The Astrophysical Journal, 559:501-506, 2001 October 1
© 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

 

Quintessence and the Separation of Cosmic Microwave Background Peaks

Michael Doran , Matthew Lilley , Jan Schwindt , and Christof Wetterich
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany; m.doran@thphys.uni-heidelberg.de, m.lilley@thphys.uni-heidelberg.de, j.schwindt@thphys.uni-heidelberg.de, c.wetterich@thphys.uni-heidelberg.de

Received 2000 December 21; accepted 2001 April 26

ABSTRACT

We propose that it should be possible to use the cosmic microwave background (CMB) to discriminate between dark energy models with different equations of state, which includes distinguishing a cosmological constant from many models of quintessence. The separation of peaks in the CMB anisotropies can be parametrized by three quantities: the amount of quintessence today, the amount at last scattering, and the averaged equation of state of quintessence. In particular, we show that CMB peaks can be used to measure the amount of dark energy present before last scattering.

Subject headings: cosmic microwave background; cosmology: theory

1. INTRODUCTION

     The idea of quintessence was born (Wetterich 1988) from an attempt to understand the vanishing of the cosmological constant. It was proposed that the cosmological evolution of a scalar field may naturally lead to an observable, homogeneous dark energy component today. This contrasts with the extreme fine-tuning needed in order for a cosmological constant to become significant at recent times only. If quintessence constitutes a major part of the energy density of the universe today, say, Ω > 0.5, structure formation tells us that this cannot always have been so in the past (Peebles & Ratra 1988; Ratra & Peebles 1988; Ferreira & Joyce 1997, 1998). Combining the phenomenology of a large quintessence component with the quest for naturalness (Hebecker & Wetterich 2000) leads to cosmologies with an equation of state for quintessence changing in time, compatible with a universe accelerating today. In several aspects of phenomenology, the models with a dynamical dark matter component resemble cosmology with a cosmological constant (Huey et al. 1999). It is therefore crucial to find possible observations that allow us to discriminate between the dynamical quintessence models and a constant dark energy theory (i.e., a cosmological constant). The detailed structure of the anisotropies in the cosmic microwave background (CMB) radiation depends upon two epochs in cosmology: the time around the emission of the radiation (last scattering) and today. The CMB may therefore serve as a test to distinguish models in which quintessence played a role at the time of last scattering from those in which it was insignificant during that epoch. It may also reveal details of the equation of state of quintessence (characterized by w = p&phis;/ρ&phis;) in the present epoch.

     The calculation of CMB spectra is in general an elaborate task (Seljak & Zaldarriaga 1996; Hu & Sugiyama 1995). However, the location of the peaks and for our purpose the spacing between the peaks can be estimated with much less detailed knowledge if adiabatic initial conditions and a flat universe are assumed. The oscillations of the primeval plasma before decoupling lead to pronounced peaks in the dependence of the averaged anisotropies on the length scale. When projected onto the sky today, the spacing between the peaks at different angular momenta l depends, in addition, on the geometry of the universe at later time. It is given, to a good approximation, by the simple formula (Hu & Sugiyama 1995; Hu, Sugiyama, & Silk 1997)



Here τ0 and τls are the conformal time today and at last scattering (which are equal to the particle horizons), and τ = dta-1(t), with cosmological scale factor a. The sound horizon at last scattering s is related to τls by s = sτls, where the average sound speed before last scattering s ≡ τ dτcs obeys c = 3 + , with ρb/ρr as the ratio of baryon to photon energy density. We note a direct dependence of Δl on the present geometry through τ0 as well as an indirect one through the dependence of τls on the amount of dark energy today (see eq. [7]).

     The location of the mth peak can be approximated by (Hu et al. 2001)



where the phase shift &phis; is typically less than 0.4 and is determined predominantly by recombination physics. By taking the ratio of two peak locations (say, l1/l2), the factor Δl and with it the dependence on postrecombination physics drop out, and we are in principle able to probe prerecombination dark energy directly. If the other cosmological parameters were known, the dependence of &phis; on the amount of dark energy at last scattering could provide a direct test of this aspect of quintessence models. Unfortunately, &phis; also depends on other cosmological parameters including the baryon density and spectral index, and there is no known analytic formula for &phis; (and in fact &phis; does have some m-dependence). We first concentrate on the peak spacing Δl, for which an analytic formula can be given.

