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Using the Comoving Maximum of the Galaxy Power Spectrum to Measure Cosmological Curvature


Draft version February 1, 2008
A Preprint typeset using L TEX style emulateapj v. 04/03/99

USING THE COMOVING MAXIMUM OF THE GALAXY POWER SPECTRUM TO MEASURE COSMOLOGICAL CURVATURE
Tom Broadhurst, Andrew H. Jaffe
Department of Astronomy, University of California, Berkeley, CA 94720

Draft version February 1, 2008

arXiv:astro-ph/9904348v1 26 Apr 1999

ABSTRACT The large-scale maximum at k ? 0.05 h/Mpc identi?ed in the power-spectrum of galaxy ?uctuations provides a co-moving scale for measuring cosmological curvature. In shallow 3D surveys the peak is broad, but appears to be well resolved in 1D, at ? 130 h?1 Mpc (k = 0.048 h/Mpc), comprising evenly spaced peaks and troughs. Surprisingly similar behaviour is evident at z ? 3 in the distribution of Lyman-break galaxies, for which we ?nd a 5σ excess of pairs separated by ?z ? 0.22 ± 0.02, equivalent to 85 h?1 Mpc for ? = 1, increasing to 170 h?1 Mpc for ? = 0, with a number density contrast of 30% averaged over 5 independent ?elds. The combination, 3.2?m ? ?Λ ? 0.7, matches the local scale of 130 h?1 Mpc, i.e. ? = 0.2±0.1 or ?f lat = 0.4±0.1 for the matter-dominated and ?at models respectively, m with an uncertainty given by the width of the excess correlation. The consistency here of the ?at model with SNIa based claims is encouraging, but overshadowed by the high degree of coherence observed in 1D compared with conventional Gaussian models of structure formation. The appearance of this scale at high redshift and its local prominence in the distribution of Abell clusters lends support to claims that the high-z ‘spikes’ represent young clusters. Finally we show that a spike in the primordial power spectrum of δρ/ρ ? 0.3 at k = 0.05 h/Mpc has little e?ect on the CMB, except to exaggerate the ?rst Doppler peak in ?at matter-dominated models, consistent with recent observations. Subject headings: cosmology: large scale structure
1. INTRODUCTION

Methods for measuring cosmological parameters are more readily conceived than realized in practice. Recently however, it has become clear that both SNIa lightcurves and CMB temperature ?uctuations provide reasonably model-independent and complimentary means of constraining cosmological curvature (Goobar & Perlmutter 1995, Eisenstein, Hu & Tegmark 1998). The angular scale of the ?rst CMB Doppler peak limits combinations of ? and Λ. At smaller angles more geometric information may be extracted but is subject to greater physical uncertainty and complex foregrounds. The utility of distant Type Ia supernovae has been convincingly demonstrated (Perlmutter et al. 1998,1999), but concerns remain. Local supernovae are largely discovered in luminous dusty late-type galaxies, deliberately targeted for convenience. In the distant ?eld, supernovae are weighted towards lower luminosity galaxies, sometimes undetected because of the volume-limited nature of the selection. Hence it is not clear that both samples should share the same family of light curves. The physical origin of the ?15% luminosity residual remaining after correction is not understood, nor indeed the light-curve correlation itself. Furthermore, some evolution is natural via progenitor metalicity, although the redshift independence of both the residual variance and the distribution of event durations is reassuring. An upward revision of the distant SNIa luminosities of only 15% would render a cosmological constant unnecessary (Perlmutter et al. 1999). Here we attempt a simple model-independent measurement of curvature, using the spatial scale of the maximum in the power spectrum of galaxy perturbations. A broad maximum is naturally expected below 1

