Planck 2015 constraints on the non-flat XCDM inflation model
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PLANCK 2015 CONSTRAINTS ON THE NON-FLAT XCDM INFLATION MODEL
Junpei Ooba, ∗ Bharat Ratra, and Naoshi Sugiyama
1, 3, 4 Department of Physics and Astrophysics, Nagoya University, Nagoya 464-8602, Japan Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya, 464-8602, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo, Chiba 277-8582, Japan (Dated: October 31, 2018)
ABSTRACTWe examine the Planck 2015 cosmic microwave background (CMB) anisotropy data by using a physically-consistentenergy density inhomogeneity power spectrum generated by quantum-mechanical fluctuations during an early epochof inflation in the non-flat XCDM model. Here dark energy is parameterized using a fluid with a negative equation ofstate parameter but with the speed of fluid acoustic inhomogeneities set to the speed of light. We find that the Planck2015 data in conjunction with baryon acoustic oscillation distance measurements are reasonably well fit by a closedXCDM model in which spatial curvature contributes a percent of the current cosmological energy density budget. Inthis model, the measured non-relativistic matter density parameter and Hubble constant are in good agreement withvalues determined using most other data. Depending on cosmological parameter values, the closed XCDM model hasreduced power, relative to the tilted, spatially-flat ΛCDM case, and can partially alleviate the low multipole CMBtemperature anisotropy deficit and can help partially reconcile the CMB anisotropy and weak lensing σ constraints,at the expense of somewhat worsening the fit to higher multipole CMB temperature anisotropy data. However, theclosed XCDM inflation model does not seem to improve the agreement much, if at all, compared to the closed ΛCDMinflation case, even though it has one more free parameter. Keywords: cosmic background radiation — cosmological parameters — large-scale structure of universe— inflation — observations ∗ [email protected] INTRODUCTIONWe recently found that cosmic microwave background (CMB) anisotropy measurements do not require flat spatialhypersurfaces in the ΛCDM scenario (Ooba et al. 2018a; Park & Ratra 2018a,b), provided one uses a physically con-sistent non-flat model power spectrum of energy density inhomogeneities (Ratra & Peebles 1995; Ratra 2017) in theanalysis of the CMB data.In the standard ΛCDM model (Peebles 1984), dark energy, taken to be the cosmological constant Λ, dominatesthe current cosmological energy budget and powers the currently accelerating cosmological expansion. Cold darkmatter (CDM) is the next largest contributor to the current energy budget, followed by baryonic matter and smallcontributions from neutrinos and photons. The standard ΛCDM model assumes flat spatial geometry. For reviews ofthe standard and related scenarios, see Ratra & Vogeley (2008), Martin (2012), Joyce et al. (2016), Huterer & Shafer(2017) and references therein.It is conventional to parameterize the standard flat-ΛCDM model in terms of six variables: Ω b h and Ω c h , thecurrent values of the baryonic and cold dark matter density parameters multiplied by the square of the Hubble constant(in units of 100 km s − Mpc − ); θ , the angular diameter distance as a multiple of the sound horizon at recombination; τ ,the reionization optical depth; and A s and n s , the amplitude and spectral index of the (assumed) power-law primordialscalar energy density inhomogeneity power spectrum, (Planck Collaboration 2016a). The predictions of the flat-ΛCDMmodel are largely consistent with most available observational constraints (Planck Collaboration 2016a, and referencestherein).There are suggestions that flat-ΛCDM might not be as compatible with more recent, larger compilations of measure-ments (Sol`a et al. 2017a,b, 2018, 2017c; Zhang et al. 2017) that might be more consistent with dynamical dark energymodels. These include the simplest, physically consistent, seven parameter flat- φ CDM model in which a scalar field φ with potential energy density V ( φ ) ∝ φ − α is the dynamical dark energy (Peebles & Ratra 1988; Ratra & Peebles1988) and α > For closedspatial hypersurfaces Hawking’s prescription for the quantum state of the universe (Hawking 1984) can be used toconstruct a closed inflation model (Ratra 1985, 2017). The initial closed inflation epoch linear perturbation solutionsconstants of integration are determined from closed de Sitter invariant quantum mechanical initial conditions in theLorentzian section of the closed de Sitter space that follow from Hawking’s prescription that the quantum state ofthe universe only include field configurations regular on the Euclidean (de Sitter) sphere sections (Ratra 1985, 2017).These initial conditions are the unique de Sitter invariant ones. Zero-point quantum-mechanical fluctuations duringclosed inflation provide a late-time energy density inhomogeneity power spectrum that is not a power law (Ratra 2017);it is a generalization to the closed case (White & Scott 1996; Starobinsky 1996; Zaldarriaga et al. 1998; Lewis et al.2000; Lesgourgues & Tram 2014) of the flat-space scale-invariant spectrum. Observational consequences of the open inflation model are discussed in Kamionkowski et al. (1994), G´orski et al. (1995), G´orski et al.(1998), Ratra et al. (1999), and references therein.
