# Brans-Dicke theory and primordial black holes in early matter-dominated era

###### Abstract

We show that primordial black holes can be formed in the matter-dominated era with gravity described by the Brans-Dicke theory. Considering an early matter-dominated era between inflation and reheating, we show that the primordial black holes formed during that era evaporate quicker than those of early radiation-dominated era. Thus, in comparison with latter case, less number of primordial black holes could exist today. Again the constraints on primordial black hole formation become stronger than their radiation-dominated era counterparts indicating a significant enhancement in the formation of primordial black holes during the matter-dominaed era.

###### pacs:

98.80.-k, 97.60.Lf###### Contents

## I Introduction

Einstein’s formulation of General Theory of Relativity(GTR) ein in 1916 takes gravitational constant() to be a time-independent quantity. It is a pure tensor theory of gravity. Following Einstein’s lead, many scalar tensor theories have been developed as the extensions of GTR. In all these theories G is a time dependent quantity. Among them Brans-Dicke(BD) theory bdt is the simplest one. In BD theory the gravitational constant is set by the inverse of a time-dependent scalar field which couples to gravity with a coupling parameter . GTR can be recovered from BD theory in the limit bam . BD theory also admits simple expanding solutions mj for scalar field and scale factor which are compatible with solar system observations prg . BD theory is also sucessful in explaining many cosmological phenomena such as inflation ls , early and late time behaviour of the Universe ss , cosmic acceleration and structure formation bm , cosmic acceleration and coincidence problem bn etc.

It was first predicted by Zeldovich and Novikov zn in and later by Hawking haw in that black holes could be formed in the early Universe which are known as Primordial Black Holes (PBHs). PBHs may be formed as a result of density fluctuation carr1 , inflation kmz , phase transition kp , bubble collision kss and decay of cosmic loops polzem etc. These black holes are of special interest because their masses could be small enough to evaporate by the present epoch as a result of quantum emission hawk . Again PBHs could act as seeds for structure formation mor and could also form a significant component of dark matter bkp .

From standard picture of Cosmology, we know that the Universe is radiation-dominated just after inflation and it becomes matter-dominated much later. So PBHs were expected to be only formed in radiation-dominated era. However, the detailed analysis ky shows strong enhancement in probability of PBH formation in the matter-dominated era compared with the radiation-dominated epoch. It has been conjectured by Khlopov et al. kp and Carr et al. cgl that there may be an early matter-dominated era between the end of the inflation and the onset of reheating during which significant PBH formation could occur. It is, therefore, an open and interesting problem to investigate PBH formation and their evolution in matter-dominated era within the context of BD theory, although a number of similar studies have been done in radiation-dominated era bc ; nsm ; mgs . In this work, we have undertaken such an analysis and show that PBHs can indeed be formed in early matter-dominated era. We have also studied how it affects various astrophysical constraints through the evaporation of PBHs.

## Ii PBH in matter-dominated era

The gravitational field equations for a spatially flat FRW Universe with scale factor ’’ using BD theory are

(1) |

(2) |

The wave equation for BD scalar field is

(3) |

Using the above three equations and the perfect fluid equation of state , energy conservation equation can be written as

(4) |

For matter-dominated era which implies . Barrow and Carr bc have found that for matter-dominated era, the solutions of above equations are

(5) |

where is the present time, is the present value of and . But Solar system
observations require bit .
Taking , we found .

Integrating equation (4), one gets

(6) |

which in conjuction with equation (5) leads to

(7) |

We now proceed to discuss about the PBH formation in matter-dominated era. Following the analysis of Khlopov ky , we assume that the density fluctuation is responsible for forming PBHs in matter dominated era. This density fluctuation grows to a sufficiently homogeneous and isotropic configuration which separates itself from cosmological expansion and contracts within its gravitational radius.

Let be the time when contraction starts,
be the size of the configuration at time ,
be the deviation of configuration from the spherical form at time
which can be defined as ,
where define the deformation along the three main orthogonal
axes of the configuration,
( ) be the inhomogeneity of the density distribution inside the configuration,
and be the mean cosmological density at time .

Now equation (7) implies,

(8) |

The mean density of the primordial black holes formed as the result of contraction is

(9) |

where with
is the gravitational radius of considered configuration .

So

(10) |

where is the time at which PBH formation completes.

