Assessing Cosmic Acceleration with the Alcock-Paczynski Effect in the SDSS-IV Quasar Catalog
aa r X i v : . [ a s t r o - ph . C O ] A ug Mon. Not. R. Astron. Soc. , 1–5 (2020) Printed 31 August 2020 (MN L A TEX style file v2.2)
Assessing Cosmic Acceleration with the Alcock-Paczy ´nski E ff ect inthe SDSS-IV Quasar Catalog Fulvio Melia ⋆ , Jin Qin and Tong-Jie Zhang Department of Physics, The Applied Math Program, and Department of Astronomy, The University of Arizona, AZ 85721, USA Department of Astronomy, Beijing Normal University, Beijing 100875, China
ABSTRACT
The geometry of the Universe may be probed using the Alcock-Paczy´nski (AP) e ff ect, inwhich the observed redshift size of a spherical distribution of sources relative to its angularsize varies according to the assumed cosmological model. Past applications of this e ff ect havebeen limited, however, by a paucity of suitable sources and mitigating astrophysical factors,such as internal redshift-space distortions and poorly known source evolution. In this Letter ,we introduce a new test based on the AP e ff ect that avoids the use of spatially bound systems,relying instead on sub-samples of quasars at redshifts z . . Λ CDM, which predicts a transition from deceleration to acceleration at z ∼ .
7; Einstein-deSitter, in which the Universe is always decelerating; and the R h = ct universe, which expandsat a constant rate. Λ CDM is consistent with these data, but R h = ct is favoured overall. Key words: cosmological parameters, cosmology: observations, cosmology: theory, distancescale, galaxies: active, quasars: supermassive black holes
The expansion history of the Universe has been probed using adiverse set of observations, including those of the Cosmic Mi-crowave Background Radiation (CMB) and the large-scale struc-ture of galaxy clusters. These approaches have been limited by pro-cesses other than those in the baseline cosmological model, how-ever, due to possible source evolution in the latter, or the genera-tion and modification of anisotropies in the former (Narlikar et al.2007; Angus & Diaferio 2011; L´opez-Corredoira 2013; Melia2014; L´opez-Corredoira 2007). Though large surveys of galaxiesmay constrain the geometry of the Universe, e.g., via the construc-tion of a Hubble diagram or the implementation of an angular-sizetest, one must typically adopt specific astrophysical models, suchas the growth of dark-matter halos, in order to extract useful cos-mological information.The geometry of the Universe can be assessed morecleanly via the Alcock-Paczy´nski (AP) (Alcock & Paczynski 1979;L´opez-Corredoira 2014) e ff ect, in which the ratio of observedradial / redshift size to angular size of a spherical distribution ofsources, such as a galaxy cluster, changes from one cosmology tothe next. AP tests largely avoid contamination from source evolu-tion because the characteristics of individual sources do not impactthe ratio of projected sizes of their distribution. Of course, to fullyutilize the AP e ff ect, one must have access to bound systems that ⋆ John Woodru ff Simpson Fellow. E-mail: [email protected] are large enough to be measurable over a broad range of redshifts,and this tends to be a principal mitigating factor.In this
Letter , we introduce a new test based on the AP ef-fect designed to probe the cosmic expansion rate as a function ofredshift, though using the very large sample of quasars in the SloanDigital Sky Survey IV (SDSS-IV) (Ahumada et al. 2019), spanningredshifts z . .
5. The novel feature with this approach is that it doesnot rely on spatially confined source distributions. As we shall see,the use of quasars can greatly improve the statistics for the purposeof model selection, especially when future surveys will grow thiscatalog by two or more orders of magnitude.Our modified AP test can be used for any cosmology but,given our focus on the impact of acceleration on the geometry,we shall here restrict our attention to three highly pertinent mod-els:
Planck - Λ CDM (Planck Collaboration et al. 2018), which pre-dicts di ff erent epochs of acceleration and deceleration; Einstein-de Sitter, which has been strongly ruled out as a viable modelof the Universe by many other kinds of data, but is included be-cause it provides a well-known example of a Universe that onlydecelerates; and another FLRW cosmology based on the zero ac-tive mass equation-of-state, ρ + p =
0, in terms of the total en-ergy density ρ and pressure p in the cosmic fluid (Melia 2007,2016, 2017; Melia & Shevchuk 2012; Melia 2020). Known as the R h = ct universe, this cosmology exhibits a constant rate of expan-sion throughtout its history. (See Table 2 in ref. Melia 2018 for amore detailed comparison of this model with Λ CDM.) c (cid:13) Melia, Qin & Zhang
Longitude (degrees) L a t i t u d e ( d e g r ee s )
0 50 100 150 200 250 300 350806040200-20
Figure 1.
