Determinazione parametrimarconi/Lezioni/Cosmo12/Lezione09.pdf8.5 The Deceleration Parameter q0 259...

Determinazione parametri

cosmologiciLezione 9-10

A. Marconi Cosmologia (2011/2012)

Relazione r-z

2

220 7 The Friedman World Models

r in unità di c/H0

dipendenza da Ω0

Cosmological Distance Ladder8.3 Hubble’s Constant H0 247

Fig. 8.1. Illustrating the ‘cosmological distance ladder’ (Rowan-Robinson, 1985, 1988). Thediagram shows roughly the range of distances over which different classes of object can beused to estimate astronomical distances. The diagram has been redrawn and updated fromRowan-Robinson’s presentation

sensitivity for faint star-like objects to enable the light curves of Cepheid variablesin the Virgo cluster to be determined precisely and so estimate the value of Hubble’sconstant to 10% accuracy. This programme was raised to the status of an HST KeyProject in the 1990s with a guaranteed share of observing time to enable a reliableresult to be obtained.

The Key Project team, led by Freedman, carried out an outstanding programmeof observations and analysis of these data. Equally important was the fact thatthe team used not only HST data, but also all the other distance measurementtechniques to ensure internal self-consistency of the distance estimates. For example,the improved determination of the local distance scale in our own Galaxy from theparallax programmes of the Hipparcos astrometric satellite improved significantlythe reliability of the calibration of the local Cepheid distance scale. The great advanceof the 1990s was that the distances of many nearby galaxies became known very

Rowan-Robinson 1988

Cosmological Distance Ladder

Età degli ammassi

Ammasso globulare Messier 5 (M5)

4.1 Introduction and Fundamental Observations

143

Fig. 4.2. (a) Color–magnitude diagram of the globular clus-ter M 5. The different sections in this diagram are labeled.A: main sequence; B: red giant branch; C: point of heliumflash; D: horizontal branch; E: Schwarzschild gap in the hori-zontal branch; F: white dwarfs, below the arrow. At the pointwhere the main sequence turns over to the red giant branch(called the “turn-off point”), stars have a mass correspond-ing to a main-sequence lifetime which is equal to the age ofthe globular cluster (see Appendix B.3). Therefore, the age ofthe cluster can be determined from the position of the turn-off point by comparing it with models of stellar evolution.

(b) Isochrones, i.e., curves connecting the stellar evolution-ary position in the color–magnitude diagram of stars of equalage, are plotted for different ages and compared to the stars ofthe globular cluster 47 Tucanae. Such analyses reveal that theoldest globular clusters in our Milky Way are about 13 billionyears old, where different authors obtain slightly differing re-sults – details of stellar evolution may play a role here. Theage thus obtained also depends on the distance of the clus-ter. A revision of these distances by the HIPPARCOS satelliteled to a decrease of the estimated ages by about 2 billionyears

Fig. 4.3. CMB spectrum, plotted as intensity vs. frequency,measured in waves per centimeter. The solid line showsthe expected spectrum of a blackbody of temperature T =2.728 K. The error bars of the data, observed by the FIRASinstrument on-board COBE, are so small that the data pointswith error bars cannot be distinguished from the theoreticalcurve

Main Sequence

Red Giant Branch

Helium Flash

Horizontal branch

gap

white dwarfs

Turn off point

Ammasso globulare 47 Tucanae

t0 =1

H0= 13.2Gyr

✓H0

74 km s�1 Mpc�1

◆


La Legge di Hubble m-z per BCGLegge di Hubble per le “brightest cluster galaxies” (BCG) in ammassi ricchi (Sandage 1968)

6

2.3 Hubble’s Law and the Expansion of the Universe 45

Fig. 2.11. A modern version of the velocity–distance relation for galaxies for the brightestgalaxies in rich clusters of galaxies. This correlation indicates that the brightest galaxies inclusters have remarkably standard properties and that their velocities of recession from ourown Galaxy are proportional to their distances (Sandage, 1968)

This was the approach adopted by Hubble and Humason in their pioneering analysisof 1934 (Hubble and Humason, 1934) – they assumed that the 5th brightest galaxyin a cluster would have more or less the same intrinsic luminosity (Fig. 1.5b). InFig. 2.11, the corrected apparent magnitude in the V waveband is plotted againstthe logarithm of the redshift of the brightest galaxies in a number of rich clustersof galaxies which span a wide range of redshifts. The redshift z is defined by theformula

z = λobs − λemλem

, (2.13)

where λem is the emitted wavelength of some spectral feature and λobs is the wave-length at which is it observed. In the limit of small velocities, v " c, if the redshiftis interpreted in terms of a recessional velocity v of the galaxy, v = cz and thisis the type of velocity plotted in the velocity–distance relation. It is an unfortunatetradition in optical astronomy that the splendidly dimensionless quantity, the redshiftz, is converted into a velocity by multiplying it by the speed of light. As we will seebelow, interpreting the redshift in terms of a recessional velocity leads to confusionand misunderstanding of its real meaning in cosmology. It is best if all mention ofrecessional velocities are expunged in developing the framework of cosmologicalmodels.

m-z per Radio galassieRelazione m-z per radiogalassie (narrow line); m è a 2.2 μm.Linea unita è fit per ΩΛ=0 e tien conto dell’evoluzione delle sorgenti: q0~0.5-1 (risultato di una cosmic conspiracy ...)