     The equation of state of a hypothetical dark energy component influences the expansion rate of the universe and thus the locations of the CMB peaks (Ferreira & Joyce 1997, 1998; Coble, Dodelson, & Frieman 1997; Caldwell, Dave, & Steinhardt 1998; Amendola 2000). In particular, the horizons at last scattering and today are modified, leaving an imprint in the spacing of the peaks. The influence of dark energy on the present horizon and therefore on the CMB has been discussed in Brax, Martin, & Riazuelo (2000). A likelihood analysis on combined CMB, large-scale structure, and supernova data (Efstathiou 1999; Bond et al. 2001) can also give limits on the equation of state. Several of these analyses concentrate on models in which the dark energy component is negligible at last scattering. In contrast, we are interested particularly in getting information about dark energy in early cosmology. Therefore, the amount of dark energy at last scattering is an important parameter in our investigation.

     We present here a quantitative discussion of the mechanisms that determine the spreading of the peaks. A simple analytic formula permits us to directly relate Δl to three characteristic quantities for the history of quintessence, namely, the fraction of dark energy today, Ω; the averaged ratio between dark pressure and dark energy, 0 = &angl0;p&phis;/ρ&phis;&angr0;0; and the averaged quintessence fraction before last scattering, (for details of the averaging see below). We compare our estimate with an explicit numerical solution of the relevant cosmological equations using CMBFAST (Seljak & Zaldarriaga 1996). For a given model of quintessence, the computation of the relevant parameters Ω, 0, and requires the solution of the background equations. Our main conclusion is that future high-precision measurements of the location of the CMB peaks can discriminate between different models of dark energy if some of the cosmological parameters are fixed by independent observations. It should be noted here that a likelihood analysis of the kind performed in Bond et al. (2001), where w is assumed to be constant throughout the history of the universe, would not be able to extract this information as it does not allow to vary. We point out that for time-varying w, there is no direct connection between the parameters 0 and , i.e., a substantial (say, 0.1) can coexist with rather large negative 0. We perform therefore a three-parameter analysis of quintessence models, and our work goes beyond the investigation of constant w in Huey et al. (1999).

2. COSMIC MICROWAVE BACKGROUND PEAKS IN QUINTESSENCE MODELS

     We wish first to illustrate the impact of different dark energy models on the fluctuation spectrum of the CMB by comparing three examples. The first corresponds to a "leaping kinetic term quintessence" (1) (Hebecker & Wetterich 2000), the second to "inverse power-law quintessence" (2) (Peebles & Ratra 1988; Ratra & Peebles 1988), and the third to a cosmological constant (3). The three examples, whose parameters are chosen such that Ω = 0.6 for each, give similar predictions for many aspects of cosmological observation (we assume everywhere a flat universe Ωtotal = 1). Details of the models can be found below in § 5. We solve the cosmology using CMBfast for a flat initial spectrum with parameters specified in Table 3. Figure 1 clearly demonstrates that the fluctuation spectra of the three models are distinguishable by future high-precision measurements. This can be traced back to different values of and 0, namely, = (0.13, 0, 0) and 0 = (-0.45, - 0.37, - 1) for models 1, 2, and 3, respectively. These quantities enter a simple analytic formula (derived below in § 3) for the spacing between the peaks:



with



and today's radiation components Ω = 9.89 × 10-5, a = 1100, and s = 0.52. In Table 1 we evaluate equation (3) for quintessence models with various parameters, together with the locations l1 and l2 of the first two peaks computed by CMBFAST. The last entry contains the peak spacing averaged over six peaks for the numerical solution. This demonstrates that an accurate measurement of the peak spacing Δl is a powerful tool for the discrimination between different dark matter models!


Fig. 1   CMB spectrum for ΛCDM (model 3), leaping kinetic term (model 1), and inverse power-law (model 2) quintessence universes with Ω = 0.6. Data points from the BOOMERANG (de Bernadis et al. 2000) and MAXIMA (Hanany et al. 2000) experiments are shown for reference.

Table 1   Location and Spacing of the CMB Peaks for Several Models

3. ANALYTIC ESTIMATE OF PEAK SPACING

     We next derive equation (3). Our first task is to estimate the sound horizon at decoupling. We assume that the fraction of quintessential energy Ω&phis;(τ) does not change rapidly for a considerable period before decoupling and define an effective average ≡ τ Ω&phis;(τ)dτ. We note that this average is dominated for τ near τls whereas very early cosmology is irrelevant. Approximating Ω&phis; by the constant average for the period around last scattering, the Friedmann equation for a flat universe reads



Here M = -1/2 is the reduced Planck mass, H(t) is the Hubble parameter, and ρ and ρ are the matter and relativistic (photons and three species of neutrinos) energy densities today.

