The time evolution of cosmological redshift as a test of dark energy
aa r X i v : . [ a s t r o - ph ] N ov Mon. Not. R. Astron. Soc. , 1–7 (2007) Printed 1 November 2018 (MN L A TEX style file v2.2)
The time evolution of cosmological redshift as a test ofdark energy
A. Balbi , ⋆ and C. Quercellini † Dipartimento di Fisica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy INFN Sezione di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy
ABSTRACT
The variation of the expansion rate of the Universe with time produces an evolution inthe cosmological redshift of distant sources (for example quasar Lyman- α absorptionlines), that might be directly observed by future ultra stable, high-resolution spectro-graphs (such as CODEX) coupled to extremely large telescopes (such as EuropeanSouthern Observatory’s Extremely Large Telescope, ELT). This would open a newwindow to explore the physical mechanism responsible for the current acceleration ofthe Universe. We investigate the evolution of cosmological redshift from a variety ofdark energy models, and compare it with simulated data. We perform a Fisher matrixanalysis and discuss the prospects for constraining the parameters of these models andfor discriminating among competing candidates. We find that, because of parameterdegeneracies, and of the inherent technical difficulties involved in this kind of observa-tions, the uncertainties on parameter reconstruction can be rather large unless strongexternal priors are assumed. However, the method could be a valuable complementarycosmological tool, and give important insights on the dynamics of dark energy, notobtainable using other probes. Key words:
Cosmology: theory - cosmological parameters - Cosmology: observations- Galaxies: distances and redshifts
The discovery of the current accelerated expansion of theUniverse (Riess et al. 1998; Perlmutter et al. 1999) hasprompted a renewed interest towards classic cosmologicaltests. The measurement of the expansion rate of the Uni-verse at different redshifts is crucial to investigate the causeof the accelerated expansion, and to discriminate candidatemodels. Until now, a number of cosmological tools have beensuccessfully used to probe the expansion and the geome-try of the Universe. The position of acoustic peaks in thecosmic microwave background (CMB) angular power spec-trum provides a powerful geometrical test, showing that thespace curvature of the Friedmann-Robertson-Walker (FRW)metric is nearly flat (Spergel et al. 2006). A similar test isperformed through the detection of baryon acoustic oscil-lations (BAO) in the power spectrum of matter extractedfrom galaxy catalogues. The luminosity distance of type Iasupernovae and other standard candles allows to constrainthe value of the expansion rate at different redshifts, typi-cally up to z ∼ ⋆ E-mail: [email protected] † E-mail: [email protected]
Currently, however, very little is known about the de-tailed dynamics of the expansion. Depending on the underly-ing cosmological model, one expects the redshift of any givenobject to exhibit a specific variation in time. An interestingissue, then, is to study whether the observation of this vari-ation, performed over a given time interval, could provideuseful information on the physical mechanism responsiblefor the acceleration, and be able to constrain specific mod-els. This is the main goal of this paper. In addition to beinga direct probe of the dynamics of the expansion, the methodhas the advantage of not relying on a determination of theabsolute luminosity of the observed source, but only on theidentification of stable spectral lines, thus reducing the un-certainties from systematic or evolutionary effects. The pos-sible application of this kind of observation as a cosmologicaltool was first pointed out by Sandage (1962). However, thetininess of the expected variation (typically, a shift of lessthan a cm/s over a period of a year) was deemed impos-sible to observe at the time. The importance of this testwas stressed again over the past decades by other authors(e.g. Lake 1981, R¨udiger 1980); more recently, Loeb (1998)re-assessed its feasibility and concluded that, given the ad-vancement in technology occurred over the last forty years,it is conceivable to expect that a measurement of the redshift c (cid:13) A. Balbi and C. Quercellini variation in the spectra of some suitable source (most no-tably the Lyman- α absorption lines of distant QSOs) couldbe detected in the not too distant future. Recently, Lake(2007) showed that measuring the time evolution of redshiftwould be a way to check the internal consistency of the un-derlying cosmological model, and to map the equation ofstate of dark energy.With the foreseen development of extremely large obser-vatories, such as the European ELT, with diameters in therange 30-100 m, and the availability of ultra-stable, high-resolution spectrographs, the perspectives for the observa-tion of redshift variations look very promising. For example,Pasquini et al. (2005, 2006) pointed out that the CODEX(COsmic Dynamics Experiment) spectrograph should havethe right accuracy to detect the expected signal by monitor-ing the shift of Lyman- α lines of distant ( z &
