^{1}

^{*}

^{1}

We present a Dark Energy (DE) model based on a scalar field with an inverse power law potential (IPL)
V(∅)=M^{4+n}∅^{−n}. We consider three different models n=1/2, n=3/4 and n=1 and we vary the value of M and the initial amount of energy density
Ω
_{∅} at the scale factor a
_{c}. We obtain a time dependent equation of state (EoS)
, with w
_{∅}=1/3 at early times for a scale factor a<a
_{c} with a steep transition to
w
_{∅}=1 at
,
,
, lasting a long period of time and a subsequent descent
w
_{∅}
=-1 to for
to finally grow to
w
_{∅}= -0.906,
w
_{∅}=-0.932,
w
_{∅}=-0.924 for n=1/2, n=3/4 and n=1 respectively. The values of
*M* and
Ω_{∅}(a_{c}) are
M(eV)= 4.63,127.31,2465.46 and
Ω
_{∅}(a
_{c})=
0.038,0.148,0.227 for
n= 1/2, n= 3/4 and n=1 respectively. We show the differences in the evolution of
*H*, the CMB and Matter power spectra, and the redshift space distortion (RSD)
parameter. Precision cosmological data allow us to test the dynamics of Dark Energy and we obtain in all three cases a reduction of
compared to
∧CDM with and an equivalent fit for CMB and SNIa data.

It has passed almost twenty years since the accelerated expansion of the universe was first observed by distance measurements using type Ia supernovae (SNeIa) [

In the standard cosmological model (LCDM), the energy density of the universe at present day is made up of 69% of DE described by a cosmological constant L, 26% of dark matter (DM) while only 5% corresponds to the Standard Model (SM) particles consisting mainly of photons, neutrinos, and ordinary matter (protons, neutrons, and electrons). However, although the LCDM model has proved to agree very well with the observations [^{120} orders of magnitude below conservative estimations based on quantum field theory, while the “coincidence” problem inquires why the energy densities of the DE and the matter are of the same order of magnitude precisely at present time. This scenario has enforced the quest of other mechanisms to explain the nature of DE. Alternative to L, scalar fields ϕ have been extensively explored as a source of DE [

The outline of the paper is the following. In Section 2 we discuss the basic picture of the model and present the dynamical equations. The constraints set by the data are presented in Section 3. We study the cosmological consequences of the model in Section 4 and give our conclusions in Section 5.

Scalar field ϕ theories have been proposed as possible sources to describe DE and a wide range of models have been studied in recent years [

V ( ϕ ) = M 4 + n ϕ − n (1)

proposed by RP [

The evolution of ϕ in a homogeneous flat universe described by the Friedmann-Lemaître-Robertson-Walker metric is completely determined by the Klein-Gordon equation

ϕ ¨ + 3 H ϕ ˙ + d V d ϕ = 0, (2)

where the dots stand for cosmic time derivatives, and the Hubble expansion rate:

H 2 ≡ ( a ˙ a ) 2 = 8 π G 3 ρ t o t = 8 π G 3 ( ρ m o a − 3 + ρ r o a − 4 + ρ ϕ ) (3)

with ρ t o t ( a ) = ρ m o a − 3 + ρ r o a − 4 + ρ ϕ the total energy density, ρ m o , ρ r o the present day energy densities of matter and radiation and the redshift z is given by a = ( 1 + z ) − 1 with a o = 1 . The energy density and pressure for the scalar field ϕ are

ρ ϕ = 1 2 ϕ ˙ 2 + V ( ϕ ) , p ϕ = 1 2 ϕ ˙ 2 − V ( ϕ ) , (4)

with an equation of state (EOS)

w ϕ = p ϕ ρ ϕ = 1 2 ϕ ˙ 2 − V ( ϕ ) 1 2 ϕ ˙ 2 + V ( ϕ ) (5)

and a mass given by

m ϕ 2 ≡ ∂ 2 V ∂ ϕ 2 = n ( n + 1 ) M 2 ( M ϕ ) 2 + n . (6)

Since the potential V in Equation (1) is an inverse power of ϕ , the evolution of ϕ is an increasing function of time, i.e. of the expansion of the universe, while the mass m ( ϕ ) is a decreasing function.

