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Exploring the Dark Energy Redshift Desert with the Sandage-Loeb Test Pier-Stefano Corasaniti1, Dragan Huterer2, Alessandro Melchiorri3 1LUTH, CNRS UMR 8102, Observatoire de Paris-Meudon, 5 Place Jules Janssen, 92195 Meudon Cedex, France 2Kavli Institute for Cosmological Physics and Astronomy and Astrophysics Department, University of Chicago, Chicago, IL 60637 3Dipartimento di Fisica e Sezione INFN, Universita’ degli Studi di Roma “La Sapienza”, Ple Aldo Moro 5,00185, Rome, Italy (Dated: February 5, 2008) WestudytheprospectsforconstrainingdarkenergyatveryhighredshiftwiththeSandage-Loeb (SL)test – ameasurement of theevolution of cosmic redshift obtained bytakingquasar spectraat sufficientlyseparatedepochs. Thistestisuniqueinitscoverageofthe“redshiftdesert”,correspond- ingroughlytoredshiftsbetween2and5,whereotherdarkenergyprobesareunabletoprovideuseful information about the cosmic expansion history. Extremely large telescopes planned for construc- 7 tion in the near future, with ultra high resolution spectrographs (such as the proposed CODEX), 0 will indeed be able to measure cosmic redshift variations of quasar Lyman-α absorption lines over 0 a period as short as ten years. We find that these measurements can constrain non-standard and 2 dynamical dark energy models with high significance and in a redshift range not accessible with n future dark energy surveys. As the cosmic signal increases linearly with time, measurements made a over several decades by a generation of patient cosmologists may provide definitive constraints on J the expansion history in the era that follows the dark ages but precedes the time when standard 5 candles and rulers come intoexistence. 1 1 I. INTRODUCTION techniques developed for detecting the reflex motion of v stars induced by unseen orbiting planets could be used 3 to detect the redshift variationof quasar (QSO) Lyman- 3 Measurements of luminosity distance to Type Ia su- α absorption lines. A sample of a few hundred QSOs 4 pernovae (SNe; [1]) in combination with the location of observed with high resolution spectroscopy with a ∼ 10 1 theacousticpeaksintheCosmicMicrowaveBackground metertelescopecouldinfactdetectthecosmologicalred- 0 (CMB) power spectrum [2], as well as the scale of the 7 shift variationat ∼1σ in a few decades. In what follows baryon acoustic oscillations (BAO) in the matter power 0 we will therefore refer to this method as the “Sandage- spectrum[3]provideanaccuratedeterminationofthege- / Loeb” (SL) test. h ometryandmatter/energycontentoftheuniverse. These The astronomicalcommunityhassince entertainedin- p measurementsarealmostexclusivelysensitivetothecos- - creasingly ambitious ideas with proposals for building a o mologicalparametersthroughatimeintegraloftheHub- new generation of extremely large telescopes (30−100 r ble parameter (or, the expansion rate). Although direct t meter diameter) [9–12]. Equipped with high resolution s measurementsoftheHubbleparameterarefeasible,they spectrographs, these powerful machines could provide a are typically difficult. For instance, the BAO probe ra- : spectacularadvancesinastrophysicsandcosmology. The v dial modes are directly sensitive to H(z) [4], but require Cosmic Dynamics Experiment (CODEX) spectrograph i exceedingly precise knowledge of individual galaxy red- X hasbeenrecentlyproposedtoachievesuchagoal[13,14]. shifts. Inaddition,the fewproposalstodirectlymeasure r The large number of absorption lines typical of the a the expansion rate all propose to determine H(z) at a Lyman-α forest provide an ideal method for measuring few specific epochs: z <∼ 2 (from, say, the BAO, or by the shift velocity. The latter can be detected by sub- measuringtherelativeagesofpassivelyevolvinggalaxies tracting the spectral templates of a quasar taken at two [5]), z ∼1000(fromthe CMB [6])andz ∼109 (fromthe different times. As quasar systems are now readily tar- Big Bang Nucleosynthesis). In particular, a new cosmo- geted and observable in the redshift range 2<∼z <∼5, we logical window would open if we could directly measure have a new test