Anthropic likelihood for the cosmological constant and the primordial density perturbation amplitude

Weinberg et al. calculated the anthropic likelihood of the cosmological constant $\Lambda$ using a model assuming that the number of observers is proportional to the total mass of gravitationally collapsed objects, with mass greater than a certain threshold, at $t\to \infty$. We argue that Weinberg’s model is biased toward small $\Lambda$, and to try to avoid this bias we modify his model in a way that the number of observers is proportional to the number of collapsed objects, with mass and time equal to certain preferred mass and time scales. The Press-Schechter formalism, which we use to count the collapsed objects, identifies our collapsed object at the present time as the Local Group, making it inconsistent to choose the preferred mass scale as that of the Milky Way at the present time. Instead, we choose an earlier time before the formation of the Local Group and this makes it consistent to choose the mass scale as that of the Milky Way. Compared to Weinberg’s model (${\mathcal{T}}_{+}({\Lambda }_0)\sim 23\%$), this model gives a lower anthropic likelihood of ${\Lambda }_0$ (${\mathcal{T}}_{+}({\Lambda }_0)\sim 5\%$). On the other hand, the anthropic likelihood of the primordial density perturbation amplitude $Q_0$ from this model is high (${\mathcal{T}}_{+}(Q_0)\sim 63\%$), while the likelihood from Weinberg’s model is low (${\mathcal{T}}_{+}(Q_0)\ll 0.1\%$). Furthermore, observers will be affected by the history of the collapsed object, and we introduce a method to calculate the anthropic likelihoods of $\Lambda$ and $Q$ from the mass history using the extended Press-Schechter formalism. The anthropic likelihoods for $\Lambda$ and $Q$ from this method are similar to those from our single mass constraint model, but, unlike models using the single mass constraint which always have degeneracies between $\Lambda$ and $Q$, the results from models using the mass history are robust even if we allow both $\Lambda$ and $Q$ to vary. In the case of Weinberg’s flat prior distribution of $\Lambda$ (pocket-based multiverse measure), our mass history model gives ${\mathcal{T}}_{+}({\Lambda }_0)\sim 10\%$, while the scale factor cutoff measure and the causal patch measure give ${\mathcal{T}}_{+}({\Lambda }_0)\gtrsim 30\%$.

Figure 1

Normalized probability distribution $\Lambda P(\Lambda |☺)$ for the scale factor cutoff measure (red) and the causal patch measure (blue), assuming both $P_{\varnothing }(\Lambda )$ and $L_c(\Lambda |☺)$ are constant, and $t_{☺}=t_0$. The pocket-based measure is not shown here because it is very small. We use $\Lambda P(\Lambda |☺)$ to make the area inside the curve to be the probability.

Figure 2

Anthropic likelihoods for the anthropic models using a single mass constraint. Left: anthropic likelihoods for the cosmological constant $L_{\mathrm{c}}(\Lambda |☺)$. Anthropic model: $M\ge M_{\mathrm{MW}}$ at $t\to \infty$ (red), $M\ge M_{\mathrm{MW}}$ at $t=6\mathrm{Gyr}$ (green) and $M=M_{\mathrm{MW}}$ at $t=6\mathrm{Gyr}$ (blue). Right: anthropic likelihoods for the primordial density perturbation amplitude $L_{\mathrm{c}}(Q|☺)$, for $\Lambda ={\Lambda }_0$. $L_{\mathrm{c}}(Q|☺)$ for anthropic models with $M\ge M_{\mathrm{MW}}$ is not shown because it is very small.

Figure 3

Probability of an observer observing $\Lambda$, $P(\Lambda |☺)$. Left: pocket-based measure; Middle: scale factor cutoff measure with $t_{☺}=t_0$; Right: causal patch measure with $t_{\mathrm{obs}}=t_0$. Anthropic model: $M\ge M_{\mathrm{MW}}$ at $t\to \infty$ (red), $M\ge M_{\mathrm{MW}}$ at $t=6\mathrm{Gyr}$ (green) and $M=M_{\mathrm{MW}}$ at $t=6\mathrm{Gyr}$ (blue).

Figure 4

$Q(\Lambda ,t_{*})$ for which the population of galaxies and clusters at $t=t_{*}$ is independent of $\Lambda$, for $t_{*}=6\mathrm{Gyr}$ (red) and $t_{*}\to \infty$ (blue).

