Differing Manifestations of Spatial Curvature in Cosmological FRW Models

Abstract

We found statistical evidence for a mismatch between the (global) spatial curvature parameter K in the geodesic equation for incoming photons and the corresponding parameter in the Friedmann equation that determines the time evolution of the background spacetime and its perturbations. The mismatch, hereafter referred to as ‘curvature slip’, was especially evident when the SH0ES prior of the current expansion rate was assumed. This result is based on joint analyses of cosmic microwave background (CMB) observations with the PLANCK satellite (P18), the first year results of the Dark Energy Survey (DES), baryonic oscillation (BAO) data, and at a lower level of significance, the Pantheon SNIa (SN) catalog as well. For example, the betting odds against the null hypothesis were greater than ${10}^7$:1, 1400:1 and 1000:1 when P18+SH0ES, P18+DES+SH0ES and P18+BAO+SH0ES were considered, respectively. Datasets involving SNIa weakened this curvature slip considerably. Notably, even when the SH0ES prior was not imposed, the betting odds for the rejection of the null hypothesis were 70:1 and 160:1 in cases where P18+DES and P18+BAO were considered. When the SH0ES prior was imposed, the global fit of the modified model (that allows for a nonvanishing ‘curvature slip’) strongly outperformed that of $\Lambda$CDM, being manifested by significant deviance information criterion (DIC) gains ranging between 7 and 23, depending on the dataset combination considered. Even in comparison with K$\Lambda$CDM, the proposed model resulted in significant, albeit smaller, DIC gains when SN data were excluded. Our finding could possibly be interpreted as an inherent inconsistency between the (idealized) maximally symmetric nature of the FRW metric and the dynamical evolution of the GR-based homogeneous and isotropic $\Lambda$CDM models. As such, this implies that there is apparent tension between the metric curvature and the curvature-like term in the time evolution of the redshift.

1. Introduction

On the largest observable scales, the Universe appears to be extremely isotropic around us. Augmented by the Cosmological Principle, this (‘local’) isotropy is endowed to every observer at rest in the cosmic microwave background (CMB) frame. The corresponding spacetime is uniquely described by the Friedmann–Robertson–Walker (FRW) metric. Space in the latter has either closed, flat or open geometry. If, in addition, it is assumed that gravitation is governed by general relativity (GR), then this background spacetime is shaped by its homogeneous and isotropic matter content. Incoming photons propagating in curved space trace its global geometry, and observational determination of various (angular diameter, luminosity distance, etc.) distance-redshift relations enable inference of the matter content of the Universe. The CMB anisotropy and polarization depend on both the matter content and geometry.

The currently favored cosmological model, $\Lambda$CDM, is spatially flat, and its energy content consists of ∼$69\%$ dark energy (DE), $31\%$ non-relativistic (NR) matter—[$5\%$ baryons and $26\%$ cold dark matter (CDM)]—as well as a residual amount $\lesssim 0.1\%$ CMB radiation and relativistic (and possibly also NR) neutrinos. The consensus ‘vanilla’ $\Lambda$CDM model, characterized by only six parameters, emerged from joint analyses of various cosmological datasets that include mainly CMB anisotropy and polarization, galaxy clustering and galaxy shear maps, type Ia supernovae luminosity distance measurements and baryonic oscillations (BAOs). However, these models are not fully consistent with each other. Perhaps the most glaring example is the disparity between the Planck satellite CMB data, which favor a spatially closed cosmological model at ∼$3\sigma$ statistical significance on the one hand, and BAO data, which strongly favor flat space on the other hand (e.g., [1,2,3,4]). Their combination considerably weakens the case for a closed Universe (e.g., [5]), resulting in the best fit ‘model of choice’: the flat $\Lambda$CDM model.

A spatially non-flat Universe would seem to be unnatural [6] (i.e., fine-tuned) in either case of a curvature radius much larger or much smaller than the present horizon size of the observable Universe. For example, if the initial conditions of the Universe are set to the Grand Unified Theories (GUT) scale, then the curvature radius must have been fine-tuned at a precision level of one part in a trillion so as to be either a quadrillion times larger or smaller than the horizon size of the observable Universe at the present time. This is the consensus understanding of the ‘flatness problem’ [6], although there are counterarguments that lend some support to the claim that this is not really problematic for the original Hot Big Bang model [7,8,9,10,11,12]. This, as well as the ‘horizon problem’, have provided the main motivation for the inflationary scenario. Observationally, whether space is flat or closed is still an unsettled issue (e.g., [1,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]), but the far-reaching implications of a closed Universe cannot be overstated (e.g., [3]).

