One parameterisation to fit them all
Gustavo Arciniega, Mariana Jaber, Luisa G. Jaime, Omar A. Rodríguez-López
MMNRAS , 1–15 (2021) Preprint 18 February 2021 Compiled using MNRAS L A TEX style file v3.0
One parameterisation to fit them all
Gustavo Arciniega, , ∗ Mariana Jaber, † Luisa G. Jaime, ‡ and Omar A. Rodríguez-López, § Centro Tecnológico Aragón, Universidad Nacional Autónoma de México, Av. Rancho Seco S/N, Bosques de Aragón,Nezahualcóyotl, Estado de México, 57130 México Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, A. P. 50-542, México, CDMX 04510, México Institute for Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Toruń, Poland Departamento de Fisica, Instituto Nacional de Investigaciones Nucleares, Apartado Postal 18-1027, Col. Escandón, Ciudad de México,11801, México. Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000, Ciudad de México, México
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Perhaps the most explored hypothesis for the accelerated cosmic expansion rate arise in the context of extra fields or modificationsto General Relativity. A prevalent approach is to parameterise the expansion history through the equation of state, 𝜔 ( 𝑧 ) . Wepresent a parametric form for 𝜔 ( 𝑧 ) that can reproduce the generic behaviour of the most widely used physical models foraccelerated expansion with infrared corrections. The present proposal has at most 3 free parameters which can be mapped backto specific archetypal models for dark energy. We analyze in detail how different combinations of data can constrain the specificcases embedded in our form for 𝜔 ( 𝑧 ) . We implement our parametric equation for 𝜔 ( 𝑧 ) to observations from CMB, luminousdistance of SNeIa, cosmic chronometers, and baryon acoustic oscillations identified in galaxies and in the Lymann- 𝛼 forest. Wefind that the parameters can be well constrained by using different observational data sets. Our findings point to an oscillatorybehaviour which is consistent with an 𝑓 ( 𝑅 ) -like model or a unknown combination of scalar fields. When we let the threeparameters vary freely, we find an EOS which oscillates around the phantom-dividing line, and, with over 99% of confidence,the cosmological constant solution is disfavored. Key words: – dark energy – cosmology: theory – cosmology: observations – cosmological parameters
Ever since the discovery of the acceleration of the Universe (Perlmut-ter et al. 1999; Riess et al. 1998) (hinted previously in Roukema &Yoshii (1993)), cosmology has tried to answer the question of whatmakes the Universe accelerate. Currently, the most accepted expla-nation by the scientific community is the cosmological constant, Λ ,in the frame of General Relativity with an FLRW metric, which hasbecome the concordance model known as Λ CDM (Aghanim et al.2018). Several phenomena can be explained by using such a simplemodel; nevertheless, the physical nature of Λ remains unaddressed.In the past few years, observations of different astrophysicalsources have been used to measure the acceleration of the Universe.The results have brought with them even more uncertainty about thenature of dark energy. They show a discrepancy on the present valueof the Hubble parameter derived when local measurements are used(Riess et al. (2019), Wong et al. (2019), Shajib et al. (2020)) with the 𝐻 value when derived by fitting the cosmological parameters as-suming the concordance model in the cosmic microwave background(CMB) (Aghanim et al. (2018)). Different estimations of the discor-dance place the discrepancy as high as 4.4- 𝜎 Riess et al. (2019) or ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] even at the level of 5.3- 𝜎 , according to (Wong et al. 2019) (see Verdeet al. (2019) for a summary plot).It is possible that a systematic miscalculation is behind this co-nundrum; nevertheless, the possibility of having some new physicsis provocative. Many alternatives to the standard concordance modelhave been proposed (for a review of several of these alternatives,see Zumalacarregui (2020)). One alternative to explore the evolutionof the Universe in a model-independent fashion way is by setting aparametric form for the equation of state (EOS), 𝜔 ≡ 𝑃 / 𝜌 of the darkenergy component.In this framework, several parameterisations of the EOS have beenproposed with the idea of simplifying the analysis of observations;however, we find either a lack of physical motivation or a stringentdependence of a particular model.Interestingly enough, in an observational effort carried out byZhao et al. (2017), a reconstruction of the EOS was presented. Theevolution found by the authors shows oscillating behaviour around thephantom line. It is well known that such evolution can not be providedby using a single scalar field (either phantom or quintessence like).Nevertheless, modified gravity or some unknown combination ofmultiple scalar fields could provide the reconstructed EOS.Currently, different kinds of parameterisations are used to providedynamical dark energy, some of them are motivated by scalar fieldsand, since the work on Jaime et al. (2018), there is a proposal inspiredby modified gravity. We present a different parameterisation that hasthe advantage of reproducing the generic behaviour for both cases, © a r X i v : . [ a s t r o - ph . C O ] F e b G. Arciniega, et al. depending on the choice of the parameters. Using this parameterisa-tion with current and future data, we could test the generic evolutionof the equation of state of the accelerating mechanism.
The parameterisations of 𝜔 ( 𝑧 ) in the literature are either mere mathe-matical descriptions, polynomials or Taylor expansions around 𝑎 (or 𝑧 ), or an attempt to capture distinctive features for particular models.In Chevallier & Polarski (2001) the authors perform a Hamiltoniananalysis that provides physical tools to build their proposal for 𝜔 ( 𝑧 ) .Their aim is to explore small deviations of the cosmological constant.In the light of the recent Hubble tension, our motivation is reproduc-ing predictions given by different alternative physical scenarios whileavoiding inherent theoretical complications for the implementationof certain models. This way, the parameterisations that can mimicwell supported physical evolution of the EOS will provide a simplerway to study complicated theories.The standard approach to this aim, is to take model by model andconstrain the introduced free parameters against data to obtain con-clusions for the chosen physical scenario. Instead of choosing a para-metric form for each different model and performing the statisticalanalysis case by case, we propose a single framework for analyzingthe generic behaviour of the most widely used physical models foraccelerated expansion with, at most, three parameters. Our proposalincludes alternative models that make modifications for late timewhile maintaining the EOS 𝜔 = − 𝜔 ( 𝑧 ) , explain its mathematicalproperties and describe its capabilities to mimic archetypal modelsfor the accelerated expansion. A particular interesting question for usis: do observations point to a 𝜔 ( 𝑧 ) which crosses the phantom-line?Our proposal is engaging in mimic and analysing two paradigmaticscenarios: 𝑓 ( 𝑅 ) modified gravity and quintessence/phantom models,but we will see that the parameterisation is not restricted to only thesetwo cases.We focus on 𝑓 ( 𝑅 ) theories of gravity because they are verystraightforward modifications of General Relativity, and have beenwidely studied over the past twenty years (see for instance Jaimeet al. (2012) and references therein). In this kind of modification, thedependence of the Ricci scalar 𝑅 in the Hilbert-Einstein action is notlinear, it is replaced by an arbitrary function of 𝑅 . Several 𝑓 ( 𝑅 ) mod-els have been proposed to provide an alternative explanation to theacceleration of the Universe. In Jaime et al. (2014) the definition ofthe EOS for the geometric dark energy is discussed and it was shownthat, in general, the generic evolution of 𝜔 ( 𝑧 ) for the 𝑓 ( 𝑅 ) modelsthat are considered candidates for dark energy is oscillatory. In Jaimeet al. (2018) some of us presented a parameterisation for the EOSin 𝑓 ( 𝑅 ) that can reproduce in a very high precision the numericalresults for some 𝑓 ( 𝑅 ) models, nevertheless if we try to go at highredshift the oscillatory behaviour given by the parameterisation willbring errors if it is implemented it into Boltzmann codes. By usingthe present proposal such problems can be avoided while the genericbehaviour is maintained.Regarding scalar fields, the most popular proposal to provide an alter-native explanation to the acceleration of the Universe is quintessencemodels. Such models can be separated into two kinds: “thawing out”and “freezing in”, depending on if the slope when is going to 𝑧 = The parameterisation we are proposing for the EOS, 𝜔 ( 𝑧 ) , is thefollowing: 𝜔 ( 𝑧 ) = − − 𝐴 exp (− 𝑧 ) (cid:0) 𝑧 𝑛 − 𝑧 − 𝐶 (cid:1) , (1)where 𝐴 , 𝑛 and 𝐶 are real numbers that can take positive or negativevalues.A quick inspection let us notice that the present-day value is givenby 𝜔 ( 𝑧 = ) ≡ 𝜔 = − + 𝐴𝐶 , while the high-redshift value rapidlyconverges to 𝜔 ( 𝑧 (cid:29) ) = −
1, corresponding to a cosmologicalconstant Λ scenario avoiding high redshift divergences that can bepresent in other parameterisations (Huterer & Turner (2001); Weller& Albrecht (2002), Jaber & de la Macorra (2018)).Given the form of equation (1), it is possible to mimic differentdynamical dark-energy EOS which can be characterized by, at most,a single oscillation of the 𝜔 ( 𝑧 ) at low redshifts.We identify four well-posed cases that we name: exponential,quintessence/phantom, 𝑓 ( 𝑅 ) , and general, besides the standard caseof a cosmological constant. In the next subsections, we provide ananalytical analysis in order to explain the flexibility of the model. 𝑛 = 𝐶 = (one-free parameter). The simplest case of (1) is when 𝑛 =
1, so that the parameterisationreduce to the expression: 𝜔 ( 𝑧 ) = − + 𝐴𝐶𝑒 − 𝑧 . (2)Without lost of generality, we can fix 𝐶 =
1, that is equivalent torename ˜ 𝐴 ≡ 𝐴𝐶 . In this case, if we do not perform the redefinitionof ˜ 𝐴 , the parameterisation will present a degeneration effect givenby the product 𝐴𝐶 .Nevertheless, it is possible to avoid the degeneration if the param-eterisation, for this particular case, takes just one parameter insteadof two. In this case, the generic behaviour is exponential. Figure 1shows the evolution of the EOS for different values of the amplitude:˜ 𝐴 = ± . , ± . , ± , ±
5, where the black lines represent the positivevalues of ˜ 𝐴 , while the grey lines show the negative ˜ 𝐴 values.From 1 we see that the value of 𝜔 at 𝑧 = − 𝐴 ≠
0. Also, the evolution of 𝜔 ( 𝑧 ) goes monotonically to theasymptotic value 𝜔 = − 𝑧 >
0. How fast it convergesto − 𝐴 . This parameter controls both, thepresent value of the equation of state, and the epoch 𝑧 , for which 𝜔 ( 𝑧 ) is practically −
1. In order to formally express the value of 𝑧 where we can consider that the EOS has reached the value −
1, let usconsider | 𝜖 | (cid:28) 𝜔 ( 𝑧 ) = − + | 𝜖 | = − + ˜ 𝐴𝑒 − 𝑧 . From here,we define ˜ 𝑧 ≡ 𝑙𝑛 ( ˜ 𝐴 /| 𝜖 |) as the redshift such that 𝜔 ( ˜ 𝑧 ) (cid:39) −
1, up toan | 𝜖 | for large 𝑧 .This characteristic will play a role in distinguishing this from othercases of study. In particular, we anticipate that although the evolutionwithin this model (Exponential) can be similar to the one obtainedin the particular cases II and IV (compare figure 1 to figures 3 and7), it is the value of 𝜔 and the asymptotic relaxation to 𝜔 ( 𝑧 ) = − MNRAS , 1–15 (2021) ne parameterisation to fit them all - - - - Figure 1.
