Probing a Steep EoS for Dark Energy with latest observations
We present a parametrization for the Dark Energy Equation of State “EoS” which has a rich structure, performing a transition at pivotal redshift between the present day value to an early time with a steepness given in terms of parameter. The proposed parametrization is , with , , and constant parameters. It reduces to the widely used EoS for . This transition is motivated by scalar field dynamics such as for example quintessence models. We study if a late time transition is favored by BAO measurements combined with local determination of and information from the CMB. According to our results, an EoS with a present value of and a high redshift value , featuring a transition at with an exponent was favored by data coming from local dynamics of the Universe (BAO combined with determination). We find that a dynamical DE model allows to simultaneously fit from local determinations and Planck CMB measurements, alleviating the tension obtained in a CDM model. Additionally to this analysis we solved numerically the evolution of matter over-densities in the presence of dark energy both at background level and when its perturbations were considered. We show that the presence of a steep transition in the DE EoS gets imprinted into the evolution of matter overdensities and that the addition of an effective sound speed term does not erase such feature.
We live in a particular epoch of the cosmic history characterized by the acceleration in the expansion rate of the Universe. Although its cause is unknown this acceleration is described as the consequence of a Cosmological Constant, , with density constant in space and time. Despite its simplicity, there is no fundamental understanding of its origin and this framework has serious theoretical issues namely the coincidence and fine-tunning problems(RevModPhys.61.1 ; Sahni:2002kh ). For this reason alternative models that either modify gravity at large scales as prescribed by General Relativity or introduce a dynamical Dark Energy (DE) component have arisen. Dynamical dark energy models are often characterized by the DE equation of state (EoS), , which is the ratio of the DE pressure to its density. Since DE properties are still unknown, several models to parametrize its EoS as a function of time, , have arisen in the literature (Doran:2006kp ; Rubin:2008wq ; 2009ApJ…703.1374S ; 2010PhRvD..81f3007M ; Hannestad:2004cb ; Jassal:2004ej ; Ma:2011nc ; Huterer:2000mj ; Weller:2001gf ; Huang:2010zra ; delaMacorra:2015aqf ). One of the most popular among them is the CPL parametrization (Chevallier:2000qy ,Linder:2002et ), widely used in cosmological observational analysis. The present value of DE EoS is restricted by observations to be close to ( according to the 95 limits imposed by Planck data combined with other astrophysical measurements Ade:2015xua ). Nevertheless, the DE behavior and its properties at different cosmic epochs are much poorly constrained by current cosmological observations. According to astrophysical observations our Universe is flat and dominated at present time by the DE component (Ade:2015xua ), so data coming from late-time, low-redshift measurements such as Baryon Acoustic Oscillations (BAO) from Large Scale Structure surveys are those best suited for its analysis.
The aim of this work is to determine the late time dynamics of DE through its EoS, and in particular we are interested in studying if a transition in takes place. To that end the parametrization used is , with , , , and constant parameters. This EoS allows for a steep transition for a large value of at the pivotal point , which is prompted by scalar field dynamics such as quintessence models delaMacorra:1999ff and motivated in delaMacorra:2015aqf , where a new parametrization that captures the dynamics of DE is presented.
The scientific community is devoting a large amount of time and resources in the quest to understand the dynamics and nature of DE, working on current (SDSS-IV Dawson:2015wdb , DES Abbott:2005bi ) and future (DESI Levi:2013gra ; Aghamousa:2016zmz ; Aghamousa:2016sne , Euclid 2011arXiv1110.3193L , LSST 2009arXiv0912.0201L ) experiments to study with very high precision the expansion history of the Universe and thus be able to test interesting models beyond a Cosmological Constant or Taylor expansions of the EoS of DE.
This article is organized as follows: we introduce our theoretical framework and the data sets used in Section II, Section III details the analysis performed at Background level and the results obtained, Section IV discusses the analysis at perturbative level while Section V summarizes our Conclusions.
Ii Method and Data
Within the General Relativity framework for a flat Universe and a FLRW metric we have
where is the Hubble parameter, the scale factor of the Universe and is the Hubble parameter at redshift zero in units of . The present value of matter, radiation, and DE fractional densities are given by , , , respectively. The function in equation (1) encodes the evolution of DE component in terms of its EoS, , according to
where specifies the evolution of the DE fluid and accordingly, the Universe expansion rate at late times, following the dynamics set by equation (1).
with where and represent the value for at large redshifts and at present day, respectively, whereas the term modulates the dynamics of this parametrization in between both values, and takes the values , and , i.e. . This EoS makes a transition between the two regimes: , , at redshift , taking a value of . The parameter modulates the steepness of the transition featured: a larger value for has a steeper transition, as figure 1 shows.
