Connecting the Higgs Potential and Primordial Black Holes

It was recently demonstrated that small small black holes can act as seeds for nucleating decay of the metastable Higgs vacuum, dramatically increasing the tunneling probability. Any primordial black hole lighter than $4.5\times 10^{14}$g at formation would have evaporated by now, and in the absence of new physics beyond the standard model, would therefore have entered the mass range in which seeded decay occurs, however, such true vacuum bubbles must percolate in order to completely destroy the false vacuum; this depends on the bubble number density and the rate of expansion of the universe. Here, we compute the fraction of the universe that has decayed to the true vacuum as a function of the formation temperature (or equivalently, mass) of the primordial black holes, and the spectral index of the fluctuations responsible for their formation. This allows us to constrain the mass spectrum of primordial black holes given a particular Higgs potential and conversely, should we discover primordial black holes of definite mass, we can constrain the Higgs potential parameters.


I. INTRODUCTION
One of the most fascinating implications of the measurement of the Higgs mass at the LHC [1,2] is that the standard model vacuum appears to be metastable [3][4][5][6][7][8][9][10][11]. Initially, this was not thought to be a problem for our universe, as standard techniques for computing vacuum decay [12][13][14][15] indicated that the half-life was many order of magnitude greater the age of the universe. However, vacuum decay represents a first order phase transition, and in nature these typically proceed via catalysis: a seed or impurity acts as a nucleus for a bubble of the new phase to form. In [16][17][18][19][20], the notion that a black hole could act as such a seed was explored, with the finding that black holes can dramatically shorten the lifetime of a metastable vacuum (see also [21][22][23][24][25][26]). Interestingly, before the discovery of the Higgs particle, the electroweak phase transition was usually described as a second order transition, and in [27] the idea that the usual second order electroweak phase transition might be followed by a first order phase transition was explored.
For a black hole to seed vacuum decay, we must be sure that the half-life for decay is less than the evaporation rate of the black hole. This means that the branching ratio of tunnelling to decay must be greater than one. In [18,19] this was found to occur for black holes of order 10 6−9 M p or so, by which point the half-life for decay is of order 10 −23 s! Clearly this process is not relevant for astrophysical black holes, however, it has been hypothesised that there exist very light black holes, formed from extreme density fluctuations in the early universe [28][29][30] dubbed primordial black holes. Such black holes have been proposed as a source for dark matter [31], and although this has now been ruled out [32], they could still constitute a component of the dark matter of the universe. Indeed, it has even been proposed that the Higgs vacuum instability could generate primordial black holes in the early universe [33].
Given that we are in a current metastable Higgs vacuum, we can be sure that there has been no primordial black hole that has evaporated in our past lightcone, however, how strong a constraint on primordial black holes can we place? For the universe to have decayed, the black hole must not only have evaporated sufficiently to reach the mass range in which catalysis spectacularly dominates, but the consequent bubble (or bubbles) of true vacuum must have percolated to engulf the current Hubble volume. Thus, this is a statement about the relative volume in the percolated bubble, which is itself a statement on the primordial black hole density and mass. In this paper, we draw together all these aspects of the problem, linking the primordial black hole spectral index and formation epoch to the standard model parameters.
The outline of the paper is as follows. In section II we review the physics of the Higgs vacuum decay in the presence of gravity. In section III we relate the primordial black hole masses that can trigger vacuum decay with the parameters in the effective Higgs potential.
In section IV we put this scenario in the cosmological context: Every black hole that can trigger the vacuum decay will create a bubble of true vacuum. These bubbles then expand with the speed of light, but their number density decreases due to the expansion of the universe. For a successful phase transition, the bubbles have to percolate, so we define a quantity P, which represents the portion of the universe that has already transitioned to the new vacuum. For P ≥ 1, the universe would be destroyed, thus the associated range of parameters is excluded. We summarise and discuss our findings in section V.

