THE ASTROPHYSICAL JOURNAL, 491:L67–L71, 1997 December 20
© 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.
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Cosmology in a String-Dominated Universe

DAVID SPERGEL

Princeton University Observatory, Princeton, NJ 08544

AND

UE-LI PEN

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138

Received 1997 October 6; accepted 1997 October 17; published 1997 November 6


ABSTRACT

     The string-dominated universe is a flat universe that locally resembles an open universe and fits dynamical measures of power spectra, cluster abundances, redshift distortions, lensing constraints, luminosity, angular diameter distance relations, and microwave background observations. We show examples of networks that might give rise to recent string-domination when formed by a phase transition near the electro-weak symmetry breaking scale. We discuss how future observations can distinguish this model from other cosmologies.

Subject headings: cosmology: theory—large-scale structure of universe

CONTENTS

§1. INTRODUCTION

     Most theoretical cosmologists prefer flat universe models. While this preference was initially based on extensions of the Copernican principle (Dicke 1970), it has been strengthened by the theoretical successes of the inflationary universe paradigm (see Linde 1990 for discussion). While it is possible to construct inflationary models with Ω<1 (see, e.g., Linde & Mezhlumian 1994), these models are less aesthetically appealing than the flat universe models.

     Observations, however, suggest that the matter density of the universe is not sufficient to make Ω=1: measurements of the Hubble constant (Freedman, Madore, & Kennicutt 1997), and estimates of the age of the universe (Bolte & Hogan 1995) suggest that H$\mathstrut{_{0}}$t$\mathstrut{_{0}}$>2/3; measurements of the baryon-to-dark matter ratio in clusters, together with estimates of the baryon density from big bang nucleosynthesis imply Ω0, the energy density in matter, is much less than 1 (White et al. 1993); and the power spectrum of large-scale structure is best fitted by models with Ω$\mathstrut{_{0}}$h$\mathstrut{_{0}}$=0.25 (Peacock & Dodds 1994). Here h$\mathstrut{_{0}}$=H$\mathstrut{_{0}}$/(100 km s-1 Mpc-1). For several decades, it has been observed that the mass-to-light ratio in clusters of galaxies suggests Ω$\mathstrut{_{0}}$∼0.2 (Bahcall, Lubin, & Dorman 1995). The simplest COBE normalized parameter-free Harrison-Zeldovich-Peebles power spectrum slope n=1 predicts local peculiar velocities and cluster abundances in Ω$\mathstrut{_{0}}$=1 and Λ universes that are significantly higher than observed (Strauss & Willick 1995; Eke, Cole, & Frenk 1996; Pen 1996; Viana & Liddle 1996), which can be resolved by lowering the matter density Ω0.

     This contradiction has led cosmologists to consider exotic equations of state for the universe. The most studied modification of the standard matter-dominated cosmology is the vacuum-dominated model. While there is no particle physics motivation for positing a vacuum energy of 10$\mathstrut{^{-124}}$M$\mathstrut{^{4}_{{\rm Planck}}}$ (Weinberg 1997), the model does appear to be consistent with a number of observations (see, e.g., Ostriker & Steinhardt 1995; Krauss & Turner 1995; Bagla et al. 1996; Liddle et al. 1996, and references therein). However, recent measurements of q0 using distant supernovae (Perlmutter et al. 1996) and limits based on the statistics of gravitational lensing (Kochanek 1996) are encouraging cosmologists to consider alternative models. A novel technique of distance determination using cluster hydrostatic equilibrium measurements also indicates positive values of q0 (Pen 1997).

     A string-dominated cosmology is an intriguing alternative to the standard model. In this model, the energy density in strings scales with the expansion factor, a, as a-2, decaying faster than a vacuum energy term, but slower than the energy density in matter (which decays as a-3). In this model, strings form at near the electro-weak symmetry breaking scale. Unlike the much heavier GUT scale strings (see, e.g., Vilenkin & Shellard 1993), these light strings do not seed structure formation. Individual strings in this model have too low a density to be observable. A typical string density would be 10-5 kg m-1. However, their cumulative effect is to alter the expansion of the universe. Locally, they make a flat universe appear to have many of the properties of an open universe model. The energy density of such a string network arises naturally to be near the critical energy density today.