For , equation (3) includes the well known CPL parametrization (Chevallier:2000qy ; Linder:2002et ) as a particular case but it allows for a richer physical behavior. CPL EoS written in terms of scale factor reads:
from where we see that its slope is constant and with a value , meaning that the late time dynamics of DE is fixed from the present and initial values of the EoS.
Clearly, taking in (3), the Cosmological Constant solution, , is recovered.
ii.1 BAO as a cosmological probe
Ever since its first detection (Cole et al. 2005 Colless:2003wz , Eisenstein et al 2005, Eisenstein:2005su ) the Baryon Acoustic Oscillation feature has been widely used as a powerful tool for cosmology becoming the standard ruler of choice. It has become the best way to probe late time dynamics of the Universe and in consequence that of DE. For that reason it is the cosmological tool used by several experiments like 6dF Beutler:2011hx , WiggleZ Kazin:2014qga , SDSS-III Anderson:2013oza ,(and most recently Alam:2016hwk ), SDSS-IV Dawson:2015wdb and Dark Energy Survey (DES) Abbott:2005bi and the main probe to be implemented in future experiments like the Dark Energy Spectroscopic Instrument (DESI) Levi:2013gra ; Aghamousa:2016zmz ; Aghamousa:2016sne and Euclid 2011arXiv1110.3193L .
The corresponding size, , is obtained by performing a spherical average of the galaxy distribution both along and across the line of sight (Bassett and Hlozek 2010 Bassett:2009mm ):
The comoving sound horizon at the baryon drag epoch is represented by and the dilation scale, , contains information about the cosmology used in :
In this way, the BAO standard ruler which is set by a particular size in the spatial distribution of matter, can be used to constrain the parameters in equation (3).
While the sound horizon, , depends upon the physics prior to the recombination era, given by Ade:2015xua and the baryon to photon ratio, , the dilation scale, , is sensitive to the physics of much lower redshifts, particularly to those probed by Large Scale Structure experiments.
In this work we make use of the observational points from the six-degree-field galaxy survey (6dFGS Beutler:2011hx ), Sloan Digital Sky Survey Data Release 7 (SDSS DR7 Ross:2014qpa ) and the reconstructed value (SDSS(R) Padmanabhan2pc ), as well as the latest result from the complete BOSS sample SDSS DR12 (Alam:2016hwk ), and the Lymann- Forest (Ly-F) measurements from the Baryon Oscillation Spectroscopic Data Release 11 (BOSS DR11 Font-Ribera:2013wce , Delubac:2014aqe ). Table 1 summarizes them all. Since the volume surveyed by BOSS and WiggleZ Kazin:2014qga partially overlap we do not use data from the latter in this work (see details in Beutler:2015tla ).
ii.2 Local value of the Hubble Constant
The present value of Hubble constant has been determined observationally from direct measurement of the local dynamics, as in the latest work of A. Riess et al in localhubble , but also from BAO measurements either from galaxy surveys or from the Lyman- forest and it can be derived as well from CMB experiments such as Planck.
Regarding the work of A. Riess et al (AR16), their best estimate in units of reports a value of
the accuracy of which was achieved in great deal due to the utilization of maser system in NGC1258 both to calibrate and as an independent anchor for the cosmic distance ladder.
ii.3 Cosmic Microwave Background
The Cosmic Microwave Background is the most precise cosmological data set. The angle subtended by the first peak is determined with exquisite precision (Ade:2015xua ):
Following the latest report of the Planck collaboration (P15) planck15DE and galli2014 we use Planck TT+TE+EE+lowP which denotes the combination of likelihood at using TT, TE, and EE spectra with the low- temperature+polarization likelihood.
However, it has been shown (wangmukherjee , mukherjee , planck15DE ) that the information of CMB power spectra can be compressed within few observables such as the angular scale of sound horizon at last scattering, , and the scaled distance to last scattering surface, .
