Hints for possible low redshift oscillation around the best-fitting ΛCDM model in the expansion history of the Universe

ABSTRACT

We search for possible deviations from the expectations of the concordance ΛCDM model in the expansion history of the Universe by analysing the Pantheon Type Ia Supernovae (SnIa) compilation along with its Monte Carlo simulations using redshift binning. We demonstrate that the redshift binned best-fitting ΛCDM matter density parameter Ω_0m and the best-fitting effective absolute magnitude |$\cal M$| oscillate about their full data set best-fitting values with considerably large amplitudes. Using the full covariance matrix of the data taking into account systematic and statistical errors, we show that at the redshifts below z ≈ 0.5 such oscillations can only occur in 4 to 5 |${{\ \rm per\ cent}}$| of the Monte Carlo simulations. While statistical fluctuations can be responsible for this apparent oscillation, we might have observed a hint for some behaviour beyond the expectations of the concordance model or a possible additional systematic in the data. If this apparent oscillation is not due to statistical or systematic effects, it could be due to either the presence of coherent inhomogeneities at low z or due to oscillations of a quintessence scalar field.

cosmological parameters, transients: supernovae

1 INTRODUCTION

In 1998, two independent groups (Riess et al. 1998; Perlmutter et al. 1999) confirmed that the Universe is undergoing a phase of accelerated expansion, which has been attributed to the cosmological constant (Carroll 2001), thus establishing ΛCDM as the concordance model of modern cosmology. Despite its simplicity and consistency with most cosmological observations for almost two decades (Betoule et al. 2014; Aubourg et al. 2015; Baxter et al. 2016; Alam et al. 2017; Efstathiou & Lemos 2018; Scolnic et al. 2018; Aghanim et al. 2020), ΛCDM faces some challenges at the theoretical level (Weinberg 1989; P.J. 1997; Sahni 2002; Velten, vom Marttens & Zimdahl 2014), as well as at the observational one, since recent observations revealed some inconsistencies between the measured values of the basic parameters of ΛCDM (Sahni, Shafieloo & Starobinsky 2014; Solà, Gómez-Valent & de Cruz Pérez 2017; Zhao et al. 2017a; Di Valentino, Melchiorri & Silk 2019; Handley 2019; Li et al. 2019; Arjona & Nesseris (2020b, a).

The most prominent tension in the context of ΛCDM is the so-called ‘H₀ tension’, which describes the discrepancy between the Planck indirect measurement of the Hubble parameter H₀, from Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), and uncalibrated Type Ia supernovae (SnIa) data using the inverse distance ladder method (Aghanim et al. 2020) with the direct measurement published from SnIa data, using the standard distance ladder method (i.e. calibrated SnIa; Riess 2019; Riess et al. 2019). This discrepancy is currently at a 4.4σ level. Moreover, a tension that is currently at a 2–3σ level, is the so-called ‘growth tension’, which refers to the mismatch between the σ₈ (density rms matter fluctuations in spheres of radius of about |$8 \, h^{-1} \textrm {Mpc}$|⁠) and/or |$\Omega _{0 \rm m}$| (matter density parameter) measurement of the Planck mission (Aghanim et al. 2020) with Weak Lensing (WL) (Hildebrandt et al. 2017; Köhlinger et al. 2017; Abbott et al. 2018; Joudaki et al. 2018; Heymans et al. 2020), Redshift Space Distortion (RSD) data (Macaulay, Wehus & Eriksen 2013; Solá 2016; Basilakos & Nesseris 2017; Nesseris, Pantazis & Perivolaropoulos 2017; Kazantzidis & Perivolaropoulos 2018, 2019; Perivolaropoulos & Kazantzidis 2019; Skara & Perivolaropoulos 2020) as well as cluster count data (which report consistently lower values of σ₈) (Böhringer, Chon & Collins 2014; Ade et al. 2016; de Haan et al. 2016).