From the cosmic inflation perspective (e.g., [34]), a spatially flat Universe is strongly favored (although open- (e.g., [35,36,37,38]) and closed-inflation (e.g., [39]) models were proposed). If so, then the PLANCK dataset stands out and is especially at odds with the BAO dataset in favoring a non-flat Universe. The challenge is that only a probe of an extremely large scales can result in a credible inference of (or bounds on) the curvature radius. Perhaps the most precise and well-understood cosmological probe is measurements of CMB anisotropy and polarization, and thus only satellite-based measurements are capable of meaningfully constraining the curvature radius. At present, the Planck database is most optimally suited for this. While joint analyses of the current CMB+BAO datasets tilt the balance toward a spatially flat space, it is arguably problematic to put too much weight on this conclusion, mainly because the current CMB+BAO datasets are internally inconsistent (e.g., [1]). Clearly, global geometry is a critical property of the Universe; whereas values of the other basic parameters of the cosmological standard model (SM) affect the (temporal) evolution of physical processes, only the curvature and DE parameters determine its global geometry and future evolution.

In this paper, we explore certain hitherto unexplored extensions to the time redshift $t\left(z\right)$, the relation of the standard FRW-based treatment, and the $r\left(t\right)$ relation that, in general, departs from the geodesic equation. Basically, we allow for $t\left(z\right)$ to differ from the relation obtained from the Friedmann equation (i.e., it does not depend solely on the matter content of the Universe). In other words, we employ $t\left(z\right)$ relations in our analysis that, under the assumption of isotropy and homogeneity, represent a clear violation of local energy-momentum conservation (i.e., we basically abandon the GR-based framework). Energy-momentum non-conservation is a generic feature of various theories of gravitation [40], the most common (and well known) of which are scalar-tensor theories. Other examples are Rastall gravity [41], unimodular gravity [42,43], and other theories [44,45,46,47]. Another possibility is that the degree of large-scale isotropy of the Universe is inconsistent [48] with the underlying assumption characterizing the FRW spacetime. Perhaps unexpectedly, our analysis results point toward phenomenological parameters in the $r\left(z\right)$ relation that depart by up to ∼$4\sigma$ from their canonical values, depending on the dataset combination used and the assumed priors. This result adds to other anomalies or tensions that have been long recognized to afflict the cosmological SM (e.g., the ‘lensing anomaly’), the ${\sigma }_8/S_8$ tension of the CMB with large scale clustering probes, and the ‘Hubble tension’ (e.g., [2,31]).

We employ the deviance information criterion (DIC) in assessing the relative likelihood of each proposed model in comparison with both the standard flat $\Lambda$CDM and its ‘curved’ extension K$\Lambda$CDM models. We find that depending on the specific dataset combinations, the proposed models can be mildly or even strongly favored over $\Lambda$CDM, thereby substantiating our finding that the $t\left(z\right)$ relation does not depend solely on the matter content from a global parameter analysis perspective.

This paper is organized as follows. In Section 2, we summarize a few basic results of the SM for reference in future sections and for setting the notation. In Section 3, we describe our proposed modifications, model comparison analysis and results. Our conclusions are discussed in Section 4, and our work is summarized in Section 5. Throughout, we adopt units such that the speed of light, where $c\equiv 1$.

2. The Benchmark Standard Model

On the largest observable scales, the Universe has been found to be highly isotropic to within one part in ${10}^5$. The implication of this observation is greatly widened by adoption of the Cosmological Principle, essentially stating that the Universe appears isotropic (and therefore uniform) to every ‘fundamental’ observer. The FRW metric of such a (maximally symmetric) space is expressed in terms of the line element

$$\begin{array}{c}\hfill d{s}^2=-d{t}^2+a^2\left[{\displaystyle \frac{d{r}^2}{1-K{r}^2}}+r^2(d{\theta }^2+{sin}^2\theta d{\phi }^2)\right],\end{array}$$

(1)

where $a=a\left(t\right)$ is the scale factor, K is the spatial curvature parameter, and (with no loss of generality) the origin of spatial coordinates is set to be at the observer. Incoming radial null geodesics in this spacetime integrate to

$$\begin{array}{c}\hfill r\left(\eta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{sin\left[\sqrt{K}({\eta }_0-\eta )\right]}{\sqrt{K}}}\hfill & ;K>0\hfill \\ {\eta }_0-\eta \hfill & ;K=0\hfill \\ {\displaystyle \frac{sinh\left[\sqrt{-K}({\eta }_0-\eta )\right]}{\sqrt{-K}}}\hfill & ;K<0,\hfill \end{array}\right.\end{array}$$

(2)

where $\eta \equiv \int {\displaystyle \frac{dt}{a\left(t\right)}}$ is the conformal time and ${\eta }_0$ is its present value.