We recently analyzed the Planck 2015 CMB anisotropy data in the non-flat ΛCDM inflation model, using theconsistent power spectrum for the non-flat case (Ooba et al. 2018a). In this paper we examine what constraints thesedata place on a model in which dark energy is parameterized in terms of a fluid, the so-called XCDM parameterization.Here the dark energy fluid pressure and energy density are related via p X = w ρ X , where w is the equation of stateparameter and is taken to be < − /
3. It is well known that such a fluid is unstable and to stabilize it we arbitrarilyrequire that spatial inhomogeneities in the fluid propagate at the speed of light.Compared to the six parameter flat-ΛCDM inflation model discussed above, in the non-flat case there is no simpletilt option, so n s is no longer a free parameter and is replaced by the current value of the curvature density parameterΩ k which results in the six parameter non-flat ΛCDM model (Ooba et al. 2018a). Here we replace the cosmologicalconstant Λ by the dark energy fluid parameterized by w , which is a new free parameter, resulting in the sevenparameter non-flat XCDM inflation model. Of course, this is not a physical model, and it is also unable to properlymimic the dark energy evolution of the scalar field dynamical dark energy φ CDM model (Podariu & Ratra 2001).Nevertheless, it is a simple parameterization of dynamical dark energy that is relatively straightforward to use in acomputation and that is worth exploring as a first, and hopefully not misleading, attempt to gain some insight intohow dark energy dynamics and non-zero spatial curvature jointly influence the CMB anisotropy data constraints.We note here that there is a physically consistent non-flat seven parameter scalar field dynamical dark energy model(Pavlov et al. 2013); because the scalar field inhomogeneities must be accounted for in this model, the computationaldemands in φ CDM are much more significant than in the XCDM case we study here.In this paper we use the Planck 2015 CMB anisotropy data to constrain this seven parameter non-flat XCDMinflation model. We find in this model that the Planck 2015 CMB anisotropy data used jointly with baryon acousticoscillation (BAO) distance measurements do not require that spatial hypersurfaces be flat. The data favor a slightlyclosed model. These results are consistent with our earlier analysis of the six parameter non-flat ΛCDM inflationmodel (Ooba et al. 2018a).In our analyses here we use different CMB anisotropy data combinations (Planck Collaboration 2016a). For CMBdata alone, we find that the best-fit non-flat XCDM model, for the TT + lowP + lensing Planck 2015 data, hasspatial curvature density parameter Ω k = − . +0 . − .
