We show that cosmic acceleration can arise due to very tiny corrections to the usual gravitationa... more We show that cosmic acceleration can arise due to very tiny corrections to the usual gravitational action of general relativity of the form R n , with n < 0. This eliminates the need for dark energy, though it does not address the cosmological constant problem. Since a modification to the Einstein-Hilbert action of the form R n , with n > 0, can lead to early-time inflation, our proposal provides a unified and purely gravitational origin for the early and late time accelerating phases of the Universe.
A nearly-massless, slowly-rolling scalar field φ may provide most of the energy density of the cu... more A nearly-massless, slowly-rolling scalar field φ may provide most of the energy density of the current universe. One potential difficulty with this idea is that couplings to ordinary matter, even if suppressed by the Planck scale, should lead to observable long-range forces and time dependence of the constants of nature. I explore the possibility that an approximate global symmetry serves to suppress such couplings even further. Such a symmetry would allow a coupling of φ to the pseudoscalar F µν F µν of electromagnetism, which would rotate the polarization state of radiation from distant sources. This effect is fairly well constrained, but it is conceivable that future improvements could lead to a detection of a cosmological scalar field.
Measurements of the polarization angle and orientation of cosmological radio sources may be used ... more Measurements of the polarization angle and orientation of cosmological radio sources may be used to search for unusual effects in the propagation of light through the universe. Recently, Nodland and Ralston have claimed to find evidence for a redshiftand direction-dependent rotation effect in existing data. We re-examine these data and argue that there is no statistically significant signal present. We are able to place stringent limits on hypothetical chiral interactions of photons propagating through spacetime.
The universe appears to be accelerating, but the reason why is a complete mystery. The simplest e... more The universe appears to be accelerating, but the reason why is a complete mystery. The simplest explanation, a small vacuum energy (cosmological constant), raises three difficult issues: why the vacuum energy is so small, why it is not quite zero, and why it is comparable to the matter density today. I discuss these mysteries, some of their possible resolutions, and some issues confronting future observations.
We propose a class of field theories featuring solitonic solutions in which topological defects c... more We propose a class of field theories featuring solitonic solutions in which topological defects can end when they intersect other defects of equal or higher dimensionality. Such configurations may be termed "Dirichlet topological defects", in analogy with the D-branes of string theory. Our discussion focuses on defects in scalar field theories with either gauge or global symmetries, in (3+1) dimensions; the types of defects considered include walls ending on walls, strings on walls, and strings on strings.
We suggest that spontaneous eternal inflation can provide a natural explanation for the thermodyn... more We suggest that spontaneous eternal inflation can provide a natural explanation for the thermodynamic arrow of time, and discuss the underlying assumptions and consequences of this view. In the absence of inflation, we argue that systems coupled to gravity usually evolve asymptotically to the vacuum, which is the only natural state in a thermodynamic sense. In the presence of a small positive vacuum energy and an appropriate inflaton field, the de Sitter vacuum is unstable to the spontaneous onset of inflation at a higher energy scale. Starting from de Sitter, inflation can increase the total entropy of the universe without bound, creating universes similar to ours in the process. An important consequence of this picture is that inflation occurs asymptotically both forwards and backwards in time, implying a universe that is (statistically) time-symmetric on ultra-large scales.
We present an exact solution for a factorizable brane-world spacetime with two extra dimensions a... more We present an exact solution for a factorizable brane-world spacetime with two extra dimensions and explicit brane sources. The compactification manifold has the topology of a two-sphere, and is stabilized by a bulk cosmological constant and magnetic flux. The geometry of the sphere is locally round except for conical singularities at the locations of two antipodal branes, deforming the sphere into an American-style football. The bulk magnetic flux needs to be fine-tuned to obtain flat geometry on the branes. Once this is done, the brane geometry is insensitive to the brane vacuum energy, which only affects the conical deficit angle of the extra dimensions. Solutions of this form provide a new arena in which to explore brane-world phenomenology and the effects of extra dimensions on the cosmological constant problem. *
Rotational invariance is a well-established feature of low-energy physics. Violations of this sym... more Rotational invariance is a well-established feature of low-energy physics. Violations of this symmetry must be extremely small today, but could have been larger in earlier epochs. In this paper we examine the consequences of a small breaking of rotational invariance during the inflationary era when the primordial density fluctuations were generated. Assuming that a fixed-norm vector picked out a preferred direction during the inflationary era, we explore the imprint it would leave on the cosmic microwave background anisotropy, and provide explicit formulas for the expected amplitudes a lm a * l ′ m ′ of the spherical-harmonic coefficients. We suggest that it is natural to expect that the imprint on the primordial power spectrum of a preferred spatial direction is approximately scale-invariant, and examine a simple model in which this is true.
