It is useful to begin the discussion about the significance of the equivalence principle of general relativity (GR) for understanding kinematically determined universes by describing the historical context. Einstein (1905) presented his special relativity theory (SRT), which connects space and time and applies to inertial systems. Some years later, based on the equivalence of inertial mass and gravitational mass, Einstein (1915) presented the theory of GR with its geometric interpretation of gravity, where gravity curves space. This indicates that in the absence of a gravitating mass (or more precisely, gravitating energy density), space is flat, and Euclidean geometry applies. Gravitating masses curve space, and the curvature of space depends on the spatial distribution of the masses. The mathematical framework of the theory is based on Riemannian spaces, which led to Einstein’s field equations for gravity, introduced in Section 2, Equation 1. Solving these equations for a specific distribution of energy or masses, respectively, yields the corresponding metric g μ ν , describing the curvature of space. For example, the well-known Schwarzschild metric describes the curvature of space due to a single-point mass (Schwarzschild, 1916).

In 1917, Einstein applied his field equations to the universe, assuming a homogeneous and isotropic distribution of matter, according to the cosmological principle (Einstein, 1917). To provide a static solution to the field equations, he added the cosmological constant Λ to the left-hand side of the equations, which can be regarded as a modification of the law of gravity. In contrast, in the current Λ cold dark matter ( Λ CDM) concordance model, Λ is considered an additive type of energy, contributing to the total energy content of the universe, i.e., to the energy–momentum tensor on the right-hand side, as any other cosmic component of the model.

In the same year, de Sitter (1916), de Sitter (1917) found the expanding solution H = Λ / 3 in empty space. This surprising result displayed, apart from the expansion, some strange properties which were later explained by Lemaître, because of de Sitter’s choice to apply the Schwarzschild metric. In 1924, Friedmann derived his solutions to Einstein’s field equations for universes with constant curvature, without restriction to specific physical or astronomical assumptions, apart from the cosmological principle, and found two differential equations, known as the first and second Friedmann equations (Friedmann, 1924), which describe the expansion history of model universes. Friedmann deduced that the “size” of the universe is not constant, but either expanding or contracting, depending on the amount of matter. Without the knowledge of Friedmann’s works, in 1927, Lemaître developed the most comprehensive and systematic set of cosmological solutions to the general relativistic field equations, also assuming the cosmological principle. It was the first time that the energy content of the universe was divided into matter and radiation components. Furthermore, his work constituted a systematic compilation of possible world models with different values for Λ and curvature. He concluded that the universe originated from a structure he called the “primeval atom,” in a unique event, which nowadays is called the big bang. Yet, already in 1927, Lemaître had predicted an expanding universe, 2 years before Hubble observed the expansion of the universe (Hubble, 1929). Additionally, Lemaître (1927) postulated that there is no center of gravity in the universe, a fact rarely mentioned in the literature, even though it is only this assumption which leads to a homogeneous and isotropic gravitational field of a universe of finite size, under the premise of the cosmological principle.

(Milne, 1932) presented the idea of a kinematically determined universe, which was based on SRT and where the recession velocities of galaxies, meanwhile discovered by Hubble (1929), were assumed to be a physical velocity. Later, the Milne model has been ruled out for several reasons and hence is not being considered a viable model (e.g., Davis and Lineweaver, 2004; Chodorowski, 2005) as it does not agree with observations. Nevertheless, the Milne model inspired, independently of each other, Robertson and Walker to transfer the idea of a kinematically determined universe into GR.

The key concept of a kinematically determined universe is that starting with an initial (or in the words of Lemaître, the primeval) expansion rate, gravity is working against the momentum of expansion and decelerates the expansion rate. In fact, Lemaître’s original postulation of the absence of a center of gravity in the universe lends the expansion rate H as the appropriate quantity to describe the expansion of the universe. In contrast to “the radius,” which is only defined by referencing to “the center of the universe,” the expansion rate is well defined for every point in the universe without any choice of a particular coordinate system. Finally, the concept of a kinematically determined universe perfectly explains the surprising de Sitter (1916), de Sitter (1917) solution for empty space as the absence of gravity in empty space keeps the expansion rate constant, just in the same way as the negative pressure exposed by Λ balances the attractive force of gravity. The term “accelerated expansion” for the observed late stages of the evolution of the universe is thus a “misnomer” since the rationale is the constant expansion rate—either due to the absence of gravity or due to balancing of gravity. As we elaborate below, both scenarios are described using an identical mathematical formalism, which can be readily observed by inspecting the second Friedmann equation; see Section 2, Equation 6.

