funzioni di correlazione e spettro ßuttuazioni iniziali -...

Funzioni di correlazione e spettro

fluttuazioni inizialiLezione 19-20

A. Marconi Cosmologia (2012/2013)

Correlazione angolare 2pt galassie

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36 2 The Large-Scale Structure of the Universe

Fig. 2.5a,b. The two-point correlation function for galaxies over a wide range of angularscales. a The scaling test for the homogeneity of the distribution of galaxies can be performedusing the correlation functions for galaxies derived from the APM surveys at increasinglimiting apparent magnitudes in the range 17.5 < m < 20.5. The correlation functions aredisplayed in intervals of 0.5 magnitudes. b The two-point correlation functions scaled to thecorrelation function derived from the Lick counts of galaxies (Maddox et al., 1990)

Fig. 2.6a,b. The two-point correlation function for galaxies determined from the Sloan DigitalSky Survey (SDSS) (Connolly et al., 2002; Scranton et al., 2002). a The angular two-pointcorrelation function determined in a preliminary analysis of 2% of the galaxy data containedin the Sloan Digital Sky Survey. b Comparison of the scaled angular two-point correlationfunctions found by Maddox and his colleagues from the APM galaxy survey (solid line) withthat found from the SDSS analysis


Correlazione angolare 2pt ammassi

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4.2 The Distribution of Galaxies in Clusters of Galaxies 101

Fig. 4.3. The two-point spatial correlation function for four richness thresholds (Ngal ! 10,! 13, ! 15, ! 20) for clusters selected from the SDSS (Bahcall et al., 2003a). Best-fitfunctions with slope 2 and correlation-scale r0 are shown by the dashed lines. The error barsshow the 1 ! uncertainties in the estimates. The values of d for the four panels are 26.2, 35.6,41.5 and 58.1 Mpc with increasing richness

4.2 The Distribution of Galaxies in Clusters of Galaxies

Clusters of galaxies come in a variety of shapes and forms and various schemes havebeen developed to put some order into this diversity. Just as in the case of galaxies,modern computer-based systems of classification bring new, quantitative insightsinto the wealth of detail contained in the visual classification of clusters.

4.2.1 The Galaxy Content and Spatial Distribution of Galaxies in Clusters

Abell classified clusters as regular if they are more or less circularly symmetrical witha central concentration, similar in structure to globular clusters (Abell, 1962). These

linee tratteggiate: fit con pendenza -2, r0 scala di riferimento


Crescita perturbazioni

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394 14 Correlation Functions and the Spectrum of the Initial Fluctuations


Funzione trasferimento Spettro potenza


Crescita perturbazioni

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Table 14.1. Properties of adiabatic cold dark matter perturbations which entered the particlehorizon at the epoch of equality of matter and radiation energy densities

!0 = 0.3 !0 = 1World model !" = 0.7 !" = 0

h = 0.7 h = 0.7

zeq 3,530 11,760teq 47,500 years 4,277 yearsComoving horizon scalereq = 2cteq/aeq 100 Mpc 26 MpcMeq = (#/6)r3

eq$0 2.3 ! 1016 M" 1.2 ! 1015 M"

It can be seen that the form of the spectrum is in agreement with the physicalarguments presented above. The transfer function shown in Fig. 14.1a indicates thatthe curvature of the spectrum between the two asymptotic relations is very gradual,reflecting the rather slow change in the growth rate of the perturbations accordingto the Meszaros formula (12.67). Notice that primordial perturbations on all scalesand masses survive into the post-recombination era in the cold dark matter picture.

This form of processed power spectrum is of particular importance and hasdominated much of the discussion of structure formation. Part of the reason forthis is the fact that the scale-invariant Harrison–Zeldovich spectrum appears rathernaturally in the preferred inflationary scenario for the early Universe. Nonetheless,let us consider two other dark matter models which have already been mentioned.