     If the universe today is string-dominated, then the strings must be produced near the electro-weak scale, a scale at which there must be new physics. These electro-weak strings are very light and would be undetected through their gravitational lensing since their bending angle is only (M/M$\mathstrut{_{{\rm Pl}}}$)$\mathstrut{^{2}}$∼10$\mathstrut{^{-32}}$ rad. Here M is the symmetry-breaking scale associated with string formation and MPl is the Planck scale. While these strings are light, they are expected to be numerous. The characteristic separation between strings is expected to be the bubble size during the phase transition, which in the case of the electro-weak phase transition is typically 10-3 of the horizon size (Moore & Prokopec 1995), 0.1 A.U. (comoving). Thus, there would be many light strings in our own solar system. If these light strings are associated with baryogenesis (Starkman & Vachaspati 1996) or are superconducting (Vilenkin 1989), then they may be directly detectable. For these noncommuting strings to be of cosmological interest, they must form at temperatures below 10 TeV or they will dominate the universe at too high a redshift.

     Only some cosmic string models lead to a string-dominated universe. In theories where cosmic strings can intercommute, their evolution obeys a "scaling solution": their energy density scales as a-3 during matter domination and as a-4 during radiation domination. In these theories, strings never dominate the energy density of the universe. On the other hand, if strings do not intercommute nor pass through each other, then the network can "freeze out" and the energy density in strings can scale as a-2 (Vilenkin 1984). Initial interest in string-dominated universes was spurred by the possibility that Abelian strings might not intercommute effectively and might dominate the energy density of the universe (Kibble 1976; Vilenkin 1984; Kardashev 1986). However, numerical simulations showed that even complicated Abelian string networks (Vachaspati & Vilenkin 1987) intercommuted effectively and rapidly approached the scaling solution. Despite the lack of a model that had nonintercommuting strings, the interesting astrophysical implications of a string-dominated universe led to a number of papers investigating their cosmological properties (Turner 1985; Charlton & Turner 1987; Gott & Rees 1987; Dabrowski & Stelmach 1989; Tomita & Watanabe 1990; Stelmach, Dabrowski, & Byrka 1994; Stelmach 1995) and the properties of cosmologies with similar equations of state (Steinhardt 1996; Coble, Dodelson, & Frieman 1997). We review some of these results in § 2 and compare string-dominated flat cosmologies to observations of large-scale structure, microwave background fluctuations, observations of rich clusters, and other cosmological probes. We show that a string-dominated universe with H$\mathstrut{_{0}}$∼60–65 km s-1 and Ω$\mathstrut{_{0}}$∼0.4–0.6 agrees remarkably well with a broad class of observations.

     In theories in which a non-Abelian symmetry is broken to a discrete subgroup, multiple types of cosmic strings can be produced (Mermin 1979). Topological constraints prevents these different types of strings from intercommuting (Toulouse 1977; Poenaru & Toulouse 1977), which led to the speculation that they could potentially dominate the energy density of the universe (Kibble 1980). There are a number of phenomenologically interesting particle physics models that utilize these non-Abelian symmetries (Chkareuli 1991; Dvali & Senjanovic 1994). These complex string networks are not merely flights of theoretical fantasy: they can be seen in biaxial nematic liquid crystals (De'Neve, Kleman, & Navard 1992).

