Dimensionless constants, cosmology, and other dark matters

We identify 31 dimensionless physical constants required by particle physics and cosmology, and emphasize that both microphysical constraints and selection effects might help elucidate their origin. Axion cosmology provides an instructive example, in which these two kinds of arguments must both be taken into account, and work well together. If a Peccei-Quinn phase transition occurred before or during inflation, then the axion dark matter density will vary from place to place with a probability distribution. By calculating the net dark matter halo formation rate as a function of all four relevant cosmological parameters and assessing other constraints, we find that this probability distribution, computed at stable solar systems, is arguably peaked near the observed dark matter density. If cosmologically relevant weakly interacting massive particle (WIMP) dark matter is discovered, then one naturally expects comparable densities of WIMPs and axions, making it important to follow up with precision measurements to determine whether WIMPs account for all of the dark matter or merely part of it.

Figure 1

The shifting boundary (horizontal lines) between fundamental laws and environmental laws/effective laws/initial conditions. Whereas Ptolemy and others sought to explain roughly spherical planets and circular orbits as fundamental laws of nature, Kepler and Newton reclassified such properties as initial conditions which we now understand as a combination of dynamical mechanisms and selection effects. Classical physics removed from the fundamental law category also the initial conditions for the electromagnetic field and all other forms of matter and energy (responsible for almost all the complexity we observe), leaving the fundamental laws quite simple. A prospective theory of everything (TOE) incorporating a landscape of solutions populated by inflation reclassifies important aspects of the remaining “laws” as initial conditions. Indeed, those laws can differ from one post-inflationary region to another, and since inflation generically makes each such region enormous, its inhabitants might be fooled into misinterpreting regularities holding within their particular region as Universal (that is, multiversal) laws. Finally, if the level IV multiverse of all mathematical structures [124] exists, then even the “theory of everything” equations that physicists are seeking are merely local bylaws in Rees’s terminology [12], that vary across a wider ensemble. Despite such retreats from ab initio explanations of certain phenomena, physics has progressed enormously in explanatory power.

Figure 2

Many selection effects that we discuss are conveniently summarized in the plane tracking the virial temperatures and densities in dark matter halos. The gas cooling requirement prevents halos below the heavy black curve from forming galaxies. Close encounters make stable solar systems unlikely above the downward-sloping line. Other dangers include collapse into black holes and disruption of galaxies by supernova explosions. The location and shape of the small remaining region (unshaded) is independent of all cosmological parameters except the baryon fraction.

Figure 3

Same as the previous figure, but including the banana-shaped contours showing the halo formation/destruction rate. Solid/greenish contours correspond to positive net rates (halo production) at 0.5, 0.3, 0,2, 0.1, 0.03 and 0.01 of maximum, whereas dashed/reddish contours correspond to negative net rates (halo destruction from merging) at -0.3, -0,2, -0.1, -0.03 and -0.01 of maximum, respectively. The heavy black contour corresponds to zero net formation rate. The cosmological parameters $({\xi }_{\mathrm{b}},{\xi }_{\mathrm{c}},{\xi }_{\nu },,Q,{\rho }_{\Lambda })$ have a strong effect on this banana: ${\xi }_{\mathrm{b}}$ and ${\xi }_{\mathrm{c}}$ shift this banana-shape vertically, $Q$ shifts if along the parallel diagonal lines and ${\rho }_{\Lambda }$ cuts it off below $16{\rho }_{\Lambda }$ (see the next figure). Roughly speaking, there are stable habitable planetary systems only for cosmological parameters where a greenish part of the banana falls within the allowed white region from Fig. 2.

Figure 4

Same as Fig. 3, but showing the effect of changing the cosmological parameters. Increasing the matter density parameter $\xi$ shifts the banana up by a factor ${\xi }^4$ (top left panel), increasing the fluctuation level $Q$ shifts the banana up by a factor $Q^3$ and to the right by a factor $Q$ (top right panel), increasing the baryon fraction $f_{\mathrm{b}}={\xi }_{\mathrm{b}}/\xi$ shifts the banana up by a factor $f_{\mathrm{b}}$ and affects the cooling and 2nd generation constraints (bottom left panel), and increasing the dark energy density ${\rho }_{\Lambda }$ bites off the banana below ${\rho }_{\mathrm{vir}}=16{\rho }_{\Lambda }$ (bottom right panel). In these four panels, the parameters have been scaled relative to their observed values as follows: up by $1/4,2/4,\dots ,9/4$ orders of magnitude ($\xi$), up by $1/3$, $2/3,\dots ,9/3$ orders of magnitude ($Q$), down by $1$ and $2$ orders of magnitude ($f_{\mathrm{b}}$), and down by $0$, $7$, $12$ and $\infty$ orders of magnitude (${\rho }_{\Lambda }$; other panels have ${\rho }_{\Lambda }=0$).

Figure 5

Halo density distribution for various values of ${\widehat{\rho }}_{*}/{\rho }_{\Lambda }$, where ${\widehat{\rho }}_{*}\equiv [s(\mu )/A{]}^3{\rho }_{*}\sim Q^3{\xi }^4$. The dashed curves show the cumulative distribution $F=\mathrm{erfc}[({\rho }_{\Lambda }/{\widehat{\rho }}_{*}{)}^{1/3}/G_{\Lambda }(x[{\rho }_{\mathrm{vir}}])]$ and the solid curves show the probability distribution $-\partial F/\partial \lg {\rho }_{\mathrm{vir}}$ for (peaking from right to left) ${\widehat{\rho }}_{*}/{\rho }_{\Lambda }={10}^5$, ${10}^4$, ${10}^3$, ${10}^2$, $10$, $1$ (heavy curves), $0.1$ and $0.05$. Note that no halos with ${\rho }_{\mathrm{vir}}<16{\rho }_{\Lambda }$ are formed.