14.3.2 Adiabatic Hot Dark Matter

In the case of the adiabatic hot dark matter model with massive neutrinos, small-scale perturbations are damped by the free-sreaming of neutrinos as soon as theycome through the horizon during the radiation-dominated era. The spectrum cutsoff exponentially below the critical mass given by (13.15). An analytic expressionfor the transfer function P(k) quoted by Peacock, following Bond and Szalay andBardeen and his colleagues (Bond and Szalay, 1983; Bardeen et al., 1986) is

Tk = exp!#3.9q # 2.1q2" , (14.38)

where q = k/!!0h2 Mpc#1

". Notice the exponential cut-off of the transfer function

in Fig. 14.1a and the corresponding cut-off in the power spectrum in Fig. 14.1b.Again the power spectrum has a maximum on scales greater than those of clustersof galaxies, but all small-scale structure has been washed out by the free-streamingof the massive neutrinos.

14.3.3 Isocurvature Cold Dark Matter

The isocurvature modes behave quite differently from the adiabatic modes discussedabove. As we asserted earlier, in the early radiation-dominated Universe when the


Le prime simulazioni ...

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14.4 Biasing 401

Fig. 14.2a–c. Simulations of the expectations of a the cold dark matter model with !0 = 0.2and !" = 0 and b the hot dark matter model with !0 = 1 and !" = 0 for origin oflarge-scale structure of the Universe (Frenk, 1986). c These simulations can be comparedwith the large-scale distribution of galaxies observed in the Harvard–Smithsonian Center forAstrophysics Survey of Galaxies (Fig. 2.7). The unbiased cold dark matter model does notproduce sufficient large-scale structure in the form of voids and filaments of galaxies, whereasthe unbiased hot dark matter model produces too much clustering

successes for the cold dark matter picture. In particular, it could account for theobserved two-point correlation function of galaxies #(r) ! r"1.8 over a widerange of physical scales. This form of correlation function resulted from furthernon-linear interactions once the perturbations had developed in amplitude to$k > 1. The cold dark matter picture was favoured by many of the investigators,but it was not without its problems. For example, in realisations of the cold darkmatter model with !0 = 1, the velocity dispersion of galaxies chosen at randomfrom the field was found to be too large (Efstathiou, 1990).

In both cases, the match to observation could be improved if it was assumed thatthe galaxies provided a biased view of the large-scale distribution of mass in theUniverse, and this is the topic we have to tackle next.

14.4 Biasing

So far, it has been implicitly assumed that the visible parts of galaxies trace thedistribution of dark matter, but one can imagine many reasons why this might notbe so. The generic term for this phenomenon is biasing, meaning the preferential


Crescita non lineare

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r ! r0, the perturbations become non-linear and it might seem more difficult torecover information about the processed power spectrum on these scales. An im-portant insight was provided by Hamilton and his colleagues who showed how it ispossible to relate the observed spectrum of perturbations in the non-linear regime,!(r) " 1, to the processed initial spectrum in the linear regime (Hamilton et al.,1991).

The idea is that the evolution of the perturbations into the non-linear regime can befollowed using the types of argument used in Sect. 11.4.2 and illustrated in Fig. 11.2.The perturbation behaves like a little closed universe which reaches maximum sizeat some epoch, known as the ‘turnround’ epoch, after which it collapses to form abound structure. According to the arguments of Sect. 16.1, a bound structure whichsatisfies the virial theorem is formed when a perturbation has collapsed to half thedimension it achieved at the turnround epoch. By the time the virialised structurehas formed, the density contrast reaches values greater than 100.

Hamilton and his colleagues showed that the evolution from the linear to thenon-linear regime closely follows a self-similar solution which can be found fromthe pioneering numerical computations of Efstathiou and his colleagues (Efstathiouet al., 1988). Figure 14.5a shows how the amplitude of the spatial two-point cor-relation function changes between the linear and non-linear regimes for differentvalues of the index of the initial power spectrum. The form of the relations can beunderstood from the results already derived. In the linear regime "#/# # a, and

Fig. 14.5. a Variation of the spatial two-point correlation function with the square of the scalefactor as perturbations evolve from linear to non-linear amplitudes. b Evolution of the spatialtwo-point correlation function as function of redshift. The function has been normalised toresult in a correlation function which resembles the observed two-point correlation functionfor galaxies which has slope $1.8. Non-linear clustering effects, as represented by the functionshown in a, are responsible for steepening the processed initial power spectrum (Hamiltonet al., 1991)

Crescita lineare e non della funzione di correlazione a due punti ℥(r) al variare di z


Il ruolo dei barioniFormazione dei picchi acustici nel contrasto di densità (da Sunyaev e Zeldovich)

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12.7 Baryonic Theories of Galaxy Formation 361

Universe. Although much of this discussion has been superseded by the !CDMmodels, it is of more than historical interest since the models contain many of theideas which are built into the current picture of structure formation.