In order to explain the aforementioned challenges a plethora of theories have been proposed in the literature to solve the theoretical (Armendariz-Picon, Mukhanov & Steinhardt 2000; Zimdahl et al. 2001; Grande, Sola & Stefancic 2006; Moffat 2006; Caldera-Cabral, Maartens & Urena-Lopez 2009; Benisty & Guendelman 2018; Anagnostopoulos et al. 2019) and the observational challenges of ΛCDM. In particular, for the observational challenges the mechanisms that have been proposed and can alleviate one or even both of these tensions simultaneously include early (Karwal & Kamionkowski 2016; Agrawal et al. 2019; Hazra, Shafieloo & Souradeep 2019; Poulin et al. 2019; Keeley et al. 2020) and late dark energy models (Di Valentino et al. 2017; Li & Shafieloo 2019; Yang et al. 2019a, b; Alestas, Kazantzidis & Perivolaropoulos 2020; Li & Shafieloo 2020; Vagnozzi 2020), interacting dark energy models (Yang et al. 2018a, b; Di Valentino et al. 2020a, b; Gómez-Valent, Pettorino & Amendola 2020; Lucca & Hooper 2020), metastable dark energy models (Shafieloo et al. 2018; Li et al. 2019; Szydlowski, Stachowski & Urbanowski 2020; Yang et al. 2020), modified gravity theories (Ballardini et al. 2016; Lin, Raveri & Hu 2019; Rossi et al. 2019; Ballesteros, Notari & Rompineve 2020; Braglia et al. 2020; Escamilla-Rivera & Levi Said 2020; Kazantzidis & Perivolaropoulos 2020), as well as modifications of the basic assumptions of ΛCDM such as non-zero spatial curvature (Ooba, Ratra & Sugiyama 2018; Park & Ratra 2019), and many more (Joudaki et al. 2017; Zhao et al. 2017b; Solá, Gómez-Valent & de Cruz Pérez 2017; Gómez-Valent & Solá Peracaula 2018; Colgáin 2019; van Putten 2019; Camarena & Marra 2020) (see also the reviews Huterer & Shafer 2018; Ishak 2019; Kazantzidis & Perivolaropoulos 2019, and references within).

The measurement of H₀ that has been published by the SnIa data leading to the ‘H₀ tension’ is based on the assumption that SnIa can be considered as standard candles, thus allowing to probe the Hubble parameter through the apparent magnitude

$$\begin{eqnarray*} m(z)=M+5 \log _{10} \left[\frac{d_L (z)}{1 \textrm {Mpc}} \right]+25, \end{eqnarray*}$$

(1)

where d_L(z) is the luminosity distance, which in a flat Universe can be expressed as

$$\begin{eqnarray*} d_L(z)=c (1+z) \int _0^z \frac{\mathrm{ d}z^{\prime }}{H(z^{\prime })}, \end{eqnarray*}$$

(2)

while M corresponds to the corrected, over stretch and colour, absolute magnitude.

Alternatively, the apparent magnitude can be expressed in terms of the dimensionless Hubble-free luminosity distance |$D_L \equiv H_0 \, d_L/c$| as

$$\begin{eqnarray*} m(z)=M +5 \log _{10}\left[D_L(z)\right] + 5 \log _{10}\left(\frac{c/H_0}{1\textrm {Mpc}}\right)+25. \end{eqnarray*}$$

(3)

Clearly, from equation (3) it is evident that the parameters H₀ and M are degenerate and since in the context of ΛCDM both of these are assumed to be constant, usually, a marginalization process is performed (Conley et al. 2011; Betoule et al. 2014; Scolnic et al. 2018), over the degenerate combination

$$\begin{eqnarray*} {\cal M} &\equiv & M +5 \log _{10}\left[\frac{c/H_0}{1 \textrm {Mpc}}\right]+25\nonumber \\ &=& M -5 \log _{10}(h)+42.38, \end{eqnarray*}$$

(4)

where |$h \equiv H_0/100 \,\,\textrm {km} \, \textrm {s}^{-1}\, \textrm {Mpc}^{-1}$|⁠. However, in our analysis we choose to keep |$\cal {M}$| in order to avoid any loss of crucial information.

The latest (and largest thus far) compilation of SnIa that has been published is the Pantheon data set (Scolnic et al. 2018), consisting of 1048 SnIa in the redshift range 0.01 < z < 2.3. Using equations (1)–(4), the corresponding χ² function reads

$$\begin{eqnarray*} \chi ^2 ({\cal M}, \Omega _{0 \rm m})=V^i_\textrm {SnIa} \, C_{ij}^{-1} \, V^j_\textrm {SnIa}, \end{eqnarray*}$$

(5)

where |$V^i_\textrm {SnIa}\equiv m_{\rm obs}(z_i)-m(z)$| and |$C_{ij}^{-1}$| is the inverse covariance matrix. The covariance matrix can be considered as the sum of two matrices: a diagonal matrix that is associated with the statistical uncertainties of the apparent magnitude m_obs of each SnIa and a non-diagonal part that is connected with the systematic uncertainties due to the bias correction method (Scolnic et al. 2018).