Exponential: 𝑛 =
1, equation (2) taking ˜ 𝐴 ≡ 𝐴𝐶 . Solid black line˜ 𝐴 =
5, dashed black line ˜ 𝐴 =
1, dotted black line ˜ 𝐴 = .
5, and dot-dashedblack line ˜ 𝐴 = .
2. Gray lines are the same as black lines but with ˜ 𝐴 → − ˜ 𝐴 . - - - - Figure 2.
Equation of state 𝜔 ( 𝑧 ) for 𝑛 =
0, Quintessence/Phantom-like case:The amplitude for all the curves is 𝐴 = .
2. The solid, dashed, dotted, and dot-dashed black lines correspond to 𝐶 = . 𝐶 = 𝐶 = . 𝐶 = − . 𝐴 → − 𝐴 . 𝑛 = (two-free parameters). In order to mimic the shape of the EOS for Quintessence or Phantommodels we fix the parameter 𝑛 = 𝜔 ( 𝑧 ) = − − 𝐴𝑒 − 𝑧 [( − 𝐶 ) − 𝑧 ] , (3)where 𝜔 ( 𝑧 ) has a single minimum/maximum value located at 𝑧 = − 𝐶 . The generic evolution of the EOS is depicted in figure (2),where it can be seen how the parameterisation is able to transitfrom a Quintessence-like profile (Roy et al. (2018)) to a mixtureof Quintessence and phantom like fields, known as Quintom (for areview of this models, see Cai et al. (2010)). In order to visualizethis case, in figure 2 we have fixed the amplitude for all the curvesto 𝐴 = .
2. The solid black line is when 𝐶 = .
45, dashed black lineis when 𝐶 =
1, dotted black line is when 𝐶 = .
5, and dot-dashedblack line corresponds to 𝐶 = − .
3. Gray lines are the same as blacklines with 𝐴 → − 𝐴 . When 𝐶 ≥ 𝜔 ( 𝑧 ) looks monotonic in therange 𝑧 >
0, the minimum (maximum) will be located somewherein the future 𝑧 <
0. This way, what we are observing, at 𝑧 > - - - - Figure 3.
Equation of state 𝜔 ( 𝑧 ) for the Quintessence/Phantom-like case, 𝑛 =
0: The amplitude 𝐴 is fixed for all the curves, 𝐴 = − .
2. The solid blackline is with 𝐶 = −
9, dashed black line is with 𝐶 = −
5, dotted black line iswith 𝐶 = −
2, and dot-dashed black line corresponds with 𝐶 = .
2. Graylines are the same as black lines but with 𝐴 → − 𝐴 . evolution going up (down) from that minimum (maximum). In thiscase the behaviour will be similar to the one presented in the case 𝑛 =
1. It is important to remark that although the profiles depicted infigure 1, and 3 look similar, the analytic form is completely differentso it is guaranteed that there is no degeneration with the Exponential-like parameterisation case. Even more, in this case 𝜔 = − 𝐶 =
1, so any deviation of 𝐶 = 𝜔 around −
1. In the case that 𝐶 ≤
2, the minimum (maximum) will belocated at 𝑧 > 𝑓 ( 𝑅 ) -like: 𝑛 ∈ ( , ) and 𝐶 = (two-free parameters). In this case the parameter 𝑛 in (1) takes any value between 0 < 𝑛 < 𝐶 is fixed to 𝐶 =
0. In this way we are imposing 𝜔 = −
1. Equation (1) can be written as 𝜔 ( 𝑧 ) = − − 𝐴𝑒 − 𝑧 ( 𝑧 𝑛 − 𝑧 ) . (4)If 𝑛 ≠ { , } , equation (4) will have at most two real roots. This allowsan oscillatory behaviour for 𝑧 > ( 𝜔 = − ) thatconverges to 𝜔 ( 𝑧 ) = − 𝑧 >>
1. In order to depict a clearidea of this case, we make two plots of the generic behaviour ofequation (4). In figure 4 we fix 𝑛 = . 𝐴 = ± . , ± , ± , ±
30. In figure 5, the amplitude is fixedto 𝐴 = ± .
2, and the parameter 𝑛 takes different values. It can benoticed that, in both cases, the evolution has the oscillatory featurethat we are looking for in order to mimic 𝑓 ( 𝑅 ) gravity dark energyEOS with 𝜔 = − 𝑛 ∈ ( , ) (three-free parameters) We will now consider the EOS (1), for 𝑛 taking values in betweenthe previous cases, i.e. 𝜔 ( 𝑧 ) = − − 𝐴𝑒 − 𝑧 (cid:0) 𝑧 𝑛 − 𝑧 − 𝐶 (cid:1) , < 𝑛 < . (5) MNRAS000
2, and the parameter 𝑛 takes different values. It can benoticed that, in both cases, the evolution has the oscillatory featurethat we are looking for in order to mimic 𝑓 ( 𝑅 ) gravity dark energyEOS with 𝜔 = − 𝑛 ∈ ( , ) (three-free parameters) We will now consider the EOS (1), for 𝑛 taking values in betweenthe previous cases, i.e. 𝜔 ( 𝑧 ) = − − 𝐴𝑒 − 𝑧 (cid:0) 𝑧 𝑛 − 𝑧 − 𝐶 (cid:1) , < 𝑛 < . (5) MNRAS000 , 1–15 (2021)
G. Arciniega, et al. - - - - Figure 4.
Equation of state for the 𝑓 ( 𝑅 ) − 𝑙𝑖𝑘𝑒 case: 𝐶 = 𝑛 = . 𝐴 = . 𝐴 = 𝐴 =
20, and 𝐴 =
30 respectively. Gray lines are the same as blacklines with 𝐴 → − 𝐴 . - - - - Figure 5.
Equation of state for the 𝑓 ( 𝑅 ) − 𝑙𝑖𝑘𝑒 case: 𝐶 =
0. The amplitudeis 𝐴 = . 𝑛 = . , . , . , . 𝐴 → − 𝐴 . This equation (5) is the one we will be referring to as ’General-model’ from now on.In general, the EOS will present two characteristic behaviours: (1)two critical points with a local maximum and minimum (figure 6),and (2) a monotone function (figure 8). For all cases, the criticalvalues are given by the two roots of the following expression: 𝑧 𝑛 − 𝑛𝑧 𝑛 − − 𝑧 + ( − 𝐶 ) = . (6)It is not surprising that the critical 𝑧 values obtained from equation(6) depend only on 𝑛 and 𝐶 , because 𝐴 acts only as a homothety factorfor the 𝜔 ( 𝑧 ) + 𝜔 ( 𝑧 ) has at most two real critical points, that we will name 𝑧 and 𝑧 , nomatter the value of 𝑛 ∈ ( , ) .It could happen that the roots, 𝑧 and 𝑧 , are complex. Thisis the case when 𝜔 ( 𝑧 ) is a monotonic function (figure 8), thatresembles the exponential case ( 𝑛 =
1) (see figure 1), and theQuintessence/Phantom case ( 𝑛 =
0) for 𝐶 ≥ 𝑧 and 𝑧 , are real numbers, we get - - - - Figure 6.
General-model, equation (5). The amplitude and the constant valueare fixed for all curves, 𝐴 = . 𝑐 = .
08. The solid black line correspondswith 𝑛 = .
2, dashed black line is when 𝑛 = .
4, dotted black line is when 𝑛 = .
6, and dot-dashed black line is when 𝑛 = .
8. Gray lines are the sameas black lines but with 𝐴 → − 𝐴 . - - - - Figure 7.
General-model, equation (5). The monotonic behaviour of 𝜔 ( 𝑧 ) is achieved when the roots of 𝜔 (cid:48) ( 𝑧 ) are complex. In this case, 𝐴 = . 𝑛 = . 𝐶 = .
5, dashed black linehas 𝐶 =
2, dotted black line has 𝐶 =
3, and dot-dashed black line has 𝐶 = 𝐴 → − 𝐴 . the generic behaviour of the EOS depicted in figure 6 with a localmaximum and minimum. It will be helpful to understand how 𝜔 ( 𝑧 ) ismodified by the parameter 𝐶 . As is shown in figure 8, the parameter 𝐶 can either increase or lower the general 𝜔 ( 𝑧 ) value alongside with asubtle displacement into the 𝑧 -axis direction. This feature allows us toput an anchor to the first critical point. Let us define 𝑧 𝑐 = min ( 𝑧 , 𝑧 ) and demand that 𝜔 ( 𝑧 ) must be -1 at 𝑧 𝑐 . Under that condition, 𝑧 𝑐 and 𝐶 are forced to be 𝑧 𝑐 = 𝑛 /( − 𝑛 ) and 𝐶 = 𝑛 𝑛 /( − 𝑛 ) − 𝑧 𝑐 , where 𝑧 𝑐 acts as the anchor point (figure 9), and can be compared with case II(figure 2), when 𝑛 = 𝐶 ∼
1, for example.It is worth to mention that 𝜔 ( 𝑧 ) = − − 𝐴𝑒 − 𝑧 ( 𝑧 𝑛 − 𝑧 − 𝐶 ) , for 𝑛 >
1, has equivalent qualitative attributes to 𝑛 ∈ ( , ) case: (1)at most two critical points, and (2) monotone behaviour when 𝑧 , 𝑧 ∈ C . In particular, the 𝑛 > 𝑛 ∈ ( , ) case. This is why we limit our analysis to 𝑛 ∈ ( , ) .This will be reflected in the choice of priors for 𝑛 parameter.By constraining these sub-cases separately, we can statistically MNRAS , 1–15 (2021) ne parameterisation to fit them all - - - - Figure 8.
General-model, equation (5). In this figure is depicted the generalbehaviour of 𝜔 ( 𝑧 ) when 𝐶 is varying. In this case, 𝐴 = . 𝑛 = . 𝐶 = .
08, dashed black line is when 𝐶 = .
3, dotted black line is when 𝐶 = .
6, and dot-dashed black line iswhen 𝐶 = .
9. Gray lines are the same as black lines but with 𝐴 → − 𝐴 . - - - - Figure 9.
General-model, equation (5). The amplitude and the constant valueare fixed for all the curves, 𝐴 = . 𝐶 = 𝑛 𝑛 /( − 𝑛 ) − 𝑛 /( − 𝑛 ) . Solidblack line is when 𝑛 = .
2, dashed black line is for 𝑛 = .
4, dotted black lineis when 𝑛 = .
6, and dot-dashed black line is when 𝑛 = .
8. Gray lines arethe same as black lines but with 𝐴 → − 𝐴 . study the constraints on a wide variety of models. By fitting thegeneral form of our EOS against a collection of data ranging from 𝑧 cmb all the way down to 𝑧 ≈ .
01, we can answer the question ofwhich one of the allowed scenarios is preferred by observations. Isthe simplest case of a cosmological constant the favored model? Dodifferent data sets point towards different dynamics of DE?