II. FALSE VACUUM DECAY WITH BLACK HOLES
The high energy effective Higgs potential has been determined by a two-loop calculation in the standard model as [7] where λ eff (φ) is the effective coupling constant that runs with scale. We now review the calculations in [19], adopting the same conventions. The running of the coupling constant can be excellently modelled over a large range of scales by the three parameter fit: where M −2 p = 8πG. By fitting the two-loop calculation with a simple analytic form, we can easily investigate not only the standard model, but also beyond the standard model potentials, allowing us to explore possible future corrections to the standard model results.
The Higgs potential supports a first order phase transition mediated via nucleation of bubbles of new vacuum inside the old, false, vacuum. The nucleation rate in the presence of gravity is determined by a saddle point 'bounce' solution of the Euclidean (signature +, +, +, +) action: The spacetime geometry is taken to have SO(3) × U (1) symmetry, in other words, it is spherically symmetric "around" the black hole, and has time translation symmetry along the Euclidean time direction, τ : with We can think of µ(r) as the local mass parameter, however caution must be used in pushing this analogy. For an asymptotic vacuum of Λ = 0, then µ(∞) truly is the ADM mass of the black hole, however, locally, µ also includes the effect of any vacuum energy: for a pure Schwarzschild-(A)dS solution, µ(r) = M + Λr 3 /6G. Since we are interested in seeding the decay of our current SM vacuum, we will take Λ + = 0, so that the asymptotic value of µ is indeed the seed black hole mass, M + , responsible for triggering the phase transition.
The remnant mass, which is a leftover from the seed black hole after some of its energy is invested into the bubble formation, may not be precisely µ(r h ), however, since we will be interested only in the area of the remnant black hole horizon, it turns out that µ(r h ) is in fact the desired quantity.
The Higgs and gravitational field equations of motion are where V ,φ ≡ ∂V /∂φ. The black hole horizon is at r = r h , at which f (r h ) = 0. We have to solve these equations of motion numerically in order to get the function φ(r), and to do this, we start from the horizon r h with a particular remnant parameter, r h = 2Gµ − , and some value for the Higgs field φ h . At the horizon therefore the fields satisfy the boundary and as r → ∞, We use a shooting method starting at r h with φ = φ h and integrate out, altering φ h until a solution is obtained with φ tending to 0 for very large values of r. In practise, rather than setting the asymptotic mass µ(∞) = M + , we set the initial (remnant) value of µ − and deduce the seed mass from (8), repeating the integration for a range of values of µ − . We The decay rate of the Higgs vacuum, Γ D , is then determined by computing the difference in entropy between the seed and remnant black holes: where As pointed out in [17][18][19], a black hole can also radiate and lose mass, eventually disappearing in Hawking radiation, at a rate initially estimated by Page [34], see also [35][36][37][38][39]: Thus, we define the branching ratio between the tunneling and evaporation rate as This equation contains all the information we need. In the next two sections, we will study the consequences of the gravitationally induced false Higgs vacuum decay.

III. THE VACUUM DECAY RATE AND THE HIGGS EFFECTIVE POTENTIAL
If the branching ratio given by Eq. (12) is larger than one, then the tunneling rate is faster than evaporation rate, and the black hole can catalyze false vacuum decay. Note that the branching ratio depends on three parameters: M + , λ * , and b: fitting the form of λ eff in (2) to the standard model value at the electroweak scale fixes c in terms of λ * and b, and M + is the primordial black hole seed.
Let us first illustrate the results for some sample choices of the potential parameters. If we set λ * = −0.004, b = 1.5 × 10 −5 , c = 0, then Fig. 1 shows that the branching ratio is larger than one for This means that primordial black holes with masses within this range can initiate Higgs vacuum decay for the associated values of the Higgs potential parameters. A black hole mass with the lifetime of the current age of the universe is approximately 4.5 × 10 14 grams, meaning that all black holes lighter than this value would have already evaporated. Along the way, they will inevitably end up in the range given by Eq. (13). This however does not automatically imply that all the primordial black holes lighter than 4.5 × 10 14 grams are excluded for this choice of parameters. To destroy the universe the bubbles of the true vacuum have to percolate, which takes time. We will study this in the next section.