We keep the flat geometry and the baryon density fixed and thus we can add the CMB information to BAO and measurements by means of the observables . The corresponding covariance matrix is
The angle of horizon at last scattering is defined to be
where is the horizon size at the decoupling epoch ( according to Planck Ade:2015xua ), defined by the integral in equation (6) evaluated from to , and is the comoving distance to last scattering surface:
The reported value for the Hubble constant by P15 is Ade:2015xua , which assumes a CDM universe and is known to be in tension with AR16 at the level.
Iii Background Evolution
Additionally to parameters in equation (3) we also investigate the constraints on the physical density of cold dark matter and (or equivalently ), resulting in the set and consider uniform priors on these: [0.5, 1], [0.001, 0.99], [-1, 0], [-1, 0], [1, 10] and [0, 3]. To determine the best-fitting values (BFV), we minimize the goodness-of-fit estimator,
where are theoretical values for each observable (namely , , , ) and the data. The corresponding covariance matrix is represented by .
The joint analysis of the different data sets is done by adding their respective functions. Further details can be found in the Appendix A.
The reported value of , and in tables summarizing our results was obtained by taking the BFV for and for each model, as well as the fixed value from P15 Ade:2015xua .
|6dF||0.106 Beutler:2011hx||0.336 0.015|
|SDSS DR7||0.15 Ross:2014qpa||0.2239 0.0084|
|SDSS(R) DR7||0.35 Padmanabhan2pc||0.1137 0.0021|
|0.38 Alam:2016hwk||0.100 0.0011|
|0.61 Alam:2016hwk||0.0691 0.0007|
|2.34 Delubac:2014aqe||0.0320 0.0013|
|2.36 Font-Ribera:2013wce||0.0329 0.0009|
Local measurements (labeled model A in table 2) point to a dynamical DE presenting a very late and abrupt transition (, ) from an initial value to a present value . This behavior is portrayed in figure 2 and corresponds to the black dot-dashed curve. The value for holds in agreement with the reported measurement from AR16 used as prior for this calculation.
The dynamics for DE resulting from the use of BAO data and CMB reduced likelihood (outcome B, Table 2) indicates the preference for a steep transition () from the initial value to the present value at a pivotal redshift . This corresponds to the dotted line in figure 2. The value for lies within the range imposed by CMB priors and the BFV for is lower, in agreement with P15 (Ade:2015xua ).
Model C in table 2 shows that a late time and smooth transition (=1.31, ) was preferred by data, with an initial value to a present value . The blue dashed line in 2 displays this particular dynamics. The amount of matter is very similar in DE and CDM models (cases C and ), however we obtain a larger amount of DE at present time and therefore a larger . We see that the dynamics of DE allows to consistently fit the variables from CMB along with the local value of , since the inclusion of in model C only increased by 0.2% compared to model B. However, from Table 3 we see that the addition of in CDM model ( and ) severely penalizes the fit by increasing by , showing a tension in the value for from CMB and local measurements.
In this case, the DE density at early times is not negligible since it has . Figure 4 shows that its contribution at decoupling is of order , adding an extra component that behaves like dust () at large redshifts. The ratio of DE density to ordinary matter () is nearly constant from and has a value (Figure 3(b)). This changes several cosmological parameters, for instance the equivalence epoch, , is smaller modifying the distance to the last scattering surface and the sound horizon at recombination. This is an interesting toy model worthwhile of further studies, and it will also impact CMB power spectrum and Large Scale Structure formation.
Having a non-negligible DE at earlier times, allows to put better constraints on its parameters: .
A DE component which is non-negligible at early times as been studied in the literature and is known as Early Dark Energy (see for example Linder:2008nq ).
From both, table 2 and figure 3 we can draw the following general results. The value for is tightly constrained by observations. The scenario is included within 1 error for all the cases. Generally speaking, for the outcomes where DE density becomes negligible at earlier times, we obtained weak or no constraints for the initial value of the EoS, , the transition time, , and the exponent . In all outcomes, the values and are contained within 1 of significance Figure 3 shows that is highly degenerated with . The results for CDM are summarized in table 3.
|Steep Equation of State for DE|
|Alias||Data sets used|
|Alias||Data sets used|
||BAO + CMB||11.74||0.3504|
|BAO + CMB +||13.98||0.3614|
Iv Growth of Perturbations
We are interested in studying the effect that the transition featured by the EoS (3) has in the evolution of matter overdensities well inside the horizon in the matter-DE domination era. We do so by means of the following system of linearized equations:
where and represent the matter and DE density contrasts, respectively; is the equation of state (3) as function of scale factor, and are the DE and matter fractional densities, , the Hubble function and the Fourier wave number.