If our universe underwent inflation, its entropy during the inflationary phase was substantially ... more If our universe underwent inflation, its entropy during the inflationary phase was substantially lower than it is today. Because a low-entropy state is less likely to be chosen randomly than a high-entropy one, inflation is unlikely to arise through randomlychosen initial conditions. To resolve this puzzle, we examine the notion of a natural state for the universe, and argue that it is a nearly-empty spacetime. If empty space has a small vacuum energy, however, inflation can begin spontaneously in this background. This scenario explains why a universe like ours is likely to have begun via a period of inflation, and also provides an origin for the cosmological arrow of time.
It goes without saying that we are stuck with the universe we have. Nevertheless, we would like t... more It goes without saying that we are stuck with the universe we have. Nevertheless, we would like to go beyond simply describing our observed universe, and try to understand why it is that way rather than some other way. Physicists and cosmologists have been exploring increasingly ambitious ideas that attempt to explain why certain features of our universe aren't as surprising as they might first appear.
This is a review of the physics and cosmology of the cosmological constant. Focusing on recent de... more This is a review of the physics and cosmology of the cosmological constant. Focusing on recent developments, I present a pedagogical overview of cosmology in the presence of a cosmological constant, observational constraints on its magnitude, and the physics of a small (and potentially nonzero) vacuum energy.
A brief review is offered of the theoretical background concerning dark energy: what is required ... more A brief review is offered of the theoretical background concerning dark energy: what is required by observations, what sort of models are being considered, and how they fit into particle physics and gravitation. Contribution to the SNAP (SuperNova Acceleration Probe) Yellow Book.
We propose a simple model in which the cosmological dark matter consists of particles whose mass ... more We propose a simple model in which the cosmological dark matter consists of particles whose mass increases with the scale factor of the universe. The particle mass is generated by the expectation value of a scalar field which does not have a stable vacuum state, but which is effectively stabilized by the rest energy of the ambient particles. As the universe expands, the density of particles decreases, leading to an increase in the vacuum expectation value of the scalar (and hence the mass of the particle). The energy density of the coupled system of variable-mass particles ("vamps") redshifts more slowly than that of ordinary matter. Consequently, the age of the universe is larger than in conventional scenarios.
We consider the gravitational effects of a single, fixed-norm, Lorentz-violating timelike vector ... more We consider the gravitational effects of a single, fixed-norm, Lorentz-violating timelike vector field. In a cosmological background, such a vector field acts to rescale the effective value of Newton's constant. The energy density of this vector field precisely tracks the energy density of the rest of the universe, but with the opposite sign, so that the universe experiences a slower rate of expansion for a given matter content. This vector field similarly rescales Newton's constant in the Newtonian limit, although by a different factor. We put constraints on the parameters of the theory using the predictions of primordial nucleosynthesis, demonstrating that the norm of the vector field should be less than the Planck scale by an order of magnitude or more.
If spacetime possesses extra dimensions of size and curvature radii much larger than the Planck o... more If spacetime possesses extra dimensions of size and curvature radii much larger than the Planck or string scales, the dynamics of these extra dimensions should be governed by classical general relativity. We argue that in general relativity, it is nontrivial to obtain solutions where the extra dimensions are static and are dynamically stable to small perturbations. We also illustrate that intuition on equilibrium and stability built up from non-gravitational physics can be highly misleading. For all static, homogeneous solutions satisfying the null energy condition, we show that the Ricci curvature of space must be nonnegative in all directions. Much of our analysis focuses on a class of spacetime models where space consists of a product of homogeneous and isotropic geometries. A dimensional reduction of these models is performed, and their stability to perturbations that preserve the spatial symmetries is analyzed. We conclude that the only physically realistic examples of classically stabilized large extra dimensions are those in which the extra-dimensional manifold is positively curved. *
These notes provide a brief introduction to modern cosmology, focusing primarily on theoretical i... more These notes provide a brief introduction to modern cosmology, focusing primarily on theoretical issues. Some attention is paid to aspects of potential interest to students of string theory, on both sides of the two-way street of cosmological constraints on string theory and stringy contributions to cosmology. Slightly updated version of lectures at the 1999 Theoretical Advanced Study Institute at the University of Colorado, Boulder.