The works of Robertson (1935), Robertson (1936a), Robertson (1936b), and Walker (1937) were based on preceding works by Friedmann (1922), Friedmann (1924), and Lemaître (1927) and applied the Riemannian formalism of curved surfaces to describe the dynamics of expansion of the universe in the reference frame of a free-falling observer, moving on a geodesics, by a metric that can be applied to Einstein’s field equations. The metric is therefore called the Friedmann–Lemaître–Robertson–Walker (FLRW) metric; see Section 2, Equation 2. It includes the curvature index k , whose value determines the geometry of a model universe: + 1 (closed universe), 0 (flat universe), and − 1 (open universe). The FLRW metric is the foundation of modern cosmology and the Λ CDM concordance model, contributing to its tremendous success. It is important to note that the definition of the curvature index refers to the critical density, although the density is not part of the metric: + 1 (density > critical density), 0 (density = critical density), and − 1 (density < critical density).

Applying the FLRW metric to the Einstein equations yields the Friedmann equation; see Section 2, Equation 4 and Section 3. In this equation, the curvature index reappears in a term called the curvature term, describing the geometry of a model universe. Customarily, this is also interpreted as the curvature of space in the model universe. In the Friedmann equation, the density also appears, but there is no “recipe” that guarantees the physically correct correspondence between the choice of k and the density in the Friedmann equation. The Friedmann equation normalized to critical density (Equation 14) ¹ does not avoid the definition of physically implausible model universes as we exemplify by the following examples.

The first example is the Einstein–de Sitter universe, which includes matter at critical density as the only component. According to Equation 14, no curvature term appears, and the universe is assumed to be flat. Interpreting the geometry as the curvature of space indicates that there is no gravitating mass in the universe. This is contradicting the definition of the mass density in the model universe. The second example is an empty model universe, which according to Equation 14 includes a curvature term ² at critical density. Again, interpreting the geometry as the curvature of space indicates that there is curved space in the model universe, although the universe is empty—again a contradiction to its original definition.

Let us turn to a more realistic model and discuss the Λ CDM concordance model, the most important representative of FLRW universes. The components of the model are radiation, matter—baryonic and CDM—and the cosmological constant Λ . Curvature does not appear as a flat geometry of space is suggested by observations, mainly of the cosmic microwave background (CMB). Figure 1 displays the well-known evolution of the expansion rate of the Λ CDM concordance model computed using the Planck 2020 values (Planck-Collaboration, 2020).

We observe a deceleration in the expansion rate H in the course of the expansion of the universe, caused by gravity of the (attractive) components (see e.g., Peacock, 1999). It is important to note that no curvature term appears in Λ CDM and space is considered flat. Hence, there should be no deceleration. On the other hand, there are gravitating energy densities in the universe, which should cause a curvature of space. This suggests that there is some “inconsistency” in Λ CDM’s flat universe interpretation. We present an approach to resolve this inconsistency and discuss the consequences in the following sections. In a follow-up paper, we apply the concepts presented here onto nonlinear structure formation and investigate the impact of the formation of the cosmic web.

This paper is organized as follows. In Section 2, we recapitulate the basic equations for the evolution of the background universe in FLRW models. Section 3 investigates the flat universe interpretation for the general case of FLRW universes, followed by a discussion of the initial conditions (ICs) of the background universe. Section 4 proposes a Λ CDM extension such that the cosmological model incorporates the post-big bang initial conditions of the early universe. Finally, in Section 5, we summarize the presented concepts, results, and implications, also in light of cosmological observations.

First, we recapitulate the well-known equations describing the evolution of the homogeneous and isotropic background universe that we need in our model. As gravity is the only force acting on cosmological length scales, it determines the evolution of the background universe and is described using Einsteins’ field equations: R μ ν − 1 2 g μ ν R = 8 π G c 4 T μ ν , (1) with the Ricci tensor R μ ν and the Ricci scalar R . The left-hand side (lhs) of the equation is often summarized as E μ ν , the Einstein tensor. The right-hand side (rhs) contains the energy–momentum tensor T μ ν , which includes the cosmic inventory of given cosmological models. The energy–momentum tensor T μ ν determines the curvature of space, expressed by the metric g μ ν , which is also included in the Ricci tensor R μ ν and the Ricci scalar R . Cosmological models differ in their assumptions on the nature and amount of the cosmic components encoded in T μ ν . As such, all models are subject to observational constraints.