005 +0 . − . (1 and 2 σ error bars) and is mildly closed. Whenwe include the BAO data that Planck 2015 used, we find for the TT + lowP + lensing CMB anisotropy case thatΩ k = − . ± . ± . ℓ CMB temperature anisotropy C ℓ power than does the best-fit six parameter tilted,spatially-flat ΛCDM model, and so appear to be in slightly better agreement with the low- ℓ temperature C ℓ observations(less so when the BAO data are included in the analysis). Overall, however, the best-fit seven parameter closed XCDMinflation model does not do better, and probably does a little worse, than the best-fit six parameter closed ΛCDMinflation model we studied earlier (Ooba et al. 2018a).In Sec. II we summarize the methods we use. Our parameter constraints are plotted, tabulated, and discussed inSec. III, where we also attempt to determine how well the best-fit closed-XCDM case fits the data. Conclusions aregiven in Sec. IV. METHODSFor our non-flat model analyses we use the open and closed inflation model energy density inhomogeneity powerspectrum (Ratra & Peebles 1995; Ratra 2017). Figure 1 compares closed XCDM and ΛCDM inflation model powerspectra and a tilted flat-ΛCDM inflation model power spectrum. We use the public numerical code CLASS (Blas et al.2011) to compute the angular power spectra of the CMB temperature, polarization, and lensing potential anisotropies.Our parameter estimations are carried out using the Markov chain Monte Carlo method program Monte Python(Audren et al. 2013).We assume flat priors for the cosmological parameters over the ranges100 θ ∈ (0 . , , Ω b h ∈ (0 . , . , Ω c h ∈ (0 . , . ,τ ∈ (0 . , . , ln(10 A s ) ∈ (0 . , , Ω k ∈ ( − . , . , w ∈ ( − , . . (1) A similar analysis, with similar conclusions, has been carried out for the non-flat φ CDM inflation model (Ooba et al. 2018b;Park & Ratra 2018c). We study the seven parameter flat XCDM inflation model elsewhere (Ooba et al. 2018c). Zhao et al. (2017) and Di Valentino et al. (2017) have recently studied other variants of flat XCDM designed to resolve cosmologicaltensions. -5 -4 -3 -2 -1 q [Mpc −1 ] -13 -12 -11 -10 -9 -8 P ( q ) [ M p c ] Flat TT + lowP + lensingnonflat TT + lowP + lensingnonflat XCDM TT + lowP + lensing
Figure 1.
Best-fit (see text) gauge-invariant fractional energy density inhomogeneity power spectra. The blue line correspondsto the tilted flat-ΛCDM model of Planck Collaboration (2016a). In the closed case, wavenumber q ∝ A + 1 where the eigenvalueof the spatial Laplacian is − A ( A + 2), A is a non-negative integer, A = 0 corresponds to the constant zero-mode on the threesphere, the power spectrum vanishes at A = 1, and the points on the red and green curves correspond to A = 2 , , , ... , seeeqns. (8) and (203) of Ratra (2017). On large scales the power spectra for the best-fit closed XCDM (red curve) and ΛCDM(green curve) models are suppressed relative to that of the best-fit tilted flat-ΛCDM model. The P ( q ) are normalized using thebest-fit values of A s at the pivot scale k = 0 .
05 [Mpc − ]. The CMB temperature and the effective number of neutrinos are taken to be T CMB = 2 . N eff = 3 .
046 with one massive (0.06 eV) and two massless neutrino species in a normal hierarchy. Theprimordial helium fraction Y He is inferred from standard Big Bang nucleosynthesis as a function of the baryon density.We constrain model parameters by comparing model predictions to the CMB angular power spectrum data fromthe Planck 2015 mission (Planck Collaboration 2016a) and the BAO distance measurements of the 6dF Galaxy Survey(Beutler et al. 2011), the Baryon Oscillation Spectroscopic Survey (LOWZ and CMASS) (Anderson et al. 2014), andthe Sloan Digital Sky Survey main galaxy sample (MGS) (Ross et al. 2015). RESULTSIn this section we present results of our parameter estimation computations and our attempt to determine how wellthe best-fit seven parameter closed-XCDM inflation case does relative to the best-fit six parameter tilted flat-ΛCDMinflation model (Planck Collaboration 2016a) and the best-fit six parameter closed-ΛCDM inflation model (Ooba et al.2018a). Table 1 lists central values and 68 .
27% (1 σ ) limits on the cosmological parameters from the various CMB andBAO data sets we use. Figure 2 shows two-dimensional constraint contour and one-dimensional likelihood (determinedby marginalizing over all other parameters) plots, derived from the two CMB anisotropy data sets, both excludingand including the BAO data. Figure 3 plots the CMB temperature anisotropy angular power spectra for the best-fitnon-flat XCDM cases determined from the two different CMB anisotropy data sets (as well as for the non-flat andtilted spatially-flat ΛCDM models), excluding and including the BAO data, Figure 4 shows 68 .
27% and 95 .
