The cluster abundance in cosmic string models for structure formation

Abstract

We use the present observed number density of large X-ray clusters to constrain the amplitude of matter density perturbations induced by cosmic strings on the scale of 8 h⁻¹ Mpc (σ₈), in both open cosmologies and flat models with a non-zero cosmological constant. We find a slightly lower value of σ₈ than that obtained in the context of primordial Gaussian fluctuations generated during inflation. This lower normalization of σ₈ results from the mild non-Gaussianity on cluster scales, where the one-point probability distribution function is well approximated by a χ² distribution and thus has a longer tail than a Gaussian distribution. We also show that σ₈ normalized using cluster abundance is consistent with the COBE normalization.

cosmic strings, galaxies: clusters: general, cosmology: theory, large-scale structure of Universe

Introduction

Current theories for structure formation can be divided into two broad categories: inflation and cosmic defects. While inflation predicts primordial and Gaussian fluctuations (in the simplest inflationary models), topological defects induce active and non-Gaussian perturbations (for a review see Vilenkin & Shellard 1994).

One of the most important constraints on models of structure formation is the observed abundance of galaxy clusters. Although the cluster abundance has been widely used to constrain cosmological models with primordial Gaussian fluctuations (White, Efstathiou & Frenk 1993; Eke, Cole & Frenk 1996; Kitayama & Suto 1997; Wang & Steinhardt 1998; Viana & Liddle 1999), there have been few studies in the context of non-Gaussian perturbations such as those generated by topological defects. This is due to the difficulty of making robust predictions in topological defect scenarios, owing to their non-linear effects which are difficult to model and require large-scale numerical simulations (e.g. Allen & Shellard 1990; Bennett & Bouchet 1990).

The present work relies on high-resolution numerical simulations of cosmic-string-seeded structure formation (Avelino et al. 1998a,b, 1999; Wu et al. 1998), which are first used to estimate the power spectrum and one-point probability distribution functions (PDFs) of the induced density perturbations (see also Albrecht, Battye & Robinson 1997, Avelino, Caldwell & Martins 1997, Battye, Robinson & Albrecht 1998 and Contaldi, Hindmarsh & Magueijo 1999 for different approaches). We then employ a simple generalization of the Press-Schechter formalism (Press & Schechter 1974), which is suitable for non-Gaussian perturbations with a general one-point PDF (Chiu, Ostriker & Strauss 1998), in order to obtain the expected number density of collapsed objects with a mass greater than a given threshold. This generalized Press-Schechter formalism has been used to constrain the Gaussianity of the density fluctuations in the Universe (Robinson, Gawiser & Silk 1998; Koyama, Soda & Taruya 1999), and has been verified for a particular set of non-Gaussian structure formation models, including a simplified flat-space cosmic string model (Robinson & Baker 2000). We finally estimate the amplitude of matter density perturbations induced by cosmic strings on the standard scale of 8 h⁻¹ Mpc, using the presently observed number density of large X-ray clusters. We do this for cosmic string models in open universes without a cosmological constant (SOCDM), and also in flat universes with a non-zero cosmological constant (SΛCDM). We compare the σ₈ constrained by cluster abundance with σ₈ constrained by COBE observations, and show that they are consistent within present uncertainties.

Power Spectrum and PDF

In a previous work (Avelino et al. 1998a; Wu et al. 1998) we described the results of high-resolution numerical simulations of string-induced structure formation. The power spectrum of CDM perturbations induced by long strings can be approximately described by  

(1)

where  

(2)

 

(3)

This fit to our numerical results has an accuracy of better than 10 per cent in the range k=0.01–100 h Mpc⁻¹, provided that the baryon-to-dark matter ratio is relatively small Note, however, that there are specified uncertainties in the underlying numerical results which are significantly larger (Wu et al. 1998). This result also does not include the contribution from small loops, which significantly enhance the overall power while leaving the overall shape unchanged (Avelino et al. 1999), as we will discuss later. A χ² analysis using the observational power spectrum reconstructed by Peacock & Dodds (1994) gives a best fit to the long-string power spectrum (1) with Γ=0.074×10^±0.1 at the 95 per cent confidence level. For a baryon energy density Ω_B=0.05, this implies that string models provide a consistent match to observations in the acceptable cosmological parameter range

We have used equation (1) to obtain a numerical fit for the standard deviation of matter density perturbations at the present time as a function of the smoothing scale R (in units of h⁻¹ Mpc):  

(4)

where  

(5)

The redshift dependence of σ for arbitrary Ω_m, Ω_Λ is described accurately by  

(6)

where  

(7)

