Dark energy: investigation and modeling

Dark energy: investigation and modeling

Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
August 19, 2019
Abstract

Constantly accumulating observational data continue to confirm that about 70 % of the energy density today consists of dark energy responsible for the accelerated expansion of the Universe. We present recent observational bounds on dark energy constrained by the type Ia supernovae, cosmic microwave background, and baryon acoustic oscillations. We review a number of theoretical approaches that have been adopted so far to explain the origin of dark energy. This includes the cosmological constant, modified matter models (such as quintessence, k-essence, coupled dark energy, unified models of dark energy and dark matter), modified gravity models (such as gravity, scalar-tensor theories, braneworlds), and inhomogeneous models. We also discuss observational and experimental constraints on those models and clarify which models are favored or ruled out in current observations.

I Introduction

The discovery of the late-time cosmic acceleration reported in 1998 Riess ; Perlmutter based on the type Ia Supernovae (SN Ia) observations opened up a new field of research in cosmology. The source for this acceleration, dubbed dark energy Huterer , has been still a mystery in spite of tremendous efforts to understand its origin over the last decade Sahnireview ; Carroll ; Peebles ; Paddy ; SahniLecture ; review ; Durrerreview ; Caldwell09 ; Tsujibook . Dark energy is distinguished from ordinary matter in that it has a negative pressure whose equation of state is close to . Independent observational data such as SN Ia SNLS ; Gold1 ; Gold2 ; Essence1 ; Essence2 ; Kowalski , Cosmic Microwave Background (CMB) WMAP1 ; WMAP3 ; WMAP5 ; WMAP7 , and Baryon acoustic oscillations (BAO) Eisenstein ; Percival1 ; Percival2 have continued to confirm that about 70 % of the energy density of the present Universe consists of dark energy.

The simplest candidate for dark energy is the so-called cosmological constant whose equation of state is . If the cosmological constant originates from a vacuum energy of particle physics, its energy scale is significantly larger than the dark energy density today Weinberg ( GeV). Hence we need to find a mechanism to obtain the tiny value of consistent with observations. A lot of efforts have been made in this direction under the framework of particle physics. For example, the recent development of string theory shows that it is possible to construct de Sitter vacua by compactifying extra dimensions in the presence of fluxes with an account of non-perturbative corrections KKLT .

The first step toward understanding the property of dark energy is to clarify whether it is a simple cosmological constant or it originates from other sources that dynamically change in time. The dynamical dark energy models can be distinguished from the cosmological constant by considering the evolution of . The scalar field models of DE such as quintessence quin1 ; Ford ; quin2 ; quin3 ; quin4 ; Ferreira1 ; Ferreira2 ; CLW ; quin5 ; Zlatev ; Paul99 and k-essence kes1 ; kes2 ; kes3 predict a wide variety of variations of , but still the current observational data are not sufficient to provide some preference of such models over the -Cold-Dark-Matter (CDM) model. Moreover, the field potentials need to be sufficiently flat such that the field evolves slowly enough to drive the present cosmic acceleration. This demands that the field mass is extremely small ( eV) relative to typical mass scales appearing in particle physics Carrollqui ; Kolda . However it is not entirely hopeless to construct viable scalar-field dark energy models in the framework of particle physics. We note that there is another class of modified matter models based on perfect fluids–so-called (generalized) the Chaplygin gas model Kamen ; Bento . If these models are responsible for explaining the origin of dark matter as well as dark energy, then they are severely constrained from the matter power spectrum in galaxy clustering Waga .

There exists another class of dynamical dark energy models that modify General Relativity. The models that belong to this class are gravity fR1 ; fR1d ; fR2 ; fR3 ; Nojiri03 ( is a function of the Ricci scalar ), scalar-tensor theories st1 ; st2 ; st3 ; st4 ; st5 , and Dvali, Gabadadze and Porrati (DGP) braneworld model DGP . The attractive feature of these models is that the cosmic acceleration can be realized without recourse to a dark energy component. If we modify gravity from General Relativity, however, there are stringent constraints coming from local gravity tests as well as a number of observational constraints such as large-scale structure (LSS) and CMB. Hence the restriction on modified gravity models is in general very tight compared to modified matter models. We shall construct viable modified gravity models and discuss their observational and experimental signatures.