We model the accelerated expansion of the Universe in terms ofa barotropic fluid, 𝜌 𝑥 , described in terms of the equation of state 𝜔 𝑥 ≡ 𝑝 𝑥 / 𝜌 𝑥 .Within the validity of General Relativity for a flat Universe and aFLRW metric, we can express the Friedmann equation as: 𝐻 ( 𝑧 )/ 𝐻 = Ω 𝑚 ( + 𝑧 ) + Ω 𝑟 ( + 𝑧 ) + Ω 𝐷𝐸 𝐹 𝐷𝐸 ( 𝑧 ) , (7) where Ω 𝑚 + Ω 𝑟 + Ω 𝐷𝐸 = 𝐻 = ℎ𝑘𝑚 / 𝑠 / 𝑀 𝑝𝑐 is the Hubbleconstant, and 𝐹 𝐷𝐸 ( 𝑧 ) is a function of redshift involving the specificform of 𝜔 𝑥 . For a Cosmological Constant, 𝜔 𝑥 = −
1, and 𝐹 Λ ( 𝑧 ) = 𝐹 𝐷𝐸 ( 𝑧 ) = exp (cid:18)∫ 𝑧 ( + 𝜔 𝑥 ( 𝑧 (cid:48) )) + 𝑧 (cid:48) 𝑑𝑧 (cid:48) (cid:19) , (8)where 𝜔 𝑥 is the one given by (1).From equations (7) and (8) it is clear how data coming fromcosmological distances can be used to constrain the free parametersin (1). In order to probe the parameters in equation (1) we use different cos-mological distance measurements, covering a wide range of redshifts:0 . (cid:46) 𝑧 (cid:46) The Baryon Acoustic Oscillations feature is an imprint on the spatialdistribution of galaxies and luminous tracers. It was detected for thefirst time by (Colless et al. 2003; Eisenstein et al. 2005) and has beenexplored with increasing detail becoming a powerful tool for cosmol-ogy. It has consolidated as one of the most robust ways to probe latetime dynamics of the Universe, as shown in several observational ef-forts like those carried by experiments like 6dF (Beutler et al. 2011),WiggleZ (Kazin et al. 2014), Dark Energy Survey (DES) (Abbottet al. 2005) and the SDSS consortium (Anderson et al. 2014; Alamet al. 2016; Dawson et al. 2016), finalizing with their latest and finalreport on (Alam et al. 2020). BAO is also one of the main featuresto be probed by experiments like the Dark Energy SpectroscopicInstrument (DESI) (Levi et al. 2013; Aghamousa et al. 2016a,b) andin the near future, Euclid (Laureijs et al. 2011).In this work we use the spherically averaged BAO signature, interms of the size 𝑟 𝐵𝐴𝑂 ( 𝑧 ) : 𝑟 𝐵𝐴𝑂 ( 𝑧 ) ≡ 𝑟 𝑠 ( 𝑧 𝑑 ) 𝐷 𝑉 ( 𝑧 ) , (9)where the comoving sound horizon at the baryon drag epoch is repre-sented by 𝑟 𝑠 ( 𝑧 𝑑 ) , and the dilation scale, 𝐷 𝑉 ( 𝑧 ) , contains informationabout the cosmology used in 𝐻 ( 𝑧 ) : 𝑟 𝑠 ( 𝑧 𝑑 ) ≡ ∫ ∞ 𝑧 𝑑 𝑑𝑧𝐻 ( 𝑧 ) √︁ ( ˜ 𝑅 ( 𝑧 ) + ) , (10) 𝐷 𝑉 ( 𝑧 ) ≡ (cid:20) 𝑧 ( + 𝑧 ) 𝐻 ( 𝑧 ) 𝐷 𝐴 ( 𝑧 ) (cid:21) / , (11)where ˜ 𝑅 ( 𝑧 ) is the baryon to photon ratio, defined by ˜ 𝑅 ( 𝑧 ) ≡ Ω 𝛾 ( 𝑧 ) Ω 𝑏 ( 𝑧 ) ,and the angular diameter distance, 𝐷 𝐴 ( 𝑧 ) , given by: 𝐷 𝐴 ( 𝑧 ) = + 𝑧 ∫ 𝑧 𝑑𝑧 (cid:48) 𝐻 ( 𝑧 (cid:48) ) , (12)where we can see clearly how to use the BAO standard ruler toconstrain the parameters in equation (1). The sound horizon, 𝑟 𝑠 ( 𝑧 𝑑 ) ,depends upon the physics prior to the recombination era, given by 𝑧 𝑑 ≈ 𝑅 ( 𝑧 ) .However, the dilation scale, 𝐷 𝑉 ( 𝑧 ) , is sensitive to the physics of MNRAS000
1, and 𝐹 Λ ( 𝑧 ) = 𝐹 𝐷𝐸 ( 𝑧 ) = exp (cid:18)∫ 𝑧 ( + 𝜔 𝑥 ( 𝑧 (cid:48) )) + 𝑧 (cid:48) 𝑑𝑧 (cid:48) (cid:19) , (8)where 𝜔 𝑥 is the one given by (1).From equations (7) and (8) it is clear how data coming fromcosmological distances can be used to constrain the free parametersin (1). In order to probe the parameters in equation (1) we use different cos-mological distance measurements, covering a wide range of redshifts:0 . (cid:46) 𝑧 (cid:46) The Baryon Acoustic Oscillations feature is an imprint on the spatialdistribution of galaxies and luminous tracers. It was detected for thefirst time by (Colless et al. 2003; Eisenstein et al. 2005) and has beenexplored with increasing detail becoming a powerful tool for cosmol-ogy. It has consolidated as one of the most robust ways to probe latetime dynamics of the Universe, as shown in several observational ef-forts like those carried by experiments like 6dF (Beutler et al. 2011),WiggleZ (Kazin et al. 2014), Dark Energy Survey (DES) (Abbottet al. 2005) and the SDSS consortium (Anderson et al. 2014; Alamet al. 2016; Dawson et al. 2016), finalizing with their latest and finalreport on (Alam et al. 2020). BAO is also one of the main featuresto be probed by experiments like the Dark Energy SpectroscopicInstrument (DESI) (Levi et al. 2013; Aghamousa et al. 2016a,b) andin the near future, Euclid (Laureijs et al. 2011).In this work we use the spherically averaged BAO signature, interms of the size 𝑟 𝐵𝐴𝑂 ( 𝑧 ) : 𝑟 𝐵𝐴𝑂 ( 𝑧 ) ≡ 𝑟 𝑠 ( 𝑧 𝑑 ) 𝐷 𝑉 ( 𝑧 ) , (9)where the comoving sound horizon at the baryon drag epoch is repre-sented by 𝑟 𝑠 ( 𝑧 𝑑 ) , and the dilation scale, 𝐷 𝑉 ( 𝑧 ) , contains informationabout the cosmology used in 𝐻 ( 𝑧 ) : 𝑟 𝑠 ( 𝑧 𝑑 ) ≡ ∫ ∞ 𝑧 𝑑 𝑑𝑧𝐻 ( 𝑧 ) √︁ ( ˜ 𝑅 ( 𝑧 ) + ) , (10) 𝐷 𝑉 ( 𝑧 ) ≡ (cid:20) 𝑧 ( + 𝑧 ) 𝐻 ( 𝑧 ) 𝐷 𝐴 ( 𝑧 ) (cid:21) / , (11)where ˜ 𝑅 ( 𝑧 ) is the baryon to photon ratio, defined by ˜ 𝑅 ( 𝑧 ) ≡ Ω 𝛾 ( 𝑧 ) Ω 𝑏 ( 𝑧 ) ,and the angular diameter distance, 𝐷 𝐴 ( 𝑧 ) , given by: 𝐷 𝐴 ( 𝑧 ) = + 𝑧 ∫ 𝑧 𝑑𝑧 (cid:48) 𝐻 ( 𝑧 (cid:48) ) , (12)where we can see clearly how to use the BAO standard ruler toconstrain the parameters in equation (1). The sound horizon, 𝑟 𝑠 ( 𝑧 𝑑 ) ,depends upon the physics prior to the recombination era, given by 𝑧 𝑑 ≈ 𝑅 ( 𝑧 ) .However, the dilation scale, 𝐷 𝑉 ( 𝑧 ) , is sensitive to the physics of MNRAS000 , 1–15 (2021)
G. Arciniega, et al. much lower redshifts, particularly to those probed by large scalestructure experiments.In this work we make use of the observational points from thesix-degree-field galaxy survey (6dFGS Beutler et al. (2011)), SloanDigital Sky Survey Data Release 7 (SDSS DR7 Ross et al. (2015))and the reconstructed value (SDSS(R) Padmanabhan et al. (2012)),as well as the uncorrelated values reported in the complete BOSSsample SDSS DR12 (Alam et al. (2016)). We included the mea-surement done in the auto and cross-correlation of the Lymann- 𝛼 Forest (Ly 𝛼 -F) measurements from the quasars sample of the 11thData Release of the Baryon Oscillation Spectroscopic (BOSS DR11)Delubac et al. (2015); Font-Ribera et al. (2014). In total we coverthe redshift range 0 . < 𝑧 < .
36. Since the volume surveyedby BOSS and WiggleZ Kazin et al. (2014) partially overlap Beutleret al. (2016), we do not use data from the latter in this work. As inthis case all the measurements we are using are independent, we canwrite the 𝜒 𝐵𝐴𝑂 in terms of the observed values 𝑟 𝑜𝑏𝑠𝐵𝐴𝑂 with theircorresponding errors 𝜎 𝑖 , and the predicted values 𝑟 𝑡ℎ𝐵𝐴𝑂 as: 𝜒 𝐵𝐴𝑂 = ∑︁ 𝑖 (cid:16) 𝑟 𝑜𝑏𝑠𝐵𝐴𝑂 ( 𝑧 𝑖 ) − 𝑟 𝑡ℎ𝐵𝐴𝑂 ( 𝑧 𝑖 ) (cid:17) 𝜎 𝑖 (13) In Jimenez & Loeb (2002), the use of the relative ages of galaxieswas proposed to track the expansion of the universe. This method wascoined “cosmic chronometers”. In Moresco et al. (2011) the authorspresented a new methodology by using the spectral properties ofearly-type galaxies and showed that including the effect of metallicityimpacts their results by less than 2 − 𝐻 ( 𝑧 ) , that does not depend on the cosmology model,unlike the case of BAO or Supernovae measurements. Another ad-vantage lies in the fact that, unlike the distance measurements, wedo not rely on the integral of 𝐻 ( 𝑧 ) to constrain the parameters in theEOS (1). It is convenient to write the expansion rate as 𝐻 ( 𝑧 ) = (cid:164) 𝑎𝑎 = − + 𝑧 𝑑𝑧𝑑𝑡 . (14)With the relation of the Hubble parameter written in this way, it ispossible to use the redshift of the galaxies that are taken as chronome-ters because its redshift can be measured with high accuracy. Thedifferential expression for 𝑑𝑧 and 𝑑𝑡 helps to cancel out systematicerrors and the possible effects given by the bias (see Moresco et al.(2012) for a detailed revision of the method).In this work we use the sample compiled in Farooq & Ratra (2013),which covers the redshift range 0 . < 𝑧 < .
