Constraints on power law cosmology from cosmic chronometer, standard ruler, and standard candle data
CConstraints on power law cosmology from cosmic chronometer, standard ruler, andstandard candle data
Joseph Ryan ∗ Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66502
In this paper I compare the quality of the fit of a simple power law model of cosmic expansion tothe standard ΛCDM model. I analyze a data set consisting of cosmic chronometer, standard ruler,and standard candle measurements, finding that the ΛCDM model provides a better fit to mostcombinations of these data than does the power law model.PACS numbers: 98.80.-k
I. INTRODUCTION
There is now a broad consensus that the ΛCDM modelis sufficient to describe the dynamics of the Universe, onlarge scales, throughout most of its history [1]. Somegroups [2–15] have found evidence of small departuresfrom the ΛCDM model (such as dynamical dark energyor large-scale spatial curvature), but these findings havegenerally not risen to the level of significance necessary todethrone ΛCDM from its position as the standard modelof cosmology. More radical alternatives to the standardmodel are conceivable, however. One such alternative isthe power law model, in which the scale factor takes theform of a power law a ( t ) ∝ t β with a constant exponent β .One of the virtues of this model is its simplicity: it onlydepends on the single parameter β , and the functionalform t β is easy to integrate analytically when it appearsin the form (cid:82) dta ( t ) (as in the computation of the co-movingdistance scale). Additionally, the power law model with β ≥ β ≈ H ( z )) [16],gravitational lensing statistics [18], Type Ia supernova(SN Ia) [17, 19–21], baryon acoustic oscillation (BAO)[22, 23], quasar angular size (QSO) [24], and cluster gasmass fraction data [25] (though other data sets favor β ≈ β = 1 can produce the right amount of primordialhelium to match current observations [27, 28], and somay be able to account for the synthesis of other lightelements. These conclusions are challenged by the resultsof other studies, which find that β ≈ β favored by ∗ [email protected] primordial nucleosynthesis are clearly disjoint with thosefavored by low redshift measurements, and it is difficultto see how they can be reconciled without introducingextra complexity to the power law model (such as theaddition of a mechanism that forces β to change its valuebetween the two eras; see e.g. [31–35]). Given that thepower law model is intended to be a simpler alternativeto the ΛCDM model, such additional complexity seemsunjustified, and the power law model appears to be ruledout on these grounds.A defender of the power law model who does not wishto make the model more complex by introducing a time-variable β could attempt to save it by arguing that:1.) The findings of [20, 29, 30] are simply incorrect,and the power law exponent has the value β ≈ β ≈ We must be careful not to push this argument too far,however, because any scale factor a ( t ) can presumablybe approximated by a power law over some arbitrarilyshort time period. What is at issue is not whether theUniverse follows power law expansion during some (rela-tively) brief portion or portions of its history, but whetherit follows power law expansion throughout all (or most)of its history. If it can be shown that the power lawmodel fits low redshift observational data as well as orbetter than ΛCDM over some appreciable range of red-shifts, then option (2) is validated (and option 1 maybe validated as well, if one can marshal a strong argu- For recent efforts to provide an account of primordial nucleosyn-thesis within the power law model, see [36, 37]. In [38], one of the earliest papers on the subject, the authorproposes that, if a hypothetical form of matter called “K-matter”were to dominate the energy budget at late times, this would leadto a “coasting” cosmic expansion with β = 1 (with β taking ondifferent values in earlier eras). a r X i v : . [ a s t r o - ph . C O ] F e b TABLE I. Fits to power law exponent from other low redshift measurements.Reference β Data type(s) used[16] 1 . +0 . − . H ( z )1 . +0 . − . SN Ia[18] 1 . ± . . +0 . − . [19] 1 . ± .
043 SN Ia[26] 1 . ± .
15 SN Ia1 . ± . . . ± . . +0 . − . H ( z )1 . +0 . − . SN Ia[21] 1 . +0 . − . H ( z )1 . +0 . − . SN Ia[17] 1 . +0 . − . SN Ia[22] 0 .