2) QSOs overa period of some decades.A previous investigation of the expected cosmologicalconstraints from this kind of observations was performedby Corasaniti et al. (2007). That work, however, only madepredictions for a very restrictive set of models and assump-tions: firstly, it only explored the case when the dark en-ergy is a standard cosmological constant (i.e. a componentwith constant equation of state w = p/ρ = −
1) plus a fewnon-standard models (the Chaplygin gas, and a model that,although named interacting dark energy, in fact only gen-eralises a constant dark matter equation of state, leavingthe cosmological constant untouched); secondly, as it willbe shown in more detail later, the observational strategyconsidered in that paper seems rather optimistic.In this paper, we aim to give an overview of the theoret-ical predictions for the most popular and still viable modelsintroduced in the recent past to explain the observed accel-erated expansion. These models either fall within the broadcategory of “dark energy”, introducing an unknown, smoothand gravitationally repulsive component, or invoke a mod-ification of the theory of gravity (for a review of possibleexplanation for the accelerated expansion see, e.g., Peebles& Ratra (2003)). Our first goal is to investigate whether thedynamics of any of these standard and non-standard darkenergy models shows interesting features that could be con-strained by future observations of the redshift variation. Wedo not restrict ourselves a priori to models with flat geom-etry, and we fully take into account correlations betweenparameters when assessing the expected uncertainties.This paper is organised as follows. In Sect. 2 we reviewthe basic equations that describe the redshift variation withtime of a source, in an expanding universe. We comment onthe possibility of detecting the effect in the near future inSect. 3. In Sect. 4 we study the expected apparent veloc-ity shift for a number of non-standard dark energy models.Finally, we discuss the main results and conclusions of ourwork in Sect. 5 and 6.
We start by reviewing the basic theory necessary to de-rive the expected redshift variation in a given cosmologi-cal model. We assume that the metric of the Universe isdescribed by the FRW metric. The observed redshift of a given source, which emitted its light at a time t s , is, today(i.e. at time t ), z s ( t ) = a ( t ) a ( t s ) − , (1)and it becomes, after a time interval ∆ t (∆ t s for the source) z s ( t + ∆ t ) = a ( t + ∆ t ) a ( t s + ∆ t s ) − . (2)The observed redshift variation of the source is, then,∆ z s = a ( t + ∆ t ) a ( t s + ∆ t s ) − a ( t ) a ( t s ) , (3)which can be re-expressed, after an expansion at first orderin ∆ t/t , as:∆ z s ≃ ∆ t (cid:18) ˙ a ( t ) − ˙ a ( t s ) a ( t s ) (cid:19) . (4)Clearly, the observable ∆ z s is directly related to a changein the expansion rate during the evolution of the Uni-verse, i.e. to its acceleration or deceleration, and it is thena direct probe of the dynamics of the expansion. It van-ishes if the Universe is coasting during a given time in-terval (i.e. neither accelerating nor decelerating). We canrewrite the last expression in terms of the Hubble parame-ter H ( z ) = ˙ a ( z ) /a ( z ):∆ z s = H ∆ t (cid:18) z s − H ( z s ) H (cid:19) . (5)The function H ( z ) contains all the details of the cosmologi-cal model under investigation. Finally, the redshift variationcan also be expressed in terms of an apparent velocity shiftof the source, ∆ v = c ∆ z s / (1 + z s ). The latest studies on the feasibility of detecting a time evolu-tion of the redshift are those by Pasquini et al. (2005, 2006).The most promising approach is to look at the spectra ofLyman- α forest absorption lines, which are very stable andbasically immune from peculiar motions. This is a scenariothat might be achieved in the next decades, when extremelylarge telescope (such as the ELT) will collect a large num-ber of photons, and high-resolution spectrographs (such asCODEX) will be able to measure tiny shifts in spectral linesover a reasonable amount of time, typically of order of fewdecades.According to Monte Carlo simulations discussed byPasquini et al. (2005, 2006), the accuracy of the spectro-scopic velocity shift measurements expected by CODEX canbe modelled as: σ ∆ v = 1 . (cid:18) S/N (cid:19) (cid:18) N QSO (cid:19) / (cid:18)
51 + z QSO (cid:19) . cm / s; (6)here, S/N is the signal to noise ratio for pixels of 0.0125 ˚A, N QSO is the number of QSOs spectra observed and z QSO istheir redshift. Based on the currently known QSOs brighterthan m . . z .