8.5 The Deceleration Parameter q0 255

Fig. 8.4. The K magnitude–redshift relation for a complete sample of narrow line radio galaxiesfrom the 3CR catalogue. The infrared apparent magnitudes were measured at a wavelengthof 2.2 µm. The dashed lines show the expectations of world models with q0 = 0 and 12 . Thesolid line is a best-fitting line for standard world models with ΩΛ = 0 and includes the effectsof stellar evolution of the old stellar population of the galaxies (Lilly and Longair, 1984)

showed that they are shock-excited gas clouds, probably associated with the strongshocks created by the passage of the radio jets through the intergalactic mediumsurrounding the radio galaxy (Best et al., 2000).

Using a combination of surface photometry of these galaxies in the optical andinfrared wavebands, we were able to show that the alignment effect does not havea strong influence upon the K magnitude–redshift relationships (Best et al., 1998).More serious was the fact that surveys of fainter samples of 6C radio galaxies byEales, Rawlings and their colleagues found that, although the K magnitude–redshiftrelation agreed with our relation at redshifts less than 0.6, their sample of radiogalaxies at redshifts z ∼ 1 were significantly less luminous than the 3CR galaxiesby about 0.6 magnitudes (Eales et al., 1997). Our most recent analysis of thesedata for a preferred cosmological model with Ω0 = 0.3 and ΩΛ = 0.7, includingcorrections for the evolution of their stellar populations, have demonstrated that3CR radio galaxies at redshifts z ≥ 0.6 are indeed significantly more luminous thantheir nearby counterparts (Inskip et al., 2002). Our apparent success in accountingfor the K magnitude–redshift relation for 3CR radio galaxies in the 1980s was anunfortunate cosmic conspiracy.

The lesson of this story is that the selection of galaxies as standard objects at largeredshifts is a hazardous business; we generally learn more about the astrophysics andastrophysical evolution of the galaxies rather than about cosmological parameters.

Curva di luce supernova tipo Ia5.5 Observations in Cosmology 165

Fig. 5.6. The average time variation of the brightness of a Type 1a supernova from a large sam-ple of supernovae observed in the Calán-Tololo and Supernova Cosmology Program projects.The light curves have been corrected for the effects of time dilation and the luminosity–widthcorrelation (Goldhaber et al., 2001)

Another way of testing the time dilation relation using the remarkably standardproperties of the Type 1a supernovae is to use their spectral evolution as a clock tocompare the time evolution of low and high redshift supernovae. This test has beencarried out for the supernova SN 1997ex, which had redshift 0.361, by members ofthe Supernova Cosmology Program team (Foley et al., 2005). The time between thefirst two spectra was 24.88 days and between the first and third spectra 30.95 days.The amount of aging in the supernova rest frame should be a factor of 1/(1 + z)smaller corresponding to ages of 18.28 and 22.74 days. The spectral feature agetechnique applied to the Keck spectra observed for the supernova showed that thecorresponding elapsed times in the supernova rest frame were 16.97 ± 2.75 and18.01 ± 3.14 days, respectively, in excellent agreement with the expectations ofcosmological time dilation. Similar results are found from the ESSENCE programmewhich involves a large consortium of the key players in the Type 1a supernova area(Wood-Vasey et al., 2007).

5.5.2 Hubble’s Law

In terms of proper distances, Hubble’s law can be written v = H! and sod!dt

= H! . (5.48)

L(t)

w

LMAX

LMAX/2

Curve di luce di Supernovae normalizzate al picco, corrette per Δt rest frame e relazione L-W; dispersione residua ~0.21 mag.

m = M +DM = M + 5[log10(DL/pc)� 1]

m-z per Supernovae Ia

Linea verde: modello ΛCDM con w = -1, ΩΛ=0.73, ΩM=0.27

258 8 The Determination of Cosmological Parameters

Fig. 8.6. The luminosity distance–redshift relation for supernovae of Type 1a from the com-bined ESSENCE and Supernova Legacy Survey data. For comparison the overplotted solidline and residuals are for a ΛCDM model with w = −1, Ω0 = 0.27 and ΩΛ = 0.73.The dotted and dashed lines are for models with ΩΛ = 0, as indicated in the figure legend(Wood-Vasey et al., 2007)

the density parameter of the matter content of the Universe Ω0 is plotted againstΩΛ. The results of the Supernova Cosmology Project are shown in Fig. 8.7.

There are various ways of interpreting Fig. 8.7, particularly when taken in con-junction with independent evidence on the mean mass density of the Universe and theevidence from the spectrum of fluctuations in the Cosmic Microwave Background

m-z per Supernovae Ia

11 (17)

The Observations Figure 1 shows the supernova data from [28] plotted in terms of brightness (bolometric magnitude) versus redshift.