The geometry of a universe with constant curvature is described by applying the Riemannian formalism of curved surfaces and was developed by Robertson (1935), Robertson (1936a), Robertson (1936b), and Walker (1937) based on Milne’s idea of a kinematically determined universe (Milne, 1932) and preceding works by Friedmann (1922), Friedmann (1924), and Lemaître (1927). In spherical coordinates ( r , θ , ϕ ) , the line element of the metric reads as d s 2 = c 2 d t 2 − R 2 ( t ) d r 2 1 − k r 2 + r 2 d Ω 2 , (2a) d Ω 2 = d θ 2 + sin 2 ⁡ θ d ϕ 2 , (2b)

with the curvature index k and its values + 1 (closed universe), 0 (flat universe), and − 1 (open universe). The spatial coordinates ( r , θ , ϕ ) are defined in the co-moving frame, where r , θ , and ϕ remain “fixed,” corresponding to the assumption of spaces of constant global curvature (the geometry of the universe). R ( t ) is the “radius of curvature,” as described by Kolb and Turner (1990), at cosmic time t (for k = ± 1 ), with the dimension of length. Robertson calls this space an “auxiliary space,” and Walker calls it a “Riemannian space,” in contrast to Friedmann, who identifies it as the physical space of the universe.

Applying the FLRW metric Equation 2 to the metric tensor g μ ν in Einstein Equation 1, the energy–momentum tensor T μ ν then takes a perfect-fluid form, which reads T μ ν = ρ + p c 2 u μ u ν − g μ ν ⁡ p , (3) where u μ and u ν is the four-velocity. The time–time component of the solution to Einstein Equation 1 yields the (first) Friedmann equation in the classical version, as derived by Friedmann (1922), see Section 3 and, for example, Kolb and Turner (1990): H 2 t = 8 π G 3 c 2 ρ − k c 2 a 2 t , (4) which describes the dynamics of the evolution of the background universe in the reference frame of a free-falling observer, co-moving with the expansion ³ , moving on a geodesic (in a possibly curved space). Here, ρ refers to the entire energy density of the universe; k is the curvature index, defined in Equation 2 determining the geometry, with k = + 1 for a closed (supercritical, density greater than the critical density) universe, k = − 1 for an open (subcritical, density less than the critical density) universe, and k = 0 for a flat geometry with critical density. In the following sections, we use the notion “geometry” for the curvature of the universe as determined by the FLRW metric (Equation 2) and the Friedmann Equation 4, respectively. a denotes the scale factor, with the dimension of length, which owing to the symmetries imposed by the isotropy and homogeneity of the background universe is only a function of cosmic time t and is defined as the “size” of an expanding or contracting universe, relative to its present-day size | a 0 | = 1 . H is the Hubble parameter (we use the term expansion rate interchangeably) defined as H t ≡ a ̇ a , (5) where the dot refers to the derivative with respect to cosmic time t . The space–space component yields the second Friedmann equation as a ̈ a = − 4 π G 3 c 2 ρ + 3 p , (6) which Friedmann called the deceleration equation. In the recent literature, it is referred to as the acceleration equation.

Now, let us introduce the cosmic inventory that features the current concordance Λ CDM model. In addition, we introduce some standard notions and equations that we need in the paper. The energy densities of interest include “CDM,” baryons (“b”), and radiation (“r”) (including photons and neutrinos). In the formulae, we also include the cosmological constant Λ , empirically added to the cosmic inventory to explain the flatness of space, confirmed by the observations of the CMB, using the balloon-based BOOMERanG experiment (de Bernardis et al., 2000; MacTavish et al., 2006) as well as observations with increasing accuracy by the space missions COBE (Smoot et al., 1992), WMAP (Hinshaw et al., 2013), and Planck (Planck-Collaboration, 2020).