Sky map in Galactic coordinates ( l , b ), of the projected locations (blue dots) of the 526,356 SDSS-IV quasars. The colored boxes show sub-samplesselected for the new AP test (see text). For example, the yellow box has a size of 10 ◦ × ◦ and is centered at Galactic coordinates (135 ◦ , ◦ ). All of the data we use in this
Letter are taken from the DataRelease 16 Quasar catalog (DR16Q), based on the extendedBaryon Oscillation Spectroscopic Survey (eBOSS) of the SDSS-IV (Ahumada et al. 2019). This collection includes all SDSS-IV / eBOSS objects spectroscopically confirmed as quasars. Withthe inclusion of previously confirmed quasars from SDSS-I, IIand III, this combined catalog encompasses an overall sample of526,356 objects, though possibly subject to an estimated contam-ination of about 0 . ∼ .
5, covering approximatey 9376 deg of the sky (see fig. 1).As we shall see, however, the quasar number density per comovingvolume decreases non-negligibly at z & b . ◦ ) compared to b & ◦ , duein part to galactic foreground e ff ects. As such, we shall here restrictour analysis to the sub-samples shown as colored boxes in figure 1to maximize the statistics.Our methodology utilizes the geometric construction shown infigure 2. In short, we select two adjacent comoving boxes, each withits four lateral sides inclined at a fixed angle ∆ θ/ b and longitude l ), relative to the line-of-sight (LOS).We then count the total number of quasars, N Q , in box 1, basedon its pre-selected length, ∆ z , in redshift space (more on this be-low), and find from the quasar catalog the value of ∆ z for which N Q = N Q . For a fixed angular size ∆ θ × ∆ θ , the redshift interval ∆ z is a unique function of ∆ z and the redshift-dependent comov-ing distance predicted by each given cosmology. A comparison ofthe ‘measured’ and theoretical relations between ∆ z and ∆ z thenprovides a likelihood of that being the correct model.In order for this method to provide a meaningful comparison,however, several conditions need to be satisfied. Box 1 has its basecentered at coordinates ( z , l , b ), with an angular size ∆ θ × ∆ θ in theplane of the sky. For convenience, we predict from the chosen cos-mological model the redshift size, ∆ z —or, equivalently, the angle ∆ θ —such that all three dimensions, which we shall call L k ( z ) (inthe radial direction) and L ⊥ ( z ) (in the two transverse directions atthe base), are all equal in the comoving frame. As it turns out, theangular size L ⊥ ( z ) does not appear in the final analysis because ∆ θ remains constant throughout the region ( z , z + ∆ z + ∆ z ). L ⊥ ( z )serves only to establish the area at the base of the first box, but be- D z D q D q D z LOS (z,l,b)(z+ D z ,l,b) D q D q Figure 2.
Schematic diagram showing two adjacent boxes along the line ofsight (LOS), each with angular dimensions ∆ θ × ∆ θ in the plane of the sky.Box 1 has its base centered at ( z , l , b ), and has a length ∆ z in redshift space.Box 2, whose base is centered at ( z +∆ z , l , b ), has the corresponding length ∆ z . cause ∆ θ remains constant, changing L ⊥ represents a change in itsarea that is mirrored in proportion by the area of the second box.Thus, the ratio ∆ z / ∆ z is independent of L ⊥ ( z ), as one may seemore formally in the derived condition shown in Equation (13) be-low. Choosing ∆ θ in this fashion (if ∆ z is fixed) merey provides aconvenient sub-sample of quasars with which to calculate N Q and N Q , and may be used for all the cosmologies being tested.We make the key assumption that the average quasar comov-ing number density, n Q ( z ), within the two boxes is uniform on a c (cid:13) , 1–5 lcock-Paczy´nski E ff ect with SDSS-IV Quasars n / M p c Q
0 0.5 1.0 1.5 2.0Redshift Figure 3.
