百年征程

$v\approx H_0D ,$

$z=\frac{\lambda_o-\lambda_e}{\lambda_e}=\sqrt{\frac{1+v/c}{1-v/c}}-1\approx\frac{v}{c} .$

$D$ 是星系到我们的固有距离，它可以由星系的光度距离近似，

$D\approx\frac{c}{H_0}\left[z+\frac12(1-q_0)z^2+\cdots\right]$

$v_p(t)\equiv\dot{d}_p(t)\equiv\dot{a}(t)d_c=H(t)d_p(t)$

距离阶梯

$m=M+25+5 \log_{10}D_L(z)=M+25-5 \log_{10}H_0+5 \log_{10}c d_L(z)$

$d_L(z)\equiv\frac{H_0}{c}D_L(z)=z\left[1+\frac12(1-q_0)z-\frac16(1-q_0-3q_0^2+j_0)z^2+\cdots\right] ,$

非距离阶梯测距

强引力透镜时间延迟

$\Delta t=\frac{1}{c}D_{\Delta t}\Delta\phi ,$

$D_{\Delta t}=(1+z_\mathrm{d})\frac{D_A(z_d)D_A(z_s)}{D_A(z_s)-\frac{1+z_d}{1+z_s}D_A(z_d)} ,$

引力波标准汽笛

$h_\times=\frac{4}{D_L(z)}\left(\frac{GM_c(z)}{c^2}\right)^{5/3}\left(\frac{\pi f}{c}\right)^{2/3}\cos\iota\sin[\Phi(f)] ,$

宇宙微波背景辐射

$P_R(k)=A_s\left(\frac{k}{k_0}\right)^{n_s-1} ,$

$H(a)^2=\omega_\Lambda+(\omega_b+\omega_c)a^{-3}+\omega_r(a)a^{-4} ,$

$\omega_r(a)=\frac{g(a)}{2}\left(\frac{4}{11}\right)^\frac{4}{3}\omega_\gamma ,$

$\omega_r(a)=\left[1+\frac{7}{8}\left(\frac{4}{11}\right)^\frac{4}{3}N_\mathrm{eff}\right]\omega_\gamma ,$

$\omega_\gamma=2.473\times10^{-5}\left(\frac{T_0}{2.7255}\right) ,$

$\theta_*\equiv \frac{r_s(z_*)}{D_A(z_*)}=\frac{\int_{z_*}^\infty\frac{\mathrm{d}z}{H(z)}c_s(z)}{\frac{c}{1+z_*}\int_0^{z_*}\frac{\mathrm{d}z}{H(z)}} ,$

$c_s^2(a)\equiv\frac{1}{3\left(1+\frac34\frac{\rho_b(a)}{\rho_\gamma(a)}\right)} .$

观测危机

BAO+BBN

$\theta_\mathrm{BAO}(z)=\frac{\int_{z_D}^\infty\frac{c_s(\omega_b,z')\mathrm{d}z'}{H(z')}}{\int_0^z\frac{\mathrm{d}z'}{H(z')}}=\frac{\int_{z_D}^\infty \frac{c_s(\omega_b,z')\mathrm{d}z'}{\sqrt{\Omega_r/\Omega_m(1+z')^4+(1+z')^3}}}{\int_0^z\frac{\mathrm{d}z'}{\sqrt{(1-\Omega_m)/\Omega_m+(1+z')^3}}} ,$

反向距离阶梯

$\frac{D_A(z)}{r_d}=\frac{\alpha}{1+\epsilon}\frac{D_{A,f}(z)}{r_{d,f}} ,$

$H(z)r_d=\frac{H_f(z)r_{d,f}}{\alpha(1+\epsilon)^2} ,$

$D_A(z)=\frac{D_L(z)}{(1+z)^2} ,$

理论机遇

扩展模型

CMB 和BAO 本质上测量的主要是声学视界的角尺度大小 $\theta_*$，它是声学视界在最后散射面的大小 $r_s(z_*)$ 与最后散射面到我们现在的角直径距离 $D_A(z_*)$ 之比。一方面，由于 $D_A(z_*)$ 反比于 $H_0$，因此为了增大 $H_0$，在不改变角尺度 $\theta_*$ 大小的情况下，需要减小声学视界 $r_s(z_*)$ 的大小。另一方面，在保持 BAO 的其中一个各向异性测量 $H_0r_s=\mathrm{const.}$ 不变的情况下，为了增大 $H_0$，也需要减小声学视界的大小。