In addition to the above mentioned models, there are attempts to explain the cosmic acceleration without dark energy. One example is the void model in which an apparent accelerated expansion is induced by a large spatial inhomogeneity Tomita1 ; Tomita2 ; Celerier ; Iguchi ; Alnes . Another example is the so-called backreaction model in which the backreaction of spatial inhomogeneities on the Friedmann-Lemaître-Robertson-Walker (FLRW) background is responsible for the real acceleration Rasanen ; Kolb1 ; Kolb2 . We shall discuss these models as well.

This review is organized as follows. In Sec. II we provide recent observational constraints on dark energy obtained by SN Ia, CMB, and BAO data. In Sec. III we review theoretical attempts to explain the origin of the cosmological constant consistent with the low energy scale of dark energy. In Sec. IV we discuss modified gravity models of dark energy–including quintessence, k-essence, coupled dark energy, and unified models of dark energy and dark matter. In Sec. V we review modified gravity models and provide a number of ways to distinguish those models observationally from the CDM model. Sec. VI is devoted to the discussion about the cosmic acceleration without using dark energy. We conclude in Sec. VII.

We use units such that , where is the speed of light and is reduced Planck’s constant. The gravitational constant is related to the Planck mass  GeV via and the reduced Planck mass  GeV via , respectively. We write the Hubble constant today as  km sec Mpc, where describes the uncertainty on the value . We use the metric signature .

Ii Observational constraints on dark energy

The late-time cosmic acceleration is supported by a number of independent observations–such as (i) supernovae observations, (ii) Cosmic Microwave Background (CMB), and (iii) Baryon acoustic oscillations (BAO). In this section we discuss observational constraints on the property of dark energy.

ii.1 Supernovae Ia observations

In 1998 Riess et al. Riess and Perlmutter et al. et al. Perlmutter independently reported the late-time cosmic acceleration by observing distant supernovae of type Ia (SN Ia). The line-element describing a 4-dimensional homogeneous and isotropic Universe, which is called the FLRW space-time, is given by Weinbergbook

 ds2=gμνdxμdxν=−dt2+a2(t)[dr21−Kr2+r2(dθ2+sin2θdϕ2)], (1)

where is the scale factor with cosmic time , and correspond to closed, open and flat geometries, respectively. The redshift is defined by , where is the scale factor today.

In order to discuss the cosmological evolution in the low-redshift regime (), let us consider non-relativistic matter with energy density and dark energy with energy density and pressure , satisfying the continuity equations

 ˙ρm+3Hρm=0, (2) ˙ρDE+3H(ρDE+PDE)=0, (3)

which correspond to the conservation of the energy-momentum tensor for each component (=0, where represents a covariant derivative). Note that a dot represents a derivative with respect to . The cosmological dynamics is known by solving the Einstein equations

 Gμν=8πGTμν, (4)

where is the Einstein tensor. For the metric (1) the (00) component of the Einstein equations gives Weinbergbook

 H2=8πG3(ρm+ρDE)−Ka2, (5)

where is the Hubble parameter. We define the density parameters

 Ωm≡8πGρm3H2,ΩDE≡8πGρDE3H2,ΩK≡−K(aH)2, (6)

which satisfy the relation from Eq. (5). Integrating Eqs. (2) and (3), we obtain

 ρm=ρ(0)m(1+z)3,ρDE=ρ(0)DEexp[∫z03(1+wDE)1+~zd~z], (7)

where “0” represents the values today and is the equation of state of dark energy. Plugging these relations into Eq. (5), it follows that

 (8)

The expansion rate can be known observationally by measuring the luminosity distance of SN Ia. The luminosity distance is defined by , where is the absolute luminosity of a source and is an observed flux. It is a textbook exercise review ; Tsujibook ; Weinbergbook to derive for the FLRW metric (1):

 dL(z)=1+zH0√Ω(0)Ksinh(√Ω(0)K∫z0d~zE(~z)), (9)

where . The function can be understood as (for ), (for ), and (for ). For the flat case (), Eq. (9) reduces to , i.e.

 H(z)=[ddz(dL(z)1+z)]−1. (10)

Hence the measurement of the luminosity distance of SN Ia allows us to find the expansion history of the Universe for .

The luminosity distance is expressed in terms of an apparent magnitude and an absolute magnitude of an object, as

 m−M=5log10(dL10 pc). (11)

The absolute magnitude at the peak of brightness is the same for any SN Ia under the assumption of standard candles, which is around Riess ; Perlmutter . The luminosity distance is known from Eq. (11) by observing the apparent magnitude . The redshift of an object is known by measuring the wavelength of light relative to its wavelength in the rest frame, i.e. . The observations of many SN Ia provide the dependence of the luminosity distance in terms of .

