arXiv:astro-ph/0701433v1 15 Jan 2007 3 e pcfi epochs: specific few h xaso aealpooet determine to propose measure all directly rate to proposals expansion few red- the the galaxy addition, individual In of shifts. knowledge precise exceedingly esrn h eaieae fpsieyeovn galaxies evolving passively [5]), of ages relative the measuring r yial icl.Frisac,teBOpoera- to probe sensitive BAO directly the are instance, modes For dial they difficult. feasible, typically direct are are Although parameter Hubble rate). the expansion of measurements the Hub- (or, the of parameter integral ble time a cos- through the parameters to mological sensitive exclusively These almost are universe. measurements the the of ge- content the of matter/energy of power and determination scale ometry matter accurate the an the provide as [3] in spectrum (BAO) well oscillations as acoustic [2], baryon spectrum Background of Microwave power Cosmic location the (CMB) the in with peaks combination acoustic the in [1]) (SNe; pernovae g vial tta ie[] n19 hs da have spectroscopic ideas that these argued He 1998 In [8]. Loeb technol- [7]. by the revisited time with been that signal at cosmic available the 10 ogy detect than to less failed by have performed at separated intervals measurements rate However, time expansion at the redshift. to tempo- source related this the directly section, extra-galactic next is the of variation in ral redshift explained As measuring the [7]. directly of sources of variation possibility temporal a the studied Sandage lan h omcepninwti h rdhf eet,ideally measure desert”, 2 directly “redshift exploring the could within we expansion cosmo- if cosmic the new open a would particular, window In logical Nucleosynthesis). Bang Big iatmnod iiaeSzoeIF,Uiest’dgiSt degli Universita’ INFN, Sezione e Fisica di Dipartimento esrmnso uioiydsac oTp asu- Ia Type to distance luminosity of Measurements uigteeryyaso i-agcsooyAl- cosmology Big-Bang of years early the During z ∼ 00(rmteCB[] and [6]) CMB the (from 1000 xlrn h akEeg esitDsr ihteSandage- the with Desert Redshift Energy Dark the Exploring ∼ < ilide eal omauecsi esitvrain of variations redshift spectrog cosmic resolution measure high to ultra able with be future, indeed Extremely near will the history. ene in expansion dark tion cosmic other where the 5, about cove and 2 its information between redshift in redshifts unique cosmic to is of roughly test ing evolution This the epochs. of separated measurement sufficiently a – test (SL) h xaso itr nteeata olw h akae bu ages dark the follows existence. into that come era rulers cosmologist the and patient in candles of l r history generation increases a expansion a signal in the by cosmic measurements and the decades these As several significance that over surveys. high find energy We with dark models years. future energy ten as dark short dynamical as period a z esuytepopcsfrcntann akeeg tvr h very at energy dark constraining for prospects the study We ∼ < .INTRODUCTION I. 5. z ∼ < irSeaoCorasaniti Pier-Stefano srpyisDprmn,Uiest fCiao Chicago, Chicago, of University Department, Astrophysics fo,sy h A,o by or BAO, the say, (from, 2 2 1 al nttt o omlgclPyisadAtooyand Astronomy and Physics Cosmological for Institute Kavli UH NSUR80,Osraor eParis-Meudon, de Observatoire 8102, UMR CNRS LUTH, lc ue ase,915Muo ee,France Cedex, Meudon 92195 Janssen, Jules Place 5 H ( z z 4,btrequire but [4], ) ∼ 7 10 er would years Dtd etme 5 2018) 15, September (Dated: 9 H fo the (from 1 ( rgnHuterer Dragan , z ta at ) d iRm L aina,PeAd oo5015 oe Italy Rome, 5,00185, Moro Aldo Ple Sapienza”, “La Roma di udi ee n bevbei h esitrne2 tar- range readily redshift now the sub- are in two observable systems by at and quasar taken detected geted As quasar be a times. can of different templates latter spectral The the tracting velocity. shift the 14]. Lyman- [13, goal a such achieve to spectrograph proposed recently (CODEX) been has Experiment Dynamics provide The cosmology. Cosmic could and astrophysics machines resolution in advances high powerful spectacular with these Equipped spectrographs, [9–12]. diameter) meter clrfil hni infiatysed pteexpan- at the universe up the speeds of significantly rate rolling it slowly sion then a — or field vacuum the scalar of energy zero-point objects small first the when forming. during ages are history universe dark expansion the the in cosmic past the just of epoch the test new a have bevdwt ihrslto pcrsoywt a with spectroscopy resolution high with Lyman- observed (QSO) used quasar be of could variation α redshift planets the of orbiting detect motion unseen to reflex by the induced detecting stars for developed techniques ee eecp ol nfc eettecsooia red- cosmological at the variation detect shift fact in could telescope meter upin akeeg ssboiata