The DR16Q quasar number density as a function of redshift, as-suming
Planck - Λ CDM as the background cosmology (see Eqns. 5 and 6). scale of several degrees. The average density is more likely to beuniform across the boxes as their size increases, of course, thoughredshift evolution in n Q ( z ) could invalidate this approach if dn Q / dz from Box 1 to Box 2 is too large to ignore. Any potential evolutionin n Q may therefore be mitigated by choosing as small a box as pos-sible, in order to minimize the ratio ( dn Q / dz ) ∆ z / n Q . Unfortunately,these two requirements conflict each other, but we have found bydirect inspection of the SDSS-IV catalog that any dispersion aris-ing from these e ff ects is well within the final calculated errors aslong as ∆ z / z . / z .
1. One may see this graphically in fig-ure 3, which shows the estimated DR16Q quasar number densityper unit comoving volume, assuming
Planck - Λ CDM as the back-ground cosmology (see Eqns. 5 and 6 below). (Note that this plot ismerely used to gauge how reliable our assumption of a constant n Q is, and is not included in the comparative analysis, which needs tobe carried out independently for each individual cosmology.) Evi-dently, n Q is very nearly constant up to z ∼ .
2, and then decreasesmonotonically towards higher redshifts.Of the three cosmologies we shall test here, the geometry in R h = ct is the easiest to understand because all of its integratedmeasures—such as the comoving distance—have simple analyticalforms. We shall therefore start with this model, and then summarizethe corresponding expressions for the other two. For Box 1 alongthe LOS, we have in the R h = ct universe L R h k ( z ) = cH Z z +∆ z z duE ( u ) , (1)where H ( z ) ≡ H E ( z ), E ( z ) = (1 + z ) and H is the Hubble con-stant. For our actual calculations, we shall employ the full integralexpressions for these quantities. It will also be helpful, however, forus to understand the results by finding approximations in the limitwhere ∆ z ≪ z . In this limit, Equation (1) may be written L R h k ( z ) ≈ c ∆ z H (1 + z ) . (2)In the plane of the sky, the corresponding comoving size is L R h ⊥ ( z ) ≈ ∆ θ cH ln(1 + z ) . (3)Thus, to estimate a reasonable size ∆ z for Box 1 when ∆ θ is cho-sen, or ∆ θ if ∆ z is fixed, we may simply set these two expressionsequal to each other, L R h k ( z ) = L R h ⊥ ( z ), and find that ∆ z ≈ ∆ θ (1 + z ) ln(1 + z ) . (4) h Einstein-de SitterPlanck- L CDM0.90.7 D z / D z Figure 4.
The ratio ∆ z / ∆ z as a function of z , assuming ∆ z = z /
3: (blue)
Planck - Λ CDM; (black) the R h = ct universe; and (red) Einstein-de Sitter.The standard model transitions from acceleration to deceleration across z ∼ .
7, while Einstein-de Sitter always decelerates, and R h = ct expands at aconstant rate. For Λ CDM, the corresponding quantities are L Λ k ( z ) = cH Z z +∆ z z du p Ω m (1 + z ) + Ω Λ , (5)and L Λ ⊥ ( z ) = ∆ θ cH Z z du p Ω m (1 + u ) + Ω Λ . (6)In these expressions, Ω m and Ω Λ are today’s matter and dark-energy densities, respectively, scaled to the critical density ρ c ≡ c H / π G . Radiation may be ignored for z . .
5. Theparametrization shown in Equations (5) and (6) is based onthe
Planck optimization (Planck Collaboration et al. 2018): Ω m = . ± . Ω Λ = . − Ω m (given that Ω k = . ± . w de = − . ± .
03, where the pressure of darkenergy is written as p de = w de ρ de , in terms of its corresponding en-ergy density ρ de . This points to a cosmological constant, for which p de = − ρ de . In our analysis, we shall therefore assume flat Λ CDMwith w de = −
1, though we shall allow Ω m to vary in order to opti-mize the fit. This procedure is independent of all other parameters,such as H . Finally, in the case of Einstein-de Sitter, we have L EdS k ( z ) ≈ c ∆ z H (1 + z ) / (7)and L EdS ⊥ ( z ) = ∆ θ cH − √ + z ! , (8)which together yield ∆ z ≈ ∆ θ (1 + z ) (cid:16) √ + z − (cid:17) . (9)Were we to ignore the δ z -dependence of L ⊥ ( z + δ z ) over theredshift range δ z ∈ (0 , ∆ z ) and ( ∆ z , ∆ z + ∆ z ), the comovingvolumes 1 and 2 would be equal if ∆ z were chosen to satisfy thecondition Z z +∆ z +∆ z z +∆ z duE ( z ) = Z z +∆ z z duE ( z ) . (10) c (cid:13) , 1–5 Melia, Qin & Zhang D z / D z D z = 0.1 Figure 5.