45% (2 σ )confidence contours in the σ –Ω m plane, after marginalizing over the other parameters, for the non-flat XCDM inflationcases as well as for one spatially-flat tilted ΛCDM inflation model, without and with the BAO data. Table 1. b h . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . θ . ± . . ± . . ± . . ± . τ . ± .
031 0 . ± .
022 0 . ± .
017 0 . ± . A s ) 3 . ± .
062 3 . ± .
044 3 . ± .
035 3 . ± . k − . ± . − . ± . − . ± . − . ± . w − . ± . − . ± . − . ± . − . ± . H [km/s/Mpc] 55 . ± .
76 73 . ± .
29 69 . ± .
16 68 . ± . m . ± .
24 0 . ± .
14 0 . ± .
02 0 . ± . σ . ± .
134 0 . ± .
162 0 . ± .
034 0 . ± . S ≡ σ p Ω m / . . ± .
104 0 . ± .
084 0 . ± .
019 0 . ± . Figure 2. .
27% and 95 .
45% confidence level contours for the non-flat XCDM inflation model using various data sets, withthe other parameters marginalized.
From the analysis without the BAO data, the spatial curvature density parameter is measured to beΩ k = − .
021 +0 . − .
050 (95 . , TT + lowP + lensing) . (2)The left hand panels of Fig. 3 show the CMB temperature anisotropy C ℓ of the best-fit non-flat XCDM inflationcases for the 2 different CMB data sets. It appears that these models fit the low- ℓ C ℓ observations better than doesthe spatially-flat tilted ΛCDM case of Planck Collaboration (2016a), with the higher- ℓ C ℓ data not as well fit by thenon-flat models. On large scales the fractional energy density inhomogeneity power spectrum for the best-fit closedXCDM and ΛCDM inflation models are suppressed relative to that of the best-fit tilted flat-ΛCDM model, as shownin Fig. 1 for the TT + lowP + lensing data. The low- ℓ C ℓ of Fig. 3 depend on this small wavenumber part of thepower spectrum, but other effects, such as the usual and integrated Sachs-Wolfe effects, also play an important rolein determining the C ℓ shape. The left panel of Fig. 4 shows σ –Ω m constraint contours for the 2 non-flat XCDMcases (as well as for one spatially-flat tilted ΛCDM model). With CMB lensing included, we find that our non-flatseven parameter XCDM inflation model reduces the tension between the CMB observations and the weak lensing data,compare Fig. 4 here to Fig. 18 of Planck Collaboration (2016a). From the analysis also including the BAO data, the spatial curvature density parameter is measured to beΩ k = − . ± .
006 (95 . , TT + lowP + lensing + BAO) . (3)In contrast to the Planck 2015 results (Planck Collaboration 2016a), our physically-consistent non-flat XCDM inflationmodel is not forced to be flat even with the BAO data included in the analysis. This case is about 3 σ away from flat.The right panels of Fig. 3 show temperature C ℓ plots for the best-fit non-flat XCDM models analyzed using the 2different CMB data sets and including the BAO data. Including the BAO data does somewhat degrade the fit in thelow- ℓ region compared with results from the analyses without the BAO data. We find that including the BAO dataalso worsens the σ –Ω m plane tension between the CMB and weak lensing constraints, see the right hand panel of Fig.4. It is interesting that the Ω m and H constraints listed in Table 1 are very consistent with estimates for theseparameters from most other data. See Chen & Ratra (2003) for the density parameter. The most recent medianstatistics analysis of H measurements gives H = 68 ± . − Mpc − (Chen & Ratra 2011), consistent with earlierestimates (Gott et al. 2001; Chen et al. 2003). Many recent H determinations from BAO, Type Ia supernovae, Hubbleparameter, and other measurements are consistent with these results (Calabrese et al. 2012; Hinshaw et al. 2013;Sievers et al. 2013; Aubourg et al. 2015; L’Huillier & Shafieloo 2017; Lukovi´c et al. 2016; Chen et al. 2017; Wang et al.2017; Lin & Ishak 2017). It is however well known that local measurements of the expansion rate find a higher H value. Freedman et al. (2012) give H = 74 . ± . − Mpc − while Riess et al. (2016) report H = 73 . ± . − Mpc − , somewhat higher than the H = 68 . ± .