In Refs. Kazantzidis & Perivolaropoulos (2020), Sapone, Nesseris & Bengaly (2020), it was shown that the best-fitting ΛCDM parameter values for the best-fitting parameters |$\cal {M}$| and |$\Omega _{0 \rm m}$| of redshift binned Pantheon data oscillate around the full data set best fit at a level that is consistently larger than 1σ for the first three out of four redshift bins. A similar effect was observed in Colgáin (2019), where the best-fitting values of Ω_0m and h for various maximum redshift cutoffs z_max were studied instead. Here we should emphasize that any realization of a data based on a given model would have its own specific features and characteristics that might look unusual but in reality they are effects of random fluctuations. So to assign statistical significance to unusual features or behaviours seen in a given data (to evaluate if they can be real) it is necessary to compare the real data with a large number of Monte Carlo simulations. This comparison with 1000 random Monte Carlo simulations is a key part of our current analysis to evaluate how statistically significant such variations are.

If the variation we see in the real data is due to statistical fluctuations, then the same variation is anticipated to be evident in simulated Pantheon-like data sets. In this analysis we will address the following questions:

• How likely is this behaviour of the data in the context of the ΛCDM model?

• In how many realizations we can see more than the σ deviations of the real data (σ^real) for both |$\cal {M}$| and |$\Omega _{0 \rm m}$| in the first three or any three out of four redshift bins?

• In how many realizations we can see more than the 1σ deviations for both |$\cal {M}$| and |$\Omega _{0 \rm m}$| in the first three or in any three out of four redshift bins?

The structure of the paper is the following: In Section 2 we describe the statistical analysis and the comparison of the constructed simulated data sets with the actual Pantheon data searching for abnormalities of the real data in the context of the reported level of Gaussian uncertainties. Finally, in Section 3 we summarize our results and discuss possible extensions of the present analysis.

2 REAL VERSUS MONTE CARLO DATA

In our Monte Carlo statistical analysis we split the Pantheon data set (Scolnic et al. 2018) into four redshift bins, consisting of equal number of datapoints (262). The number of bins is an important implicit parameter that could affect the results of our analysis. Too many bins may lead to overfitting of the data, while a very small number of bins may miss interesting signals hidden. In the present analysis we have chosen to use four bins with equal number of data. However, this choice is clearly not unique. For example, bins could have been chosen so that each bin has the same redshift interval, while it is not appropriate for the present analysis since most of SnIa in Pantheon are concentrated in the lower part of their redshift range, or has the same cumulative signal to noise (S/N) (including downweighting from systematics, which correlate points within the same bin). In fact an interesting extension of the present analysis would be the effect of the binning method on the strength and the statistical significance of the identified oscillating signal.

We then find the best-fitting parameters |$\cal {M}$| and |$\Omega _{0 \rm m}$| and 1σ uncertainties in the context of a ΛCDM model for each bin, with |$\cal {M}$| and |$\Omega _{0 \rm m}$| being allowed to vary simultaneously. We also find the corresponding best fit for the full Pantheon data set and identify the σ distance between the best-fitting parameter values in each bin and the best-fitting value of the full data set. The results of the tomography for the real data can be seen in Table 1 as well as in Fig. 1. Clearly, all first three bins of the real data best fits of |$\cal {M}$| and |$\Omega _{0 \rm m}$| differ by at least 1σ from the full data set best fits.

Figure 1.

The 1σ best-fitting values (blue dots) of |$\cal {M}$| and Ω_0m of the real data for each bin. The dashed line corresponds to the best-fitting value of the full data set, while the dot dashed line to its 1σ error. Clearly, the first three bins best-fitting differ by at least 1σ from the best-fitting values of the full data set.

Table 1.

