Exploring parameter constraints on quintessential dark energy: The pseudo-Nambu-Goldstone-boson model

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  • Exploring parameter constraints on quintessential dark energy: The pseudo-Nambu-Goldstone-boson model

    Augusta Abrahamse, Andreas Albrecht, Michael Barnard, and Brandon Bozek

    Physics Department, University of California, Davis, California, USA(Received 17 January 2008; published 5 May 2008)

    We analyze the constraining power of future dark energy experiments for pseudo-Nambu-Goldstone-

    boson (PNGB) quintessence. Following the Dark Energy Task Force (DETF) methodology, we forecast

    data for three experimental stages: Stage 2 represents in-progress projects relevant to dark energy; Stage 3

    refers to medium-sized experiments; Stage 4 comprises larger projects. We determine the posterior

    probability distribution for the parameters of the PNGB model using Markov Chain Monte Carlo analysis.

    Utilizing data generated on a CDM cosmology, we find that the relative power of the different data

    stages on PNGB quintessence is roughly comparable to the DETF results for the w0 wa parametrizationof dark energy. We also generate data based on a PNGB cosmological model that is consistent with a

    CDM fiducial model at Stage 2. We find that Stage 4 data based on this PNGB fiducial model will rule

    out a cosmological constant by at least 3.

    DOI: 10.1103/PhysRevD.77.103503 PACS numbers: 95.36.+x, 98.80.Es


    A growing number of observations indicate that theexpansion of the universe is accelerating. Given our currentunderstanding of physics, this phenomenon is a mystery. IfEinsteins gravity is correct, then it appears that approxi-mately 70% of the energy density of the universe is in theform of a dark energy. Although there are many ideas ofwhat the dark energy could be, as of yet, none of themstands out as being particularly compelling. Future obser-vations will be crucial to developing a theoretical under-standing of dark energy.A number of new observational efforts have been pro-

    posed to probe the nature of dark energy, but evaluating theimpact of a given proposal is complicated by our poortheoretical understanding of dark energy. In light of thisissue, a number of model independent methods have beenused to explore these issues (see, for example, [1,2]).Notably, the Dark Energy Task Force (DETF) produced areport in which they used the wo wa parametrization ofthe equation of state evolution in terms of the scale factor,wa wo wa1 a [3]. The constraints on the pa-rameters wo and wa were interpreted in terms of a figureof merit (FoM) designed to quantify the power of futuredata and guide the selection and planning of differentobservational programs. Improvements in the DETF FoMbetween experiments indicate increased sensitivity to pos-sible dynamical evolution of the dark energy. This iscrucial information, since current data is consistent witha cosmological constant and any detection of a deviationfrom a cosmological constant would have a tremendousimpact. There are, however, a number of questions leftunanswered when the dark energy is modeled using ab-stract parameters such as w0 wa that are perhaps betteraddressed with the analysis of actual theoretical models ofdark energy.

    First of all, the wo wa parametrization is simplisticand not based on a physically motivated model of darkenergy. Although simplicity is part of the appeal of thisparametrization, some of the most popular dark energymodels exhibit behavior that cannot be described by thewo wa parametrization. The pseudo-Nambu-Goldstone-boson (PNGB) quintessence model considered in this pa-per, for instance, allows equations of state that cannot beapproximated by the wo wa parametrization, as shownin Fig. 1.Conversely, the wo wa parametrization may allow

    solutions that do not correspond to a physically motivatedmodel of dark energy. Because of these issues, one couldwonder whether the DETF FoMs are somehow mislead-ing. Various concerns with the DETF analysis have alreadybeen explored. Albrecht and Bernstein used a more com-plex parametrization of the scale factor to check the valid-ity of the w0 wa approximation [4], and Huterer andPeiris considered a generalized class of scalar field models[5]. Additionally, it has been suggested that the DETF datamodels might be improved (see, for instance, [6]).As of yet, however, no actual proposed models of dy-

    namical dark energy have been considered in terms offuture data. Given the issues above, such an analysis isan important compliment to existing work. Of course allspecific models of dark energy are suspect for variousreasons, and one can just as well argue that it is better tomake the case for new experiments in a more abstractparameter space rather than tying our future efforts tospecific models that themselves are problematic. Ratherthan take a side in this discussion, our position is thatgiven the diversity of views on the subject, a model-basedassessment will have an important role in an overall as-sessment of an observational program.In this paper we consider the pseudo-Nambu-Goldstone-

    boson quintessence model of dark energy [7]. As one of the

    PHYSICAL REVIEW D 77, 103503 (2008)

