Approaches to understanding cosmic accel

arXiv:0904.0024v2 [astro-ph.CO] 28 Jun 2009 Approaches to Understanding Cosmic Acceleration Alessandra Silvestri1 and Mark Trodden2 Department of Physics and Kavli Institute for Astrophysics and Space Research, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA Center for Particle Cosmology, Department of Physics and Astronomy, David Rittenhouse Laboratories, University of Pennsylvania, Philadelphia, PA 19104, USA E-mail: asilvest@mit.edu, trodden@physics.upenn.edu Abstract Theoretical approaches to explaining the observed acceleration of the universe are reviewed We briefly discuss the evidence for cosmic acceleration, and the implications for standard General Relativity coupled to conventional sources of energy-momentum We then address three broad methods of addressing an accelerating universe: the introduction of a cosmological constant, its problems and origins; the possibility of dark energy, and the associated challenges for fundamental physics; and the option that an infrared modification of general relativity may be responsible for the large-scale behavior of the universe Approaches to Understanding Cosmic Acceleration Introduction The development of General Relativity almost 90 years ago provided not only a new way to understand gravity, but heralded the dawn of an entirely new branch of science - cosmology Of course, a discussion of the visible contents of the night sky was not a new development - these objects and phenomena have forever fascinated civilizations However, a coherent theoretical framework within which to explore their distribution and evolution was lacking before Einstein’s crucial insight - that space and time are themselves players in the cosmic drama Hubble’s 1921 observation that distant galaxies are receding from ours at speeds proportional to their distances, in agreement with Friedmann’s corresponding solution to General Relativity, then provided the first experimental confirmation of this new science Almost a century has passed since the beginning of this era, and in the intervening years increasingly accurate predictions of this model of the cosmos, supplemented only by the presence of a dark matter component, have been confronted with, and spectacularly passed, a host of detailed tests - the existence of the Cosmic Microwave Background Radiation (CMB); the abundances of the light elements through Big Bang Nucleosynthesis (BBN); the formation of structure under gravitational instability; the small temperature anisotropies in the CMB; the structure of gravitational lensing maps; and many more Many of these tests are highly nontrivial and provide remarkable support for the overall big bang model All this held true until approximately a decade ago, when the first evidence for cosmic acceleration was reported by two groups using the lightcurves of type Ia supernovae to construct an accurate Hubble diagram out to redshift greater than unity As we will describe in some detail, the natural expectation is that an expanding universe, evolving under the rules of GR, and populated by standard matter sources, will undergo deceleration - the expansion rate should slow down as cosmic time unfolds Amazingly, that initial data, supplemented over the last decade by dozens of further, independent observations, showed that the universe is speeding up - what we observe is not deceleration, but cosmic acceleration! The goal of this article is to provide an overview of what is meant by cosmic acceleration, some proposed theoretical approaches to understanding the phenomenon, and how upcoming observational data may cast crucial light on these We have not attempted to be completely comprehensive, but have tried to provide a flavor of the current research landscape, focusing on the better known approaches, and highlighting the challenges inherent in constructing a fundamental physics understanding of the accelerating universe There are a number of excellent recent review articles on dark energy, which the reader might find as a useful complement to the present paper [1, 2, 3, 4] The article is organized as follows In the next section we provide a brief introduction to the relevant cosmological solutions to General Relativity, with perfect fluid sources In section we review the observational evidence, primarily from Approaches to Understanding Cosmic Acceleration observations made during the last decade or so, for an accelerating universe In section we then discuss the minimal possibility that Einstein’s cosmological constant is responsible for late-time cosmic acceleration, and discuss the challenges the required magnitude of this parameter raises for particle physics In we then explore a class of dynamical dark energy models, in which the universe accelerates due to the evolution of a new component of the cosmic energy budget, under the assumption that the cosmological constant is negligibly small In a similar vein, in we then consider the possibility that the universe contains only standard model and dark matter sources of energy and momentum, but accelerates because GR is modified at large distances, leading to new self-accelerating solutions Finally, before concluding, in we then