Expanding the function (9) around , it follows that

 dL(z) = 1H0[z+{1−E′(0)2}z2+O(z3)] (12) = 1H0[z+14(1−3wDEΩ(0)DE+Ω(0)K)z2+O(z3)],

where a prime represents a derivative with respect to . Note that, in the second line, we have used Eq. (8). In the presence of dark energy ( and ) the luminosity distance gets larger than that in the flat Universe without dark energy. For smaller (negative) and for larger this tendency becomes more significant. The open Universe without dark energy can also give rise to a larger value of , but the density parameter is constrained to be close to 0 from the WMAP data (more precisely, WMAP5 ). Hence, in the low redshift regime (), the luminosity distance in the open Universe is hardly different from that in the flat Universe without dark energy.

As we see from Eq. (12), the observational data in the high redshift regime () allows us to confirm the presence of dark energy. The SN Ia data released by Riess et al. Riess and Perlmutter et al. Perlmutter in 1998 in the redshift regime showed that the luminosity distances of observed SN Ia tend to be larger than those predicted in the flat Universe without dark energy. Assuming a flat Universe with a dark energy equation of state (i.e. the cosmological constant), Perlmutter et al. Perlmutter found that the cosmological constant is present at the 99 % confidence level. According to their analysis the density parameter of non-relativistic matter today was constrained to be (68 % confidence level) in the flat universe with the cosmological constant.

Over the past decade, more SN Ia data have been collected by a number of high-redshift surveys–such as SuperNova Legacy Survey (SNLS) SNLS , Hubble Space Telescope (HST) Gold1 ; Gold2 , and “Equation of State: SupErNovae trace Cosmic Expansion” (ESSENCE) Essence1 ; Essence2 survey. These data also confirmed that the Universe entered the epoch of cosmic acceleration after the matter-dominated epoch. If we allow the case in which dark energy is different from the cosmological constant (i.e. ), then observational constraints on and (or ) are not so stringent. In Fig. 1 we show the observational contours on for constant obtained from the “Union08” SN Ia data by Kowalski et al. Kowalski . Clearly the SN Ia data alone are not yet sufficient to place tight bounds on .

In the flat Universe dominated by dark energy with constant , it follows from Eq. (8) that . Integrating this equation, we find that the scale factor evolves as for and for . The cosmic acceleration occurs for . In fact, Fig. 1 shows that is constrained to be smaller than . If , which is called phantoms or ghosts phantom , the solution corresponding to the expanding Universe is given by , where is a constant. In this case the Universe ends at with a so-called big rip singularity Star00 ; CKW at which the curvature grows toward infinity.111There are other classes of finite-time singularities studied in Refs. Barrow86 ; Barrow04 ; Barrow04d ; Ste05 ; NOT05 ; Brevik ; Dab05 ; Sami05 ; Marium . In some cases quantum effects can moderate such singularities Nojiri04 ; NOT05 ; Singh06 ; Samart . The current observations allow the possibility of the phantom equation of state. We note, however, that the dark energy equation of state smaller than does not necessarily imply the appearance of the big rip singularity. In fact, in some of modified gravity models such as gravity, it is possible to realize without having a future big rip singularity AmenTsuji .

If the dark equation of state is not constant, we need to parametrize as a function of the redshift . This smoothing process is required because the actual observational data have discrete values of redshifts with systematic and statistical errors. There are several ways of parametrizations proposed so far. In general one can write such parametrizations in the form

 wDE(z)=∑n=0wnxn(z), (13)

where ’s are integers. We show a number of examples for the expansions:

 (i) Redshift:    xn(z)=zn, (14) (ii) Scale factor:  xn(z)=(1−a)n=(z1+z)n, (15) (iii) Logarithmic: xn(z)=[ln(1+z)]n. (16)

The parametrization (i) was introduced by Huterer and Turner Huterer and Weller and Albrecht Weller02 with , i.e. . Chevalier and Polarski ChePo and Linder Linderpara proposed the parametrization (ii) with , i.e.

 wDE(z)=w0+w1(1−a)=w0+w1z1+z. (17)

This has a dependence for and for . A more general form, , was proposed by Jassal et al. Jassal . The parametrization (iii) with was introduced by Efstathiou Efstathiou . A functional form that can be used for a fast transition of was also proposed Bassett1 ; Bassett2 . In addition to the parametrization of , a number of authors assumed parametric forms of Star98 , or Saini ; Alam ; Alam03 . Many works placed observational constraints on the property of dark energy by using such parametrizations Maor ; Cora ; Wang ; Nesseris1 ; Nesseris2 ; Ishak ; GuoOhta05 ; Wang2 ; Nesseris3 ; Crittenden ; Ichikawa ; Zhao1 ; Zhao2 ; Marina ; Ichikawa2 .