In the GR-based SM, use of the FRW metric in the field equations for the adiabatically evolving Universe can be summarized in a single Friedmann equation:

$$\begin{array}{c}\hfill H^2+{\displaystyle \frac{K}{a^2}}={\displaystyle \frac{8\pi G\rho \left(a\right)}{3}},\end{array}$$

(3)

where $H\equiv \dot{a}/a$, $\dot{a}\equiv {\displaystyle \frac{da}{dt}}$, G is the Universal gravitational constant, and $\rho$ is the total energy density. Assuming that the various contributions to the cosmic energy budget do not mutually interact, then the continuity equation applied to the ith species is

$$\begin{array}{c}\hfill {\dot{\rho }}_i+3(1+w_i)H{\rho }_i=0,\end{array}$$

(4)

where $w_i$ is the equation of state (EOS) parameter of the ith species, which integrates into

$$\begin{array}{c}\hfill {\rho }_i\left(a\right)={\rho }_{i,0}exp\left[-\int 3(1+w_i\left(a\right)){\displaystyle \frac{da}{a}}\right].\end{array}$$

(5)

In addition, if it is assumed that $w_i\left(a\right)=w_i$ is fixed (an assumption that breaks down in the case of neutrinos when they turn NR once the Universe cools down below a temperature equivalent to their rest masses), an assumption that is not made in actual computations, then ${\rho }_{i}a={\rho }_{i,0}{a}^{-3(1+w_i)}$, and the total energy density is

$$\begin{array}{c}\hfill \rho \left(a\right)=\sum _i{\rho }_{i,0}{a}^{-3(1+w_i)}.\end{array}$$

(6)

Substituting Equation (6) into Equation (3) and integration yield the basic expression for $t\left(a\right)$ or, equivalently, $\eta \left(z\right)$, where $a\equiv {(1+z)}^{-1}$. Use of the latter expressions in Equation (2) results in the following $r\left(z\right)$ relation:

$$\begin{array}{c}\hfill H_{0}r\left(z\right)=\left\{\begin{array}{cc}{\displaystyle \frac{sin\left(\sqrt{-{\Omega }_k}\mathcal{D}\right)}{\sqrt{-{\Omega }_k}}}\hfill & ;{\Omega }_k<0,K>0\hfill \\ \mathcal{D}\hfill & ;{\Omega }_k=0,K=0\hfill \\ {\displaystyle \frac{sinh\left(\sqrt{{\Omega }_k}\mathcal{D}\right)}{\sqrt{{\Omega }_k}}}\hfill & ;{\Omega }_k>0,K<0,\hfill \end{array}\right.\end{array}$$

(7)

where ${\Omega }_k\equiv -K/H_0^2$ and

$$\begin{array}{ccc}\hfill \mathcal{D}(z;\left\{{\Omega }_{i,0}\right\})& \equiv & H_0({\eta }_0-\eta )={\int }_0^z{\displaystyle \frac{d{z}^{\prime }}{E\left(z^{\prime }\right)}},\hfill \\ \hfill E^2\left(z^{\prime }\right)& \equiv & \sum _i{\Omega }_{i,0}{(1+z^{\prime })}^{3(1+w_i)}.\hfill \end{array}$$

(8)

Here, $\left\{{\Omega }_{i,0}\right\}$ collectively denotes the various ${\Omega }_{i,0}$, which are the respective values of the energy densities at present, expressed in critical density units, and ${\rho }_c\equiv 3H_0^2/\left(8\pi G\right)$, including ${\Omega }_k$ with an effective EOS $w_{k,eff}=-1/3$. Specifically, in a Universe containing only photons, dust, DE, and curvature, we have

$$\begin{array}{c}\hfill E\left(z\right)\equiv \sqrt{{\Omega }_r{(1+z)}^4+{\Omega }_m{(1+z)}^3+{\Omega }_k{(1+z)}^2+{\Omega }_{\Lambda }}.\end{array}$$

(9)

It should be noted that ${\Omega }_k$ controls both the dynamics of cosmic evolution (via the Friedmann equation, shown in Equation (3), or equivalently Equation (9)) and the kinematics (or perhaps more precisely referred to as the ‘optics’) of incoming photons (via the geodesic equation, shown in Equation (2)). It is also important to keep in mind that whereas the optics only depends on the form of the metric, essentially reflecting the spatial symmetry, the dynamics depends on the underlying theory of gravitation (which relates the geometry to the underlying energy-momentum content) as well.

3. Modified Curvature SM

In the present work, we explore the implications of phenomenologically modifying the time-redshift $t\left(z\right)$ and distance-time $r\left(t\right)$ relations on cosmological scales (on which the assumption of homogeneity and isotropy is still assumed). The SM $t\left(z\right)$ relation follows from the Friedmann equation (i.e., local energy-momentum conservation). In addition, the $r\left(t\right)$ relation describes the trajectories of incoming radiation along null geodesics. The latter describes the kinematics of test particles under the premise of energy-momentum conservation. Combining $r\left(t\right)$ and $t\left(z\right)$ yields the observable $r\left(z\right)$ relation.