§2. ASTROPHYSICS OF STRING-DOMINATED UNIVERSE

§2.1. Expansion of the Universe: H$\mathstrut{_{0}}$, q$\mathstrut{_{0}}$, Ω$\mathstrut{_{0}}$, and t$\mathstrut{_{0}}$

     Noncommuting strings formed at low energies have only one basic effect on the universe: they add an additional term to the Friedmann equation that governs the expansion of the universe,



where H is the Hubble rate, a is the expansion factor, and ρm0, ρr0, and ρs0 are the current energy density in matter, radiation, and strings, respectively. Since this additional term has the same a dependence as the presence of space curvature, a string-dominated flat universe is observationally similar to matter-dominated open universe. Since we are focusing on a flat universe, we let Ω$\mathstrut{_{0}}$≡8πGρ$\mathstrut{_{m0}}$/3H$\mathstrut{^{2}_{0}}$, Ω$\mathstrut{_{s}}$≡8πGρ$\mathstrut{_{s0}}$/3H$\mathstrut{^{2}_{0}}$, Ω$\mathstrut{_{r}}$≡8πGρ$\mathstrut{_{r0}}$/3H$\mathstrut{^{2}_{0}}$, and assume Ω$\mathstrut{_{0}}$+Ω$\mathstrut{_{r}}$+Ω$\mathstrut{_{s}}$=1.

     As in a curvature-dominated universe, we can divide the history of the universe into three epochs: a radiation-dominated epoch, a matter-dominated epoch, and a string-dominated epoch. The relationship between the age of the universe, t0, the energy density in matter, and the Hubble constant is the same as in a curvature-dominated universe (see, e.g., Kolb & Turner 1990). We also recover the familiar relationship for the deceleration parameter, q$\mathstrut{_{0}}$=Ω$\mathstrut{_{0}}$/2.

     Thus, the string-dominated cosmology makes the same predictions for most of the classical cosmological tests as the open universe model.

     Because the curvature of the string-dominated universe is flat, its angular diameter-redshift relationship differs from an open universe. In a string-dominated flat universe, the angular diameter distance is



where



This altered angular diameter distance affects number count predictions, the probability of gravitational lensing, and the predictions for microwave background fluctuations.

     The statistics of gravitational lensing in this model differs significantly from the predicted statistics in a vacuum-dominated model. Current observations already place strong constraints on the vacuum-dominated model, which predicts too many small lens events, particularly with small angular separation (Turner 1990; Kochanek 1996). The absence of large number of lenses in the HST snapshot survey (Maoz et al. 1993) and in radio surveys implies that Ω$\mathstrut{_{{\Lambda}}}$<0.6 and rules out most of the interesting parameter space for vacuum-dominated models. Because of the very different relation between redshift and distance in string-dominated models, it predicts many fewer gravitational lenses than the vacuum-dominated models. A recent analysis by Bloomfield-Torres & Waga (1996) finds that string-dominated flat universes are excellent fits to the observed lens statistics in the HST snapshot survey.

     Observations of supernova at high redshift are another powerful probe of cosmology. Perlmutter et al. (1996) have already been able to rule out cosmological constant models with Ω$\mathstrut{_{0}}$<0.6 at the 95% confidence level with their supernova data. Thus, there are no cosmological constant models compatible with this observation, measurements of large-scale structure, measurements of the Hubble constant, and the constraint that the age of the universe exceed 11 Gyr. At the redshifts probed by the supernova study, the distance redshift relation in a string-dominated universe is close to, but not identical to, an open universe. Using the relations given in equation (2), the Perlmutter et al. (1996) observations imply that Ω$\mathstrut{_{0}}$>0.15 in a flat string-dominated cosmology.

§2.2. Microwave Background Fluctuations

     Because the strings only make significant contributions to the energy density of the universe at very late times, they have no effect on the physics at the surface of last scatter. However, since the strings alter the expansion rate of the universe, they have two effects on the detailed shape of the microwave background spectrum: (1) the decay of potential fluctuations at late times produces additional fluctuations on large angular scales; and (2) since the conformal distance to the surface of last scatter is smaller, the Doppler peaks are shifted to larger angular scales (Stelmach et al. 1994). Since identical effects occur in a vacuum-dominated universe, it will be difficult to distinguish a string-dominated universe from a vacuum-dominated universe based on CMB observations. On the other hand, it will be very easy to distinguish a flat cosmology from an open universe because of the large differences in the angular diameter-distance relation (Kamionkowski & Spergel 1994).

