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Supersymmetry and cosmology

Jonathan L. Feng^,

Department of Physics and Astronomy, University of California, Irvine, CA 92697, United States

Received 23 February 2004;  accepted 29 September 2004.  Available online 18 December 2004.

Abstract

Cosmology now provides unambiguous, quantitative evidence for new particle physics. I discuss the implications of cosmology for supersymmetry and vice versa. Topics include: motivations for supersymmetry; supersymmetry breaking; dark energy; freeze out and WIMPs; neutralino dark matter; cosmologically preferred regions of minimal supergravity; direct and indirect detection of neutralinos; the DAMA and HEAT signals; inflation and reheating; gravitino dark matter; Big Bang nucleosynthesis; and the cosmic microwave background. I conclude with speculations about the prospects for a microscopic description of the dark universe, stressing the necessity of diverse experiments on both sides of the particle physics/cosmology interface.

Article Outline

1. Introduction

2. Supersymmetry essentials

2.1. A new spacetime symmetry

2.2. Supersymmetry and the weak scale

2.3. The neutral supersymmetric spectrum

2.4. R-Parity

2.5. Supersymmetry breaking and dark energy

2.6. Minimal supergravity

2.7. Summary

3. Neutralino cosmology

3.1. Freeze out and WIMPs

3.2. Thermal relic density

3.2.1. Bulk region

3.2.2. Focus point region

3.2.3. A funnel region

3.2.4. Co-annihilation region

3.3. Direct detection

3.4. Indirect detection

3.4.1. Positrons

3.4.2. Photons

3.4.3. Neutrinos

3.5. Summary

4. Gravitino cosmology

4.1. Gravitino properties

4.2. Thermal relic density

4.3. Production during reheating

4.4. Production from late decays

4.5. Detection

4.5.1. Energy release

4.5.2. Big Bang nucleosynthesis

4.5.3. The cosmic microwave background

4.6. Summary

5. Prospects

5.1. The particle physics/cosmology interface

5.2. The role of colliders

5.3. Synthesis

5.4. Summary

Acknowledgements

References

1. Introduction

Not long ago, particle physicists could often be heard bemoaning the lack of unambiguous, quantitative evidence for physics beyond their standard model. Those days are gone. Although the standard model of particle physics remains one of the great triumphs of modern science, it now appears that it fails at even the most basic level—providing a reasonably complete catalog of the building blocks of our universe.

Recent cosmological measurements have pinned down the amount of baryon, matter, and dark energy in the universe [1] and [2]. In units of the critical density, these energy densities are

implying a non-baryonic dark matter component with

where h   0.71 is the normalized Hubble expansion rate. Both the central values and uncertainties were nearly unthinkable even just a few years ago. These measurements are clear and surprisingly precise evidence that the known particles make up only a small fraction of the total energy density of the universe. Cosmology now provides overwhelming evidence for new particle physics.

At the same time, the microscopic properties of dark matter and dark energy are remarkably unconstrained by cosmological and astrophysical observations. Theoretical insights from particle physics are therefore required, both to suggest candidates for dark matter and dark energy and to identify experiments and observations that may confirm or exclude these speculations.

Weak-scale supersymmetry is at present the most well-motivated framework for new particle physics. Its particle physics motivations are numerous and are reviewed in Section 2. More than that, it naturally provides dark matter candidates with approximately the right relic density. This fact provides a strong, fundamental, and completely independent motivation for supersymmetric theories. For these reasons, the implications of supersymmetry for cosmology, and vice versa, merit serious consideration.

Many topics lie at the interface of particle physics and cosmology, and supersymmetry has something to say about nearly every one of them. Regrettably, spacetime constraints preclude detailed discussion of many of these topics. Although the discussion below will touch on a variety of subjects, it will focus on dark matter, where the connections between supersymmetry and cosmology are concrete and rich, the above-mentioned quantitative evidence is especially tantalizing, and the role of experiments is clear and promising.

Weak-scale supersymmetry is briefly reviewed in Section 2 with a focus on aspects most relevant to astrophysics and cosmology. In Sections 3 and 4 the possible roles of neutralinos and gravitinos in the early universe are described. As will be seen, their cosmological and astrophysical implications are very different; together they illustrate the wealth of possibilities in supersymmetric cosmology. I conclude in Section 5 with speculations about the future prospects for a microscopic understanding of the dark universe.

2. Supersymmetry essentials

2.1. A new spacetime symmetry

Supersymmetry is an extension of the known spacetime symmetries [3]. The spacetime symmetries of rotations, boosts, and translations are generated by angular momentum operators L_i, boost operators K_i, and momentum operators P_μ, respectively. The L and K generators form Lorentz symmetry, and all 10 generators together form Poincare symmetry. Supersymmetry is the symmetry that results when these 10 generators are further supplemented by fermionic operators Q_α. It emerges naturally in string theory and, in a sense that may be made precise [4], is the maximal possible extension of Poincare symmetry.

If a symmetry exists in nature, acting on a physical state with any generator of the symmetry gives another physical state. For example, acting on an electron with a momentum operator produces another physical state, namely, an electron translated in space or time. Spacetime symmetries leave the quantum numbers of the state invariant—in this example, the initial and final states have the same mass, electric charge, etc.