Here, g(Ω_m, Ω_Λ) gives the suppression of the growth of density perturbations relative to that of a critical-density universe (Carroll, Press & Turner 1992; Avelino & de Carvalho 1998; Wu et al. 1998), and we can describe the evolution of the Ω_m with redshift as  

(8)

Using the simulation results for string-induced matter perturbations, we also found a reasonable fit to the positive side of the one-point PDF as follows  

(9)

where R is the smoothing radius in units of h⁻¹ Mpc (for R≳0.3 h⁻¹ Mpc), ν is the number of standard deviations from mean density, and g_n(χ²) is the standard chi-square (χ²) distribution, with the number of degrees of freedom n given as  

(10)

Modified Press-Schechter Formalism

The Press-Schechter formalism (Press & Schechter 1974) relates the mass fraction of collapsed objects whose masses are larger than some given threshold M with the fraction of space in which the evolved linear density field exceeds some threshold δ_c. It has been extensively tested with great success against N-body simulations in the context of primordial Gaussian fluctuations, allowing for the computation of the number density of clusters within a given background cosmology. The first attempt to extend the Press-Schechter formalism into the non-Gaussian regime was carried out by Colafrancesco, Lucchin & Matarrese (1989). Here we used the extension of the Press-Schechter formalism for non-Gaussian perturbations proposed by Chiu et al. (1998). It has been verified by Robinson et al. (1999) for a particular set of non-Gaussian structure formation models. The fraction of the total mass within collapsed objects larger than a given mass M is given by  

(11)

where M is the cluster mass defined by M=4πR³ρ_b/3, f_R=1/ δ_*δ_cσ(R,z), and δ_c=1.7±0.2 (95 per cent confidence interval) assuming spherical collapse (Bernardeau 1994).

To obtain the number density of clusters in a mass interval dM about M at a redshift z, we differentiate the Press-Schechter formula (11) to obtain (cf. Viana & Liddle 1996, 1999)  

(12)

In the derivation of equation (12) we have ignored other terms resulting from the R dependence of P_R and f_R. This is a mathematically motivated approximation in the regime of mild non-Gaussianity, and we have verified in our case that it gives rise to at most an extra 2 per cent error in the final estimate of σ₈. Substituting (6) into (12) give  

(13)

Lacey & Cole (1993, 1994) constructed a merging history for dark matter haloes based on the excursion set approach, and obtained an analytical expression for the probability that a galaxy cluster with present virial mass M would have formed at some redshift z. The probability that a galaxy cluster with the present virial mass M would have formed at a given redshift z is given b  

(14)

where  

(15)

 

(16)

and F=0.75±0.15 (95 per cent confidence interval) is the fraction of the cluster mass assembled by a redshift z (Navarro, Frenk & White 1995). We note that this result was derived in the context of primordial Gaussian fluctuations, and must ultimately be verified using N-body simulations. Although it may not be valid for generic non-Gaussian models, we still expect this to be a good approximation in the context of the cosmic string scenario for structure formation, as the departures from Gaussianity on clusters scales are relatively small (Avelino et al. 1998b).

The Mass-Temperature Relation

In order to use the generalized Press-Schechter formalism to determine the abundance of X-ray clusters with a given temperature, we need to relate the X-ray temperature of a cluster to its virial mass. Here we use the results of Viana & Liddle (1996, 1999) for the normalized virial mass-temperature relation (modified from Viana & Liddle 1999)  

(17)

where z_t is the turnaround redshift, Ω_mt=Ω_m(z_t), an  

(18)

 

(19)

For a given redshift of cluster collapse z_c, the turnaround redshift z_t is easily obtained using the fact that 2t(z_t)=t(z_c).

Hence we can now estimate the present comoving number density of galaxy clusters per temperature interval d(k_BT) with a mean X-ray temperature k_BT which were formed at a given redshift z as  

(20)

The present abundance of X-ray clusters with a temperature k_BT greater than 6.2 keV can be estimated by integrating equation (20) from zz_c=0 to infinity. A comparison between the observed cluster abundance and its theoretical prediction will give an estimate of σ₈.

We will use the observation for the number density of galaxy clusters at z=0.05 with X-ray temperatures exceeding 6.2 keV, given by Viana & Liddle (1999), based on the data set presented by Henry (1997), and updated by Henry (in preparation)  

(21)

The uncertainty in (21) is the 1σ interval, and these results have taken into account the effect of temperature measurement errors. The reasons for concentrating on galaxy clusters with temperatures greater than 6.2 keV have been extensively discussed by Viana & Liddle (1999).

Results and Discussion

By comparing (21) with the result integrated from (20), we obtain the observationally constrained σ₈ as plotted in Fig. 1. The overall error in the value of σ₈ was estimated by Monte Carlo simulations over 10⁴ realizations, treating the intrinsic uncertainties in Γ, N(>6.2 keV, 0.05) as lognormal, and those in δ_c, M_v , F as Gaussian. An accurate numerical fit to this result is

Figure 1.

