Quintessence in a quandary: prior dependence in dark energy models
David J. E. Marsh, Philip Bull, Pedro G. Ferreira, Andrew Pontzen
QQuintessence in a quandary: prior dependence in dark energy models
David J. E. Marsh, ∗ Philip Bull, Pedro G. Ferreira, and Andrew Pontzen Perimeter Institute, 31 Caroline St N, Waterloo, ON, N2L 6B9, Canada Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway Astrophysics, University of Oxford, DWB, Keble Road, Oxford OX1 3RH, UK Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK (Dated: Received September 30, 2018; published – 00, 0000)The archetypal theory of dark energy is quintessence: a minimally coupled scalar field with a canonicalkinetic energy and potential. By studying random potentials we show that quintessence imposes a restrictedset of priors on the equation of state of dark energy. Focusing on the commonly-used parametrisation, w ( a ) ≈ w + w a ( − a ) , we show that there is a natural scale and direction in the ( w , w a ) plane that distinguishesquintessence as a general framework. We calculate the expected information gain for a given survey and showthat, because of the non-trivial prior information, it is a function of more than just the figure of merit. Thisallows us to make a quantitative case for novel survey strategies. We show that the scale of the prior sets targetobservational requirements for gaining significant information. This corresponds to a figure of merit FOM (cid:38) What drives the accelerated expansion of the Universe?Anything with a sufficiently negative equation of state willdo. Consequently, there are a vast number of possible models,generically termed ‘dark energy’ (DE). The equation of statecan depend on the scale factor, a , and is used to parametrisea wide range of these theories. One is left, however, withouta clear idea of how accurate observations must be to actuallyconstrain DE.Consider the commonly-used series expansion of the equa-tion of state, w ≈ w + w a ( − a ) [1, 2]; this is the param-eterisation most commonly used by observers. Given finiteresources, what is the optimal precision to which we shouldmeasure w and w a ? To tackle this question we need sometheoretical input to identify the regions within the ( w , w a ) plane that would allow us to to have a realistic chance of ac-tually distinguishing physical models of dynamical DE froma cosmological constantThe archetypal physical model of DE is quintessence [3–5]: a scalar field with a potential energy that dominates at latetimes. If one assumes the well-motivated case of a canonicalkinetic energy term, different models consist solely of partic-ular choices of potentials. If the scalar field of quintessence issubject to the rules of effective field theory (EFT), for exam-ple, the potential is restricted to a particular functional form,with coupling constants of a particular amplitude (modulo thecosmological constant, Λ , problem). Similar restrictions arisein specific models within particle physics and string theory,such as pseudo-Nambu-Goldstone Bosons (PNGBs) [6, 7] oraxions (e.g. [8]), moduli of extra dimensional theories (e.g.[9–11]), and monodromy [12–14].In this paper we show that quintessence a priori defines anatural scale and degeneracy direction on the ( w , w a ) planewhen various physical guiding principles are taken into ac-count. This is demonstrated graphically in Fig. 1, which is anew result of this work. A typical error ellipse for a futuregalaxy survey with figure of merit (FOM) ∼
600 is shown by ∗ [email protected] w w a FIG. 1. Quintessence priors in the ( w , w a ) plane (out to 95% CL),after loose observational priors have been applied. A remarkablytight structure is observed for all physical models. (Red: EFT; Green:Axion; Blue: Modulus; Yellow: Monomial. All have Λ = ∼ the filled contours (1 and 2 σ regions), and 95% CL regionsfor the physical quintessence priors are shown by the unfilledcontours. The way that these two areas overlap allows us toquantify the information that can actually be gained about DEby undertaking a given survey. Evolution equations —
The evolution equations are3 (cid:18) ˙ aa (cid:19) = ρ r , a − (cid:0) + a / a eq (cid:1) +
12 ˙ φ + A P ( φ ) , − (cid:18) ¨ aa (cid:19) = ρ r , a − (cid:0) + a / a eq (cid:1) + (cid:2) ˙ φ − A P ( φ ) (cid:3) , ¨ φ = − φ ˙ a / a − A P , φ , where P ( φ ) is the dimensionless functional form of thequintessence potential and A is its overall scale, V ( φ ) = AM P M H P ( φ ) . We work in units of the reduced Planckmass (energy scale), M P = / √ π G = . × eV, and a r X i v : . [ a s t r o - ph . C O ] D ec the Hubble rate (time scale), M H =
100 km s − Mpc − = . × − eV. We have used the redshift of matter-radiation equality, z eq = / a eq − T CMB = .