Figure 5

$L_{\mathrm{c}}(\Lambda |☺)$ (top) and $L_{\mathrm{c}}(Q|☺)$ (bottom) for the anthropic models using the mass history with $M_{\mathrm{f}}=M_{\mathrm{MW}}$ at $t_{\mathrm{f}}=6\mathrm{Gyr}$. Left: $M_{\mathrm{i}}=0.8M_{\mathrm{MW}}$ at $t_{\mathrm{i}}=3\mathrm{Gyr}$ (red), 4 Gyr (green) and 5 Gyr (blue). Middle: $M_{\mathrm{i}}=0.2M_{\mathrm{MW}}$ (red), $0.8M_{\mathrm{MW}}$ (green) and $0.9M_{\mathrm{MW}}$ (blue) at $t_{\mathrm{i}}=4\mathrm{Gyr}$. Right: $M_{\mathrm{i}}\le$ (red), $=$ (green), and $\ge$ (blue) 0.8 at $t_{\mathrm{i}}=4\mathrm{Gyr}$.

Figure 6

$P(\Lambda |☺)$ using the mass history with different multiverse measure. Left to right: same as Fig. 5. Top: the pocket-based measure; Middle: the scale factor cutoff measure; Bottom: the causal patch measure.

Figure 7

Contour diagrams of ${\mathcal{T}}_{+}({\Lambda }_0)$ using the mass history with the usual flat prior/pocket-based measure and $M_{\mathrm{f}}=M_{\mathrm{MW}}$ at $t_{\mathrm{f}}=6\mathrm{Gyr}$. Left: $Q=Q_0$; Right: $Q=Q(\Lambda ,t_{\mathrm{f}})$, assuming $P_c(Q)=\mathrm{\text{constant}}$, see Sec. 3d. Top: $M_{\mathrm{i}}\le M_{*}$; Middle: $M_{\mathrm{i}}=M_{*}$; Bottom: $M_{\mathrm{i}}\ge M_{*}$. Typicality: 0–0.01 (violet), 0.01–0.03 (blue), 0.03–0.1 (green), 0.1–0.3 (yellow), 0.3–1 (red).

Figure 8

Contour diagrams of ${\mathcal{T}}_{+}(Q_0)$ using the mass history with $\Lambda ={\Lambda }_0$ and $M_{\mathrm{f}}=M_{\mathrm{MW}}$ at $t_{\mathrm{f}}=6\mathrm{Gyr}$. Left: $P_{\mathrm{c}}(Q)=\mathrm{\text{constant}}$; Right: $P_{\mathrm{c}}(Q)\propto Q^{-1}$. Top: $M_{\mathrm{i}}\le M_{*}$; Middle: $M_{\mathrm{i}}=M_{*}$; Bottom: $M_{\mathrm{i}}\ge M_{*}$. Typicality: 0–0.01 (violet), 0.01–0.03 (blue), 0.03–0.1 (green), 0.1–0.3 (yellow), 0.3–1 (red). White dash: ${\mathcal{T}}_{+}(Q_0)=1$.

Figure 9

Contour diagrams of ${\mathcal{T}}_{+}({\Lambda }_0)$ using the mass history with $Q=Q_0$ and $M_{\mathrm{f}}=M_{\mathrm{MW}}$ at $t_{\mathrm{f}}=6\mathrm{Gyr}$. Left: pocket-based measure; Middle: scale factor cutoff measure with $t_{☺}=t_0$; Right: causal patch measure with $t_{☺}=t_0$. Top: $M_{\mathrm{i}}\le M_{*}$; Middle: $M_{\mathrm{i}}=M_{*}$; Bottom: $M_{\mathrm{i}}\ge M_{*}$. Typicality: 0–0.01 (violet), 0.01–0.03 (blue), 0.03–0.1 (green), 0.1–0.3 (yellow), 0.3–1 (red). White dash: ${\mathcal{T}}_{+}({\Lambda }_0)=1$.

Figure 10

Examples which illustrate the difference between the mass history which makes our universe typical and that which is typical in our universe. Left: the typicality of ${\Lambda }_0$ in the case of the causal patch measure. Right: the typicality of mass history in our universe. Typicality: 0–0.01 (violet), 0.01–0.03 (blue), 0.03–0.1 (green), 0.1–0.3 (yellow), 0.3–1 (red). White dash: the maximal typicality.