# Distance Priors from Planck and Dark Energy Constraints from Current Data

###### Abstract

We derive distance priors from Planck first data release, and examine their impact on dark energy constraints from current observational data. We give the mean values and covariance matrix of , which give an efficient summary of Planck data. The CMB shift parameters are , and , where is the redshift at the last scattering surface, and and denote our comoving distance to and sound horizon at respectively.

We find that Planck distance priors are significantly tighter than those from WMAP9. However, adding Planck distance priors does not lead to significantly improved dark energy constraints using current data, compared to adding WMAP9 distance priors. This is because Planck data appear to favor a higher matter density and lower Hubble constant, in tension with most of the other current cosmological data sets. Adding Planck distance priors to current data leads to a marginal inconsistency with a cosmological constant in a flat universe.

###### pacs:

98.80.Es,98.80.-k,98.80.JkJune 28, 2021

## I Introduction

Current observational data do not yet allow us to differentiate two likely explanations for the observed cosmic acceleration Riess et al. (1998); Perlmutter et al. (1999): dark energy, and the modification of general relativity. For recent reviews, see Copeland06 ; Ruiz07 ; Ratra07 ; Frieman08 ; Caldwell09 ; Uzan09 ; Wang10 ; Li11 ; Weinberg12 . Cosmic acceleration is generally referred to as “dark energy” for convenience.

There are three vigorously studied direct probes of dark energy. Type Ia supernovae (SNe Ia) probe the Hubble parameter (i.e., the expansion history of the universe) via the measurement of luminosity distances to the SNe Ia Riess et al. (1998); Perlmutter et al. (1999). Galaxy clustering (GC) directly probes (and its integral form ) via the baryon acoustic oscillation (BAO) Blake & Glazebrook (2003); Seo & Eisenstein (2003) measurements, and the growth rate (i.e., the growth history of cosmic large scale structure) via redshift space distortion measurements. Weak lensing of galaxies probes a combination of the expansion history and growth history of the universe Hu (2002); Jain & Taylor (2003).

While these direct probes of cosmic acceleration complement each other, each with its own set of systematic uncertainties, they require the inclusion of cosmic microwave background (CMB) anisotropy data to help break the degeneracies among the dark energy and cosmological parameters. This is because CMB data provide the strongest constraints on cosmological parameters (see, e.g., Komatsu et al. (2011)).

Direct measurements of the Hubble constant (see, e.g., Riess et al. (2011)) also help break the degeneracy amongst the dark energy and cosmological parameters. Other data, e.g., gamma ray bursts Amati02 ; Bloom, Frail, & Kulkarni (2003); Schaefer03 , can help strengthen the dark energy constraints.

In this paper, we derive distance priors from the Planck first data release, and examine their impact on dark energy constraints from current observational data.

We describe our method in Sec.II, present our results in Sec.III, and conclude in Sec.IV.

## Ii Method

Our main goal is to derive distance priors from Planck data, and illustrate their impact on current observational data. For simplicity and clarity, we only use methods that give geometric constraints on dark energy in this paper. The constraints on the growth rate of cosmic large scale structure are degenerate with the geometric constraints (see, e.g., Wang (2008a); Simpson & Peacock (2010)). We adopt a conservative approach by marginalizing over the growth constraints.

Geometric constraints on dark energy are derived from the measurement of distances. The comoving distance to an object at redshift is given by:

(1) | |||

where , , for , , and respectively; and the expansion rate of the universe (i.e., the Hubble parameter) is given by

where , includes the contribution from massive neutrinos, and the dark energy density function is defined as

(3) |

Note that (with denoting the redshift at matter-radiation equality), thus the term is usually omitted in dark energy studies at , since dark energy should only be important at late times.

### ii.1 CMB data

CMB data give us the comoving distance to the photon-decoupling surface , and the comoving sound horizon at photon-decoupling epoch Page (2003). Wang & Mukherjee (2007) Wang & Mukherjee (2007) showed that the CMB shift parameters

(4) |

together with , provide an efficient summary of CMB data as far as dark energy constraints go. This has been verified by Li08 . Replacing with gives identical constraints when the CMB distance priors are combined with other data Wang08b . Using , instead of , is more appropriate in an MCMC analysis in which is a base parameter.

Note that in order to summarize the CMB data, and are defined to contain the physical parameters (matter density), (comoving distance to the photon-decoupling surface), and (angular size of the comoving sound horizon at photon-decoupling epoch), as these physical parameters are tightly constrained by CMB data and are essentially independent of model assumptions (except for the assumption on the cosmic curvature).