3, with 28 independentmeasurements of the Hubble parameter. The value of 𝜒 will beestimated as: 𝜒 𝐶𝐶 = ∑︁ 𝑖 (cid:16) 𝐻 ( 𝑧 ) 𝑜𝑏𝑠𝐶𝐶 − 𝐻 ( 𝑧 ) 𝑡ℎ𝐶𝐶 (cid:17) 𝜎 𝑖 , (15)where ( ) 𝑜𝑏𝑠 , and ( ) 𝑡ℎ , stands for observational values and predictedvalues of the theory, respectively. Type-Ia supernovae were crucial for the discovery of the acceleratedexpansion of the Universe and are angular cosmological probes. Ever since the discovery made by Perlmutter et al. (1999) and Riess et al.(1998), they had played a crucial role in discovering the cosmic ac-celeration and have consolidated as one of most useful and powerfultools to investigate the nature behind the cosmic acceleration. Sev-eral high quality samples have been released over the past decade(Kowalski et al. 2008; Hicken et al. 2009; Kessler et al. 2009; Aman-ullah et al. 2010a; Conley et al. 2011; Suzuki et al. 2012; Betouleet al. 2014; Scolnic et al. 2018).The 𝜒 function of Supernovae Ia can be expressed as 𝜒 𝑆𝑁 𝑒 = Δ 𝜇 𝑇 · 𝐶 − · Δ 𝜇 (16)where Δ 𝜇 ≡ 𝜇 𝑜𝑏𝑠 − 𝜇 𝑡ℎ . We take 𝐶 = 𝐷 𝑠𝑡𝑎𝑡 and 𝜇 𝑜𝑏𝑠 from thecompilation presented in (Suzuki et al. 2012) , and estimate 𝜇 𝑡ℎ , thedistance modulus of the luminosity distance, as: 𝜇 ( 𝑧 𝑖 ) = (cid:20) ( + 𝑧 ) 𝐻 ∫ 𝑧 𝑑𝑧 (cid:48) 𝐻 − ( 𝑧 (cid:48) ) (cid:21) + , (17)where 𝐻 ( 𝑧 ) contains the free parameters of (1) through eq. (7). Eventhough SNe provide a measurement of the luminosity distance asa function of redshift, their absolute luminosity is uncertain and ismarginalized out, which also removes any constraints on 𝐻 . Forthat reason we do not include ℎ as part of the parameter vector to beconstrained during the analysis when we use only this sample, andwe consider a given value ℎ = .
7, as was done in (Amanullah et al.2010b; Suzuki et al. 2012).This sample covers the range 0 . < 𝑧 < .
03 with a total of 557data points.
In order to add information from the CMB we follow the strategyused by (Ade et al. 2016b), suggested in (Mukherjee et al. 2008). InMukherjee et al. (2008) it was shown how to compress the informa-tion of CMB power spectra within few observable quantities such asthe angular scale of sound horizon at last scattering, 𝑙 𝐴 ≡ 𝜋 / 𝜃 ∗ , thescaled distance to last scattering surface, 𝑅 ≡ √︃ Ω 𝑀 𝐻 𝑑 𝐴 ( 𝑧 ∗ ) , thebaryon density, Ω 𝑏 ℎ , and the scalar spectral index, 𝑛 𝑠 .For correlated data, the 𝜒 estimator reads as 𝜒 = Σ 𝑖 𝑗 ( 𝐷 𝑖 − 𝑦 ( 𝑥 𝑖 | 𝜃 )) 𝑄 𝑖 𝑗 (cid:0) 𝐷 𝑗 − 𝑦 ( 𝑥 𝑗 | 𝜃 ) (cid:1) (18)where 𝑄 𝑖 𝑗 = 𝐶 − 𝑖 𝑗 , is the inverse of the covariance matrix of the data.In the particular case of 𝜒 𝐶𝑀 𝐵 , we have 𝜒 𝐶𝑀 𝐵 = (cid:174) 𝑦 𝐶𝑀 𝐵 · C − 𝐶𝑀 𝐵 · (cid:174) 𝑦 𝐶𝑀 𝐵 (19)where C − 𝐶𝑀 𝐵 is the inverse of the covariance matrix and (cid:174) 𝑦 𝐶𝑀 𝐵 = 𝐷 𝑖 − 𝑦 ( 𝑥 𝑖 | 𝜃 ) given in terms of the data vector, 𝐷 𝑖 = ( 𝑅, 𝑙 𝐴 , 𝜔 𝑏 , 𝑛 𝑠 ) ,and 𝑦 ( 𝑥 𝑖 | 𝜃 ) = ( 𝑅 ( 𝑧, 𝜃 ) , 𝑙 𝐴 ( 𝑧, 𝜃 ) , 𝜔 𝑏 , 𝑛 𝑠 ) , the theoretical predictionthat depends on the free parameters: (cid:174) 𝜃 = { 𝐴, 𝑛, 𝐶, ℎ, Ω 𝑏 ℎ , Ω 𝑐 ℎ } .In this case, the inverse of the covariance matrix, C − , is C − = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) R l A 𝜔 b n s R . − . . × . l A − . . . × − . 𝜔 b . . . × − . n s . − . − . × . (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) Data can be found in http://supernova.lbl.gov/Union/ .MNRAS , 1–15 (2021) ne parameterisation to fit them all (20)where we have chosen the more conservative compressed likelihoodvalues from Planck
TT +lowP marginalizing over the amplitude ofthe lensing power, 𝐴 𝐿 as presented in (Ade et al. 2016b).The angle of horizon at last scattering is defined to be 𝜃 ∗ ≡ 𝑟 𝑠 ( 𝑧 ∗ ) 𝑑 𝐴 ( 𝑧 ∗ ) , (21)where 𝑟 𝑠 ( 𝑧 ∗ ) is the horizon size at the decoupling epoch ( 𝑧 ∗ ≈ .
95 according to Planck (Ade et al. 2016a)), defined by theintegral in equation (10) evaluated from 𝑧 ∗ to ∞ , and 𝑑 𝐴 ( 𝑧 ∗ ) is thecomoving distance to last scattering surface: 𝑑 𝐴 ( 𝑧 ∗ ) = ∫ 𝑧 ∗ 𝑑𝑧 (cid:48) 𝐻 ( 𝑧 (cid:48) ) . (22)Introduced in this way, we are making use of the position thatcorresponds to the sharply-defined acoustic angular scale on the sky,and of the relative heights of the successive peaks seen in the CMBpower spectra. In our analysis we combine the different measurements: BAO, CC,SNe and CMB by adding their respective 𝜒 functions, as they areall independent from each other and are probing different cosmicepochs.In this manner, we write down the combination of all the data as: 𝜒 𝑇 𝑜𝑡𝑎𝑙 = 𝜒 𝐵𝐴𝑂 + 𝜒 𝐶𝑀 𝐵 + 𝜒 𝐶𝐶 + 𝜒 𝑆𝑁 𝑒 , (23)where each function is defined as explained in section 5.1.Furthermore we are interested in the sample of standard rulers,fixed in the CMB and detected in the clustering of luminous tracersvia the BAO. This will be defined as the combination: 𝜒 𝐵𝐴𝑂 − 𝐶𝑀 𝐵 = 𝜒 𝐵𝐴𝑂 + 𝜒 𝐶𝑀 𝐵 , (24)to explore the constraining power of acoustic oscillations.Additionally, we want to investigate the constrains coming fromlate time observations, and to that end we define the function: 𝜒 𝑙𝑎𝑡𝑒 = 𝜒 𝐵𝐴𝑂 + 𝜒 𝐶𝐶 + 𝜒 𝑆𝑁 𝑒 , (25)where we ignore the CMB data.Even more, we investigate the constrains in our free parametersfrom the CC, 𝜒 𝐶𝐶 (15), and the SNe samples (16), 𝜒 𝑆𝑁 𝑒 , indepen-dently.For Λ 𝐶𝐷𝑀 , the energy density fraction for DE is constant and,we know that for a flat Universe, we can simply express it by theflatness condition, Ω Λ = − Ω 𝑚 − Ω 𝑟 . However, with a differentdynamics for dark energy, this cannot be assumed to be equal to thefiducial value provided by the Planck collaboration (Aghanim et al.2018), for instance, for Λ 𝐶𝐷𝑀 . This means that, in addition to 𝐴 , 𝑛 ,and 𝐶 , the free parameters in (1), we let the physical densities, Ω 𝑐 ℎ , Ω 𝑏 ℎ , and the reduced Hubble constant, ℎ , free.The free parameters were varied within uniform priors: 𝐴 ∈[− , ] , 𝑛 ∈ [ , ] , 𝐶 ∈ [− , ] , Ω 𝑐 ℎ ∈ [ . , . ] , Ω 𝑏 ℎ ∈ [ . , . ] , and the Hubble parameter ℎ ∈ [ . , . ] .However, not all data samples have the same constraining powerover different cosmological parameters. In particular, if CMB datais not included in the fitting process, we fix Ω 𝑏 ℎ = . Planck
TT + lowP likelihood Ade et al. (2016a).We individually optimize the parameters in each case by minimiz-ing the 𝜒 statistic A . . . . n CC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC (a) Full explored parameter space 𝐴 − 𝑛 within the most general case of theparameterisation, (3.4). − − A . . . . n CC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNe (b) Close up in the region 𝐴 ∈ [− , ] Figure 10. 𝜎 confidence levels in the parameter space 𝐴 − 𝑛 for thegeneral model (3.4) using the likelihoods described in (15), (16), (23), (24),and (25). We implement our parameterisation to observational data by a nu-merical optimisation on its parameters space, using the techniquesand computational code developed for (Jaime et al. 2018), where weperform a numerical implementation of the Differential Evolutionalgorithm. This is a stochastic population based method, particu-larly useful for global optimization problems. Assuming gaussianl-distributed errors on the measurements, we treat the likelihood asa multivariate Gaussian. Then, our best fit value parameters arethese that minimize the 𝜒 estimator, and the reduced chi-squared, 𝜒 𝑟𝑒𝑑 = 𝜒 𝑚𝑖𝑛 /( 𝑑.𝑜. 𝑓 . ) , as our statistical measure of the goodness ofthe fit.Given that we obtained a good fit for all our models and likelihoods(see last column of Table 1), of order unity (close to 1), we proceeddiscussing our results as follows.We report the 1-3 𝜎 confidence intervals for different combinationsin parameter space: the parameters of equation (1) 𝐴 − 𝑛 , 𝑛 − 𝐶 (figures10-11), and the parameters, Ω 𝑏 ℎ − ℎ , and Ω 𝑐 ℎ − ℎ (figures 12 and13, respectively). Also, we report the individual uncertainties after Our implementation will be made publicly available and a version of thecode can be shared upon reasonable request to the authors.MNRAS000
TT + lowP likelihood Ade et al. (2016a).We individually optimize the parameters in each case by minimiz-ing the 𝜒 statistic A . . . . n CC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCCCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC (a) Full explored parameter space 𝐴 − 𝑛 within the most general case of theparameterisation, (3.4). − − A . . . . n CC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNeCC ( Ω m fixed) BAO − CMB − SNe − CCBAO − SNe − CCBAO − CMBSNe (b) Close up in the region 𝐴 ∈ [− , ] Figure 10. 𝜎 confidence levels in the parameter space 𝐴 − 𝑛 for thegeneral model (3.4) using the likelihoods described in (15), (16), (23), (24),and (25). We implement our parameterisation to observational data by a nu-merical optimisation on its parameters space, using the techniquesand computational code developed for (Jaime et al. 2018), where weperform a numerical implementation of the Differential Evolutionalgorithm. This is a stochastic population based method, particu-larly useful for global optimization problems. Assuming gaussianl-distributed errors on the measurements, we treat the likelihood asa multivariate Gaussian. Then, our best fit value parameters arethese that minimize the 𝜒 estimator, and the reduced chi-squared, 𝜒 𝑟𝑒𝑑 = 𝜒 𝑚𝑖𝑛 /( 𝑑.𝑜. 𝑓 . ) , as our statistical measure of the goodness ofthe fit.Given that we obtained a good fit for all our models and likelihoods(see last column of Table 1), of order unity (close to 1), we proceeddiscussing our results as follows.We report the 1-3 𝜎 confidence intervals for different combinationsin parameter space: the parameters of equation (1) 𝐴 − 𝑛 , 𝑛 − 𝐶 (figures10-11), and the parameters, Ω 𝑏 ℎ − ℎ , and Ω 𝑐 ℎ − ℎ (figures 12 and13, respectively). Also, we report the individual uncertainties after Our implementation will be made publicly available and a version of thecode can be shared upon reasonable request to the authors.MNRAS000 , 1–15 (2021)
G. Arciniega, et al. − − C . . . . n CC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCBAO − CMBCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCBAO − CMBCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCBAO − CMBCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCBAO − CMBCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCBAO − CMBCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCBAO − CMB (a) Full explored parameter space 𝑛 − 𝐶 within the most general case of theparameterisation, (3.4). . . . . C . . . . n CC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCCCC ( Ω m fixed) SNeBAO − SNe − CCBAO − CMB − SNe − CCCC (b) Close up in the region 𝐶 ∈ [− . , . ] Figure 11. 𝜎 confidence levels in the parameter space 𝑛 − 𝐶 for thegeneral model (3.4) using the likelihoods described in (15), (16), (23), (24),and (25). marginalization over the other dimensions, and these can be foundin Table 1. Once we know the constraints on individual parameters,we report the resulting dynamical behaviour of 𝜔 ( 𝑧 ) (5). These areshown Table 2.We can see how sensitive the parameterisation (5) is to different setof observational data when we fit simultaneously all three parameters, 𝐴 , 𝑛 and 𝐶 . To this end we discuss the joint constraints on the 𝐴 − 𝑛 and 𝑛 − 𝐶 parameter spaces for the unrestricted model, (5), using thedifferent data sets as described in section 5.1. 𝐴 - 𝑛 contour plots Figure 10 shows the 1-3 𝜎 joint confidence levels (CL) for the param-eters 𝑛 and 𝐴 in the general model, (5), fitted by each set of observa-tions. It is noticeable how different observations constrain differentlythe behaviour of 𝑛 . In particular, from the Cosmic Chronometers(CC) sample, its value is tightly constrained around 𝑛 ≈
0, whereasfor the joint likelihood BAO-CMB-SNe-CC, it is consistent with 𝑛 ≥ .