k ? 0.1 Mpc/h, corresponding to the Horizon at the epoch of matter-radiation equality, up to which high frequency power is suppressed relative to an initially rising power-law spectrum, bending the spectrum over at higher frequency. Observations have established a maximum at k ? 0.05 h/Mpc (? 130 h?1 Mpc) from wide angle galaxy and cluster surveys (Bough & Efstathiou 1993,1994, Einasto et al. 1997, Tadros, Efstathiou & Dalton 1998, Guzzo et al. 1998, Einasto et al. 1998, Hoyle et al. 1998, Scheuker et al. 1999). In particular a pronounced peak is evident in local cluster samples (Einsato et al. 1998) and in the deprojection of the large APM survey (Bough & Efstathiou 1994). In all these estimates the high frequency decline is steeper than expected (Feldman, Kaiser & Peacock 1995) and together with the relatively small amplitude ?uctuations implied by CMB at very low k (Smoot et al. 1992) requires the existence of a maximum between 0.01<k<0.1. The low frequency peak in the projected 3D powerspectrum and its coincidence in scale with the excess large power detected in lower dimensional redshift surveys (Broadhurst et al. 1990,1992,1999; Landy et al. 1995, Small, Sargent & Hamilton 1997, Tucker, Lin, Schectman 1998) encourages the possibility that the test proposed here can be explored already, by simply comparing pencil beam surveys at high and low redshift. Power on these large scales comoves with the expansion, providing a means of measuring curvature. For example, at z = 0 the redshift interval corresponding to a scale of 130 h?1 Mpc is ?z = 0.043, but this stretches by a factor of 4–8 in redshift to ?z ? 0.25 at z = 3, increasing with ? and Λ. Here we describe the evidence for a preferred large scale of ?z ? 0.22 in the ?elds of high redshift galaxies presented by Steidel (1998) and in the Hubble Deep Field

2 (Adelberger et al. 1998). We examine this in light of the increasing local evidence for a maximum in the power spectrum at 130 h?1 Mpc. We obtain the locus of ? and Λ, under the assumption that the preferred scale in these datasets is the same. Finally we discuss the implication of a spike in the primordial power spectrum for the CMB.
2. LOW REDSHIFT STRUCTURE

isotropic with k. A non-Gaussian distribution leads to “hotter” and “colder” spots at a given frequency, which may average out in projection but generate a larger variance in 1D pencil beams. This is particularly true of course for highly coherent structure (e.g. Voronoi foam, Icke & Van De Weygaert 1991) , where the 3D variance can be sub-Poissonian on scales larger than the coherence length.
3. HIGH REDSHIFT STRUCTURE

Several independent wide angle surveys of galaxies and clusters allow the projected 3D power spectrum, P3 (k) to be estimated on scales large enough to cover the predicted low frequency roll-over. These surveys all show a sharp decline in power at k > 0.1 Mpc/h, and if deep enough, evidence of either a peak, or a maximum at k ? 0.05 Mpc/h (Baugh & Efstathiou 1993, Einasto et al. 1995, 1998, Tadros & Efstathiou 1996, Hoyle et al. 1999, Schuecker et al. 1999). This despite the relatively sparse sampling and fairly shallow depth, so that the peak may be better resolved in larger ongoing surveys, where the binning in k can be made ?ner, and the isotropy of Fourier amplitudes examined around k = 0.05 Mpc/h for any coherent or generally non-Gaussian behaviour. Consistent with this, redshift surveys directed at the Bo¨tes void and towards the Phoenix region and the o Shapley supercluster also show a pronounced pattern of large wall/voids separated by ? 130 h?1 Mpc (Kirshner et al. 1981, 1987, 1990, Quintana et al. 1988, Small, Hamilton & Sargent 1997). Large strip surveys in two dimensions also contain excess power around ? 100 h?1 Mpc (Landy et al. 1995, Vettolani et al. 1995). This is despite the narrowness of the strips, of only a few Mpc in width, which must therefore underestimate the real radius of voids (Tucker, Lin & Shectman 1999). The transverse extent of the peaks and troughs identi?ed at the galactic poles by Broadhurst et al. (1990) are now known to span much more than the width of the original beams, which were of order only the correlation length (5 h?1 Mpc). The closest northern peak at v=7000km/s intersects the transverse ‘great wall’ structure near the Coma cluster, as revealed in the maps of De-Lapparent, Geller & Huchra (1986), which extends over 100 h?1 Mpc. Wider angle redshift surveys at the galactic poles (of diameter 100 h?1 Mpc) have clearly con?rmed and strengthened the early ?nding of a scale of 130 h?1 Mpc spanning the redshift range z < 0.3 (Broadhurst et al. 1999) revealing a sinusoidal alternating pattern of peaks and troughs spanning the redshift range z < 0.3 and hence a correspondingly narrow concentration of power in one dimension P1 (k), at k = 0.045 h/Mpc. Independent support for the reality of this pattern is found in the coincidence of these peaks with the local distribution of rich clusters (Broadhurst 1990, Bahcall 1991, Guzzo et al. 1992) and in a southern ?eld close to the pole (Ettori, Guzzo & Tarengi 1997). In comparing the 1D distribution with the projected 3D power spectrum it is important to keep in mind that the wide angle redshift surveys do not have su?cient volume and sampling density to construct the real 3D power spectrum P3 (k) on 100 h?1 Mpc scales, so interpretations are based on P3 (|k|), which is the mean power averaged over solid angle. Hence, it is only sensible to compare the amplitude of P3 (|k|) with P1 (k) if the power is known to be