93 km s − Mpc − of the last column of Table 1.In addition, we note that many analyses based on a number of different observations also do not rule out non-flat darkenergy models (Farooq et al. 2015; Sapone et al. 2014; Li et al. 2014; Cai et al. 2016; Chen et al. 2016; Yu & Wang2016; L’Huillier & Shafieloo 2017; Farooq et al. 2017; Li et al. 2016; Wei & Wu 2017; Rana et al. 2017; Yu et al. 2018;Mitra et al. 2018; Ryan et al. 2018). Table 2.
Minimum χ values for the best-fit closed-XCDM (and tilted flat-ΛCDM) inflation model.Data sets χ d.o.f.TT+lowP 11277 (11262) 188 (189)TT+lowP+lensing 11292 (11272) 196 (197)TT+lowP+BAO 11289 (11266) 192 (193)TT+lowP+lensing+BAO 11298 (11277) 200 (201) Table 3. χ values for the best-fit closed-XCDM (and tilted flat-ΛCDM) inflation model.Data TT+lowP TT+lowP+lensing TT+lowP+BAO TT+lowP+lensing+BAOCMB low- ℓ ℓ ℓ Note that, as discussed in Sec. 5 of Planck Collaboration (2016a), because of the incorrect amount of lensing in the TT powerspectrum, this tension is also alleviated when the Planck lensing reconstruction data is used (Renzi et al. 2018). For another option seeDi Valentino et al. (2018).
10 100 1000ℓ10010006000 T C M B ℓ ( ℓ + ) C TT ℓ / ( π ) [ µ K ] (a)
10 100 1000ℓ10010006000 T C M B ℓ ( ℓ + ) C TT ℓ / ( π ) [ µ K ] (b) T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π ) [ ℓ K ] δ ( T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π )) [ ℓ K ] (c) T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π ) [ ℓ K ] δ ( T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π )) [ ℓ K ] (d) T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π ) [ ℓ K ] Flat TT + lowP + lensingnonflat ΛCDM TT + lowP + lensingnonflat XCDM TT + lowPnonflat XCDM TT + lowP + lensingPlanck Data
30 500 1000 1500 2000 2500(cid:4)-80-4004080 δ ( T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π )) [ ℓ K ] (e) T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π ) [ ℓ K ] Flat TT + lowP + lensing + BAOnonflat ΛCDM TT + lowP + lensing + BAOnonflat XCDM TT + lowP + BAOnonflat XCDM TT + lowP + lensing + BAOPlanck Data
30 500 1000 1500 2000 2500(cid:4)-80-4004080 δ ( T C M B (cid:4) ( (cid:4) + ) C TT (cid:4) / ( π )) [ ℓ K ] (f) Figure 3.
Temperature C ℓ for the best-fit non-flat XCDM, non-flat ΛCDM and spatially-flat tilted ΛCDM (gray solid line)models. Linestyle information are listed in the boxes in the two bottom panels. Planck 2015 measurements are shown as blackpoints with error bars. Left panels (a), (c) and (e) are from CMB data alone analyses, while right panels (b), (d) and (f) analysesalso include BAO data. Top panels show the all- ℓ region. Middle panels show the low- ℓ region C ℓ and residuals. Bottom panelsshow the high- ℓ region C ℓ and residuals. Figure 4. .
27% and 95 .