4. In the same figure, for the case of equation (5) constrainedwith the CC sample, we see that the resulting dynamics agrees withthat of a cosmic fluid with a dust-like equation of state 𝜔 ≈ Ω 𝑚 . To further test this hy- pothesis we reanalyzed the sample fixing the value of Ω 𝑚 to the onereported by Aghanim et al. (2018) ( Ω 𝑐 ℎ = . 𝐴 ∈ [− , ] in the pa-rameter space 𝐴 − 𝑛 . Here we can appreciate better the fact that theBAO-CMB joint likelihood constraints tightly the value of 𝐴 aroundthe value 𝐴 = . ± .
01. It is important to recall that, at 1- 𝜎 level, the value 𝐴 = 𝜔 = −
1, which recovers acosmological constant model. 𝐶 - 𝑛 contour plots From the 𝑛 − 𝐶 CL, figure 11, we notice that the value 𝑛 = 𝜎 level by the SNe sample and the joint likelihoodsBAO-SNe-CC and BAO-CMB-SNe-CC, this is, all the data sets an-alyzed which included the SNe sample. On the other hand, the CCsample, imposes very tight constraints on the value of 𝑛 ≈
0. Let uspoint out again that for this particular result we recover a dust-likeEOS with almost no matter ( Ω 𝑀 ≈ . Ω 𝑀 , we rerunthe analysis Ω 𝑐 ℎ = . Planck
TT+lowP). In this case, we findthat the value of 𝑛 is not constrained, allowing an uniform variationalong the 𝑛 axis. Similarly, for the joint acoustic oscillations sample,BAO-CMB, we find that these are insensitive to the value of 𝑛 .Looking at the 𝐶 axis of figure 11 we see that the joint likelihoodBAO-CMB, constrains 𝐶 around 𝐶 ≈
2, while the sample of CCwith the prior on Ω 𝑚 from Planck, imposes the weakest constraintson this parameter around 𝐶 ≈
0. However, from the same sample,without fixing the value for Ω 𝑚 , we find very tight constraints for 𝐶 ≈ .
8. The supernovae sample, by its own, constrains 𝐶 around 𝐶 =
0, as we can see from the figure 11b.When used in combination with other data sets, as in BAO-SNe-CC, we find that the value 𝐶 = 𝜎 level. Forthe joint analysis of all the data sets, we find that, even when 𝐶 = .
7% of confidence, we obtain a value for
𝐶 < . 𝜔 close to 𝜔 = −
1. The case 𝐶 =
0, as discussed in section 4, gives a dynamics that is consistentwith an 𝑓 ( 𝑅 ) -like expansion for 𝜔 = − 𝑛 controls whether theparameterisation depicts one, two or no oscillations at all. It is worthto notice that the value 𝑛 = Ω 𝑚 fixed, do not constrain 𝑛 within the explored range, 𝑛 ∈ [ , ] . On the other hand, the Cosmic Clocks sample by itself,fixes 𝑛 = . + . − . . Ω 𝑏 ℎ - ℎ and Ω 𝑐 ℎ - ℎ contours. To explore more carefully these possibilities, we perform the sameanalysis for the other three cases of our proposal: the particular case 𝑛 = 𝑛 = 𝜔 to cross only once the phantom dividingline, 𝜔 = −
1, (II: Quintessence/Phantom or Quintom, figures 2 and3), and the case 𝐶 =
0, which presents an oscillatory behaviouraround 𝜔 = − 𝑓 ( 𝑅 ) , figures 4 and 5).By performing this analysis we can investigate if some of thefeatures marked by the value of the parameters have a statistical MNRAS , 1–15 (2021) ne parameterisation to fit them all preference. We compare the particular cases I, II, III against thegeneral model (IV), and with the concordance Λ 𝐶𝐷𝑀 scenario.Figures 12 and 13 summarize our results for this part of the analysis,along with the figures in table 2. In both figures we present theconstrains on the different models using the total likelihood, i.e. , 𝜒 𝑇 𝑜𝑡𝑎𝑙 (23), and the acoustic oscillations observations, i.e. , BAO-CMB (24). As it was detailed in section 3, each particular case of (1)is referred to as a different model since each choice is motivated bya specific dynamical behaviour.Figure 12 presents the parameter space for the physical density ofcold dark matter, Ω 𝑐 ℎ , and the Hubble parameter, ℎ . The fist thingwe notice in this case, is that the constrains are more extended forthe BAO-CMB likelihood (represented by dotted contour lines) thanfor the combination of all data sets (shown in solid contour lines).Moving away from that observation to more specific , we see thatdifferent models agree with different values of Ω 𝑐 ℎ and ℎ .Focusing first on the constraints from BAO-CMB, we see that Λ CDM gives a higher ℎ and large amount of matter while the modelQuintessence/Phantom, on the contrary, is consistent with a lower ℎ and smaller amount of matter. Using BAO-CMB data sets, onthe space of cosmological parameters Ω 𝑐 ℎ - ℎ , it is not possible todistinguish the Exponential model from the general form of the pa-rameterisation, or from the 𝑓 ( 𝑅 ) -like background expansion. Whichis to say that the Exponential ( 𝑛 = 𝐶 = 𝑓 ( 𝑅 ) -like ( 𝐶 = Ω 𝑐 ℎ − ℎ . However, we must remember that eachone is quite distinctive from the other in the space of their respectiveparameters, 𝑛 and 𝐶 . Interestingly, the Quintessence/Phantom model(II) is consistent with the low 𝐻 value reported by Planck collabora-tion, while the rest of the models (I Exponential, III 𝑓 ( 𝑅 ) , and IV, thegeneral model) have an 𝐻 value consistent with the determinationfor the Hubble parameter using the Tip of the Red Giants Branch(TRGB) done by Freedman et al. (2019), which sits midway in therange defined by the current Hubble tension (and indicated by theorange shaded area around ℎ = . Ω 𝑐 ℎ − ℎ space,as compared to those obtained only from the acoustic oscillations.In particular, we find that the general form of the EOS (model IV)is consistent with a lower Ω 𝑐 ℎ , while Λ CDM prefers a slightlyhigher value. Exponential (I) and Quintessence/Phantom (II) modelsagree with each other at the 1- 𝜎 level, as well as the Exponential(I) and 𝑓 ( 𝑅 ) -like models. The models Quintessence/Phantom (II)and 𝑓 ( 𝑅 ) -like (III) are consistent with each other only at the 2- 𝜎 level. However, all the resulting CL lie within the uncertainties of theTRGB determination of 𝐻 (Freedman et al. 2019).For the sake of clarity, we present the CL in Ω 𝑏 ℎ − ℎ spacein two separate figures. Figure 13a shows the resulting contoursfrom the acoustic oscillations, BAO-CMB, while the resulting con-straints from the combination of all data sets, BAO-CMB-SNe-CC,can be seen in figure 13b. Similarly to the Ω 𝑐 ℎ − ℎ contours, inthe Ω 𝑏 ℎ − ℎ space. From figure 13a (top panel of 13) we find thatthe Quintessence/Phantom model (case II, 𝑛 =
0) is consistent witha low value of ℎ which lies within Planck’s determination of 𝐻 .In contrast, Λ CDM is consistent with a higher value of ℎ , whichcoincides with the local determination of 𝐻 made by SH0ES (Riesset al. 2019). Models I (Exponential, 𝑛 = 𝐶 = 𝑓 ( 𝑅 ) with 𝐶 = 𝑛 ∈ [ , ] ), are consistentwithin each other with 99.7% of confidence. They share a value of ℎ in agreement with the TRGB central value. Even when these threemodels cannot be discerned in the Ω 𝑏 ℎ − ℎ space using BAO-CMBdata, they are quite distinctive in the values of their respective param- .
625 0 .
65 0 .
675 0 . .
725 0 .
75 0 . h . . . . . Ω c h TRGB SH0ESPlanck Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Figure 12.
Confidence intervals for the space Ω 𝑐 ℎ − ℎ for the models Λ CDM( 𝐴 = 𝑛 = 𝐶 = 𝑛 = 𝑓 ( 𝑅 ) ( 𝐶 = 𝑛 ∈ [ , ] ). Joint likelihoods: 𝐵𝐴𝑂 − 𝐶 𝑀 𝐵 are showed in dotted contour lines and
𝑇 𝑜𝑡𝑎𝑙 with solidcontour lines. Vertical shaded zones show the different 𝐻 values from theCMB reported by Planck (Aghanim et al. 2018), the TRGB determination(Freedman et al. 2019), and the SH0ES experiment (Riess et al. 2019). eters ( 𝑛 and 𝐶 ). All the CL shown in 13a lie around a central valuefor Ω 𝑏 ℎ ≈ . 𝜒 𝑇 𝑜𝑡𝑎𝑙 .There is more dispersion around the Ω 𝑏 ℎ value compared to theBAO-CMB constraints. However, we see that the five models agreewithin 1- 𝜎 level with each other, both in the ℎ , and in the Ω 𝑏 ℎ dimensions. At 3- 𝜎 level, the constraints for ℎ lie within the un-certainties from the TRGB determination of 𝐻 . More in detail, weobserve that, at the 1- 𝜎 level, Λ CDM (gray contour) and the generalmodel (red contour) do not overlap in figure 13b.