of the cosmic expansion history during thecosmicexpansionwithinthe“redshiftdesert”,ideally the epoch just past the dark ages when the first objects exploring 2<∼z <∼5. in the universe are forming. During the early years of Big-Bang cosmology Al- If dark energy is consistent to our simplest models — lan Sandage studied a possibility of directly measuring smallzero-pointenergyofthe vacuumoraslowlyrolling the temporal variation of the redshift of extra-galactic scalar field — then it significantly speeds up the expan- sources[7]. Asexplainedinthenextsection,thistempo- sion rate of the universe at z <∼ 1. Under the same as- ral variation is directly related to the expansion rate at sumption, dark energy is subdominant at redshift z >∼1, the source redshift. However, measurements performed and almost completely negligible at z >∼ 2. One may at time intervals separated by less than 107 years would therefore ask whether is worthwhile to probe z >∼2 any- have failed to detect the cosmic signal with the technol- way. The answer is affirmative: since we do not know ogy available at that time [7]. In 1998 these ideas have muchabout the physicalprovenanceofdark energy,it is been revisitedby Loeb[8]. He arguedthat spectroscopic useful to adopt an entirely empirical approach and look 2 for the signatures of dark energy at all available epochs regardlessoftheexpectations. Thisquestionhasrecently been studied in some detail by Linder [15], who considered toy models of dark energy that have non-negligible energy density at high redshift. In this paper we study the cosmological constraints that can be inferred from future observations of velocity shift and their impact on different classes of dark energy models. In Sec. II we review the physics behind the SL test,inSecs.IIIandIVwedescribefutureconstraintson standard and non-standard dark energy models respectively, and in Sec. V we discuss our results and future prospects. II. THE SANDAGE-LOEB TEST It is useful to firstly review a standard textbook cal- culation of the Friedmann-Robertson-Walkercosmology. Consider an isotropic source emitting at rest. Since it does not posses any peculiar motion, the comoving distance to an observer at the origin remains fixed. Then FIG.1: Cosmicvelocityshiftasfunctionofthesourceredshift waves emitted during the time interval (t , t +δt ) and in a flat universe for different values of ΩM and w. A time s s s interval of 10 years has been assumed. The signal primarily detected later during(t , t +δt ) satisfy the relation[7] o o o dependson ΩM and more weakly on w. to dt to+δto dt = . (1) a(t) a(t) Zts Zts+δts Friedmannequationtorelatea˙ tothematterandenergy where ts is the time ofemissionandto the time ofobser- content of the Universe we finally obtain (see [8]): vation. For small time intervals (δt ≪ t) this gives the well known redshift relation of the radiation emitted by a source at ts and observed at to, ∆v =H ∆t 1− E(zs) , (6) 0 0 c 1+z (cid:20) s(cid:21) 1+z (t )=a(t )/a(t ). (2) s o o s where H is Hubble constant, E(z) ≡ H(z)/H is the 0 0 Let us now consider waves emitted after a period ∆t at (scaled)Hubbleparameteratredshiftz,cisthespeedof s ts+∆ts and detected later at to+∆to. Similarly to the light,andwehavenormalizedthescalefactortoa(to)=1 previousderivationtheobservedredshiftofthesourceat and neglected the contribution from relativistic compo- t +∆t is nents. For a constant dark energy equation of state we o o have 1+z (t +∆t )=a(t +∆t )/a(t +∆t ). (3) s o o o o s s 1 Therefore an observer taking measurements at times to E(z)= Ω (1+z)3+Ω (1+z)3(1+w)+Ω (1+z)2 2 M DE K andt +∆t wouldmeasurethefollowingvariationofthe o o h (7i) source redshift whereΩ andΩ arethematteranddarkenergyden- M DE a(to+∆to) a(to) sity relative to critical, ΩK =1−ΩM −ΩDE is the cur- ∆z ≡ − . (4) s a(t +∆t ) a(t ) vature, and w is the dark energy equation of state. s s s In Figure 1 we plot ∆v as function of the source red- Inthe approximation∆t/t≪1,wecanexpandtheratio shift in the flat case for different values of ΩM and w a(to+∆to)/a(ts+∆ts) to linear order and further using assuming a time interval ∆to =10 