An intuitive explanation for the effectiveness of in summarizing CMB data is as follows. As indicated by the detailed studies in Wang & Mukherjee (2007) Wang & Mukherjee (2007), both and must be used to describe the complex degeneracies amongst the cosmological parameters that determine the CMB angular power spectrum. Models that correspond to the same value of but different values of give rise to very different CMB angular power spectra, because determines the average acoustic peak structure. Models that correspond to the same value of but different values of have the same acoustic peak structure in their CMB angular power spectra, but the overall amplitude of the acoustic peaks is different in each model because of the difference in . The inclusion of CMB lensing data in deriving the shift parameters makes little difference in their mean values, since the bulk of the information comes from the CMB angular power spectra, but it does reduce their uncertainties by reducing parameter degeneracies.

The comoving sound horizon at redshift is given by

(5) | |||||

where is the cosmic scale factor, , and , with , and . The sound speed is , with , . We take .

The redshift to the photon-decoupling surface, , is given by the fitting formula Hu & Sugiyama (1996):

(6) |

where

(7) | |||||

(8) |

The redshift of the drag epoch is well approximated by Eisenstein & Hu (1998)

(9) |

where

(10) | |||||

(11) |

Figs.1-4 show the one-dimensional marginalized probability
distributions (pdf) of from Planck Ade et al. (2013),
WMAP9 Bennett et al. (2012), and WMAP7 Komatsu et al. (2011) data,
for and one massive neutrino (with mass of 0.06eV),
and without assuming a flat universe. We have used the Planck archiv
data to obtain constraints on from Planck and WMAP9;
this archiv does not include cases without assuming a flat universe.
Fortunately, the constraints on
(including the pdf’s) are not sensitive to the assumption about dark energy Wang, Chuang, & Mukherjee (2012).
Three sets of pdf’s are shown in Figs.1-4:

(1) Planck+lensing+WP: Planck temperature data combined with Planck lensing, as well as
WMAP polarization at low multipoles (). This set represents the tightest constraints
from CMB data only at present.
Note that excluding the Planck lensing data changes the mean values of
by 0.2% or less, and increases the dispersions slightly.

(2) WMAP: WMAP 9 year temperature and polarization data.

(3) WMAP7: WMAP 7 year temperature and polarization data.

The Planck+lensing+WP pdf’s in Figs.1-4 are well fitted by Gaussian distributions with the following means and standard deviations:

(12) |

The normalized covariance matrix of is

(13) |

For comparison, we have also obtained the WMAP9 constraints on . The WMAP9 in Figs.1-4 are well fitted by Gaussian distributions with the following means and standard deviations:

(14) |

The normalized covariance matrix of is

(15) |

The WMAP7 constraints are from Wang, Chuang, & Mukherjee (2012).

Since the primary GC data we use in this paper have been marginalized
over CW12 , we should marginalize the CMB distance priors
over as well.^{2}^{2}2An improved approach is to use GC data
that retain the dependence, and combine with the CMB constraints
on .
This means dropping the 4th row and 4th column from
the normalized covariance matrix of ,
then obtain the covariance matrix for as follows:

(16) |

where . The rms variance and the normalized covariance matrix are given by Eqs.(12) and (13) for Planck+lensing+WP, and Eqs.(14) and (15) for WMAP9 respectively.

### ii.2 Analysis of SN Ia Data

SN Ia data give measurements of the luminosity distance through that of the distance modulus of each SN:

(18) |

where and represent the apparent and absolute magnitude of a SN. The luminosity distance , with the comoving distance given by Eq.(1).

Care must be taken in interpreting supernova distances in an inhomogeneous universe Clarkson11 . We use the compilation of SN Ia data by Conley et al. (2011) Conley11 , which include the SNe Ia from the first three years of the Supernova Legacy Survey (SNLS3), the largest homogeneous SN Ia data set publicly available at present, and apply flux-averaging to reduce the systematic bias due to weak lensing magnification of SNe Wang (2000); Wang & Mukherjee (2004); Wang (2005), as detailed in Wang, Chuang, & Mukherjee (2012) Wang, Chuang, & Mukherjee (2012).