To study a variety of cosmological models, it has become customary to put “curvature” and the cosmological constant “ Λ ” into the energy–momentum tensor T μ ν by operationally defining “effective” energy densities for them, namely, ρ k = − 3 k c 2 / ( 8 π G a 2 ) for curvature (“k”) and ρ Λ = Λ c 2 / ( 8 π G ) for the cosmological constant.

We stress that although ρ k represents a geometric quantity, it has morphed into a “substance” or cosmic inventory, described by T μ ν , upon this standard operational procedure ⁴ . Nevertheless, it is rightfully not regarded as a physical constituent of the universe (see Section 3.1) but simply a mathematical formalism, contributing an effective or artificial contribution to T μ ν . On the other hand, in Λ CDM, Λ is usually regarded as a real physical cosmic inventory, which contributes to T μ ν , basically in the same manner as matter and radiation.

The Friedmann equation in modern language reads as H 2 t = 8 π G 3 c 2 ρ r t + ρ b t + ρ CDM t + ρ k t + ρ Λ t , (7) with the time-dependent background energy densities for radiation ( ρ r ) , baryons ( ρ b ) , CDM ( ρ CDM ) , the curvature ( ρ k ) , and the cosmological constant ( ρ Λ ) . H ( t ) is the Hubble parameter, and its present-day value ⁵ , the Hubble constant, is denoted as H 0 . The critical density, defining a flat universe (i.e., a universe with flat geometry), is given by ρ crit , t = 3 H 2 t c 2 8 π G , (8) which is derived from Equation 4 with a vanishing curvature term. It is convenient to introduce the so-called density parameters or cosmological parameters as Ω i , t = ρ i t ρ crit , t , (9) where i = CDM, b, r, etc., which are nothing but the background energy densities relative to the critical density (Equation 8).

To customarily solve the Friedmann equation, the energy conservation equation is applied (for each component, i = CDM, b, r, …), which reads ∂ ρ i ∂ t + 3 H ρ i + p i = 0 , (10) where ρ i and p i stand for the background energy densities and pressures ⁶ , respectively. The energy densities and pressures are each related by their respective equation of state (EoS): p i t = w i t ρ i t , (11) where w i is often called the EoS parameter, which can also change with time, in general. However, in Λ CDM, w i is assumed to be a constant ⁷ for every component i = CDM, b, r, Λ , and k. Assuming a constant EoS parameter w i and substituting Equation 11 in Equation 10, it follows that ρ i ̇ / ρ i = − 3 ( 1 + w i ) a ̇ / a , which is readily integrated to yield the well-known relationship: ρ i a = Ω i , 0 ρ crit , 0 a − 3 1 + w i , (12) which describes the evolution of the background energy densities as a function of the scale factor a for the constant w i . The background evolution of the standard cosmic components is thus given as ρ r ( a ) = Ω r , 0 ρ crit , 0 / a 4 , (13a) ρ m ( a ) = Ω m , 0 ρ crit , 0 / a 3 , (13b) ρ k ( a ) = Ω k , 0 ρ crit , 0 / a 2 , (13c) ρ Λ = Ω Λ , 0 ρ crit , 0 , (13d)

where Equation 13a refers to the radiation component (its EoS parameter in Equation 11 is w r = 1 / 3 ), Equation 13b refers to baryonic matter and CDM ( w m = 0 ) , Equation 13c refers to the curvature ( w k = − 1 / 3 ) , and Equation 13d refers to the cosmological constant Λ ( w Λ = − 1 ) .

The Friedmann equation for the Λ CDM model (Equation 7) can be alternatively written as an algebraic closure condition. In the present study, it reads 1 = Ω r , 0 + Ω b , 0 + Ω CDM , 0 + Ω k , 0 + Ω Λ , 0 . (14) In other words, Equation 14 is the normalization of Friedmann Equation 7 to the critical density. In the Λ CDM model, Ω k , 0 = 0 is prescribed such that Λ closes the universe to critical density, defining it a flat universe, which we elaborate in the next section.

The Λ CDM model is a member of the broader family of FLRW cosmological models. Furthermore, a flat space in the universe is assumed on the grounds of the curvature of space, as measured, for example, by the observations of the CMB—the flat universe interpretation, where the curvature term in Friedmann Equation 4 is customarily interpreted to express the geometry of the universe. We will now reassess this interpretation.