725 K)to fix the relative matter and radiation densities, ρ r , = . π k B T M − P M − H = ρ M , / ( + z eq ) , where the leadingnumerical factor accounts for photons and three generations ofneutrinos with negligible mass.The DE equation of state is w ( a ) = P φ / ρ φ ≈ w + ( − a ) w a . The coefficients can be evaluated directly at a = w = ˙ φ − A P ( φ ) ˙ φ + A P ( φ ) , w a = A ˙ a ρ φ (cid:2) P ( φ ) ˙ φ H + P , φ ˙ φ ρ φ (cid:3) . In our units, the fractional density in a given component is Ω X ( a ) = ρ X ( a ) / H ( a ) . Where relevant, we include the cos-mological constant (c.c.) within V ( φ ) and hence w .We proceed by Monte Carlo sampling (a) various randomfunctional forms for the potential, (b) the parameters of thesefunctional forms, and (c) the initial conditions of the field.The resulting cosmologies are subjected to loose observa-tional cuts to ensure broad consistency with the real Universe. Functional forms —
We consider a number of generalquintessence potentials with functional forms P ( φ ) = c Λ ξ Λ + f ( φ ) + n max ∑ n min c n ξ n b n ( φ ) , where c n is a deterministic constant, ξ n is a random variable, b n ( φ ) is a basis function and f ( φ ) is a leading contributionto the potential [15]. The term c Λ ξ Λ takes account of the c.c.,with c Λ = , ξ i ≡ ξ ∈ N ( , ) .All potentials are truncated at finite order n max , while n min ismodel-specific. In this paper we consider various types of po-tential, summarised in Table I. Free parameters are sampledaccording to the distributions given in Table II. These distri-butions are chosen to be sufficiently broad and reasonable tocapture a wide range of quintessence behaviours. Kac/Weyl potentials are simple random polynomial func-tions [16]. These will serve as baseline random potentials, buthave no physical motivation.A
Monomial potential is an integer power law, with onlya leading order part, f ( φ ) = φ N . Although possible physi-cal motivations include possible relation to chaotic inflation[17], or as large-field limits of certain monodromy models,our chief reason for including these potentials is simplicity. EFT potentials contain a leading ‘classical contribution’[18], f ( φ ) = ε ξ φ + ε ξ φ , plus a random polynomial of‘quantum corrections’ expanded in an energy scale parame-ter, ε F . To allow quintessence-like masses and energy den-sities, one requires | φ | >
1, and therefore the EFT must becontrolled by a super-Planckian shift symmetry, F > M P [19].For ε F = M P / F <
1, this fixes c n = ε n F . In order to have theexpansion begin at some leading order beyond the classicalcontribution, n min = p E >
4. The number of quantum correc-tion terms is n Q = n max − p E + Axion/PNGB is a sum of cosines. Wechoose f ( φ ) such that the leading term contributes no c.c. in Model b n ( φ ) c n n min f ( φ ) φ i Kac φ n [ − , ] Weyl φ n / √ n ! 1 0 [ − , ] Mono. 0 – – φ N [ , ] EFT φ n ( ε F ) n p E ξ ε φ [ − ε − , ε − ]+ ξ ε φ Axion cos ( n ε F φ ) ( ε NP ) n − + cos ε F φ [ − πε F , πε F ] Modulus e α ( p D − n ) φ ( ε D ) n [ − , ] TABLE I. Model specifications for the functional form P ( φ ) . Parameter Model Dist. log A All U ( − , ) N Monomial U Z ( , ) n max Kac, Weyl, Ax., Mod. U Z ( , ) n Q , p E EFT U Z ( , ) log ε F , NP , D EFT, Ax., Mod. U ( − , − ) p D Modulus U Z ( , ) α Modulus U ( , ) TABLE II. Parameter distributions for the models in Table I. U is theuniform distribution, and subscript ‘ Z ’ indicates that the distributionis over the integers. the vacuum, as is conventional for axions, and higher-ordernon-perturbative corrections are suppressed by ε NP <
1. Theshift symmetry is controlled by the scale F > M P , so ε F < Modulus of a higher dimensional theorygenerically includes exponentials [20]. There can be leadingpositive exponentials, with higher-order negative exponentialssuppressed by the compactification scale ε D = ( lM ) − , where l (cid:46) − m is a length scale and M < M P a mass scale, giving f ( φ ) = b n ( φ ) = exp ( α ( p D − n ) φ ) , c n = ε nD and n min = Initial conditions —
Initial conditions on the field are drawnfrom a uniform distribution at a i = − a eq , well beforematter-radiation equality. Field displacement can always bereabsorbed in a shift, but total displacement is relevant to thefate of the universe [22, 23] and depends on UV completion[24]. The natural field range for each of our models is givenin Table I. For Kac/Weyl and monomial models we take φ ∈ [ − , ] and φ i ∈ [ , ] respectively from demands on energydensity, steepness, symmetry and zeros [16]. For EFT con-trolled by a super-Planckian shift symmetry, the natural rangeis [ − ε − , ε − ] ; for PNGB/axions it is [ − πε − , πε − ] ; and formoduli it is [ − , ] , emerging from ε D < / l < M < M P for sub-Planckian compactification.We observed little difference in the resulting ( w , w a ) priorsbased on the prior on ˙ φ (cid:54) = z . We have also tested ourmodels with log-flat priors on the initial conditions for thefield and field velocity, and found that this also had little effecton the ( w , w a ) priors. w w a Kac
Weyl Mono EFT Axion Moduli
No cut,
Λ 0Λ 0Λ =0
FIG. 2. The (log-scaled) density of prior samples in the ( w , w a ) plane before (top) and after (middle) the observational cuts, for modelswith a cosmological constant and ˙ φ i =
0. Models with Λ = | w ( z ) | ≤ Ω DE > Observational cuts —
Models with excessive amounts ofearly DE are discarded [25] by requiring Ω DE ( z LSS ≈ ) < .
