On the Interpretation of Cosmic Acceleration

1. Introduction

For over thirty years, cosmologists have built up conclusive evidence that cosmic expansion is accelerating. To explain such observation, we need to assume that there is a mysterious new component: Dark Energy (DE) or a Cosmological Constant, $\Lambda$. This new term is usually interpreted as a repulsive force between galaxies that opposes gravity and dominates the expansion. Such strange behaviour is often flagged as one of the most important challenges to understand the laws of Physics today and could provide an observational window to understand Quantum Gravity (e.g. see de Boer et al. 2022 and refrences therein).

Cosmic acceleration is typically measured using the adimensional coefficient q, defined as $(\ddot{a}/a)H^{-2}$, where $H\equiv \dot{a}/a$. If the universe follows an equation of state with $p=\omega \rho$, this leads to $q=-\frac{1}{2}(1+3\omega )$. For regular matter or radiation where $\omega >0$, we’d expect deceleration in the expansion ($q<0$) due to gravity. However, measurements from various sources, such as galaxy clustering, Type Ia supernovae, and CMB, consistently show an expansion asymptotically approaching $q\simeq 1$ or $\omega \simeq -1$ (e.g. see DES Collaboration 2022 and references therein for a review of more recent results, including weak gravitational lensing). This aligns well with a Cosmological Constant $\Lambda$, where $H^2$ approaches $H^2=\Lambda /3\equiv r_{\Lambda }^{-2}$ and q approaches 1. So, what’s the significance of all this?

The term Dark Energy (DE) was introduced by Huterer and Turner (1999) to refer for any component with $\omega <-1/3$. However, there is no fundamental understanding of what DE is or why we measure a term with $\omega \simeq -1$. A natural candidate for DE is $\Lambda$ , which is equivalent to $\omega =-1$ and can also be thought of as the ground state of a field (the DE), similar to the Inflaton but with a much smaller ($\simeq {10}^{-50}-{10}^{-100}$) energy scale. $\Lambda$ can also be a fundamental constant in GR, but this has some other complications (Weinberg 1989; Carroll et al. 1992; Peebles and Ratra 2003).

2. Cosmic acceleration

Current observations of the cosmos seem consistent with General Relativity (GR) with a flat FLRW (Friedmann–Lemaitre–Robertson–Walker) metric in comoving coordinates, corresponding to a homogeneous and isotropic space :

$$d{s}^2=-d{\tau }^2+a{\left(\tau \right)}^2\left[d{\chi }^2+{\chi }^2{d\Omega }^2\right],$$

(1)

where we use units of $c=1$ and $a\left(\tau \right)$ is the scale factor. For a classical perfect fluid with matter and radiation density $\rho ={\rho }_m+{\rho }_R$, the solution to Einstein’s field equations (called LCDM) is well known:

$$H^2=\frac{8\pi G}{3}\rho +\frac{\Lambda }{3}\equiv H_0^2\left[{\Omega }_m{a}^{-3}+{\Omega }_R{a}^{-4}+{\Omega }_{\Lambda }\right],$$

(2)

where ${\Omega }_X\equiv \frac{{\rho }_X}{{\rho }_c}$, where ${\Omega }_m$ and ${\Omega }_R$ represents the current ($a=1$) matter and radiation density, ${\rho }_c\equiv \frac{3H_0^2}{8\pi G}$ and ${\Omega }_m+{\Omega }_R+{\Omega }_{\Lambda }=1$. The cosmological constant ($\Lambda$) term corresponds to ${\Omega }_{\Lambda }=H_0^{-2}\Lambda /3\simeq 0.7$ where $H_0\simeq 70$Km/s/Mpc. Given ${\Omega }_m\simeq 0.3$ and ${\Omega }_R\simeq 4\times {10}^{-5}$ we can use the above equations to find $a\left(\tau \right)$.