4, Pasquiniet al. (2005, 2006) assumed to observe either 40 QSOs with c (cid:13) , 1–7 edshift evolution as a test of dark energy S/N ratio of 2000 each, or 30 QSOs with
S/N of 3000, re-spectively. In fact, again according to Pasquini et al. (2005),a CODEX-like experiment, coupled to a 60 m telescope withapproximately 20% total efficiency, would give a cumulative
S/N of 12000 for a single QSO, requiring roughly 125 hoursof observation to get a
S/N of 3000 on that spectrum. Then,starting with 10 hours of observation per night, and takinginto account a 20% use of the telescope, and a 90% of actu-ally usable data, one finds that 40 spectra can be measuredwith that
S/N in roughly 7.6 years (this time would actu-ally increase to about 15 years if the telescope aperture issmaller, e.g., 42 m instead of 60 m). Then, it seems that areasonable time span to perform a second observation of thesame spectra might be 30 years.Based on the previous considerations, our study willthen be conducted assuming a future dataset containinga total of 40 QSOs spectra (uniformly distributed over 5equally spaced redshift bins in the redshift range 2-5), witha
S/N = 3000, observed twice over a time interval of 30years. We note that a previous study performed by Corasan-iti et al. (2007) seems to make optimistic claims, i.e. that 240pairs QSO’s spectra can be observed with
S/N = 3000 overa time span of 10 years. It is easily shown (with the samearguments as above) that with these figures the required ob-serving time would actually be roughly 90 years for each ofthe two epochs. Moreover, even assuming an increase of thenumber of QSOs in the redshift range 2-4 with future largeall-sky photometric surveys, it seems quite difficult to pre-dict an order of magnitude increase over the current knownobjects, which are 25 (Pasquini et al. 2005).Using the expected error bars from Eq. 6, we can pre-dict the level of accuracy that can be reached in the recon-struction of the parameters for a comprehensive set of darkenergy models. Furthermore, we can predict whether any ofthese models can be distinguished from the standard ΛCDMscenario. We will now look into these problems.
All of the models we will consider in this paper are currentlyviable candidates to explain the observed acceleration, i.e.they have not been falsified by available tests of the back-ground cosmology. Clearly, some models may be preferredwith respect to others based on some statistical assessmentof their “economy”, i.e. the fact that they fit the data wellwith a smaller number of parameters. Given the current sta-tus of cosmological observations, there is no strong reasonto go beyond the simple, standard cosmological model withzero curvature and a cosmological constant Λ (except for theconceptual problems arising when one attempts to reconcileits observed value with some estimate derived from funda-mental arguments, see, e.g., Weinberg 1989). For the scopeof the present paper, however, it is interesting to explore asmany models as possible, since future observations of thetime variation of redshift could reach a level of accuracywhich could allow to better discriminate competing candi-dates, and to understand the physical mechanism drivingthe expansion. We refer the interested reader to the paperby Davis et al. (2007), which discusses the constraints onmost of the models that we will focus on in our study. Un- less stated otherwise, throughout our paper we assume foreach class of models the best fit values found in that work,and vary the parameters within their 2 σ uncertainties.All the predictions on the time evolution of redshift pre-sented in the following were derived assuming observationsperformed over a time interval ∆ t = 30 years. From Eq. 5it is clear that the expected velocity shift signal increaseslinearly with ∆ t , so that it is straightforward to calculatethe expected signal when a different period of observation isassumed. Fig. 1 shows our predictions for the cosmologicalmodels discussed in the following, along with simulated datapoints and error bars derived from Eq. 6. Λ CDM)
We start our analysis by first setting out the predictions forthe current standard cosmological model. In the simplestscenario, the dark energy is simply a cosmological constant,Λ, i.e. a component with constant equation of state w = p/ρ = −
1. Flatness of the FRW metric is usually assumed,but in general one can parametrize the deviation from thezero-curvature case in terms of the parameter Ω k ≡ − Ωwhere Ω is the total density of the Universe in units of thecritical value ρ c = 3 H / πG .The Hubble parameter evolves according to the Fried-mann equation, which, for this model, is: (cid:16) HH (cid:17) = Ω k a + Ω m a + Ω Λ , (7)where Ω m and Ω Λ parameterize the density of matter andcosmological constant, respectively. When flatness is as-sumed, Ω = Ω m + Ω Λ = 1, and the model has only one freeparameter, Ω Λ . The current best fit value from cosmologicalobservations is Ω Λ = 0 . ± .