93 BAO1 . .
56 SN Ia[23] 0 . ± .
019 BAO1 . ± .
13 SN Ia[25] 1 . ± .
05 Cluster gas mass fraction ment against the findings of [20, 29, 30]). If, on the otherhand, the power law model fails to provide a good fit tothe available low redshift data, then both (1) and (2) arefalsified.A few studies [21–23] have been conducted along theselines. These studies find that, when the power law modelis fitted to multiple independent data sets ( H ( z ) aloneand H ( z ) + BAO + SNe Ia + CMB in [21], BAO +SNe Ia in [22], and BAO + SNe Ia + CMB in [23]), itperforms poorly compared to ΛCDM. Here I continue inthis vein by fitting the power law and ΛCDM modelsto a data set consisting of cosmic chronometer, standardruler, and standard candle data, some of which have notyet been used to test the power law model (see Sec. III fora description of the data). I use simple model comparisonstatistics (the same as those used in [21–23]; see Sec. IV)to compare the quality of the fit in both cases. I discussmy results in Sec. V and draw my conclusions in Sec.VI. II. THEORY
In the power law model, the scale factor a ( t ) takes theform a ( t ) = kt β , (1)where k and β are constants. From the definition of red-shift, a a ( t ) := 1 + z ( a is the current value of the scalefactor and z is the redshift) and eq. (1), we can write a kt β = 1 + z, (2) from which it follows that1 t = (cid:20) ka (1 + z ) (cid:21) /β . (3)The definition of the Hubble parameter, H ( t ) := ˙ a ( t ) a ( t ) ,with the overdot denoting the time derivative, implies H ( t ) = βt . Therefore eq. (3) can be written in the form H ( z ) = H (1 + z ) /β , (4)where I have defined the present value of the Hubbleconstant to be H := β (cid:16) ka (cid:17) /β . The power law modeltherefore has two free parameters: H and β .My fiducial model in this paper is the spatially flatΛCDM model. In this model, at late times, the Hubbleparameter can be written as a function of the redshift z in the form H ( z ) = H (cid:113) Ω m (1 + z ) + 1 − Ω m , (5)where H is the Hubble constant and Ω m is the currentvalue of the non-relativistic matter density. The ΛCDMmodel also has two free parameters: H and Ω m . Be-cause of the relatively low redshifts of the data I use (seeTable II) I neglect the contribution that radiation makesto the energy budget.The data that I use depend on several kinds of dis-tance measurements (see Sec. III). These are the Hubbledistance D H ( z ) = cH ( z ) , (6)the transverse co-moving distance D M ( z ) = cH (cid:90) z dz (cid:48) E ( z (cid:48) ) , (7)where E ( z ) := H ( z ) /H , the angular diameter distance D A ( z ) = D M ( z )1 + z , (8)the volume-averaged angular diameter distance D V ( z ) = (cid:20) czH D ( z ) E ( z ) (cid:21) / , (9)and the luminosity distance D L ( z ) = (1 + z ) D M ( z ) , (10)as defined in [39, 40]. Note that D M ( z ) only has the formshown in eq. (7) in the special case that the Universe isspatially flat on large scales (which I assume in this paper,for both models I study); for open and closed universesthe integral on the right-hand side is more complicated. III. DATA
In Table II I list the types of measurements I use, thenumber of measurements of each type, and the redshiftranges within which the measurements lie. The cosmicchronometer data consist of measurements of the Hub-ble parameter as a function of the redshift z ( H ( z )), andare listed in [5]; see that paper for more details, and forreferences to the original studies from which these mea-surements were taken. To fit the power law and ΛCDMmodels to the cosmic choronometer data, I compute H ( z )theoretically using eqs. (4) and (5).I use two sets of standard ruler measurements in thispaper. The first set consists of measurements of the quan-tities H ( z ), D H ( z ), D M ( z ), D A ( z ), and D V ( z ), definedin eqs. (4-9), and scaled by the value that the soundhorizon r s takes at the baryon drag epoch. These mea-surements are the same as those listed in Table 1 of [41].See that paper, as well as [2–5] for more details, and forreferences to the original literature. To compute the sizeof the sound horizon, I use the approximate formula r s = 55 .