Models of dark energy are conveniently characterized by the equation-of-state parameter w = p/ρ, ... more Models of dark energy are conveniently characterized by the equation-of-state parameter w = p/ρ, where ρ is the energy density and p is the pressure. Imposing the Dominant Energy Condition, which guarantees stability of the theory, implies that w ≥ −1. Nevertheless, it is conceivable that a well-defined model could (perhaps temporarily) have w < −1 , and indeed such models have been proposed. We study the stability of dynamical models exhibiting w < −1 by virtue of a negative kinetic term. Although naively unstable, we explore the possibility that these models might be phenomenologically viable if thought of as effective field theories valid only up to a certain momentum cutoff. Under our most optimistic assumptions, we argue that the instability timescale can be greater than the age of the universe, but only if the cutoff is at or below 100 MeV. We conclude that it is difficult, although not necessarily impossible, to construct viable models of dark energy with w < −1; observers should keep an open mind, but the burden is on theorists to demonstrate that any proposed new models are not ruled out by rapid vacuum decay.
We study N = 1 SUSY theories in four dimensions with multiple discrete vacua, which admit soliton... more We study N = 1 SUSY theories in four dimensions with multiple discrete vacua, which admit solitonic solutions describing segments of domain walls meeting at one-dimensional junctions. We show that there exist solutions preserving one quarter of the underlying supersymmetry -a single Hermitian supercharge. We derive a BPS bound for the masses of these solutions and construct a solution explicitly in a special case. The relevance to the confining phase of N = 1 SUSY Yang-Mills and the M-theory/SYM relationship is discussed.
We show that cosmic acceleration can arise due to very tiny corrections to the usual gravitationa... more We show that cosmic acceleration can arise due to very tiny corrections to the usual gravitational action of general relativity of the form R n , with n < 0. This eliminates the need for dark energy, though it does not address the cosmological constant problem. Since a modification to the Einstein-Hilbert action of the form R n , with n > 0, can lead to early-time inflation, our proposal provides a unified and purely gravitational origin for the early and late time accelerating phases of the Universe.
A nearly-massless, slowly-rolling scalar field φ may provide most of the energy density of the cu... more A nearly-massless, slowly-rolling scalar field φ may provide most of the energy density of the current universe. One potential difficulty with this idea is that couplings to ordinary matter, even if suppressed by the Planck scale, should lead to observable long-range forces and time dependence of the constants of nature. I explore the possibility that an approximate global symmetry serves to suppress such couplings even further. Such a symmetry would allow a coupling of φ to the pseudoscalar F µν F µν of electromagnetism, which would rotate the polarization state of radiation from distant sources. This effect is fairly well constrained, but it is conceivable that future improvements could lead to a detection of a cosmological scalar field.
Measurements of the polarization angle and orientation of cosmological radio sources may be used ... more Measurements of the polarization angle and orientation of cosmological radio sources may be used to search for unusual effects in the propagation of light through the universe. Recently, Nodland and Ralston have claimed to find evidence for a redshiftand direction-dependent rotation effect in existing data. We re-examine these data and argue that there is no statistically significant signal present. We are able to place stringent limits on hypothetical chiral interactions of photons propagating through spacetime.
The universe appears to be accelerating, but the reason why is a complete mystery. The simplest e... more The universe appears to be accelerating, but the reason why is a complete mystery. The simplest explanation, a small vacuum energy (cosmological constant), raises three difficult issues: why the vacuum energy is so small, why it is not quite zero, and why it is comparable to the matter density today. I discuss these mysteries, some of their possible resolutions, and some issues confronting future observations.
