# Can the Cosmological Constant explain an accelerated expansion?

From what I've learned so far, it appears that all models that attempt to explain the expansion of the universe are either based on Lambda-CDM or quintessence. The former support a big bang with rapid expansion, then deceleration of the expansion and then expansion again (non accelerated expansion) with $w=-1$. The latter (quintessence) do not support big bang, but support accelerated expansion with $w<-1$. The two schools of thought appear to box you in one way or the other, depending whether $w=-1$ or $w<-1$.

Why doesn't Lambda-CDM have a model that explain an accelerated expansion (i.e. $w < -1$) ? Or do they have one? Does Lambda-CDM maintain that $\Lambda$ has to be constant and so you're stuck with quintessence whenever $w<-1$? If that is the case, why couldn't $\Lambda$ increase with time?

In summary, is there any model that support a universe with:

1. Big Bang

2. Inflationary period with rapid expansion

3. Deceleration of expansion

4. Linear Expansion

5. Future acceleration of expansion?

That is, we should be able to see in that model that $H_t > H_0$ for any $t_i >> t_0$ when $w<-1$.

(Disclaimer: this is a follow-up question to Equation for Hubble Value as a function of time)

You still have a few misconceptions:

First, a model with a cosmological constant does lead to accelerated expansion. Look at the second derivative of $a(t)$ in my post: $$\ddot{a}(t) = \frac{1}{2}H_0^2\left(-2\,\Omega_{R,0}\,a(t)^{-3} - \Omega_{M,0}\,a(t)^{-2} +2\,\Omega_{\Lambda,0}\,a(t)\right).$$ You see that $\ddot{a}(t)>0$ if $a(t)$ is sufficiently large. In particular, with the values $$H_0 = 67.3\;\text{km}\,\text{s}^{-1}\text{Mpc}^{-1},\\ \Omega_{R,0} \approx 0,\qquad\Omega_{M,0} = 0.315,\qquad\Omega_{\Lambda,0} = 0.685,\qquad\Omega_{K,0} = 0,$$ you can work out that $\ddot{a}(t) > 0$ for $a(t)> 0.6$, which corresponds with $t > 7.7$ billion years. That is, the expansion began to accelerate when the universe was 7.7 billion years old, and will continue to do so. See also this post for more details: Can space expand with unlimited speed?

Second, a model with quintessence does have a big bang. The most general equation for $t(a)$, which I hadn't posted in my answer to your other question, is \begin{align} t(a) &= \frac{1}{H_0}\int_0^a \frac{a'\,\text{d}a'}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a' + \Omega_{K,0}\,a'^2 + \Omega_{\Lambda,0}\,a'^{1-3w}}}. \end{align} This is a well-behaved function when $a\rightarrow 0$, because $\Omega_{R,0}>0$. This means that in the early universe, radiation was the dominant factor (the other terms in the denominator of the integrand go to zero for $a\rightarrow 0$), and we get $t(0)=0$ or conversely $a(0)=0$.

In your previous question, you used a simplified model with $\Omega_{R,0} = \Omega_{M,0} = \Omega_{K,0} =0$. Those models don't have a big bang, because then the integrand becomes infinite for $a\rightarrow 0$. But of course, those toy models do not correspond with our actual universe. You need the general model.

So, both a model with a cosmological constant and one with quintessence produce a universe with a big bang and an accelerated expansion. Does the data suggest a non-constant dark energy? It's too soon to tell, the error-bars are still too large; a cosmological constant is still consistent with the data. It's obvious that quintessence provides a better fit, because it has the extra parameter $w$, but that itself doesn't mean that this extra parameter is necesary to explain the observations. But time will tell...