154 exp[ − . ν h + 0 . ](Ω b0 h ) . (Ω m h − Ω ν h ) . Mpc , (11)where Ω m , Ω b , and Ω ν are the dimensionless energydensity parameters of non-relativistic matter, of baryons,and of neutrinos, respectively, and h := H /
100 km s − Mpc − [42]. Following [43], I set Ω ν = 0 . m and Ω b h )when the power law model is fitted to data combinationscontaining BAO data, and one additional free parame-ter (Ω b h ) when the ΛCDM model is fitted to the samedata combinations. The second set of standard ruler dataconsists of measurements of the angular sizes θ obs , in mil-liarcseconds (mas), of intermediate-luminosity quasars(QSO). The angular size of a quasar can be computedtheoretically via θ th ( z ) = l m D A ( z ) , (12) where D A ( z ) is given by eq. (8) and l m = 11 . ± . θ obs todetermine the quality of the fit of the given model tothe QSO data. The QSO angular size measurements arelisted, and l m is determined, in [44]; see that paper and[4] for discussion and details.I use two sets of standard candle data in this paper.The first set consists of measurements of the luminosities,fluxes, and velocity dispersions of HII starburst galaxies(HIIG), from which the distance moduli of these galaxiescan be computed. The HIIG data consist of a low redshift(0 . ≤ z ≤ . . ≤ z ≤ . have been used in several studies to constraincosmological parameters [2, 3, 41, 46–49]. See [3] for adetailed description of how the distance modulus can becomputed. Briefly, if one knows the luminosity L , flux f , and velocity dispersion σ of an HII galaxy, one canuse these quantities to compute a distance modulus µ obs .This quantity can then be compared to the theoreticaldistance modulus µ th = 5log D L ( z ) + 25 , (13)where D L ( z ) is given by eq. (10), to determine the qualityof the model’s fit to the data. The second set of stan-dard candle data consists of measurements of the bolo-metric fluence S bolo and observed peak energy E p , obs of119 gamma-ray bursts from [50] (GRB). Given a knowl-edge of the bolometric fluence of a source, one can com-pute the energy radiated isotropically in the source’s restframe E iso = 4 πD z S bolo . (14)GRBs can be standardized through the Amati relation[51, 52] log E iso = a + b log [(1 + z ) E p , obs ] , (15)which connects the observed peak energy of a given GRBto its isotropic radiated energy (here a and b are freeparameters which I vary when fitting the power law andΛCDM models to the GRB data). The GRB likelihoodfunction also contains a parameter which describes theextrinsic scatter of the GRBs in the sample ( σ ext ) [53]. Asin [2], I vary this parameter freely when fitting the powerlaw and ΛCDM models to the GRB data. By comparingthe value of log E iso as computed from eq. (14) to thatcomputed from eq. (15), one can determine the quality ofthe model fit. For more details about the GRB analysis,see [2, 54]. Private communications, 2019 and 2020.
TABLE II. Data used in this paper.Data type Number of data points Redshift range H ( z ) 31 0 . ≤ z ≤ . . ≤ z ≤ . . ≤ z ≤ . . ≤ z ≤ . . ≤ z ≤ . There is a little overlap between the cosmic chronome-ter data I use in this paper and those that have previouslybeen used to constrain the parameters of the simple (con-stant β ) power law model. In particular, such data wereused in [16, 20, 21]. Many of the measurements these au-thors used are the same as mine, although I use a larger,more up-to-date set. I use a different sample of QSOdata than does [24], and my BAO measurements haveall been updated relative to those of [22, 23, 26]. To myknowledge, neither HIIG data nor GRB data have beenused to constrain the parameters of power law models.Because these data are independent of the H ( z ), BAO,and QSO data sets, I obtain tight constraints on the pa-rameters of the power law model when I fit it to thesedata in combination with the H ( z ), BAO, and QSO data(see Sec. V). IV. METHODS