12.7.1 The Adiabatic Scenario

In the adiabatic picture developed by Zeldovich and his colleagues, it was assumedthat a spectrum of small adiabatic perturbations was set up in the very early Universeand their evolution was then followed according to the physical rules developedabove. Only large-scale perturbations with masses M ! MS = 1012

!"Bh2

""5/4 M#survived to the epoch of recombination, all fluctuations on smaller mass scales beingdamped out by photon diffusion, as discussed in Sect. 12.5. Perturbations withmasses greater than the Jeans mass, MJ = 3.75 $ 1015/

!"Bh2

"2 M#, continued togrow from the time they came through their particle horizons to the present epoch.Once they came through their particle horizons, those perturbations with massesless than the Jeans mass were sound waves which oscillated with a small decreasein amplitude until the epoch of recombination, when their internal pressure supportvanished and the Jeans mass dropped abruptly to MJ = 1.6 $ 105

!"0h2

""1/2 M#.Zeldovich and his colleagues realised that there would be structure in the power

spectrum of oscillations which survived to the epoch of recombination, as illustratedin Fig. 12.5a (Sunyaev and Zeldovich, 1970). Those fluctuations on a given massscale, which would eventually develop into bound structures at late epochs, werethose which had large positive amplitudes when they came through their particlehorizons. Figure 12.5a shows perturbations on two different mass scales coming

Fig. 12.5a,b. ‘Stability diagram’ of Sunyaev and Zeldovich (Sunyaev and Zeldovich, 1970).a The region of instability is to the right of the solid line. The two additional graphs illustratethe evolution of density perturbations of different masses as they come through the horizonup to the epoch of recombination. b Perturbations corresponding to different masses arrive atthe epoch of recombination with different phases, resulting in a periodic dependence of theamplitude of the perturbations upon mass


Il ruolo dei barioni

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14.6 The Acoustic Peaks in the Power Spectrum of Galaxies 411

Fig. 14.6. Four examples of transfer functions for models of structure formation with baryonsonly (top pair of diagrams) and with mixed cold and baryonic models (bottom pair ofdiagrams) (Eisenstein and Hu, 1998). Eisenstein and Hu’s primary objective was to presentfitting functions to the transfer functions derived from numerical solutions to the Boltzmannequation for the development of mixed baryonic and cold dark matter perturbations. Thenumerical results are shown as solid lines and their fitting functions by dashed lines. Thelower small boxes in each diagram show the percentage residuals to their fitting functions,which are always less than 10%

14.6 The Acoustic Peaks in the Power Spectrum of Galaxies

At last, we can tackle the power spectra of galaxies derived from the 2dF GalaxyRedshift Survey and the Sloan Digital Sky Survey. Before doing that, it is worth-while paying tribute to the astrophysicists, engineers and technologists involved inboth these very large undertakings. The Anglo-Australian Telescope Board was per-

Solo Barioni

Ωb = ΩDM


Picchi acustici da survey galassie 2dF

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14.6 The Acoustic Peaks in the Power Spectrum of Galaxies 413

Fig. 14.7. The power spectrum of the three-dimensional distribution of galaxies in the 2dFGalaxy Redshift Survey. The points with error bars are the best estimates of the observedpower spectrum once the biases and corrections for incompleteness are taken into account.In the lower panel, the data from the upper panel have been divided by a reference cold darkmatter model, with !D = 0.2, !" = 0 and !B = 0, which has a smooth power spectrum.The grey dashed line is a best fitting model before convolution with the window function forthe survey. The solid line shows the best fit once the model is convolved with the windowfunction (Cole et al., 2005)

Table 14.2. Cosmological parameters derived from analysis of 2dF Galaxy Redshift Survey

Power spectrum spectral index n = 1 assumedHubble’s constant h = 0.72 assumedNeutrino masses m# = 0 assumedOverall density parameter !0h = 0.168 ± 0.016 derivedBaryon fraction !B/!0 = 0.185 ± 0.046 derived