In Fig. 2 we show the SN Ia constraints combined with other measurements such as the WMAP 7-year WMAP7 and the BAO data Percival2 . The parametrization (17) is used in this analysis. The Gaussian prior on the present-day Hubble constant Riess09 ,  km sec Mpc (68 % confidence level), is also included in the analysis (obtained from the magnitude-redshift relation of 240 low- SN Ia at ). In Fig. 2, “” means a constraint coming from the measurement of gravitational lensing time delays Suyu . The joint constraint from WMAP+BAO+++SN gives the bound

 w0=−0.93±0.13,w1=−0.41+0.72−0.71, (18)

at the 68 % confidence level. Hence the current observational data are consistent with the flat Universe in the presence of the cosmological constant ().

ii.2 Cmb

The temperature anisotropies in CMB are affected by the presence of dark energy. The position of the acoustic peaks in CMB anisotropies depends on the expansion history from the decoupling epoch to the present. Hence the presence of dark energy leads to the shift for the positions of acoustic peaks. There is also another effect called the Integrated-Sachs-Wolfe (ISW) effect ISW induced by the variation of the gravitational potential during the epoch of the cosmic acceleration. Since the ISW effect is limited to large-scale perturbations, the former effect is typically more important.

The cosmic inflation in the early Universe Sta79 ; Kazanas ; Sato ; Guth predicts nearly scale-invariant spectra of density perturbations through the quantum fluctuation of a scalar field. This is consistent with the CMB temperature anisotropies observed by COBE COBE and WMAP WMAP1 . The perturbations are “frozen” after the scale ( is a comoving wavenumber) leaves the Hubble radius during inflation () Liddlebook ; BTW . After inflation, the perturbations cross inside the Hubble radius again () and they start to oscillate as sound waves. This second horizon crossing occurs earlier for larger (i.e. for smaller scale perturbations).

We define the sound horizon as , where is the sound speed and . The sound speed squared is given by

 c2s=1/[3(1+Rs)],Rs=3ρb/(4ργ), (19)

where and are the energy densities of baryons and photons, respectively. The characteristic angle for the location of CMB acoustic peaks is Page

 θA≡rs(zdec)d(c)A(zdec), (20)

where is the comoving angular diameter distance related with the luminosity distance via the duality relation Weinbergbook ; Tsujibook , and is the redshift at the decoupling epoch. The CMB multipole that corresponds to the angle (20) is

 ℓA=πθA=πd(c)A(zdec)rs(zdec). (21)

Using Eq. (9) and the background equation for the redshift (where and are the energy density of non-relativistic matter and radiation, respectively), we obtain HuSugi1 ; HuSugi2

where and , and is the so-called CMB shift parameter defined by Efs99

 R≡ ⎷Ω(0)mΩ(0)Ksinh(√Ω(0)K∫zdec0dzE(z)). (23)

The quantity can be expressed as

 Rs(a)=(3ωb/4ωγ)a. (24)

In Eq. (22), and correspond to the scale factor at the decoupling epoch and at the radiation-matter equality, respectively.

The change of cosmic expansion history from the decoupling epoch to the present affects the CMB shift parameter, which gives rise to the shift for the multipole . The general relation for all peaks and troughs of observed CMB anisotropies is given by Doran

 ℓm=ℓA(m−ϕm), (25)

where represents peak numbers ( for the first peak, for the first trough,…) and is the shift of multipoles. For a given cosmic curvature , the quantity depends weakly on and . The shift of the first peak can be fitted as Doran . The WMAP 5-year bound on the CMB shift parameter is given by WMAP5

 R=1.710±0.019, (26)

at the 68 % confidence level. Taking together with other values , , and constrained by the WMAP 5-year data, we obtain from Eq. (22). Using the relation (25) with we find that the first acoustic peak corresponds to , as observed in CMB anisotropies.