04 in the flat case (Davis et al.2007), while relaxing the assumption of flatness results in apreference for slightly closed models, with Ω k = − . ± . The next step is to allow for deviations from the simple w = − w < − /
3. The Hubble param-eter for this generic dark energy component with densityΩ de then becomes: (cid:16) HH (cid:17) = Ω k a + Ω m a + Ω de a − w ) . (8)The currently preferred values of w in this models still in-clude the cosmological constant case: w = − . ± . If the equation of state of dark energy is allowed to varywith time, then the Hubble parameter is: (cid:16) HH (cid:17) = Ω k a + Ω m a + Ω de e R da (1+ w ( a )) /a . (9) c (cid:13) , 1–7 A. Balbi and C. Quercellini
In this case, one has to choose a suitable functional form for w ( a ), which in general involves a parametrization. The mostcommonly used (Chevallier & Polarski 2001; Linder 2003) is: w ( a ) = w + w a (1 − a ) , (10)although different approaches can be used. Clearly, the ex-act form of w ( a ) with time will lead to completely differentevolution for the dark energy component. In interacting dark energy models the dark components in-teract through an energy exchange term. The conservationequations for matter and dark energy can be written in avery general way as˙ ρ m + 3 Hρ m = δHρ m , (11)˙ ρ de + 3 Hρ de (1 + w ) = − δHρ m , (12)so that the total energy-momentum tensor is conserved.Whenever δ is a non-zero function of the scale factor, theinteraction causes ρ m and ρ de to deviate from the standardscaling, and the mass of the particles is not conserved. Thisnon-standard behaviour has been parametrized (Dalal et al.2001; Majerotto et al. 2004) by the relation ρ de /ρ m = Aa ξ ,where A = Ω de / (1 − Ω de − Ω k ) and the density parametersare the present quantities. The Hubble parameter or thismodel then reads (cid:16) HH (cid:17) = Ω k a + a − (1 − Ω k ) (cid:0) − Ω de (cid:0) − a ξ (cid:1)(cid:1) − wξ , (13)which reduces to the uncoupled case for ξ = − w . Thismodel also includes all late-time scaling solutions. We alsonote that this model is a genuinely interacting dark energy,unlike the one discussed in Corasaniti et al. (2007) which isactually a generalised dark matter (thus more similar to themodel we discuss in 4.8). The Dvali-Gabadadze-Porrati (DGP) model (Dvali et al.2000) provides a mechanism for accelerated expansion whichis alternative to the common repulsive-gravity fluid ap-proach: within the context of brane-world scenarios, theleaking of gravity in the bulk, above a certain cosmologi-cally relevant physical scale, is responsible for the increasein the expansion rate with time. The only parameter of thisclass of models is r , the length at which the leaking occurs,which defines an adimensional parameter Ω r ≡ / (4 r H ).The Hubble parameter then reads: (cid:16) HH (cid:17) = Ω k a + r Ω m a + Ω r + √ Ω r ! , (14)where Ω m = 1 − Ω k − √ Ω r √ − Ω k . Another possibility originating from the brane-world sce-nario is that of a so-called Cardassian expansion (Freese &Lewis 2002) resulting from a modification of the Friedmannequation, with the introduction of a term that depends non linearly on the average density of the Universe, assumedto be composed only of matter. This additional term, phe-nomenologically, is equivalent to the introduction of a darkenergy component, with a scaling law ∝ a − n , where n iscompletely equivalent to the quantity w + 1 of the dark en-ergy models with constant equation of state. More inter-esting are the so-called “modified polytropic Cardassian”models, which have an extra parameter q and a Friedmannequation: (cid:16) HH (cid:17) = Ω m a (cid:0) Ω − qm − (cid:1) a q ( n − ! /q . (15) There are a few models which attempts to explain bothstructure formation and the current acceleration of the Uni-verse with a single “dark fluid”, whereas the standard sce-nario relies on two separate dark contributions to the stress-energy tensor (a dark matter and a dark energy component).A well-studied case is the so called Chaplygin gas (Kamen-shchik et al. 2001), where the unified dark component hasequation of state p = ˜ Aρ − γ with ˜ A > (cid:16) HH (cid:17) = Ω k a + (1 − Ω k ) (cid:18) A + (1 − A ) a γ ) (cid:19) / (1+ γ ) . (16)with the definition A = ˜ A/ρ γ , where ρ is the totaldensity of the Universe at the present. The so-called “stan-dard case” for the Chaplygin gas is obtained for the choice γ = 1 (which, however, is not a good fit to current data),while for γ = 0 the model recovers the standard cosmologicalconstant case with Ω m = (1 − Ω k )(1 − A ). An interesting possibility to consider is that the dark en-ergy is modelled by a generic, barotropic equation of state p = p ( ρ ), as discussed in Chiba et al. (1997); Visser (2004);Ananda & Bruni (2006). In particular, the case where theTaylor expansion of an arbitrary equation of state of thatsort is truncated to first order, e.g. p = p + αρ , has recentlybeen investigated by Balbi et al. (2007), who also derivedcosmological constraints on the parameters of the model. Itis interesting to note that such an affine equation of statecan be used to describe a simple unified dark matter model,with a time evolution of the background density given by ρ ( a ) = ρ Λ + ( ρ o − ρ Λ ) a − α ) , where ρ Λ ≡ − p / (1 + α ) and ρ is the dark energy fluid energy density at present. TheHubble parameter is given by: (cid:16) HH (cid:17) = Ω k a + ˜Ω m a α ) + Ω Λ , (17)where ˜Ω m ≡ ( ρ o − ρ Λ ) /ρ c ; for α = 0 this model recovers thestandard ΛCDM case. c (cid:13) , 1–7 edshift evolution as a test of dark energy We used the expected error bars from Eq. 6 to perform aFisher matrix analysis (Tegmark et al. 1997), leading to anestimate of the best possible constraints on the parametersof dark energy models. All our predictions are based on 30years of observation, assuming that the fiducial values forthe parameters of each dark energy model are those whichbest fit current observations (as from Davis et al. 2007, Balbiet al. 2007, Majerotto et al. 2004).The Fisher matrix formalism allows one to estimate thebest possible accuracy attainable on the determination ofthe parameters of a certain model. Specifically, given a setof cosmological parameters p i , i = 1 , ..., n , and the corre-sponding Fisher matrix F ij (that is easily calculated basedon a theoretical fiducial model and the assumed data errorsfrom Eq. 6), the best possible 1 σ error on p i is given by∆ p i ≡ C / ii , where the covariance matrix C ij is simply theinverse Fisher matrix: C ij = F − ij . It is a well-known fact(see, e.g. Bond et al. 1997) that, when estimating the ex-pected errors ∆ p i on each parameters, one has to be carefulabout existing correlations with the other parameters. Dur-ing the inversion process, non-null off-diagonal Fisher matrixelements propagate in the diagonal elements of the covari-ance matrix, giving their contribution to the estimated un-certainties. Neglecting this contribution, for example by sim-ply taking the inverse of the elements of the Fisher matrix (aprocess which is equivalent to assume perfect knowledge ofall the other parameters) usually results in severely under-estimated errors. The existence of correlations among esti-mated parameters is a manifestation of degeneracies: whenmore and more parameters are allowed to vary, and theyhave similar effects on the observable quantities, it becomesincreasingly difficult to constrain each parameter.From the above considerations, two crucial conse-quences arise: first, dark energy models with intrinsicallyless free parameters will have an advantage with respect tomodels with more free parameters; second, not taking prop-erly into account the degeneracies among parameters (forexample, by assuming perfect knowledge on some of them)will lead to wrong estimates of the errors. For this reason, weallowed all the parameters which are specific of a given darkenergy model to vary simultaneously in our analysis (i.e.we inverted the full Fisher matrix when estimating errors).However, since it would be pointless not to assume any priorknowledge, we fixed the parameters not directly related todark energy (such as, for example, the baryon density) totheir current best fit value.As an example of what we just discussed, let us considerfirst the standard ΛCDM case. Assuming that the fiducialmodel has Ω Λ = 0 . k = 0 and that both Ω Λ and Ω k can vary, we find ∆Ω Λ = 0 . k = 0 .