The term in equation (15b) represents the speed of sound for the DE. We can split it as the sum of an adiabatic contribution and an effective or non-adiabatic part:
stands for the adiabatic speed of sound, defined as
which depends solely on the equation of state for the DE fluid and its derivative, whereas represents the non-adiabatic contribution to the speed of sound. We model it as a constant and take some values, .
Figure 5 shows the effect a transition in the EoS has in . We note that the greater the value for , the sharper the bump in the adiabatic speed of sound. For this part of the analysis we take the values for as those resulting from A in Table 2.
Adiabatic initial conditions were assumed and the value for was fixed to be at the time of entrance for the k-mode into the horizon, whose value was chosen to be .
Figure 6 displays the solution from equation (15) for an DE EoS modeled by result A (table 2) and . We note that the effect of is to reduce the magnitude in the growth of , keeping the shape the same. It is also notorious that the evolution of becomes very non-linear when solved coupled to , due to the term in equation (15b).
In figure 7 we analyze in more detail the effect of a steep transition. We fix to focus on the effect of only (displayed in figure 5). For comparison we take the CPL with the same values for and as in Result A and we take the ratio of both solutions. The result is displayed in the lower panel of figure 7. We note the difference between and at the transition time, , as a sudden increase during the transition.
It is customary to take . In such case, the system of equations (15) reduces to:
In figure 9 we show the solution to equation (18) for the case of a CDM scenario, the best fit corresponding to Result A in table 2, and its CPL limit (4). The bottom panel shows the relative difference from each model to CDM, this is , where corresponds to the growth of matter contrast when we assume a Cosmological Constant solution. From we find differences from CDM of around 1 for a -mode of .
A deeper analysis on the impact of DE perturbations will be subject of a next paper Jaber2016perturbs .
V Summary and Conclusions
We presented a parametrization for the EoS of DE and found the constraints deduced by using BAO measurements contained in Table 1 combined with the latest local determination of Hubble constant (localhubble ). Additionally we used the compressed CMB likelihood from Planck (Ade:2015xua ), by means of the sound horizon at decoupling, , and .
The constraints for the free parameters, , and their 68 errors resulting from the combined analysis of the datasets were obtained.
Our results show that a dynamical DE is favored by data and that a steep transition is preferred by local measurements, i.e. BAO and , and by BAO with CMB Planck observables (figure 2).
Whereas for a CDM model, the tension between the local determination of (localhubble ) and the value derived from Planck (Ade:2015xua ) remains (table 3), we find that it is possible to simultaneously conciliate the observations from BAO, and CMB in a single model (table 2) by means of a dynamical Dark Energy.
For the perturbative analysis it was shown that the feature from the shape of due to the particular form of the EoS got imprinted in the evolution of . We modeled the speed of sound splitting it into an adiabatic contribution and an effective term, , where encapsulates the physics beyond the EoS of the dark fluid. The addition of this effective term did not erase the features from the bump in the adiabatic speed of sound but suppresses the exponential evolution of the over-densities by several orders of magnitude. This should be studied in detail and will be subject to discussion in a future paper Jaber2016perturbs .
The solution is only exact for the case of a cosmological constant. To be consistent we need to take DE perturbations into account. The solution to (15) with taken as the adiabatic contribution (17) for the model “BAO+” reported in table 2 was shown in figure 8. From this we saw that if we take , the solutions were highly unstable and became non-linear extremely fast during the evolution of over-densities.
To summarize, the study of dynamics of Dark Energy is a matter of profounds implications for our understanding of the Universe and its physical laws. Although the measurements from CMB are the most precise data sets in Cosmology, the best way to analyze the properties of DE comes from the low redshift regime, where the BAO feature is the most robust cosmic ruler. In this work we have contributed towards that direction, and we have presented the constraints for a dynamical DE model coming from the analysis of BAO distance measurements combined with the most recent determination and CMB information.
Acknowledgements.We acknowledge financial support from DGAPA-PAPIIT IN101415 and Conacyt Fronteras de la Ciencia 281 projects. MJ thanks Conacyt for financial support and to the attendees of MACSS for helpful discussions.