In the flat Universe () the CMB shift parameter is simply given by . For smaller (i.e. for larger ), tends to be smaller. For the cosmological constant () the normalized Hubble expansion rate is given by . Under the bound (26) the density parameter is constrained to be . This is consistent with the bound coming from the SN Ia data. One can also show that, for increasing , the observationally allowed values of gets larger. However, depends weakly on the . Hence the CMB data alone do not provide a tight constraint on . In Fig. 3 we show the joint observational constraints on and (for constant ) obtained from the WMAP 5-year data and the Union08 SN Ia data Kowalski . The joint observational constraints provide much tighter bounds compared to the individual constraint from CMB and SN Ia. For the flat Universe Kowalski et al. Kowalski obtained the bounds and (with statistical and systematic errors) from the combined data analysis of CMB and SN Ia. See also Refs. MMOT03 ; Hannestad02 ; Maor ; WL03 ; Cora05 ; LeeNg ; WT04 ; Rape ; Pogo ; Xia06 for related observational constraints.

ii.3 Bao

The detection of baryon acoustic oscillations first reported in 2005 by Eisenstein et. al. Eisenstein in a spectroscopic sample of 46,748 luminous red galaxies observed by the Sloan Digital Sky Survey (SDSS) has provided another test for probing the property of dark energy. Since baryons are strongly coupled to photons prior to the decoupling epoch, the oscillation of sound waves is imprinted in baryon perturbations as well as CMB anisotropies.

The sound horizon at which baryons were released from the Compton drag of photons determines the location of baryon acoustic oscillations. This epoch, called the drag epoch, occurs at the redshift . The sound horizon at is given by , where is the sound speed. According to the fitting formula of by Eisenstein and Hu EiHu , and are constrained to be around and  Mpc.

We observe the angular and redshift distributions of galaxies as a power spectrum in the redshift space, where and are the wavenumbers perpendicular and parallel to the direction of light respectively. In principle we can measure the following two ratios Shoji

 θs(z)=rs(zd)d(c)A(z),δzs(z)=rs(zd)H(z)c, (27)

where the speed of light is recovered for clarity. In the first equation is the comoving angular diameter distance related with the proper angular diameter distance via the relation . The quantity characterizes the angle orthogonal to the line of sight, whereas the quantity corresponds to the oscillations along the line of sight.

The current BAO observations are not sufficient to measure both and independently. From the spherically averaged spectrum one can find a combined distance scale ratio given by Shoji

 [θs(z)2δzs(z)]1/3≡rs(zd)[(1+z)2d2A(z)c/H(z)]1/3, (28)

or, alternatively, the effective distance ratio Eisenstein

 DV(z)≡[(1+z)2d2A(z)cz/H(z)]1/3. (29)

In 2005 Eisenstein et al. Eisenstein obtained the constraint  Mpc at the redshift . In 2007 Percival et al. Percival1 measured the effective distance ratio defined by

 rBAO(z)≡rs(zd)/DV(z), (30)

at the two redshifts: and . This is based on the data from the 2-degree Field (2dF) Galaxy Redshift Survey. These data provide the observational contour of BAO plotted in Fig. 3. From the joint data analysis of SN Ia Kowalski , WMAP 5-year WMAP5 , and BAO data Percival1 , Kowalski et al. Kowalski placed the constraints and for the constant equation of state of dark energy.

The recent measurement of the 2dF as well as the SDSS data provided the effective distance ratio to be and Percival2 . Using these data together with the WMAP 7-year data WMAP7 and the Gaussian prior on the Hubble constant  km sec Mpc Riess09 , Komatsu et al. WMAP7 derived the constraint (68 % confidence level) for the constant equation of state in the flat Universe. Adding the high- SN Ia in their analysis they found the most stringent bound: (68 % confidence level). Hence the CDM model is well consistent with a number of independent observational data.

Finally we should mention that there are other constraints coming from the cosmic age Feng05 , large-scale clustering LSS1 ; LSS2 ; LSS3 , gamma ray bursts Hooper ; Oguri ; Bai08 ; Wang08 ; Tsutsui , and weak lensing Jain:2003tb ; Takada04 ; Ishak05 ; Schimd ; Bridle ; Hollen . So far we have not found strong evidence for supporting dynamical dark energy models over the CDM model, but future high-precision observations may break this degeneracy.

Iii Cosmological constant

The cosmological constant is one of the simplest candidates of dark energy, and as we have seen in the previous section, it is favored by a number of observations. However, if the origin of the cosmological constant is a vacuum energy, it suffers from a serious problem of its energy scale relative to the dark energy density today Weinberg . The zero-point energy of some field of mass with momentum and frequency is given by . Summing over the zero-point energies of this field up to a cut-off scale  , we obtain the vacuum energy density

 ρvac=∫kmax0d3k(2π)312√k2+m2. (31)

Since the integral is dominated by the mode with large (), we find that

 ρvac≈∫kmax04πk2dk(2π)312k=k4max16π2. (32)

Taking the cut-off scale to be the Planck mass , the vacuum energy density can be estimated as . This is about times larger than the observed value .