(Eisenstein et al., 2005). In order to maximise the volume of space available forstudy, attention was restricted to a sample of 46,748 luminous red galaxies for whichuniform selection criteria were adopted in the redshift range 0.16–0.47. The selectioncriteria and homogeneity of the sample were described by Hogg and his colleagues


Picco acustico da survey galassie SDSS

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(Hogg et al., 2005). The mean redshift of the sample was about 0.3 compared with0.1 for the complete 2dF sample. Thus, although the statistics are smaller than inthe 2dF sample, the restriction to luminous galaxies meant that better statistics wereachieved over larger volumes, particularly in the crucial 50h!1 to 200h!1 Mpc rangeof scales.

The two-point correlation function is presented in Fig. 14.8 in the form s2!(s),where s is the separation of the galaxies. This form of presentation was adopted tohighlight the curvature of the power spectrum on small physical scales. The clearmaximum observed in the power spectrum at physical scale 100h!1 Mpc corre-sponds to the first acoustic peak in the power spectrum of primordial fluctuations.Its location is in good agreement with that inferred from the 2dF Galaxy RedshiftSurvey.

From the overall shape of the correlation function, the matter density was foundto correspond to "0 = 0.273 ± 0.025, if it is assumed that the dark energy isassociated with the cosmological constant and the global geometry of the Universeis flat. If the scale of the acoustic peak is included in the estimates, the constraint onthe spatial curvature was found to be "# = (c/H0)

2/"2 = !0.010 ± 0.009. Noticethat these conclusions are independent of the information derived from analyses ofthe fluctuations in the cosmic microwave background radiation.

Both the Sloan and 2dF teams recognised the central importance of the discoveryof baryon oscillations in the power spectrum of galaxies. In conjunction with the verymuch larger amplitude perturbations observed in the cosmic microwave background

Fig. 14.8. The large-scale redshift-space correlation function of the Sloan Digital Sky SurveyLuminous Red Galaxy sample plotted as the correlation function times s2. This presentationwas chosen to show the curvature of the power spectrum at small physical scales. The modelshave "0h2 = 0.12 (top), 0.13 (middle) and 0.14 (bottom), all with "Bh2 = 0.024 andn = 0.98. The smooth line through the data with no acoustic peak is a pure cold dark mattermodel with "0h2 = 0.105 (Eisenstein et al., 2005)


Vari modelli CDM

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Fig. 14.9. Examples of the predicted power spectra of galaxies for different models of structureformation (Dodelson et al., 1996). The models shown involve standard cold dark matter(sCDM), open cold dark matter (CDM), cold dark matter with a finite cosmological constant(!CDM), cold dark matter with decaying neutrinos ("CDM) and an alternative neutrino darkmatter model (#CDM) described by Dodelson and his colleagues. The models are comparedwith the power spectrum of galaxies derived by Peacock and Dodds (Peacock and Dodds,1994) and the normalisation at small wavenumbers derived from the COBE observationsof the temperature fluctuations in the cosmic microwave background radiation. The !CDMmodel has been shifted upwards for the sake of clarity. Notice that, in this presentation, thepower spectrum P(k) has dimensions Mpc3 since the authors have not included the term V inthe definition of the power spectrum (see (14.19))

spectrum to be

keq =!2$0 H2

0 zeq/c2"1/2 = 7.3 ! 10"2$0h2 Mpc"1 , (14.60)

where the temperature of the cosmic microwave background radiation has beenassumed to be 2.728 K (Hu et al., 1997; Eisenstein and Hu, 1998). It can be seen thata value of $0 = 1 results in the maximum of the power spectrum being shifted tolarger wavenumbers than the observed maximum and so to excess power on smallscales. The trick is to find ways of moving the maximum, which (14.60) showsdepends upon $0h2, to smaller wavenumbers. This is achieved in the open CDM

sCDM: Ω0 = 1.0 ΩΛ = 0. CDM: Ω0 = 0.2 ΩΛ = 0.ΛCDM: Ω0 = 0.3 ΩΛ = 0.7

ΛCDM

Simulazioni del progetto Virgo: struttura delle galassie su grande scala.Lato quadrati 240 h-1 Mpc2563 = 1.7 107 particelle

Vari modelli CDM

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