虚惊一场

$m=5\log D_L(z)+M+25 ,$

$H_0D_L(z)=c(1+z)\int_0^z\frac{\mathrm{d}z'}{E(z')} .$

$m_\mathrm{SN}^\mathrm{flow}=5\log D_L(z_\mathrm{SN}^\mathrm{flow})+M_\mathrm{SN}+25 ,$

$m_\mathrm{Ceph}^\mathrm{anch}=5\log D_L(z_\mathrm{Ceph}^\mathrm{anch})+M_\mathrm{Ceph}+25 ,$

$m_\mathrm{SN}^\mathrm{flow}-m_\mathrm{SN}^\mathrm{host}+m_\mathrm{Ceph}^\mathrm{host}-m_\mathrm{Ceph}^\mathrm{anch}=5\log D_L(z_\mathrm{SN}^\mathrm{flow})-5\log D_L(z_\mathrm{Ceph}^\mathrm{anch}) ,$

$5\log H_0=5\log[H_0D_L(z)]-5\log D_L^\mathrm{anch}+\Delta m_\mathrm{SN}-\Delta m_\mathrm{Ceph} ,$

$\frac{\delta H_0}{H_0}=\frac{\delta(H_0D_L)}{(H_0D_L)}-\frac{\delta D_L^\mathrm{anch}}{D_L^\mathrm{anch}}+\frac{\ln10}{5}[\delta(\Delta m_\mathrm{SN})-\delta(\Delta m_\mathrm{Ceph})] ,$

Semiclassical vacuum decay

【广告】半经典真空衰变的发生 arXiv:1909.11196 Phys.Rev.D100 (2019) 096019

Quantum tunnelling in non-relativistic quantum mechanics of a single particle is a distinguishing feature from the classical mechanics where surmounting a potential barrier requires large enough energy instead of quantum mechanically penetrating the potential barrier with lower energy.

This feature persists when generalized into the case of relativistic field theory with multiple potential minimums, one of which is absolute minimum in both classical and (perturbative) quantum sense, while the rest of which is still classically stable but metastable by barrier penetration in quantum field theory.

Vacuum decay in field theory proceeds via sudden nucleations of true vacuum bubbles in the false vacuum environment, which is essentially a quantum phenomenon without classical analog. The decay rate is estimated by Euclidean instanton as an analog to the WKB approximation in nonrelativistic quantum mechanics.

Recently, semiclassical vacuum decay was found in Phys. Rev. Lett. 123 no. 3, (2019) that vacuum decay in field theory could proceed via classical evolution of the equation of motion for some initial configurations of false vacuum fluctuations. In particular, flyover vacuum decay was suggested in that, a sufficiently large local initial fluctuation in field velocity could carry the field value directly flying over the potential barrier.

In our paper, we found that condition for semiclassical vacuum decay is rather loose, even allowing for the dubbed pop-up vacuum decay, where the semiclassical vacuum decay could still occur even if the initial energy density is everywhere insufficient to classically overcome the potential barrier.

gif : The time evolution of field profile for the last example of Fig.4 in the mentioned paper. The initial profile for field velocity is everywhere below the threshold for a classical surmounting over the potential barrier, however, after gathering energy through the gradient term in the equation of motion, a true vacuum bubble (red) is eventually formed out of the false vacuum background (blue), and then expands as usual but with oscillating feature inside the bubble.

It is worth noting that, the semiclassical vacuum decay is not only entirely motivated from the numerical simulations [ Phys.Rev. D100 (2019) 065016 , JHEP 1910 (2019) 174 , Phys.Rev.Lett. 123 (2019) 031601 ], but also from some theoretical considerations [ JHEP 1807 (2018) 014 , Phys.Rev. D100 (2019) 016011 ] and even experimential (cold atom) interest [ EPL 110 (2015) 56001 , J.Phys. B50 (2017) 024003].