In an exactly supersymmetric world, then, acting on any physical state with the supersymmetry generator Q_α produces another physical state. As with the other spacetime generators, Q_α does not change the mass, electric charge, and other quantum numbers of the physical state. In contrast to the Poincare generators, however, a supersymmetric transformation changes bosons to fermions and vice versa. The basic prediction of supersymmetry is, then, that for every known particle there is another particle, its superpartner, with spin differing by 1/2.

One may show that no particle of the standard model is the superpartner of another. Supersymmetry therefore predicts a plethora of superpartners, none of which has been discovered. Mass degenerate superpartners cannot exist—they would have been discovered long ago—and so supersymmetry cannot be an exact symmetry. The only viable supersymmetric theories are therefore those with non-degenerate superpartners. This may be achieved by introducing supersymmetry-breaking contributions to superpartner masses to lift them beyond current search limits. At first sight, this would appear to be a drastic step that considerably detracts from the appeal of supersymmetry. It turns out, however, that the main virtues of supersymmetry are preserved even if such mass terms are introduced. In addition, the possibility of supersymmetric dark matter emerges naturally and beautifully in theories with broken supersymmetry.

2.2. Supersymmetry and the weak scale

Once supersymmetry is broken, the mass scale for superpartners is unconstrained. There is, however, a strong motivation for this scale to be the weak scale: the gauge hierarchy problem. In the standard model of particle physics, the classical mass of the Higgs boson receives quantum corrections (see Fig. 1). Including quantum corrections from standard model fermions f_L and f_R, one finds that the physical Higgs boson mass is

where the last term is the leading quantum correction, with λ the Higgs-fermion coupling. Λ is the ultraviolet cutoff of the loop integral, presumably some high scale well above the weak scale. If Λ is of the order of the Planck scale 10¹⁹ GeV, the classical Higgs mass and its quantum correction must cancel to an unbelievable 1 part in 10³⁴ to produce the required weak-scale m_h. This unnatural fine-tuning is the gauge hierarchy problem.

Fig. 1. Contributions to the Higgs boson mass in the standard model and in supersymmetry.

In the supersymmetric standard model, however, for every quantum correction with standard model fermions f_L and f_R in the loop, there are corresponding quantum corrections with superpartners and . The physical Higgs mass then becomes

where the terms quadratic in Λ cancel, leaving a term logarithmic in Λ as the leading contribution. In this case, the quantum corrections are reasonable even for very large Λ, and no fine-tuning is required.

In the case of exact supersymmetry, where , even the logarithmically divergent term vanishes. In fact, quantum corrections to masses vanish to all orders in perturbation theory, an example of powerful non-renormalization theorems in supersymmetry. From Eq. (6), however, we see that exact mass degeneracy is not required to solve the gauge hierarchy problem. What is required is that the dimensionless couplings λ of standard model particles and their superpartners are identical, and that the superpartner masses be not too far above the weak scale (or else even the logarithmically divergent term would be large compared to the weak scale, requiring another fine-tuned cancellation). This can be achieved simply by adding supersymmetry-breaking weak-scale masses for superpartners. In fact, other terms, such as some cubic scalar couplings, may also be added without re-introducing the fine-tuning. All such terms are called “soft,” and the theory with weak-scale soft supersymmetry-breaking terms is “weak-scale supersymmetry.”

2.3. The neutral supersymmetric spectrum

Supersymmetric particles that are electrically neutral, and so promising dark matter candidates, are shown with their standard model partners in Fig. 2. In supersymmetric models, two Higgs doublets are required to give mass to all fermions. The two neutral Higgs bosons are H_d and H_u, which give mass to the down-type and up-type fermions, respectively, and each of these has a superpartner. Aside from this subtlety, the superpartner spectrum is exactly as one would expect. It consists of spin 0 sneutrinos, one for each neutrino, the spin 3/2 gravitino, and the spin 1/2 Bino, neutral Wino, and down- and up-type Higgsinos. These states have masses determined (in part) by the corresponding mass parameters listed in the top row of Fig. 2. These parameters are unknown, but are presumably of the order of the weak scale, given the motivations described above.

Fig. 2. Neutral particles in the supersymmetric spectrum. M₁, M₂, μ, , and m_3/2 are unknown weak-scale mass parameters. The Bino, Wino, and down- and up-type Higgsinos mix to form neutralinos.

The gravitino is a mass eigenstate with mass m_3/2. The sneutrinos are also mass eigenstates, assuming flavor and R-parity conservation (see Section 2.4). The spin 1/2 states are differentiated only by their electroweak quantum numbers. After electroweak symmetry breaking, these gauge eigenstates therefore mix to form mass eigenstates. In the basis the mixing matrix is

where c_W≡cosθ_W, s_W≡sinθ_W, and β is another unknown parameter defined by tanβ≡ H_u / H_d , the ratio of the up-type to down-type Higgs scalar vacuum expectation values (vevs). The mass eigenstates are called neutralinos and denoted χ ≡ χ₁, χ₂, χ₃, χ₄ , in order of increasing mass. If M₁   M₂, |μ|, the lightest neutralino χ has a mass of approximately M₁ and is nearly a pure Bino. However, for M₁   M₂   |μ|, χ is a mixture with significant components of each gauge eigenstate.