redshift at subdominant is energy dark sumption, e eeaino xrml ag eecps(30 telescopes large a extremely building of for generation proposals new with ideas ambitious “Sandage- creasingly the as method this test. to (SL) refer Loeb” therefore will we hrfr s hte swrhhl oprobe to worthwhile is whether ask therefore sflt dp netrl miia prahadlook and approach know empirical is entirely not it an do energy, adopt dark we to of since provenance useful physical affirmative: the is about much answer The way. n lotcmltl elgbeat negligible completely almost and h ag ubro bopinlnstpclo the of typical lines absorption of number large The fdr nryi ossett u ipetmdl — models simplest our to consistent is energy dark If h srnmclcmuiyhssneetrandin- entertained since has community astronomical The bopinlns apeo e ude QSOs hundred few a of sample A lines. absorption aeo h rdhf eet,correspond- desert”, “redshift the of rage 2 lsadoMelchiorri Alessandro , ah sc stepooe CODEX), proposed the as (such raphs a rvd entv osrit on constraints definitive provide may s nal ihtm,maueet made measurements time, with inearly α usrLyman- quasar g rbsaeual opoieuseful provide to unable are probes rgy ag eecpspandfrconstruc- for planned telescopes large bandb aigqaa pcr at spectra quasar taking by obtained oetpoiea da ehdfrmeasuring for method ideal an provide forest rcdstetm hnstandard when time the precedes t g esitwt h Sandage-Loeb the with redshift igh dhf ag o cesbewith accessible not range edshift a osri o-tnadand non-standard constrain can ∼ 1 σ L60637 IL α nafwdcds nwa follows what In decades. few a in bopinlnsover lines absorption z ∼ < 3 .Udrtesm as- same the Under 1. obTest Loeb z ∼ > .Oemay One 2. ∼ < z z ∼ > ∼ < any- 2 z − ,we 5, ∼ ∼ > 100 10 1, 2 for the signatures of dark energy at all available epochs regardless of the expectations. This question has recently been studied in some detail by Linder [15], who consid- ered 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, in Secs. III and IV we describe future constraints on standard and non-standard dark energy models respec- tively, 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-Walker cosmology. Consider an isotropic source emitting at rest. Since it does not posses any peculiar motion, the comoving dis- tance to an observer at the origin remains fixed. Then FIG. 1: Cosmic velocity shift as function of the source redshift in a flat universe for different values of ΩM and w. A time waves emitted during the time interval (ts, ts + δts) and interval of 10 years has been assumed. The signal primarily detected later during (to, to + δto) satisfy the relation [7] depends on ΩM and more weakly on w. to dt to+δto dt = . (1) a(t) a(t) Zts Zts+δts Friedmann equation to relatea ˙ to the matter and energy where ts is the time of emission and to the time of obser- 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 ∆v E(z ) a source at ts and observed at to, = H ∆t 1 − s , (6) c 0 0 1+ z  s  1+ zs(to)= a(to)/a(ts). (2) where H0 is Hubble constant, E(z) ≡ H(z)/H0 is the Let us now consider waves emitted after a period ∆ts at (scaled) Hubble parameter at redshift z, c is the speed of ts + ∆ts and detected later at to + ∆to. Similarly to the light, and we have normalized the scale factor to a(to)=1 previous derivation the observed redshift of the source at and neglected the contribution from relativistic compo- to + ∆to is nents. For a constant dark energy equation of state we have 1+ zs(to + ∆to)= a(to + ∆to)/a(ts + ∆ts). (3)

1 Therefore an observer taking measurements at times t 3 3(1+w) 2 2 o E(z)= ΩM (1 + z) +ΩDE(1 + z) +ΩK (1 + z) and to +∆to would measure the following variation of the source redshift h (7)i where ΩM and ΩDE are the matter and dark energy den- a(to + ∆to) a(to) sity relative to critical, ΩK =1 − ΩM − ΩDE is the cur- ∆zs ≡ − . (4) a(ts + ∆ts) a(ts) vature, and w is the dark energy equation of state. In Figure 1 we plot ∆v as function of the source red- In the approximation ∆t/t ≪ 1, we can expand the ratio 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 ∆to = [a(to)/a(ts)]∆ts (as it can be easily ∆v is positive at small redshifts and becomes negative at > inferred from Eq. (1)) we obtain z ∼ 2. Under the assumption of flatness the amplitude and slope of the signal depend mainly on ΩM , while the a˙ (to) − a˙ (ts) dependence on w is weaker. ∆zs ≈ ∆to. (5) a(ts) In spite of the tiny amplitude of the velocity shift, the   absorption lines in the quasar Ly-α provide a powerful This redshift variation can be expressed as a spectro- tool to detect such a small signal. As already pointed out scopic velocity shift, ∆v ≡ c∆zs/(1 + zs). Using the in [8] the width of these lines is of order ∼ 20 km/s, with 3 metal lines even narrower. Although these are still a few orders of magnitude larger than the cosmic signal we seek to measure, each Ly-α spectrum has hundreds of lines. 