25 at 1 σ . Whenwe fix Ω k = 0, the bound on Ω Λ becomes 0.007 (at 1 σ ).If dark energy is modelled by a constant equation of state(with a fiducial value w = −
1) and the flatness constraint isimposed we find a looser bound on the dark energy density,∆Ω de = 0 . w = 0 .
58. This clearly shows that different assump-tions on the knowledge of any parameter has an influenceon all the others. Having noted this, however, we also notethat the parameter Ω k can be much better constrained using external datasets, such as the CMB anisotropy. For the sakeof simplicity, then, we will assume flatness in the following.The prospect of detecting departures from the standardΛCDM case could in principle be one of the real assets ofobserving the time evolution of redshift, and is thus worthcloser investigation. Since the simulated data used in ouranalysis assume that QSOs are used as a tracer of the red-shift evolution, we expect that the more constrained modelswill be those that have the largest variability in the redshiftrange 2 . z .
5. Of course, additional information could beobtained using other observational windows. We will com-ment on this issue later on.The DGP model is the one for which we obtained thetightest constraints in our analysis: ∆Ω r = 0 . σ ,assuming Ω r = 0 .
13 as a fiducial value. This is not onlydue to the strong dependence of the velocity shift on Ω r (see Fig. 1), but also to the simplicity of the model, whichdepends on only one parameter (in this respect, this is thesimplest model, together with the standard flat ΛCDM). Ingeneral, as we already discussed, it is to be expected thatmodels with less parameters perform better.The issue of degeneracies is crucial when looking intothe behaviour of models having more than one free param-eters. We start by discussing the Chaplygin model. Whenboth ˜ A and γ vary freely, no interesting constraint can beobtained observing the velocity shift with the assumed QSOdata: we find ∆ ˜ A = 0 .
42 and ∆ γ = 1 . A = 0 . γ = 0 . A , on the contrary, resultsin a very tight bound on γ : ∆ γ = 0 . a − scaling,giving a distinctive signature when one looks at higher andhigher redshifts. We could then naively expect tight boundson the parameters of these models from the observation ofvelocity shift. Unfortunately, again, this is not the case, dueto strong degeneracies among the parameters. For example,when Ω Λ is allowed to vary, for the affine equation of statewe find ∆ α = 0 .
05 (for a fiducial value α = 0 . Λ is known, the affine parameter α can be reconstructed with an error ∆ α = 0 . w = − ξ = 0 .
06 for theinteracting dark energy model (for a fiducial value ξ = 3).The other models do not seem to have very interest-ing signatures to be exploited, at least in the redshift rangeconsidered in our analysis. The worst bounds are expectedfor the Cardassian models, which has not a large parameterdependence in the redshift range 2 . z . w a (already excluded by current observations) areassumed. However, alternative parametrizations might leadto different scalings, and dark energy might dominate in theredshift window probed by Lyman- α forests. Models withthis behaviour can in principle exist (see, for example, Do-delson et al. 2000, Frieman et al. 1995, Caldwell & Linder c (cid:13) , 1–7 A. Balbi and C. Quercellini χ test we canquantify how well we can exclude the competing modelsbased on their expected signal. As it is clear from Fig. 2,some models can be excluded with a high confidence level.In particular, we find that the Chaplygin gas model and theinteracting dark energy model would be excluded at morethan 99% confidence level, and that the affine model wouldbe out of the 1 σ region.As a final caveat, we note that our results were obtainedassuming an equal number (8) of QSOs for each of 5 red-shift bins in the range z =2–5. (Such a uniform distributionwas also assumed in the simulations performed by Pasquiniet al. 2006.) This explains the fact that our error bars sig-nificantly decrease with redshift (see Fig. 1). Assuming adecreasing number of QSOs at higher redshifts (undoubt-edly, a somewhat more realistic assumption) would result ina slight increase in the error bars for those bins. For exam-ple, assuming 3 QSOs instead of 8 in the highest redshiftbin would increase the error bar for that bin of a factor 1.6.However, since the largest variability in theoretical predic-tions is precisely at high redhisfts, we would not expect ourmain conclusions to change dramatically. In summary, we found that the measurement of the veloc-ity shift with future extremely large telescopes and high-resolution spectrographs could provide interesting informa-tion on the source of cosmic acceleration, which would com-plement other, more traditional cosmological tools. From ouranalysis, it is also