45% confidence contours in the σ –Ω m plane. Left panel shows Planck data alone contours whileright panel contours also account for BAO data. It is vital to understand how well the best-fit closed-XCDM inflation case does relative to the best-fit tilted flat-ΛCDM model in fitting the data. As for the closed-ΛCDM model (Ooba et al. 2018a), we are unable to resolve this in aquantitative manner, although qualitatively, overall, the best-fit closed-XCDM model does not do as well as the best-fittilted flat-ΛCDM model. It also appears to not do as well as the best-fit closed-ΛCDM inflation model (Ooba et al.2018a), which has one fewer parameter.Table 2 lists the minimum χ = − max ) determined from the maximum value of the likelihood, for the 4 datasets we consider, for both the closed-XCDM and tilted flat-ΛCDM inflation models, as well as the number of (binneddata) degrees of freedom (d.o.f.). The d.o.f. are determined from combinations of 112 low- ℓ TT + lowP, 83 high- ℓ TT,8 lensing CMB (binned) measurements, 4 BAO observations, and 7 (or 6 for the tilted flat-ΛCDM) model parameters.It is likely that the large χ values are the result of the many nuisance parameters that have been marginalized over,since the tilted flat-ΛCDM model is said to be a good fit to the data. From this table we see that ∆ χ = 20(21) forthe closed-XCDM inflation case (196 (200) d.o.f.), relative to the tilted flat-ΛCDM (197 (201) d.o.f.), for the TT +low P + lensing (+ BAO) data. This might make the closed-XCDM case much less probable, however, one can seefrom the residual panels of Fig. 3 (e) & (f) that this ∆ χ appears to be a consequences of many small deviations, andnot of a few significant outliers.Though there are correlations in the data, it is useful to also consider a standard goodness of fit χ that makes useof only the diagonal elements of the correlation matrix. These χ ’s are listed in Table 3 for the 4 data sets we studyand for both the closed-XCDM and tilted flat-ΛCDM inflation models. From Table 3 for the TT + low P + lensingdata, the χ per d.o.f. is 273/196 (227/197) for the closed-XCDM (tilted flat-ΛCDM) inflation model, and when BAOdata is included these become 294/200 (234/201). Again, while the closed-XCDM case is less favored than the tiltedflat-ΛCDM case, it is not clear how to assess the quantitative significance of this. In addition to the discussion at theend of the previous paragraph, here we have also ignored all off-diagonal information in the correlation matrix, so it ismeaningless to compute standard probabilities from such χ ’s. In summary, while the best-fit closed-XCDM inflationcase appears less favored, it might be useful to perform a more thorough analysis of the model.We additionally compute the Bayesian evidence for each of the models we consider, by using the public numericalcode MCEvidence (Heavens et al. 2017). We then compute the natural logarithm of the ratio of the Bayesian evidenceof our model to that of the spatially-flat tilted ΛCDM model, which we write as ln( B ). For the TT + low P (+ lensing)case, we get ln( B ) = +4 .
08 ( − . B ) = − .
98 ( − . | ln B | > | ln B | > CONCLUSIONWe determine Planck 2015 CMB data constraints on the physically consistent seven parameter non-flat XCDMmodel with inflation-generated energy density inhomogeneity power spectrum. This is a first attempt to examinethe effect of the interplay between dark energy dynamics and spatial curvatures on constraints derived from CMBanisotropy measurements. We again draw attention to the fact that the X-fluid dark energy part of the XCDM modelis only a parameterization, and not a physical model, and hope it does not lead to misleading conclusions.Unlike the case for the seven parameter non-flat tilted ΛCDM model with the physically-inconsistent power spectrumused in Planck Collaboration (2016a), we discover that CMB anisotropy data do not force spatial curvature to vanishin our non-flat XCDM inflation model with a physically-consistent power spectrum. This is consistent with what wefound in the six parameter non-flat ΛCDM inflation model (Ooba et al. 2018a), where spatial curvature contributesabout 2 % to the present energy budget of the closed model that best fits the Planck TT + lowP + lensing data.These closed inflation models are more consistent with the low- ℓ C ℓ temperature observations and the weak lensing σ measurements than is the best fit spatially-flat tilted ΛCDM, but they do worse at fitting the higher- ℓ C ℓ data. It might be useful to reexamine the issue of possible small differences in cosmological parameter constraints derivedfrom higher- ℓ and from lower- ℓ Planck 2015 CMB anisotropy data (Addison et al. 2016; Planck Collaboration 2016b),by using the non-flat XCDM model. In addition we note that in the tilted flat ΛCDM model there seem to beinconsistencies between the higher- ℓ Planck and the South Pole Telescope CMB data (Aylor et al. 2017). Also of greatinterest would be a method for quantitatively assessing how well the best-fit tilted spatially-flat ΛCDM model and thebest-fit non-flat XCDM case fit the Planck 2015 CMB anisotropy (and other cosmological) measurements.Unlike the analysis for the seven parameter non-flat tilted ΛCDM model in Planck Collaboration (2016a), alsoincluding the BAO data in the mix does not force our seven parameter non-flat XCDM case to be flat; in fact,Ω k = − . ± .