Appendix A Appendix
For the BAO measurements we use the function defined as
where is the difference between theoretical prediction for according to (5) and the values listed in table 1, and is the inverse of the covariance matrix containing the observational errors for the measurements. Since the data points used in this work are not correlated we have a diagonal matrix whose elements are the square-root of the errors reported in table 1.
Additionally to BAO data we make use of the local determination of by means of
Finally, for the CMB, we use the determination of and , made by the Planck Collaboration (P. A. R. Ade et al. 2015, Ade:2015xua ). In particular we use Planck TT + TE + EE + low P values from the table 4 of Ade:2015xua with the covariance matrix displayed in (11). With this matrix we can build the function:
where is the corresponding data vector defined as , and is the inverse of matrix (11).
- (1) Steven Weinberg. The cosmological constant problem. Rev. Mod. Phys., 61:1–23, Jan 1989.
- (2) Varun Sahni. The Cosmological constant problem and quintessence. Class. Quant. Grav., 19:3435–3448, 2002.
- (3) Michael Doran and Georg Robbers. Early dark energy cosmologies. JCAP, 0606:026, 2006.
- (4) D. Rubin et al. Looking Beyond Lambda with the Union Supernova Compilation. Astrophys. J., 695:391–403, 2009.
- (5) J. Sollerman, E. Mörtsell, T. M. Davis, M. Blomqvist, B. Bassett, A. C. Becker, D. Cinabro, A. V. Filippenko, R. J. Foley, J. Frieman, P. Garnavich, H. Lampeitl, J. Marriner, R. Miquel, R. C. Nichol, M. W. Richmond, M. Sako, D. P. Schneider, M. Smith, J. T. Vanderplas, and J. C. Wheeler. First-Year Sloan Digital Sky Survey-II (SDSS-II) Supernova Results: Constraints on Nonstandard Cosmological Models. Astrophys. J. , 703:1374–1385, October 2009.
- (6) M. J. Mortonson, W. Hu, and D. Huterer. Testable dark energy predictions from current data. Phys. Rev. D, 81(6):063007, March 2010.
- (7) Steen Hannestad and Edvard Mortsell. Cosmological constraints on the dark energy equation of state and its evolution. JCAP, 0409:001, 2004.
- (8) H. K. Jassal, J. S. Bagla, and T. Padmanabhan. WMAP constraints on low redshift evolution of dark energy. Mon. Not. Roy. Astron. Soc., 356:L11–L16, 2005.
- (9) Jing-Zhe Ma and Xin Zhang. Probing the dynamics of dark energy with novel parametrizations. Phys. Lett., B699:233–238, 2011.
- (10) Dragan Huterer and Michael S. Turner. Probing the dark energy: Methods and strategies. Phys. Rev., D64:123527, 2001.
- (11) Jochen Weller and Andreas Albrecht. Future supernovae observations as a probe of dark energy. Phys. Rev., D65:103512, 2002.
- (12) Zhiqi Huang, J. Richard Bond, and Lev Kofman. Parameterizing and Measuring Dark Energy Trajectories from Late-Inflatons. Astrophys. J., 726:64, 2011.
- (13) Axel de la Macorra. Dark Energy Parametrization motivated by Scalar Field Dynamics. 2015.
- (14) Michel Chevallier and David Polarski. Accelerating universes with scaling dark matter. Int. J. Mod. Phys., D10:213–224, 2001.
- (15) Eric V. Linder. Exploring the expansion history of the universe. Phys. Rev. Lett., 90:091301, 2003.
- (16) P. A. R. Ade et al. Planck 2015 results. XIII. Cosmological parameters. 2015.
- (17) A. de la Macorra and G. Piccinelli. General scalar fields as quintessence. Phys. Rev., D61:123503, 2000.
- (18) Kyle S. Dawson et al. The SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Overview and Early Data. Astron. J., 151:44, 2016.
- (19) T. Abbott et al. The dark energy survey. 2005.
- (20) Michael Levi et al. The DESI Experiment, a whitepaper for Snowmass 2013. 2013.
- (21) Amir Aghamousa et al. The DESI Experiment Part I: Science,Targeting, and Survey Design. 2016.
- (22) Amir Aghamousa et al. The DESI Experiment Part II: Instrument Design. 2016.