consider the challenge of distinguishing among these possible explanations for cosmic acceleration, and discuss how upcoming observational missions may help us to better understand this phenomena A note on conventions Throughout this article we use units in which = c = 1, adopt the (−, +, +, +) signature and define the reduced Planck mass by Mp−2 ≡ κ2 = 8πG Essential Elements of Background Cosmology Our goal here is to lay out the bare minimum for understanding the background evolution of the universe By background in this context, we mean the dynamics pertaining to the homogeneous and isotropic description of spacetime, valid on the very largest scales, without reference to spatial perturbations of either the metric or matter fields‡ The most general homogeneous and isotropic metric ansatz is the Friedmann, Robertson-Walker (FRW) form dr , (1) + r dθ2 + sin2 θdφ2 ds2 = −dt2 + a2 (t) − kr where k describes the curvature of the spatial sections (slices at constant cosmic time) and a(t) is referred to as the scale factor or the universe Without loss of generality, we may normalize k so that k = +1 corresponds to positively curved spatial sections (locally isometric to 3-spheres); k = corresponds to local flatness, and k = −1 corresponds to negatively curved (locally hyperbolic) spatial sections These local definitions say nothing about the global topology of the spatial sections, which may be that of the covering spaces – a 3-sphere, an infinite plane or a 3-hyperboloid – but it need not be, as topological identifications under freely-acting subgroups of the isometry group of each manifold are allowed As a specific example, the k = spatial geometry could apply just as well to a 3-torus as to an infinite plane Note that we have not chosen a normalization such that a0 = We are not free to this and to simultaneously normalize |k| = 1, without including explicit factors of the current scale factor in the metric In the flat case, where k = 0, we can safely choose a0 = ‡ We follow closely the brief discussion in [5] Approaches to Understanding Cosmic Acceleration At cosmic time t, the physical distance from the origin to an object at radial coordinate r is given by d(t) = a(t)r The recessional velocity of such an object due to the expansion of the universe is then given by v(t) = H(t)d(t) , (2) where H(t) ≡ a/a ˙ (an overdot denotes a derivative with respect to t) is the Hubble parameter, the present day value of which we refer to as the Hubble constant H0 The Hubble parameter is a useful function through which to parametrize the expansion rate of the universe As we shall encounter later, it is convenient to define a second function, describing the rate at which the expansion rate is slowing down or speeding up - the deceleration parameter, defined as a ăa (3) q(t) − a˙ 2.1 Dynamics: The Friedmann Equations The FRW metric is merely an ansatz, arrived at by requiring homogeneity and isotropy of spatial sections The unknown function a(t) is obtained by solving the differential equations obtained by substituting the FRW ansatz into the Einstein equation (4) Rµν − Rgµν = 8πGTµν The energy-momentum tensors Tµν describes the matter content of the universe It is often appropriate to adopt the perfect fluid form for this Tµν = (ρ + p)Uµ Uν + pgµν , (5) where U µ is the fluid four-velocity, ρ is the energy density in the rest frame of the fluid and p is the pressure in that frame Substituting (1) and (5) into (4), one obtains the Friedmann equation a˙ a H2 a ă + a a a = 8πG ρi − i k , a2 (6) and = −4πG i pi − k 2a2 (7) Here i indexes all different possible types of energy in the universe The Friedmann equation is a constraint equation, in the sense that we are not allowed to freely specify the time derivative a; ˙ it is determined in terms of the energy density and curvature The second equation an evolution equation We may combine (6) and (7) to obtain the acceleration equation 4G a ă = a (ρi + 3pi ) , i which will prove central to the subject of this review (8) Approaches to Understanding Cosmic Acceleration In fact, if we know the magnitudes and evolutions of the different energy density components ρi , the Friedmann equation (6) is sufficient to solve for the evolution uniquely The acceleration equation is conceptually useful, but rarely invoked in calculations The Friedmann equation relates the rate of increase of the scale factor, as encoded by the Hubble parameter, to the total energy density of all matter in the universe We may use the Friedmann equation to define, at any given time, a critical energy density, 3H , (9) 8πG for which the spatial sections must be precisely flat (k = 0) We then define the density parameter ρ , (10) Ωtotal ≡ ρc which allows us to relate the total energy density in the universe to its local geometry via ρc ≡ Ωtotal > ⇔ k = +1 Ωtotal = ⇔ k = (11) Ωtotal < ⇔ k = −1 It is often convenient to define the fractions of the critical energy density in each different component by ρi Ωi = (12) ρc Energy conservation is expressed in GR by the vanishing of the covariant divergence of the energy-momentum tensor, ∇µ T µν = (13) For the FRW metric (1) and a perfect-fluid energy-momentum tensor (5) this yields a single energy-conservation