006 at 2 σ and is about 3 σ away from flat. In this case the improved agreement with the low- ℓ C ℓ temperature data and the weak lensing σ constraints are not as good compared with the results from the analysesusing only the CMB observations. We note that the BAO and CMB data are from very different redshifts and it ispossible that a better model for the intervening epoch or an improved understanding of one or both sets of observationsmight alter this finding.Perhaps adding a small spatial curvature contribution, of order a percent, can improve the spatially-flat standardΛCDM model. However, it appears that the six parameter closed-ΛCDM inflation model might do a better job at this(Ooba et al. 2018a; Park & Ratra 2018a,b) than does the seven parameter closed-XCDM inflation model we studiedhere. A more thorough analysis of the non-flat ΛCDM and XCDM inflation models is needed to establish if eitheris viable and can help resolve some of the low- ℓ C ℓ issues as well as possibly the σ power issues. Perhaps non-zerospatial curvature might be more important for this purpose than is dark energy dynamics.ACKNOWLEDGMENTSWE acknowledge valuable discussions with C.-G. Park. This work is supported by Grants-in-Aid for ScientificResearch from JSPS (Nos. 16J05446 (J.O.) and 15H05890 (N.S.)). B.R. is supported in part by DOE grant DE-SC0019038. Deuterium abundance measurements mildly favor the flat case over the open one (Penton et al. 2018) while a compilation of all recentBAO, Type Ia supernova, and Hubble parameter data mildly favor closed spatial hypersurfaces (Park & Ratra 2018d).
Addison, G. E., et al. 2016, ApJ, 818, 132[arXiv:1511.00055]Anderson, L. et al. 2014, MNRAS, 441, 24 [arXiv:1312.4877]Aubourg, ´E. et al. 2015, Phys. Rev. D, 92, 123516[arXiv:1411.1074]Audren, B,, Lesgourgues, J., Benabed, K., & Prunet, S.2013, JCAP, 1302, 001 [arXiv:1210.7183]Aurich, R., & Steiner, F. 2002, MNRAS, 334, 735[arXiv:astro-ph/0109288]Aurich, R., & Steiner, F. 2003, Phys. Rev. D, 67, 123511[arXiv:astro-ph/0212471]Aurich, R., & Steiner, F. 2004, Int. J. Mod. Phys. D, 13,123 [arXiv:astro-ph/0302264]Aylor, K., Hou, Z., Knox, L., et al. 2017, ApJ, 850, 101[arXiv:1706.10286]Beutler, F. et al. 2011, MNRAS, 416, 3017[arXiv:1106.3366]Blas, D,, Lesgourgues, J., & Tram, T. 2011, JCAP, 1107,034 [arXiv:1104.2933]Cai, R.-G., Guo, Z.-K., & Yang, T. 2016, Phys. Rev. D, 93,043517 [arXiv:1509.06283]Calabrese, E., Archidiacono, M., Melchiorri, A., & Ratra,B. 2012, Phys. Rev. D, 86, 043520 [arXiv:1205.6753]Chen, G., Gott, J. R., & Ratra, B. 2003, PASP, 115, 1269[arXiv:astro-ph/0308099]Chen, G., & Ratra, B. 2003, PASP, 115, 1143[arXiv:astro-ph/0302002]Chen, G., & Ratra, B. 2011, PASP, 123, 1127[arXiv:1105.5206]Chen, Y., Kumar, S., & Ratra, B. 2017, ApJ, 835, 86[arXiv:1606.07316]Chen, Y., et al. 