Cosmic acceleration is usually defined as $\ddot{a}/a$, where the dot represents a derivative with respect to proper time $\tau$ at emission. A derivative over Eq.2 shows that:

$$q\left(z\right)\equiv \frac{\ddot{a}}{a}\frac{1}{H^2}=\left({\Omega }_{\Lambda }-\frac{1}{2}{\Omega }_M{a}^{-3}-{\Omega }_R{a}^{-4}\right)\frac{H_0^2}{H^2}.$$

(3)

For $\Lambda =0$ Eq.2-3 indicate that as time passes ($a\Rightarrow \infty$) we have that $H\Rightarrow 0$ and $q\Rightarrow -1/2$. This is because gravity opposes cosmic expansion and brings the expansion asymptotically to a halt. Including $\Lambda$ brings the expansion to accelerate so that $H\Rightarrow r_{\Lambda }^{-1}$ and $q\Rightarrow 1$. This is illustrated as black lines in Figure 1 for ${\Omega }_{\Lambda }=0.7$. The effect of $\Lambda$ is then interpreted as a mysterious new repulsive force (or Dark Energy) that opposes gravity.

3. Event Acceleration

The above interpretation is solely based on the definition for acceleration $\ddot{a}/a$ in Eq.3. We will show next, that such definition corresponds to events without a cause-and-effect connection and this lead us to the wrong picture of what is happening. We will then introduce a more physical alternative.

Consider the distance between two events corresponding to the light emission of a galaxy at $(\tau ,\chi )$ and the reception somewhere in its future $({\tau }_1,{\chi }_1)$. The photon travels following an outgoing radial null geodesic $ds=0$ which from Eq.1 implies $d\tau =a\left(\tau \right)d\chi$. This situation is depicted in Figure 2. We can define a 3D space-like distance d based in the comoving separation $\Delta \chi =\chi -{\chi }_1$:

$$d=a\left(\tau \right)\Delta \chi .$$

(4)

This is in fact the distance that corresponds to the acceleration given by $\ddot{a}/a$ in Eq.3, because $\dot{d}/d=\dot{a}/a$ and $\ddot{d}/d=\ddot{a}/a$, where the derivative is with respect to $\tau$, the time at emission. Such distance corresponds to the distance between $(\tau ,\chi )$ and $(\tau ,{\chi }_1)$, so that $d\tau =0$. These events lack causal connection and are beyond observation. While using d isn’t inherently incorrect, it involves extrapolating observed events (like luminosity distance) into non-observable realms. Essentially, d aligns with a non-local theory of Gravity or the Newtonian approximation, where action at a distance occurs with an infinite speed of light.

We can instead use the the distance traveled by the photon:

$$d_{LC}={\int }_{\tau }^{{\tau }_1}a\left(\tau \right)d\chi ={\int }_{\tau }^{{\tau }_1}d\tau =\Delta \tau =({\tau }_1-\tau )$$

(5)

Note that we use units of $c=1$, so that this should be read as $d_{LC}=c\Delta \tau$. The problem with this distance is that it does not account for the cosmic expansion that occurred between $\tau$ and ${\tau }_1$. In fact it contains no information about cosmic expansion or cosmic acceleration because ${\dot{d}}_{LC}=-1$ and ${\ddot{d}}_{LC}=0$.

So the usual definition currently use by cosmologist, in Eq.4, corresponnds to events that are space-like, i.e. at a fix comoving separation or fix cosmic time $d\tau =0$. It only takes into account the change in the distance due to the expansion of the universe. While the distance in Eq.5 corresponds to the actual distance travel by light, but it ignores the expansion altogether. To have a measurement of cosmic acceleration that is closer to actual observations, we need to use the distance between events that are causally connected, i.e. that not only takes into account how much the universe has expanded, but also how long it has taken for the two events to be causally connected.