Same as figure 4, except now for a fixed ∆ z = .
1, highlighting the variation with redshift at 0 . z . .
2. The color coding is the same as in figure 4,except that blue now corresponds to the best-fit Λ CDM model, with the optimized matter density Ω m = . + . − . . The data points are obtained by countingquasars in the DR16Q catalog. This is not a good approximation, however, even for boxes with ∆ z / z . /
3. Fortunately, it is quite straightforward to incorporatethe redshift-dependence of L ⊥ into a calculation of the comovingvolume, via the inclusion of the expression L ⊥ ( z + δ z ) = L k ( z ) " + I ( z , z + δ z ) I (0 , z ) , (11)where I ( z , z ) ≡ Z z z duE ( u ) . (12)Since ∆ θ remains constant across both boxes, it is not di ffi cult tosee that ∆ z must satisfy the condition I (0 , z + ∆ z + ∆ z ) = I (0 , z + ∆ z ) − I (0 , z ) . (13)The complete solution for ∆ z / ∆ z , taking all of these spher-ical e ff ects into account, is shown in figure 4 for each of the threecosmologies, using the criterion ∆ z = z /
3. We show in this figurethe expected model di ff erences over the extended range (0 . z . . z .
1) in this paper (see fig. 5), as an illustrationof the method.It has been noted elsewhere (see, e.g., refs. Melia 2018, 2020)that the transition from deceleration to acceleration at z ∼ . Λ CDM produces an overall expansion history very similar to thatof R h = ct , at least up to z ∼
1. The two phases e ff ectively can-cel out, producing an integrated expansion approximately equal towhat it would have been if the Universe had expanded at a constantrate from the beginning. This is reflected in the overlap betweenthe two ∆ z / ∆ z (black and blue) curves below z ∼ z & . Λ CDMexhibits the e ff ects of deceleration analogous to Einstein-de Sitter.To di ff erentiate between cosmological models with di ff erentcombinations of unknowns, one typically uses information criteria,such as the Akaike Information Criterion, AIC ≡ χ + f , where f is the number of free parameters (Akaike 1974; Liddle 2007;Burnham & Anderson 2002; Melia & Maier 2013), and the Kull-back Information Criterion, KIC ≡ χ + f (Cavanaugh 2004). Athird variant, known as the Bayes Information Criterion (Schwarz1978), is defined as BIC = χ + f ln N , where N is the numberof data points. The BIC is actually not based on information the-ory, but rather on an asymptotic ( N → ∞ ) approximation to theoutcome of a conventional Bayesian inference procedure for decid-ing between models. It suppresses overfitting much more than AICand KIC if N is large. In the application we are considering here, N .
30, for which ln(30) ∼ .
4, compared to the analogous factor2 in the case of AIC, and 3 for KIC. In this case, the BIC does notprovide a perspective very di ff erent from the other two criteria, andthere is no point in including it in this Letter .For model M α , the unnormalized confidence of it being ‘true’is the Akaike weight exp( − AIC α / P ( M α ) = exp( − AIC α / P β exp( − AIC β / . (14)The various outcomes may also be characterized via the di ff erence ∆ AIC ≡ AIC − AIC (and similarly for KIC), which determinesthe extent to which M is favoured over M . The result is judged‘positive’ when ∆ ∼ −
6, ‘strong’ for ∆ ∼ −
10, and ‘very strong’if ∆ & c (cid:13) , 1–5 lcock-Paczy´nski E ff ect with SDSS-IV Quasars Table 1.
Model Selection using AP with the SDSS-IV quasar catalogModel χ AIC Prob KIC Prob(AIC) (KIC) R h = ct .
094 1 .
095 62 .
4% 1 .
095 69 . Λ CDM ( Ω m = . + . − . ) 0 .
827 2 .
827 26 .
3% 3 .
827 17 . .
513 4 .
513 11 .
3% 4 .