We define the scale factor a c and mass parameter M as the scale when the scalar field takes the value of M (i.e. ϕ c ( a c ) = M ) and at this time we have

V ( ϕ ( a c ) ) = M 4 , m ( ϕ ( a c ) ) = n ( n + 1 ) M . (7)

The parameters a c , M and the DE energy density parameter Ω ϕ ≡ ρ ϕ / ρ t o t at a c are related by

Ω ϕ c = 3 M 4 ρ m o a c − 3 + ρ r o a c − 4 + 3 M 4 (8)

where the initial conditions at a c are ϕ c ( a c ) = M , ρ ϕ ( a c ) = 2 M 4 / ( 1 − w ϕ c ) = 3 M 4 , w ϕ c = 1 / 3 , and ϕ ˙ ( a c ) = 2 M 4 ( 1 + w ϕ c ) / ( 1 − w ϕ c ) = 2 M 2 .

For particles in thermal equilibrium as long as they are relativistic we have m ≪ T with T the temperature with an energy density given by ρ ϕ ( T ≫ m ) = ( π 2 / 30 ) g ϕ T 4 , with g ϕ = 1 the degrees of freedom of ϕ , and an EoS w ϕ = 1 / 3 . Notice that at a c we have ρ ϕ ( a c ) ~ M 4 ~ T 4 , i.e. m ~ M ~ T . Therefore, for scales a < a c we have m < T and the evolution of ϕ has an EoS w ϕ = 1 / 3 while for a > a c the scalar fields are no longer relativistic m > T and its evolution is determined by Equations ((2) and (3)) with the IPL potential in Equation (1).

The homogeneous background approximation must be refined by considering also the perturbations of the different fluids. We stay in the linear regime where the energy density and other quantities can be decomposed into a homogeneous part (commonly denoted with a bar) and a small position-dependent perturbation. We solve the perturbed equations in the syncrhonous gauge defined by the line element [

d s 2 = a 2 ( τ ) ( − d τ 2 + ( δ i j + h i j ) d x i d x j ) , (9)

where d τ ≡ d t / a denotes the conformal time. The perturbations in the energy density and pressure of the scalar field are given by:

δ ρ ϕ = ϕ ¯ ′ δ ϕ ′ a 2 + d V d ϕ δ ϕ , δ P ϕ = ϕ ¯ ′ δ ϕ ′ a 2 − d V d ϕ δ ϕ , (10)

where the primes stand for conformal time derivatives. The evolution of δ ϕ in Fourier space is determined by:

δ ϕ ″ + 2 H δ ϕ ′ + ( k 2 + a 2 d 2 V d 2 ϕ ) δ ϕ = − 1 2 ϕ ¯ ′ h ′ (11)

Here H ≡ a ′ / a is the conformal expansion rate and h = T r ( h i j ) . The scalar field enters the perturbation equations of the other fluids [

δ ″ c + H δ ′ c − 3 2 H 2 ∑ Ω i δ i ( 3 c s , i 2 + 1 ) = 0, (12)

where the sum runs over all the fluids with sound speed c s , i 2 = δ P i / δ ρ i .

We explore the parameter space using the CosmoMC [

dynamics described in section 2. In our analysis, we take n = 1 2 , 3 4 , 1 and find

the best fit point for each case varying the physical densities of baryons ( Ω b h 2 ) and cold dark matter ( Ω c h 2 ), the optical depth ( τ ), the spectral index ( n s ), the amplitude of scalar perturbations ( A s ), a c and the density parameter of the scalar field at that time ( Ω ϕ c ) and M, where Ω r h 2 and Ω m h 2 stand for the densities of radiation (photons and three massless neutrino species) and matter (baryons and CDM), respectively; Ω ϕ o is given by the solution of the system of Equations ((2) and (3)) at present time.

We see that the transition from the radiation-like state to the scalar-field state takes place well deep in the radiation era for all the cases. However, the transition occurs later as we go to smaller values of n, while the scalar field density at a c ( Ω ϕ c ) and the scale of energy M increase as we go to larger n. The same trend is observed in H o and the DE density parameter at present. Nevertheless, none of these two is larger than in LCDM.