Here, we replace Equation (7) with

$$\begin{array}{c}\hfill H_{0}r\left(z\right)=\left\{\begin{array}{cc}{\displaystyle \frac{sin\left(\sqrt{-{\omega }_k}\mathcal{D}\right)}{\sqrt{-{\omega }_k}}}\hfill & ;{\omega }_k<0\hfill \\ \mathcal{D}\hfill & ;{\omega }_k=0\hfill \\ {\displaystyle \frac{sinh\left(\sqrt{{\omega }_k}\mathcal{D}\right)}{\sqrt{{\omega }_k}}}\hfill & ;{\omega }_k>0\hfill \end{array}\right.\end{array}$$

(10)

where we introduce a new parameter ${\omega }_k$ while leaving $\mathcal{D}$ (Equation (8)) unchanged. In general, we do not require ${\omega }_k$ to be equal to ${\Omega }_k$ in Equations (9) and (10) and define $\kappa \equiv {\omega }_k-{\Omega }_k$ as a derived parameter in our modified models. Clearly, nested within these models are K$\Lambda$CDM, if indeed ${\omega }_k={\Omega }_k$, and flat $\Lambda$CDM in the particular case where ${\omega }_k={\Omega }_k=0$. In a similar vein to the addition of the cosmological constant to the SM as discussed above, we introduce the free dimensionless parameter $\kappa$ coupled to a specific redshift dependence (as shown in Equations (9) and (10)) but with neither energetic interpretation nor any perturbation imprints (i.e., it does not cluster), much like DE (assuming it is a cosmological constant). Obviously, the likelihood of $\kappa \ne 0$ will be determined by contrasting the modified model with the various datasets.

Whereas, by their very nature, cosmological observables typically probe Equation (10), the Sandage–Loeb effect (also known as the redshift drift effect [49,50,51]) is sensitive to ${\displaystyle \frac{dz}{dt}}$ and is therefore directly dependent on the Friedmann equation (Equation (9)). While Equation (10) depends on both ${\Omega }_k$ and ${\omega }_k$ (as well as ${\Omega }_m$), Equation (9) depends solely on ${\Omega }_k$. Future measurements of the redshift drift, when combined with precision measurements of $r\left(z\right)$, could in principle allow for a clean separation of ${\Omega }_k$ and ${\omega }_k$. However, undertaking such an endeavor would be challenging, given that the redshift drift signal is extremely weak over typical observational timescales.

Because no perturbations are considered in the value of ${\Omega }_k$, in our analysis, $\kappa$ is a truly anomalous parameter essentially devoid of physical meaning. In this specific sense, it plays a role similar to other phenomenological parameters that have been considered in various generalizations of the SM, such as the lensing anomaly parameter $A_L$ (e.g., [52]), the dipole and integrated Sachs–Wolfe (ISW) anomaly parameters (e.g., [53,54]), and the CMB temperature mismatch parameter when local priors on $H_0$ are assumed [55,56,57]. The parameter $\kappa$ should be consistent with zero if GR and the underlying assumption of the Cosmological Principle are not just qualitatively but also quantitatively valid. In principle, since ${\Omega }_k$ appears in Equation (9) as a contribution of an effective fluid with equations of state, where $w=-1/3$, it could be viewed as the combined effect of curvature and a ‘K-matter contribution’ (save for the fact that it is unperturbed in our analysis) (e.g., [58]). However, as we will see below, it turns out that our analysis of the (present) observational datasets yields $\kappa >0$, further reducing the likelihood for it to be a new exotic matter contribution candidate. In general terms, the ‘null hypothesis’ is rejected to the extent that $\kappa$ statistically departs from zero.

One possible explanation for the nonvanishing phenomenological parameter $\kappa$ could be that the FRW spacetime metric is an over-idealization of spacetime that fails on sufficiently small scales. This could possibly indicate that when averaging over sufficiently small scales, second-order perturbations do not vanish and can back-react on a smoothed out background. In particular, it can manifest in the form of effective spatial curvature or K-matter, DE, and also stress-like contributions in the volume-averaged Friedmann equation. This does not readily and unequivocally reflect on the effective spacetime metric. These effects could not only tilt the balance between the various terms in the volume-averaged Friedmann equation but also affect the growth of the structure, thereby biasing the inferred cosmological parameters, with the curvature parameter included (e.g., [59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76]).