The methods that I use to compare the power lawmodel to the ΛCDM model are largely the same as meth-ods that have been used previously in [2–5, 21–23, 41, 55],which I briefly summarize here. For each combination ofdata that I study, I compute the quantity χ := − L max (16)where the likelihood function L depends on the param-eters of the model under consideration. The likelihoodfunction takes a different form depending on the datacombination that is used to compute it; these forms aredescribed in [2] and [3]. For models having the samenumber of parameters, the best-fitting model to the datais that which has a smaller value of χ . As in [2, 3], Iuse the Python module emcee [56] to sample the likeli-hood function L , and I use the Python module getdist [57] both to generate the one- and two-dimensional likeli-hood contours shown in left and right panels of Fig. 1 andto compute the one-dimensional marginalized best-fittingvalues (sample means) and 68% uncertainties (two-sidedlimits) of the model parameters. Some of the BAO data that I use are correlated, so it is necessaryto take their covariance matrices into account when computing χ . See [41] and [4] for the covariance matrices of the corre-lated data. When comparing models with different numbers of pa-rameters, the χ function is not necessarily the mostinformative statistic to use, because it gives the sameweight to simple models that it gives to complex models.For this reason, I also use the corrected Akaike Informa-tion Criterion:AICc := AIC + 2 n ( n + 1) N − n − , (17)where AIC := χ + 2 n, (18)is the Akaike Information Criterion (suitable in the limitthat N >> n ), and the Bayes Information Criterion:BIC := χ + n ln N. (19)[58]. In the equations above, n is the number of modelparameters and N is the number of data points. TheAICc and BIC punish models that have a greater numberof parameters in favor of models with fewer parameters.In this sense, the AICc and BIC provide a quantitativebasis for choosing which model, among a set of models,provides the most parsimonious fit to a given set of data.
V. RESULTS AND DISCUSSION
The best-fitting values of the parameters of the powerlaw model (namely, those that minimize the χ function),are recorded in columns 2-8 of Table III. The number ofdegrees of freedom, ν := N − n (20)is recorded in column 9 of this table. Columns 10-12record, respectively, the minimum value of the reduced χ function, and the minimum values of the AICc and BIC.Similarly, the best-fitting values of the parameters of the In previous work [2–5, 41], my collaborators and I used the AICand BIC to compare the quality of cosmological model fits todata. Here I use the AICc in place of the AIC because the AICcis more appropriate for smaller data sets (like the H ( z ) and BAOsets), because it approaches the AIC in the limit that N is large,and to facilitate the comparison of my results with the results of[22, 23], both of which used the AICc in their analyses. TABLE III. Best-fitting parameters of the power law model.Data type H (km s − Mpc − ) β Ω m Ω b h a b σ ext ν χ /ν AICc BIC H ( z ) 61.92 0.9842 - - - - - 29 0.5721 21.02 23.46BAO 89.78 0.9206 0.6192 0.03819 - - - 7 1.513 25.26 20.18QSO 61.83 0.9673 - - - - - 118 2.991 357.1 362.6BAO+QSO 60.57 0.9213 0.2030 0.07690 - - - 127 2.864 372.0 383.2GRB 72.34 0.7530 - - 49.99 1.115 0.4010 114 1.138 140.3 153.7HIIG 70.99 1.251 - - - - - 151 2.725 415.5 421.5GRB+HIIG 70.31 1.158 - - 50.12 1.157 0.4066 267 2.039 554.7 572.5All Data 63.06 0.9470 0.2234 0.06706 50.16 1.144 0.4025 427 2.229 966.2 994.4 ΛCDM model are recorded in columns 2-7 of Table IV,with the number of degrees of freedom in column 8, theminimum value of the reduced χ function in column 9,and the minimum values of the AICc and BIC in columns10 and 11, respectively.In Table V, in columns 2 and 3, I record the samplemeans and two-sided uncertainties of the marginalizedparameters of the power law model (I exclude the param-eters Ω m , Ω b h , a , b , σ ext from this table because theseare nuisance parameters for the power law model). Incolumn 4 I record the sample mean and two-sided uncer-tainties (computed from the sample mean and two-sideduncertainties of β ) of the current value of the decelerationparameter q = 1 β − . (21)In column 5 I record ∆ χ , which I define as the differ-ence between the value of χ as computed within thepower law model for a given data combination, and thevalue of χ as computed within the ΛCDM model forthe same data combination. The relative probabilities e − ∆AICc / and e − ∆BIC / of the power law model I recordin columns 6 and 7, where ∆AICc and ∆BIC are definedin the same way as ∆ χ . In columns 2 and 3 of TableVI, I record the sample means and two-sided uncertain-ties of the marginalized parameters of the ΛCDM model,excluding the nuisance parameters Ω b h , a , b , and σ ext .In column 4 of Table VI I record the sample mean andtwo-sided uncertainties (computed from the sample meanand two-sided uncertainties of Ω m ) of the current valueof the deceleration parameter q = Ω m − Ω Λ = 32 Ω m − . (22)The prior probabilities of all parameters are flat, andnon-zero within the ranges 20 km s − Mpc − ≤ H ≤
100 km s − Mpc − , 0 . ≤ β ≤
4, 0 . ≤ Ω m ≤ . . ≤ Ω b h ≤ .