equation, ρ˙ + 3H(ρ + p) = (14) This equation is actually not independent of the Friedmann and acceleration equations, but is required for consistency It implies that the expansion of the universe (as specified by H) can lead to local changes in the energy density Note that there is no notion of conservation of “total energy,” as energy can be interchanged between matter and the spacetime geometry One final piece of information is required before we can think about solving our cosmological equations: how the pressure and energy density are related to each other Within the fluid approximation used here, we may assume that the pressure is a singlevalued function of the energy density p = p(ρ) It is often convenient to define an equation of state parameter, w, by p = wρ (15) Approaches to Understanding Cosmic Acceleration This should be thought of as the instantaneous definition of the parameter w; it does not represent the full equation of state, which would be required to calculate the behavior of fluctuations Nevertheless, many useful cosmological matter sources obey this relation with a constant value of w For example, w = corresponds to pressureless matter, or dust – any collection of massive non-relativistic particles would qualify Similarly, w = 1/3 corresponds to a gas of radiation, whether it be actual photons or other highly relativistic species A constant w leads to a great simplification in solving our equations In particular, using (14), we see that the energy density evolves with the scale factor according to (16) ρ(a) ∝ 3(1+w) a(t) Note that the behaviors of dust (w = 0) and radiation (w = 1/3) are consistent with what we would have obtained by more heuristic reasoning Consider a fixed comoving volume of the universe - i.e a volume specified by fixed values of the coordinates, from which one may obtain the physical volume at a given time t by multiplying by a(t)3 Given a fixed number of dust particles (of mass m) within this comoving volume, the energy density will then scale just as the physical volume, i.e as a(t)−3 , in agreement with (16), with w = To make a similar argument for radiation, first note that the expansion of the universe (the increase of a(t) with time) results in a shift to longer wavelength λ, or a redshift, of photons propagating in this background A photon emitted with wavelength λe at a time te , at which the scale factor is ae ≡ a(te ) is observed today (t = t0 , with scale factor a0 ≡ a(t0 )) at wavelength λo , obeying a0 λo = ≡1+z (17) λe ae The redshift z is often used in place of the scale factor Because of the redshift, the energy density in a fixed number of photons in a fixed comoving volume drops with the physical volume (as for dust) and by an extra factor of the scale factor as the expansion of the universe stretches the wavelengths of light Thus, the energy density of radiation will scale as a(t)−4 , once again in agreement with (16), with w = 1/3 Thus far, we have not included a cosmological constant Λ in the gravitational equations This is because it is equivalent to treat any cosmological constant as a component of the energy density in the universe In fact, adding a cosmological constant Λ to Einstein’s equation is equivalent to including an energy-momentum tensor of the form Λ gµν (18) Tµν = − 8πG This is simply a perfect fluid with energy-momentum tensor (5) with Λ 8πG pΛ = − ρΛ , ρΛ = (19) Approaches to Understanding Cosmic Acceleration so that the equation-of-state parameter is wΛ = −1 (20) This implies that the energy density is constant, ρΛ = constant (21) Thus, this energy is constant throughout spacetime; we say that the cosmological constant is equivalent to vacuum energy Similarly, it is sometimes useful to think of any nonzero spatial curvature as yet another component of the cosmological energy budget, obeying 3k ρcurv = − 8πGa2 k , (22) pcurv = 8πGa2 so that wcurv = −1/3 (23) It is not an energy density, of course; ρcurv is simply a convenient way to keep track of how much energy density is lacking, in comparison to a flat universe 2.2 Flat Universes Analytically, it is much easier to find exact solutions to cosmological equations of motion when k = We are able to make full use of this convenience since, as we shall touch on in the next section, the combined results of modern cosmological observations show the present day universe to be extremely spatially flat This can be seen, for example, from precision observations of the cosmic microwave background radiation and independent measures of the Hubble expansion rate In the case of flat spatial sections and a constant equation of state parameter w, we may exactly solve the Friedmann equation (16) to obtain a(t) = a0 t t0 2/3(1+w) , (24) where a0 is the scale factor today, unless w = −1, in which case one obtains a(t) ∝ eHt Applying this result to some of our favorite energy density sources yields table Note that the matter- and radiation-dominated flat universes begin with a = 0; this is a singularity, known as the Big Bang We can easily calculate the age of such a universe: da t0 = = (25) 3(1 + w)H0 aH(a) Unless w is close to −1, it is often useful to approximate this answer by t0 ∼ H0−1 (26) It is for this reason that the quantity