Parameter | n = 1 / 2 | n = 3 / 4 | n = 1 | LCDM |
---|---|---|---|---|

a c | 1.79 × 10 − 5 | 9.30 × 10 − 7 | 5.48 × 10 − 8 | --- |

Ω ϕ c | 0.0381 | 0.1479 | 0.2268 | --- |

M (eV) | 4.63 | 127.31 | 2465.46 | --- |

Ω b h 2 | 0.02225 | 0.02254 | 0.02266 | 0.02242 |

Ω c h 2 | 0.1179 | 0.1174 | 0.1173 | 0.1181 |

H o | 66.60 | 67.78 | 67.86 | 68.63 |

Ω D E o | 0.684 | 0.695 | 0.696 | 0.702 |

w D E o | −0.906 | -0.932 | −0.924 | −1 |

σ 8 o | 0.822 | 0.849 | 0.865 | 0.871 |

r BAO ( 0.57 ) | 0.0720 | 0.0724 | 0.0752 | 0.07230 |

f σ 8 ( 0.57 ) | 0.4697 | 0.4847 | 0.4934 | 0.5013 |

Y P | 0.2508 | 0.2628 | 0.2719 | 0.2467 |

χ 2 ( n = 1 / 2 ) = 5.737 ( BAO ) + 781.811 ( CMB ) + 697.300 ( SNeIa ) . χ 2 ( n = 3 / 4 ) = 5.674 ( BAO ) + 776.807 ( CMB ) + 695.581 ( SNeIa ) . χ 2 ( n = 1 ) = 5.589 ( BAO ) + 776.214 ( CMB ) + 695.658 ( SNeIa ) . χ 2 ( Λ CDM ) = 7.115 ( BAO ) + 776.883 ( CMB ) + 695.075 ( SNeIa ) .

The evolution of the matter, radiation and DE densities is shown in

to the scalar field occurs, the EOS leaps abruptly to 1 and stays at this value for some time, then drops to −1 around the decoupling epoch ( z * ≈ 1090 for the three models) and finally grows to w D E o > − 1 at recent times. The inner panel zooms into the late-time behaviour.

It is interesting to note that in [

The different late-time dynamics followed by the EOS leaves distinctive imprints on some quantities probed by the cosmological observables. The immediate consequence of such difference is the change in the amount of DE which modifies the size and evolution of the expansion rate H affecting the cosmological distances measured by the SNeIa, BAO and CMB observations. In this regard, we recall that SNeIa flux measurements provide an estimation of the luminosity distance given by d L ( z ) = ( 1 + z ) ∫ 0 z d z ′ / H ( z ′ ) , while BAO measurements are sensitive to the ratio r B A O ( z ) ≡ r d r a g / D V ( z ) (c.f.

At perturbation level, the IPL model also leaves important imprints on the CMB power spectrum and the evolution of matter perturbations. When we run the models with the same parameters, a change in cosmological distances is reflected in a shift of the position of the peaks of the CMB power spectrum. Moreover, although the amount of extra early radiation vanishes once the transition occurs and therefore it does not have any direct influence on the physical processes from that time onwards, a larger value of primordial helium fraction Y P produced by this extra radiation (c.f.

parameters.

As far as the evolution of matter perturbations is concerned,

Here we consider three different IPL potentials V = M 4 + n ϕ − n models with a power n = 1 / 2 , 3 / 4 and n = 1 . We have seen that they have a better cosmological fit than the standard LCDM model, reducing BAO χ 2 by 20% and leaving CMB an SNIa with an equivalent fit. With more precise measurements coming in the near future we could be at the stage to determine the dynamics of Dark Energy. In particular BAO and RSD data will be vital to achieve this goal.

We acknowledge financial support from UNAM DGAPA IN103518 and Conacyt Fronteras 281.

Almaraz, E. and de la Macorra, A. (2018) Constraints on Dynamical Dark Energy with Precision Cosmological Data. Journal of Modern Physics, 9, 302-313. https://doi.org/10.4236/jmp.2018.92021