The maximal density which may be reached in the contraction of non-spherical configuration is given by ky

(11) |

In order to form the black hole, the configuration should be nearly spherically symmetric. i.e.

(12) |

The upperbound , in conjuction with the negative value of gives

(13) |

Since , we can write where is a very small quantity i.e. .

Thus, equation (13) gives

(14) |

But , and .

Now, one obtains

So primordial black holes could be formed during matter-dominated era which the Universe has passed through after inflation. .

The sufficient condition for the PBH formation imposes constraint on the inhomogeneity of the density distribution of the configuration at time in the form ky

(15) |

which is also satisfied in our case where and .

We, thus, arrive at the conclusion that PBHs can indeed be formed in the matter-dominated era within Brans-Dicke formalism.

with

## Iii PBH evaporation

To study PBH evaporation, we consider an early matter-dominated era between the epochs of inflation and reheating.

The rate at which the PBH mass decreases due to Hawking evaporation is given by

(16) |

Using the standard expressions for the black hole radius and the Hawking temperature , one gets

(17) |

where is the black body constant.

Again in our study of PBH evaporation, we consider two possibilities. In scenario A, has the same value everywhere at a given time, so that PBH evaporation is always determined by its current value. In scenario B, the local value of within the black hole is preserved implying gravitational memory so that the evaporation is determined by the value of when the PBH is formed.

### iii.1 Scenario A

In this scenario, has the same value everywhere at a given time.

If early matter-dominated era exists upto reheating time , then the evaporation equation becomes

(18) |

where is the initial mass of PBH formed at time in matter-dominated era,

(19) |

where is the initial mass of PBH formed at time in radiation-dominated era

For different initial times (), the numerical solutions of equations (18) and (19) are exhibited in the Table-I. To arrive at the numbers, we have used s and s. The last two rows of the table give the formation times of presently evaporating PBHs formed in early radiation-dominated era and in early matter-dominated era respectively.

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

It is clear from Table-I that the PBHs which are formed during early matter-dominated era will evaporate in significantly quicker rate than those of early radiation-dominated era because of their low masses at the time of formation.

### iii.2 Scenario B

In this scenario, associated with the black hole will continue to hold its value when PBH formation started.

Considering continuation of the matter-dominated era upto reheating time , the evaporation equation takes the form

(20) |

where is the initial mass of PBH formed at time in matter-dominated era.

By integrating, we get

(21) |

which gives evaporation time of PBH as

(22) |

But for PBHs which are formed in early radiation-dominated era, the evaporating equation takes the form,

(23) |

On integration, one gets

(24) |

where is the initial mass of PBH formed at time in radiation-dominated era.

Equation (24) gives evaporation time of PBH as

(25) |

For different initial times (), the results from equations (22) and (25) are shown in the Table-II.

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

s | g | g | s | s |

Here, again, one finds that the PBHs which are formed during early matter-dominated era will evaporate in significantly quicker rate than those of early radiation-dominated era.

From Table-I and Table-II, one finds that the gravitational memory does not significantly affect the longevity of the PBHs.

In recent papers nsm ; mgs , it is shown that the accretion of radiation prolongates the lifetime of PBHs. Now one can consider the accretion of radiation for this case also. Here the initial mass of PBH is times the horizon mass and PBH undergoes evaporation during early matter-dominated era. So during the radiation-dominated era, the initial mass of previously formed PBH becomes much smaller than . But we know that the accretion of radiation is negligible if initial mass of PBH is less than ns2 . Hence the accretion of radiation is ineffective for the PBHs which are formed in early matter-dominated era.