The most puzzling aspect of the distant dropout galaxies is the appearance of sharp peaks in the redshift distribution (Adelberger et al. 1998, Steidel 1998), resembling the situation at low redshift, implying at face value little growth of structure. A conventional interpretation of these peaks resorts to “bias” (Wechsler et al. 1998) which has come to represent a ?exible translation between the observed structure and the relatively smooth mass distribution of standard simulations, so that the observed peaks are interpreted as rare events destined to become massive clusters by today (Wechsler et al. 1998). High biases have been claimed for the Lyman-break population on the basis of the amplitude of small scale clustering at z ? 3 (Adelberger et al. 1997, Giavalisco et al. 1998). The occurrence of such peaks is enhanced with a steep spectrum by a reduction of high frequency ‘noise’ (Wechsler et al 1998) but this gain is o?set if the steepness is attributed to low ?, since then a given redshift bin corresponds to a larger volume, and hence a greater proper density contrast (Wechsler et al 1998). We may regard the existence of regular spikes at low and high redshift as evidence for a revision in our understanding of large scale structure, indicating perhaps that initial density ?uctuations are not Gaussian distributed such as may be implied by the observed lack of evolution of the number density of X-ray selected clusters (Rosati et al. 1998). The baryon isocurvature model (Peebles 1997) more naturally accommodates both the early formation and the frequent non-Gaussian occurrence of high density regions (Peebles 1998a,b). We analyze the ?elds of Lyman-Break galaxies of Steidel (1998) and Adelberger(1998). These include 4 ?elds of ? 100 ? 200 galaxies and a smaller sample of redshifts in the Hubble Deep Field direction. The ?elds are ? 10Mpc in width by ? 400Mpc in the redshift direction and include over 600 redshifts in the range 2.5 < z < 3.5 histogrammed in bins of ?z = 0.04. In Figure 1 we plot the pair counts and correlation function, assuming that galaxies are evenly distributed within the narrow redshift bins. A clear excess is apparent on a large scale, corresponding to a preference for separations of ?z = 0.22. A pair excess is also seen at twice this separation (Fig 1) indicating phase coherence along the redshift direction. This behaviour is an obvious consequence of the regular peaked structure visible in the redshift histograms of four of the ?ve ?elds (Fig 1). In a bin of 10Mpc, the number of pairs at the peak is ? 1220 compared with an expected ? 1060, representing a 4.6σ departure from random (Fig 1). The observed redshift interval may be related to comoving scale by the usual formula for a universe with negligible radiation energy-density (Peebles 1993),
z+?z

?w = 3000 h?1 Mpc
z

dz E(z)