H0−1 is known as the Hubble time, and provides a useful estimate of the time scale for which the universe has been around Approaches to Understanding Cosmic Acceleration Type of Energy Dust Radiation Cosmological Constant ρ(a) a−3 a−4 constant a(t) t2/3 t1/2 eHt Table A summary of the behaviors of the most important sources of energy density in cosmology The behavior of the scale factor applies to the case of a flat universe; the behavior of the energy densities is perfectly general Observational Evidence for Cosmic Acceleration An important breakthrough in cosmology occurred in the late ’90s with the measurements of type Ia Supernovae (SNeIa) by the High-Z Supernova team [6] and the Supernova Cosmology Project [7] As we review in this section, the data from these surveys, as well as from complementary probes, provide strong evidence that the universe has recently entered a phase of accelerated expansion 3.1 Type Ia Supernovae Type Ia supernovae are stellar explosions that occur as a white dwarf, onto which mass is gradually accreting from a companion star, approaches the Chandrasekhar limit [8] (of about 1.4 solar masses) and the density and temperature in its core reach the ignition point for carbon and oxygen This begins a nuclear flame that fuses much of the star up to iron They are extremely bright events, with luminosities a significant fraction of that of their host galaxies during the peak of their explosions Therefore they are relatively easy to detect at high redshift (z ∼ 1) This specific type of supernova is characterized by the absence of a hydrogen line (or, indeed, a helium one) in their spectrum (which is typical of type I SNe) and instead by the presence of a singly-ionized silicon line at 615nm, near peak light Despite a significant scatter, type Ia supernovae peak luminosities have been found to be very closely correlated with observed differences in the shapes of the light curves: dimmer SNe are found to decline more rapidly after maximum brightness, while brighter SNe decline more slowly [9, 10, 11] This difference seems to be traceable to the amount of 56 Ni produced in the explosion; more nickel implies both a higher peak luminosity and a higher temperature, and thus opacity, leading to a slower decline After one adjusts for the difference in the light-curves, the scatter in peak luminosity can be reduced to 15% In this sense, SNeIa are referred to as “standardizable candles” and are very good candidates for distance indicators, since one would expect any remaining difference in their peak luminosity to be due to a difference in distance Another important aspect of this uniformity is that it provides standard spectral and light curve templates that offer the possibility of singling out those SNe that deviate slightly from the norm [12] After the necessary adjustments, all SNeIa have the same absolute magnitude M, Approaches to Understanding Cosmic Acceleration and any difference in their apparent magnitude m is attributable to a difference in their distance The apparent and intrinsic magnitudes are related via the luminosity distance dL m = M + log dL 10pc +K, (27) where K is a correction for the shifting of the spectrum [6, 7, 9, 11, 13, 14] necessary because only a part of the spectrum emitted is actually observed In an expanding universe, the expression of the luminosity distance as a function of redshift is z dL (z) = (1 + z) · dz ′ H(z ′ ) (28) By independently measuring dL (z) and z, one can then constrain the expansion history H(z) Following pioneering work reported in [15], the High-Z Supernova team [6] and the Supernova Cosmology Project [7] measured the apparent magnitudes of many SNeIa (of redshift z 1) and directly determined their distances They then compared these distances to those inferred from the redshifts of the host galaxies (measured from the spectra of the galaxies when possible, otherwise the spectra of the SNe themselves) The most distant SNe appeared dimmer than expected in a universe currently dominated by matter, and a careful comparison with low redshift supernovae allowed one to rule out the possibility that this dimming is due to intervening dust Assuming a flat universe described by GR, and homogeneous and isotropic on large scales, the data are best fit by a universe which has recently entered a phase of accelerated expansion, i.e a universe for which the deceleration parameter (3) is currently negative (q0 ≃ −1) The simplest model fitting the data is a universe in which matter accounts for only about a quarter of the critical density, while the remaining 70% of the energy density is in the cosmological constant Λ This model is commonly referred to as the ΛCDM model In fact, the SNe data by themselves allow a range of possible values for the matter and cosmological constant density parameters (Ω0M and Ω0Λ respectively) However, as we will discuss in the following subsections, we can use what we know from other observations, such as the Cosmic Microwave Background (CMB) and the large scale structure (LSS), to further constrain these parameters In more recent years, distant SNe with redshifts up to z ∼ 1.7 have been observed by the Hubble Space Telescope (HST) [17, 18] These observations