Figure 1: The Hubble diagram for 42 high redshift type Ia supernovae from SCP and 18 low redshift supernovae from the Calan/Tololo Supernova Survey. The solid curves represent a range of cosmological models with = 0 and M = 0, 1 and 2. The dashed curves show a range of ”flat” models where M + = 1. Note the linear redshift scale. The larger the magnitude, the fainter is the object. On the redshift scale, z = 1 corresponds to a light travel time of almost 8 billion light years. The data is compared to a number of cosmological scenarios with and without vacuum energy (or cosmological constant). The data at z < 0.1 is from [26]. At redshifts z > 0.1 (i.e., distances greater than about a billion light years), the cosmological predictions start to diverge. Compared to an unrealistic empty Universe ( M = = 0) with a constant expansion rate, the SNe for a given high redshift are observed to be about 10 - 15% fainter. If the Universe were matter dominated ( M = 1), the high-z supernovae should have been about 25% brighter than what is actually observed. The conclusion is that the

Ω0 e ΩΛ da Supernovae IaRisultati del Supernova Cosmologi Project (2003)

8.5 The Deceleration Parameter q0 259

Fig. 8.7. The 68%, 90%, 95%and 99% confidence limits for thevalues of Ω0 and ΩΛ determinedby the Supernova CosmologyProject. Also shown the diagramis the condition Ω0 + ΩΛ =1 which corresponds to flatgeometry (Knop et al., 2003)

Radiation. Perhaps the most conservative approach is to note that the matter densityin the Universe must be greater than 0 and, as discussed in Sect. 8.7, all the data areconsistent with values of Ω0 ≈ 0.25 − 0.3. Consequently, ΩΛ must be non-zero.The data would be consistent with Ω0 + ΩΛ = 1 if Ω0 ≈ 0.25 − 0.3. We will comeback to these results in Chap. 15.

8.5.4 The Number Counts of Galaxies

In his assessment of approaches to the determination of cosmological parameters,Sandage was not optimistic about the use of the number counts of galaxies (Sandage,1961a):

Galaxy counts are insensitive to the model . . . There seems to be no hope offinding q0 from the N(m) counts because the predicted differences betweenthe models are too small compared with the known fluctuations of thedistribution.

These concerns have been fully justified by subsequent studies. The determinationof precise counts of galaxies has proved to be one of the more difficult areas ofobservational cosmology. Ellis has provided an excellent account of the problems ofdetermining and interpreting the counts of faint galaxies (Ellis, 1997). The reasonsfor these complications are multifold. First of all, galaxies are extended objects, oftenwith complex brightness distributions, and great care must be taken to ensure that

Vincoli da fluttuazioni CMB

Vincoli su Ω0 dagli ammassi e moti galassie universo locale

Ω0 e ΩΛ da Supernovae Ia

12 (17)

deceleration parameter q0 is negative, and that the expansion at the present epoch unexpectedly accelerates (see above). The result of the analyses of the two collaborations, showing that = 0 is excluded with high significance, and that the expansion of the Universe accelerates, is shown in Fig. 2.

Figure 2. The left-hand panel shows the results of fitting the SCP supernova data to cosmological models, with arbitrary M and [28]. The right-hand panel shows the corresponding results from HZT [27]. Could the dimness of the distant supernovae be the effect of intervening dust? Or might the SNe Ia in the early Universe have had different properties from the nearby, recent ones? Such questions have been extensively addressed by both collaborations, indicating that dust is not a major problem and that the spectral properties of near and distant SNe are very similar. Although not as evident at the time of the discovery, later studies of SNe beyond z = 1 [29], from the time when the Universe was much denser and M dominated, indicate that at that early epoch, gravity did slow down the expansion as predicted by cosmological models. Repulsion only set in when the Universe was about half its present age.

Confronto tra i risultati dei due gruppi che hanno vinto il Nobel.

Conteggi di galassieConteggi di galassie in banda H fino a m=28.

Modelli diversi di evoluzione fittano i dati con q0 diversi!


the same types of object are compared at different magnitude limits and redshifts.Furthermore, the distribution of galaxies is far from uniform on scales less thanabout 50 h−1 Mpc, as illustrated by the large voids and walls seen in Figs. 2.7and 2.8. Even at the faintest magnitudes, this ‘cellular’ structure in the distributionof galaxies results in fluctuations in the number counts of galaxies which exceed thestatistical fluctuations expected in a random distribution (see Sect. 17.7). In addition,the probability of finding galaxies of different morphological types depends uponthe galaxy environment. Finally, the luminosity function of galaxies is quite broad(Figs. 3.14 to 3.16) and so the differences between models are masked by theconvolution of the predictions of the world models with this function.