For a set of 472 SNe Ia, Conley et al. (2011) Conley11 give the apparent magnitude, , and the covariance matrix for , with

(19) |

where is the luminosity distance multiplied by for a given set of cosmological parameters , is the stretch measure of the SN light curve shape, and is the color measure for the SN. is a nuisance parameter representing some combination of the absolute magnitude of a fiducial SN Ia, , and the Hubble constant . Since the time dilation part of the observed luminosity distance depends on the total redshift (special relativistic plus cosmological), we have Hui06

(20) |

where and are the CMB restframe and heliocentric redshifts of the SN.

For a set of SNe with correlated errors, we have Conley11

(21) |

where is a vector with components, and C is the covariance matrix of the SNe Ia.

Note that is equivalent to , since

(22) |

The total covariance matrix is Conley11

(23) |

with the diagonal part of the statistical uncertainty given by Conley11

(24) | |||||

where , , and are the covariances between , , and for the -th SN. Note also that includes a peculiar velocity residual of 0.0005 (i.e., 150km/s) added in quadrature Conley11 .

The statistical and systematic covariance matrices, and , are generally not diagonal Conley11 , and are given in the form:

(25) |

where , , , , , and are matrices given by the SNLS data archive at https://tspace.library.utoronto.ca/handle/1807/24512/. includes the uncertainty in the SN model. includes the uncertainty in the zero point. Note that and do not depend on , since the relative distance moduli are independent of the value of Conley11 .

We refer the reader to Conley et al. (2011) Conley11 for a detailed discussion of the origins of the statistical and systematic errors. As an example, we note that the correlation of errors on different SNe arises from a statistical uncertainty in the zero point of one passband, e.g., . This directly affects all SNe with measurements due to K-corrections (restframe to ), and indirectly affects even the SNe without measurements through the empirical SN models by changing the templates and the measured color-luminosity relationship.

For statistics using MCMC or a grid of parameters, here are the steps in flux-averaging Wang, Chuang, & Mukherjee (2012):

(1) Convert the distance modulus of SNe Ia into “fluxes”,

(26) |

(2) For a given set of cosmological parameters , obtain “absolute luminosities”, {}, by removing the redshift dependence of the “fluxes”, i.e.,

(27) |

(3) Flux-average the “absolute luminosities” {} in each redshift bin to obtain :

(28) |

(4) Place at the mean redshift of the -th redshift bin, now the binned flux is

(29) |

(5) Compute the covariance matrix of and :

where is the covariance of the measured distance moduli of the -th SN Ia in the -th redshift bin, and the -th SN Ia in the -th redshift bin. is defined by Eqs.(26) and (27).

(6) For the flux-averaged data, , compute

(31) |

where

(32) |

with .

For the sample of SNe we use in this study, we flux-averaged the SNe with , to ensure that almost all redshift bins contain at least one SN. Our SN flux-averaging code is available at http://www.nhn.ou.edu/wang/SNcode/.

### ii.3 Galaxy Clustering Data

For GC data, we use the measurements of and (where is the Hubble parameter, is the angular diameter distance, and is the sound horizon at the drag epoch) from the two-dimensional two-point correlation function measured at z=0.35 CW12 and z=0.57 Chuang et al. (2013). The measurement was made by Chuang & Wang (2012) CW12 using a sample of the SDSS DR7 Luminous Red Galaxies (LRGs). The measurement was made by Chuang et al. (2013) Chuang et al. (2013) using the CMASS galaxy sample from BOSS.

Using the two-dimensional two-point correlation function of SDSS DR7 in the scale range of 40-120Mpc/, Chuang & Wang (2012) CW12 found that

(33) |

where is the normalized correlation coefficient between and , and is the sound horizon at the drag epoch (given by Eqs.(5) and (9).

In a similar analysis using the CMASS galaxy sample from BOSS, Chuang et al. (2013) found that

(34) |

We marginalize over the growth rate measurement made by Chuang et al. 2013 Chuang et al. (2013) for a conservative approach.

GC data are included in our analysis by adding , with and , to the of a given model. Note that

(35) |

where and , with .

### ii.4 Gammay-ray Burst Data

We add gammay-ray burst (GRB) data to our analysis, since these are
complementary in redshift range to the SN Ia data.
We use GRB data in the form of the model-independent GRB distance
measurements from Wang (2008c) Wang08c , which were
derived from the data of 69 GRBs with
from Schaefer (2007) Schaefer07 ^{3}^{3}3The proper
calibration of GRBs is an active area of research. For
recent studies on the impact of detector thresholds,
spectral analysis, and unknown selection effects, see, e.g.,
Butler07 ; Butler09 ; Petrosian09 ; Shahmoradi11 ..