show that the trend toward fainter SNe at moderate redshifts has reversed; therefore they play a key role in disregarding dust as a plausible explanation of the dimming of intermediate SNe, and in firmly establishing [19, 20] acceleration More generally, SNe at lower and higher redshift are both important in the study of the expansion history Nearby SNe (z 0.3) are useful for defining characteristics of Type Ia SNe, understanding the explosion and exploring systematics; all combined, they are important for establishing distance indicators and are currently [21, 22] being measured, with more to come in the future [23, 24] Intermediate redshift SNe (0.3 z 0.8) measure the strength of cosmic acceleration and are being measured by the ESSENCE project [25], the Approaches to Understanding Cosmic Acceleration 10 Figure Hubble diagram for Type Ia Supernovae, plotting the effective magnitude mB versus redshift z (Knop et al [16]) The solid line is the best-fit cosmology and corresponds to a universe with Ω0M = 0.25 and Ω0Λ = 0.75 SuperNova Legacy Survey (SNLS) [26] and the Sloan Digital Sky Survey-II (SDSSII) Supernova Survey [27] Finally, distant SNe (z 0.8) are important to break the degeneracy among the cosmological parameters and they are currently being measured by the Higher-z Supernova Search Team (HZT) [28] and the GOODS team [29] In the near future, the space-based Joint Dark Energy Mission (JDEM) [30], (a wide-field optical-infrared telescope), will offer precision measurements with the aim of determining the nature of cosmic acceleration This mission may incorporate the Baryon Acoustic Oscillation (BAO), Supernovae (SNe) and Weak Lensing (WL) techniques Also, upcoming and future surveys such as the earth-based Dark Energy Survey (DES) [31] and the Large Synoptic Survey Telescope (LSST) [32] will provide observations useful for constraining the expansion history of the universe 3.2 Complementary Probes Supernovae are the first and most direct evidence for the late-time acceleration of the universe However, independent evidence comes from complementary probes Data from different observations are important not only to confirm the results from SNe, Approaches to Understanding Cosmic Acceleration 42 have been introduced in [197, 212, 214, 215] From the previous examples we can easily infer that a quite generic parametrization would be + β1 λ21 k as + λ21 k as + β2 λ22 k as , γ(a, k) = + λ22 k as µ(a, k) = (102) (103) where the parameters λ2i have dimensions of length squared, while the βi represent dimensionless couplings Finally, from a scalar-tensor point of view, the parameter s encodes the time-dependence of the scalar field mass The expressions in (102) coincide with the scale-dependent parametrization introduced in [197] In [201], the authors performed a Fisher matrix analysis to forecast the constraints on the parameters {λi , βi , s} of (102) from a combination of power spectra from galaxies, Weak Lensing and the Integrated Sachs-Wolfe effect on the CMB Such an analysis reveals the extent to which one can constrain these theories and also allows one to reconstruct the shape of the slip and effective Newton’s constant based on the chosen form In [201], it was found that these data, even at the linear level, are quite powerful in constraining the modified growth parameters The results, of course, depend on the choice of the parametrization However, they are good indicators of the power of current and upcoming surveys to constrain departures from GR An alternative approach, which can work for certain estimators of the slip, is a direct reconstruction from data In [199, 200], it was proposed to consider the ratio of the peculiar velocity-galaxy correlation with the Weak Lensing-galaxy correlation Specifically, the authors propose the following estimator for the gravitational slip P∇2 (Ψ−Φ)g − 1, (104) γˆ = P∇2 Ψg where P∇2 (Ψ−Φ)g and P∇2 ψg are the cross-power spectrum between the galaxy number overdensity and, respectively, weak lensing and the potential Ψ (which is related to peculiar velocities via the continuity equation) In such a ratio, the dependence on the galaxy bias cancels out Such a ratio, if appropriately constructed, would directly probe any difference between Φ and Ψ This is a more direct and model-independent way of testing GR with the growth of structure; however its power will depend on how well future experiments will be able to measure peculiar velocities From the analysis in [199, 200], it appears however that the ratio measured by future surveys might have strong discriminating power for some dark universe scenarios Finally, another, non-parametric approach, consists of performing a Principal Component Analysis (PCA) to determine the eigenmodes of the slip and the Newton constant, that can be constrained by data, in the same way as has been done for the dark energy equation of state [216, 217, 218] This method allows one to compare different experiments and their combinations, according to the relative gain in information about the functions PCA can also point to the “sweet spots” in redshift and scale where data Approaches to Understanding Cosmic Acceleration 43 is most sensitive to variations in the slip and