Up till about 1980, the deepest counts extended to apparent magnitudes of about22 to 23 and, although there were disagreements between the results of differentobservers, there was no strong evidence that the counts of galaxies departed fromthe expectations of uniform world models. Since that time, much deeper numbercounts have been determined thanks to the use of large area CCD cameras on largetelescopes, as well as the spectacular images obtained from the Hubble Deep Field

Fig. 8.8. Galaxy number counts in the infrared H waveband (1.65 µm) to H = 28 magnitudecompiled by Metcalfe and his colleagues. The predictions of the various evolving and non-evolving models discussed by them are also shown (Metcalfe et al., 2006)


Relazione DA-z

14

222 7 The Friedman World Models

Fig. 7.8. a The variation of the angular diameter of a rigid rod of unit proper length withredshift for world models with ΩΛ = 0. b The variation of the angular diameter of a rigid rodof unit proper length with redshift for world models with finite values of ΩΛ and flat spatialgeometry, Ω0 + ΩΛ = 1. In both diagrams, c/H0 has been set equal to unity

a larger fraction of the celestial sphere at a large redshift, by the factor (1 + z) whichappears in (5.54).

Metric angular diameters are different from the types of angular diameter whichare often used to measure the sizes of galaxies. The latter are often defined to somelimiting surface brightness and so, since bolometric surface brightnesses vary with

Diametro angolare di una stecca rigida di lunghezza unitaria (c/H0=1)


Sorgenti radio doppieImmagini ottiche (HST, sinistra; da terra a destra)Contorni radio.Immagini disegnate con la stessa scala fisica.

15


Fig. 8.5. HST (left) and UKIRT (right) images of the radio galaxies 3C 266, 368, 324, 280and 65 with the VLA radio contours superimposed (Best et al., 1996). The images are drawnon the same physical scale. The angular resolution of the HST images is 0.1 arcsec while thatof the ground-based infrared images is about 1 arcsec

8.5.3 The Redshift–Magnitude Relation for Type 1a Supernovae

The discussion of Sect. 7.4.5 makes it clear that what is required is a set of standardobjects which are not susceptible to poorly understood evolutionary changes withcosmic epoch. The use of supernovae of Type 1a to extend the redshift–apparentmagnitude relation to redshifts z > 0.5 has a number of attractive features. Firstof all, it is found empirically that these supernovae have a very small dispersion inabsolute luminosity at maximum light (Branch and Tammann, 1992). This dispersioncan be further reduced if account is taken of the correlation between the maximumluminosity of Type 1a supernovae and the duration of the initial outburst. Thiscorrelation, referred to as the luminosity–width relation, is in the sense that thesupernovae with the slower decline rates from maximum light are more luminousthan those which decline more rapidly (Phillips, 1993). Secondly, there are goodastrophysical reasons to suppose that these objects are likely to be good standardcandles, despite the fact that they are observed at earlier cosmological epochs. Thepreferred picture is that these supernovae result from the explosion of white dwarfswhich are members of binary systems which accrete mass from the other memberof the binary. Although the precise mechanism which initiates the explosion has notbeen established, the favoured picture is that mass accreted onto the surface of thewhite dwarf raises the temperature of the surface layers to such a high temperaturethat nuclear burning is initiated and a deflagration front propagates into the interior of


Relazione DA-z per radio sorgenti

16


Fig. 8.9. a The angular diameter–redshift relation for double radio sources, in which themedian angular separation of the double radio source components θm is plotted againstredshift (Kapahi, 1987). The observed relation follows closely the relation θm ∝ z−1. Theleft-hand panel shows fits to the observations for a world model with q0 = 0 and the right-handpanel for a model with q0 = 0.5, in both cases, the median separation of the components beingassumed to change with redshift as lm ∝ (1 + z)−n . b The mean angular diameter–redshiftrelation for 82 compact radio sources observed by VLBI (Kellermann, 1993). In addition tothe standard Friedman models, the relation for steady state cosmology (SS) as well as therelation θ ∝ z−1 (dashed line) are shown

in the past, for example, the ambient interstellar and intergalactic gas may well havebeen greater in the past and so the source components could not penetrate so farthrough the surrounding gas. Again, we learn more about astrophysical changes withcosmic epoch of the objects studied rather than about cosmological parameters.

8.6 ΩΛ and the Statistics of Gravitational Lenses 263

Fig. 8.10. The median angulardiameter–redshift relation for145 high luminosity compactradio sources observed by VLBI(Gurvits et al., 1999). The nota-tion is the same as Fig. 8.9b

Another version of the same test was described by Kellermann and involvedusing only compact double radio structures studied by Very Long Baseline Inter-ferometry (Kellermann, 1993). He argued that these sources are likely to be lessinfluenced by changes in the properties of the intergalactic and interstellar gas, sincethe components are deeply embedded within the central regions of the host galaxy.In his angular diameter–redshift relation, there is evidence for a minimum in therelation, which would be consistent with a value of q0 ∼ 0.5 (Fig. 8.9b). A problemwith this analysis is that the sources at small redshifts are less luminous that those atlarge redshift. Gurvits and his colleagues repeated the analysis with a much largersample of 330 compact radio quasars from which they selected a subsample of 145high luminosity quasars with L ≥ 1026 W Hz−1 (Gurvits et al., 1999). The resultingangular diameter–redshift relation shown in Fig. 8.10 shows the large scatter whenthe data were binned into 12 bins, each with 12–13 sources. As they comment:

None of the solid lines represents the best fit.

Presumably, the properties of the sources are determined by local physical conditionsclose to the quasar nucleus. In addition, there is a clear lack of high luminosity sourcesat small redshifts which would ‘anchor’ the relation.