We propose a class of field theories featuring solitonic solutions in which topological defects c... more We propose a class of field theories featuring solitonic solutions in which topological defects can end when they intersect other defects of equal or higher dimensionality. Such configurations may be termed "Dirichlet topological defects", in analogy with the D-branes of string theory. Our discussion focuses on defects in scalar field theories with either gauge or global symmetries, in (3+1) dimensions; the types of defects considered include walls ending on walls, strings on walls, and strings on strings.
We suggest that spontaneous eternal inflation can provide a natural explanation for the thermodyn... more We suggest that spontaneous eternal inflation can provide a natural explanation for the thermodynamic arrow of time, and discuss the underlying assumptions and consequences of this view. In the absence of inflation, we argue that systems coupled to gravity usually evolve asymptotically to the vacuum, which is the only natural state in a thermodynamic sense. In the presence of a small positive vacuum energy and an appropriate inflaton field, the de Sitter vacuum is unstable to the spontaneous onset of inflation at a higher energy scale. Starting from de Sitter, inflation can increase the total entropy of the universe without bound, creating universes similar to ours in the process. An important consequence of this picture is that inflation occurs asymptotically both forwards and backwards in time, implying a universe that is (statistically) time-symmetric on ultra-large scales.
We present an exact solution for a factorizable brane-world spacetime with two extra dimensions a... more We present an exact solution for a factorizable brane-world spacetime with two extra dimensions and explicit brane sources. The compactification manifold has the topology of a two-sphere, and is stabilized by a bulk cosmological constant and magnetic flux. The geometry of the sphere is locally round except for conical singularities at the locations of two antipodal branes, deforming the sphere into an American-style football. The bulk magnetic flux needs to be fine-tuned to obtain flat geometry on the branes. Once this is done, the brane geometry is insensitive to the brane vacuum energy, which only affects the conical deficit angle of the extra dimensions. Solutions of this form provide a new arena in which to explore brane-world phenomenology and the effects of extra dimensions on the cosmological constant problem. *
Rotational invariance is a well-established feature of low-energy physics. Violations of this sym... more Rotational invariance is a well-established feature of low-energy physics. Violations of this symmetry must be extremely small today, but could have been larger in earlier epochs. In this paper we examine the consequences of a small breaking of rotational invariance during the inflationary era when the primordial density fluctuations were generated. Assuming that a fixed-norm vector picked out a preferred direction during the inflationary era, we explore the imprint it would leave on the cosmic microwave background anisotropy, and provide explicit formulas for the expected amplitudes a lm a * l ′ m ′ of the spherical-harmonic coefficients. We suggest that it is natural to expect that the imprint on the primordial power spectrum of a preferred spatial direction is approximately scale-invariant, and examine a simple model in which this is true.
If our universe underwent inflation, its entropy during the inflationary phase was substantially ... more If our universe underwent inflation, its entropy during the inflationary phase was substantially lower than it is today. Because a low-entropy state is less likely to be chosen randomly than a high-entropy one, inflation is unlikely to arise through randomlychosen initial conditions. To resolve this puzzle, we examine the notion of a natural state for the universe, and argue that it is a nearly-empty spacetime. If empty space has a small vacuum energy, however, inflation can begin spontaneously in this background. This scenario explains why a universe like ours is likely to have begun via a period of inflation, and also provides an origin for the cosmological arrow of time.
It goes without saying that we are stuck with the universe we have. Nevertheless, we would like t... more It goes without saying that we are stuck with the universe we have. Nevertheless, we would like to go beyond simply describing our observed universe, and try to understand why it is that way rather than some other way. Physicists and cosmologists have been exploring increasingly ambitious ideas that attempt to explain why certain features of our universe aren't as surprising as they might first appear.
This is a review of the physics and cosmology of the cosmological constant. Focusing on recent de... more This is a review of the physics and cosmology of the cosmological constant. Focusing on recent developments, I present a pedagogical overview of cosmology in the presence of a cosmological constant, observational constraints on its magnitude, and the physics of a small (and potentially nonzero) vacuum energy.
A brief review is offered of the theoretical background concerning dark energy: what is required ... more A brief review is offered of the theoretical background concerning dark energy: what is required by observations, what sort of models are being considered, and how they fit into particle physics and gravitation. Contribution to the SNAP (SuperNova Acceleration Probe) Yellow Book.