     We have calculated the predicted CMB spectrum in a string-dominated universe using a modified version of a Boltzmann code developed by Seljak & Zaldarriaga (1996). Figure 1 shows the predicted multipole spectrum for various string-dominated cosmologies. While the three spectra cannot yet be distinguished by current observations, future CMB maps should easily be able to distinguish between the curves in Figure 1.

Fig. 1

     Most observations of large-scale structure are effectively measurements of the galaxy power spectrum. Peacock & Dodds (1994) have shown that most galaxy surveys are consistent with a standard CDM power spectrum with Γ≡Ω$\mathstrut{_{0}}$h$\mathstrut{{\rm exp}}$(-Ω$\mathstrut{_{b}}$-Ω$\mathstrut{_{b}}$/Ω$\mathstrut{_{0}}$)=0.25±0.05. The Las Campanas redshift survey is also compatible with a power spectrum with Γ=0.2–0.3 (Lin et al. 1996). Figure 2 shows that this constraint alone is sufficient to rule out much of parameter space. We use the CMB spectrum to normalize the standard inflationary model (scale-invariant, Ω$\mathstrut{_{b}}$h$\mathstrut{^{2}}$=0.0125) in this cosmology to the COBE observations. Once this normalization is fixed, there are no free parameters left in the model, so that it can be compared directly to observations of matter power spectrum.

Fig. 2

     Observations of clusters are powerful probes of the matter power spectrum. Gravitational lensing observations, X-ray observations of hot gas, and studies of galaxy kinematics in clusters, all probe the velocity distribution in clusters. Thus, they can constrain the distribution of mass rather than the distribution of light. A number of studies (Eke et al. 1996; Viana & Liddle 1996; Pen 1996) have concluded that these cluster observations place very strong constraints on the amplitude of mass fluctuations on the 8 h-1 Mpc scale: σ$\mathstrut{_{8}}$=0.6±0.1Ω$\mathstrut{^{-0.6}_{0}}$. Roughly speaking, COBE normalized CDM implies σ$\mathstrut{_{8}}$∼2.4hΩ$\mathstrut{_{0}}$. The details of the transfer function are incorporated in Figure 2, which shows that most string-dominated models fit all the constraints.

§2.3. String Dynamics

     To simplify the simulation while capturing the essentials of a wide range of non-Abelian string dynamics, we chose a modification of the nonlinear σ model from Pen, Spergel, & Turok (1995, hereafter PST). In this model, we have a classical field &phis; defined at every lattice point $\mathstrut{{\bmi x}}$, which takes on values in the range [0,π)×Z$\mathstrut{_{N}}$. The field has both a continuous component, and a discrete index in the range 0, ..., N-1. The multiple leaves of semicircles are to be thought of as a rolodex filer: Whenever we examine the dynamics of two leaves, we open the system such that the two leaves form a full circle, and treat the dynamics as lying in a unit circle. Since the evolution equation only requires the pairwise force, any two points always lie on some such circle.

     For N=2, we recover the standard global strings from PST. For N=3, a system very similar to the biaxial nematics and the Z3 string system is obtained, with three different types of strings and three point vertices that annihilate pairwise. Two strings get stuck when they try to pass through each other, just like the biaxial nematic liquid crystals. We show such a network in Figure 3, where we have represented the string corresponding to each of the three generators by a different color. Each of the three semicircle leaves are either red, green, or blue. Since each string contains a complete rotation that covers two leaves, the strings appear as composite colors, green+red=yellow, etc. There are three such possible pairs.

Fig. 3

     In general, we have a system of N(N-1)/2 strings. Strings join at three-point vertices, of which there are N(N-1)(N-2)/6 different types. When two vertices join, there is a one in 3/N chance that they can annihilate and result in two disconnected strings. Otherwise, the two vertices can pass through one another and result in a new configuration that still contains the same number of vertices. We can now vary the number of string generators N to correspond to a one parameter class of non-Abelian strings. We expect strings to become more strongly tangled as we increase N. This is indeed observed, as shown in Figure 4. The global field dynamics differs systematically from gauged strings in the fact that global strings exert long-range forces on each other that can cause the network to move even when the configuration is neutrally stable.