0.2 Therefore spectroscopic measurements with a resolution R > 20000 for a sample of ∼ 100 QSOs observed ∼ 10 years apart can lead to a positive detection of the cosmic 0.1 signal. Moreover astrophysical systematic effects such as H Sandage- peculiar velocities and accelerations can lead to negligible 0 BAO Loeb CMB corrections [8]. Local accelerations may indeed be more / H 0 important, but due to their direction dependence they can be determined from velocity shift measurements of δΗ QSOs sampled in different directions on the sky. -0.1 SNe In [13] the authors have performed Monte Carlo simu- Weak Lensing lations of Lyman-α absorption lines to estimate the un- Number Counts certainty on ∆v as measured by the CODEX spectro- -0.2 graph. The statistical error can be parametrized in terms 0.01 0.1 1 10 100 1000 of the spectral signal-to-noise S/N, the number of Ly-α redshift quasar systems NQSO, and the quasar’s redshift FIG. 2: Fractional accuracy in the measurement of the Hubble −1.8 parameter H(z) as a function of redshift for a sample of cos- 2350 30 1+ zQSO cm mological probes. The accuracy and redshift ranges shown are σ∆v =1.4 , (8) S/N NQSO 5 s best-guess values for the future surveys, and assume a single   s   measurement of the Hubble parameter for each probe. Since (at z < 4) where S/N is defined per 0.0125A pixel. The SNe, number counts of clusters, and weak gravitational lens- source redshift dependence becomes flat at z > 4. The ing do not measure the Hubble constant directly, but rather numerical factor slightly changes with the source redshift, some combination of distances and (in the case of the lat- ter two) growth of density perturbations, we only indicate varying from 1.4 at z =4to2at z ≈ 2. This small vari- their redshift range with the shaded region. The Sandage- ation arises because the number of observed absorption Loeb overall constraint assumes roughly a 30 year survey and lines decreases with zQSO. For simplicity we assume its 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. We assume upcoming measurements of H0 accu- installed on the ESO Extremely Large Telescope; a ∼ 50 rate to 2%, the overall BAO measurements of the 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. and the SL test measurement of H using the assumptions In the next sections we describe the cosmological win- outlined above. For clarity we do not show the Big Bang 9 dow that such observations could open, with particular Nucleosynthesis constraint on H(z) with z = zbbn ≈ 10 , focus on dark energy models. which is accurate to ∼ 10% (e.g. [17]) and will improve as soon as the baryon density and deuterium abundance are more accurately determined. Since Type Ia super- III. COSMOLOGICAL PARAMETERS AND novae, number counts of clusters, and weak gravitational 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 shaded region. Clearly, the SL test is probing an era not from velocity shift measurements using the Fisher ma- covered with any other reliable cosmological probe. trix method for a LCDM fiducial cosmology. We assume In Figure 3 we plot the 1σ contours in the ΩM −w plane experimental configuration and uncertainties similar to as expected from type Ia supernovae, weak lensing power those expected from CODEX [13, 14]. Namely we con- spectrum, power spectrum plus the bispectrum, and con- sider a survey observing a total of 240 QSOs uniformly straints expected from Planck’s measurement of distance distributed in 6 equally spaced redshift 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 ΩM − w plane. It is worth analytically. noticing that such measurements are mostly sensitive to In Figure 2 we show the near-future status of measure- 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, ΩK, 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-year survey and the number of quasars as in the text. We also show constraints expected from type Ia supernovae, weak FIG. 4: Cosmic velocity shift as function of the source redshift . lensing power spectrum, power spectrum in combination with for a flat ΛCDM model with Ωm = 0 3 (solid line), Chaplygin bispectrum, and constraints expected from Planck’s measure- gas model (long dashed line) and an interactive dark energy- ment of distance to the last scattering surface. SNa and weak dark matter model (short dashed line) with 2% deviation from lensing estimates are based on the SNAP mission, and SNe standard redshift scaling of matter (see text). The error bars also include systematic errors. The SL test contour is comple- on the ΛCDM model assume a 10 year survey and other spec- mentary to other constraints, and is mostly sensitive to the ifications as in the text. matter density, represented here by ΩM .