The best-fitting values with the 1σ error of |$\cal {M}$| and |$\Omega _{0 \rm m}$| for the four redshift bins with equal number of data points for the real data. Notice that for first three redshift bins the σ distance (Δσ) of the best fit from the full data set best fit is at least 1σ and on the average it is larger than 1.2σ. In the simulated Pantheon data such large simultaneous deviations for the first three bins occurs for about 2 per cent of the data sets.

Bin	z Range	|${\cal {M}} \pm 1 \sigma \text{ error}$|	|$\Delta \sigma _{{\cal {M}}}$|	Ω0m ± 1σ error	|$\Delta \sigma _{\Omega _{0 \rm m}}$|
Full data	0.01 < z < 2.26	23.81 ± 0.01	–	0.29 ± 0.02	–
1st	0.01 < z < 0.13	23.78 ± 0.03	1.14	0.07 ± 0.17	1.35
2nd	0.13 < z < 0.25	23.89 ± 0.06	1.48	0.56 ± 0.19	1.34
3rd	0.25 < z < 0.42	23.75 ± 0.06	0.99	0.18 ± 0.11	1.05
4th	0.42 < z < 2.26	23.85 ± 0.06	0.69	0.33 ± 0.06	0.50

In order to estimate the likelihood of such a σ deviation of best-fitting values in the first three bins, we construct 1000 simulated Pantheon-like data sets, with random apparent magnitudes m obtained from a multivariate normal distribution with a mean value equal to the best-fitting ΛCDM value of the real data using the full covariance matrix of the real data. The corresponding probability distribution is of the form

$$\begin{eqnarray*} f_{\boldsymbol {m}} \left(m_1, \ldots , m_k \right)=\frac{\exp \left[-\frac{1}{2} \left(\boldsymbol {m}-\bar{\boldsymbol {m}} \right)^T \boldsymbol {C}^{-1} \left(\boldsymbol {m}-\bar{\boldsymbol {m}} \right) \right]}{ \sqrt{(2 \pi)^k \, |\boldsymbol {C}|}}, \end{eqnarray*}$$

(6)

where |$\boldsymbol {C}$| is the full non-diagonal covariance matrix including both statistical and systematic errors, |$\boldsymbol {m}$| is the vector {m₁, m₂, …m_k}, and |$\bar{\boldsymbol {m}}$| corresponds to the mean value of the apparent magnitude vector. Using this multivariate normal distribution, we construct the simulated data sets and find the percent fraction of them where all first three redshift bins have best fit ΛCDM parameter values |$\cal {M}$| and |$\Omega _{0 \rm m}$| that have simultaneously σ distance from the real data best fit more than k σ ≡ σ_k σ. These results for the parameters |$\cal {M}$| (⁠|$\sigma _k=\sigma _{k {\cal {M}}})$| and |$\Omega _{0 \rm m}$| (σ_k = σ_kΩ) are shown in Fig. 2.

Figure 2.

Left-hand panel: Percent of simulated Pantheon data set (including systematics) where the first three out of four bins all differ simultaneously more than |$\sigma _{k {\cal {M}}}\,\, \sigma$| from the best fit of the full data set. The red dotted line corresponds to the real data that differ more than 1σ (1.14, 1.48, and 0.99σ for the first three bins respectively) from the full data set best fits. Right-hand panel: Same as the left-hand panel for the parameter |$\Omega _{0 \rm m}$| instead of |$\cal {M}$|⁠.

According to Fig. 2, the probability that all three first bins differ simultaneously more than 1σ from the best fit of each simulated full data set in the context of ΛCDM is less than |$5{{\ \rm per\ cent}}$|⁠. This is an effect approximately at 2σ level.

In fact, this probability is even smaller if we consider the exact σ differences that are shown in Table 1 and find the fraction of simulated data sets with simultaneous σ differences larger that the exact corresponding σ differences of the real data. In particular we find that the probability to have simultaneously 1.14σ difference (or larger) in the first bin, 1.48σ difference (or larger) in the second bin, and 0.99σ difference (or larger) in the third bin for |$\cal {M}$|⁠, is |$1.3 \pm 0.7 {{\ \rm per\ cent}}$|⁠. Similarly, for |$\Omega _{0 \rm m}$| we find the same probability to be |$1.4 \pm 2 {{\ \rm per\ cent}}$|⁠. Even though this decrease of probability is interesting to note, it is not generic as it is based on the fine tuned σ deviations of the real data bins from the full data best fits (1.14σ, 1.48σ, and 0.99σ). Thus, this is an aposteriori statistic constructed after looking at the data.