3 ? 3000 h?1 Mpc with E(z) = ?(1 + z)3 + ?Λ + (1 ? ? ? ?Λ )(1 + z)2 . (2) ?z E(z) (1) the possibility of non-Gaussian ?uctuations which would certainly diminish the angle-averaged power. If the value of the power spectrum at the k = 0.05 h?1 Mpc peak is already large, as with COBEnormalized Λ models, an additional peak of the above strength has negligible e?ect. However, COBE-normalized models with ? = 1 and Λ = 0 have comparatively low power at this scale, hence, the e?ect on the ?nal C? and P (k) is signi?cant. Of course, the precise width and location of the peak, not yet at all well-constrained by the CMB data, also a?ects the relative power in the peak versus the underlying “smooth power-law” spectrum. This is illustrated in Figure 2. The spike is seen to raise the amplitude of the ?rst Doppler peak by nearly a factor of two for the sCDM model. Note, although it is premature to interpret the claims of the various CMB experiments for and excess of power at C? = 200 (Netter?eld et al 1997, Tegmark 1998) without proper treatment of the covariance matrix of errors and foreground subtraction, the indications of a large amplitude for this peak are not inconsistent with the degree of boosting (Fig. 2). Others, Gawiser and Silk 1998, have noted that the whole aggregate of current C? and P (k) data is extremely well-?t—much better than the ?t to standard models—by an adiabatic in?ationary power spectrum with sharp bump like that considered here.
5. CONCLUSIONS

Whereas standard candles at moderate redshift (SNIa) measure ? (? ? ?Λ ), and CMB anisotropies measure ? (? + ?Λ ), these observations considered here measure ? E(3) or ? (3? ? ?Λ ) which lies between these two locii, thus adding complimentary information. With this, the ?z = 0.22 scale of the peak corresponds to 85 h?1 Mpc for ? = 1, doubling to 170 h?1 Mpc for ? = 0. Flat cosmologies with a positive Λ fall between these limits. If we constrain ?w = 130 h?1 Mpc we ?nd 48? ? 15?Λ ? 10.5. This gives ? = 0.2 for an open universe (?Λ ≡ 0) or ? = 0.4 for a ?at universe (? + ?Λ ≡ 1), with an uncertainty of only 0.1 in ?m , given by the 15% width of the excess correlation (Figure 1). This scale, if borne out by subsequent redshift data, is certainly consistent with the ?at model preferred by SNIa data with ?f lat = 0.3 (Perlmutter et al. 1998,1999). m
4. EXCESS POWER AND THE CMB

One can imagine two broad classes of physical mechanisms that might be responsible for the excess power required to ?t this data. The power might be a truly primordial feature in the power spectrum. In in?ationary scenarios, for example, this could be generated by the proliferation of super-horizon bubbles in a suitably conspiratorial in?aton potential (La 1991, Occhinero & Amendola 1994). On the other hand, excess power could be due to causal microphysics in the universe after the 130 h?1 Mpc scales enter the horizon. In other words, the power could be added by the transfer function. A high baryon density naturally imparts large scale “bumps and wiggles” to the power spectrum (Peebles 1998a,b, Eisenstein et al. 1997, Meiksin et al. 1998) in particular if matter density ?uctuations are created at the expense of radiation (Peebles 1998a,b). An even more radical possibility is that the universe is topologically compact, and that we are seeing evenly-spaced copies of a small universe with an extent of only ? 130 h?1 Mpc although this scale seems too small to accommodate the unique and relatively distant (z=0.18) cluster A1689 (Gott 1980). In light of this it is interesting to explore the implications of any excess power on CMB temperature ?uctuations. Eisenstein et al. (1997) has examined in what way conventional adiabatic models maybe stretched to match the power spectrum of Broadhurst et al. (1990), demonstrating that such models do not naturally account for a scale of 130 h?1 Mpc in the mass distribution. Our approach is to simply add power in the primordial spectrum at a ?xed wave number, using a modi?ed version of CMBFAST code (Seljak & Zaldarriaga 1999), to simulate temperature ?uctuation spectra. A primordial spike of power will e?ect the CMB directly through the projection of three-dimensional features. A narrow band of power is added, ?2 (k) = k 3 P (k)/(2π 2 ). The amplitude is set so that d ln k ?2 (k) ? 0.1, the value of the correlation function at the peak and harmonics equivalent to at 30% density contrast. We ignore bias, which may conceivably be large, lowering the peak amplitude; we also ignore