Fig. 4

     We have found that for N=3 strings, the solution scales much like the Z3 monopole-string network, which would suggest that the biaxial nematic liquid crystal system would also exhibit scaling behavior (Pen & Spergel 1998). We also see that the network does seem to stop disentangling for large N. While no current model of the electro-weak phase transition predicts cosmic strings, electro-weak baryogenesis calculations have argued for the presence of more complicated symmetry structures. It would be conceivable that both the baryon asymmetry and the present-day vacuum energy be caused by the electro-weak symmetry breaking, which may have testable laboratory consequences in the near future.

§3. CONCLUSIONS

     String-dominated cosmologies have a number of very attractive features. For Ω$\mathstrut{_{0}}$∼0.4–0.6 and H$\mathstrut{_{0}}$∼60–70 km s-1 Mpc-1, the model is consistent with current observations. It fits observations of the CMB, measurements of the shape of galaxy power spectrum, and measurements of the amplitude of the mass power spectrum, and it is compatible with age limits. Unlike cosmological constant models, string-dominated cosmology is also consistent with observations of high-redshift supernova and gravitational lensing statistics. They suppose the introduction of new physics at the TeV scale, where unitarity arguments in the standard model require new physics (Wilzcek 1997).

     The observational predictions of the string-dominated model are intermediate between the open universe model and the vacuum energy (cosmological constant) model. CMB observations can easily distinguish between an open universe model and the flat universe models (string-dominated, matter-dominated, or vacuum energy-dominated): the open universe model predicts that the Doppler peak should occur at l∼220Ω$\mathstrut{^{-1{/}2}}$. Low-redshift measurements can distinguish between the matter-, vacuum energy-, and string-dominated models: the string-dominated model predicts q$\mathstrut{_{0}}$=Ω$\mathstrut{_{0}}$/2, while the vacuum-dominated model predicts q$\mathstrut{_{0}}$=3Ω$\mathstrut{_{0}}$/2-1. Thus, future observations should be able to determine the equation of state of the universe.

REFERENCES

FIGURES


Full image (47kb) | Discussion in text

     FIG. 1.—Predicted multipole spectrum for three different models: a flat standard CDM model with Ω$\mathstrut{_{0}}$=1.0 and H$\mathstrut{_{0}}$=50 km s-1 Mpc-1 (solid line) and a string-dominated flat cosmology with Ω$\mathstrut{_{0}}$=0.4 and Ω$\mathstrut{_{s}}$=0.6.


Full image (269kb) | Discussion in text

     FIG. 2.—Constraints from various astrophysical measurements. The vertically shaded region lies outside the best determinations of the Hubble constant, H$\mathstrut{_{0}}$=73±6±8 km s-1 Mpc-1 (Freedman, Madore, & Kennicutt 1997); the horizontally shaded regions do not agree with measurements of the shape of the galaxy power spectrum, Γ=0.25±0.05 (Peacock & Dodds 1994; dashed line) and with measurements of the fluctuation amplitude from clusters, σ$\mathstrut{_{8}}$Ω$\mathstrut{^{0.6}_{0}}$=0.6±0.1 (Eke et al. 1996; Viana & Liddle 1996; Pen 1996; solid line); and the region shaded with 45° lines corresponds to cosmic ages less than 11 Gyr. For all lines, Ω$\mathstrut{_{b}}$h$\mathstrut{^{2}}$=0.0125.


Full image (229kb) | Discussion in text

     FIG. 3.—The network of a N=3 string system, which exhibits dynamics very similar to biaxial nematic liquid crystals. The strings are color coded according to the generator they belong to.


Full image (40kb) | Discussion in text

     FIG. 4.—Evolution of the string density (normalized to the scaling density) as a function of time in two different models. In scaling solutions, the string density should asymptote to a constant value in this plot. Note that the large N models do not scale and become tangled. In the Abelian N=2 case, they disappear completely once the horizon size exceeds the box size (in this case 3003).

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