In Section 3.1, we argue that a prospective observation of flat space does not necessarily imply a universe at critical density since the curvature terms ρ k and Ω k in Equations 7, 14, respectively, vanish in the local inertial frame of co-moving FLRW observers, giving the reasons for the flatness of space as measured by the observations of the CMB. To interpret the curvature term in Equation 4, we associate the general concept of dark energy (DE) with the geometry of the FLRW metric (Equations 2a,b ) and use the subscript “de” for the quantities describing the dynamics of expansion in the following section.

In Section 3.2, we argue that the ICs of the background universe are given by the initial expansion rate H ini and the initial densities. Based on the CMB measurements, within the Λ CDM model, the initial densities are determined to high precision (see also Foidl and Rindler-Daller, 2024). For this reason, it is sufficient to incorporate H ini into the Λ CDM formalism. Let us proceed in our approach by considering the following parameters: 1 = Ω r , 0 + Ω b , 0 + Ω CDM , 0 + Ω k , 0 + Ω de , 0 , (25) where the operationally defined density parameter of the geometrical curvature Ω de takes the place of Ω Λ . For the sake of completeness, we include Ω k , 0 ( = 0 ) in the equation to express the perceived flatness of space. In the same way as in Λ CDM, Ω k , 0 = 0 describes the flatness of space but based on novel arguments as it appears to us in our local reference frame as co-moving FLRW observers.

To proceed with our approach, in an inflationary big bang cosmology, we allow for the following simplification. We analyze the evolution of cosmological models by the time inflation has ended, and we call the expansion rate at the end of inflation “primordial expansion rate” (in analogy to the primordial power spectrum in structure formation). Detailed information of the exact evolution of H prior to this point is not required.

We recognize from Equation 25 that Ω de , 0 = 1 − Ω phys , 0 , (26) where Ω phys , 0 denotes the sum total of the density parameters ( Ω CDM , 0 , Ω b , 0 , and Ω r , 0 ) of all physical contributions to the energy budget of the universe (CDM, baryons, and radiation; without considering the operationally defined contributions to the energy–momentum tensor T μ ν ).

We want to use the EoS parameter w de of ρ de to parameterize the dynamics of the expansion, determined by the curvature term in Equation 4, as a function of the sum total of the energy densities of the cosmic components in the universe Ω phys , that is, relate it to Equation 25. This retains the customary Λ CDM formalism, although we are only left to adapt the computation of w de , instead of using the constant EoS parameter w Λ = − 1 of the Λ CDM model.

To this end, we carried out a change in variable ρ in Equation 4 to Ω phys : we multiply Equation 4 by a 2 ( t ) , divide it by 2, and use the total amount of energy ( 4 π / 3 ) ρ phys a 3 in the sphere of “radius” (scale factor) a , in units of the critical density (see Equation 26), instead of the respective energy density ρ phys , which yields 1 2 a ̇ 2 t − G Ω phys a = κ , (27) with the constant κ .

We now use Equation 27 to determine w de as follows. Since κ is a constant, the evolution of the first term is determined by the evolution of the second term, given by d d a G Ω phys a ∝ − Ω phys a 2 , (28) where we use the fact that all the cosmic components of interest evolve smoothly with respect to the scale factor a , as observed by the power laws in Equation 13, just as in Λ CDM. Moreover, this is unsurprisingly equivalent to Equation 13c, which is derived from Equation 12. In what follows, we do not need detailed prefactors. We use the right-hand side of Equation 28 in Equation 12, which yields ρ de ∝ a − 2 Ω phys , 0 . (29) To transform the variables back to the customarily used energy density ρ de , we equate the exponents in Equation 12, 29 by − 3 ( 1 + w de ) = − 2 Ω phys , 0 . Rearranged to express w de , it reads as w de = 2 3 Ω phys , 0 − 1 , (30) where w de is a constant.

The significant property of Equation 30 is that in general, it does not yield the EoS of a cosmological constant. Only for an empty universe, we get exactly w de = − 1 . In fact, this is in good accordance with the original empty de Sitter universe solution (de Sitter, 1917) and the expectations from a kinematically determined universe: if the universe is empty, there is no global curvature of space and hence no deceleration but rather H = c o n s t , resulting in an exponential growth of the scale factor.