years. As we can see the relation ∆t = [a(t )/a(t )]∆t (as it can be easily ∆v ispositiveatsmallredshiftsandbecomesnegativeat o o s s inferred from Eq. (1)) we obtain z >∼ 2. Under the assumption of flatness the amplitude and slope of the signal depend mainly on Ω , while the M a˙(t )−a˙(t ) dependence on w is weaker. o s ∆z ≈ ∆t . (5) s a(t ) o In spite of the tiny amplitude of the velocity shift, the (cid:20) s (cid:21) absorption lines in the quasar Ly-α provide a powerful This redshift variation can be expressed as a spectro- tooltodetectsuchasmallsignal. Asalreadypointedout scopic velocity shift, ∆v ≡ c∆z /(1 + z ). Using the in[8]thewidthoftheselinesis oforder∼20km/s,with s s 3 metal lines evennarrower. Although these arestill a few ordersofmagnitudelargerthanthecosmicsignalweseek 0.2 to measure, each Ly-α spectrum has hundreds of lines. Therefore spectroscopic measurements with a resolution R > 20000 for a sample of ∼ 100 QSOs observed ∼ 10 yearsapartcanleadtoapositivedetectionofthecosmic 0.1 signal. Moreoverastrophysicalsystematic effects suchas H Sandage- peculiarvelocitiesandaccelerationscanleadtonegligible H 0 BAO Loeb CMB corrections [8]. Local accelerations may indeed be more / 0 important, but due to their direction dependence they Η δ can be determined from velocity shift measurements of QSOs sampled in different directions on the sky. SNe -0.1 In [13] the authors have performedMonte Carlo simu- Weak Lensing lations of Lyman-α absorption lines to estimate the un- Number Counts certainty on ∆v as measured by the CODEX spectrograph. Thestatisticalerrorcanbeparametrizedinterms -0.2 0.01 0.1 1 10 100 1000 of the spectral signal-to-noise S/N, the number of Ly-α redshift quasar systems N , and the quasar’s redshift QSO FIG.2: FractionalaccuracyinthemeasurementoftheHubble −1.8 parameter H(z) as a function of redshift for a sample of cos- 2350 30 1+z cm σ =1.4 QSO , (8) mologicalprobes. Theaccuracyandredshiftrangesshownare ∆v (cid:18)S/N(cid:19)sNQSO (cid:18) 5 (cid:19) s best-guess values for the future surveys, and assume a single measurement of the Hubble parameter for each probe. Since (at z <4) where S/N is defined per 0.0125A pixel. The SNe, numbercounts of clusters, and weak gravitational lensing do not measure the Hubble constant directly, but rather source redshift dependence becomes flat at z > 4. The some combination of distances and (in the case of the lat- numericalfactorslightlychangeswiththesourceredshift, ter two) growth of density perturbations, we only indicate varying from 1.4 at z =4 to 2 at z ≈2. This small vari- their redshift range with the shaded region. The Sandage- ation arises because the number of observed absorption Loeboverallconstraintassumesroughlya30yearsurveyand lines decreases with z . For simplicity we assume its QSO other specifications as in the text. value to be fixed to 1.4. The large S/N necessary to detect the cosmic signal implies that a positive detec- tion is not feasible with current telescopes. However the CODEX spectrograph is currently being designed to be epochs. WeassumeupcomingmeasurementsofH accu- 0 installed onthe ESO Extremely LargeTelescope;a ∼50 rateto 2%,the overallBAOmeasurementsofthe expan- meter giant that can reach the necessary signal-to-noise sion rate to 1% [16], the CMB measurement of 1.4% [6], with just few hours integration. andtheSLtestmeasurementofH usingtheassumptions In the next sections we describe the cosmologicalwin- outlinedabove. Forclaritywedonotshowthe BigBang dow that such observations could open, with particular NucleosynthesisconstraintonH(z)withz =z ≈109, bbn focus on dark energy models. which is accurate to ∼ 10% (e.g. [17]) and will improve as soon as the baryondensity and deuterium abundance are more accurately determined. Since Type Ia super- III. COSMOLOGICAL PARAMETERS AND novae,numbercountsofclusters,andweakgravitational STANDARD DARK ENERGY MODELS lensing do not measure the Hubble constant directly, we only indicate their approximate redshift range with the We forecast constraints on cosmological parameters shadedregion. Clearly,the SL test is probing an era not from