3.1. Models and Datasets

We carried out an extensive statistical analysis in order to determine the most probable ranges of the extended parameter set of the proposed modifications to the SM. To the ‘vanilla’ fundamental model parameters ${\mathit{\theta }}_{\mathit{MC}}$, ${\Omega }_b{h}^2$, ${\Omega }_c{h}^2$, $\tau$, $A_s$, and $n_s$, we add both ${\Omega }_k$ and ${\omega }_k$. The parameter $\kappa$ is a derived parameter. Additional relevant derived parameters that will be of interest to us here are $H_0$, $S_8$, and ${\sigma }_8$, especially in light of the well-known tensions involving these parameters when inferred from various dataset combinations. The models considered in this work are summarized in

Table 1. Models explored in this work.

Model 1 ($\Lambda$CDM)	${\Omega }_k=0={\omega }_k$
Model 2 (K$\Lambda$CDM)	${\Omega }_k\equiv {\omega }_k$
Model 3	${\Omega }_k\ne 0$, ${\omega }_k\ne 0$
Model 4	${\Omega }_k\ne 0$
Model 5	${\omega }_k\ne 0$

.

The datasets employed in this analysis constitute the standard commonly adopted set (in model testing), which is included (along with the corresponding likelihood functions) as part of the CosmoMC 2021 package. These include the CMB Planck 2018 data [77], DES 1 yr [78], BAO (data compilation from BOSS DR12 [79], MGS [80], and 6DF [81]), and Pantheon data (catalog of 1048 SNIa in the redshift range of $z\lesssim 2$ [82]). The entire Planck dataset was included, with a multipole range of $2<l<2500$ covered by the plikHM_TTTEEE, lowl, and lowE likelihood functions.

As in our previous (somewhat) related work [57], we considered the P18, P18+DES, P18+BAO, P18+SN, and P18+BAO+SN dataset combinations, each of which we performed the analysis with and without the SH0ES prior on $H_0$ for. Sampling from posterior distributions was performed using the fast-slow dragging algorithm with a Gelman–Rubin convergence criterion $R-1<0.02$ (where R is the scale reduction factor). Testing the curved space model (with the new parameters included) using only the Planck dataset (without the SH0ES prior) did not satisfy this convergence criterion. Therefore, the results for this case were not considered in this work.

3.2. Results

For each of the five models described above, we computed the DIC for each dataset combination. The DIC is defined as [83]

$$\begin{array}{c}\hfill DIC\equiv 2\overline{{\chi }^2}\left(\theta \right)-{\chi }^2\left(\overline{\theta }\right),\end{array}$$

(11)

where $\theta$ is the vector of free model parameters and bars denote averages over the posterior distribution $P\left(\theta \right)$. According to the ‘Jeffreys scale’ convention, a model characterized by $\Delta$DIC that is lower with respect to a reference model by <1, 1.0–2.5, 2.5–5.0, and >5 would be considered inconclusively, weakly-to-moderately, moderately-to-strongly, or decisively favored [83] over the reference model, respectively.

Our quantitative DIC results are summarized in

Table 2. DIC values for the various models and dataset combinations. The references, Model 1 and Model 2, are the flat $\Lambda$CDM and K$\Lambda$CDM models, respectively.

Datasets	Model 1	Model 2	Model 3	Model 4	Model 5
P18	2799.4	2791.0	–	2800.1	2791.3
		−8.3	–	0.7	−8.1
P18+DES	3339.8	3338.4	3331.6	3338.4	3339.2
		−1.3	−8.1	−1.3	−0.6
P18+BAO	2805.9	2806.5	2805.9	2806.8	2806.4
		0.6	0	0.9	0.5
P18+SN	3834.8	3835.0	3827.7	3835.9	3835.7
		0.2	−7.0	1.1	0.9
P18+SN+BAO	3840.8	3841.3	3841.7	3841.3	3841.4
		0.5	0.9	0.5	0.6

and

Table 3. Same as Table 2, but with the SH0ES prior included.

Datasets	Model 1	Model 2	Model 3	Model 4	Model 5
P18+SH0ES	2819.3	2809.3	2795.7	2804.8	2807.8
		−10.0	−23.6	−14.5	−11.6
P18+DES+SH0ES	3354.6	3342.0	3331.9	3338.4	3340.1
		−12.6	−22.7	−16.2	−14.5
P18+BAO+SH0ES	2824.8	2821.5	2812.2	2818.4	2820.9
		−3.3	−12.6	−6.4	−3.9
P18+SN+SH0ES	3853.8	3844.9	3843.1	3845.0	3844.2
		−8.9	−10.7	−8.8	−9.6
P18+SN+BAO+SH0ES	3859.4	3856.8	3852.4	3854.3	3856.0
		−2.6	−7.0	−5.2	−3.4