Before the observational discovery of dark energy in 1998, most people believed that the cosmological constant is exactly zero and tried to explain why it is so. The vanishing of a constant may imply the existence of some symmetry. In supersymmetric theories the bosonic degree of freedom has its Fermi counter part which contributes to the zero point energy with an opposite sign222The readers who are not familiar with supersymmetric theories may consult the books Bailin ; Green .. If supersymmetry is unbroken, an equal number of bosonic and fermionic degrees of freedom is present such that the total vacuum energy vanishes. However it is known that supersymmetry is broken at sufficient high energies (for the typical scale  GeV). Therefore the vacuum energy is generally non-zero in the world of broken supersymmetry.

Even if supersymmetry is broken there is a hope to obtain a vanishing or a tiny amount of . In supergravity theory the effective cosmological constant is given by an expectation value of the potential for chiral scalar fields Bailin :

 V(φ,φ∗)=eκ2K[DiW(Kij∗)(DjW)∗−3κ2|W|2], (33)

where and are the so-called Kähler potential and the superpotential, respectively, which are the functions of and its complex conjugate . The quantity is an inverse of the derivative , whereas the derivative is defined by .

The condition corresponds to the breaking of supersymmetry. In this case it is possible to find scalar field values leading to the vanishing potential (), but this is not in general an equilibrium point of the potential . Nevertheless there is a class of Kähler potentials and superpotentials giving a stationary scalar-field configuration at . The gluino condensation model in superstring theory proposed by Dine Dine belongs to this class. The reduction of the 10-dimensional action to the 4-dimensional action gives rise to a so-called modulus field . This field characterizes the scale of the compactified 6-dimensional manifold. Generally one has another complex scalar field corresponding to 4-dimensional dilaton/axion fields. The fields and are governed by the Kähler potential

 K(T,S)=−(3/κ2)ln(T+T∗)−(1/κ2)ln(S+S∗), (34)

where and are positive definite. The field couples to the gauge fields, while does not. An effective superpotential for can be obtained by integrating out the gauge fields under the use of the -invariance Affleck :

 W(S)=M3pl[c1+c2exp(−3S/2c3)], (35)

where , and are constants.

Substituting Eqs. (34) and (35) into Eqs. (33), we obtain the field potential

 V = 1(T+T∗)3(S+S∗)(DSW)KSS∗(DSW)∗ (36) = M4pl(T+T∗)3(S+S∗)∣∣∣c1+c2exp(−3S/2c3){1+32c3(S+S∗)}∣∣∣2,

where, in the first line, we have used the property for the modulus term. This potential is positive because of the cancellation of the last term in Eq. (33). The stationary field configuration with is realized under the condition . The derivative, , does not necessarily vanish. When the supersymmetry is broken with a vanishing potential energy. Therefore it is possible to obtain a stationary field configuration with even if supersymmetry is broken.

The discussion above is based on the lowest-order perturbation theory. This picture is not necessarily valid to all finite orders of perturbation theory because the non-supersymmetric field configuration is not protected by any symmetry. Moreover some non-perturbative effect can provide a large contribution to the effective cosmological constant Kolda . The so-called flux compactification in type IIB string theory allows us to realize a metastable de Sitter (dS) vacuum by taking into account a non-perturbative correction to the superpotential (coming from brane instantons) as well as a number of anti D3-branes in a warped geometry KKLT . Hence it is not hopeless to obtain a small value of or a vanishing even in the presence of some non-perturbative corrections.

Kachru, Kallosh, Linde and Trivedi (KKLT) KKLT constructed dS solutions in type II string theory compactified on a Calabi-Yau manifold in the presence of flux. The construction of the dS vacua in the KKLT scenario consists of two steps. The first step is to freeze all moduli fields in the flux compactification at a supersymmetric Anti de Sitter (AdS) vacuum. Then a small number of the anti D3-brane is added in a warped geometry with a throat, so that the AdS minimum is uplifted to yield a dS vacuum with broken supersymmetry. If we want to use the KKLT dS minimum derived above for the present cosmic acceleration, we require that the potential energy at the minimum is of the order of  GeV. Depending on the number of fluxes there are a vast of dS vacua, which opened up a notion called string landscape Susskind .