Renormalization group improvement

【Note】Tommi Markkanen, Arttu Rajantie, Stephen Stopyra, Cosmological Aspects of Higgs Vacuum MetastabilityarXiv: 1809.06923

As we known, the classical potential of a field after quantization receives quantum corrections and thus becomes the effective potential

$V_\mathrm{eff}(\phi; g_i)=V_\mathrm{cl}(\phi(\mu); g_i(\mu))+\delta V(\phi(\mu); g_i(\mu), \mu)$

where $\mu$ is the so-called renormalization scale, and the dependences of running coupling parameters $g_i(\mu)$ on the renormalization scale are given by the corresponding beta function $\beta_{g_i}=\mu\frac{\partial g_i}{\partial\mu}$, and the renormalized one-point function of a quantized field is related to the bare field $\phi_\mathrm{bare}=\sqrt{Z(\mu)}\phi(\mu)$ by the field-renormalization-factor $Z(\mu)$ determined by the anomalous dimension $\gamma=\mu\frac{\partial\sqrt{Z}}{\partial\mu}$.

The full effective potential including all orders of loop-diagram quantum-corrections $\delta V_1, \delta V_2, \cdots$ does not depend on the choice of renormalization scale, leading to so-called Callan-Symazik equation

$\mu\frac{\mathrm{d}}{\mathrm{d}}V_\mathrm{eff}(\phi; g_i)=\left(\mu\frac{\partial}{\partial\mu}+\beta_{g_i}\frac{\partial}{\partial g_i}-\gamma\phi\frac{\partial}{\partial\phi}\right)V_\mathrm{eff}(\phi; g_i)=0$

If there is such a choice of function for the renormalization scale $\mu=\mu^*(\phi)$ so that the total quantum correction $\delta V(\phi(\mu^*); g_i(\mu^*), \mu^*(\phi))$ is vanished, then we have a renormalization-group (RG) improved effective potential

$V_\mathrm{eff}^\mathrm{RGI}(\phi; g_i)=V_\mathrm{cl}(\phi(\mu^*(\phi)); g_i(\mu^*(\phi)))$

However, the practical calculations normally could only contain finite orders of loop-diagram quantum-corrections, for example, one-loop correction $\delta V_1$ is the dubbed Coleman-Weinberg potential, and the corresponding effective potential would explicitly depend on the choice of renormalization scale and impact the effectiveness and precision of perturbative expansion. Then an important question is how to choose  the renormalization scale.

Firstly, we could be conservative by requiring some function form $\mu=\mu^*_1(\phi)$ for the renormalization scale so that $\delta V_1$ is vanished, thus the corresponding effective potential is given by the RG-improved classical potential

$V^\mathrm{classical}_\mathrm{RGI-eff}(\phi(\mu^*_1); g_i(\mu^*_1))=V_\mathrm{cl}(\phi(\mu^*_1); g_i(\mu^*_1))$

Nevertheless, starting from the condition of a vanishing $\delta V_1$ usually could not solve for an explicit function form of renormalization scale. Even such a explicit solution is obtained, it would be considerably complicated.

Therefore, we can be even more conservative by requiring $\delta V_1$ to be nonzero but as small as possible for its logarithmic part, for example, we could choose the renormalization scale as $\mu^{\#}_1\propto\phi$, and the corresponding effective potential is given by the RG-improved 1-loop-level effective potential

$V_\mathrm{RGI-eff}^\mathrm{1-loop}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1), \mu^{\#}_1)=V_\mathrm{cl}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1))+\delta V_1(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1), \mu^{\#}_1)$

We could be even bolder by throwing away the whole 1-loop corrections and obtain the RG-improved tree-level effective potential

$V_\mathrm{RGI-eff}^\mathrm{tree}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1))=V_\mathrm{cl}(\phi(\mu^{\#}_1); g_i(\mu^{\#}_1))$