- (23) R. Laureijs, J. Amiaux, S. Arduini, J. . Auguères, J. Brinchmann, R. Cole, M. Cropper, C. Dabin, L. Duvet, A. Ealet, and et al. Euclid Definition Study Report. arXiv:1110.3193, October 2011.
- (24) LSST Science Collaboration, P. A. Abell, J. Allison, S. F. Anderson, J. R. Andrew, J. R. P. Angel, L. Armus, D. Arnett, S. J. Asztalos, T. S. Axelrod, and et al. LSST Science Book, Version 2.0. ArXiv e-prints, December 2009.
- (25) Matthew Colless et al. The 2dF Galaxy Redshift Survey: Final data release. 2003.
- (26) Daniel J. Eisenstein et al. Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies. Astrophys. J., 633:560–574, 2005.
- (27) F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson. The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant. MNRAS, 416:3017–3032, October 2011.
- (28) Eyal A. Kazin et al. The WiggleZ Dark Energy Survey: improved distance measurements to z = 1 with reconstruction of the baryonic acoustic feature. Mon. Not. Roy. Astron. Soc., 441(4):3524–3542, 2014.
- (29) Lauren Anderson et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: measuring and H at z = 0.57 from the baryon acoustic peak in the Data Release 9 spectroscopic Galaxy sample. Mon. Not. Roy. Astron. Soc., 439(1):83–101, 2014.
- (30) Shadab Alam et al. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. Submitted to: Mon. Not. Roy. Astron. Soc., 2016.
- (31) Bruce A. Bassett and Renee Hlozek. Baryon Acoustic Oscillations. 2009.
- (32) Ashley J. Ross, Lado Samushia, Cullan Howlett, Will J. Percival, Angela Burden, and Marc Manera. The clustering of the SDSS DR7 main Galaxy sample – I. A 4 per cent distance measure at . Mon. Not. Roy. Astron. Soc., 449(1):835–847, 2015.
- (33) Nikhil Padmanabhan, Xiaoying Xu, Daniel J. Eisenstein, Richard Scalzo, Antonio J. Cuesta, Kushal T. Mehta, and Eyal Kazin. A 2 per cent distance to z = 0.35 by reconstructing baryon acoustic oscillations – i. methods and application to the sloan digital sky survey. Monthly Notices of the Royal Astronomical Society, 427(3):2132–2145, 2012.
- (34) Andreu Font-Ribera et al. Quasar-Lyman Forest Cross-Correlation from BOSS DR11 : Baryon Acoustic Oscillations. JCAP, 1405:027, 2014.
- (35) Timothée Delubac et al. Baryon acoustic oscillations in the Lyman Forest of BOSS DR11 quasars. Astron. Astrophys., 574:A59, 2015.
- (36) Florian Beutler, Chris Blake, Jun Koda, Felipe Marin, Hee-Jong Seo, Antonio J. Cuesta, and Donald P. Schneider. The BOSS?WiggleZ overlap region ? I. Baryon acoustic oscillations. Mon. Not. Roy. Astron. Soc., 455(3):3230–3248, 2016.
- (37) Adam G. Riess et al. A 2.4% determination of the local value of the hubble constant. Astrophys. J., 826(1):56, 2016.
- (38) P. A. R. Ade et al. Planck 2015 results. XIV. Dark energy and modified gravity. Astron. Astrophys., 594:A14, 2016.
- (39) Silvia Galli, Karim Benabed, Fran çois Bouchet, Jean-Fran çois Cardoso, Franz Elsner, Eric Hivon, Anna Mangilli, Simon Prunet, and Benjamin Wandelt. Cmb polarization can constrain cosmology better than cmb temperature. Phys. Rev. D, 90:063504, Sep 2014.
- (40) Yun Wang and Pia Mukherjee. Observational constraints on dark energy and cosmic curvature. Phys. Rev. D, 76:103533, Nov 2007.
- (41) Pia Mukherjee, Martin Kunz, David Parkinson, and Yun Wang. Planck priors for dark energy surveys. Phys. Rev. D, 78:083529, Oct 2008.
- (42) Eric V. Linder and Georg Robbers. Shifting the Universe: Early Dark Energy and Standard Rulers. JCAP, 0806:004, 2008.
- (43) Mariana Jaber and Axel de la Macorra. Dark energy perturbations. in preparation.