Strong limits on the possible decay of the vacuum energy into CDM or CMB photons

Abstract

We investigate models that suggest that the vacuum energy could decay into cold dark matter (CDM) or into a homogeneous distribution of thermalized cosmic microwave background (CMB) photons. We show that the agreement of the density fluctuation spectra obtained from the CMB and galaxy distribution data puts strong limits on the rate of vacuum energy decay. A vacuum energy decaying into CDM increases the density of the CDM rho, diluting the CDM density fluctuations (<FONT FACE=Symbol>dr</FONT>/rho)². The temperature fluctuations of the CMB photons (deltaT/T)² are approximately proportional to (<FONT FACE=Symbol>dr</FONT>/rho)², at the recombination epoch. We define F as the predicted increase of (<FONT FACE=Symbol>dr</FONT>/rho)² (or (deltaT/T)²) at the recombination epoch. Since the present observed (<FONT FACE=Symbol>dr</FONT>/rho)² derived from the CMB and galaxy distribution data agree to ~ 10%, the maximum value for F is Fmax <FONT FACE=Symbol>@</FONT> 1.1. Our results indicate that the rate of decay of the vacuum energy into CDM or CMB photons is extremely small.

Strong limits on the possible decay of the vacuum energy into CDM or CMB photons

Reuven Opher; Ana Pelinson

IAG, Universidade de São Paulo, Rua do Matão, 1226, Cidade Universitária, CEP 05508-900. São Paulo, S.P., Brazil

ABSTRACT

We investigate models that suggest that the vacuum energy could decay into cold dark matter (CDM) or into a homogeneous distribution of thermalized cosmic microwave background (CMB) photons. We show that the agreement of the density fluctuation spectra obtained from the CMB and galaxy distribution data puts strong limits on the rate of vacuum energy decay. A vacuum energy decaying into CDM increases the density of the CDM r, diluting the CDM density fluctuations (dr/r)². The temperature fluctuations of the CMB photons (dT/T)² are approximately proportional to (dr/r)², at the recombination epoch. We define F as the predicted increase of (dr/r)² (or (dT/T)²) at the recombination epoch. Since the present observed (dr/r)² derived from the CMB and galaxy distribution data agree to ~ 10%, the maximum value for F is F_max@ 1.1. Our results indicate that the rate of decay of the vacuum energy into CDM or CMB photons is extremely small.

I. INTRODUCTION

Recent observations indicate that the universe is spatially flat and undergoing a late time acceleration. This acceleration has been attributed to a dark energy component with negative pressure which can induce repulsive gravity. The simplest and most obvious candidate for this dark energy is the cosmological constant L (which can be interpreted as vacuum energy) with the equation of state w = p / r = -1, where p is the pressure and r is the energy density. A decaying vacuum energy is very attractive since it may link the present vacuum energy that is accelerating the universe today, with perhaps the large vacuum energy that created the inflation epoch in the past.

We analyze how the observed cosmic microwave background (CMB) and large galaxy survey data constrain the decaying vacuum energy models into cold dark matter (CDM) from the recombination era (redshift z ~ 1070) to the present (z ~ 0) [1]. A vacuum energy decaying into CDM increases the CDM density, diluting its (dr/r)². In order to evaluate (dr/r)² at the recombination era, when it created the dT/T of the CMB, its present measured value obtained from the galaxy distribution data, extrapolated back to the recombination era, must be increased by a factor F. The density fluctuations derived from the CMB data were compared with those derived from the 2dF Galaxy Redshift Survey (2dFGRS) [2, 3]. Since the present (dr/r)² derived from the CMB and galaxy distribution data agree to ~ 10 per cent, the maximum value for F is F_max@ 1.1 (see [4] for the final data set of the 2dFGRS).

We made a similar analysis for the possible decay of the vacuum energy into CMB photons [5]. In this scenario, the temperature fluctuations, created at the recombination epoch (dT/T)_rec, were diluted by the photons created by the vacuum energy decay making the temperature fluctuations at present smaller.

II. VACUUM ENERGY DECAYING INTO CDM

A vacuum energy decaying into CDM increases the CDM density, diluting the CDM fluctuations (dr/r)². Consequently, a larger density fluctuation spectrum (dr/r)² is predicted at the recombination era (z_rec = 1070) by the factor

where

is the matter density for a constant vacuum energy density, where º 3/(8pG) ~ 1.88× 10^-29g cm^-3 is the critical density, and is the normalized matter density, = / (~0.3). The difference between the matter density _M and the matter density predicted by the model in which the vacuum energy decays into matter, r_Mv, is

The density r_Mv(z) is normalized at redshift z = 0 [r_Mv(z = 0) º r⁰_M]. In order to describe the transfer of the vacuum energy density, r_L, into matter r_Mv [6], we use the conservation of energy equation,

where P_Mv is the pressure due to r_Mv. For CDM, we have P_Mv = 0.

There exists an extensive list of phenomenological L-decay laws. Several models in the literature are described by a power law dependence

where º r_L(z = 0), which we investigate here.

The solution for the matter density has the form [6, 7]

where A and B are unknown constants. Using Eqs.(6) and (5) in Eq.(4), the dependence of r_Mv as a function of n is

Using Eqs.(2) and (7) in Eq.(3), we find from Eq.(1) that

If, as discussed in section I, the density power spectrum from observations can be increased by no more than approximately 10% due to the decay of the vacuum energy, we then have a maximum value for the F factor F_max@ 1.1. This maximum value gives n_max@ 0.06.