We propose a simple model in which the cosmological dark matter consists of particles whose mass ... more We propose a simple model in which the cosmological dark matter consists of particles whose mass increases with the scale factor of the universe. The particle mass is generated by the expectation value of a scalar field which does not have a stable vacuum state, but which is effectively stabilized by the rest energy of the ambient particles. As the universe expands, the density of particles decreases, leading to an increase in the vacuum expectation value of the scalar (and hence the mass of the particle). The energy density of the coupled system of variable-mass particles ("vamps") redshifts more slowly than that of ordinary matter. Consequently, the age of the universe is larger than in conventional scenarios.
We consider the gravitational effects of a single, fixed-norm, Lorentz-violating timelike vector ... more We consider the gravitational effects of a single, fixed-norm, Lorentz-violating timelike vector field. In a cosmological background, such a vector field acts to rescale the effective value of Newton's constant. The energy density of this vector field precisely tracks the energy density of the rest of the universe, but with the opposite sign, so that the universe experiences a slower rate of expansion for a given matter content. This vector field similarly rescales Newton's constant in the Newtonian limit, although by a different factor. We put constraints on the parameters of the theory using the predictions of primordial nucleosynthesis, demonstrating that the norm of the vector field should be less than the Planck scale by an order of magnitude or more.
If spacetime possesses extra dimensions of size and curvature radii much larger than the Planck o... more If spacetime possesses extra dimensions of size and curvature radii much larger than the Planck or string scales, the dynamics of these extra dimensions should be governed by classical general relativity. We argue that in general relativity, it is nontrivial to obtain solutions where the extra dimensions are static and are dynamically stable to small perturbations. We also illustrate that intuition on equilibrium and stability built up from non-gravitational physics can be highly misleading. For all static, homogeneous solutions satisfying the null energy condition, we show that the Ricci curvature of space must be nonnegative in all directions. Much of our analysis focuses on a class of spacetime models where space consists of a product of homogeneous and isotropic geometries. A dimensional reduction of these models is performed, and their stability to perturbations that preserve the spatial symmetries is analyzed. We conclude that the only physically realistic examples of classically stabilized large extra dimensions are those in which the extra-dimensional manifold is positively curved. *
These notes provide a brief introduction to modern cosmology, focusing primarily on theoretical i... more These notes provide a brief introduction to modern cosmology, focusing primarily on theoretical issues. Some attention is paid to aspects of potential interest to students of string theory, on both sides of the two-way street of cosmological constraints on string theory and stringy contributions to cosmology. Slightly updated version of lectures at the 1999 Theoretical Advanced Study Institute at the University of Colorado, Boulder.
Models of dark energy are conveniently characterized by the equation-of-state parameter w = p/ρ, ... more Models of dark energy are conveniently characterized by the equation-of-state parameter w = p/ρ, where ρ is the energy density and p is the pressure. Imposing the Dominant Energy Condition, which guarantees stability of the theory, implies that w ≥ −1. Nevertheless, it is conceivable that a well-defined model could (perhaps temporarily) have w < −1 , and indeed such models have been proposed. We study the stability of dynamical models exhibiting w < −1 by virtue of a negative kinetic term. Although naively unstable, we explore the possibility that these models might be phenomenologically viable if thought of as effective field theories valid only up to a certain momentum cutoff. Under our most optimistic assumptions, we argue that the instability timescale can be greater than the age of the universe, but only if the cutoff is at or below 100 MeV. We conclude that it is difficult, although not necessarily impossible, to construct viable models of dark energy with w < −1; observers should keep an open mind, but the burden is on theorists to demonstrate that any proposed new models are not ruled out by rapid vacuum decay.
We study N = 1 SUSY theories in four dimensions with multiple discrete vacua, which admit soliton... more We study N = 1 SUSY theories in four dimensions with multiple discrete vacua, which admit solitonic solutions describing segments of domain walls meeting at one-dimensional junctions. We show that there exist solutions preserving one quarter of the underlying supersymmetry -a single Hermitian supercharge. We derive a BPS bound for the masses of these solutions and construct a solution explicitly in a special case. The relevance to the confining phase of N = 1 SUSY Yang-Mills and the M-theory/SYM relationship is discussed.
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Papers by Sean Carroll