. Specifically, when considering dataset combinations with the SH0ES prior, the fits for Model 3 yielded DIC values lower by 7–23 in comparison with those for Model 1. When the SH0ES prior was excluded, Model 3 was significantly favored for the dataset combinations P18+DES and P18+SN, with DIC gains of eight and seven, respectively, but did not outperform the Model 1 fits with the dataset combinations P18+BAO, and P18+SN+BAO. Even when compared with Model 2, the Model 3 fits were better when the SH0ES prior was included. When the SH0ES prior was excluded, Model 1 and Model 2 yielded quite similar DIC values (except in the case of P18 data alone, when Model 2 was strongly favored over Model 1, as is shown in Table 2). Therefore, Model 3 improved over Model 2 to about the same extent that it improved over Model 1. Model 3 was found to be favored over Model 4 and Model 5 for its fits for all dataset combinations with the SH0ES prior.

Whereas the ultimate model comparison test must consider the overall model fitness to data, as quantified by the corresponding DIC values in Table 2 and Table 3, it is sometimes useful to compare the results for ‘anomaly’ parameters, which should be identically zero in the reference model (flat $\Lambda$CDM). Significant evidence in the data for the addition of an ‘anomaly’ parameter is indicative of underperformance of the reference model in comparison with the new modified models. The significance of statistical evidence for such an anomaly parameter is usually reported in terms of percentiles or equivalently in terms of ‘betting odds’ against a finite value for the parameter. We chose the latter option, and the results are listed in

Table 4. Betting odds against ${\Omega }_k\ne 0$, ${\omega }_k\ne 0$, and $\kappa \ne 0$ in Model 3 for various dataset combinations. Whereas either ${\Omega }_k\ne 0$ or ${\omega }_k\ne 0$ ruled out flat $\Lambda$CDM, $\kappa \ne 0$ ruled out both flat $\Lambda$CDM and K$\Lambda$CDM.

Dataset	${\mathsf{\Omega }}_{\mathit{k}}$	${\mathit{\omega }}_{\mathit{k}}$	$\mathit{\kappa }$
P18+SH0ES	1:500	1:20	1:10,000,000
P18+DES+SH0ES	1:300	1:5	1:1400
P18+BAO+SH0ES	1:1250	1:25	1:1000
P18+SN+SH0ES	1:7	fair	1:14
P18+BAO+SN+SH0ES	1:20	fair	1:25
P18	–	–	–
P18+DES	1:50	1:7	1:70
P18+BAO	1:4	fair	1:5
P18+SN	1:200	1:200	1:160
P18+BAO+SN	fair	fair	fair

and

Table 5. Betting odds against either ${\Omega }_k\ne 0$ (Model 4) or ${\omega }_k\ne 0$ (Model 5) in the data. For reference, betting odds against ${\Omega }_k\ne 0$ in K$\Lambda$CDM (Model 2) are shown.

Dataset	Model 2	Model 4	Model 5
P18+SH0ES	1:300	1:100,000	1:100,000
P18+DES+SH0ES	1:2500	1:100,000	1:1250
P18+BAO+SH0ES	1:14	1:100	1:20
P18+SN+SH0ES	1:300	1:500	1:500
P18+BAO+SN+SH0ES	1:16	1:50	1:20
P18	1:500	1:4	1:1100
P18+DES	1:4	1:12	1:5
P18+BAO	fair	fair	fair
P18+SN	fair	fair	fair
P18+BAO+SN	fair	fair	fair

. For example, in Table 5, we report the corresponding results for the two fundamental parameters ${\Omega }_k$ and ${\omega }_k$, as well as for the derived parameter $\kappa$. It is clear from the table that $\kappa$ was, in general, a better diagnostic of the departures from the reference model than either ${\Omega }_k$ or ${\omega }_k$. It is also evident from Table 5 that whereas K$\Lambda$CDM outperformed flat $\Lambda$CDM in virtually all dataset combinations considered in the present work when the SH0ES prior was imposed (open geometry), as well as in the case of P18 and P18+DES even with the SH0ES prior excluded (closed geometry), Model 5 and Model 4 did so even more significantly. We stress again that focusing on a single parameter might be misleading, given that the ultimate test is the ‘volume’ of the likelihood function in parameter space, where the smaller the volume, the better the fit to the data. Indeed, this was gauged by the DIC values reported in Table 2 and Table 3. The general conclusions and trends deduced from Table 4 and Table 5 bode well given the results that are shown in Table 2 and Table 3.