Newton’s constant, which can be a useful guide for designing future observing strategies The PCA method does not allow one to reconstruct the shape of the functions from data However, one can still reproduce the errors on parameters of any parametrization from the eigenvectors and eigenvalues found using PCA [217] Thus, it is a promising method, but computationally challenging 7.2.1 The observables As we mentioned above, in the ΛCDM model the sub-horizon evolution of gravitational potentials and the matter density fluctuations are described by a single function of time – the scale-independent growth factor g(a) In models of modified gravity, on the other hand, the dynamics of perturbations can be richer and, generically, the evolution of Φ, Ψ and δ (the matter density contrast), will be described by different functions of scale and time Therefore, we expect that different observables will be described by different functions, and by combining different types of measurements, one can try to reconstruct these functions, or at least put a limit on how different they can be In what follows, we shall give a brief review of the relation between the different types of observables and the gravitational potentials they probe For a more thorough overview of the various ways of looking for modifications in the growth of perturbations we refer the reader to [213] Galaxy Counts (GC) probe the distribution and growth of matter inhomogeneities However, to extract the matter power spectrum, one needs to account for the bias, which typically depends on the type of galaxies and can be both time- and scale-dependent On large scales, where non-linear effects are unimportant, one can use a scale-independent bias factor to relate galaxy counts to the total matter distribution This relation becomes increasingly complicated and scale-dependent as one considers smaller and smaller scales In principle, the bias parameters can be determined from higher order correlation functions [219, 220, 221] On sub-horizon linear scales, the evolution of matter density contrast is determined by equation (95) Hence, measurements of GC over multiple redshifts can provide an estimate of Ψ as a function of space and time, up to a bias factor A more direct probe of the potential Ψ, would be a measurement of peculiar velocities, which follow the gradients of Ψ Such measurements would be independent of uncertainties associated with modeling the bias Peculiar velocity surveys typically use redshift-independent distance indicators to separate the Hubble flow from the local flow, and nearby SNeIa are therefore good candidates; a number of surveys, like the 6dFGS [222] and the 2MRS [223], use galaxies An interesting alternative is offered by the kinetic Sunyaev-Zel’dovich effect in clusters [224], that arises from the inverse Compton scattering of CMB photons off high-energy electrons in the clusters This effect provides a useful way of measuring the bulk motion of electrons in clusters, hence the peculiar velocity of clusters, but it is limited by low signal-to-noise ratio Current measurements of peculiar velocities are limited in accuracy, and at this point it is not clear how to forecast the accuracy of future observations Therefore we did not include them in our observables, even though they are a potentially powerful probe [225] Approaches to Understanding Cosmic Acceleration 44 In contrast to galaxy counts and peculiar velocities, which respond to one of the metric potentials, namely Ψ, Weak Lensing of distant light sources by intervening structure is determined by spatial gradients of the sum (Φ+Ψ) Hence, measurements of the weak lensing shear distribution over multiple redshift bins can provide an estimate of the space and time variation of the sum of the two potentials In the ΛCDM and minimally coupled models of dark energy, the two metric potentials coincide, and therefore WL probes essentially the same growth function that controls the evolution of galaxy clustering and peculiar velocities In models of modified gravity, however, there could be a difference between the potentials, corresponding to an effective shear component Measurements of the Integrated Sachs-Wolfe effect in the CMB probe the time dependence of the sum of the potentials: d (Φ + Ψ) /dt By combining multiple redshift information on GC, WL and CMB, and their crosscorrelations, one can constrain the differences between the metric potentials and the space-time variation of the effective Newton constant defined in the previous section Ideally, experimentalists would measure all possible cross-correlations, between all possible pairs of observables, in order to maximize the amount of information available to us In practice, however, it can be difficult to obtain these cross-correlations, since their measurements require that each of the individual fields (CMB, GC, WL) be measured on the same patch of sky This will be addressed with near and distant future tomographic large scale structure surveys (such as DES [31], LSST [32] and PAN-STARRS [226]) Even with conservative assumptions about the data, (i.e considering only linear scales), it is hoped [201], that DES will produce non-trivial constraints on modified growth, and