clear that the observation of velocity shiftalone can be affected by strong parameter degeneracies, lim-iting its ability to constrain cosmological models. Contrarilyto other analyses, then, our conclusion is that the uncer-tainties on parameter reconstruction (particularly for non-standard dark energy models with many parameters) canbe rather large unless strong external priors are assumed.When combined with external inputs, however, the time evo-lution of redshift could discriminate among otherwise indis-tinguishable models.We also found that a reasonable time span to performthe comparison on the redshift evolution seems to be roughlythree decades. Shorter time spans seem unrealistic, giventhe time needed to observe QSOs spectra with the neces-sary
S/N . A shorter interval would also make the redshift difference too small to produce any interesting constraint oncosmological parameters.Despite its inherent difficulties, the method has manyinteresting advantages. One is that it is a direct probe ofthe dynamics of the expansion, while other tools (e.g. thosebased on the luminosity distance) are essentially geometricalin nature. This could shed some light on the physical mecha-nism driving the acceleration. For example, even if the accu-racy of future measurements will turn out to be insufficientto discriminate among specific models, this test would bestill valuable as a tool to support the accelerated expansionin an independent way, or to check the dynamical behaviourof the expansion expected in general relativity compared toalternative scenarios. Furthermore, despite being observa-tionally challenging, the method is conceptually extremelysimple. For example, it does not rely on the calibration ofstandard candles (as it is the case of type Ia SNe) or on astandard ruler which originates from the growth of perturba-tions (such as the acoustic scale for the CMB) or on effectsthat depend on the clustering of matter (except on scaleswhere peculiar accelerations start to play a significant role).Moreover, the errors on the measured data points decreaselinearly with time and can become significantly smaller overonly a few decades of observations. Finally, it is at least con-ceivable that suitable sources at lower redshifts than thoseconsidered in this work could be used to monitor the veloc-ity shift in the future. This would be extremely valuable,since some non-standard models have a stronger parameterdependence at low and intermediate redshifts (see Fig. 1),that could be exploited as a discriminating tool. In Loeb(1998) speculative possibilities of using other sources havebeen indicated, like masers in galactic nuclei, extragalacticpulsars or gravitationally lensed galaxy surveys: this wouldfurther extend the lever arm in redshift space and increasethe ability of constraining models. Exploring the feasibilityof such proposals is beyond the scope of this paper, but itmay certainly be an interesting topic for further studies fromobservers.
ACKNOWLEDGEMENTS
We thank Luca Pasquini for pointing out the right referencesto the simulated CODEX observations. We are also gratefulto an anonymous referee for comments and suggestions thatgreatly improved the final manuscript.
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Ξ = Ξ = s - - - D v H c m (cid:144) s L W r = W r = W r = W r = s - - - D v H c m (cid:144) s L q = = = = = s - - - D v H c m (cid:144) s L n = = = = = s - - - D v H c m (cid:144) s L Γ = -
Γ = Γ =
Γ = = s - - - D v H c m (cid:144) s L Α = -
Α =
Α =
Α = -
Figure 1.
The apparent spectroscopic velocity shift over a period ∆ t = 30 years, for a source at redshift z s , for the models describedin the text. From top to bottom and from left to right: the ΛCDM (Sect. 4.1), dark energy with a constant equation of state (Sect.4.2), dark energy with varying equation of state (Sect. 4.3), interacting dark energy (Sect. 4.4), DGP (Sect. 4.5), Cardassian (Sect. 4.6),generalized Chaplygin gas (Sect. 4.7), dark fluid with an affine equation of state (Sect. 4.8). All the other parameters are fixed at theirbest fit value. The decreasing error bars are due to the assumption of a uniform distribution of QSOs over the entire redshift range (seeSect. 5 for a discussion). c (cid:13) , 1–7 edshift evolution as a test of dark energy s - - - D v H c m (cid:144) s L L CDMCHAPLYGININTERACTINGDGPCARDASSIANW H a L AFFINE
Figure 2.
The predicted velocity shift for the models explored in this work, compared to simulated data as expected from the CODEXexperiment. The simulated data points and error bars are estimated from Eq. 6, assuming as a fiducial model the standard ΛCDM model.The other curves are obtained assuming, for each non-standard dark energy model, the parameters which best fit current cosmologicalobservations.c (cid:13)000