corrections to the Sheth-Tormen mass function,
                        as indicated by the Bolshoi simulations <cit.>. Thus, once we resolve the
                        question of whether or not this ratio evolved with redshift, the inferred halo
                        mass distribution can clearly distinguish between the R_ h=ct and
                        ΛCDM cosmologies.
                
                        The timeline in R_ h=ct allows both massive galaxies and
                        supermassive black holes to form at very high redshifts without invoking
                        exotic physics. It should also be noted that, while ΛCDM
                        must rely on the unproven and as yet unverified physics of inflation
                        to account for the generation of scale-invariant primordial fluctuations
                        and a mechanism for driving the modes to exit and re-enter the horizon, thus
                        creating an intricate mechanism for producing different growth rates at
                        different epochs, no such complicated, fine-tuned mechanism is necessary
                        in R_ h=ct. This model does not have a horizon problem and does
                        not incorporate inflation into its expansion history. As explained in
                        more detail in ref. <cit.>, the growth of structure
                        in R_ h=ct is simple, streamlined and does not require a different
                        handling of small modes compared to the larger ones. Such simplicity,
                        particularly when viewed in the context of the excellent agreement between
                        theory and observations (figs. 2-7), adds considerable support for the
                        viability of this cosmology.
                
                        Looking forward to upcoming surveys and further theoretical developments,
                        it is already clear that observations, e.g., with JWST, will play a crucial
                        role in determining the quasar distribution and the rate of gamma ray
                        bursts from Pop III stars, both heavily dependent on the growth rates
                        we have been discussing in this paper. There is therefore significant
                        promise of improving the comparison we have made here even further, perhaps
                        strongly ruling out one or other of these two models.
                
                
                        We are grateful to Charles Steinhardt and Peter Behroozi
                        for very informative discussions. FM is grateful to the Instituto de Astrofísica de
                        Canarias in Tenerife and to Purple Mountain Observatory in Nanjing, China for their
                        hospitality while part of this research was carried out.
                99Sheth2001 R. K. Sheth, H. J. Mo & G. Tormen, MNRAS  323 (2001) 1
                                Springel2005 V. Springel et al., Nature  435 (2005) 629
                                Vogelsberger2014 M. Vogelsberger et al., MNRAS  444 (2014) 1518
                                Holder2001 G. Holder, Z. Haiman & J. Mohr, ApJ  560 (2001) L111
                                Furlanetto2006 S. R. Furlanetto, M. McQuinn, L. Hernquist, MNRAS  365 (2006) 115
                                HaimanLoeb2001 Z. Haiman & A. Loeb, ApJ  552 (2001) 459
                                Grogin2011 N. A. Grogin et al., APJ  197 (2011) 35G
                                Capak2013 P. Capak et al., A&A  558 (2013) 67A
                                Steinhardt2016 C. L. Steinhardt, P. Capak, D. Masters & J. S. Speagle, ApJ  824 (2016) 1
                                Behroozi2013a P. S. Behroozi, R. H. Wechsler & C. Conroy, APJL  762 (2013) L31
                                Behroozi2013b P. S. Behroozi, R. H. Wechsler, C. Conroy, APJ  770 (2013) 57
                                BehrooziSilk2015 P. S. Behroozi & J. Silk, ApJ  799 (2015) 32
                                Finkelstein2015 S. L. Finkelstein et al., ApJ  814 (2015) 95
                                RodriguezPuebla2017 A. Rodríguez-Puebla, J. R. Primack, V. Avila-Reese &
                                          S. M. Faber, MNRAS  470 (2017) 651
                                Stefanon2017 M. Stefanon et al., ApJ  843 (2017) 36
                                YennapureddyMelia2019 M. K. Yennapureddy & F. Melia, PRD (to be submitted) (2019).
                                Melia2007 F. Melia, MNRAS  382 (2007) 1917
                                Melia2013a F. Melia, A&A  553 (2013) A76
                                Melia2016a F. Melia, Frontiers of Physics  11 (2016) 119801 
                                Melia2016b F. Melia, Frontiers of Physics  12 (2017) 121980
                                MeliaAbdelqader2009 F. Melia & M. Abdelqader, IJMP-D  18 (2009) 1889
                                MeliaShevchuk2012 F. Melia & A.S.H. Shevchuk, MNRAS  419 (2012) 2579
                                Melia2018 F. Melia, MNRAS  481 (2018) 4855
                                MeliaFatuzzo2016 F. Melia & M. Fatuzzo, MNRAS  456 (2016) 3422
                                John:2000 M. V. John & K. B. Joseph, PRD  61 (2000) id.087304
                                John:2019 M. V. John, MNRAS  484 (2019) L35
                                Shafer2015 D. L. Shafer, PRD  91 (2015) 103516
                                Kim2011 A. G. Kim, Publ. Astron. Soc. Pac.  123 (2011) 230
                                Wei2015 J.-J. Wei, X. Wu, F. Melia & R. S. Maier, AJ  149 (2015) 102
                                Melia2018e F. Melia et al., EPL  123 (2018) 59002
                                Lin2018 H.-N. Lin, X. Li & Y. Sang, Chinese Physics C  42 (2018) 095101
                                MeliaLopez2017 F. Melia & M. López-Corredoira, IJMP-D  26 (2017) id. 1750055-265
                                MaBertschinger1995 C. P. Ma & E. Bertschinger, ApJ  455 (1995) 7
                                Bardeen1980 J. M. Bardeen, PRD  22 (1980) 1882B
                                YennapureddyMelia2018 M. K. Yennapureddy & F. Melia, PDU  20 (2018) 65Y
                                Wong2009 T. Wong, ApJ  705 (2009) 650
                                PressSchechter1974 W. Press & P. Schechter, ApJ  187 (1974) 425
                                Klypin2011 A. A. Klypin, S. Trujillo-Gomez & J. Primack, ApJ  740 (2011) 102
                                Hildebrandt2009 H. Hildebrandt, J. Pielorz, T. Erben, L. Van Waerbeke,
                                          P. Simon & P. Capak, A&A  498 (2009) 725
                                Ilbert2013 O. Ilbert et al., A&A  556 (2013) A55
                                Leauthaud2012 A. Leauthaud et al., ApJ  744 (2012) 159
                                Planck2016 Planck Collaboration et al., A&A  594 (2016) A13 
                                Lee2012 K. S. Lee et al., ApJ  752 (2012) 66
                                Caputi2015 K. I. Caputi et al., ApJ  810 (2015) 73
                                Melia2013b F. Melia, ApJ  764 (2013) 72
                                MeliaMcClintock2015 F. Melia & T. M. McClintock, Proc. R. Soc. A  471 (2015) 20150449 
                                Melia2014 F. Melia, JCAP  01 (2014) 027
                                Dias2010 B. Dias, P. Coelho, B. Barbuy, L. Kerber, T. Idiart, A&A  520 (2010)  A85
                                Melia2017 F. Melia, MNRAS  464 (2017) 1966