velocity shift measurements using the Fisher ma- covered with any other reliable cosmologicalprobe. trix method for a LCDM fiducial cosmology. We assume InFigure3weplotthe1σcontoursintheΩ −wplane M experimental configuration and uncertainties similar to asexpectedfromtypeIasupernovae,weaklensingpower those expected from CODEX [13, 14]. Namely we con- spectrum,powerspectrumplusthebispectrum,andcon- sider a survey observing a total of 240 QSOs uniformly straintsexpectedfromPlanck’smeasurementofdistance distributed in 6 equallyspacedredshift bins in the range to the last scattering surface. Supernova and weak lens- 2 <∼ z <∼ 5, with a signal-to-noise S/N = 3000, and the ing estimates are based on the SNAP mission [18], and expected uncertainty as given by Eq. (8). Since there is SNe include systematic errors. The SL contour are com- no time integration involved in the computation of ∆v plementary to those of other tests since ∆v probes a dif- the Fisher matrix components can be easily determined ferent degeneracy line in the Ω −w plane. It is worth M analytically. noticing that such measurements are mostly sensitive to InFigure2weshowthenear-futurestatusofmeasure- the matter density, as expected from the plot shown in ments of the Hubble parameter at different cosmological Fig. 1. Allowing also for variation of the curvature, Ω , K 4 -0.5 -0.6 Sandage-Loeb 10, 30 and 50 yrs PLANCK -0.7 WL(PS) w -0.8 -0.9 SNe WL (PS+BS) -1 0.1 0.2 0.3 0.4 0.5 Ω M FIG. 3: Constraints in the ΩM −w plane for the SL test assuming (with increasingly smaller contours) a 10, 30 and 50-yearsurvey andthenumberof quasars asin thetext. We FIG.4: Cosmicvelocityshiftasfunctionofthesourceredshift alsoshowconstraintsexpectedfromtypeIasupernovae,weak lensingpowerspectrum,powerspectrumincombinationwith foraflatΛCDMmodelwithΩm =0.3(solidline),Chaplygin gas model (long dashed line) and an interactive dark energy- bispectrum,andconstraintsexpectedfromPlanck’smeasure- mentofdistancetothelastscatteringsurface. SNaandweak darkmattermodel(shortdashedline)with2%deviationfrom lensing estimates are based on the SNAP mission, and SNe standardredshift scaling ofmatter(seetext). Theerrorbars alsoincludesystematicerrors. TheSLtestcontouriscomple- ontheΛCDMmodelassumea10yearsurveyandotherspec- ifications as in the text. mentary to other constraints, and is mostly sensitive to the matter density,represented here byΩM. IV. NON-STANDARD DARK ENERGY MODELS A unique advantage of the SL test is that it probes the redshift range 2 <∼ z <∼ 5 which is very difficult to wefindthattheSLtestalonedeterminesthematterden- accessotherwise. AsmentionedinSectionI,probingthis sity about four times better than the curvature. As an redshiftrangeisimportantfortestingnon-standarddark example, for the 30-year survey the marginalized errors energymodels thatwouldotherwisebe indistinguishable are σ(Ω )=0.03 and σ(Ω )=0.13. from those with a smooth, nearly or exactly constant M K equation of state function w(z). WMAP observations in combinationwith low-redshift WehavealsofoundthatlimitedaccuracyintheHubble limits from SNe Ia impose a weak upper bound on the constantmeasurementdoesnotdegradethepowerofthe amount of dark energy deep during matter domination SL test. For example, if h is known to 0.04 (or to about to be Ω (z) < 0.1 [19, 20]. Let us suppose that dark 5%), the accuracy in w is degraded by only 2% relative DE energy re-emerged at 3 < z < 5 with Ω (z) = 0.1 to the case when h is perfectly known. DE in this range, while essentially zero at larger redshifts and that it behaved as a standard ΛCDM at z < 3 (i.e. As far as dark energy is concerned, assuming a Gaus- assuming standard Ω = 0.75, w = −1 values). We DE sian prior on h with σ = 0.04, we obtain σ = 0.8 for wouldliketodistinguishsuchamodelfromapureΛCDM h w ∆t = 10 years time interval and 0.3 for 30 years. Thus with Ω and w as above. Clearly, low-redshift probes 0 DE the constraints on w are not competitive with those in- (SNe, BAO, galaxy clustering) cannot distinguish these ferred from other, well-established tests such as SNe Ia, twomodelssincetherequiredredshiftistoohighevenfor