The GRB distance measurements are given in terms of Wang08c

(36) |

where is the comoving distance at .

The GRB data are included in our analysis by adding the following term to the of a given model:

(37) |

where is defined by Eq.(36). The covariance matrix is given by

(38) |

where is the normalized covariance matrix from Table 3 of Wang (2008c) Wang08c , and

(39) |

where and are the 68% C.L. errors given in Table 2 of Wang (2008c) Wang08c .

### ii.5 Dark energy parametrization

Since we are ignorant of the true nature of dark energy, it is useful to measure the dark energy density function as a free function of redshift Wang & Garnavich (2001); Wang & Tegmark (2004); Wang & Freese (2006). This has the advantage of allowing dark energy models in which becomes negative in the future, e.g., the “Big Crunch” models Linde (1987); WangLinde04 , which are precluded if we parametrize dark energy with an equation of state Wang & Tegmark (2004).

Here we parametrize by cubic-splining its values at , , and 1.0, and assume that . For simplicity of notation, we define , , and . Fixing reflects the limit of current data, and avoids making assumptions about early dark energy that can be propagated into artificial constraints on dark energy at low Wang & Tegmark (2004); Wang & Mukherjee (2007).

For comparison with the work of others, we also consider a dark energy equation of state linear in the cosmic scale factor , Chev01 (2001). A related parametrization is Wang08b

(40) |

where . Wang (2008b) Wang08b showed that (, ) are much less correlated than (, ), thus are a better set of parameters to use. We find that (, ) converge much faster than (, ) in a Markov Chain Monte Carlo (MCMC) likelihood analysis for the same data.

## Iii Results

We perform a MCMC likelihood analysis Lewis02 (2002) to obtain () samples for each set of results presented in this paper. We assume flat priors for all the parameters, and allow ranges of the parameters wide enough such that further increasing the allowed ranges has no impact on the results. We process the MCMC chains following the standard practice of ensuring convergence and thinning using CosmoMC.

In addition to the SN Ia, CMB, GC, and GRB data discussed in Sec.II, we impose a prior of kmsMpc, from the HST measurements by Riess et al. (2011) Riess et al. (2011).

We do not assume a flat universe. In addition to the dark energy parameters described in Sec.II.5, we also constrain cosmological parameters (), where . In addition, we marginalize over the SN Ia nuisance parameters . We only use flux-averaged SN Ia data (with ), as flux-averaging reduces the impact of systematic uncertainties on dark energy and cosmological parameter constraints Wang, Chuang, & Mukherjee (2012).

We will present results for dark energy density at , , and 1, as well as () and (), and a constant dark energy equation of state .

### iii.1 Constraints on dark energy density function

Figs.5-8 summarize our constraints on parametrized by its value at , , and 1. Planck data give very similar results as WMAP9 data on , even although Planck data favor higher . However, note that adding Planck priors leads to a marginal inconsistency with a cosmological constant in a flat universe (see bottom right panel in Fig.6).

### iii.2 Constraints on a linear dark energy equation of state

We have studied the constraints on both () and (), as these have different base parameters (assumed to have flat priors). In order to compare with previous work, and to display the impact of replacing WMAP9 priors with Planck+lensing+WP priors, we do not include GC data from BOSS in this comparison.

Fig.9 shows the joint 68% and 95% confidence contours for () (left panel) and () (right panel), for SNe++GRB+GC(CW12), combined with Planck+lensing+WP (solid) and WMAP9 (dotted) data respectively. Fig.10 shows the corresponding joint 68% and 95% confidence contours for () (left panel) and () (right panel), for the same data and with the same line types. Fig.9 indicates that Planck priors do not have a significant impact on the constraints on a linear dark energy equation state; this means that the DETF FoM remains approximately the same compared to that found by Wang, Chuang, & Mukherjee (2012) using WMAP7 priors (using WMAP9 priors gives similar results as using WMAP7 priors). It is interesting to note that the right panel of Fig.10 shows that the combined data with Planck priors rule out and a flat universe at . Planck data favor a small but positive (i.e., a slightly open universe).

### iii.3 Constraints on a constant dark energy equation of state

In order to understand better the difference between the Planck+lensing+WP priors and the WMAP9 priors, we now study a constant dark energy equation of state, for the minimal combination of SNe+ data with the CMB priors.