Triangle plots for the selected cosmological parameters inferred from fitting Model 3 to various dataset combinations are shown in Figure 1. There is a clear correlation between $\kappa$ and ${\Omega }_b{h}^2$, ${\Omega }_{\Lambda }$, and ${\sigma }_8$ and anticorrelation with ${\Omega }_c{h}^2$, ${\Omega }_k$, and ${\omega }_k$. For $H_0$, ${\Omega }_m$, and $S_8$, both correlations and anticorrelations can be seen, depending on the data combination used. If $S_8$ rather than ${\sigma }_8$ better captured the impact of growth of the structure on the cosmological observables considered in this work, especially the DES dataset, then it is evident from Figure 1 that once $\kappa$ was allowed to freely vary, the data favored lower $S_8$ and higher $H_0$ values, which were even well beyond those obtained with K$\Lambda$CDM and surely those obtained with flat $\Lambda$CDM. Specifically, these parameters inferred from fitting Model 3 to P18+DES resulted in $S_8=0.773\pm 0.017$ and $H_0=76.5\pm 3.0$ km/sec/Mpc at a 68% confidence level. For reference, when the same dataset was fitted with K$\Lambda$CDM (flat $\Lambda$CDM), we obtained $S_8=0.795\pm 0.016$ ($0.8018\pm 0.0066$) and $H_0=70.1\pm 1.7$ ($68.16\pm 0.48$) km/s/Mpc.

4. Discussion

The standard GR-based flat $\Lambda$CDM model has been remarkably successful in describing a broad spectrum of cosmological phenomena probed with a wide variety of CMB projects, optical and IR telescopes, and extensive galaxy surveys, aiming for spectral and spatial mapping of the CMB and tracing the large-scale structure and evolution of the Universe. The K$\Lambda$CDM model was a better fit for all dataset combinations than $\Lambda$CDM when the SH0ES prior was assumed, whereas the two models performed equally well when the SH0ES prior was not adopted. However, when the models were contrasted only with the P18 dataset without the SH0ES prior, there was a strong preference for K$\Lambda$CDM. When the SH0ES prior was not included, the dataset combinations P18+BAO, P18+SN, and P18+SN+BAO showed no preference for K$\Lambda$CDM over the simpler flat $\Lambda$CDM model. Consequently, if the SH0ES prior was ignored in the analysis, then the concordance model converged on flat $\Lambda$CDM by default.

The unique spacetime metric that describes a maximally symmetric spatial hypersurface is the FRW metric, possibly having a non-vanishing (constant) spatial curvature. The curvature parameter appeared in both the Friedmann equation and geodesic equations, a single parameter that appeared in both the $t\left(z\right)$ and $r\left(t\right)$ relations and thus also in the $r\left(z\right)$ relation. In this work, we explored the consequences of adding an additional degree of freedom by allowing two independent curvature parameters in the $t\left(z\right)$ and $r\left(t\right)$ relations, as described in Section 3. We defined a derived ‘anomaly’ (the ‘curvature slip’) parameter $\kappa \equiv {\omega }_k-{\Omega }_k$ that quantified the degree of mismatch between these two parameters, which we inferred by determining the statistical significance of its departure from its canonical value of zero. The reference model, Model 1, was $\Lambda$CDM, and Model 2 was K$\Lambda$CDM. Our modified models were Model 3, in which both ${\Omega }_k$ and ${\omega }_k$ are free parameters. In Model 4, only ${\Omega }_k$ is a free parameter, whereas in Model 5, only ${\omega }_k$ is allowed to vary.

When the SH0ES prior was imposed, K$\Lambda$CDM was clearly a better fit to the data than $\Lambda$CDM, as is evident from Table 3. Model 3 performed much better with a DIC gain of $\gtrsim 20$ when neither BAO nor SN data were included. When these datasets were included, the gain reduced to ∼7–12, which was still better than K$\Lambda$CDM. Remarkably, with just the P18+SH0ES data, Model 3 resulted in $\kappa \ne 0$ having extremely high statistical significance, with betting odds worse than $1:{10}^7$ against $\kappa \ne 0$ (Table 4). A similar conclusion applies to P18+DES+SH0ES and P18+BAO+SH0ES but with lower statistical significance. Model 4 and Model 5 were also found to be statistically acceptable, though at a lower level of significance, as is apparent from the results listed in Table 4 and Table 5.

For dataset combinations with no SH0ES prior, we found that none of Models 3–5 provided a better fit for the data than Model 2. With these dataset combinations, Model 3 did not provide a better fit over either Model 4 or Model 5, except for the cases of P18+DES and P18+SN, for which there was strong evidence (supported by a DIC gain $\gtrsim 7$) in favor of this model over both K$\Lambda$CDM and $\Lambda$CDM. We conclude that a model that allows for simultaneous variation of both ${\Omega }_k$ and ${\omega }_k$ is, in general, not warranted by data combinations that include BAO.