However, as soon as we have physical components, Ω phys , 0 > 0 , the EoS parameter in Equation 30 fulfills w de > − 1 . If we choose the same matter content as the Λ CDM concordance model, that is, using Ω phys , 0 ≈ Ω m , 0 ≃ 0.3 , Equation 30 yields w de ≃ − 0.8 ; thus, there is a weak deceleration compared to a model with w Λ = − 1 . Still, the two EoS parameters are close numerically and in terms of their phenomenological impact onto the expansion history (which we present in detail in a follow-up paper).

On the other hand, a universe at critical density, that is, Ω phys , 0 = 1 , yields w de = − 1 / 3 , which is the EoS parameter of the spatial curvature used in the FLRW formalism and restricted to the fine-tuned case of a universe at critical density. This is exactly what we expect.

To retain Λ CDM’s formalism and Equation 12, we make a distinction of cases. We apply the constant value of w de = − 1 / 3 to flat and closed geometries, as is also the case in Λ CDM (with w k = − 1 / 3 ). For open geometries, we apply Equation 30. In summary, we have w de = − 1 3 − Θ ( Ω de , 0 ) 2 3 Ω de , 0 , (31a) Ω de , 0 = 1 − Ω phys , 0 , (31b)

where in Equation 31a Θ is the Heaviside function. The EoS parameter w de is a function of Ω de , 0 (see Equation 31b) and therefore a constant for a given sum total of physical energy densities. The Heaviside function separates the two regimes of super- and subcritical model universes. The first term corresponds to deceleration due to the critical density. The factor after the Heaviside function applies to subcritical universes only, where w de falls below − 1 / 3 ; that is, a decreasing EoS parameter implies less deceleration as it should. In addition, we retain Ω k , 0 = 0 to express the perceived flatness of space in our local inertial frame as co-moving FLRW observers, just the same way as in Λ CDM. This is essential to the linear perturbation theory applied in Λ CDM, for example, in the calculation of the CMB temperature spectrum, as the notion “curvature” herein refers to spatial curvature in the Einstein equations (see, e.g., Ma and Bertschinger, 1995; Weinberg, 2008; Coles and Lucchin, 2002; Mukhanov 2005; Dodelson, 2003; Peebles, 1993), and not to the geometry of the FLRW metric. A description of how the spatial curvature affects the CMB temperature spectrum can be found in many textbooks covering structure formation (see the aforementioned references). However, there is a degeneracy for the impact of spatial curvature on the CMB spectrum with the cosmological constant Λ or dark energy, respectively (see, e.g., Hu and Dodelson, 2002). We will explain this point in a follow-up paper.

Finally, the Friedmann Equation 32a reads H 2 ( t ) = 8 π G 3 c 2 ρ r ( t ) + ρ b ( t ) + ρ CDM ( t ) + ρ de ( t ) , (32a) ρ de ( a ) = Ω de , 0 ρ crit , 0 a − 3 ( 1 + w de ) , (32b)

where Equation 32b now describes the evolution of ρ de as a function of scale factor a for a constant w de , given by Equation 31. The present-day critical density ρ crit , 0 is defined in Equation 8.

Figure 2 finally displays the time evolution of the scale factors of model universes with various matter densities, color-coded by density parameter Ω m , whose value refers to the present, within our Λ CDM extension. The red curves indicate universes with closed geometry; the yellow curve has exactly the critical density, that is, the EdS universe, which separates the supercritical from the subcritical universes, which go from light green to deep blue for the empty universe.

The curves between the yellow and the dark blue curves depict the evolution of subcritical models with matter densities between the EdS model with critical density (solid yellow curve) and the empty model (solid dark blue curve). We can observe that the cosmological models transition uniformly between these two limiting model cases, corresponding to decreasing mass density and “approaching” the exponential curve of the empty model, exactly as expected for kinematically determined universes in GR. In fact, a universe filled with a cosmological constant to critical density displays the same evolution as the expectations for an empty universe, in the former by negative pressure balancing gravity ¹¹ and in the latter by vanishing gravity in an empty universe.