    1550-7998=2008=77(10)=103503(13) 103503-1 2008 The American Physical Society

  • most well-motivated quintessence models from a particlephysics perspective, it is a worthwhile one to study. We usethe data models forecasted by the DETF and generate twotypes of data sets, one based on a CDM backgroundcosmology and one based on a background cosmologywith PNGB dark energy using a specific fiducial set ofPNGB parameters. We determine the probability space forthe PNGB parameters given the data using Markov ChainMonte Carlo analysis. This paper is part of a series ofpapers in which a number of quintessence models areanalyzed in this manner [8,9].We show that the allowed regions of parameter space

    shrink as we progress from Stage 2 to Stage 3 to Stage 4data in much the same manner as was seen by the DETF inthe w0 wa space. This result holds for both CDM andthe PNGB data models. Additionally, with our choice ofPNGB fiducial background model, we demonstrate theability of Stage 4 data to discriminate between a universedescribed by a cosmological constant and one containingan evolving PNGB field. As cosmological data continues toimprove, careful analysis of specific dark energy modelsusing real data will become more and more relevant.MCMC analysis can be computationally intensive andtime consuming. Since future work in this area is likelyto encounter similar challenges, we discuss some of thedifficulties we discovered and solutions we implemented inour MCMC exploration of PNGB parameter space.


    Quintessence models of dark energy are popular con-tenders for explaining the current acceleration of the uni-verse [1012]. Although the cosmological constant isregarded by many to be the simplest theory of the darkenergy, the required value of the cosmological constantappears to be many orders of magnitude too small in naiveparticle theory estimates. In quintessence models this prob-lem is not solved. Instead it is generally sidestepped byassuming some unknown mechanism sets the vacuum en-

    ergy to exactly zero, and the dark energy is due to a scalarfield evolving in a pressure dominated state. As such fieldscan appear in many proposed fundamental theories, andas the mechanism mimics ideas familiar from cosmicinflation, quintessence models are regarded by many (butcertainly not by everyone [13]) to be at least as plausible asa cosmological constant [14,15].Here the quintessence field is presumed to be homoge-

    neous in space, and is described by some scalar degree offreedom and a potential V which governs the fieldsevolution. In a Friedmann-Robertson-Walker (FRW)spacetime, the fields evolution is given by

    3H _ dVd

    0; (1)


    H _aa



    H2 13M2p

    r m k; (3)

    where MP is the reduced Planck mass, r is the energydensity of radiation, m is the energy density of nonrela-tivistic matter, and k is the effective energy density ofspacetime curvature. The energy density and pressure as-sociated with the field are

    12 _2 V; P 12 _2 V (4)and the equation of state w is given by

    w P

    : (5)

    If the potential energy dominates the energy of the field,then as can be seen in Eq. (4) the pressure will be negativeand in some cases can be sufficiently so to give rise toacceleration as the universe expands.















    FIG. 1 (color online). Examples of possible equations of state for PNGB quintessence (left panel). Attempts to imitate this behaviorwith wz curves for the w0 wa parametrization are depicted on the right.



  • The PNGB model of quintessence is considered com-pelling because it is one of the few models that seemsnatural from the perspective of 4D effective field the-ory. In order to fit current observations, the quintessencefield must behave at late times approximately as a cosmo-logical constant. It must be rolling on its potential with-out too much contribution from kinetic energy, and thevalue of the potential must be close to the observed valueof the dark energy density, which is on the order of thecritical density of the universe c3H20m2p1:881026h2 kg=m3 or 2:3 10120h2 in reduced Planck units,where h H0=100. These considerations require the fieldto be nearly massless and the potential to be extraordinarilyflat from the point of view of particle physics. In general,radiative corrections generate large mass renormalizationsat each order of perturbation theory unless symmetriesexist to suppress this effect [1618]. In order for such