042 [26]; we require that the present Hubble rate, h = H / M H , is between 0 . < h < .
8, and the fractional DEdensity is between 0 . < Ω DE < .
8. We also put a weakprior on the present-day matter density by sampling z eq ∼ U [ , ] . We hold T CMB fixed and do not vary the neu-trino density. We reject any cosmologies that do not reach a = −
10% of the samples remain after the various cutsare applied, so we draw ∼ samples for each model to en-sure sufficient statistics. Fig. 2 shows the Monte Carlo priorsamples before and after cuts for each model. Results —
There is a strong correlation between the equa-tion of state values at different redshifts in quintessence mod-els [21], which is observed as restrictive joint priors on ( w , w a ) once our broad priors on other cosmological pa-rameters are imposed, as shown in Figs. 1 and 2. Themore typical assumption of independent uniform priors onall of { H , Ω M h , w , w a } is not valid for generic physicalquintessence models.Quintessence models define a narrow strip in the ( w , w a ) plane, with which certain values (such as the reference point ( − . , ) [27]) are inconsistent. Most of the allowed priorregion has w a < w → − z , while friction lessens at low- z , allowing w to become larger. Thus the prior lies near (but not exactlyon) the ‘thawing’ region of Ref. [29] (see also Ref. [30]). Therandom quintessence models studied by Ref. [21] constructedusing priors in the flow equations were found to be almostentirely ‘freezing’: random evolution constructs arbitrary po-tentials, distinct from our random physical models (see alsoRefs. [31, 32]).Because of this asymptotic behaviour at high- z , the ( w , w a ) parametrisation can fail at z (cid:38) w ( z ) < −
1. Some of our models lie in this ‘ap-parent phantom’ region of the ( w , w a ) plane, but are actuallynon-phantom for all z . The ( w , w a ) fit should always be takento have broken down as a description of quintessence above agiven z if it predicts w ( z ) < −
1. Our priors for this parametri-sation are suitable for forthcoming low- z tests of DE; for (e.g.)the CMB, the full scalar field evolution or a different asymp-totic fit for w ( z ) should be used.To quantify the effect of a non-trivial prior we calculatethe information gain over the prior from conducting a givenDE survey. The information is a uniquely-motivated quantityfor describing the constraining power of a given probabilitydistribution [33]. The gain in information from conductinga set of observations is sometimes known as the Kullback-Leibler divergence, and for discrete (binned) probability dis-tributions is defined as ∆ S = ∑ k P k log ( P k / Q k ) . Here, k la-bels the bins, Q k is the prior (i.e. the normalised 2D his-togram in ( w , w a ) -space), P k = C L k Q k is the normalised pos-terior, and L k is the likelihood. In the limit that the like-lihood is completely uniform (i.e. uninformative), ∆ S → FIG. 3. Relative entropy as function of figure of merit for a typicalfuture galaxy survey (solid lines), and for the same but with errorellipse rotated by 90 ◦ (dashed lines). The red and grey lines are forlikelihoods fixed at given fiducial values of ( w , w a ) , while the blacklines are for ∆ S marginalised over all fiducial values, (cid:104) ∆ S (cid:105) . The thickblue line shows ∆ S for a uniform prior over ( w , w a ) -space, and doesnot depend on the fiducial point. hood with the covariance given by the inverse of a Fisher ma-trix for a future galaxy redshift survey, F , centred on somefiducial point ( w , w a ) | fid . , and marginalised over all otherparameters [34]. We consider the cases where the fiducialpoint is fixed and where it is marginalised, defining (cid:104) ∆ S (cid:105) = (cid:82) Q ( x , y ) ∆ S ( x , y ) dx dy / (cid:82) Q ( x , y ) dx dy , where ( x , y ) run overall fiducial values of ( w , w a ) . We rescale the covariance ma-trix by the figure of merit, FOM = / (cid:112) det F − | w , w a , which(loosely) increases with the increasing accuracy of distancemeasurements from a survey. We also consider the possibilityof