The black solid curve indicates the evolution of the Λ CDM model with Ω m = 0.3 and the cosmological constant Ω Λ = 0.7 . Comparing it to the solution obtained by our Λ CDM extension with Ω de = 0.7 and w de = − 0.8 for a model universe with an identical amount of matter Ω m = 0.3 (thick solid light blue curve), we recognize the similarity between both models. Both display the typical s-shape in the evolution of the scale factor, indicating the transition from decelerated to accelerated expansion, known from Λ CDM. However, there is a difference in the age between the two world models. We discuss this finding and other features, along with the results of further calculations in detail in a follow-up paper. Moreover, the evolution of supercritical model universes with matter densities above the critical density also matches the expectation. They display no deviations from conventional computations, with respect to the evolution of their scale factors.

Our approach shows a significant difference compared to the cosmological constant Λ , which is considered a physical content of the universe. However, unlike the cosmological constant Λ , ρ de here does not contribute to the energy budget of the universe; instead, it refers to a kinematic effect, described by an effective DE component emerging from the geometry of the FLRW metric. As such, it is considered an operational contribution to the energy–momentum tensor and is not regarded as a physical contribution to the universe, as depicted in Equation 19, just in the same way as ρ k in Λ CDM obviously. Although it seems to be only a minor modification, it brings a significant difference compared to the Λ CDM model as the universe, in this approach, is considered an open universe with subcritical energy density and the sum total of the energy densities of the physical contributions (radiation and matter) to the energy budget of the universe is below critical density. Finally, Ω k , 0 = 0 refers to the perceived flatness of space in the local inertial system of the co-moving FLRW observer, in contrast to Λ CDM, where it is interpreted as the global flatness of space.

We first presented the historical context of the foundation of modern cosmology: the Einstein equations and the FLRW metric. The geometric interpretation of gravity describes it as the dynamical curvature of space by gravitating masses (more precisely, gravitating energy densities ¹² ). Based on Milne’s idea of a kinematically determined universe, Robertson and Walker derived the FLRW metric. Applying this metric to the Einstein equations yields the Friedmann equation, which describes the evolution of the expansion history of a kinematically determined universe in the reference frame of observers co-moving with the expansion: they move on geodesics; i.e., they are free-falling. The expansion of the universe started with a very high expansion rate, which is continuously decelerated due to the action of gravity: the kinematic determination of the evolution of the universe. We presented three examples of model universes, which suggested that there might be some “incompleteness” in Λ CDM’s flat universe interpretation. Our proposal to solve this incompleteness includes the following novelties:

a. We take the concept of a kinematically determined universe, the equivalence principle, and the concept of the co-moving FLRW observer at face value and find that the FLRW metric and the curvature of space are two individual aspects of the geometry of the universe, where we identify the geometry given by the FLRW metric with a kinematic DE component.

Owing to the different evolution of ρ de in our approach versus Λ in Λ CDM, there is a difference in the age of the models, namely, 13.04 Gyr versus 13.8 Gyr, respectively. We checked to see that the younger age of our model is not in conflict with age estimates of the oldest known stars. It is also not in conflict with the very early galaxies found through the James Webb Space Telescope since the redshift determinations of these galaxies require a cosmological model to convert redshift into age. In our model, these galaxies would be correspondingly younger than those in a Λ CDM universe. In a follow-up paper, we present an in-depth comparison with observations.

We first motivated our approach presented in this article by placing the key concepts of modern cosmology in a historical context. We now want to complete this view of the historical context. Although Robertson and Walker showed that observers co-moving with the expansion of the universe move on geodesics, they neither emphasized that they are in a locally flat space nor discussed the consequences. Unlike Friedmann, however, they describe the curvature of an auxiliary Riemannian space, not the physical space of the universe. In the closing statement of his works (Friedmann, 1922; Friedmann, 1924), Friedmann concluded that it is not possible to determine, based on the Einstein equations alone, whether the universe is finite (i.e., has supercritical density) or infinite (i.e., has critical or subcritical density) and that supplementary assumptions are required. Lemaître concluded that the universe originated from an event nowadays known as the big bang and has been expanding ever since; this is precisely this supplementary assumption Friedmann referred to. Considering the concept of a kinematically determined universe, we presented the idea that not only the initial density but also its relationship with the initial expansion rate determines the expansion history of the universe. This led to a very natural explanation for the phenomenology of a late-time accelerated expansion as a kinematic effect, which we incorporated into the FLRW formalism as a kinematical DE component.