To this end, we should use the proper future light-cone distance $R\left(\tau \right)$ (see e.g. Ellis and Rothman 1993):

$$R(\tau ,{\tau }_1)=a\left(\tau \right){\int }_{\tau }^{{\tau }_1}d\chi =a\left(\tau \right){\int }_{\tau }^{{\tau }_1}\frac{d{\tau }^{\prime }}{a\left({\tau }^{\prime }\right)}$$

(6)

Note that the term with the integral is not $\Delta \chi$, but it corresponds to the coordinate distance $d_{\chi }$ travel by light between the two events including the effect of cosmic expansion. Thus, we argue that we should use R instead of d in Eq.4 as a measure of distance in cosmology to define cosmic acceleration and expansion rate. The difference between this 3 distances is illustrated in Figure 2.

Using R as a distance is equivalent to a simple change of coordinates in the FLRW metric of Eq.1, from comoving coordinates $d\chi$ to physical coordinates $dR=ad\chi$:

$$d{s}^2=-d{\tau }^2+d{R}^2+R^2\left(\tau \right)d{\Omega }^2,$$

(7)

which is just Minkowski’s metric in spherical coordinates with a radius $R=R\left(\tau \right)$.

We then have that: $\dot{R}=HR-1$ and define the expansion rate between null events as:

$$H_E\left(\tau \right)\equiv \frac{\dot{R}}{R}=H\left(1-\frac{1}{HR}\right)$$

(8)

where the additional term, $\frac{1}{HR}$ corresponds to a friction term. There is an ambiguity in this definition because R in Eq.6 depends also on the time ${\tau }_1$ use to define R. To break this ambiguity we arbitrarily fix R to be the distance to ${\tau }_1\Rightarrow \infty$ (which corresponds to a possible future Event Horizon):

$$H_E\left(\tau \right)=H\left(1-\frac{1}{H{R}_{EH}}\right)$$

(9)

where $R_{EH}\equiv R[\tau ,{\tau }_1=\infty ]$. As we will see in next section, this choice implies that $\frac{1}{H{R}_{EH}}$ is zero unless $\Lambda \ne 0$. So this new invariant way to define cosmic expansion reproduces the standard definition when $\Lambda =0$. But for $\Lambda >0$ we have that the event expansion halts $H_E\Rightarrow 0$ (blue line in in Figure 1) due to the friction term (red line) for $a>>1$, while the standard Hubble rate definition approaches a constant $H\Rightarrow \sqrt{\Lambda /3}\equiv r_{\Lambda }^{-1}$(black line). This might seem irrelevant at first look, but the resulting physical interpretation is quite different. In the standard definition, H, the expansion with $\Lambda$ becomes asymptotically exponential (or inflationary expansion). While in our new definition, $H_E$, the expansion becomes static (as in the static de-Sitter metric).

The event acceleration can then be measured as:

$$q_E\equiv \frac{\ddot{R}}{R}\frac{1}{H_E^2}=\left(q-\frac{1}{H{R}_{EH}}\right){\left[1-\frac{1}{H{R}_{EH}}\right]}^{-2}.$$

(10)

The correct way to define a 4D acceleration in relativity is based on the geodesic deviation equation Eq.A1. The relation to q and $q_E$ will be discussed in the Appendix.

As before, for $\Lambda =0$ the friction term, $\frac{1}{H{R}_{EH}}$, makes little difference between q and $q_E$. For $\Lambda >0$ the friction term asymptotically cancels the $\Lambda$ term in $\frac{\ddot{a}}{a}$ (i.e. Eq.3) so that $\frac{\ddot{R}}{R}$ is always negative, no matter how large is $\Lambda$ ($H{R}_{EH}\Rightarrow 1$ and $q_E=-\infty$). The net effect of the $\Lambda$ term is to bring the expansion of events to a faster stop ($H_E\Rightarrow 0$) that in the case with gravity alone. This is illustrated in Figure 1. The $\Lambda$ term produces a faster deceleration (than with gravity alone). This corresponds to an attracting (and not repulsive) force, as explained in more detail in the Appendix.