weak lensing or BAO once we take into accountthat the themostambitioussurveys. Furthermore,thedistanceto latter probes will provide strong constraints by the time thelastscatteringsurfacebetweenthetwomodelsdiffers the SL test is undertaken. However, one should note only by 0.5%, which is too low to be observable even thattheconstraintsobtainedbySLdecreaselinearlywith withPlanck,sincethe1-σuncertaintyisabout0.4%with time. For measurements made over a century, and with temperatureandpolarizationinformation[21]. However, the expectedly larger number of QSOs, the SL limits on the two models can be distinguished via the SL test at w can easily be at the few percent level. about3-σlevel,assumingonlya10-yearsurveyandother 5 specifications as in the previous section. current energy density of the gas ρ = 3H2 is the total 0 0 While the aforementioned scenario with dark energy energy density and α > −1 is a dimensionless parame- emerging in the specific window at 3 <∼ z <∼ 5 may seem ter. This correspondstoa fluidwhichbehavesasdustin contrived, it is easy to find physically motivated models the past and as cosmologicalconstant in the future. For whoseidentificationcansignificantlybenefitfromdatain α=0 the model reduces to ΛCDM [38]. The Chaplygin this “desert”. One example is given by scalar field mod- gas energy density evolves with redshift according to: els which predict the periodic emergence of DE at var- ious epochs during the history of the universe [22, 23]. 1 Even though for these particular models one has to go Ω (z)=Ω0 −w +(1+w )(1+z)3(α+1) 1+α . (9) Ch Ch 0 0 to a much higher redshift to see the next phase where h i Ω = O(1), it is plausible, and certainly currently ob- DE As shown in [39] this model can provide a good fit to servationally allowed (e.g. [24]) that such a phase could currentcosmologicalobservableswithΩ ∼0.95(witha have occurred somewhere within 2<∼z <∼5. baryonic component of Ω =0.05), w ∼Ch−0.75 and α= b 0 Anotherexampleisgivenbymodelswithdarkenergy- 0.2. FromFig.4wecanseethatalthoughthismodelhas darkmatterinteraction(seee.g. [25–31]). Inthesimplest avelocityshiftatz <1similartothatofaΛCDM,itcan realization where the scalar field only couples to dark be tested with high redshift measurements. For instance matter,itmediatesalongrangeinteractionwhichcauses we find that, for this particular model, the Chaplygin two separate effects. First, dark matter particles, unlike parameters can be determined with uncertainties σ = w0 the baryons, experience a scalar-tensor type of gravity, 0.03 and σ = 0.04 respectively and thus distinguished α which modifies the Newtonian regime (see [32]). The from the ΛCDM values at a high confidence level. timeandscaleofwhensuchtypeofmodificationbecomes cosmologically relevant depend on the particular model considered. Second,darkmatter particlesacquirea time V. DISCUSSION dependent mass whose evolution is determined by the specifics of the scalar field dynamics. As a consequence In this paper we have analyzed the prospects for con- of this, the redshift evolution of the dark matter density straining dark energy at high redshift (2 <∼ z <∼ 5) deviatesfromtheusual(1+z)3. Atlowredshift,whenthe by direct measurements of the temporal shift of the universe is dark energy dominated, these models cannot quasar Lyman-α absorption lines (the Sandage-Loeb ef- be distinguished from the standard ΛCDM . Therefore, fect). While the signal is extremely small, the physics is thebestwaytoprobemodelswithsuchdarkmatter-dark straightforward,andthemeasurementiscertainlywithin energyinteractionistomapoutcosmicexpansionduring reachoffuturelargetelescopeswithhighresolutionspec- thematterdominatedphase(seeFigure4). TheSLtests trographs. offers a unique tool to do just that. AstheSLtestmostlyprobesthematterdensityathigh In order to forecast how well a deviation of the dark redshift, the constraints on standard dark energy mod- matter densityfromthe (1+z)3 lawcanbe detected, we els with a nearly or exactly constant equation of state w parametrizeitsredshiftevolutionas∝(1+z)3(1−b)inthe are weaker than those that observations of SN