that LSST will even better (with ≈ 10% relative errors in the parameters) From the discussions in the previous section, it is clear that the predictions for the growth of structure in models of modified gravity and coupled dark energy can significantly differ from those in ΛCDM or uncoupled quintessence models Therefore, LSS and CMB data offer a valuable testing ground for gravity and cosmological tests are expected to play an important role in determining the physics of cosmic acceleration In principle, a combination of galaxy number counts and weak lensing measurements, along with CMB and SNe data, will allow a distinction between modified gravity/exotic dark energy models and the standard ΛCDM model of cosmology However, as we have already mentioned, it will be harder to break the degeneracy between models of modified gravity and generalized models of dark energy, where the dark fluid is allowed to cluster and carry anisotropic stress As discussed in [213], the latter task will require the independent measurements of at least three observables Conclusions That the expansion rate of the universe is accelerating is now, just over a decade after the first evidence from observations of type Ia supernovae, a firmly established aspect of cosmology The rapid progress in establishing this fact is a testament to the breathtaking convergence of techniques and technology that has emerged in observational cosmology Approaches to Understanding Cosmic Acceleration 45 In turn, cosmic acceleration has introduced new wrinkles into almost every part of theoretical cosmology, ranging from the details of structure formation, through the CMB and gravitational lensing Beyond the fascinating new problems in theoretical cosmology, cosmic acceleration has presented an enormous and, as yet, unmet challenge to fundamental physics A perfectly good fit to all known cosmological data is that cosmic acceleration is driven by a cosmological constant However, accounting for the necessary magnitude of such an object seems at least as difficult as attempts to understand why it should be precisely zero Anthropic arguments, fueled by results from the landscape of string theory and the idea of eternal inflation, may yield a way to understand such a small number However, it is, at present, far too early to know if this is a sensible outcome of string theory, and there are no developed ideas of how such proposals might be tested This is not to say, of course, that they are incorrect; merely that there are immense obstacles to establishing such ideas as a tested answer If cosmic acceleration is not due to a cosmological constant, then an exotic component of the energy budget of the universe may be the culprit It is important to recognize up front that any such understanding of cosmic acceleration assumes that the cosmological constant itself is either precisely zero, or at least negligibly small an unsolved problem in itself If we make such an assumption, then proposals for a dynamical component of the cosmic energy budget, driving the accelerating expansion, come under the heading of dark energy In their simplest forms, ideas of dark energy can be thought of as attempts to construct a model for late-time cosmological inflation, with the key differences being the extremely low energy scales involved, and the fact that this period of acceleration need not end in the way that inflation must As with early universe inflation, a major challenge is finding a home for the abstract idea within a well-motivated and technically natural particle physics model In fact, in a number of ways, this problem is more acute for dark energy models, since they must operate in a regime in which we already have a supremely well-tested low-energy effective field theory of much of the matter content of the universe As we have described, some tentative ideas exist to tackle this problem, but as yet no convincing particle physics implementation of dark energy exists, in contrast to the somewhat natural occurrence of dark matter candidates in theories of physics beyond the standard model But dark energy is not the only possible solution to the problem of cosmic acceleration Instead we might imagine leaving the material contents of our theories untouched, comprised only of the standard model and dark matter, and instead revisiting the dynamical relationship between these components and the evolution of the background geometry of the universe In other words, we might consider that cosmic acceleration is our first evidence for a modification of General Relativity in the far infrared One might imagine that this approach would hold out the possibility of a solution to the cosmological constant problem itself, as well as providing an understanding of cosmic acceleration However, to date this possibility remains unrealized, and just as for dark energy, modified gravity approaches to cosmic Approaches to Understanding Cosmic Acceleration 46 acceleration must also merely assume that the cosmological constant is either zero, or negligibly small Making this assumption, a number of authors have considered how one might modify gravity However, to date all attempts have run into problems, either with matching constraints