Constraining the Properties of Dark Energy
a r X i v :a s t r o -p h /0103175v 1 12 M a r 2001Constraining the properties of dark energy
Dragan Huterer ?and Michael S.Turner ?
?
Physics Department,University of Chicago,Chicago,IL ?Physics and Astronomy and Astrophysics Departments,University of Chicago,Chicago,IL Abstract.The presence of dark energy in the Universe is inferred directly from the accelerated expansion of the Universe,and indirectly,from measurements of cosmic microwave background (CMB)anisotropy.Dark energy contributes about 2/3of the critical density,is very smoothly distributed,and has large negative pressure.Its nature is very much unknown.Most of its discernible consequences follow from its effect on evolution of the expansion rate of the Universe,which in turn affects the growth of density perturbations and the age of the Universe,and can be probed by the classical kinematic cosmological tests.Absent a compelling theoretical model (or even a class of models),we describe dark energy by an effective equation of state w =p X /ρX which is allowed to vary with time.We describe and compare different approaches for determining w (t ),including magnitude-redshift (Hubble)diagram,number counts of galaxies and clusters,and CMB anisotropy,focusing particular attention on the use of a sample of several thousand type Ia supernova with redshifts z <1.7,as might be gathered by the proposed SNAP satellite.Among other things,we derive optimal strategies for constraining cosmological parameters using type Ia supernovae.While in the near term CMB anisotropy will provide the ?rst measurements of w ,supernovae and number counts appear to have the most potential to probe dark energy.INTRODUCTION Three major lines of evidence point to the existence of a smooth energy component in the universe.Various measurements of the matter density indicate ?M ?0.3±0.1(e.g.,[1]).Recent cosmic microwave background (CMB)results strongly favor a ?at (or nearly ?at)universe,with the total energy density ?0?1.1±0.07[2].Finally,there is direct evidence coming from type Ia supernovae (SNe Ia)that the universe is accelerating its expansion,and that it is dominated by a component with strongly negative pressure,
w =p X /ρX
Even before the direct evidence from SNe Ia,there was a dark energy candidate:the energy density of the quantum vacuum (or cosmological constant)for which p =?ρ.However,the inability of particle theorists to compute the energy of the quantum vacuum –contributions from well understood physics amount to 1055times critical density –casts a dark shadow on the cosmological constant.Another important issue is the coincidence problem:dark energy seems to start dominating the energy budget,and accelerating the expansion of the universe,just around the present time.A number of other candidates have been proposed:rolling scalar ?eld (or quintessence),and network of frustrated topological defects,to name a couple.While these and other models have some motivation and attractive features,none are compelling.
In this work we discuss the cosmological consequences of dark energy that allow its
0123z *10?3
10?210?1100δlnf/δw δlnr/δw 0123
z 0
0.20.40.6
d V /d ?d
z
FIGURE 1.Left panel:relative sensitivity of the comoving distance r (z )and the comoving volume element f (z )≡dV /d ?dz to a localized change in the value of w at redshift z ?.Right panel:comoving volume element dV /d ?dz vs.redshift for constant w =?1,?0.8,?0.6,?0.4(from top to bottom).
nature to be probed.We also discuss relative merits of various cosmological probes,focusing particular attention to a large,well-calibrated sample of SNe Ia that would be obtained by the proposed space telescope SNAP 1.We parameterize dark energy by its scaled energy density ?X and equation of state w ,and assume a ?ducial cosmological model with ?X =1??M =0.7and w =?1,unless indicated otherwise.
COSMOLOGICAL CONSEQUENCES OF THE DARK ENERGY Dark energy is smooth,and if does clump at all,it does so at very large scales only (k ～H 0).All of its consequences therefore follow from its modi?cation of the expansion rate
H (z )2=H 20 ?M (1+z )3+?X exp [3 z 0
(1+w (x ))d ln (1+x )] .For a given rate of expansion today H 0,the expansion rate in the past,H (z ),was smaller in the presence of dark energy.Therefore,dark energy increases the age of the universe.The comoving distance r (z )= dz /H (z )also increases in the presence of dark energy.The same follows for the comoving volume element dV /d ?dz =r 2(z )/H (z ).
At small redshifts r (z )is insensitive to w for the simple reason that all cosmological models reduce to the Hubble law (r =H ?10z )for z ?1:
00.20.4?M ?1.0?0.8
?0.6
?0.4
w 00.51 1.5
z ?1?0.50
FIGURE 2.Left panel:present and projected future constraints (68%C.L.)on constant w .The Sunyaev-Zeldovich effect (SZE)constraint roughly corresponds to the estimate from [5],while for the halo counts we assumed a random uncertainty of 10%in each of eight redshift bins.Right panel:projected constraints (68%and 95%C.L.)on time-varying w using simulated data expected from SNAP.The ?ducial model is w (z )=?0.8+0.1z and ?M is considered to be known.The broken lines show the effect of assuming a Gaussian uncertainty of 0.05in ?M .
r (z )→H ?10 z ?3
4?X wz 2+··· for z ?1.(1)
At redshift greater than about ?ve,the sensitivity of r (z )to a change in w levels off because dark energy becomes an increasingly smaller fraction of the total energy density,ρX /ρM ∝(1+z )3w .Note also that the volume depends upon dark energy parameters through H (z ),which contains one integral less than the distance r (z ).This is the reason why number-count surveys (which effectively measure dV /d ?dz )are potentially a strong probe of dark energy.