The question why the vacuum we live in has a very small energy density among many possible vacua has been sometimes answered with the anthropic principle ant1 ; ant2 . Using the anthropic arguments, Weinberg put the bound on the vacuum energy density Weinbergbound

 −10−123m4pl \raisebox{-4.3pt}{<∼} ρΛ \raisebox{-4.3pt}{<∼} 3×10−121m4pl. (37)

The upper bound comes from the requirement that the vacuum energy does not dominate over the matter density for the redshift . Meanwhile the lower bound comes from the condition that does not cancel the present cosmological density. Some people have studied landscape statistics by considering the relative abundance of long-lived low-energy vacua satisfying the bound (37) Garriga00 ; Denef ; GLV ; Blum . These statistical approaches are still under study, but it will be interesting to pursue the possibility to obtain high probabilities for the appearance of low-energy vacua.

Even in 1980s there were some pioneering works for finding a mechanism to make the effective cosmological constant small. For example, let us consider a 4-form field expressed by a unit totally anti-symmetric tensor , as ( is a constant). Then the energy density of the 4-form field is given by . Taking into account a scalar field with a potential energy , the total energy density is . In 1984 Linde Linde84 considered the quantum creation of the Universe and claimed that the final value of can appear with approximately the same probability because can take any initial value such that .

In 1987-1988 Brown and Teltelboim Brown1 ; Brown2 studied the quantum creation of closed membranes by totally antisymmetric tensor and gravitational fields to neutralize the effective cosmological constant with small values. The constant appearing in the energy density of the 4-form field can be quantized in integer multiples of the membrane charge , i.e. . If we consider a negative bare cosmological constant (as in the KKLT model) in the presence of the flux energy density , then the effective gravitational constant is given by . The field strength of the 4-form field is slowly discharged by a quantum Schwinger pair creation of field sources []. However, in order to get a tiny value of consistent with the dark energy density today, the membrane change is constrained to be very small (for natural choices of ) Boussoreview .

In 2000 Bousso and Polchinski BoussoPol considered multiple 4-form fields that arise in M-theory compactifications and showed that the small value of can be explained for natural choices of . More precisely, if we consider 4-form fields as well as membrane species with charges and the quantized flux , the effective cosmological constant is given by

 Λ=−Λb+J∑i=1n2iq2i/2. (38)

Bousso and Polchinski BoussoPol showed that, for natural values of charges () there exists integers such that with . This can be realized for and .

There are some interesting works for decoupling from gravity. In the cascading gravity scenario proposed in Ref. Rham the cosmological constant can be made gravitationally inactive by shutting off large-scale density perturbations. In Ref. Afshordi an incompressible gravitational Aether fluid was introduced to degraviate the vacuum. In Refs. Paddy1 ; Paddy2 Padmanabhan showed an example to gauge away the cosmological constant from gravity according to the variational principle different from the standard method. See also Refs. Hawking84 ; Kachru00 ; Kaloper00 ; Feng01 ; Tye ; Garriga01 ; Yokoyama02 ; Burgess03 ; Burgess04 ; Mukoh04 ; Sorkin ; Kane ; Dolgov08 for other possibilities to solve the cosmological constant problem. If the cosmological constant is completely decoupled from gravity, it is required to find alternative models of dark energy consistent with observations.

In the subsequent sections we shall consider alternative models of dark energy, under the assumption that the cosmological constant problem is solved in such a way that it vanishes completely.

Iv Modified matter models

In this section we discuss “modified matter models” in which the energy-momentum tensor on the r.h.s. of the Einstein equations contains an exotic matter source with a negative pressure. The models that belong to this class are quintessence, k-essence, coupled dark energy, and generalized Chaplygin gas.

iv.1 Quintessence

A canonical scalar field responsible for dark energy is dubbed quintessence quin5 ; Zlatev (see also Refs. quin1 ; Ford ; quin2 ; quin3 ; quin4 ; Ferreira1 ; Ferreira2 for earlier works). The action of quintessence is described by

 S=∫d4x√−g[12κ2R−12gμν∂μϕ∂νϕ−V(ϕ)]+SM, (39)

where is a Ricci scalar, and is a scalar field with a potential . As a matter action , we consider perfect fluids of radiation (energy density , equation of state ) and non-relativistic matter (energy density , equation of state ).