on gravity at small (solar system) scales, or with the existence of ghost degrees of freedom, raising the possibility of problems similar to those we mentioned when discussing phantom fields Nevertheless, if one views these models either as a long-wavelength approximation to a better-behaved theory of gravity, or as a very low energy effective theory, in need of an ultraviolet completion to cure its problems, one may extract a variety of predictions for signatures in upcoming cosmological observations Perhaps the biggest question to be answered is binary in nature - is cosmic acceleration due to a cosmological constant or not? In many ways, an affirmative answer is the most depressing, since a true cosmological constant varies in neither space nor time, and thus, if it is the driver of acceleration, we already know all that we will ever know about it As such, we would not expect further observational insights into the required unnaturally small value More generally, we would like to be able to distinguish among this minimal explanation, a possible dark energy component and the option of an infrared modification of gravity It has become increasingly clear that our best hope for this is to compare geometrical measures, such as the distance to the surface of last scattering, with measures of the growth of large-scale structure, which depend in detail, in a redshift-dependent manner on the gravitational force law underlying the collapse of overdensities In this article we have endeavored to provide an overview of the major approaches to cosmic acceleration and to the techniques through which we hope to further our understanding What should be clear is that the issues involved here are far from settled, and that the field is hungry for new ideas It is unclear where such an idea will come from, and for this reason it seems important to pursue all possible avenues, no matter how unpromising they may seem We hope to have provided a snapshot of some of these avenues of enquiry It remains to be seen which, if any, of these will provide the key to the problem of cosmic acceleration Acknowledgements We would like to thank Rachel Bean, Eanna Flanagan, Justin Khoury and Levon Pogosian for useful discussions during the writing of this article, and all our collaborators for helping develop our understanding The work of MT is supported in part by National Science Foundation grants PHY-0653563 and PHY-0930521, by Department of Energy grant DE-FG05-95ER40893-A020 and by NASA ATP grant NNX08AH27G The work of AS is supported by the National Science Foundation under grant AST-0708501 Approaches to Understanding Cosmic Acceleration 47 References [1] Edmund J Copeland, M Sami, and Shinji Tsujikawa Dynamics of dark energy Int J Mod Phys., D15:1753–1936, 2006 [2] Eric V Linder Mapping the Cosmological Expansion Rept Prog Phys., 71:056901, 2008 [3] Joshua Frieman, Michael Turner, and Dragan Huterer Dark Energy and the Accelerating Universe Ann Rev Astron Astrophys., 46:385–432, 2008 [4] Robert R Caldwell and Marc Kamionkowski The Physics of Cosmic Acceleration 2009 [5] Mark Trodden and Sean M Carroll TASI lectures: Introduction to cosmology 2004 [6] Adam G Riess et al Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant Astron J., 116:1009–1038, 1998 [7] S Perlmutter et al Measurements of Omega and Lambda from 42 High-Redshift Supernovae Astrophys J., 517:565–586, 1999 [8] Subrahmanyan Chandrasekhar The maximum mass of ideal white dwarfs Astrophys J., 74:81– 82, 1931 [9] M M Phillips The absolute magnitudes of Type IA supernovae Astrophys J., 413:L105–L108, 1993 [10] Adam G Riess, William H Press, and Robert P Kirshner A Precise distance indicator: Type Ia supernova multicolor light curve shapes Astrophys J., 473:88, 1996 [11] Mario Hamuy et al The Absolute Luminosities of the Calan/Tololo Type Ia Supernovae Astron J., 112:2391, 1996 [12] Saul Perlmutter Supernovae, dark energy, and the accelerating universe: The status of the cosmological parameters Talk given at 19th International Symposium on Lepton and Photon Interactions at High-Energies (LP 99), Stanford, California, 9-14 Aug 1999 [13] Peter Nugent, Alex Kim, and Saul Perlmutter K-corrections and Extinction Corrections for Type Ia Supernovae 2002 [14] Saul Perlmutter and Brian P Schmidt Measuring Cosmology with Supernovae 2003 [15] H.E Jorgensen A Aragon Salamanca H.H Norgaard-Nielsen, L Hansen and R.S Ellis [16] Robert A Knop et al New Constraints on ΩM , ΩΛ , and w from an Independent Set of Eleven High-Redshift Supernovae Observed with HST Astrophys J., 598:102, 2003 [17] Pierre Astier et al The Supernova Legacy Survey: Measurement of OmegaM , OmegaL ambda and w from the First Year Data Set Astron Astrophys., 447:31–48, 2006 [18] Adam G Riess et al New Hubble Space Telescope Discoveries of Type Ia Supernovae at z > 1: Narrowing Constraints on the Early Behavior of Dark Energy Astrophys J., 659:98–121, 2007 [19] Glenn Starkman, Mark Trodden, and Tanmay Vachaspati Observation of cosmic acceleration and determining the fate of the universe Phys Rev Lett., 83:1510–1513, 1999 [20] Dragan Huterer, Glenn D Starkman, and Mark Trodden Is the universe inflating? 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