     Today, neglecting radiation, we have 3MH(1 - Ω) = ρ, which we insert into equation (5) to obtain



where we have changed from coordinate time t to conformal time τ. Separating the variables and integrating gives



which is well known for vanishing . For fixed H0, Ω, Ω, and als (see Table 3 for the values used in this paper), we see that τls = τ(1 - )1/2, where τ is the last scattering horizon for a Λ cold dark matter (ΛCDM) universe (which we treat here to be just a special realization of dark energy with w = -1). To estimate the sound horizon, we need also s, which may be obtained numerically, and in our model universe is equal to 0.52.

     Turning to the horizon today, we mimic the steps of above, this time assuming some equation of state p&phis;(t) = w(t)ρ&phis;(t) for quintessence.

     We define an averaged value 0 by



It is Ω&phis;-weighted, reflecting the fact that the equation of state of the dark energy component is more significant if the dark energy constitutes a higher proportion of the total energy of the universe (see Fig. 2).


Fig. 2   Equation of state w(τ), (τ) ≡ Ω&phis;(τ)w(τ)τ0/ Ω&phis;(τ′)dτ′, and averaged equation of state 0 for the leaping kinetic term model with = 0.13 and 0.22.

     In the limiting case that the equation of state did not change during the recent history of the universe, the average is of course equal to w today. Nevertheless, the difference between the average 0 and today's value, w0, can be substantial for certain models as can be seen from Table 2.

Table 2   Horizons in Mpc at Last Scattering and Today for Various Kinds of Quintessence

     Integrating the cosmological equation with constant w0,



gives



with F given by equation (4). Substituting equations (7) and (10) into equation (1), we obtain the final result (eq. [3]).

     The integral F of equation (4) can be solved analytically for special values of 0, e.g.,



Since the integral in equation (4) is dominated by a close to 1 (typically 0 ≤ 0), only the present epoch matters, consistent with the averaging procedure from equation (8). From this we regain upon inserting into equation (10) the trivial result that the age of the universe is the same for a CDM and a pressureless dark energy universe. We plot F(Ω,0) for various values of Ω in Figure 3.


Fig. 3   F(Ω,0) as a function of 0 of the dark energy component for Ω between 0.2 and 0.7. Between the limiting cases of 0 = -1 (cosmological constant) and 0 = 0 (corresponding to pressureless dust), the age of the universe varies considerably.

     For Ω ≲ 0.6, equation (3) to good approximation (better than 1%) can be written as



The precision of our analytic estimate for Δl can be inferred from Table 1. Similarly, we show in Table 2 the accuracy of the estimates of τls (eq. [7]) and τ0 (eq. [10]) by comparison with the numerical solution. This demonstrates that our averaging prescriptions are indeed meaningful. We conclude that the influence of a wide class of different quintessence models (beyond the ones discussed here explicitly) on the spreading of the CMB peaks can be characterized by the three quantities Ω, , and 0.

4. RATIOS OF PEAK LOCATIONS

     An alternative to the spacing between the peaks is the ratio of any two peak (or indeed trough) locations. After last scattering, the CMB anisotropies simply scale according to the geometry of the universe; taking the ratio of two peak locations factors out this scaling and leaves a quantity that is sensitive only to pre–last-scattering physics. As can be seen in Table 1 (spatially flat) models with negligible all have l2/l1 ≈ 2.41 for the parameters given in Table 3. The dependence of this ratio on the other cosmological parameters can be computed numerically (Doran & Lilley 2001). If the other parameters can be fixed by independent observations, the ratios of peak locations are fixed uniquely for models with vanishing . A deviation from the predicted value would be a hint of time-varying quintessence. It may also be possible to make a direct measurement of from ratios of successive peak locations.