Goodness of the fit
Table 1 shows the BFV for all models separated by observationaldata set. Columns 2, 3, 4 give the BFV for the parameters 𝐴 , 𝑛 and 𝐶 respectively with uncertainties at 3- 𝜎 , while columns 5, 6, 7, showthe values 𝜔 , Ω 𝑚 , and 𝐻 within 1- 𝜎 . The last column of this tableshows the 𝜒 𝑟𝑒𝑑 .From the 𝜒 𝑟𝑒𝑑 values, we point out that all fits were very closeor approximately of order unity. Also, we find little discrepanciesbetween each other and, moreover, when analyzed data sets individ-ually, we find better fits for our model than for Λ 𝐶𝐷𝑀 .On the other hand, we notice that from all cases, the best fitwas obtained for model IV ( 𝑓 ( 𝑅 ) -like) fitting the full likelihood, 𝜒 𝑇 𝑜𝑡𝑎𝑙 , equation (23). From Table 1 we see that this has a value of 𝜒 𝑟𝑒𝑑 = . 𝜒 𝐵𝐴𝑂 − 𝐶𝑀 𝐵 forwhich we find 𝜒 𝑟𝑒𝑑 = . 𝜒 𝑟𝑒𝑑 = . i.e. Λ 𝐶𝐷𝑀 , MNRAS000
Table 1 shows the BFV for all models separated by observationaldata set. Columns 2, 3, 4 give the BFV for the parameters 𝐴 , 𝑛 and 𝐶 respectively with uncertainties at 3- 𝜎 , while columns 5, 6, 7, showthe values 𝜔 , Ω 𝑚 , and 𝐻 within 1- 𝜎 . The last column of this tableshows the 𝜒 𝑟𝑒𝑑 .From the 𝜒 𝑟𝑒𝑑 values, we point out that all fits were very closeor approximately of order unity. Also, we find little discrepanciesbetween each other and, moreover, when analyzed data sets individ-ually, we find better fits for our model than for Λ 𝐶𝐷𝑀 .On the other hand, we notice that from all cases, the best fitwas obtained for model IV ( 𝑓 ( 𝑅 ) -like) fitting the full likelihood, 𝜒 𝑇 𝑜𝑡𝑎𝑙 , equation (23). From Table 1 we see that this has a value of 𝜒 𝑟𝑒𝑑 = . 𝜒 𝐵𝐴𝑂 − 𝐶𝑀 𝐵 forwhich we find 𝜒 𝑟𝑒𝑑 = . 𝜒 𝑟𝑒𝑑 = . i.e. Λ 𝐶𝐷𝑀 , MNRAS000 , 1–15 (2021) G. Arciniega, et al. A ( . ) n ( . ) C ( . ) 𝜔 ( ) Ω ( ) 𝑀 ( ) 𝐻 ( ) 𝜒 𝑟𝑒𝑑 SNeI (Exponential) − . + . − . − . ± .
035 0 . + . − . − . + . − . . + . − . -1.039 + . − . . ± .
011 70.00 0.978III ( 𝑓 ( 𝑅 ) , 𝜔 = −
1) 6 . + . − . . + . − . . ± .
013 70.00 0.979IV (General) 39 . + . − . . ± .
015 0 . ± . − . + . − . . ± .
012 70.00 0.980 Λ CDM 0.00 − − -1.00 0 . + . − . − . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . ± .
001 77 . ± .
52 0.594III ( 𝑓 ( 𝑅 ) , 𝜔 = −
1) 3 . + . − . . + . − . . ± .
003 70 . + . − . . + . − . ( . × − ) + . − . . + . − . . + . − . . + . − . . ± .
52 0.620IV ∗ (General) 0 . + . − . . + . − . − . + . − . -1.10 + . − . . ± .
019 74 . + . − . Λ CDM 0.00 − − -1.00 0 . ± .
002 68 . + . − . . + . − . ± .
006 0 . ± .
001 69 . ± .
21 1.763II (Quint./Phantom) − . + . − . . + . − . − . + . − . . ± .
001 66 . ± .
23 2.076III ( 𝑓 ( 𝑅 ) , 𝜔 = − − . ± .
272 0 . + . − . . ± .
001 69 . ± .
21 2.094IV (General) 0 . + . − . . + . − . . + . − . − . ± .
012 0 . ± .
001 69 . + . − . Λ CDM 0.00 − − -1.00 0 . ± . . ± .
19 1.563SNe-CC-BAOI (Exponential) − . + . − . ± .
026 0 . + . − . . + . − . . + . − . . + . − . − . ± .
085 0 . + . − . . ± .
26 0.973III ( 𝑓 ( 𝑅 ) , 𝜔 = −
1) 0 . + . − . . + . − . . ± .
007 69 . ± .
26 0.974IV (General) 1 . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . Λ CDM 0.00 − − -1.00 0 . ± .
007 69 . ± .
26 0.971SNe-CC-BAO-CMBI (Exponential) 0 . ± .
019 1.00 1.00 -1.05 ± .
006 0 . ± . . ± .
16 0.988II (Quint./Phantom) 0 . + . − . .
00 1 . ± . − . + . − . . ± . . ± .
16 0.989III ( 𝑓 ( 𝑅 ) , 𝜔 = − − . + . − . . + . − . . ± . . ± .
16 0.991IV (General) 2 . + . − . . ± .
011 0 . ± . − . + . − . . ± . . ± .
17 0.984 Λ CDM 0.00 − − -1.00 0 . ± . . ± .
16 0.989
Table 1.
Best fit values by data set. Column 1 refers the model: I-Exponential, II-Quintessence/Phantom, III- 𝑓 ( 𝑅 ) with 𝜔 ( ) = −
1, IV-General case and wecompare with Λ CDM in the last row for each observational set. Columns 2, 3 and 4 show the best fit values for the 𝐴 , 𝑛 and 𝐶 parameters. Columns 5, 6 and 7correspond to the values 𝜔 , Ω 𝑚 and 𝐻 respectively. Column 8 shows the value of the reduced 𝜒 the 𝜒 𝑟𝑒𝑑 is of the same order. This is to be expected, given the sizeor the error bars for this sample (see section 6.1).Taking a closer look at results from Table 1 each data set at a time,we find that: • The model IV (General) was the best fit for data sets SNe andthe CC sample. • The III ( 𝑓 ( 𝑅 ) -like) model, was the best fit for local data, BAO-SNe-CC, and also for the combination of all data sets. • Λ 𝐶𝐷𝑀 was the best fit only in the case of the acoustic oscilla-tions sample, BAO-CMB.To conservatively report our uncertainties for 𝐴 , 𝑛 , and 𝐶 , wequote them within 3- 𝜎 level, whereas the cosmological parameters Ω ( ) 𝑀 and ℎ are reported at 1 𝜎 to facilitate the comparison with otherworks. The full 1,2,3 𝜎 contours have been already discussed.The particularly compact constraints obtained from the CC sampleare explained due to the fact that we obtained a profile for the EOS that behaves as dust ( 𝜔 ∼
0) during its evolution, making this fluidnot negligible and hence, being able to constrain the values of theparameters in equation (5) very strictly. We can notice this in thefigure portrayed in the last column and second row of Table 2, wherewe notice clearly 𝜔 ( 𝑧 ) ≈ 𝑧 ∈ [ . , . ] . As the densityfor a dust-like component is non-negligible during this epoch, theirdynamics can be better constrained. As a counter example, we pointout the dynamics we obtained for the supernovae sample, undermodel II (Quintessence/Phantom). This can be seen in detail in thefirst row and second column of Table 2. In this case, the resultingdynamics for the EOS is that of a phantom component: for 𝑧 ≥ 𝜔 ( 𝑧 ) < −
1. Since this results in a highly sub-dominantcomponent, 𝜌 ≈ ( 𝜌 ( ) )( 𝑎 / 𝑎 ) − ( + 𝜔 ) , the involved parameters aremuch less tightly constrained.To directly show the resulting dynamics of our EOS for a givenmodel and how it is constrained by different data sets, in Table 2 we MNRAS , 1–15 (2021) ne parameterisation to fit them all .
64 0 .
66 0 .
68 0 . .
72 0 .
74 0 . h . . . . Ω b h Planck TRGB
SH0ES Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) (a) CL in Ω 𝑏 ℎ − ℎ for the models I-IV and Λ CDM from BAO-CMB jointlikelihood. .
67 0 .
68 0 .
69 0 . .
71 0 . h . . . . . Ω b h Planck TRGB Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) Λ CDM
I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) (b) CL in Ω 𝑏 ℎ − ℎ for the models I-IV and Λ CDM from BAO-CMB-SNe-CCjoint likelihood.
Figure 13.
Confidence intervals for the space Ω 𝑏 ℎ − ℎ for the models Λ CDM( 𝐴 = 𝑛 = 𝐶 = 𝑛 = 𝑓 ( 𝑅 ) ( 𝐶 = 𝑛 ∈ [ , ] ). Vertical shaded zonesshow the different 𝐻 values from the CMB reported by Planck (Aghanimet al. 2018), the TRGB determination (Freedman et al. 2019), and the SH0ESexperiment (Riess et al. 2019). depict the evolution of the EOS for each model according to its bestfit values.Table 2 shows all the different profiles for 𝜔 ( 𝑧 ) that we obtainwithin its 3- 𝜎 uncertainties (see appendix A). It is organized asfollows: Columns 1, 2, 3, and 4, show the evolution for cases Expo-nential, Quintessence/Phantom, 𝑓 ( 𝑅 ) with 𝜔 = −
1, and the Generalmodel, respectively. Each row shows the resulting constraints fromthe different data sets and their combinations.We particularly stress the general case and how the evolution ofthe EOS is shaped by the observational data sets. (Column 4) In thefirst row we notice that, for the 𝑧 − range of the SNe sample, the EOSprefers values 𝜔 ( 𝑧 ) < −
1. In the second row, interestingly, the CCallow to the EOS to take values very close to 𝜔 ≈
0, this way theparameterisation could mimic the EOS for dust, i.e. matter, and weobtain Ω 𝑚 = . 𝐴 vs 𝑛 (Fig 10) and 𝑛 vs 𝐶 (Fig. 11) we include one case where we fix thevalue of Ω 𝑚 = . 𝜔 ( 𝑧 ) = − 𝜔 ≠ −
1. For the late-timecollection (BAO-CC-SNe) the evolution of 𝜔 ( 𝑧 ) crosses twice thephantom-line, and shows an oscillatory behaviour. When the CMB is - - - - - Figure 14. (Color on line) Equation of state for the four models constrainedby the full observational data set at 99.7% of CL. Dotted (green) line showsthe evolution of model I (Exponential). Dotted-dashed (blue) line depictsthe behaviour of model II (Quint./Phantom). Dashed (orange) line shows theevolution for model III, 𝑓 ( 𝑅 ) , 𝜔 = −
1. Finally, the full model is depictedin the solid (red) line. included (last row) the behaviour is very similar and the uncertaintyis dramatically reduced.
Figure 14 shows the EOS for all the models constrained by the differ-ent observations used at 99 .