In the flat FLRW background radiation and non-relativistic matter satisfy the continuity equations and , respectively. The energy density and the pressure of the field are and , respectively. The continuity equation, , translates to

 ¨ϕ+3H˙ϕ+V,ϕ=0, (40)

where . The field equation of state is given by

 wϕ≡Pϕρϕ=˙ϕ2−2V(ϕ)˙ϕ2+2V(ϕ). (41)

From the Einstein equations (4) we obtain the following equations

 H2 = κ23[12˙ϕ2+V(ϕ)+ρm+ρr], (42) ˙H = −κ22(˙ϕ2+ρm+43ρr). (43)

Although during radiation and matter eras, the field energy density needs to dominate at late times to be responsible for dark energy. The condition to realize the late-time cosmic acceleration corresponds to , i.e. from Eq. (41). This means that the scalar potential needs to be flat enough for the field to evolve slowly. If the dominant contribution to the energy density of the Universe is the slowly rolling scalar field satisfying the condition , we obtain the approximate relations and from Eqs. (40) and (42), respectively. Hence the field equation of state in Eq. (41) is approximately given by

 wϕ≃−1+2ϵs/3, (44)

where is the so-called slow-roll parameter Liddlebook . During the accelerated expansion of the Universe, is much smaller than 1 because the potential is sufficiently flat. Unlike the cosmological constant, the field equation of state deviates from ().

Introducing the dimensionless variables , , and , we obtain the following equations from Eqs. (40), (42), and (43) CLW ; Macorra ; Nunes ; review :

 x′1=−3x1+√62λx22+12x1(3+3x21−3x22+x23), (45) x′2=−√62λx1x2+12x2(3+3x21−3x22+x23), (46) x′3=−2x3+12x3(3+3x21−3x22+x23), (47)

where a prime represents a derivative with respect to , and is defined by . The density parameters of the field, radiation, and non-relativistic matter are given by , , and , respectively. One has constant for the exponential potential CLW

 V(ϕ)=V0e−κλϕ, (48)

in which case the fixed points of the system (45)-(47) can be derived by setting (). The fixed point that can be used for dark energy is given by

 (x1,x2,x3)=(λ/√6,√1−λ2/6,0),wϕ=−1+λ2/3,Ωϕ=1. (49)

The cosmic acceleration can be realized for , i.e. . One can show that in this case the accelerated fixed point is a stable attractor CLW . Hence the solutions finally approach the fixed point (49) after the matter era [characterized by the fixed point ].

If varies with time, we have the following relation

 λ′=−√6λ2(Γ−1)x1, (50)

where . For monotonically decreasing potentials one has and for and and for . If the condition

 Γ=VV,ϕϕV2,ϕ>1, (51)

is satisfied, the absolute value of decreases toward 0 irrespective of the signs of Paul99 . Then the solutions finally approach the accelerated “instantaneous” fixed point (49) even if is larger than 2 during radiation and matter eras Macorra ; Nunes . In this case the field equation of state gradually decreases to , so the models showing this behavior are called “freezing” models Caldwell . The condition (51) is the so-called tracking condition under which the field density eventually catches up that of the background fluid.

A representative potential of the freezing model is the inverse power-law potential () quin3 ; Paul99 , which can appear in the fermion condensate model as a dynamical supersymmetry breaking Binetruy . In this case one has and hence the tracking condition is satisfied. Unlike the cosmological constant, even if the field energy density is not negligible relative to the background fluid density around the beginning of the radiation era, the field eventually enters the tracking regime to lead to the late-time cosmic acceleration Paul99 . Another example of freezing models is , which has a minimum with a positive energy density at which the field is eventually trapped. This potential is motivated in the framework of supergravity Brax .

There is another class of quintessence potentials called “thawing” models Caldwell . In thawing models the field with mass has been frozen by the Hubble friction (i.e. the term ) until recently and then it begins to evolve after drops below . At early times the equation of state of dark energy is , but it begins to grow for . The representative potentials that belong to this class are (a) () and (b) . The potential (a) with was originally proposed by Linde Linde87 to replace the cosmological constant by a slowly evolving scalar field. In Ref. Kallosh03 this was revised to allow for negative values of . The universe will collapse in the future if the system enters the region with . The potential (b) is motivated by the Pseudo-Nambu-Goldstone Boson (PNGB), which was introduced in Ref. Frieman in response to the first tentative suggestions for the existence of the cosmological constant. The small mass of the PNGB model required for dark energy is protected against radiative corrections, so this model is favored theoretically. In fact there are a number of interesting works to explain the small energy scale  eV required for the PNGB quintessence in supersymmetric theories Nomura ; Choi ; Kim ; Hall . See Refs. CNR ; Townsend ; Heller ; Kallosh1 ; Kallosh2 ; Fre ; Piazza1 ; Piazza2 ; Albrecht02 ; Burgess03d ; Burgess for the construction of quintessence potentials in the framework of supersymmetric theories.