## Iv Constraints on PBH

We now discuss the different cosmological constraints associated with PBHs formed in matter-dominated era. PBH whose lifetime exceeds present era will contribute to the overall energy density. As the present observable Universe is nearly flat and, therefore, possesses critical density, the PBH mass density can be constrained on the ground that it should not overdominate the Universe. PBHs evaporate by producing bursts of evaporation products. Limits can be obtained by imposing that they should not interfere disastrously with esablished processes such as those of nucleosynthesis. Shorter-lived PBHs will have evaporated completely at an earlier stage. If this happened well before photon decoupling time, then their Hawking radiation will thermalize with the surroundings, boosting the photon-to-baryon ratio za . In the case of evaporation after photon decoupling, the radiation spectrum is affected and subsequently redshifts in a monotonic manner. Thus, constraints arise from the cosmic background radiation at high frequencies mc ; ph ; carr . Further, if the PBHs evaporate close to the time of photon decoupling, it cannot be fully thermalised and will produce distortion in the cosmic microwave background spectrum. Generally speaking, at a given epoch, the constraint on various physical observables is usually dominated by those PBHs with a lifetime of order of the epoch in question. Hence, the observational constraint can be translated into an upper limit on the initial mass fraction of PBHs. In terms of mass fraction, the different astrophysical constraints associated with PBHs formed in the early radiation-dominated era have recently been analysed by Nayak, Majumdar and Singh nms . In this work, we analyse the corresponding constraints for PBHs formed in the early matter-dominated era by using the result of our previous work nms and conversion formula.

The fraction of the Universe’s mass going into PBHs of mass M at a time in radiation-dominated era within BD formalism is given by

(26) |

If there exists an early stage of matter-domination immediately after inflation, then the constraints on the fraction of the Universe going into PBHs during the matter-dominated era and are related via the equation pk

(27) |

with .

But for early radiation-dominated era, . So becomes

(28) |

where is the initial mass fraction of PBH defined as

(29) |

So for as found in our earlier work nms , .

Cause of the constraint | in gm | ||
---|---|---|---|

Present Density | |||

Photon Spectrum | |||

Distortion of CMB | |||

Helium abundance | |||

Deuterium abundance |

We can obtain the bounds on using the translation formula (27) and the constraints we have obtained in our previous work nms . We have presented the constraints for both early radiation-dominated era and early matter-dominated era in Table-III for easy comparison. Here we have used which comes from our assumptions of early matter domination i.e. and Brans-Dicke gravity i.e. . Here one finds a significant change in the value of different constraints from their early radiation-dominated era counterparts. All known constraints associated with evaporation of PBHs become stronger than those of early radiation-dominated era. Nucleosynthesis constraints (Helium abundance constraint and Deuterium abundance constraint) are strengthened by three orders in magnitude whereas constraints associated with presently evaporating PBHs are strengthened by less than one order in magnitude.

It may be mentioned here that the recent paper by Carr et al. carretal has reanalyzed the standard constraints on primordial black holes by considering the effects of emission of quarks and gluons and the resultant secondary emission of photons. It was shown that the effect of secondary photon emission could alter the standard constraints on PBH fraction by a couple of orders of magnitude in certain cases, e.g., deuterium constraint, while leaving the standard constraints more or less unaltered in certain other cases, e.g., distortion of CMB spectrum. The results obtained by Carr et al. carretal are based on detailed numerical analysis. Of course, such a scenario of emission would also impact constraints on Brans-Dicke primordial black holes in more or less similar ways as they impact PBHs in the standard cosmology. However, the lack of analytical results describing the effect of such emission on the constraint formalism makes it considerably harder to perform a similar analysis in the context of an altered gravitational scenario. So following the arguments of our previous paper nms , here we can only write that the emission of quarks and gluons and the resultant secondary emission of photons may affect the constraints associated with matter-dominated era PBHs in similar fashion as they affect their radiation-dominated era counterparts.

## V Conclusions

We have shown that PBHs can be formed in matter-dominated era of cosmic evolution with gravity described by BD theory. For realisation of our result, we consider an early matter-dominaed era existing between end of inflation and begining of reheating. We found that the evaporation period of those PBHs are substantially shorter than those of the early radiation-dominated era. Thus, in comparison with latter case, less number of PBHs could exist today. Further, we find that the constraints on PBH formation become stronger in the present formalism compared with their radiation-dominated counterparts, which provides a strong evidence for the enhancement of PBH formation in matter-dominated era than the radiation-dominated era. It may be recalled that in matter-dominated phase, cosmic matter remains in a nearly zero pressure state. Absence of an opposing force to gravitational pull, thus, increases the probability of PBH formation in this era. Our work presents yet another application of BD theory as a viable alternate theory of gravity in addition to the various ones mentioned in the introduction.

## Acknowledgements

We are thankful to Institute of Physics, Bhubaneswar, India, for providing the library and computational facility. B.Nayak would like to thank the Council of Scientific and Industrial Research, Government of India, for the award of SRF, F.No. .

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