513 12 . A direct comparison of these three curves with the ‘measurement’of ∆ z / ∆ z using quasar counts in the DR16Q catalog is shown infigure 5. The errors associated with measuring the position of eachquasar, ( z , l , b ), are insignificant compared to the dispersion in theircomoving number density n Q . To estimate the errors shown here,we therefore assemble as many boxes as possible within the sameredshift bin ∆ z (though clearly not at the same angular position),and then calculate the population variance from this distribution.The error bars shown in figure 5 represent the 1 σ uncertainties in-ferred from this variance.The fits in this plot, and the summary of results given in Ta-ble 1, show that the data are consistent with Λ CDM, but actuallysomewhat favour the R h = ct cosmology, which expands at a con-stant rate at all redshifts, while strongly rejecting Einstein-de Sit-ter. The optimized matter density Ω m = . + . − . for Λ CDM isfully consistent with the
Planck measurement (i.e., 0 . ± . R h = ct is favouredover Λ CDM is judged positive, with an outcome ∆ AIC ∼ . ∆ KIC ∼ .
7, based on the DR16Q catalog.Unlike many other kinds of observation, these data are e ff ec-tively independent of any model because they are constrained by afixed ∆ θ throughout the redshift range ( z , z + ∆ z ) and ( z + ∆ z , z +∆ z + ∆ z ), and they do not depend on the actual value of L ⊥ ( z ).Thus, the same set of data apply to all three curves in figure 5. Fora fixed ∆ θ , however, the ratio L k ( z ) / L ⊥ ( z ) of the boxes does changein redshift space according to each model’s prediction of the angu-lar diameter distance. As we may see from this figure, the statisticalweight of the DR16Q quasar catalog is already su ffi cient for us tostart distinguishing between the three models we have examined inthis paper.But clearly, the probative power of this technique will growconsiderably when future surveys will allow us to extend this testto redshifts as high as ∼ . n Q ( z ) individually for each cosmol-ogy over redshifts where the comoving density is not constant (seefig. 3), thereby attaining an even higher level of precision for the ∆ z / ∆ z curves. ACKNOWLEDGMENTS
We are grateful to the anonymous referee for several suggestionsto improve the presention in our manuscript. FM is also gratefulto Amherst College for its support through a John Woodru ff Simp-son Lectureship. This work was partially supported by the National Science Foundation of China (Grants No.11929301,11573006) andthe National Key R & D Program of China (2017YFA0402600).
DATA AVAILABILITY STATEMENT
All of the data underlying this paper are taken from the DataRelease 14 Quasar catalog (DR16Q), based on the extendedBaryon Oscillation Spectroscopic Survey (eBOSS) of the SDSS-IV, which is fully described in Ahumada et al. (2019). The DR16Qcatalog may be downloaded in fits format from the websitehttps: // / dr16 / . REFERENCES
Ahumada, R., Allende Prieto, C., Almeida, A., et al. 2019, arXive-prints, arXiv:1912.02905Akaike, H. 1974, IEEE Transactions on Automatic Control, 19,716Alcock, C., & Paczynski, B. 1979, Nat, 281, 358Angus, G. W., & Diaferio, A. 2011, MNRAS, 417, 941Burnham, K. P., & Anderson, D. R. 2002, Model Selection andMultimodel Inference (Springer-Verlag)Cavanaugh, J. E. 2004, Aust N Z J Stat, 46, 257Izevic, Z. 2017, Proceedings of the International AstronomicalUnion, IAU Symposium, 324, 330Liddle, A. R. 2007, MNRAS, 377, L74L´opez-Corredoira, M. 2007, Journal of Astrophysics and Astron-omy, 28, 101—. 2013, International Journal of Modern Physics D, 22, 1350032—. 2014, ApJ, 781, 96Melia, F. 2007, MNRAS, 382, 1917—. 2014, A&A, 561, A80—. 2016, Frontiers of Physics, 11, 119801—. 2017, Frontiers of Physics, 12, 129802—. 2018, MNRAS, 481, 4855—. 2020, The Cosmic Spacetime (Taylor & Francis)Melia, F., & Maier, R. S. 2013, MNRAS, 432, 2669Melia, F., & Shevchuk, A. S. H. 2012, MNRAS, 419, 2579Narlikar, J. V., Burbidge, G., & Vishwakarma, R. G. 2007, Journalof Astrophysics and Astronomy, 28, 67Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2018, arXive-prints, arXiv:1807.06209Schwarz, G. 1978, Annals of Statistics, 6, 461 c (cid:13)000