Two interesting results emerge from the above comparison of structure in the local and distant pencil beam data. Firstly both samples of galaxies show regular large scale power con?ned to a narrow range of frequency. Secondly, by matching these scales we obtain values of ? and Λ in good agreement with the SNIa claims. These ?ndings may of course be regarded as remarkable coincidences, unlikely to occur by chance in a clumpy galaxy distribution (Kaiser & Peacock 1991). Distinguishing between the open and ?at solutions found here requires pencil beam data at a third redshift. Optimally this redshift turns out to be convenient for observation, z ? 0.7, where the ratio of ?z between the open ? = 0.2 and ?f lat = 0.4 cases is maximal, di?ering m by 13%, and may be explored with existing data. Since locally the excess power at 130 h?1 Mpc is most prominent in cluster selected samples, we may conclude that the peaks in the high-z sample correspond to protoclusters at z ? 3. This conclusion is independent of the high-bias interpretation which also regards the spikes as young rich clusters. Understanding the physics behind these spikes is complicated by a conspiracy of scales. We have seen in Fig. 2 that the k ? 0.05 Mpc/h spike produces a feature in the CMB power spectrum at nearly exactly the position of the ?rst doppler peak. In the simple model presented here—a primordial spike at this position—this is merely a coincidence, however unlikely. That these scales are so closely matched seems yet more unlikely in the light of another fact, that the required peak in the matter power spectrum is at nearly the position of the expected peak due to the passage from radiation to matter-domination. If the acoustic peaks in the CMB power spectrum are instead a signature of physics at a somewhat later epoch:

4 √ the sound horizon at recombination (set by c/ 3) then that these scales are nearly coincident (within an order of magnitude or so of 100 h?1 Mpc) is a consequence of the particular values of the cosmological and physical parameters. Thus, if we change the initial conditions to increase the amplitude of the matter power spectrum near the equality scale, we also increase the CMB temperature power spectrum at roughly the scale of the ?rst acoustic peak. That is, in neither case do we add an unexpected peak, but merely increase the amplitude and “sharpness” of an expected one. Of course, the inexplicable coincidence is that the feature in the power spectrum appears close to the expected matter-radiation equality peak, but with an amplitude much too large to be explained by standard theories. In addition, we must also note that merely adding a peak to the mean power spectrum cannot account for the nearly periodic structure observed in 1D. A real theoretical explanation must account for the complicated non-Gaussian properties of this distribution. The excess 1D power identi?ed by Broadhurst et al. (1990) will soon be subject to easy dismissal with large 3D redshift surveys. However, whether or not it transpires that universal large scale coherence exists, the test proposed here is still viable in principle using the general predicted turnover of the 3D power spectrum, requiring larger volume redshift surveys. In particular comparisons of local and distant clusters over a range of redshift will be particularly useful given their sharply peaked and high amplitude power spectrum.. We thank Richard Ellis, Alex Szalay, Gigi Guzzo, Alvio Renzini, Eric Gawiser, Saul Perlmutter and Martin White for useful conversations.

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0.2

1000 0.1

500 0

0 0 100 200 S Mpc/h 300

-0.1 0 100 200 S Mpc/h 300

Fig. 1.— The co-added pair counts of 5 ?elds of galaxies at z ? 3 compared with the random expectation of a modi?ed selection function based on (solid curve) Adelberger et al. 1998 (dotted curve). The contributions of the ?ve ?elds are also histogrammed, ordered by number of redshifts for clarity. The right hand panel shows the pair weighted correlation function with obvious excesses at 130 h?1 Mpc and at double this scale, 260 h?1 Mpc, (? = 0.2) implying large scale coherence.

6

120

100

80

60

40

20 10 100 1000

Fig. 2.— Models for the CMB temperature ?uctuation spectrum including ?at (with no cosmological constant) standard CDM models (green) and a Λ dominated model (black). Standard n = 1 primordial power spectra are shown with solid lines and the addition of a peak at k = 0.05 h/Mpc with dashed lines. The highest dashed and the lowest solid curves are the SCDM model; the middle pair or the Λ model. Note that the peak generates a relatively large enhancement of the ?rst doppler peak in the sCDM case.



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