Therefore, we adopt the more generic and conservative statistical level of significance of |$5{{\ \rm per\ cent}}$| corresponding to the simultaneous deviation of at least 1σ for all three lowest z bins. Note that a similar oscillating effect was also observed in Refs. Kazantzidis & Perivolaropoulos (2020), Sapone et al. (2020) even though its statistical significance was not quantified using simulated data as in the present analysis.

Moreover, it is interesting to check if this behaviour is also evident for any three out of four bins. In 1000 Monte Carlo realizations we find that the number of simulated data sets where the derived |$\Omega _{0 \rm m}$| in any 3 bins is more than 1σ away from the best fit |$\Omega _{0 \rm m}$| to the whole (random) data sample is |$10.4 \pm 2.2 {{\ \rm per\ cent}}$| while the corresponding number of cases for |$\cal {M}$| is |$11.1 \pm 2.4 {{\ \rm per\ cent}}$| as it is demonstrated in Fig. 3. The probability is smaller if we consider the exact σ difference of Table 1. In particular, we derive the number of cases where the derived |$\Omega _{0 \rm m}$| in any 3 bins is more than |$\sigma ^{real}_{\Omega _{0 \rm m}}$| away from the best fit |$\Omega _{0 \rm m}$| to the whole (random) data sample is |$7.5 \pm 1.5 {{\ \rm per\ cent}}$|⁠, while the corresponding number of cases for |$\cal {M}$| is |$7.4 \pm 1.5 {{\ \rm per\ cent}}$|⁠. A summary of the results can be seen in Table 2. These results indicate that the aforementioned oscillating effect is much more prominent at low z ≲ 0.5 where the dark energy density is more prominent than in the fourth bin, which involves higher z. This fact favours the possibility that the effect has a physical origin since a systematic effect would probably affect equally all four redshift bins.

Figure 3.

Left-hand panel: Percent of simulated Pantheon dataset (including systematics) where any three out of four bins all differ simultaneously more than |$\sigma _{k {\cal {M}}}\,\, \sigma$| from the best fit of the full data set. Right-hand panel: Same as the left-hand panel for the parameter |$\Omega _{0 \rm m}$| instead of |$\cal {M}$|⁠.

Table 2.

Summary of the Monte Carlo deviations from the simulated and real data. The obtained results considering the exact σ_real differences should be treated with care, since this decrease of the probabilities is not generic and it is based on a fine tuned σ value.

Number of cases	Probability
|$\Omega _{0 \rm m}$| in the first 3 bins >1σ away from the best fit |$\Omega _{0 \rm m}$| to the whole data sample	|$4.8 \pm 2 $| per cent
|$\cal {M}$| in the first 3 bins >1σ away from the best fit |$\cal {M}$| to the whole data sample	|$4 \pm 2.5 $| per cent
|$\Omega _{0 \rm m}$| in any 3 bins >1σ away from the best fit |$\Omega _{0 \rm m}$| to the whole data sample	|$10.4 \pm 2.2 $| per cent
|$\cal {M}$| in any 3 bins >1σ away from the best fit |$\cal {M}$| to the whole data sample	|$11.1 \pm 2.4 $| per cent
|$\Omega _{0 \rm m}$| in the first 3 bins >σreal away from the best fit |$\Omega _{0 \rm m}$| to the whole data sample	|$1.4 \pm 2 $| per cent
|$\cal {M}$| in the first 3 bins >σreal away from the best fit |$\cal {M}$| to the whole data sample	|$1.3 \pm 0.7$| per cent
|$\Omega _{0 \rm m}$| in any 3 bins >σreal away from the best fit |$\Omega _{0 \rm m}$| to the whole data sample	|$7.5 \pm 1.5$| per cent
|$\cal {M}$| in any 3 bins >σreal away from the best fit |$\cal {M}$| to the whole data sample	|$7.4 \pm 1.5$| per cent

3 CONCLUSION – OUTLOOK

We performed a redshift tomography of the Pantheon data dividing them into four redshift bins of equal number of datapoints and searched for hints of abnormal oscillation behaviour for the best-fitting parameter values of |$\cal {M}$| and |$\Omega _{0 \rm m}$| in these bins with respect to the corresponding best fits of the full Pantheon data set.