8.6 ΩΛ and the Statistics of Gravitational Lenses

One way of testing models with finite values of ΩΛ is to make use of the factthat the volume enclosed by redshift z increases as ΩΛ increases, as illustrated byFig. 7.11b for the case Ω0 +ΩΛ = 1. The statistics of gravitationally lensed imagesby intervening galaxies therefore provides an important test of models with finite ΩΛ.

Relazione DA-z per due diversi campioni di radiosorgenti. Le sorgenti a destra sono a L maggiore.


Relazione Vcomov-z

17

7.4 Observations in Cosmology 227

Fig. 7.11. a The variation of the comoving volume within redshift z for world models withΩΛ = 0. b The variation of the comoving volume within redshift z for flat world models withfinite values of ΩΛ


Statistica delle lenti gravitazionali

Probabilità di trovare una lente gravitazionale di un quasar a z=2 relativamente al modello Einstein-de Sitter.

18


Fig. 8.11. The probability of observing strong gravitational lensing relative to that of thecritical Einstein–de Sitter model, Ω0 = 1,ΩΛ = 0 for a quasar at redshift zS = 2 (Carrollet al., 1992). The contours show the relative probabilities derived from the integral (8.30) andare presented in the same format as in Fig. 7.4

have to be averaged over the luminosity functions of lensing galaxies and the distri-bution of background sources. Models have to be adopted for the lenses which canaccount for the observed structures of the lensed images. The amplification of thebrightness of the images as well as the detectability of the distorted structures needto be included in the computations. These complications are considered in somedetail in the review by Carroll and his colleagues and by Kochanek (Carroll et al.,1992; Kochanek, 1996).

The largest survey to date designed specifically to address this problem has beenthe Cosmic Lens All Sky Survey (CLASS) in which a very large sample of flatspectrum radio sources was imaged by the Very Large Array (VLA), the Very LongBaseline Array (VLBA) and the MERLIN long baseline interferometer. The sourceswere selected according to strict selection criteria and resulted in the detection of13 sources which were multiply imaged out of a total sample of 8958 radio sources(Chae et al., 2002). More recently, the CLASS collaboration has reported the point-source lensing rate to be one per 690 ± 190 targets (Mitchell et al., 2005). Theanalysis of these data used the luminosity functions for different galaxy types foundin the AAT 2dF survey as well as models for the evolution of the population of flat-spectrum radio sources. The CLASS collaboration found that the observed fractionof multiply lensed sources was consistent with flat world models, Ω0 + ΩΛ = 1, in

Redshift drift e misura Ω0, ΩΛ

11The Messenger 133 – September 2008

Sandage (1962) first discussed an effect that suggests an extremely direct meas-urement of the expansion history. He showed that the evolution of the Hubble expansion causes the redshifts of distant objects partaking in the Hubble flow to change slowly with time. Just as the redshift, z, is in itself evidence of the ex-pansion, so is the change in redshift ( ż = (1 + z)H0 – H(z)), evidence of its de- or acceleration be-tween the epoch z and today, where H is the Hubble parameter and H0 its present-day value. This equation implies that it is remarkably simple (at least in principle) to determine the expansion history: one simply has to monitor the redshifts of a number of cosmologically distant sources over several years.

This simple equation has two remarkable features. The first is the stunning simplic-ity of its derivation. For this equation to be valid all one needs to assume is that the Universe is homogeneous and isotropic on large scales, and that gravity can be described by a metric theory. That’s it. One does not need to know or assume anything about the geometry of the Uni-verse or the growth of structure. One does not even need to assume a specific theory of gravity. The redshift drift is an entirely direct and model-independent measure of the expansion history of the Universe which does not require any cos-mological assumptions or priors what-soever.

The other remarkable feature of the equa-tion is that it involves observations of the same objects at different epochs (albeit separated only by a few years or dec-ades). Other cosmological observations, such as those of SN Ia, weak lensing and BAO also probe different epochs but use different objects at each epoch. In other words, these observations seek to deduce the evolution of the expansion by mapping out our present-day past light-cone. In contrast, the redshift drift directly measures the evolution by comparing our past light-cones at different times. In this sense the redshift drift method is qualita-tively different from all other cosmological observations, offering a truly independent and unique approach to the exploration of the expansion history of the Universe.

Measuring the redshift drift with E-ELT

The trouble with the redshift drift is that it is exceedingly small. From Figure 1 we see that at z = 4 the redshift drift is of the order of 10–9 or 6 cm/s per decade! Putting meaningful data points onto Fig-ure 1 will clearly require an extremely stable and well-calibrated spectrograph as well as a lot of photons. Let us as-sume that the first requirement has been met, i.e. that we are in possession of a spectrograph capable of delivering radial velocity measurements that are only lim-ited by photon-noise down to the cm/s level. In this best possible (but by no means unrealistic) scenario, how well can we expect the E-ELT to measure the red-shift drift, and hence constrain the cos-mic expansion history?