Constraints on (constant)w
Supernovae are perhaps the strongest probes of dark energy,as the relevant observable r (z )depends only on ?M ,?X and w ;moreover,SNe probe the optimal redshift range.A powerful supernova program,such as SNAP,with about 2000SNe distributed at 0.2 Number-count surveys are potentially also a very strong probe of dark energy.To estimate the expected number density of objects (galaxies,galaxy clusters etc.),one typically uses the Press-Schechter formalism [6].For example,number counts of galaxy clusters either from an X-ray survey or from Sunyaev-Zeldovich survey can serve as a probe of dark energy [5].The number of these objects depends upon the growth of structure as well as on cosmological volume,and the former makes the overall constraint complementary to that of SNe Ia.Another example is using halos of ?xed rotational speed[7]to infer the dark energy dependence of the volume element.In order to obtain constraints on dark energy using number-count surveys,control over modeling and systematic errors will be critical. CMB anisotropy,which mainly probes the epoch of recombination(z～1000)is weakly sensitive to the presence of dark energy because dark energy becomes incon-sequential at such high redshift.The small dependence comes through the distance to the surface of last scattering which slightly increases in the presence of dark energy, moving the acoustic peaks to smaller scales.However,this effect is small: ?l1 ?M h2+0.09 ??B h2 ?M ?1.25 ??0 z MAX 00.20.40.60.8σ w 00.10.20.30.40.5 ?M ?2?1.5?1?0.5 w FIGURE 3.Left panel:upper curve shows the constraint upon (constant)w using the (nearly uniform)distribution of 2000SNe out to z max .Lower curve shows uncertainties using the same number of SNe with mathematically optimal distribution in redshift.Right panel:constraints on ?M and w using 100SNe with uniform (dark)and optimal (light)distribution in redshift. this analysis may be useful in constraining dark energy models.The parameterization w (z )=w 1?αln [(1+z )/(1+z 1)]yields comparable constraints. OPTIMAL STRATEGIES Given the importance and dif?culty of probing dark energy,it is worthwhile to consider how to optimize data sets in order to obtain tighter constraints.We address the following question:what is the optimal redsh?t distribution of SNe Ia in order to best constrain the cosmological parameters? We choose to minimize the area of uncertainty ellipsoid for P parameters,which (in the Fisher matrix formalism [9])is equivalent to maximizing det F ,where F is the Fisher matrix.Making a few simplifying but reasonable assumptions,in [10]we show that P parameters are always best determined if SNe are distributed as P +1delta-functions in redshifts,two of which are always at the lowest and highest redshifts available.For example,to best constrain ?X and w (assuming a ?at universe),SNe should be located at z =0,z ≈2/5z max and z =z max in equal numbers,where z max is the maximum redshift available.In Fig.3we show the merits of optimal distribution.Note,however,that there are other considerations that may favor a more uniform distribution of SNe in redshift –for example,ability to perform checks for dust and evolution. CONCLUSIONS We discussed prospects for constraining dark energy.Dark energy is best probed at redshifts roughly between 0.2and 1.5,where it is dynamically important.Probes of the low-redshift Universe(supernovae and number counts)seem the most promising,as they only depend upon three cosmological parameters(?M,?X and w),which will be effectively reduced to two(?X and w)when precision CMB measurements determine ?0=?M+?X to better than1%.CMB is weakly sensitive to the presence of dark energy,but is very important as a complementary probe.Alcock-Paczynski test and the age of the universe seem less promising. Constraining the equation of state w is the?rst step in revealing the nature of the dark energy.We show that future surveys will be able to determine constant w quite accu- rately.A high-quality sample of2000supernovae out to redshift z～1.7could determine w to a precision ofσw=0.05.A similar accuracy might be achieved by number counts of galaxies and clusters of galaxies out to z～1.5,provided systematics are held in check (e.g.,in the case of galaxies,the evolution of the comoving number density needs to be known to better than5%).Time-varying w will be considerably more dif?cult to pin down because of additional degeneracies that arise in this case.Nevertheless,interest-ing constraints may be achieved using high-quality data and assuming a slowly-varying w(z).Consequently,use of complementary measurements and further development of techniques to constrain dark energy will be of crucial importance. ACKNOWLEDGMENTS We would like to thank the organizers for the wonderful conference and the“black hole”drink recipe.This work was supported by DOE at Chicago. REFERENCES 1.Turner,M.S.,Physica Scripta,85,210(2000). 2.Jaffe,A.H.,et al.,astro-ph/0007333(2000). 3.Riess,A.,et al.,Astron.J.,116,1009(1998). 4.Perlmutter,S.,et al.,Astrophys.J.,517,545(1999). 5.Haiman,Z.,Mohr,J.J.,and Holder,G.P.,astro-ph/0002336(2000). 6.Press,W.H.,and Schechter,P.L.,Astrophys.J.,187,425(1974). 7.Newman,J.,and Davis,M.,Astrophys.J.Lett.,534,11(2000). 8.Cooray,A.R.,and Huterer,D.,Astrophys.J.Lett.,513,95(1999). 9.Tegmark,M.,Taylor,A.N.,and Heavens,A.F.,Astrophys.J.,480,22(1997). 10.Huterer,D.,and Turner,M.S.,astro-ph/0012510(2000).