Figure 6

Qualitatively different parts of the $({\xi }_{\mathrm{b}},{\xi }_{\mathrm{c}})$ plane. Our observed universe corresponds to the star, and altering the parameters to leave the white region corresponds to qualitative changes. The hatched regions delimit baryon fractions $fb={\xi }_{\mathrm{b}}/{\xi }_{\mathrm{c}}>300$ and $fb<1$, which may suppress galaxy formation by impeding disk fragmentation and by Silk damping out Galaxy-scale matter clustering, respectively. The shaded regions delimit interesting ranges of constant total matter density parameter $\xi ={\xi }_{\mathrm{b}}+{\xi }_{\mathrm{c}}$—to what extent these qualitative transitions are detrimental to galaxy formation deserves further work.

Figure 7

The fraction of all protons that end up in halos with density in the range ${\rho }_{\min }<{\rho }_{\mathrm{vir}}<{\rho }_{\max }$ is shown as a function of the cosmological parameters ${\rho }_{\Lambda }$ and ${\rho }_{*}\equiv Q^3{\xi }^4$. The contours show fractions $0.3$, $0.05$ and $0.003$ for the example ${\rho }_{\min }={10}^{-128}$, ${\rho }_{\max }={10}^{-120}$ (the two dashed lines) and $\mu ={10}^{-5}$ [corresponding roughly to ${10}^{12}{M}_{\odot }$ halos and giving $s(\mu )\approx 28$]. When ${\rho }_{\Lambda }\gg Q^3{\xi }^4$, essentially no halos at all are formed. When $Q^3{\xi }^4\lesssim {10}^{-3}{\rho }_{\min }$, essentially all halos have ${\rho }_{\mathrm{vir}}<{\rho }_{\min }$. When $Q^3{\xi }^4\gtrsim {10}^2{\rho }_{\max }$, essentially all halos have ${\rho }_{\mathrm{vir}}>{\rho }_{\max }$. The star shows our observed parameter values.

Figure 8

The probability factor $f_h({\rho }_{*})$ after marginalizing over ${\rho }_{\Lambda }$. The black line of slope $1$ is for ${\rho }_{\min }=0$, ${\rho }_{\max }=\infty$. Increasing ${\rho }_{\min }$ shifts the exponential cutoff on the left side progressively further the right: ${\rho }_{\min }={10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$ and ${10}^{-2}$ times the density at the dotted horizontal line gives the four curves on the left. Decreasing ${\rho }_{\max }$ shifts left the break to a $-1/3$ slope further to the right: ${\rho }_{\max }=100$, $10$, $1$ and $0.1$ times the density at the dotted horizontal line gives the four downward-sloping curves on the right, respectively.

Figure 9

Probability distribution for the axion dark matter density parameter $\xi$ measured from a random ${10}^{12}{M}_{\odot }$ halo with virial density below 5000 times the present cosmic matter density. Green/light shading indicates the 95% confidence interval. The dotted vertical line shows our observed value $\xi \approx 4\mathrm{eV}$, in good agreement with the prediction. The dashed curves show the analytic asymptotics ${\xi }^{9/2}$ and ${\xi }^{-5/6}$, respectively.

Figure 10

Same as previous figure, but showing the probability distribution for ${\xi }^{1/2}$, the quantity which has a uniform prior in our axion model. The shape of this curve therefore reflects the selection effect alone.

Figure 11

Probability distribution for the quantity $R\equiv {\rho }_{\Lambda }/{\xi }^4{Q}^3$ measured from a random ${10}^{12}{M}_{\odot }$ halo, using a uniform prior for $R$ and ignoring other selection effects. This is equivalent to treating $\xi$ and $Q$ as fixed. Green/light shading indicates the 95% confidence interval, the dotted line indicates the observed value $R\approx 15$.

Figure 12

Crude summary of constraints $(\xi ,Q)$-plane constraints from Table . The star shows our observation $(\xi ,Q)\approx (4\mathrm{eV},2\times {10}^{-5})$. The parallel dotted lines are lines of constant characteristic halo density, so the constraint that dark matter halos form (that they go nonlinear before vacuum domination freezes fluctuation growth) rules out everything to the lower left of these lines, which from left to right correspond to ${\rho }_{\Lambda }/{\rho }_{\Lambda }^{\mathrm{obs}}={10}^{-3}$, $1$, ${10}^3$. This means that if $\xi$ is fixed, marginalizing over ${\rho }_{\Lambda }$ pushes things to the triangle at larger $Q$ (the “smoothness problem” [3]), but if $\xi$ too can vary, marginalizing over ${\rho }_{\Lambda }$ instead pushes things to the square at smaller $Q$.

Figure 13

The top panel shows the turnaround (minimum) matter density ${\rho }_{\mathrm{turn}}$ of a top hat halo as a function of the time $x$ when its collapses. The virial density of the resulting dark matter halo is ${\rho }_{\mathrm{vir}}\sim 8{\rho }_{\mathrm{turn}}$. The solid curve is the exact result, with the asymptotic behavior ${\rho }_{\mathrm{turn}}\to 9{\pi }^2{\rho }_{\mathrm{m}}/4x$ as $x\to 0$ and ${\rho }_{\mathrm{turn}}\to 2{\rho }_{\Lambda }$ as $x\to \infty$ (dotted lines). The three analytic fits correspond to Eq. (a18) (short-dashed), Eq. (a19) (dotted) and that of 121, 122, which in our notation is $(18{\pi }^2/x+18{\pi }^2-82-39x/(1+x))/8$. The first two are seen to be accurate to better than 18% and 4%, respectively (bottom panel).