Ia, BAO, range 2<∼z <∼5, where |b|<1 is a constant free param- weak lensing and number counts will be able to achieve eter. The scalar field, on the other hand, can be treated in the future (although the SL test becomes competi- asadarkenergycomponentwithw ≈−1,sincethe field tive for measurements spread over a period of several slowly rolls toward the minimum of its effective poten- decades or more). This is mostly because the sensitivity tial at late times [30]. Assuming a flat fiducial model of cosmological probes to standard dark energy models with ΩM = 0.3 and w = −1, we find that the SL test is exhausted at z <∼ 2 and higher redshift data do not can detect deviations from the standard matter scaling improvethemsignificantly(e.g.[40]). However,theprin- as small as 1% (i.e. b=0.01) over10 years and 0.3% for cipalpowerofSLmeasurementsat2<∼z <∼5comesfrom 30 years. Therefore, SL test can provide constraints an theirabilityto constrainnon-standarddarkenergymod- order of magnitude tighter than those inferred using fu- els, where the dark energy density is non-negligible at ture SNe Ia or the Alcock-Paczynskytest [33]. Since the higher redshifts — or equivalently, models where the to- deviationbisgenerallyafunctionofredshift,onecanuse talenergydensitydoesnotscalewithredshiftas(1+z)3 the velocity shift measurements to reconstruct the red- at z >∼2. In particular, as discussed in the previous sec- shift dependence of b, and then determine the strength tion, in only one decade the SL measurements will allow and functional form of the scalar interaction. to test the redshift scaling of the matter density with an The Chaplygin gas is yet another dark energy candi- accuracy one order of magnitude greater than standard datethatcanbetestedintherangeofredshiftprobedby cosmologicaltests. the SL test. Proposed as a phenomenological prototype Tighter constraints on standard dark energy models of unified dark energy and dark matter model [34–36], it could be obtained if observations of Ly-α systems were describes an exotic fluid with an inverse power law ho- feasible at z <∼ 2. What are the prospects for perform- mogeneous equation of state, P = −|w |Ω0 ρ /ρα (e.g. ing the SL test at lower redshift? For this to be possible 0 Ch 0 [37]),wherew isthepresentequationofstate,Ω0 isthe UV space-based instruments are necessary. Space-based 0 Ch 6 UVLyman-αastronomyis possibleandhasalreadypro- degrades their power to constrain cosmological models duced remarkable results (see e.g [41]), though it gener- [44]. Conversely, the SL test is based on extremely sim- allylacksthespectralresolutionandwavelengthcoverage ple physics and involves controllable systematic errors, of the higher redshift studies. However during the past but it does require a powerful instrument and patience few years high quality observations of several low red- to wait at least a decade before repeating the measure- shift QSOs have been obtained with the Hubble Space ments in order to produce interesting results. Telescope (HST) and its Space Telescope Imaging Spec- We finally point out that the velocity shift sig- trograph (STIS). It is therefore conceivable that future nal increases linearly with time, thus amply rewarding space based experiments will be able to measure the SL increased temporal separation between measurements. effect at low redshift. Therefore observations made over a period of several Other astrophysical probes that can potentially ex- decadesbyagenerationofpatientcosmologistsmaypro- ploretheredshiftdesertarenotyetwellunderstood. For videdefinitiveconstraintsontheexpansionhistoryinthe instance, it might be possible to measure the angulardi- erabeforetheusualstandardcandlesandrulers,TypeIa ameterdistanceatredshiftof10-20fromtheacousticos- supernovaeandacousticoscillationsinthedistributionof cillations in the power-spectrum of the 21cm brightness galaxies, become readily available. fluctuations [42]. Further, gamma ray bursts (GRBs) (e.g. [43]) have been proposed as alternative standard candles. Thesecanproberoughlythesameredshiftrange astheSL(z <∼6). 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