Even though the above analysis was based on the addition of a phenomenologically motivated parameter, it is nonetheless of interest to consider its possibly physical interpretations. As discussed above, the proposed models were better fits for the current datasets than K$\Lambda$CDM and particularly $\Lambda$CDM when the SH0ES prior was imposed. Since $H_0$ and ${\Omega }_k$ were correlated in K$\Lambda$CDM, imposing the SH0ES prior drove ${\Omega }_k$ to positive values (i.e., toward a preference for a hyperbolic topology). The $H_0-{\Omega }_k$ correlation was intuitively clear; in a spatially closed Universe (i.e., ${\Omega }_k<0$), incoming light rays are bent inward, the crossover time of a photon is longer, and consequently, $H_0$ is smaller. Therefore, a higher value of $H_0$ resulted in a higher value for ${\Omega }_k$. A sufficiently high value of $H_0$ necessarily led to ${\Omega }_k>0$ (i.e., an open universe). Now, when both ${\omega }_k$ and ${\Omega }_k$ are allowed to simultaneously vary, even a flat geometry ${\omega }_k\approx 0$ (which favors a higher $H_0$ than in closed space due to the above-mentioned correlation) is possible, but it would still have an extremely negative ${\Omega }_k$ value that allows for a longer fly time for incoming photons. (In the extreme case where ${\Omega }_k$ is sufficiently negative (i.e., the Universe is strongly closed), the fly time could be infinite.) In the process, we can allow for a large $H_0$ value to still be consistent with an extremely negative ${\Omega }_k$ value without affecting the angular diameter distance and without having to compensate (by lowering the value of $H_0$) for the closed geometry simply because even though ${\Omega }_k$ can be significantly negative, ${\omega }_k$ can still satisfy $|{\omega }_k|\ll 1$.

Whereas the curvature slip considered in the present work was examined within the $\Lambda$CDM framework, the concept is clearly applicable to broader classes of cosmological models. For instance, the Dark Energy Spectroscopic Instrument (DESI) collaboration has recently reported strong evidence for evolving dark energy within a spatially flat universe. Specifically, the equation of state (EOS) of the DE component $w\left(a\right)=w_0+w_a(1-a)$, where $w_0$ and $w_a$ are free model parameters, evolves with the scale factor in such a way that it behaves as a ‘phantom’ DE in the past [84]. Notably, this behavior has not been hitherto observed in other datasets. While the DESI collaboration has already considered models with nonzero spatial curvature, incorporating curvature slip could potentially further extend the space of viable models. Given that nonzero curvature slip is independently favored by several datasets, as illustrated in the present work, it would be natural to introduce this additional degree of freedom when analyzing the DESI data as well, potentially even providing an alternative explanation for what the DESI currently interprets as evidence for exotic dark energy evolution.

5. Summary

With the ever-increasing precision of cosmological observations, the challenge for $\Lambda$CDM is correspondingly higher. As the number and statistical significance of SM anomalies increase, so does the interest in alternative models. The SM is based on our best-tested theory of gravitation, GR, which withstood the test of time. However, GR is well tested only within our solar system, whereas on galactic and supergalactic scales, it has faired poorly, as manifested by the need to invoke the existence of CDM and DE. Whereas on the largest cosmological scales, these two mass-energy forms were described by only two new parameters and therefore added only slight complexity to the model, this is far from being the case on galactic scales, where CDM profiles have to be fit separately for each and every galaxy or cluster of galaxies. As argued here—and, to the best of our knowledge, nowhere else in the context of the present work—what is known as ‘dark energy’ (assuming its simplest form: a cosmological constant) could by itself represent a gravitational anomaly on the largest cosmological scales, rather than a genuine form of energy.

The P18 data strongly favor a spatially closed space but only with a relatively low value of $H_0$, which is at odds with local measurements of $H_0$. Other datasets (e.g., P18+DES) even favored an open Universe; only P18+SN supported the closed Universe model but at an extremely weak $\sim 1\sigma$ confidence level. The BAO dataset has played a (somewhat surprisingly) significant role in the acceptance of flat $\Lambda$CDM as the SM. It would be remarkable if indeed the Universe, with its (post-inflation) continuous growth of structure, has no appreciable departure from global spatial flatness. Yet, this is what the current varied datasets, interpreted within the framework of the GR-based SM, seem to suggest. In the present work, we relaxed the ‘stiffness’ of K$\Lambda$CDM to test for the possibility that a less ‘rigid’ model better fit the data. In the process, we found that especially when local measurements of $H_0$ were adopted as a prior (in the statistical analysis), this generalized model better fit the data than K$\Lambda$CDM and notably better than $\Lambda$CDM. This new anomaly, the ‘curvature slip’ (gauged by the departure of a newly introduced $\kappa$ parameter from its vanishing canonical value), is yet another in a growing list of SM anomalies that enhance the need for either new physics or more nuanced analyses within the realm of standard physics.