Fig.11 shows the joint 68% and 95% confidence contours for () (left panel) and () (right panel), for SNe+, combined with Planck+lensing+WP (solid) and WMAP9 (dotted) data respectively. Here, we see even more clearly the trend of Planck+lensing+WP priors favoring a small but postive (a slightly open universe), and a somewhat higher , compared to WMAP9 priors. Again, we find that adding Planck priors to other data leads to a marginal inconsistency with a cosmological constant in a flat universe.

## Iv Discussion and Summary

We have derived the distance priors from Planck first data release, in the form of the mean values and covariance matrix of , which give an efficient summary of Planck data in the context of dark energy constraints. As a test of the accuracy of this approach, we compare the constraints on a constant dark energy equation of state in a flat universe using Planck+lensing+WP data combined with SNLS SNe (no flux-averaging) in the form of MCMC chains from the Planck data archiv, with the Planck+lensing+WP data summarized by combined with the same SN data. We find for Planck+lensing+WP data summarized by , in excellent agreement with from the full Planck+lensing+WP data. Not surprisingly, the full CMB data give slightly tighter constraints, but the difference is not statistically significant.

We have used constraints on from Planck data in combination with other data to probe dark energy in a conservative geometric approach. We have considered three different dark energy parametrizations: (1) Dark energy density , parametrized by splining its values at , 2/3, and 1.0. (2) Dark energy equation of state linear in the cosmic scale factor, , as well as the alternative parametrization with much less correlated parameters and , . (3) Constant dark energy equation of state .

In addition to CMB priors, we used SNe compiled by Conley et al. (2011) Conley11 , flux-averaged to reduce systematic errors, the Hubble constant measurement by Riess et al. (2011) Riess et al. (2011), the GRB data as summarized in Wang (2008) Wang08c , and GC data from SDSS DR7 derived by Chuang & Wang (2012) CW12 . We have chosen to use the same data as in Wang, Chuang, & Mukherjee (2012), except for the CMB priors (Planck+lensing+WP and WMAP9, versus WMAP7), in order to compare the impact of the various CMB priors. For completeness, we have added the GC data from BOSS derived by Chuang et al. (2013) Chuang et al. (2013) in constraining .

We find that when dark energy density is allowed to be a free function, current data (excluding GC data from BOSS, with either Planck+lensing+WP or WMAP priors) are fully consistent with a cosmological constant and a flat universe at 95% C.L., but deviate from a cosmological constant in a flat universe at 68% C.L. (see Fig.6). The addition of BOSS data leads to a deviation from a cosmological constant at the redshift near that of the BOSS data at confidence independent of cosmic curvature (see Fig.6). In general, adding Planck+lensing+WP priors leads to a preference for a small positive (i.e., a slightly open universe) for a cosmological constant, or a flat universe with dark energy deviating from a cosmological constant, compared to adding WMAP9 priors (see Fig.6).

When dark energy equation of state is assumed to be a linear function in the cosmic scale factor , the dark energy constraints depend on the base parameters used, the highly correlated , or the much less correlated . The constraints on are consistent with a cosmological constant and a flat universe for both Planck+lensing+WP and WMAP9 priors, while that on are marginally inconsistent with a cosmological constant in a flat universe (see Fig.9 and Fig.10) for Planck+lensing+WP priors, similar to our findings in the case.

We find that the above trend in the Planck+lensing+WP versus WMAP9 comparison becomes more pronounced when we assume a constant dark energy equation of state. Here we only combine CMB priors with a minimal set of other data, SNe and , to highlight the difference between Planck+lensing+WP and WMAP9 priors. We find that adding the Planck+lensing+WP priors to the SNe and measurement leads to an inconsistency with a cosmological constant in a flat universe at % confidence level (see Fig.11).

To conclude, we find that Planck distance priors are significantly tighter than those from WMAP9 (see Figs.1-3). However, adding Planck distance priors does not lead to significantly improved dark energy constraints using current data, compared to adding WMAP9 distance priors. This is because Planck data appear to favor a higher matter density and lower Hubble constant Ade et al. (2013), in tension with most of the other current cosmological data sets.

In order to understand the nature of dark energy, we will need to improve our understanding of the systematic uncertainties of all data used. Future dark energy measurements from space jedi ; Cimatti09 ; euclid ; wfirst that minimize systematic uncertainties by design will enable us to make dramatic progress in our quest to shed light on dark energy.

Acknowledgments We acknowledge the use of Planck data archiv and CosmoMC. This work is supported in part by DOE grant DE-FG02-04ER41305.

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