1, 40 ≤ a ≤
60, 0 ≤ b ≤
5, and0 ≤ σ ext ≤ H ( z ) data are shown as dotted blue curves, those associated with the BAO + QSO data combination areshown as dash-dotted red curves, those associated withthe GRB + HIIG combination are shown as dashed greencurves, and those associated with the combination of allthe data are shown as solid black curves (I combine thestandard ruler and standard candle data in these plotsto reduce visual clutter). The right panel of Fig. 1shows the two-dimensional confidence contours and one-dimensional likelihoods of the ΛCDM model, for the samedata combinations.From the marginalized parameter fits in Table V, Ifind that the best-fitting value of β from the H ( z ), QSO,GRB, and HIIG data is consistent with β = 1 to within1-2 σ , in agreement with many of the studies quoted inTable I. This translates to the best-fitting value of q being within 1-2 σ of q = 0 for each of these data sets,consistent with a coasting universe. The BAO data, how-ever, are not consistent with β = 1, the best-fitting valueof β for this data set being more than 4 σ away from unity.This means, as reflected in the best-fitting q value, thatwhen the power law model is fitted to the BAO data,these data favor a slowly decelerating universe (ratherthan a coasting one) to more than 4 σ . The BAO + QSOcombination also favors a slowly decelerating universe tomore than 4 σ . When these data are combined with the H ( z ), GRB, and HIIG data, the error bars on β and q tighten, and the central values of these parametersmove slightly closer to β = 1 and q = 0, respectively,though the best-fitting value of q is still inconsistentwith a coasting universe to more than 3 σ (see also Fig.1).From Tables III and IV, we can see that the best-fittingpower law model has greater χ /ν , AICc, and BIC valuesthan the best-fitting ΛCDM model across all data com-binations, except when these models are fitted to GRBdata alone. In this case, the power law model providesa slightly better fit to the data. When we examine therelative probabilities e − ∆AICc / and e − ∆BIC / in TableV, we find that the power law model produces a slightlybetter fit to the GRB data than does ΛCDM. This pref-erence for the power law model over the ΛCDM modelis unique to the GRB data, however, as all other datacombinations favor the ΛCDM model, with the relativeprobability of the power law model ranging from a highof 0 . . × − (full dataset). TABLE IV. Best-fitting parameters of the ΛCDM model.Data type H (km s − Mpc − ) Ω m Ω b h a b σ ext ν χ /ν AICc BIC H ( z ) 68.15 0.3196 - - - - 29 0.5000 18.93 21.37BAO 74.01 0.2967 0.03133 - - - 8 1.124 18.43 16.19QSO 68.69 0.3154 - - - - 118 2.983 356.1 361.6BAO+QSO 69.51 0.2971 0.02459 - - - 128 2.821 367.3 375.7GRB 75.65 0.7000 - 49.98 1.108 0.4012 114 1.141 140.6 154.0HIIG 71.81 0.2756 - - - - 151 2.720 414.8 420.8GRB+HIIG 71.45 0.2950 - 50.17 1.136 0.4035 267 2.031 552.5 570.3All Data 70.07 0.2949 0.02542 50.19 1.135 0.4040 428 2.148 931.6 955.8TABLE V. Marginalized best-fitting parameters and model comparison statistics for the power law model. The BAO dataalone do not place a tight upper limit on the best-fitting value of H , and the GRB data do not constrain H at all, so theselimits are omitted from the table.Data type H (km s − Mpc − ) β q ∆ χ e − ∆AICc / e − ∆BIC / H ( z ) 62 . +2 . − . . +0 . − . − . +0 . − . . − . . +0 . − . . +0 . − . . +4 . − . . +0 . − . − . +0 . − . . ± .