The same Fig. 1 indicates that if we set λ * = −0.00045, and keep b = 1.5 × 10 −5 and c = 0, then the branching ratio is always smaller than one (these values are not consistent with a pure standard model effective coupling, however, indicate the principle of model dependence of the branching ratio). In that case, the primordial black holes of any mass (i.e. M p < M + < ∞) cannot stimulate the false vacuum to decay into true vacuum, and our universe is safe. We excluded the black hole seed masses less than M p from the discussion, as the semi-classical approximation used in computing the decay rate is no longer expected to be valid at the Planck scale, where presumably a full theory of quantum gravity is required.
It is now instructive to systematically analyze the range of parameters for the effective coupling (2). Fig. 2 shows the threshold curve Γ D Γ H = 1 in (b, λ * ) parameter space for two values of the parameter c. The region of parameter space with Γ D Γ H < 1, for which the universe is safe, is above the curve. Below the curve, the branching ration will be greater than one for some range of black hole masses (similar to that shown in Eq. (13)) below the quantum gravity scale. This range is different for differing λ * , b, and c (so not easy to plot) however, it can easily calculated by substituting the concrete values for λ * , b, and c, in Eq. (12). The boundary with c = 6.3 × 10 −8 is lower than that with c = 0 because of the contribution from the quartic terms in the Higgs potential. However, numerical experiments indicate that the curves do not change significantly as we vary the parameter c.
According to [19] the standard model parameter space corresponding to the allowed range

IV. PRIMORDIAL BLACK HOLE MASSES AND PERCOLATING BUBBLES
In the previous section, we saw that any primordial black hole that had enough time to evaporate sufficiently to fit into an appropriate mass range for the corresponding choice of the parameters λ * and b, could initiate false vacuum decay. The bubbles of true vacuum then expand with the speed of light, but the background universe expands as well. Successful completion of the first order phase transition depends on the number density of the created bubbles. In our scenario, every black hole that can initiate the false vacuum decay will create a bubble, so the number of the bubbles is equal to the number of such primordial where T is the temperature of radiation. Obviously, the earlier the black holes are formed, the lighter they are, hence their lifetime is shorter. Their lifetime is given as [34][35][36][37][38][39].
Black holes of mass M 4.5 × 10 14 g have a lifetime greater than 1.38 × 10 10 years, or the age of the universe. Therefore only lighter primordial black holes will have the potential to destroy the universe. We focus on these lighter black holes which, according to Eq. (14), are created at temperatures higher than T F 4.7 × 10 8 GeV.
After primordial black holes are formed at T F , their number density changes with temperature as where β i is the mass fraction of the universe in black holes at formation, while ρ r,i = π 2 30 g F T 4 F is the radiation energy density at that time, with g F ≈ 100 being the number of degrees of freedom of radiation species at T F . M F is the mass of the primordial black holes at formation, and we take M F = M H (T F ) as usual. The mass fraction β i can be found assuming a Gaussian perturbation spectrum of fluctuations that lead to black hole formation (see e.g. [40,46,47]) The parameter δ min ≈ 0.3 is the minimum density contrast required for black hole creation, while σ H (T ) is the mass variance evaluated at horizon crossing at the temperature T defined as [46] σ H ( Here, T eq ≈ 0.79eV is the temperature at the matter/radiation equilibrium, T 0 = 2.725K = 2.35 × 10 −4 eV is the present temperature of the universe, while n is the spectral index of the fluctuations that lead to black hole formation, i.e. P (k) ∝ k n . Note, the cosmic microwave background data indicate that the value of the spectral index of the inflaton field is n ≈ 1, however the CMB data probe the scales between 10 45 and 10 60 times larger than those probed by primordial black holes. It is expected that primordial black holes are formed by fluctuations of fields other than the inflaton (e.g. during phase transitions), and the typical value of n used in this context is between 1.23 and 1.31 [40,47,48]. To normalize Eq. (18) we use the mass variance evaluated at the horizon crossing σ H (T 0 ) = 9.5 × 10 −5 .