4. Event Horizon

What is more relevant to understand the meaning of $\Lambda$ is that the additional deceleration brings the expansion to a halt within a finite proper distance between the events, creating an Event Horizon (EH). The EH is the maximum distance that a photon emitted at time $\tau$ can travel following the outgoing radial null geodesic:

$$R_{EH}=a{\int }_a^{\infty }\frac{d{a}^{\prime }}{H\left(a^{\prime }\right)a^{\prime 2}}<\frac{1}{H[a=\infty ]}=r_{\Lambda }\equiv \sqrt{3/\Lambda }$$

(11)

For $\Lambda =0$ we have $R_{EH}=\infty$, so there is no EH. But for $\Lambda >0$ we have that $R_{EH}\Rightarrow r_{\Lambda }$ (red line in Figure 1). We can then see that $\Lambda$ corresponds to a causal horizon or boundary term. The analog force behaves like a rubber band between observed galaxies (null events) that prevents them crossing some maximum stretch (i.e. the EH). We can interpret such force as a boundary term that just emerges from the finite speed of light (see the Appendix).

The FLRW metric with $\Lambda$, asymptotically tends to the de-Sitter metric, which can be written as:

$$d{s}^2=-\left(1-r^2/r_{\Lambda }^2\right)d{t}^2+\frac{d{r}^2}{1-r^2/r_{\Lambda }^2}+r^{2}d{\Omega }^2$$

(12)

This form corresponds to a static 4D hyper-sphere of radius $r_{\Lambda }$. So in this (rest) frame, events can only travel a finite distance $R<r_{\Lambda }$ within a static 3D surface of the imaginary 4D hyper-sphere. Thus there is a frame duality that allows us to equivalently describe de-Sitter space as either static (in proper or physical coordinates $r=a\left(\tau \right)\chi$ and t) or exponentially expanding (in comoving coordinates $\chi$ and $\tau$).

This frame duality can be understood as a Lorentz contraction $\gamma =1/\sqrt{1-{\dot{r}}^2}$, where $\dot{r}=Hr=r/r_H$. An observer in the rest frame, sees the moving fluid element $ad\chi$ contracted by the Lorentz factor $\gamma$. Therefore, the FLRW metric becomes de-Sitter:

$$d{R}^2=a^{2}d{\chi }^2={\gamma }^{2}d{r}^2=\frac{d{r}^2}{1-r^2/r_H^2},$$

(13)

This radial element corresponds to the metric of a hypersphere of radius $r_H\equiv 1/H$ that expands towards a constant event horizon $r_H\Rightarrow r_{\Lambda }$ (see also Mitra 2012 and the Appendix in Gaztanaga 2022). This duality is better understand using our new measures for the expansion rate $H_E$ and cosmic deceleration $q_E$ based on the distance between causal events.

5. Comparison to data

We show next how to estimate the new measure of cosmic acceleration, $q_E$, using direct astrophysical observations. As an example consider the Supernovae Ia (SNIa) data as given by the ’Pantheon Sample’ compilation (Scolnic et al. 2018) consisting on 1048 SNIas between $0.01<z<2.3$. Each SNIa provides a direct estimate of the luminosity distance $d_L\left(z\right)$ at a given measured redshift z. This corresponds to the comoving look-back distance:

$$\chi \left(z\right)=a{d}_L\left(z\right)={\int }_0^z\frac{dz}{H\left(z\right)}$$

(14)

so that ${\chi }^{\prime }\equiv d\chi /dz$ gives us directly ${\chi }^{\prime }=r_H=H^{-1}$. The second derivative gives us the acceleration:

$$q=1+\frac{{\chi }^{″}}{a{\chi }^{\prime }};q_E=\frac{q-{\chi }^{\prime }/R_{EH}}{{(1-{\chi }^{\prime }/R_{EH})}^2}.$$

(15)