First of all, we need to define where we want to measure the redshift drift. There are several reasons to believe that the so-called Lyman- forest is the most suit-able target, as first suggested by Loeb (1998). These H I absorption lines are seen in the spectra of all QSOs and arise in the intervening intergalactic medium. Using hydrodynamical simulations we have explicitly shown that the peculiar motions of the gas responsible for the absorption are far too small to interfere with a red-shift drift measurement (Liske et al., 2008). Similarly, other gas properties, such as the density, temperature or ionisation state, also evolve too slowly to cause any headaches. Furthermore, QSOs exist over a wide redshift range, they are the brightest objects at any redshift, and each QSO spectrum displays hundreds of lines. These are all very desirable fea-tures.

The next question is how the properties of the Lyman- forest (the number and sharpness of the absorption features), and the signal-to-noise (S/N) at which it is recorded, translate to the accuracy, , with which one can determine a radial velocity shift. In order to obtain this trans-lation we have performed extensive Monte Carlo simulations of Lyman- for-est spectra. Mindful of the forest’s evolu-tion with redshift, we have derived a quantitative relation between the of the Lyman- forest on the one hand, and the spectral S/N and the background QSO’s redshift on the other hand (Liske et al., 2008).

Now in a photon-noise limited experiment the S/N only depends on the flux density of the source, the size of the telescope (D), the total combined telescope/instru-ment throughput ( ) and the integration time (tint). Unfortunately, the photon flux from QSOs is not a free parameter that can be varied at will. In Figure 2 we show the fluxes and redshifts of all known high-z QSOs. Assuming values for D, and tint we can calculate the expected S/N for any given Nphot. Combining this with a given zQSO and using the relation derived above, we can calculate the value of that would be achieved if all of the time tint were invested into observing a single QSO with the given values of Nphot and zQSO. The background colour image and solid contours in Figure 2 show the result of this calculation, where we have assumed D = 42 m, = 0.25, and tint = 2 000 h. Note that tint denotes the total integration time, summed over all epochs.

Figure 1. The solid lines and left axis show the redshift drift ż as a function of redshift for standard relativistic cos-mology and various combinations of

M and as indicated. The dotted lines and right axis show the same in velocity units. The dashed line shows ż for the case of an alternative dark energy model with a different equation of state parameter wDE (and M, DE = 0.3, 0.7).

M’

= 1.0, 0.0 M’

= 0.3, 0.0 M’

= 0.3, 0.7

wDE = –2/3

0 2 3 41

–2

–1

0

–2

–1.5

–1

–0.5

0

z

dz/

dt (1

0–1

0 h 7

0 yr

–1)

dv/d

t (h 7

0 cm

/s y

r–1 )

Esempio di Ly forest (z~3)


Redshift drift e misura Ω0, ΩΛSimulazioni per osservazioni di 3 campioni di oggetti diversi (Lyα forest di ~10-20 quasar brillanti) per la misura del redshift drift conD=40 m (E-ELT), ε=0.25 (efficienza sistema), tint=4000 h durata 20-30 anni.Necessità di misura accurata e stabile di lunghezza d’onda → laser combs.Area grigia: variazione di H0 di 8 km/s/Mpc.

21

12 The Messenger 133 – September 2008

Telescopes and Instrumentation

We can see that, although challenging, a reasonable measurement of the redshift drift appears to be possible with a 42-m telescope. The best object gives = 1.8 cm/s and there exist 18 QSOs that are bright enough and/or lie at a high enough redshift to put them at < 4 cm/s.

Figure 2 tells us which QSO delivers the best accuracy and is hence the most suitable for a redshift drift experiment. However, for many practical reasons it will be desirable to include more than just the best object in the experiment. Doing so comes at a penalty though: the more objects that are included into the experiment the worse the final result will be because some of the fixed amount of observing time will have to be redistrib-uted from the ‘best’ object to the less suited ones.

The dependence of the full experiment’s final, overall on the telescope diameter, system throughput, total integration time and number of QSOs is shown in Fig- ure 3. We can see that an overall accu-racy of 2–3 cm/s is well within reach of the E-ELT, even when 20 or so objects are targeted for the experiment. However, the figure also shows that for a 30-m tele-scope it would be very time consuming indeed to achieve an accuracy better than 3 cm/s.

To further illustrate what can be achieved we show in Figure 4 three different si-mulations of the redshift drift experiment. The blue dots show the results that can be expected from monitoring the 20 best QSOs over a 20-year period, investing a total of 4 000 h of observing time. By construction these points represent the most precise measurement of ż that is possible with a set of 20 QSOs and the given set-up. However, since many of the selected QSOs lie near the redshift where ż = 0 this experiment does not actually result in a positive detection of the effect. If we want to detect the effect with the highest possible significance we need to choose a different set of QSOs. The yellow squares show the result of selecting the 10 best QSOs according to this criterion.

