Cosmological constant and quintessence

But DM does not seem the end of the story regarding the energy of the universe. There is a growing evidence that the energy is the critical one

$$\Omega _0=\Omega _M+\Omega _{\wedge }=1,$$ (32)

with $\Omega \simeq 0.6$ corresponding to a cosmological constant.

This constant $\wedge$ had been introduced by Einstein in his equations in a way equivalent to add to the energy-momentum tensor Eq.(2) a term

$$T_{vac}^{\mu \nu }=\rho _{vac}g^{\mu \nu },$$ (33)

with the definition $\rho _{vac}=\wedge$ $/(8\pi G_N)$ . Note that Eq.(33) gives a negative pressure $p=-\rho _{vac}$ . This justifies the equivalence of $\rho _{vac}$ with a vacuum energy density because thermodynamically this system increases its energy with volume in agreement with a negative pressure.

Now, with Eqs.(3), (4) and (32), the deceleration parameter is

$$q_o=-\frac{(\ddot{R}/R)_o}{H_o^2}=\frac 12+\frac 32\Sigma _i\Omega _iw_i ,$$ (34)

where $p_i=e_i\rho _i$ for each component, so that $w_i=\frac 13,0,-1$ for radiation, nonrelativistic matter and vacuum respectively. For the case of Eq.(32), $q_o=\frac 12-\frac 32\Omega _{\wedge }=-0.4$, and the universe now would be actually accelerating! This is precisely what is emerging from the observation of distant supernovae(13).

The existence of a cosmological constant leads to an agreement between the relative large age of the universe and large present Hubble parameter $H_o$ because it acts as opposing the effect of gravitational attraction. Another impressive evidence is the fit of the so-called acoustic peak of the anisotropy of CBR which favours $\Omega _o=1$ and $\Omega _o=0.6$, issue that will be completely settled by the future satellites MAP and Planck.

It is interesting(14) that also the reionization of intergalactic matter, corresponding to the evolution of universe, since a redshift $z\sim 5$ is in favour of CDM plus a cosmological constant contribution $\Omega _{\wedge}\sim 0.6$.

From the point of view of quantum field theory it is very hard to explain why the vacuum energy should have a value of the order of Eq.(7) which, using the Appendix, can be rewritten as

$$\rho _c\sim (0.001eV)^4.$$ (35)

In fact, since if SUSY were exact, the vacuum energy should be zero, one might expect that $\rho _c$ would depend on the scale of SUSY breaking, $i.e.$, $\rho _c\sim (1\ TeV)^4$, i.e., an estimation wrong by 60 orders of magnitude!

One may prefer to think that for some unexplained reason Einstein cosmological constant and vacuum energy exactly compensates and that what one observes today is a sort of dynamical cosmological constant due to a uniform field(15) $\phi$ called ''quintessence''. Things would go in the following way:

Since the energy-momentum tensor for a scalar field is

$$T^{\mu \nu }=\frac{\partial \mathcal{L}}{\partial \partial _\mu \phi }%% \partial ^v\phi -g^{\mu \nu }\mathcal{L}$$ (36)

for the spatially homogeneous $\phi$ with a potential $V$, comparing with Eq.(2)

$$\rho =\frac 12\dot{\phi}^2+V, \hspace{0.4cm} p=\frac 12\dot{\phi}-V.$$ (37)

On the other hand, the equation of motion, similar to that of axion Eq.(30), is

$$\ddot{\phi}+3H(t)\dot{\phi}+V^{\prime }(\phi )=0.$$ (38)

Therefore, as discussed for the axion, when $H>m_\phi$, friction dominates, $%% \phi$ is constant and from Eq.(37) $p=-\rho$ as for a cosmological constant. The difference is that the rolling down would occur in the present age because

$$m_\phi \sim H_o\sim 10^{-33}eV,$$ (39)

using Eq.(5) and the Appendix. If the potential is also of the type corresponding to the axion

$$V(\phi ) = \rho _c(1-\cos\frac \phi {f_\phi })$$ (40)

where $f_\phi \sim m_{pl}$ is the order of the mass of the fermion to which $\phi$ is coupled, when $\phi \simeq$ constant $\sim f_\phi .$ $V$ gives a false vacuum energy, and for $\phi \sim 0$ the mass Eq.(39) appears.

An explanation of why $m_\phi$ has this particular value, may be that $\phi$ gets its mass through the mixing with pion, as for axion, but with an interaction between the superheavy fermion and ordinary quarks of gravitational order instead of strong one(16).

In conclusion, the components of the energy of universe seem reasonably determined, but the identification of their nature requires still a great astrophysical and theoretical effort beyond the standard model of fundamental interactions.