Table 3   Variables Used in this Paper: Their Definitions and Numerical Values

5. SPECIFIC QUINTESSENCE MODELS

     Different models of quintessence may be characterized by the potential V(&phis;) and the kinetic term of the scalar "cosmon" field &phis;:



For practical purposes, a variable transformation allows us to work either with a standard kinetic term k(&phis;) = 1 or a standard potential, (&phis;) = M exp(-&phis;/M). A cosmological constant corresponds to the limit V(&phis;) = λ, k(&phis;) = 0. It is also mimicked by k(&phis;) → ∞. We consider four types of models:

  1. Leaping kinetic term model (Hebecker & Wetterich 2000), with



    and kinetic term



    We have taken kmin = 0.05, 0.1, 0.2, and 0.26, and &phis;1 is adjusted to ≈277 in order to obtain Ω = 0.6. The value of is determined by these parameters.
  2. Inverse power-law potential models (Peebles & Ratra 1988; Ratra & Peebles 1988), with k(&phis;) = 1 and



    We have chosen α = 6, 22, and 40 and adjusted A such that Ω = 0.6. Once again, follows.
  3. Cosmological constant model, tuned such that Ω ≡ Ω = 0.6.
  4. Original exponential potential model (Wetterich 1988, 1995), with k(&phis;) = 1 and



    where α = 1/2.

     For models 1 and 4, quintessence is not negligible at last scattering. The pure exponential potential requires Ω ≤ 0.2 for consistency with nucleosynthesis and structure formation. It does not lead to a presently accelerating universe. We quote results for Ω = 0.6 for comparison with other models and in order to demonstrate that a measurement of Δl can serve as a constraint for this type of model, independent of other arguments. The inverse power-law models (2) are compatible with a universe accelerating today only if is negligible. Again, our parameter list includes cases that are not favored by phenomenology. As an illustration, we quote in Table 1 the value of σ8, which should typically range between 0.6 and 1.1 for the models considered. For example, the exponential potential model with large is clearly ruled out by its tiny value of σ8.1 The main interest for listing also phenomenologically disfavored models arises from the question of to what extent the location of the peaks can give independent constraints. From the point of view of naturalness, only models 1 and 4 do not involve tiny parameters or small mass scales.

     The peak spacing Δl and horizons for the models considered are shown in Tables 1 and 2, respectively. We note that the estimates and the exact numerical calculations are in very good agreement. A different choice of als, say, a = 1150, would have affected the outcome on the low-percent level. Also, the average spacing obtained from CMBfast varies slightly (at most 2%) when averaging over four, five, or six peaks. For a fixed value of the equation of state, 0 = -0.7, we plot the peak spacing as a function of Ω and in Figure 4.


Fig. 4   Contours of equal peak spacing Δl as a function of Ω and . Average equation of state is kept fixed, 0 = -0.7. Increasing leads to a pronounced stretching of the spacing.

     For fixed 0 and Ω, we see from equation (3) that Δl ∝ (1 - )-1/2. Hence, when combining bounds on Ω and 0 from the structure of the universe, supernova redshifts, and other sources with CMB data, the amount of dark energy in a redshift range of z ∼ 105 to last scattering z ∼ 1100 may be determined.

     From Figure 2 we see that the averaged equation of state of the quintessence field for the present epoch is in principle a very influential quantity in determining the spreading of the peaks. Since combined large-scale structure, supernova, and CMB analyses in Bond et al. (2001) suggest 0 ≲ -0.7, the difference between a cosmological constant and quintessence may be hard to spot if is negligible. However, even with the data currently available, the first peak is determined to be at l = 212 ± 7 (Bond et al. 2001). Once the third and fourth peak have been measured, the measurement of the spacing between the peaks becomes an averaging process with high precision. We can then hope to distinguish between different scenarios.

     1 Of course σ8 itself also depends on other cosmological parameters, and so it alone cannot be used to determine .

6. CONCLUSIONS

     The influence of quintessence on the spacing between the CMB peaks is determined by three quantities: Ω, , and 0. When the location of the third peak is accurately measured, we can hope to be able to discriminate between a pure cosmological constant and a form of dark energy with a nontrivial equation of state that is possibly—and most likely—changing in time. The peak ratios will help to determine , which in principle can also be extracted from Δl if Ω and 0 are measured by independent observations. With fixed, the peak spacing can be used to constrain Ω and 0. This permits consistency checks for the quintessence scenario. Together with bounds on Ω&phis; for the period of structure formation (5 ≲ z ≲ 104) and the bound Ω < 0.2 from big bang nucleosynthesis (z ∼ 109; Wetterich 1988, 1995; Birkel & Sarkar 1997), we will post a few milestones in our attempt to trace the cosmological history of quintessence.

     M. Doran would like to thank Luca Amendola for his kind and ongoing support.

REFERENCES