However, neither of these datasets is par-ticularly well suited to proving the exist-ence of accelerated expansion, i.e. of a

region where ż > 0. Ideally, this would be achieved by obtaining a ż measurement at z ≈ 0.7 – were it not for the atmosphere that restricts observations of the Lyman- forest to z > 1.7. The best thing to do is to combine a measurement at the lowest possible redshift with a second measure-ment at the highest possible redshift, thereby gaining the best possible con-

straint on the slope of ż (z). The brown tri-angles in Figure 4 show the result of a simulation using appropriately selected QSOs: clearly, given these data one could confidently conclude that ż must turn positive at z ≈ 2 for any reasonably well- behaved functional form of ż (z), i.e. re-gardless of the cosmological model. Thus we find that a redshift drift experiment on

Figure 3. The colour image and the contours show the final, overall value of achieved by targeting the NQSO best objects and by employing a given combination of telescope size, effi-ciency and total integration time. The contour levels are at tot = 2, 3, 4 and 5 cm/s.

Figure 2. The dots show the known, bright, high-redshift QSO population as a function of redshift and estimated photon flux. The right-hand vertical axis shows the photon flux converted to a corresponding Johnson V-band magnitude. The background colour image and solid contours show the value of that can be achieved for a given photon flux and redshift, as-suming D = 42 m, = 0.25, and t int = 2 000 h. The contour levels are at

= 2, 3, 4, 6, 8 and 10 cm/s. The dotted contours show the same as the solid ones, but for D = 35 m or, equiv-alently, for = 0.17 or t int = 1389 h.

Figure 4. The three sets of ‘data’ points show simulations of three differ-ent implementations of the redshift drift experiment. In each case we have assumed D = 42 m, = 0.25, t int = 4 000 h and a total experiment duration of 20 years. Blue dots: best overall , NQSO = 20 (binned into four redshift bins). Yellow squares: most significant redshift drift detec-tion, NQSO = 10 (in two redshift bins). Brown triangles: best constraint on

, NQSO = 2. The solid lines show the expected redshift drift for different parameters as indicated. The grey shaded areas result from varying H0 by ± 8 km/s Mpc–1.

2 3 4 5

log

Nph

ot (s

–1 m

–2 µ

m–1

)

Equi

vale

nt V

-ban

d m

agni

tude2 4 6

(cm/s)8 10

zQSO

5 000

104

2 × 104

5 × 104

105

2 × 105

18

17

16

15

SDSSVeron-SSSVeron

0.5 1 1.5 2

10

20

30

40

[D/(42 m)]2 × /0.25 × tint/(2 000 h) N

QS

O 2

3

4

5

tot (

cm/s

)

0 1 2 3 4 5z

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cm

/s y

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–1

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0

0.5

M’ = 0.3, 0.0

M’ = 0.3, 0.7

Liske J. et al., E-ELT and the Cosmic Expansion History – A Far Stretch?


Redshift drift e misura Ω0, ΩΛVincoli dai due esperimenti cerchiati nella figura precedente

22

CODEX Phase A Science Case: E-TRE-IOA-573-0001 Issue 1 16

Figure 7: Expected constraints in the ⌦⇤-⌦M plane from two measurements of the redshift drift at two differentredshifts as indicated for a duration of the redshift-drift experiment of �t0 = 30 yr and a total integration time of4000 h. The two measurements correspond to the brown triangles in Fig. 6. The red and blue solid lines and thegrey shaded bands show the individual constraints provided by each of the two objects assuming a fixed h70 = 1.The coloured ellipses show the joint 68 and 90 per cent confidence regions that result from combining the twomeasurements, marginalising overH0 using an external prior ofH0 = (70± 8) km s�1 Mpc�1. The hashed regionindicates the 95 per cent lower limit on ⌦⇤. Flat cosmologies and the boundary between current deceleration andacceleration are marked by solid black curves. The dark shaded region in the upper left corner designates the regimeof ‘bouncing universe’ cosmologies which have no big bang in the past.

E-ELTEuropean Extremely Large TelescopeOttico ed infrarosso, specchio da 39 m di diametro (~1500 segmenti!).Oltre 16 volte l’area dei telescopi più grandi esistenti. Progetto ESO.

E-ELT mirror mock

The Astrophysical Journal Supplement Series, 192:18 (47pp), 2011 February Komatsu et al.

Table 1Summary of the Cosmological Parameters of ΛCDM Modela

Class Parameter WMAP Seven-year MLb WMAP+BAO+H0 ML WMAP Seven-year Meanc WMAP+BAO+H0 Mean

Primary 100Ωbh2 2.227 2.253 2.249+0.056−0.057 2.255 ± 0.054Ωch2 0.1116 0.1122 0.1120 ± 0.0056 0.1126 ± 0.0036ΩΛ 0.729 0.728 0.727+0.030−0.029 0.725 ± 0.016ns 0.966 0.967 0.967 ± 0.014 0.968 ± 0.012τ 0.085 0.085 0.088 ± 0.015 0.088 ± 0.014

∆2R(k0)d 2.42 × 10−9 2.42 × 10−9 (2.43 ± 0.11) × 10−9 (2.430 ± 0.091) × 10−9

Derived σ8 0.809 0.810 0.811+0.030−0.031 0.816 ± 0.024H0 70.3 km s−1 Mpc−1 70.4 km s−1 Mpc−1 70.4 ± 2.5 km s−1 Mpc−1 70.2 ± 1.4 km s−1 Mpc−1Ωb 0.0451 0.0455 0.0455 ± 0.0028 0.0458 ± 0.0016Ωc 0.226 0.226 0.228 ± 0.027 0.229 ± 0.015