We now have all the elements to calculate the black hole abundance for any set of desired parameters. After formation, primordial black holes evaporate, and at some stage of their life they will trigger false vacuum decay. When exactly this will happen depends on the specific parameters of the Higgs potential; we must be above the threshold value of the branching ratio, or in the range of parameters below the curve in Fig. 2, where it is guaranteed that the phase transition will be initiated for some black hole mass range. To illustrate the procedure, we calculate the excluded primordial black hole parameter space for the example from Section II, i.e. for the values of the potential parameters λ * = −0.004, b = 1.5 × 10 −5 , c = 0. As shown in section II, the branching ratio is larger than one for the seed black hole masses M p M + 10 6 M p , therefore all the black holes that have evaporated down to 10 6 M p or less by the present time will trigger false vacuum decay for this set of parameters. We note that this number is effectively the same as the number of the black holes that have evaporated completely by the present time, since it takes only a fraction of the second for a black hole to evaporate from 10 6 M p to zero.
The scenario is as follows. Suppose that primordial black holes are formed at a temperature T F with mass M F (lighter than 4.5 × 10 14 g). They then evaporate until they reach a mass of 10 6−9 M p , which is essentially equivalent to a complete evaporation, given the scale of the lifetimes involved. At that moment (which depends on the initial black hole mass) they seed vacuum decay and form a bubble of true vacuum that then expands at the speed of light. For a successful phase transition, the bubbles have to percolate, so we compute the overall volume of true vacuum in the expanding universe from the volume of an individual bubble and the number density of black holes.
The present time number density of the bubbles, n b (T 0 ), is shown in Fig. 3. It is calculated from Eq. (16) following the procedure outlined above. The present time radius of the bubble depends on the time it was created. If an object (in this case a bubble of true vacuum) is created at a cosmological redshift Z, its present age, t, is given by where Here Ω m , Ω rad , Ω k and Ω Λ are the present values of the dark matter, radiation, curvature, and dark energy density respectively. We take their numerical values from Planck results [49], Since dr = cdt/a = cdt(1 + Z), the current physical radius of the true vacuum bubble where E is given by Eq. (20). The redshift, Z B , is calculated at the moment when a black hole of a certain seed mass (formed at the temperature T F ) evaporates enough to fit into the appropriate mass window where it can trigger the false vacuum decay.
Thus, the portion of the universe which is already in the new vacuum at the present time Fig. 4 shows the boundary of the P =1 region. For the range of parameters in the upper part of the plot the universe today is destroyed, since the bubbles percolate. In contrast, for the range of parameters in the lower part of the plot, the universe is safe, though the primordial black holes may initiate false vacuum decay.
With the help of Eq. (14), we can convert the temperature of the universe at the time of the primordial black hole formation to the primordial black hole mass. This is shown in Fig. 5. We can see that lighter black holes are more dangerous than the more massive ones because they evaporate quickly, form the true vacuum bubbles earlier, and the bubbles have

V. CONCLUSIONS
We demonstrated here that it is possible to connect the parameters of the Higgs potential with the primordial black hole masses and physics of their formation (in our case the spectral index of perturbations that leads to their formation). We used the recent result that corrections due to black hole seeds can significantly increase the tunneling probability from the false to true Higgs vacuum. Any primordial black hole that had enough time to lose its mass from its formation till today to fit into an appropriate mass range for the correspond- However, just triggering the decay is not enough to destroy the universe, and automatically exclude associated black hole range. For a successful completion of the first order phase transition the bubbles have to percolate, which in turn depends on the number density of the created bubbles. Since every black hole that can initiate the false vacuum decay will create a bubble, the number of the bubbles is equal to the number of such primordial black holes. We then trace evolution of the bubbles. The bubbles of the true vacuum expand with the speed of light, but the background universe expands as well, so their number density decreases. We define a quantity P, which represents a portion of the universe which already transitioned to the new vacuum at the current time. For P ≥ 1, the universe is destroyed, and the associated range of parameters is excluded.
Our procedure can be used in two ways. If we use the Higgs potential parameters as an input, we can constrain the black hole masses and the physics of formation (e.g. the spectral index of perturbations). In turn, if we ever discover primordial black holes of definite mass, we can use it to constrain the Higgs potential parameters, or indeed the presence of extra dimensions [50,51].