7% CL of the BFV. The evolution in theexponential case (green dotted line) goes very close to 𝜔 ( 𝑧 ) = − 𝑧 =
0, the EOS goes toward higher val-ues going to 𝜔 = − . ± . 𝜔 = − . + . − . .In this case, the EOS does not cross the phantom line, which is inconsistence with a single scalar field as in standard quintessencemodels. In the case of Model III, 𝑓 ( 𝑅 ) , 𝜔 = − 𝜔 = −
1, the evolution hasthe characteristic behaviour of 𝑓 ( 𝑅 ) and the 𝜒 𝑟𝑒𝑑 value is closer to1 than any other model of the parameterisation, including the case 𝐴 = Λ CDM model. Finally,the general model (red solid line), with three free parameters, showsan oscillatory behaviour that goes to 𝜔 = − . + . − . at 𝑧 = 𝜔 ( 𝑧 ) is not directly observable. Hence, to explore howdistinguishable the models are from each other, we compare their fitsto each set of data individually. Figure 15 shows the prediction forthe cosmic distances 𝜇 ( 𝑧 ) = 𝑚 − 𝑀 , for supernovae, and 𝑟 𝐵𝐴𝑂 ( 𝑧 ) according to the best fit values obtained constraining the respectivedata sample assuming each one of the models from table 1. In eachcase we show the direct prediction for the observable quantity (either 𝜇 ( 𝑧 ) , 𝑟 𝐵𝐴𝑂 ( 𝑧 ) , or 𝐻 ( 𝑧 )/( + 𝑧 ) ), and the ratio between the best fit formodels I-IV to Λ 𝐶𝐷𝑀 , Δ 𝐷 = ( 𝐷 − 𝐷 Λ 𝐶𝐷𝑀 )/ 𝐷 Λ 𝐶𝐷𝑀 . In figure15a we show 𝜇 ( 𝑧 ) = 𝑚 − 𝑀 for the best fit obtained of all cases.We focus only on the fits done to Union 2.1 supernovae sample. Theupper panel of the figure shows the predictions for 𝜇 = 𝑚 − 𝑀 vs 𝑧 according to the best fit values obtained for the models, along with theobservational points with error bars. The bottom panel shows the ratiobetween each model’s prediction for 𝜇 ( 𝑧 | 𝐴, 𝑛, 𝐶 ) , and 𝜇 ( 𝑧 ) Λ 𝐶𝐷𝑀 ,this is, Δ 𝜇 ( 𝑧 ) = ( 𝜇 ( 𝑧 ) − 𝜇 ( 𝑧 ) Λ 𝐶𝐷𝑀 )/ 𝜇 ( 𝑧 ) Λ 𝐶𝐷𝑀 . In this case wefind differences of the order Δ 𝜇 ( 𝑧 ) = − . 𝑟 𝐵𝐴𝑂 ( 𝑧 ) vs 𝑧 for our models, along with the observations MNRAS000
1, the evolution hasthe characteristic behaviour of 𝑓 ( 𝑅 ) and the 𝜒 𝑟𝑒𝑑 value is closer to1 than any other model of the parameterisation, including the case 𝐴 = Λ CDM model. Finally,the general model (red solid line), with three free parameters, showsan oscillatory behaviour that goes to 𝜔 = − . + . − . at 𝑧 = 𝜔 ( 𝑧 ) is not directly observable. Hence, to explore howdistinguishable the models are from each other, we compare their fitsto each set of data individually. Figure 15 shows the prediction forthe cosmic distances 𝜇 ( 𝑧 ) = 𝑚 − 𝑀 , for supernovae, and 𝑟 𝐵𝐴𝑂 ( 𝑧 ) according to the best fit values obtained constraining the respectivedata sample assuming each one of the models from table 1. In eachcase we show the direct prediction for the observable quantity (either 𝜇 ( 𝑧 ) , 𝑟 𝐵𝐴𝑂 ( 𝑧 ) , or 𝐻 ( 𝑧 )/( + 𝑧 ) ), and the ratio between the best fit formodels I-IV to Λ 𝐶𝐷𝑀 , Δ 𝐷 = ( 𝐷 − 𝐷 Λ 𝐶𝐷𝑀 )/ 𝐷 Λ 𝐶𝐷𝑀 . In figure15a we show 𝜇 ( 𝑧 ) = 𝑚 − 𝑀 for the best fit obtained of all cases.We focus only on the fits done to Union 2.1 supernovae sample. Theupper panel of the figure shows the predictions for 𝜇 = 𝑚 − 𝑀 vs 𝑧 according to the best fit values obtained for the models, along with theobservational points with error bars. The bottom panel shows the ratiobetween each model’s prediction for 𝜇 ( 𝑧 | 𝐴, 𝑛, 𝐶 ) , and 𝜇 ( 𝑧 ) Λ 𝐶𝐷𝑀 ,this is, Δ 𝜇 ( 𝑧 ) = ( 𝜇 ( 𝑧 ) − 𝜇 ( 𝑧 ) Λ 𝐶𝐷𝑀 )/ 𝜇 ( 𝑧 ) Λ 𝐶𝐷𝑀 . In this case wefind differences of the order Δ 𝜇 ( 𝑧 ) = − . 𝑟 𝐵𝐴𝑂 ( 𝑧 ) vs 𝑧 for our models, along with the observations MNRAS000 , 1–15 (2021) G. Arciniega, et al.
I II III IV(Exponential) (Quint./Phantom) ( 𝑓 ( 𝑅 ) , 𝜔 = −
1) (General)SNe - - - - - - - - - - - - - - - - - - - - CC - - - - - - - - - - - - - - - - - - - - BAO-CMB - - - - - - - - - - - - - - - - - - - - BAO-CC-SNe - - - - - - - - - - - - - - - - - - - - Total - - - - - - - - - - - - - - - - - - - - Table 2.
Equation of state of each model by data set. Column 1 corresponds to model I-Exponential, column 2 shows the EOS for model II-Quintessence/Phantom,column 3 is model III- 𝑓 ( 𝑅 ) with 𝜔 = − 𝜔 ( 𝑧 ) and the contours around them are the 3 𝜎 error propagation (see appendix A). The colorvertical shadow show the range of 𝑧 for each data set. In the plots vertical axis is 𝜔 ( 𝑧 ) ∈ [− . , ] and the horizontal axis corresponds to the redshift 𝑧 ∈ [ , ] . we used in this work. The bottom panel contains the ratio Δ 𝑟 𝐵𝐴𝑂 ( 𝑧 ) ,from where we see that the Quintessence/Phantom model differs themost from the Λ 𝐶𝐷𝑀 prescription. Nevertheless the difference isbelow Δ 𝑟 𝐵𝐴𝑂 ≈
1% in all cases, reaching the maximum discrepancyaround 𝑧 ≈ .
1. This range of redshift (0 . < 𝑧 < .
4) will beaccurately measured using BAO in the Bright Galaxy Survey (BGS)(Ruiz-Macias et al. 2020) by DESI, making it possible to accuratelydifferentiate between these models at low redshifts.Figure 16 shows the evolution of 𝐻 ( 𝑧 )/( + 𝑧 ) vs 𝑧 for our modelsand their best fit to the CC sample. We superimpose the observationswith error bars. In this case, we notice bigger discrepancies betweenour models’ predictions for the quantity 𝐻 ( 𝑧 )/( + 𝑧 ) , and Λ 𝐶𝐷𝑀 . Δ 𝐻 ( 𝑧 ) reaches a value of ≈ 𝑓 ( 𝑅 ) -like models (case III) differ from Λ CDM in the same proportion, with up to a Δ 𝐻 ( 𝑧 ) ≈
3% at 𝑧 = . Δ 𝐻 ( 𝑧 ) ≈ − .
5% around 𝑧 ≈ .
5, and Δ 𝐻 ( 𝑧 ) ≈ + .
5% at 𝑧 ≈
3. On the other hand, Quintessence/Phantom(case II) and the General model show the same pattern in their dis-crepancies from Λ CDM, but with Δ 𝐻 ( 𝑧 ) ≈ 𝐻 ( 𝑧 )
5% at 𝑧 = . Δ 𝐻 ( 𝑧 ) ≈ −
3% around 𝑧 ≈ . Δ 𝐻 ( 𝑧 ) ≈ +
5% at 𝑧 ≈ .
5, and peak-ing at 𝑧 ≈
3, with Δ 𝐻 ( 𝑧 ) ≈ − Δ 𝐻 ( 𝑧 ) , so even when the errors for this par-ticular observable are systematic dominated and hence, bigger thanfor the other type of observations, the oscillatory behaviour in Δ 𝐻 ( 𝑧 ) can potentially help distinguishing them. MNRAS , 1–15 (2021) ne parameterisation to fit them all m − M SNeIa .
01 0 . z − . − . ∆ ( m − M )( % ) I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) (a) 𝜇 ( 𝑧 ) . r B A O ( z ) α SDSS-III DR11 z − − − ∆ r B A O ( % ) I (Exponential)II (Quint./Phantom)III ( f ( R ) , ω (0) = − )IV (General model) (b) 𝑟 𝐵𝐴𝑂 ( 𝑧 ) Figure 15.
Cosmic distances, 𝜇 ( 𝑧 ) = 𝑚 − 𝑀 , and 𝑟 𝐵𝐴𝑂 ( 𝑧 ) , used to con-straint the model and the best fitting models. H ( z ) / ( + z ) Cosmic Chronometers z − − ∆ H ( z )( % ) I (Exponential)III ( f ( R ) , ω (0) = − )IV (General model)II (Quint./Phantom) Figure 16. 𝐻 ( 𝑧 )/( + 𝑧 ) We present a new parameterisation that can reproduce the genericbehaviour of the most widely used physical models for acceleratedexpansion with infrared corrections. Our mathematical form for 𝜔 ( 𝑧 ) has at most three free parameters which can be mapped back tospecific archetypal models for dark energy. We analyze in detailhow different combinations of data can constrain the specific casesembedded in our form for 𝜔 ( 𝑧 ) , and report: the confidence intervals,individual uncertainties, resulting dynamics and statistical indicatorof the goodness of the fit. We show that the parameters can be wellconstrained by different observational data sets and that all caseswere good fits to the data.With only one free parameter ( 𝑛 = 𝐶 = 𝐴 free), we can pa-rameterise the expansion rate of the variety of models described in(Roy et al. 2018). We call this case, Exponential model. With this,we obtain not only a much richer dynamics for dark energy than thesimplified 𝜔 = const. ≠ − 𝜒 𝑟𝑒𝑑 = .