$R_{EH}$ is given by the model prediction in Eq.11 (arbitrarily fixed at ${\Omega }_{\Lambda }=0.85$ in both data and models). We adopt here the approach presented in Liu et al. (2023), who used an empirical fit to the luminosity distance measurements, based on a third-order logarithmic polynomial:

$$\chi \left(a\right)=a{d}_L=a\left(x+A{x}^2+B{x}^3\right)H_0^{-1}ln10$$

(16)

where $x=-{log}_{10}a$. Liu et al. (2023) find a good fit to: $A=3.15\pm 0.12$ and $B=3.27\pm 0.41$ to the full SNIa ’Pantheon Sample’. We use these values of A and B and its corresponding errors to estimate H, q and $q_E$ using the above relations. Results for H and q are shown as shaded cyan regions in the left panel Figure 3. They are compared to the LCDM predictions in Eq.2 and 3.

There is a very good agreement in $H\left(z\right)$ for ${\Omega }_{\Lambda }\simeq 0.65$. At $z<1$, the $q\left(z\right)$ estimates are also consistent with the ${\Omega }_{\Lambda }\simeq 0.65$ predictions. But the detailed $q\left(z\right)$ evolution with redshift in the SNIa data does not seem to follow any of the model predictions, specially for $z>1$. The $q\left(z\right)$ estimates are too steep compare to the different models predictions. If we compare instead the $q_E$ estimates (see right panel of Figure 3) we find a much better agreement with the model predictions. This seems to validate our $q_E$ approach, but it is not clear from this comparison alone if this is caused by the fitting function use in Eq.16.

To test this further we use measurements of the radial BAO data to estimate $q_E$. Such measurements give us a direct estimate of $H\left(z\right)$ (as first demonstrated by Gaztañaga et al. 2009a, 2009b) so they have the advantage over SNIa that we only need to do a first order derivative, to estimate q or $q_E$:

$$q=1+\frac{1}{aH}\frac{dH}{dz};q_E=\frac{q-r_H/R_{EH}}{{(1-r_H/R_{EH})}^2}.$$

(17)

As an illustration we use $H\left(z\right)$ measurements presented in Table 2 in Niu et al. (2023). This compilation of $H\left(z\right)$ is shown as red points with $2\sigma$ errorbars in the left panel Figure 3. The compilation include values from the clustering of galaxies ($z<1$) and Ly-alpha forest in QSO ($z>2$). The combination of two separate ranges of redshift allows for a very good measurement of $dH/dz$ at the intermediate redshift ($1<z<2$), where we found the discrepancies in SNIa for q and $q_E$ model comparison (see above). The radial BAO provides a very good constraint on cosmic acceleration, independent of possible calibration errors in $H_0$ or sampling errors (from small area samples). This is something that we can not yet do with the current SNIa data, but will be very interesting to see in the near future with upcoming data from wider and deeper surveys.

We fit a quadratic polynomial to the radial BAO data:

$$H\left(z\right)=H_0+H_{1}z+H_2{z}^2.$$

(18)

We have checked that the results presented here are very similar if we use a cubic polynomial. In units of Km/s/Mpc, we find $H_0\simeq 68\pm 3$, $H_1\simeq 39\pm 8$ and $H_2\simeq 12\pm 3$, with strong covariance between the errors (the cross-correlation coefficient between $H_1$ and $H_2$ is $-0.99$). The value of $H_0$ is in good agreement with the Planck CMB fit (Planck Collaboration 2020) but in some tension with the SNIa local calibration: $H_0=73\pm 1$ (see Riess 2019). This corresponds to either a local calibration problem (in SNIa, in radial BAO or in both) or a tension in the $\Lambda$CDM model at different times or distances (see e.g. Abdalla et al. 2022). We ignore this normalization problem here and just focus on the evolution of $H/H_0$ to measure cosmic acceleration q or $q_E$ (which are fairly independent of $H_0$).