108 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . -0.3534 1.162 1.162HIIG 71 . +1 . − . . +0 . − . − . +0 . − . . ± .
755 1 . +0 . − . − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . × − . × − It is interesting that the best case to be made for thepower law model comes from the standard candle data,as the GRB data favor the power law model and theHIIG do not strongly disfavor it. In a similar fashion,[26], [21], and [17] find that standard candle data (inthe form SN Ia measurements) alone do not rule outor do not strongly disfavor the power law model. How-ever, when the GRB and HIIG data are combined, witheach other and with the cosmic chronometer and stan-dard ruler data, it is the ΛCDM model that comes outon top. Cosmic chronometer ( H ( z )) data alone also donot favor the power law model, and neither does the stan-dard ruler (BAO + QSO) combination. Of these threedata sets, the QSO set has the least discriminating power,perhaps because of the wide dispersion of the measure-ments it contains (see the lower left panel of Fig. 2); aswith the HIIG data, the power law model is not stronglyruled out by QSO data alone. The BAO data have themost discriminating power of any solo data set, the fit ofthe power law model to these data having the smallestrelative probabilities compared to ΛCDM. This is alsotrue of the standard candle set (BAO + QSO), whichgives a smaller relative probability than either the cos-mic chronometer or standard candle (GRB + HIIG) setwhen the power law model is fitted to this data combi-nation. When the power law model is fitted to the fulldata set, the relative probabilities decrease drastically, tothe point that the power law model appears to be verystrongly ruled out, at z (cid:46)
8, in favor of the ΛCDM model.These results are in broad agreement with the findings of[21–23], although they differ somewhat in the details. Inparticular, using a set of H ( z ) data that is slightly differ- ent from mine, [21] finds much stronger evidence againstthe power law model than I do. Both [22] and [23] useBAO data (a smaller set than mine) to evaluate the powerlaw model. Contrary to my results, the BAO measure-ments they use favor the power law model, although theyboth find that the power law model is strongly disfavoredwhen BAO data are combined with independent probes(SN Ia in [22, 23] and SN Ia + CMB in [23]).That the power law model is ruled out in favor of theΛCDM model, from an analysis of H ( z ), BAO, QSO,GRB, and HIIG data, is a strong statement, and shouldnot be accepted uncritically. Though I believe I havemade a good case against the power law model, a fewcaveats must also be mentioned:1.) The results that are shown in Tables III-VI do nottake the finite detection significance of the BAO datainto account. As discussed in [22, 23, 59], for a weak BAOsignal, one must account for the possibility that the BAOfeature in the large-scale matter power spectrum has notactually been detected. To do this, one must replace thestandard gaussian χ function χ := − L G with χ := χ (cid:113) (cid:0) SN (cid:1) − χ , (23)where S/N , the signal-to-noise ratio, is the detection sig-nificance of the BAO feature. As described in [41], three They quote χ /ν = 1 . H ( z ) data, and χ /ν = 0 . χ /ν = 1 . χ /ν = 0 . TABLE VI. Marginalized best-fitting parameters of the ΛCDM model. The BAO data alone do not place a tight upper limiton the value of H , and the GRB data do not constrain H at all, so these limits are excluded from the table.Data type H Ω m q H ( z ) 67 . +3 . − . . +0 . − . − . +0 . − . BAO 83 . − . . +0 . − . − . +0 . − . QSO 67 . +4 . − . . +0 . − . − . +0 . − . BAO+QSO 69 . ± .