764 in the worst case, and 𝜒 𝑟𝑒𝑑 = . 𝑛 = 𝐴 and 𝐶 , free), we are able toreproduce the generic expansion rate of minimally coupled scalarfields, such as quintessence. Depending on the sign of 𝜔 ( 𝑧 ) , thegeneric behaviour of the so called Phantom models could be de-scribed with this subset of parameters. This comprises one of themost explored models for DE which we can model with the samenumber of free parameters as in the widely used CPL (Chevallier &Polarski 2001; Linder 2003) parameterisation. Our fits to the data arecompetitive, and both parameters can be simultaneously constrained.Using a different subset of only two free parameters from ourmodel ( 𝐶 = 𝐴 , and 𝑛 free), we are able to mimic the genericexpansion rate provided by cosmologically viable 𝑓 ( 𝑅 ) theories ofgravity (as shown in Jaime et al. (2014), and previously attempted MNRAS000
764 in the worst case, and 𝜒 𝑟𝑒𝑑 = . 𝑛 = 𝐴 and 𝐶 , free), we are able toreproduce the generic expansion rate of minimally coupled scalarfields, such as quintessence. Depending on the sign of 𝜔 ( 𝑧 ) , thegeneric behaviour of the so called Phantom models could be de-scribed with this subset of parameters. This comprises one of themost explored models for DE which we can model with the samenumber of free parameters as in the widely used CPL (Chevallier &Polarski 2001; Linder 2003) parameterisation. Our fits to the data arecompetitive, and both parameters can be simultaneously constrained.Using a different subset of only two free parameters from ourmodel ( 𝐶 = 𝐴 , and 𝑛 free), we are able to mimic the genericexpansion rate provided by cosmologically viable 𝑓 ( 𝑅 ) theories ofgravity (as shown in Jaime et al. (2014), and previously attempted MNRAS000 , 1–15 (2021) G. Arciniega, et al. in (Jaime et al. 2018)). For this case in particular, referred to as 𝑓 ( 𝑅 ) -like and characterized by an oscillatory behaviour for the EOSaround 𝜔 = −
1, we find the best fit of the whole sample, BAO-SNe-CC-CMB, with a 𝜒 𝑟𝑒𝑑 = . 𝜔 ( 𝑧 ) , we can answer which dynamical behaviour is favored byobservations. In this case, we find as result, an EOS which oscillatesaround the phantom-dividing line, and, with over 99% of confidence,the cosmological constant solution is disfavored.The strength of our proposal lies in its independence of a specifictheoretical model. Hence, even when we argued that the simplest,theoretically-sustained, explanation behind an oscillatory profile for 𝜔 ( 𝑧 ) , can arise in the context of 𝑓 ( 𝑅 ) theories of gravity, as opposedto a convoluted mixture of scalar fields, a tantalising alternative tothis could come from a unknowingly biased radial selection of theextra-galactic targets in the samples we use.We analyze in detail how different combinations of data can con-strain the specific cases embedded in our form for 𝜔 ( 𝑧 ) , and reportthe confidence intervals, individual uncertainties, resulting dynamicsand statistical indicator of the goodness of our fits, as well as a com-parison against the required increase in precision for observationsof the cosmic distances to be able to differentiate among particularcases. We find that all cases are good fits to the data.It is interesting to note that our best fit values for 𝐻 lie in betweenthe values to be known in tension.To summarise, in this work we have presented a single equationwhich is able to reproduce a variety of well motivated physical sce-narios for cosmic expansion at late times, we probed its adequacyto be implemented to data and aim to provide the community witha simple framework to incorporate physically-motivated models intosurveys and clustering analyses and better link observational phe-nomena and theoretical hypotheses for testing the nature of cosmicacceleration. ACKNOWLEDGEMENTS
The authors thank to E. Almaraz and M. Rodríguez-Meza for help-ful discussions, and to B. Roukema for helpful suggestions to im-prove this document. GA acknowledges the postdoctoral fellow-ship from DGAPA-UNAM. MJ acknowledges the support of thePolish Ministry of Science and Higher Education MNiSW grantDIR/WK/2018/12. Part of this work was supported by the “A next-generation worldwide quantum sensor network with optical atomicclocks” project, which is carried out within the TEAM IV programmeof the Foundation for Polish Science co-financed by the EuropeanUnion under the European Regional Development Fund. LGJ thanksthe financial support of SNI, CONACyT-140630 and the hospitalityof the ININ. GA and LGJ acknowledge the support from PAPIITIN120620.
DATA AVAILABILITY
All the observational data used in this work is of public knowledge.The cosmic chronometers sample we used can be founds in Farooq& Ratra (2013), as a compiled table of 𝑧 , 𝐻 ( 𝑧 ) , and the related errors, 𝜎 𝐻 ( 𝑧 ) . Our BAO data points can be found in the respective reference. Forthe six-degree-field galaxy survey (6dFGS) (Beutler et al. 2011), theSloan Digital Sky Survey Data Release 7 (SDSS DR7) (Ross et al.2015), the reconstructed value SDSS(R) presented in (Padmanabhanet al. 2012), and the uncorrelated values of the complete BOSS sam-ple SDSS DR12 are repoted in (Alam et al. 2016). The measurementof the auto and cross correlation of the Lymann- 𝛼 Forest (Ly 𝛼 -F)measurements from quasars of the 11th Data Release of the BaryonOscillation Spectroscopic (BOSS DR11) can be found in Delubacet al. (2015); Font-Ribera et al. (2014).The compressed CMB likelihood with Planck TT+lowP valuescan be found in (Ade et al. 2016b), and we have given the full formof the reduced matrix in section 5.1.Our chosen Supernovae compilation was Union 2.1, presentedin Suzuki et al. (2012) and which can be downloaded from http://supernova.lbl.gov/Union/ .Our numerical implementation will soonbe made publicly available in the repository https://github.com/oarodriguez/cosmostat , but aversion of the code can be shared upon reasonable request to theauthors.
REFERENCES
Abbott T., et al., 2005, preprint (arXiv:0510346)Ade P., et al., 2016a, A&A, 594, A13Ade P. A. R., et al., 2016b, A&A, 594, A14Aghamousa A., et al., 2016a, preprint (arXiv:1611.00036)Aghamousa A., et al., 2016b, preprint (arXiv:1611.00037)Aghanim N., et al., 2018, A&AAlam S., et al., 2016, MNRASAlam S., et al., 2020, preprint (arXiv:2007.08991)Amanullah R., et al., 2010a, ApJ, 716, 712Amanullah R., et al., 2010b, The Astrophysical Journal, 716, 712–738Anderson L., et al., 2014, MNRAS, 439, 83Betoule M., et al., 2014, A&A, 568, A22Beutler F., et al., 2011, MNRAS, 416, 3017Beutler F., Blake C., Koda J., Marin F., Seo H.-J., Cuesta A. J., SchneiderD. P., 2016, MNRAS, 455, 3230Cai Y.-F., Saridakis E. N., Setare M. R., Xia J.-Q., 2010, Phys. Rept., 493, 1Caldwell R., Linder E. V., 2005, PRL, 95, 141301Chevallier M., Polarski D., 2001, IJMPD, 10, 213Colless M., et al., 2003, preprint (arXiv:0306581)Conley A., et al., 2011, ApJS, 192, 1Dawson K. S., et al., 2016, AJ, 151, 44Delubac T., et al., 2015, A&A, 574, A59Eisenstein D. J., et al., 2005, ApJ, 633, 560Farooq O., Ratra B., 2013, ApJL, 766, L7Font-Ribera A., et al., 2014, JCAP, 1405, 027Freedman W. L., et al., 2019, ApJ, 882, 34Hicken M., Wood-Vasey W., Blondin S., Challis P., Jha S., Kelly P. L., RestA., Kirshner R. P., 2009, ApJ, 700, 1097Huterer D., Turner M. S., 2001, Phys. Rev., D64, 123527Jaber M., de la Macorra A., 2018, Astropart. Phys., 97, 130Jaime L. G., Patino L., Salgado M., 2012, preprint (arXiv:1206.1642)Jaime L. G., Patiño L., Salgado M., 2014, PRD, 89, 084010Jaime L. G., Jaber M., Escamilla-Rivera C., 2018, PRD, 98, 083530Jimenez R., Loeb A., 2002, ApJ, 573, 37Kazin E. A., et al., 2014, MNRAS, 441, 3524Kessler R., et al., 2009, ApJS, 185, 32Kowalski M., et al., 2008, ApJ, 686, 749Laureijs R., et al., 2011, preprint (arXiv:1110.3193)Levi M., et al., 2013, preprint (arXiv:1308.0847)Linder E. V., 2003, PRL, 90, 091301Moresco M., Jimenez R., Cimatti A., Pozzetti L., 2011, JCAP, 03, 045MNRAS , 1–15 (2021) ne parameterisation to fit them all Moresco M., et al., 2012, JCAP, 2012, 006–006Mukherjee P., Kunz M., Parkinson D., Wang Y., 2008, PRD, 78, 083529Padmanabhan N., Xu X., Eisenstein D. J., Scalzo R., Cuesta A. J., MehtaK. T., Kazin E., 2012, MNRAS, 427, 2132Perlmutter S., et al., 1999, ApJ, 517, 565Riess A. G., et al., 1998, AJ, 116, 1009Riess A. G., Casertano S., Yuan W., Macri L. M., Scolnic D., 2019, ApJ, 876,85Ross A. J., Samushia L., Howlett C., Percival W. J., Burden A., Manera M.,2015, MNRAS, 449, 835Roukema B. F., Yoshii Y., 1993, ApJ, 418, L1Roy N., Gonzalez-Morales A. X., Urena-Lopez L. A., 2018, PRD, 98, 063530Ruiz-Macias O., et al., 2020, Research Notes of the AAS, 4, 187Scolnic D., et al., 2018, ApJ, 859, 101Shajib A., et al., 2020, MNRAS, 494, 6072Suzuki N., et al., 2012, ApJ, 746, 85Verde L., Treu T., Riess A., 2019. ( arXiv:1907.10625 ),doi:10.1038/s41550-019-0902-0Weller J., Albrecht A., 2002, Phys. Rev., D65, 103512Wong K. C., et al., 2019, MNRASZhao G.-B., et al., 2017, Nature Astron., 1, 627Zumalacarregui M., 2020, PRD, 102, 023523
APPENDIX A: PROPAGATION OF UNCERTAINTIES INTHE EOS
From equation (1) we note that 𝜔 ( 𝑧 ) is a function of parameters 𝐴 , 𝑛 , and 𝐶 , and of redshift, 𝑧 . From here, it follows that the uncer-tainty 𝛿𝜔 ( 𝑧 ) depends, in the same way, on the parameters, and theiruncertainties 𝛿𝐴 , 𝛿𝑛 , and 𝛿𝐶 .When the uncertainties are independent of each other, 𝛿𝜔 ( 𝑧 ) hasa broader dispersion around the central point at every 𝑧 .From the individual errors, 𝛿𝐴 , 𝛿𝑛 , and 𝛿𝐶 , we estimate the prop-agated uncertainty in the resulting 𝜔 ( 𝑧 ) , computed as: 𝛿𝜔 ( 𝑧 ) = ( 𝜔 + ) (cid:20) 𝛿𝐴𝐴 − 𝛿𝐶 ( 𝑧 𝑛 − 𝑧 − 𝐶 ) + 𝑧 𝑛 ln ( z ) 𝛿 n ( 𝑧 𝑛 − 𝑧 − 𝐶 ) (cid:21) . (A1)We use equation (A1), 𝛿𝜔 ( 𝑧 ) , considering the uncertainties pa-rameters 𝛿𝐴 , 𝛿𝑛 , and 𝛿𝐶 as independent of 𝑧 .In the case that the uncertainties depend explicitly on 𝑧 , equation(A1) will be an overestimation of 𝛿𝜔 ( 𝑧 ) , which guarantees that thedynamical range for each EOS lies inside our estimated errors.Using the uncertainties for 𝐴 , 𝑛 , and 𝐶 , reported in column 5 fromtable 1, into equation (A1) for 𝛿𝜔 ( 𝑧 ) , we calculated the 99.7% CL infigures of table 2. This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000