In the right panel of Figure 3 we show (as shaded regions) the measurements for $q_E$ given by combining Eq.16 with Eq.15 and Eq.18 with Eq.17. The measurements clearly favour models with large negative cosmic event acceleration $q_E<0$, which supports our interpretation of $\Lambda$ as a friction term.

Comparing left and right panels in Figure 3 we see that both q and $q_E$ are rougthly consistent with models with ${\Omega }_{\Lambda }\simeq 0.7$ ( or $\Lambda \simeq 2H_0^2$) in good concordance with $H\left(z\right)$ in the upper left panel of Figure 3.

Even when the underlying model for q and $q_E$ is the same, note how the measured q and $q_E$ data have different tensions with the model predictions as a function of redshift. In particular, the radial BAO and SNIa data sets show inconsistencies among them for q around $1.5<z<2.5$. This is a well known tension (see e.g. Fig.17 in Bautista et al. 2017). This tension disappears when we use the corresponding estimates for $q_E$. Thus, data is more consistent with the $q_E$ than with the q description.

One would expect that a perfect realizations of the LCDM model in Eq.2 would produce consistent results in both q and $q_E$. But deviations from LCDM and systematic effects can produce tensions in data, specially if we use a parametrization, like q, which refers to events that we never observe. The q and $q_E$ parametrization of acceleration are more general than the particular LCDM model and the fact that data prefers $q_E$ is an important indication. Data lives in the light-cone, which corresponds to $q_E$ rather than q. At $z\simeq 2$ the difference between a light-cone and space-like separations is very significant and any discrepancies in the data or model will show more pronounced in the q modeling.

We conclude that the data shows some tensions with LCDM predictions (as indicated by q) but confirms that cosmic expansion is clearly decelerating (as indicated by $q_E$) so that events are trapped inside an Event Horizon ($R_{EH}$).

6. Discussion & Conclusion

In our exploration, we’ve demonstrated that the commonly interpreted $\Lambda$ term, thought to drive cosmic acceleration (as discussed in Section 2), actually leads to a quicker cosmic deceleration of events compared to the influence of gravity alone (as explained in Section 3). This explains the origin of the Event Horizon (EH, see Section 4) that results from an expansion dominated by $\Lambda$. It suggests that $\Lambda$ might not be a new form of vacuum energy (Weinberg 1989; Carroll et al. 1992; Peebles and Ratra 2003) but rather a boundary or surface term in the field equations.

The EH measured in our cosmic expansion behaves like the interior of a Schwarzschild Black Hole (BH) and it is identical to it if we just assume that the space outside $R_{EH}$ is approximately empty (see Gaztañaga 2022). We can also conjecture from this, that the interior dynamics of the Schwarzschild BH radius, $r_S=2GM$, could also have a similar $\Lambda$ surface term $r_{\Lambda }=r_S$, as part of its field equations, at least for expanding interior BH solutions. The $\Lambda$ term becomes the actual mechanism that prevents anything from escaping the BH interior (see Gaztañaga 2023a).

The Event Horizon $R_{EH}$ measured with $q_E$ (i.e. Eq.11, which is equivalent to the presence of $\Lambda$) also tell us that there is a finite mass $M_T$ trapped within $R_{EH}$. If we assume that the space outside $R_{EH}$ is approximately empty, such finete mass $M_T$ provides the explanation for the observed $R_{EH}$ and therefore for $\Lambda$: i.e. $2G{M}_T=\sqrt{3/\Lambda }$ (see Gaztañaga 2023b).

That $\Lambda$ is fixed by the total mass $M_T$ of our universe is in good agreement with the physical interpretation presented here that $\Lambda$ corresponds to a friction (attractive) force that decelerates cosmic events. In the Appendix we elaborate in this idea and revisit the Newtonian limit to show that $\Lambda$ corresponds to an additional (attractive) Hooke’s term to the inverse square gravitational law. A rubber band Universe.