We also considered a recent model suggested by the renormalization group equation of the effective quantum field theory which has a L-decay dependence [8]

where r_L(z = 0) º , k = 0, and

The dimensionless parameter n comes from the renormalization group

where sM² is the sum of all existing particles (fermions with s = -1 and bosons with s = +1). The range of n is n Î (0,1) [9].

Using Eqs.(9) and (10), the matter density as a function of z and n, in the matter era, is

Using Eqs.(12) and (2) in Eq.(3), we find from Eq.(1), the factor F modifying the density power spectrum:

Using F_max@ 1.1 we place an upper limit on the n parameter: n_max@ 2.3 × 10^-3.

III. VACUUM ENERGY DECAYING INTO CMB PHOTONS

According to the standard model, the temperature fluctuations observed today are given by the expression

where is approximately constant and the temperature dependence of T(z) is

where T₀~ 2.75 K is the present CMB temperature [10]. The present value of (dr/r)² is gotten from the relation

where (z_rec® z = 0) is the growth factor from the recombination era until the present time.

When we assume that the decay is adiabatic, the vacuum energy decays into a homogeneous distribution of thermalized black body CMB photons and the standard linear temperature dependence becomes modified [11]. The decay can be described by a generic temperature dependence,

of the CMB photons. In principle, the possible range of b is b Î [0,1] [11].

There are two effects due to the decaying vacuum energy into CMB photons:

1) Since the temperature fluctuations at the recombination epoch (dT/T)_rec should be diluted by the photons created, the temperature fluctuations at present become smaller; and

2) The value of the recombination redshift

_rec is higher than that of the standard model z_rec since the universe is cooler at any given redshift.

Due to the dilution of dT/T, instead of Eq.(14) of the standard model, we must use the relation

where F₁ is defined by

DT(z) is the difference between the recombination temperature T(z_rec) predicted by the standard model and that of the model in which the vacuum energy decays into photons at temperature (z_rec):

Using Eqs.(17), (19), and (20), we obtain

From Eqs.(17) and (20), (z) was lower than T(z) by DT at z_rec. Thus, the resultant recombination redshift _rec was higher than that of the standard model z_rec. Instead of Eq.(16), (dr/r) at z ~ 0 is now given by

where (_rec® z = 0) is the density fluctuation growth factor from the recombination era at _rec until the present epoch. Therefore, instead of Eq.(14), we have

Using Eqs.(16) and (18), we have

and from Eqs.(22) and (23),

Equations (24) and (25) give the correction factor F₂ due to the change in the value of the recombination redshift,

The growth of a perturbation in a matter-dominated Einstein-de Sitter universe is dr/r µ a = (1 + z)^-1, where a is the cosmic scale factor [12]. Thus, the growth factor is

~ (1 + z).

We then find from Eq.(26)

The temperature at z_rec in the standard model is

In order for the temperature at the recombination epoch rec to be the same as the standard model T(z_rec), when the vacuum energy is decaying into CMB photons, we must have, from Eq.(17),

From Eq.(27), we then have

The total factor F is composed of F₁, due to the dilution of the CMB as a result of vacuum energy decay, and F₂, due to the change in the redshift of the recombination epoch. Assuming that the effects described by and are independent and that the total factor F is the product of and , we have

Thus, from Eqs.(21), (30) and (31), the condition for the maximum value of b Î [0,1] is

where

As noted above, the maximum value of F from observations is F_max@ 1.1. For z_rec~ 1070, we find a very small maximum value of the b parameter, b_max~ 3.4 × 10^-3 .

IV. CONCLUSIONS

We showed that the CMB and large galaxy survey data agreement puts strong limits on the rate of a possible decay of the vacuum energy into CDM and CMB photons.

When the vacuum energy decays into CDM, dr/r is diluted and the density fluctuation spectrum is amplified by a factor F at the recombination era. The (dr/r)² derived from the CMB and galaxy distribution data agree to ~ 10%, implying a maximum value for F: F_max@ 1.1.

We found that the decay of the vacuum energy into CDM as a scale factor power law r_L µ (1 + z)ⁿ, gives a maximum value for the exponent n_max@ 0.06. For a parametrized vacuum decay into a CDM model with the form r_L(z,n) = r_L(z = 0) + [n/(1 – n)][(1 + z)^3(1-n) - 1] , where is the present critical density, an upper limit on the n parameter was found to be n_max@ 2.3 × 10^-3.

We made a similar analysis for the possibility of the decay of the vacuum energy into CMB photons. When photon creation due to the vacuum energy decay takes place, the standard linear temperature dependence, T(z) = T₀(1 + z), where T₀ is the present CMB temperature, is modified. We can place an upper limit on the b parameter for the decay of the vacuum energy into CMB photons, parametrized by a change in the CMB temperature at a given redshift z: (z) = T₀(1 + z)^1-b. We find that b_max@ 3.4 × 10^-3.

Our results indicate that the rate of decay of the vacuum energy into CDM or CMB photons is extremely small. Since the results show that the vacuum energy can only decay to a negligible extent into cold dark matter or CMB photons, we conclude that if the vacuum energy is decaying, it is probably decaying, for example, into hot dark matter (e.g., high energy neutrinos) or exotic matter (e.g., scalar fields), since they do not affect the (dr/r)² or the dT/T CMB spectra.

Acknowledgments

R.O. thanks the Brazilian agencies FAPESP (grant 00/06770-2) and CNPq (grant 300414/82-0) for partial support. A.P. thanks FAPESP for financial support (grants 03/04516-0 and 00/06770-2).

References

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Received on 15 October, 2005

Publication Dates

• Publication in this collection
  16 Jan 2006
• Date of issue
  Dec 2005

History

• Received
  15 Oct 2005
• Accepted
  15 Oct 2005