We constructed 1000 simulated Pantheon-like data sets and found that including both systematic and statistical uncertainties, the percentage of the simulated Pantheon data set with a similar amplitude oscillating behaviour is |$\simeq\! 5{{\ \rm per\ cent}}$|⁠. Considering only statistical uncertainties in the construction of the simulated data sets this probability decreases to about |$2.7{{\ \rm per\ cent}}$|⁠.

While the statistical significance of the oscillations reduces when we consider any three bins out of four bins, we emphasize that the first three bins covering the |$75{{\ \rm per\ cent}}$| of the total data points are all at relatively low redshifts (z < 0.42) where dark energy is dominant. Hence, concerning the physical origin of the aforementioned effect, we anticipate that the importance of the first three bins is amplified compared to any other three bin combination.

The important issue here is how generic is the identified effect and also if it would have been expected in the context of a particular physical context. We argue that larger than expected oscillations around the standard model is a simple generic effect, especially if it is prominent at low redshifts where the effects of dark energy are more important. Thus, even though the look-elsewhere effect is hard to quantify in the context of the Monte Carlo statistical analysis, the generic nature of the oscillating effect as well as the fact that it is more prominent at low z where dark energy dominates, indicates that the statistical significance of the identified signal will not be significantly affected by the look elsewhere effect.

Plausible physical causes for such low z oscillating behaviour of the data include the following

• The presence of large-scale inhomogeneities at low z including voids or superclusters (Grande & Perivolaropoulos 2011; Shanks, Hogarth & Metcalfe 2019).

• Dark energy with oscillating density in redshift (Xia et al. 2006; Lazkoz, Salzano & Sendra 2011; De Felice, Nesseris & Tsujikawa 2012; Pace et al. 2012; Pan, Saridakis & Yang 2018). Such oscillations may be induced e.g. by scalar field potentials with a local minimum (Cicoli et al. 2019; Ruchika et al. 2020).

Finally, some interesting extensions of the present analysis include the following:

• Further investigation for a similar oscillating behaviour in other data (e.g. BAO or H(z) cosmic chronometer data; Marcondes & Pan 2017; Ishak 2019; Raveri 2020). Regarding the cosmic chronometer data (Marcondes & Pan 2017), even though no oscillating signal is evident, the errors are significantly larger than other probes and could well hide any interesting signal evident in other higher quality probes. On the contrary, regarding the BAO data (Raveri 2020) an interesting descending trend is evident, which could be interpreted as a hint for oscillations. Clearly, if such oscillations are observed in other cosmological data sets, the overall statistical significance of such an effect would be considerably boosted.

• Construction of physical models that naturally lead to such an oscillating low z behaviour of the data.

• Forecasts with future SnIa compilations, e.g. by the LSST survey, to ascertain whether this oscillatory effect would be more prominent in upcoming data.

• Making some internal consistency checks such as using ‘Robustness’ criterion (Amendola, Marra & Quartin 2013) or/and looking for redshift evolution in the light-curve parameters of the data (Koo et al. 2020) to determine whether the Pantheon sample is statistically consistent or is contaminated with systematics.

ACKNOWLEDGEMENTS

The research of LK is co-financed by Greece and the European Union (European Social Fund – ESF) through the Operational Programme ‘Human Resources Development, Education and Lifelong Learning’ in the context of the project ‘Strengthening Human Resources Research Potential via Doctorate Research – 2nd Cycle’ (MIS-5000432), implemented by the State Scholarships Foundation (IKY). The research of LP is co-financed by Greece and the ESF through the Operational Programme ‘Human Resources Development, Education and Lifelong Learning 2014–2020’ in the context of the project No. MIS 5047648. SN acknowledges support from the Research Projects PGC2018-094773-B-C32, the Centro de Excelencia Severo Ochoa Program SEV-2016-0597 and the Ramón y Cajal program through Grant No. RYC-2014-15843. AS would like to acknowledge the support of the Korea Institute for Advanced Study (KIAS) grant funded by the government of Korea.

DATA AVAILABILITY

The data access to the Pantheon compilation of SnIa is provided by https://github.com/dscolnic/Pantheon. Description of the Pantheon compilation is in https://archive.stsci.edu/prepds/ps1cosmo/index.html and Scolnic et al. (2018).