379 0 . +0 . − . − . +0 . − . GRB - 0 . − . − . − . HIIG 71 . +1 . − . . +0 . − . − . +0 . − . GRB+HIIG 71 . +1 . − . . +0 . − . − . +0 . − . All Data 70 . ± . . +0 . − . − . +0 . − . . . . . . H (km s Mpc ) H ( z )BAO+QSOGRB+HIIGAll Data . . . . m H (km s Mpc ) m H ( z )BAO+QSOGRB+HIIGAll Data FIG. 1. The left panel shows one- and two-dimensional constraints on the parameters of the power law model from severalcombinations of data, and the right panel shows one- and two-dimensional constraints on the ΛCDM model from the samecombinations of data (nuisance parameters excluded). of the BAO measurements I use in this paper are un-correlated, and the rest are correlated. To test the ro-bustness of my results, I replaced the gaussian likelihoodsof the uncorrelated BAO measurements with their coun-terparts defined by eq. (23) and performed the BAOanalysis again. I found no significant change in the re-sults when I did this, which is perhaps not surprising;in Fig. 11 of [59], the authors show that accounting forthe finite detection significance of BAO data only has theeffect of widening the confidence contours (primarily the3 σ contour) a little, and that this widening almost disap-pears when BAO data are combined with other probes.With that said, I did not investigate the effect of the de-tection significance of the correlated BAO data on themodel fits, and I do not know how large the effect is forthese data. However, based on the above considerationsas they apply to the uncorrelated BAO data, I do notexpect the effect of the detection significance of the un-correlated BAO data to be a significant factor affecting the validity of my results.2.) The fit to the QSO and HIIG data, across bothmodels, gives reduced χ values that are all >
2. Thefit to the H ( z ) data gives, for both the power law andΛCDM models, reduced χ values comparable to 0.5.The larger reduced χ values suggest that neither thepower law model nor the ΛCDM model is a particularlygood fit to the QSO or the HIIG data (though the re-duced χ values of the ΛCDM model are consistentlylower for these data than those of the power law model),or that the uncertainties of these data have been underes-timated, or both. The reduced χ values of the H ( z ) datasuggest, on the other hand, that the uncertainties of thesedata have been overestimated. The possible overestima-tion of the H ( z ) uncertainties has previously been notedby myself and my collaborators (see [2]), and is perhapsapparent in Fig. 2. My collaborators and I have also pre-viously noted the possible underestimation of the QSOand HIIG error bars [3, 4]. That the power law model hasconsistently higher values of χ for all data sets (GRBexcepted) alone and in combination (with the measure-ments presumably having mostly independent systemat-ics), argues against its validity as a model of cosmic ex-pansion for z (cid:46)
8, though the argument could be madestronger with a better understanding of the error bars onthe measurements.3.) The H ( z ) data are somewhat correlated with theQSO data. These data are correlated because some cos-mic chronometer data were used to obtain the charac-teristic angular size l m of the QSO data. As describedin [44], using the Gaussian Process method [60], 24 H ( z ) measurements at z ≤ . H ( z ). This function was then integratedto produce the angular diameter distances used, in con-junction with angular size measurements θ obs , to obtain l m = 11 . ± .
25 pc. This correlation has been noted inthe literature [41] and I currently believe that the param-eter constraints from QSO data alone are wide enoughthat the correlation between these data and H ( z ) data isnot significant. With that said, the magnitude of this cor-relation is not currently known in detail, and a defenderof the power law model could point to this as a weaknessof my study. One could solve this problem by treating l m as a free parameter in the cosmological model fits, al-though this tends to produce parameter constraints thatare so wide as to be nearly uninformative. My collab-orators and I are currently working to understand thisissue better.
VI. CONCLUSION
In this paper, I analyzed a set of cosmic chronometer( H ( z )), standard ruler (BAO and QSO), and standardcandle (GRB and HIIG) data to find out whether or notthe power law model fits these data as well as or betterthan the standard ΛCDM model. Using simple modelcomparison statistics, similar to what I and many othershave used to test alternatives to ΛCDM, I found that thepower law model does not provide a good fit to the data,compared to ΛCDM. The power law model is thereforenot a viable candidate to replace the ΛCDM model at z (cid:46) β does not adequately describe theevolution of the Universe over the course of its history. Shulei Cao, private communication, 2021.
VII. ACKNOWLEDGMENTS
The computing for this project was performed onthe Beocat Research Cluster at Kansas State Univer-sity, which is funded in part by NSF grants CNS-1006860, EPS-1006860, EPS-0919443, ACI-1440548,CHE-1726332, and NIH P20GM113109.