2016, ApJ, 829, 61 [arXiv:1603.07115]Clarkson, C., Cortˆes, M., & Bassett, B. 2007, JCAP, 0708,011 [arXiv:astro-ph/0702670]Crooks, J. L., et al. 2003, Astropart. Phys., 20, 361[arXiv:astro-ph/0305495]Di Valentino, E., Bøehm, C., Hivon, E., & Bouchet, F. R.2018, PhRvD, 97, 043513 [arXiv:1710.02519]Di Valentino, E., Melchiorri, A., Linder, E. V., & Silk, J.2017, PhRvD, 96, 023523 [arXiv:1704.00762]Farooq, O., Madiyar, F. R., Crandall, S., & Ratra, B. 2017,ApJ, 835, 26 [arXiv:1607.03537]Farooq, O., Mania, D., & Ratra, B. 2015, ApSS, 357, 11[arXiv:1308.0834]Fixsen, D. J. 2009, ApJ, 707, 916 [arXiv:0911.1955]Freedman, W. L., et al. 2012, ApJ, 758, 24[arXiv:1208.3281]Gong, Y., Wu, Q., & Wang, A. 2008, ApJ, 681, 27[arXiv:0708.1817] G´orski, K. M., et al. 1998, ApJS, 114, 1[arXiv:astro-ph/9608054]G´orski, K. M., Ratra, B., Sugiyama, N., & Banday, A. J.1995, ApJ, 444, L65 [arXiv:astro-ph/9502034]Gott, J. R. 1982, Nature, 295, 304Gott, J. R., Vogeley, M. S., Podariu, S., & Ratra, B. 2001,ApJ, 549, 1 [arXiv:astro-ph/0006103]Hawking, S. W. 1984, Nucl. Phys. B, 239, 257Heavens, A., Fantaye, Y., Mootoovaloo, A., et al. 2017,arXiv:1704.03472Hinshaw, G., et al. 2013, ApJS, 208, 19 [arXiv:1212.5226]Hlozek, R., Cortˆes, M., Clarkson, C., & Bassett, B. 2008,Gen. Relativ. Gravit., 40, 285 [arXiv:0801.3847]Huterer, D., & Shafer, D. L. 2017, arXiv:1709.01091Ichikawa, K., Kawasaki, M., Sekiguchi, T., & Takahashi, T.2006, JCAP, 0612, 005 [arXiv:astro-ph/0605481]Ichikawa, K., & Takahashi, T. 2006, Phys. Rev. D, 73,083526 [arXiv:astro-ph/0511821]Ichikawa, K., & Takahashi, T. 2007, JCAP, 0702, 001[arXiv:astro-ph/0612739]Ichikawa, K., & Takahashi, T. 2008, JCAP, 0804, 027[arXiv:0710.3995]Joyce, A., Lombriser, L., & Schmidt, F. 2016, Ann. Rev.Nucl. Part. Sci., 66, 95 [arXiv:1601.06133]Kamionkowski, M., Ratra, B., Spergel, D. N., & Sugiyama,N. 1994, ApJ, 434, L1 [arXiv:astro-ph/9406069]Kass, R. E., & Raftery, A. E. 1995, Journal of theAmerican Statistical Association, 90, 773Lesgourgues, J., & Tram, T. 2014, JCAP, 1409, 032[arXiv:1312.2697]Lewis, A., Challinor, A, & Lasenby, A, 2000, ApJ, 538, 473[arXiv:astro-ph/9911177]L’Huillier, B., & Shafieloo, A. 2017, JCAP, 1701, 015[arXiv:1606.06832]Li, Y.-L., Li, S.-Y., Zhang, T.-J., & Li, T.-P. 2014, ApJ,789, L15 [arXiv:1404.0773]Li, Z., Wang, G.-J., Liao, K., & Zhu, Z.-H. 2016, ApJ, 833,240 [arXiv:1611.00359]Lin, W., & Ishak, M. 2017, PhRvD, 96, 083532[arXiv:1708.09813]Lukovi´c, V. V., D’Agostino, R., & Vittorio, N. 2016,Astron. Astrophys., 595, A109 [arXiv:1607.05677]Martin, J., 2012, C. R. Physique, 13, 566 [arXiv:1205.3365]Mitra, S., Choudhury, T. R., & Ratra, B. 2018, MNRAS,479, 4566 [arXiv:1712.00018]Ooba, J., Ratra, B., & Sugiyama, N. 2018a, ApJ, 864, 80[arXiv:1707.03452]Ooba, J., Ratra, B., & Sugiyama, N. 2018b, ApJ, 866, 68[arXiv:1712.08617]1