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 we find that the SL test alone determines the matter den- access otherwise. As mentioned in Section I, probing this sity about four times better than the curvature. As an redshift range is important for testing non-standard dark example, for the 30-year survey the marginalized errors energy models that would otherwise be indistinguishable are σ(ΩM )=0.03 and σ(ΩK)=0.13. from those with a smooth, nearly or exactly constant equation of state function w(z). WMAP observations in combination with low-redshift We have also found that limited accuracy in the Hubble limits from SNe Ia impose a weak upper bound on the constant measurement does not degrade the power of the 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

2 specifications as in the previous section. current energy density of the gas ρ0 = 3H0 is the total 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 corresponds to a fluid which behaves as dust in contrived, it is easy to find physically motivated models the past and as cosmological constant in the future. For whose identification can significantly benefit from data in 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 0 3(α+1) 1+α Even though for these particular models one has to go ΩCh(z)=ΩCh −w0 +(1+ w0)(1 + z) . (9) to a much higher redshift to see the next phase where h i ΩDE = O(1), it is plausible, and certainly currently ob- As shown in [39] this model can provide a good fit to servationally allowed (e.g. [24]) that such a phase could current cosmological observables with ΩCh ∼ 0.95 (with a have occurred somewhere within 2 < z < 5. ∼ ∼ baryonic component of Ωb =0.05), w0 ∼−0.75 and α = Another example is given by models with dark energy- 0.2. From Fig. 4 we can see that although this model has dark matter interaction (see e.g. [25–31]). In the simplest a velocity shift at z < 1 similar to that of a ΛCDM , it can realization where the scalar field only couples to dark be tested with high redshift measurements. For instance matter, it mediates a long range interaction which causes we find that, for this particular model, the Chaplygin two separate effects. First, dark matter particles, unlike parameters can be determined with uncertainties σw0 = the baryons, experience a scalar-tensor type of gravity, 0.03 and σα = 0.04 respectively and thus distinguished which modifies the Newtonian regime (see [32]). The from the ΛCDM values at a high confidence level. time and scale of when such type of modification becomes cosmologically relevant depend on the particular model considered. Second, dark matter particles acquire a 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) 3 ∼ ∼ deviates from the usual (1+z) . At low redshift, when the 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 the best way to probe models with such dark matter-dark straightforward, and the measurement is certainly within energy interaction is to map out cosmic expansion during reach of future large telescopes with high resolution spec- the matter dominated phase (see Figure 4). The SL tests trographs. offers a unique tool to do just that. As the SL test mostly probes the matter density at high In order to forecast how well a deviation of the dark redshift, the constraints on standard dark energy mod- matter density from the (1 + z)3 law can be detected, we els with a nearly or exactly constant equation of state w − parametrize its redshift evolution as ∝ (1+z)3(1 b) in the 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- as a dark energy component with w ≈−1, since the 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 improve them significantly (e.g. [40]). However, the prin- < < as small as 1% (i.e. b =0.01) over 10 years and 0.3% for cipal power of SL measurements at 2 ∼ z ∼ 5 comes from 30 years. Therefore, SL test can provide constraints an their ability to constrain non-standard dark energy mod- 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-Paczynsky test [33]. Since the higher redshifts — or equivalently, models where the to- deviation b is generally a function of redshift, one can use tal energy density does not scale with redshift as (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 date that can be tested in the range of redshift probed by cosmological tests. 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- 0 α mogeneous equation of state, P = −|w0|ΩChρ0/ρ (e.g. ing the SL test at lower redshift? For this to be possible 0 [37]), where w0 is the present equation of state, ΩCh is the UV space-based instruments are necessary. Space-based 6

UV Lyman-α astronomy is possible and has already pro- 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- ally lacks the spectral resolution and wavelength coverage 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- decades by a generation of patient cosmologists may pro- plore the redshift desert are not yet well understood. For vide definitive constraints on the expansion history in the instance, it might be possible to measure the angular di- era before the usual standard candles and rulers, Type Ia ameter distance at redshift of 10-20 from the acoustic os- supernovae and acoustic oscillations in the distribution of 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. These can probe roughly the same redshift range as the SL (z < 6). Similarly gravitational wave “standard ∼ Acknowledgments sirens” can measure distances to GW sources all the way out to z ∼ 20 [44]. However, the prospects of turning the GRBs into standard candles is extremely uncertain It is a pleasure to thank Paolo Molaro and Matteo Viel at this time, and it is not clear that they can be used as for useful discussions, and Avi Loeb for comments on the cosmological probes. The GW sirens, while potentially manuscript. PSC is supported by CNRS with contract providing amazingly accurate distance measurement, suf- No. 06/311/SG. DH has been supported by NSF As- fer from gravitational lensing of the signal, which adds an tronomy and Astrophysics Postdoctoral Fellowship under effective noise term to distance measurements and greatly Grant No. 0401066.

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