Ωmh2 0.1338 0.1347 0.1345+0.0056−0.0055 0.1352 ± 0.0036zreion

e 10.4 10.3 10.6 ± 1.2 10.6 ± 1.2t0f 13.79 Gyr 13.76 Gyr 13.77 ± 0.13 Gyr 13.76 ± 0.11 Gyr

Notes.a The parameters listed here are derived using the RECFAST 1.5 and version 4.1 of the WMAP likelihood code. All the other parameters in the other tablesare derived using the RECFAST 1.4.2 and version 4.0 of the WMAP likelihood code, unless stated otherwise. The difference is small. See Appendix A forcomparison.b Larson et al. (2011). “ML” refers to the maximum likelihood parameters.c Larson et al. (2011). “Mean” refers to the mean of the posterior distribution of each parameter. The quoted errors show the 68% confidence levels (CLs).d ∆2R(k) = k3PR(k)/(2π2) and k0 = 0.002 Mpc−1.e “Redshift of reionization,” if the universe was reionized instantaneously from the neutral state to the fully ionized state at zreion. Note that these values aresomewhat different from those in Table 1 of Komatsu et al. (2009a), largely because of the changes in the treatment of reionization history in the Boltzmanncode CAMB (Lewis 2008).f The present-day age of the universe.

Table 2Summary of the 95% Confidence Limits on Deviations From the Simple (Flat, Gaussian, Adiabatic, Power-law) ΛCDM Model Except for Dark Energy Parameters

Section Name Case WMAP Seven-year WMAP+BAO+SNa WMAP+BAO+H0Section 4.1 Grav. waveb No running ind. r < 0.36c r < 0.20 r < 0.24Section 4.2 Running index No grav. wave −0.084 < dns/d ln k < 0.020c −0.065 < dns/d ln k < 0.010 −0.061 < dns/d ln k < 0.017Section 4.3 Curvature w = −1 N/A −0.0178 < Ωk < 0.0063 −0.0133 < Ωk < 0.0084Section 4.4 Adiabaticity Axion α0 < 0.13c α0 < 0.064 α0 < 0.077

Curvaton α−1 < 0.011c α−1 < 0.0037 α−1 < 0.0047Section 4.5 Parity violation Chern–Simonsd −5.◦0 < ∆α < 2.◦8e N/A N/ASection 4.6 Neutrino massf w = −1

∑mν < 1.3eVc

∑mν < 0.71eV

∑mν < 0.58eVg

w $= −1∑

mν < 1.4eVc∑

mν < 0.91eV∑

mν < 1.3eVh

Section 4.7 Relativistic species w = −1 Neff > 2.7c N/A 4.34+0.86−0.88 (68% CL)iSection 6 Gaussianityj Local −10 < f localNL < 74

k N/A N/AEquilateral −214 < f equilNL < 266 N/A N/AOrthogonal −410 < f orthogNL < 6 N/A N/A

Notes.a “SN” denotes the “Constitution” sample of Type Ia supernovae compiled by Hicken et al. (2009a), which is an extension of the “Union” sample (Kowalski et al. 2008)that we used for the five-year “WMAP+BAO+SN” parameters presented in Komatsu et al. (2009a). Systematic errors in the supernova data are not included. While theparameters in this column can be compared directly to the five-year WMAP+BAO+SN parameters, they may not be as robust as the “WMAP+BAO+H0” parameters,as the other compilations of the supernova data do not give the same answers (Hicken et al. 2009a; Kessler et al. 2009). See Section 3.2.4for more discussion. The SNdata will be used to put limits on dark energy properties. See Section 5 and Table 4.b In the form of the tensor-to-scalar ratio, r, at k = 0.002 Mpc−1.c Larson et al. (2011).d For an interaction of the form given by [φ(t)/M]Fαβ F̃ αβ , the polarization rotation angle is ∆α = M−1

∫dta φ̇.

e The 68% CL limit is ∆α = −1.◦1 ± 1.◦4(stat.) ± 1.◦5(syst.), where the first error is statistical and the second error is systematic.f ∑ mν = 94(Ωνh2)eV.g For WMAP+LRG+H0,

∑mν < 0.44eV.

h For WMAP+LRG+H0,∑

mν < 0.71eV.i The 95% limit is 2.7 < Neff < 6.2. For WMAP+LRG+H0, Neff = 4.25 ± 0.80 (68%) and 2.8 < Neff < 5.9 (95%).j V+W map masked by the KQ75y7 mask. The Galactic foreground templates are marginalized over.k When combined with the limit on f localNL from SDSS, −29 < f

localNL < 70 (Slosar et al. 2008), we find −5 < f

localNL < 59.

temperature polarization fluctuations rules out any causalmodels as the primary mechanism for generating the CMBfluctuations (Spergel & Zaldarriaga 1997). This implies that

the fluctuations were either generated during an accelerat-ing phase in the early universe or were present at the timeof the initial singularity.

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Determinazione che combina i dati della CMB, BAO (Baryon Acoustic Oscillation, struttura a grande scala delle galassie, Supernovae, Ammassi, ...)

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