Appendix: Direct comparison of models to data
Here I plot the predictions of the power law and ΛCDMmodels together with the various data sets I use. In theupper left panel of Fig. 2 I plot H ( z )1+ z versus z , wherethe blue dots represent the H ( z ) measurements and thecurves represent the predicted value of H ( z )1+ z , as a func-tion of redshift, for the power law and ΛCDM modelswhen these models are fitted either to the full data set orto the H ( z ) data alone. From the figure, we can see thatthe power law model fails to account for deceleration-acceleration transition which occurs around z ∼ . This is backed up by the analyses of [20] and [21] (al-though a stronger case for this could be made using H ( z )data with smaller error bars).The upper right panel of Fig. 2 shows a plot of themeasured value of the volume-averaged angular diameterdistance D V ( z ), at three different redshifts, from the un-correlated BAO measurements shown in Table 1 of [41]. To obtain the central value and error bars of D V ( z ) at z = 0 .
81, I computed D V ( z ) = (cid:20) cz (1 + z ) D A ( z ) H ( z ) (cid:21) / (A.1)from the central value and uncertainty of the D A ( z ) mea-surement at z = 0 .
81, along with the median centralvalue and median uncertainty of the two H ( z ) measure-ments at z = 0 .
70 and z = 0 .
90 from Table 2 of [5]. The curves shown in the upper right panel of Fig. 2represent the predicted values of D V ( z ) for the ΛCDMand power law models. Although the power law modelappears to be ruled out when fitted to the BAO dataalone (as it severely under-predicts the values of D V ( z )at all redshifts), when it is fitted to the full data set itspredictions are nearly indistinguishable (within the er-ror bars of the measurements) from those of the ΛCDM For a discussion of the deceleration-acceleration transition, seee.g. [61]. I did not use the correlated measurements because these do nothave independent error bars. Although the sound horizon r s (which sets the scale of the BAOmeasurements) is a function of the model parameters, I foundthat both the ΛCDM and power law models predict r s = 144 . D V ( z )data are effectively model-independent. z H ( z ) / ( + z ) ( k m s M p c ) CDM, h = 0.7007, m = 0.2949CDM, h = 0.6815, m = 0.3196PL, h = 0.6306, = 0.9470PL, h = 0.6192, = 0.9842 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 z D V ( z ) ( M p c ) CDM, h = 0.7007, m = 0.2949CDM, h = 0.7401, m = 0.2967PL, h = 0.6306, = 0.9470PL, h = 0.8978, = 0.92060.5 1.0 1.5 2.0 2.5 z ( z ) ( m a s ) PL, h = 0.6306, = 0.9470PL, h = 0.6813, = 0.9673CDM, h = 0.7007, m = 0.2949CDM, h = 0.6869, m = 0.3153 z ( z ) ( m a g ) CDM, h = 0.7007, m = 0.2949CDM, h = 0.7148, m = 0.2950PL, h = 0.6306, = 0.9470PL, h = 0.7031, = 1.158CDM, h = 0.7007, m = 0.2949CDM, h = 0.7148, m = 0.2950PL, h = 0.6306, = 0.9470PL, h = 0.7031, = 1.158 FIG. 2. In all panels, the abbreviation “PL” denotes the power law model and h := H / (100 km s − Mpc − ). In the lowerright panel the HIIG data are represented by blue dots, and each GRB datum is represented by a purple “x”. model. A stronger (or perhaps weaker) case against thepower law model, however, could presumably be madewith more independent measurements, as the dispersionof the values of D V ( z ) can not be readily inferred fromsuch a small data set.Large dispersion is a particular problem for the QSOdata, as the angular size measurements θ ( z ) do not showa clear trend with increasing redshift. This, coupled withthe fact that the ΛCDM and power law model predictions of the angular size are very similar over the range of theQSO data, means that these data do not clearly favor onemodel over the other (see the lower left panel of Fig. 2).Similarly, the ΛCDM and power law model predictionsof the distance modulus µ ( z ) are almost identical overthe redshift range containing the HIIG and GRB data.Although these data show a clear trend with increasingredshift, the predictions of the ΛCDM and power lawmodels only begin to diverge around z ≈
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