The σ₈ in cosmic string scenarios normalized to cluster abundance. The upper panel is the SOCDM models, while the lower panel is the SΛCDM models.

 

(22)

The 95 per cent confidence limits are ±32 per cent in the SOCDM case, and +35 and-32 per cent in the SΛCDM case. We note that the overall shape of σ₈ in Fig. 1 is similar to that obtained in the context of inflationary models with primordial Gaussian fluctuations (Viana & Liddle 1996, 1999). This is expected because both the string-induced and inflationary power spectra used in the calculation of cluster abundance are constrained by the same observations (Peacock & Dodds 1994), and have roughly the same shape within the scales of interest (though with quite different choices of Γ). However, we also notice that the amplitude of σ₈ here is about 10–20 per cent lower than that of inflationary perturbations. This is due to a slightly larger right-hand-side tail of the PDF in the cosmic string scenario.

In addition, we have verified that both the finite size and limited dynamic range of our numerical simulations do not result in significant uncertainties in our PDF. The chosen dynamic range around the matter-radiation transition includes the primary contributions to perturbations on cluster scales. The epochs missing from the simulations (both deep radiation and later matter eras) induce only weak perturbations which, in principle, should not affect the high-density tail on the right-hand side of the PDF which is associated with clusters. The finite simulation size is again tailored to study cluster formation, with a grid resolution 0.25 h⁻¹ Mpc well below cluster scales up to a box-size of 64 h⁻¹ Mpc. Note that on the largest scales the PDF is very nearly Gaussian, so we do not expect to gain further information from larger simulations.

The value of σ₈ in the context of the cosmic string model for structure formation was also investigated by van de Bruck (1998). In his work he assumed the one-point PDF to be independent of scale by taking the distribution at the non-Gaussian scale [∼ from Avelino et al. (1998b) to be valid up to scales relevant for the cluster abundance calculation. Although this assumption can give the right qualitative results, for small values of the R dependence of the one-point PDF needs to be taken into account in order to obtain more accurate results. This improvement has been incorporated in our work, which also took into consideration the merging history of dark matter haloes.

To see the difference between the COBE-normalized long-string-induced σ₈ and the cluster-abundance constraint, we define a paramete  

(23)

where is the cluster result (22), and is inferred from the COBE normalised long-string-induced power spectrum (1). In the calculation of we have used and Gμ₆Gμ×10⁶=1.7 (Allen et al. 1997) for and Λ=0, with the open and Λ-parameter dependence for Gμ₆ given by for SOCDM and for SΛCDM (Avelino et al. 1997).

We can observe in Fig. 2 for the flat Λ-model that ranges from 1.3±0.5 to 2.3±0.8 (95 per cent confidence level) for different choices of and For small B is approximately one, meaning that and seem to be consistent. However, for large B is higher than one. Nevertheless, as recently shown by Avelino et al. (1999), the inclusion of perturbations induced by cosmic string loops can boost the string-induced power spectrum by a factor of about 2 under reasonable assumptions. Hence this may help remove the small discrepancy between the COBE and cluster-abundance constraints on σ₈, which we see in Fig. 2 even for high values of We also note that the exact contribution of the background of gravitational radiation emitted by cosmic string loops remains a significant uncertainty (Avelino & Caldwell 1996).

Figure 2.

B, the ratio of σ₈ normalized by cluster abundance and COBE, is plotted as a function of and The hatched (SOCDM) and shaded (SΛCDM) areas are within a 95 per cent confidence level.

Conclusions

We have constrained the amplitude of matter density perturbations induced by cosmic strings on the scale of 8 h⁻¹ Mpc in both open cosmologies and flat models with a non-zero cosmological constant, using the currently observed number density of large X-ray clusters. Because string-seeded matter perturbations are mildly non-Gaussian on cluster scales, we obtained a slightly lower normalization of σ₈ than that found for cosmological models with primordial Gaussian fluctuations. We compared this σ₈ with that constrained by COBE, and found that they are consistent when current uncertainties in the normalization of the power spectrum are taken into account.

Acknowledgments

We thank Pedro Viana and Andrew Liddle for useful conversations. PPA is funded by JNICT (Portugal) under the ‘Program PRAXIS XXI’ (PRAXIS XXI/BPD/9901/96). JHPW is funded by NSF KDI Grant (9872979) and NASA LTSA Grant (NAG5-6552). This work was performed on COSMOS, the Origin2000 owned by the UK Computational Cosmology Consortium, supported by Silicon Graphics/Cray Research, HEFCE and PPARC.