having an error ellipse that is orthogonal to that of a galaxysurvey, which could be achieved in practise by (e.g.) cosmicshear [35] or redshift drift measurements [36]. It is also pos-sible to partially rotate the error ellipse of a redshift survey bymaking an appropriate choice of target redshift and binning.In Fig. 3 we show ∆ S as a function of FOM for physicalquintessence as a whole (i.e. combining, with equal weights,the normalised prior distributions for all but the unmotivatedKac, Weyl, and Monomial models), and compare this to the ∆ S that would be obtained if uniform priors on ( w , w a ) wereassumed. The value of ∆ S is larger for the uniform prior –since quintessence disfavours large regions of the ( w , w a ) plane a priori , there is less information to be gained from agiven survey than if all regions have equal prior probability.With quintessence priors, we also observe features in ∆ S as afunction of FOM as the observational error shrinks inside theprior region about a fixed fiducial point (c.f. the results for thepoint ( − . , − . ) in Fig. 3). The function marginalised over all fiducial points, (cid:104) ∆ S (cid:105) , does not show such a featurehowever; this is because the prior is still dominated by the Λ -like peak at ( w , w a ) = ( − , ) .The value (cid:104) ∆ S (cid:105) = ∆ S = σ than the prior, and istherefore related to a 5 σ detection threshold). This occurs forFOM ≈
200 for our quintessence priors applied to a galaxyredshift survey. Our reference DETF Stage IV experimentsurpasses this requirement. Even if they merely tighten con-straints around the c.c. case, observations with this precisionare valuable since they can start to rule out significant portionsof the prior space of quintessence.The orientation of the error ellipse, though unimportantin the uniform prior case, has a substantial effect for thequintessence prior; with the orthogonal ellipse, one alwaysfinds a greater information gain. For example, a survey withan orthogonal ellipse and FOM ∼
100 offers an equivalent (cid:104) ∆ S (cid:105) to a standard galaxy redshift survey with a much higherFOM of ∼ Letter we considered random, physically-motivatedmodels of quintessence, which were found to impose a spe-cific structure on the DE equation of state. The resulting prioron ( w , w a ) is only weakly sensitive to the details of how themodels are constructed, and is therefore suitable as a guide toregions of observational interest. The value of FOM where (cid:104) ∆ S (cid:105) = Acknowledgements —
We acknowledge C. Burgess, E.Copeland, A. Liddle, J. March-Russell, M. C. D. Marsh, P.Marshall and H. Peiris for useful discussions. DJEM acknowl-edges Oxford University for hospitality. PB is supported byEuropean Research Council grant StG2010-257080, and ac-knowledges Oxford Astrophysics, Caltech/JPL, and Perime-ter Institute for hospitality. PGF acknowledges support fromLeverhulme, STFC, BIPAC and the Oxford Martin School.AP acknowledges a Royal Society University Research Fel-lowship. This research was supported in part by PerimeterInstitute for Theoretical Physics. Research at Perimeter In-stitute is supported by the Government of Canada throughIndustry Canada and by the Province of Ontario throughthe Ministry of Economic Development & Innovation. Thecomputer code used in this paper is publicly available at gitorious.org/random-quintessence . [1] M. Chevallier and D. Polarski, Int. J. Mod. Phys. D10 , 213(2001), arXiv:gr-qc/0009008 [gr-qc].[2] E. V. Linder, Phys. Rev. Lett. , 091301 (2003), arXiv:astro- ph/0208512 [astro-ph].[3] C. Wetterich, Nuclear Physics B , 668 (1988).[4] P. J. E. Peebles and B. Ratra, Astrophys. J. , L17+ (1988). [5] R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. ,1582 (1998), arXiv:astro-ph/9708069 [astro-ph].[6] C. Hill and G. Ross, Nuclear Physics B , 253 (1988).[7] J. A. Frieman, C. T. Hill, A. Stebbins, and I. Waga, Phys. Rev.Lett. , 2077 (1995).[8] N. Kaloper and L. Sorbo, Phys. Rev. D79 , 043528+ (2009).[9] D. J. E. Marsh, Phys. Rev.
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