• Ok, that clarifies it a lot. However, one of the factors of my confusion is that when I tried a very large $t$ in your equation for a future time, to my surprise, I got $H=H_0$. No matter what number I tried in the future $H=H_0$ is persistent, which led me to believe that the ΛCDM does not support the accelerated expansion. So the equation holds for the past and it holds for the present. But what about the future? Is that what you mean by time will tell? The equation says that $H$ will be constant in the future, until we know otherwise and we'll figure out how to change it. Correct? – Luis Jan 10 '14 at 4:09
• And one more questions. How do I add calculate $w$ from the above equations. i – Luis Jan 10 '14 at 4:12
• How do I calculate $w$ from the above equation? Is it as simple as applying $p=\rho w c^2$ after I calculate $\rho$ and $P$ from $\Omega_{\Lambda,0}$ ? – Luis Jan 10 '14 at 4:23
• @Luis The Hubble parameter will indeed approach a constant value in the ΛCDM model: $H_\infty \approx H_0\sqrt{0.685} < H_0$. Remember that $H = \dot{a}/a$. So $\dot{a}$ increases (accelerated expansion), but $a$ increases as well, and their ratio $H$ actually decreases. Note that $\dot{a}/a=\text{const}$ means $a\sim\exp{t}$, so the ΛCDM model evolves towards an exponential expansion. The value of $w$ is derived from observations, so it's an extra input parameter. For a given $w$ and $\Omega_{\Lambda,0}$, you can calculate $\rho$ and $p = \rho w c^2$. – Pulsar Jan 12 '14 at 20:22
• As you said, I was able to confirm that $H_\infty \approx H_0\sqrt{0.685} < H_0$. This is contrary to my expectations (or understanding of $H$). I would think that as the universe expands, galaxies are further away from the mutual gravitational force, to a point that there is nothing to hold the acceleration leading to a higher $H$. Where am I wrong on this? But interestingly enough, with $w<-1$, $H$ begins to pick up for $a>1$. What is the meaning of that? If you like, I can formulate this in new question so that you can better elaborate. – Luis Jan 15 '14 at 17:39

Pure $\Lambda$ spacetimes have, trivially $w = -1$. So long as expansion isn't capped at some upper value, as expansion continues, all of the content of matter with nonnegative pressure will get more and more diffuse. As the matter gets more diffuse, it's contribution to the stress-energy tensor decreases. Any $\Lambda$ term, however, will not get more diffuse, it will remain constant. Therefore, at late times, any non-recollapsing $\Lambda$CDM model will asymptote to a $w = -1$ model.

And really, you can write down any model you want. It's trivial to use Einstein's equation as a way to just predict what the stress-energy tensor is for some general spacetime. The question is whether you can use a model with a small number of assumptions to do this. $\Lambda$CDM is not too far from a bare-minimum set of assumptions.

• Understood. However, if tomorrow the Type 1a SN survey shows w = -1.3, what do we do then? Do we have to disregard ΛCDM and look for something totally different (e.g. quintessence) or can we still attempt to write an equation using ΛCDM that accounts for $w=-1.3$? Would it be possible, for example, to augment the Friedman equation that explains w=-1.5 with a big bang and everything else in it? As far as you know, is there any model available that shows this scenario? I trust you can see my confusion. – Luis Jan 9 '14 at 22:07
• @Luis: You'd have to add in other matter to your model to make the expansion right. So long as observations don't brake homogeniety and isotropy, Robertson-walker models will always be right, but the matter content will depend on how $a(t)$ evolves. If you measure some value for one of the parameters, then either $\Lambda$CDM needs to be augmented with other matter, or it is wrong. – Jerry Schirmer Jan 9 '14 at 23:06
• Makes sense. Do you know of any model (or paper) that shows ΛCDM being augmented as a possible solution for accelerated expansion? – Luis Jan 9 '14 at 23:51

Ok, that clarifies it a lot. However, one of the factors of my confusion is that when I tried a very large t in your equation for a future time, to my surprise, > I got H=H0. No matter what number I tried in the future H=H0 is persistent, which led me to believe that the ΛCDM does not support the accelerated expansion. So the equation holds for the past and it holds for the present. But > what about the future? Is that what you mean by time will tell? The equation says that H will be constant in the future, until we know otherwise and we'll figure out how to change it. Correct?

I think some of the confusion is coming from what "accelerated expansion" means. It does not mean increasing H0. It means ä > 0. Exponential expansion, as in inflation and Λ-dominated cosmology, does indeed have constant H – and is indeed undergoing accelerated expansion. The two statements don't conflict.